Exogenous Brain Tumour Model

• Exogenous brain tumour models are strategies that prescribe external structure—such as reaction-diffusion laws, PDE priors, and experimental surrogates—to enable tractable tumour prediction and delineation.
• They are applied in radiotherapy planning, physics-informed segmentation, experimental tumour induction, and treatment-aware interventions to enhance workflow integration.
• Key limitations include sensitivity to segmentation quality, challenges in accounting for patient-specific dynamics, and the potential mismatch between prescribed models and true tumour behaviour.

An exogenous brain tumour model is a modelling construct in which decisive structure is supplied externally rather than inferred solely from image intensities or endogenous patient-specific biophysics. In the cited literature, this external structure takes several forms: a phenomenological growth law with tissue-class constraints for radiotherapy planning, a PDE prior imposed on a segmentation network, an experimentally induced tumour in animals, a treatment input acting as an explicit control variable, or an ex vivo fluorophore distribution injected into resected brain tissue for device assessment. This suggests that “exogenous” is best understood as a family of modelling strategies in which tumour behaviour, tumour surrogates, or tumour interventions are prescribed by external laws, priors, or manipulations to make prediction, delineation, or measurement tractable (Unkelbach et al., 2013, Zhang et al., 2024, Ezhov et al., 2019, Wang et al., 8 Mar 2026, Raschke et al., 14 Jul 2025).

1. Conceptual scope

The literature uses the term in multiple technical senses. In radiotherapy planning for glioblastoma, “exogenous” denotes a phenomenological reaction–diffusion law with simple, externally set parameters and tissue constraints rather than patient-specific biophysical calibration. In biophysics-informed segmentation, it denotes a tumour growth PDE imposed as an external prior on an otherwise image-driven network. In rat studies, it can denote tumours induced by intracerebral injection and then calibrated from MRI. In treatment-aware world models and immunotherapy systems, it denotes interventions introduced as explicit decision variables or source terms. In fluorescence-guided surgery testbeds, it denotes an externally constructed “tumoral” PpIX gradient inside resected brain tissue (Unkelbach et al., 2013, Zhang et al., 2024, Ezhov et al., 2019, Ballatore et al., 20 Dec 2025, Raschke et al., 14 Jul 2025).

Usage in the literature	Exogenous element	Representative papers
Phenomenological tumour growth	Governing PDE, tissue classes, barriers, global parameters	(Unkelbach et al., 2013, Agosti et al., 2022, Subramanian et al., 2018, Kumar et al., 2020)
Learning regularisation	PDE residual and boundary losses imposed on training	(Zhang et al., 2024)
Experimental induction or construction	Injected rat glioma or injected ex vivo PpIX region	(Ezhov et al., 2019, Raschke et al., 14 Jul 2025)
Controlled intervention	Treatment plan tokens, CTL infusion, T11TS pulses	(Wang et al., 8 Mar 2026, Ballatore et al., 20 Dec 2025, Banerjee et al., 2014)

A recurrent distinction is between exogenous specification and personalization. Exogenous models generally encode accepted macroscopic mechanisms—diffusion-driven infiltration, logistic proliferation, no-flux barriers, chemotaxis, or treatment inputs—but do not necessarily infer patient-specific kinetics, fibre tensors, or observation laws from data. That distinction is explicit in the radiotherapy and segmentation papers, and it reappears as a limitation in the animal-calibration and intervention papers.

2. Anatomy-constrained phenomenological growth models

A canonical exogenous formulation is the Fisher–Kolmogorov reaction–diffusion model used to extrapolate tumour cell density beyond MRI-visible abnormality. In the radiotherapy-planning setting, tumour cell density $c(\mathbf{x},t)$ obeys

$$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$

with piecewise isotropy by tissue class,

$$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$

and homogeneous Neumann conditions on the brain boundary and along barriers such as CSF, falx cerebri, tentorium cerebelli, ventricles, dura, and skull. The T1-Gd boundary is taken as an isoline near $c\approx 0.7K$, and the full time-dependent PDE is replaced by a static Hamilton–Jacobi/eikonal approximation solved by Fast Marching on geodesic distances through segmented WM and GM. In homogeneous tissue, the resulting profiles decay approximately exponentially with infiltration lengths $\lambda_w=\sqrt{D_w/\rho}$ and $\lambda_g=\sqrt{D_g/\rho}$, so target volumes can be defined as isolines of simulated $c(\mathbf{x})$ rather than uniform geometric margins (Unkelbach et al., 2013).

