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Individualizing Glioma Radiotherapy Planning by Optimization of Data and Physics-Informed Discrete Loss (2312.05063v3)

Published 8 Dec 2023 in physics.med-ph, cs.NA, math.NA, and q-bio.QM

Abstract: Brain tumor growth is unique to each glioma patient and extends beyond what is visible in imaging scans, infiltrating surrounding brain tissue. Understanding these hidden patient-specific progressions is essential for effective therapies. Current treatment plans for brain tumors, such as radiotherapy, typically involve delineating a uniform margin around the visible tumor on pre-treatment scans to target this invisible tumor growth. This "one size fits all" approach is derived from population studies and often fails to account for the nuances of individual patient conditions. We present the GliODIL framework, which infers the full spatial distribution of tumor cell concentration from available multi-modal imaging, leveraging a Fisher-Kolmogorov type physics model to describe tumor growth. This is achieved through the newly introduced method of Optimizing the Discrete Loss (ODIL), where both data and physics-based constraints are softly assimilated into the solution. Our test dataset comprises 152 glioblastoma patients with pre-treatment imaging and post-treatment follow-ups for tumor recurrence monitoring. By blending data-driven techniques with physics-based constraints, GliODIL enhances recurrence prediction in radiotherapy planning, challenging traditional uniform margins and strict adherence to the Fisher-Kolmogorov partial differential equation (PDE) model, which is adapted for complex cases.

