From Brain Scans to Therapy Response: PDE Modelling of Immunotherapy for Glioblastoma

Abstract: Glioblastoma Multiforme (GBM) is a highly aggressive brain tumour with limited therapeutic options and poor prognosis. This study presents a mathematical framework to investigate the efficacy of immunotherapy strategies based on cytotoxic T-lymphocyte (CTL) infusion. The model couples tumour and immune dynamics through a system of partial differential equations (PDEs), incorporating cell proliferation, diffusion, and chemotactic migration in response to TGF-$β$, a tumour-secreted signalling molecule. A reduced ordinary differential equation (ODE) model is first analysed to derive threshold conditions for tumour eradication, identifying critical infusion levels consistent with clinical data. Numerical bifurcation analysis explores the impact of parameter variations. The full PDE model is solved using the finite element method on simplified 2D domains, followed by sensitivity analyses to quantify parameter influence on tumour mass and volume. The model is then applied to a realistic 3D brain geometry reconstructed from patient-specific MRI and DTI data, accounting for anatomical anisotropy and tissue heterogeneity. Therapeutic scenarios are simulated with spatially localised lymphocyte infusion. Results highlight spatial variations in tumour growth and treatment response, with infusion intensity and tumour location critically influencing therapeutic outcomes. These findings emphasise the importance of personalised, spatially informed modelling in optimising immunotherapy protocols for GBM.