In that workflow, multimodal MRI is rigidly registered with FLIRT, tissue segmentation is performed by an EM-based multimodal method following Menze et al., with falx and tentorium augmentation after Unkelbach et al., and the simulated density is transferred into planning CT coordinates. In a retrospective cohort of 10 glioblastoma patients treated with IMRT to 60 Gy in 30 fractions, manual and model-based CTVs had mean Dice coefficient $0.79$ with range $0.74$–$0.84$, while the Dice coefficient for the $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$0 Gy $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$1 isodose between IMRT plans based on manual and model targets was $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$2 with range $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$3–$$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$4. The detailed falx/corpus-callosum case showed constrained spread across the falx, extension through the corpus callosum into the contralateral hemisphere, and qualitative agreement between pre-therapy isolines and later T2-FLAIR progression at $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$5 and $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$6 months post-diagnosis (Unkelbach et al., 2013).

A broader mechanistic extension replaces single-species Fisher–Kolmogorov dynamics by multiphase or multispecies systems while preserving the same exogenous logic. The four-phase Cahn–Hilliard Keller–Segel framework introduces viable tumour $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$7, necrotic tumour $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$8, angiogenetic phase $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$9, liquid phase $$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$0, nutrient $$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$1, and angiogenic factor $$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$2, with the saturation constraint $$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$3, DTI-derived preferential-direction tensor $$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$4, diffusion tensors $$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$5 and $$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$6, rCBV-informed vascular source terms, and homogeneous Neumann boundary conditions on a patient-specific mesh. Its thermodynamic structure yields an energy-dissipation law and permits pre-surgical high- and low-nutrient scenarios as well as post-surgical simulations under Stupp-type therapy profiles (Agosti et al., 2022).

3. Exogenous priors in segmentation and parameter inference

In physics-informed segmentation, the exogenous tumour model is not used to generate contours directly but to regularise a discriminative network. The segmentation framework $$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$7 outputs class probabilities for normal, tumour core, whole tumour, and enhancing tumour, while a dedicated tumour-cell-density estimator $$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$8 is constrained to satisfy

$$D(\mathbf{x}) = \begin{cases} D_w I, & \mathbf{x}\in \text{white matter}\ D_g I, & \mathbf{x}\in \text{gray matter}, \end{cases} \qquad D_w>D_g,$$9

with no-flux Neumann conditions on the feature-map boundary. The total loss is

$c\approx 0.7K$0

Here the estimator uses sine activations, high-level feature maps are cropped to $c\approx 0.7K$1, spatial derivatives are computed by finite differences, and $c\approx 0.7K$2 is obtained by automatic differentiation with respect to a synthetic time variable. The implementation assumes isotropic diffusion across voxels, samples $c\approx 0.7K$3 and $c\approx 0.7K$4 as stochastic regularisers, and trains for 175 epochs in PyTorch 2.1.0 and MONAI v1.3.0 on an NVIDIA A10. On BraTS 2023, the regulariser improved multiple backbones; for example, UNet improved from Dice $c\approx 0.7K$5 and HD95 $c\approx 0.7K$6 to Dice $c\approx 0.7K$7 and HD95 $c\approx 0.7K$8, while ablations reported robustness to reduced modalities, smaller training sets, and alternative segmentation losses (Zhang et al., 2024).

A different exogenous strategy appears in neural posterior estimation for experimentally induced rat tumours. There the tumour is introduced by F98 glioma cell injection, MRI is acquired at days $c\approx 0.7K$9, $\lambda_w=\sqrt{D_w/\rho}$0, $\lambda_w=\sqrt{D_w/\rho}$1, and $\lambda_w=\sqrt{D_w/\rho}$2, and the Fisher–KPP model is coupled to an observation model in which T1w-Gd and T2w abnormalities are thresholded isolines of simulated density $\lambda_w=\sqrt{D_w/\rho}$3. In synthetic experiments the thresholds are fixed at $\lambda_w=\sqrt{D_w/\rho}$4 and $\lambda_w=\sqrt{D_w/\rho}$5, while for real rats the initial condition at day $\lambda_w=\sqrt{D_w/\rho}$6 is built from inverse ADC inside the T2w lesion,

$\lambda_w=\sqrt{D_w/\rho}$7

and the parameter vector $\lambda_w=\sqrt{D_w/\rho}$8 is inferred by a mixture-density network with posterior

$\lambda_w=\sqrt{D_w/\rho}$9

Synthetic tests recovered $\lambda_g=\sqrt{D_g/\rho}$0, $\lambda_g=\sqrt{D_g/\rho}$1, $\lambda_g=\sqrt{D_g/\rho}$2, $\lambda_g=\sqrt{D_g/\rho}$3, and $\lambda_g=\sqrt{D_g/\rho}$4 accurately, and NPE achieved accurate estimates after 4 posterior-update iterations versus about 20 generations for the MCMC baseline; in real rats, predictions matched at calibration day $\lambda_g=\sqrt{D_g/\rho}$5 but tended to underestimate later T2w growth at days $\lambda_g=\sqrt{D_g/\rho}$6 and $\lambda_g=\sqrt{D_g/\rho}$7 (Ezhov et al., 2019).