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References (26)
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Radiother Oncol (2023) Frosina [2023] Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Louis, D.N., Perry, A., Wesseling, P., Brat, D.J., Cree, I.A., Figarella-Branger, D., Hawkins, C., Ng, H.K., Pfister, S.M., Reifenberger, G., Soffietti, R., Deimling, A., Ellison, D.W.: The 2021 who classification of tumors of the central nervous system: a summary. Neuro Oncol (2021) https://doi.org/10.1093/neuonc/noab106 Weller [2021] Weller, M.: Eano guidelines on the diagnosis and treatment of diffuse gliomas of adulthood. Nature Reviews Clinical Oncology 18, 170–186 (2021) Niyazi et al. [2023] Niyazi, M., Andratschke, N., Bendszus, M., Chalmers, A.J., Erridge, S.C., Galldiks, N., et al.: Estro-eano guideline on target delineation and radiotherapy details for glioblastoma. Radiother Oncol (2023) Frosina [2023] Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Weller, M.: Eano guidelines on the diagnosis and treatment of diffuse gliomas of adulthood. Nature Reviews Clinical Oncology 18, 170–186 (2021) Niyazi et al. [2023] Niyazi, M., Andratschke, N., Bendszus, M., Chalmers, A.J., Erridge, S.C., Galldiks, N., et al.: Estro-eano guideline on target delineation and radiotherapy details for glioblastoma. Radiother Oncol (2023) Frosina [2023] Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Niyazi, M., Andratschke, N., Bendszus, M., Chalmers, A.J., Erridge, S.C., Galldiks, N., et al.: Estro-eano guideline on target delineation and radiotherapy details for glioblastoma. Radiother Oncol (2023) Frosina [2023] Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. 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[2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. 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[2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
  2. Louis, D.N., Perry, A., Wesseling, P., Brat, D.J., Cree, I.A., Figarella-Branger, D., Hawkins, C., Ng, H.K., Pfister, S.M., Reifenberger, G., Soffietti, R., Deimling, A., Ellison, D.W.: The 2021 who classification of tumors of the central nervous system: a summary. Neuro Oncol (2021) https://doi.org/10.1093/neuonc/noab106 Weller [2021] Weller, M.: Eano guidelines on the diagnosis and treatment of diffuse gliomas of adulthood. Nature Reviews Clinical Oncology 18, 170–186 (2021) Niyazi et al. [2023] Niyazi, M., Andratschke, N., Bendszus, M., Chalmers, A.J., Erridge, S.C., Galldiks, N., et al.: Estro-eano guideline on target delineation and radiotherapy details for glioblastoma. Radiother Oncol (2023) Frosina [2023] Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Weller, M.: Eano guidelines on the diagnosis and treatment of diffuse gliomas of adulthood. Nature Reviews Clinical Oncology 18, 170–186 (2021) Niyazi et al. [2023] Niyazi, M., Andratschke, N., Bendszus, M., Chalmers, A.J., Erridge, S.C., Galldiks, N., et al.: Estro-eano guideline on target delineation and radiotherapy details for glioblastoma. Radiother Oncol (2023) Frosina [2023] Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Niyazi, M., Andratschke, N., Bendszus, M., Chalmers, A.J., Erridge, S.C., Galldiks, N., et al.: Estro-eano guideline on target delineation and radiotherapy details for glioblastoma. Radiother Oncol (2023) Frosina [2023] Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. 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[2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Niyazi, M., Andratschke, N., Bendszus, M., Chalmers, A.J., Erridge, S.C., Galldiks, N., et al.: Estro-eano guideline on target delineation and radiotherapy details for glioblastoma. Radiother Oncol (2023) Frosina [2023] Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. 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In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. 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Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. 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Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. 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[2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. 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J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. 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[2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
  4. Niyazi, M., Andratschke, N., Bendszus, M., Chalmers, A.J., Erridge, S.C., Galldiks, N., et al.: Estro-eano guideline on target delineation and radiotherapy details for glioblastoma. Radiother Oncol (2023) Frosina [2023] Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. 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SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Frosina, G.: Radiotherapy of high-grade gliomas: dealing with a stalemate. Crit Rev Oncol Hematol 190, 104110 (2023) https://doi.org/10.1016/j.critrevonc.2023.104110 Cristini and Lowengrub [2010] Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Cristini, V., Lowengrub, J.: Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, 1st edn. Cambridge University Press, Cambridge (2010) Mang et al. [2020] Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. 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IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. 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[2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. 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[2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. 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J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. 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Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. 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IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. 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[2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. 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[2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. 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J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. 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In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. 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Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. 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[2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
  7. Mang, A., Bakas, S., Subramanian, S., Davatzikos, C., Biros, G.: Integrated biophysical modeling and image analysis: application to neuro-oncology. Annual review of biomedical engineering 22, 309 (2020) Hogea et al. [2008] Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Hogea, C., Davatzikos, C., Biros, G.: An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. Journal of mathematical biology 56(6), 793–825 (2008) Lê et al. [2015] Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 424–432 (2015). Springer Knopoff et al. [2017] Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. 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SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. 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In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. 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[2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Knopoff, D., Fernández, D., Torres, G., Turner, C.: A mathematical method for parameter estimation in a tumor growth model. Computational and Applied Mathematics 36(1), 733–748 (2017) Subramanian et al. [2019] Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. 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[2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., Gholami, A., Biros, G.: Simulation of glioblastoma growth using a 3d multispecies tumor model with mass effect. J Math Biol 79(3), 941–967 (2019) https://doi.org/10.1007/s00285-019-01383-y Scheufele et al. [2020] Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. 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Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. 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[2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. 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Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. 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[2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. 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  12. Scheufele, K., Subramanian, S., Mang, A., Biros, G., Mehl, M.: Image-driven biophysical tumor growth model calibration. SIAM journal on scientific computing: a publication of the Society for Industrial and Applied Mathematics 42(3), 549 (2020) Subramanian and et al [2020] Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. 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[2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Subramanian, S., al: Where did the tumor start? an inverse solver with sparse localization for tumor growth models. Inverse Problems 36(045006) (2020) https://doi.org/10.1088/1361-6420/ab649c Lipková et al. [2019] Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., et al.: Personalized radiotherapy design for glioblastoma: Integrating mathematical tumor models, multimodal scans, and bayesian inference. IEEE transactions on medical imaging 38(8), 1875–1884 (2019) Petersen et al. [2019] Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Jäger, P.F., Isensee, F., Kohl, S.A., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., Kickingereder, P., et al.: Deep probabilistic modeling of glioma growth. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Conference, Shenzhen, China, October 13–17, 2019, Proceedings, Part II 22, pp. 806–814 (2019). Springer Petersen et al. [2021] Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. 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Journal of Neuropathology and Experimental Neurology (2007) Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. 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[2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. 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[2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. 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J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. 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Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. 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  16. Petersen, J., Isensee, F., Köhler, G., Jäger, P.F., Zimmerer, D., Neuberger, U., Wick, W., Debus, J., Heiland, S., Bendszus, M., et al.: Continuous-time deep glioma growth models. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part III 24, pp. 83–92 (2021). Springer Mascheroni et al. [2021] Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
  17. Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L., Meyer-Hermann, M., Hatzikirou, H.: Improving personalized tumor growth predictions using a bayesian combination of mechanistic modeling and machine learning. Communications medicine 1(1), 19 (2021) Wang et al. [2022] Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Wang, H., Argenziano, M.G., Yoon, H., Boyett, D., Save, A., Petridis, P., Savage, W., Jackson, P., Hawkins-Daarud, A., Tran, N., et al.: Biologically-informed deep neural networks provide quantitative assessment of intratumoral heterogeneity in post-treatment glioblastoma. bioRxiv, 2022–12 (2022) Nardini et al. [2020] Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Nardini, J.T., Lagergren, J.H., Hawkins-Daarud, A., Curtin, L., Morris, B., Rutter, E.M., Swanson, K.R., Flores, K.B.: Learning equations from biological data with limited time samples. Bulletin of mathematical biology 82, 1–33 (2020) Ezhov et al. [2021] Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. 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  20. Ezhov, I., Mot, T., Shit, S., Lipkova, J., Paetzold, J.C., Kofler, F., Pellegrini, C., Kollovieh, M., Navarro, F., Li, H., et al.: Geometry-aware neural solver for fast bayesian calibration of brain tumor models. IEEE Transactions on Medical Imaging 41(5), 1269–1278 (2021) Ezhov et al. [2022] Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
  21. Ezhov, I., Scibilia, K., Franitza, K., Steinbauer, F., Shit, S., Zimmer, L., Lipkova, J., Kofler, F., Paetzold, J., Canalini, L., Waldmannstetter, D., Menten, M.J., Metz, M., Wiestler, B., Menze, B.: Learn-morph-infer: a new way of solving the inverse problem for brain tumor modeling. Medical Image Analysis, 102672 (2022) Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
  22. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) Karnakov et al. [2023] Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
  23. Karnakov, P., Litvinov, S., Koumoutsakos, P.: Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. The European Physical Journal E 46(7), 59 (2023) https://doi.org/10.1140/epje/s10189-023-00313-7 Kofler et al. [2020] Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
  24. Kofler, F., Berger, C., Waldmannstetter, D., Lipkova, J., Ezhov, I., Tetteh, G., Kirschke, J., Zimmer, C., Wiestler, B., Menze, B.H.: Brats toolkit: translating brats brain tumor segmentation algorithms into clinical and scientific practice. Frontiers in neuroscience, 125 (2020) Harpold et al. [2007a] Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
  25. Harpold, H.L.P., Alvord, E.C., S., K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol (2007) Harpold et al. [2007b] Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007) Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
  26. Harpold, H.L.P., Alvord, E.C., Swanson, K.R.: The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology (2007)
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