4. Exogenous interventions as control variables

The most explicit control-theoretic use of the term appears in treatment-aware world models. Brain-WM represents the patient state at clinical time $\lambda_g=\sqrt{D_g/\rho}$8 as $\lambda_g=\sqrt{D_g/\rho}$9, where $c(\mathbf{x})$0 contains FLAIR, T1CE, and T2W, and $c(\mathbf{x})$1 contains treatment history and demographics. The exogenous decision variable is the next-step treatment plan $c(\mathbf{x})$2, encoded through five special tokens: [SUR], [CRT], [RT], [TMZ], and [AM]. The model uses a Y-shaped Mixture-of-Transformers with 28 layers, split into 14 shared and 14 task-specific layers, jointly learning autoregressive treatment prediction and flow-matched future MRI generation, while MM-Align injects tumour-mask supervision at shared layers $c(\mathbf{x})$3. On internal multi-institutional data comprising 527 subjects and 1,659 timepoints, Brain-WM reported treatment-planning accuracy $c(\mathbf{x})$4, F1 $c(\mathbf{x})$5, specificity $c(\mathbf{x})$6, and precision $c(\mathbf{x})$7; internal SSIM for future MRI generation was $c(\mathbf{x})$8 for FLAIR, $c(\mathbf{x})$9 for T1CE, and $0.79$0 for T2W, with lower but still substantial external performance on 61 subjects and 128 timepoints (Wang et al., 8 Mar 2026).

Therapy-coupled PDEs make exogeneity even more literal by inserting treatment as a source term. In the CTL immunotherapy model for GBM, tumour density $0.79$1, activated CTLs $0.79$2, and TGF-$0.79$3 concentration $0.79$4 evolve by anisotropic diffusion, logistic tumour growth, Michaelis–Menten killing, chemotaxis, and a spatially localised infusion term $0.79$5 in the CTL equation. The corresponding ODE reduction yields a critical total CTL production threshold

$0.79$6

so eradication in the well-mixed approximation requires endogenous recruitment plus exogenous infusion to exceed that threshold. In 3D DTI-informed simulations with the infusion centred 14 mm from the tumour, $0.79$7 cells/h produced noticeable slowing and $0.79$8 cells/h produced effective control, while no-therapy growth remained elongated along fibre tracts (Ballatore et al., 20 Dec 2025).

An older ODE immunotherapy model uses T11 target structure as an exogenous pulse input $0.79$9 added directly to the macrophage and CTL equations, with

$0.74$0

The simulations report that the first dose mainly boosts macrophages, the third dose is decisive for CTL cytotoxic activity, and the three-dose regimen drives tumour burden to zero while lowering TGF-$0.74$1 and increasing IFN-$0.74$2 (Banerjee et al., 2014).

5. Multiphase, biomechanical, and microenvironmental generalisations

A major development in exogenous modelling has been the replacement of scalar diffusion by coupled phase fields, mechanics, metabolism, and environmentally driven taxis. The image-informed Cahn–Hilliard Keller–Segel model with angiogenesis derives a four-phase mixture from variational principles consistent with the second law of thermodynamics, uses DTI and spherical deconvolution to define $0.74$3, uses diffusion MRI for $0.74$4 and $0.74$5, and uses leakage-corrected DSC-MRI rCBV maps to specify nutrient supply and angiogenic drivers. In pre-surgical low-nutrient scenarios, angiogenesis intensifies at the tumour boundary, infiltrates the core via chemotaxis, and raises viable-cell density by additional nutrient supply; in the post-surgical case, simulated $0.74$6 isosurfaces co-localise with high-rCBV regions and the $0.74$7 isosurface shows good spatial overlap with enhancing tumour while also indicating infiltrative regions beyond segmentation (Agosti et al., 2022).

Another exogenous extension couples tumour growth to lactate, viscoelasticity, and reversible tissue damage. The unknowns are tumour phase $0.74$8, intracellular lactate $0.74$9, displacement $0.84$0, and damage variable $0.84$1, with no-flux conditions for $0.84$2 and $0.84$3, Robin exchange for $0.84$4, and homogeneous Dirichlet boundary condition for $0.84$5. Exogenous influence enters through the lactate boundary datum $0.84$6, body force $0.84$7, damage drive $0.84$8, and initial data. Under hypotheses (A), the system admits at least one global-in-time weak solution; under hypotheses (A) and (B), it enjoys higher regularity, continuous dependence, and uniqueness. This moves the notion of exogeneity from a fixed tumour-growth law toward a broader class of externally driven initial-boundary value problems (Cavalleri et al., 4 Feb 2025).

Biomechanical exogeneity also appears in the multispecies reaction–advection–diffusion model with mass effect. That framework couples proliferating, invasive, and necrotic tumour species with oxygen, edema, and screened linear elasticity on a $0.84$9 grid with $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$00 resolution. Diffusion is isotropic but spatially varying through tissue composition, elasticity depends on local Lamé coefficients derived from species concentrations, and the forcing term $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$01 drives displacement. The same paper explicitly discusses adaptation to extra-axial, compressive tumours such as meningiomas by suppressing infiltration, increasing tumour stiffness, localising screening, and shifting edema toward a mechanically driven source (Subramanian et al., 2018).

At the microenvironmental scale, pseudopalisade models treat vascular occlusion, hypoxia, proton accumulation, and pH gradients as the decisive exogenous drivers. In the KTAP-to-PDE derivation, a mesoscopic kinetic model with fast subcellular proton sensing upscales to a macroscopic reaction–diffusion–taxis system in which glioma density is driven by anisotropic “myopic” diffusion and repellent pH-taxis, while proton concentration follows diffusion, tumour production, and buffering. Simulations show that pseudopalisade-like rings arise in the parabolic, effectively undirected-tissue limit, particularly when buffering is weak relative to proton production, whereas the hyperbolic, transport-dominated limit does not form such patterns. The paper interprets this as evidence that, on the pseudopalisade scale, glioma migration behaves as if the tissue were undirected (Kumar et al., 2020).

6. Experimental surrogates, validation practices, and recurring limitations

Not all exogenous brain tumour models are computational. The ex vivo PpIX model constructs a controllable fluorescence surrogate by injecting a $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$02 bolus of $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$03 PpIX in DMSO approximately $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$04 beneath the cortical surface of a freshly resected adult Wistar rat brain. Under two $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$05 LEDs and fibre spectroscopy at a working distance of about 5 mm, the model reproduces a central $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$06 PpIX peak, a brain autofluorescence peak at about $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$07, and sometimes a $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$08 photoproduct peak. The “tumoral” region is defined by the ratio $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$09 and extends from about $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$10 mm to $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$11 mm along a scan line, giving an approximate width of $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$12 mm. Cross-study comparison after AF normalisation gave a ratio of $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$13 for the exogenous model, versus $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$14, $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$15, and $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$16 in cited in vivo studies, with reported Pearson “$$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$17” above $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$18–$$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$19 and $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$20. DMSO did not alter AF, and storage in HBSS at about $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$21C preserved AF for about 20 hours, whereas refrigerated storage without moisture caused AF decay (Raschke et al., 14 Jul 2025).

Across computational and experimental settings, several limitations recur. Radiotherapy contouring is highly sensitive to segmentation quality, especially falx, tentorium, ventricles, and corpus callosum labelling, and prospective validation against recurrence maps remains limited (Unkelbach et al., 2013). Physics-informed segmentation assumes isotropic diffusion, omits mass effect, uses static residual evaluation rather than explicit forward integration, and samples rather than infers $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$22 and $$\frac{\partial c}{\partial t} = \nabla \cdot \big(D(\mathbf{x}) \nabla c\big) + \rho c\left(1-\frac{c}{K}\right),$$23 per patient (Zhang et al., 2024). Rat parameter inference underestimates later progression when necrosis, mass effect, or time-varying kinetics become important (Ezhov et al., 2019). Brain-WM loses correlation between MM-Align focus and ground-truth tumour volume as the forecasting horizon lengthens, and surgical cases and CRT are more difficult because of postoperative variability and pseudoprogression confounders (Wang et al., 8 Mar 2026). These are not contradictions so much as boundary markers of the exogenous approach: it gains tractability, controllability, and workflow integration by prescribing external structure, but it also inherits whatever mismatch exists between that prescribed structure and the true tumour–host system.

The cumulative literature therefore defines exogenous brain tumour modelling less by a single equation than by a methodological stance. Whether the external element is a Fisher–Kolmogorov law, a PDE residual in a segmentation loss, an injected tumour in a rat, a treatment token in a world model, a CTL source term, a lactate boundary condition, or an ex vivo PpIX bolus, the common purpose is to impose biologically or experimentally motivated structure on a problem that imaging alone does not determine. Within that stance, the field spans radiotherapy target delineation, segmentation robustness, uncertainty-aware calibration, intervention planning, mechanics and angiogenesis, microenvironment-driven histopathology, and fluorescence-device benchmarking (Unkelbach et al., 2013, Zhang et al., 2024, Ezhov et al., 2019, Wang et al., 8 Mar 2026, Agosti et al., 2022, Raschke et al., 14 Jul 2025).