Mathematical Formulation and Computational Implementation of a Fisher-KPP Model for Glioblastoma Invasion under Fractionated Radiotherapy

Abstract

Glioblastoma (GBM) is a highly infiltrative primary brain tumor, and reaction-diffusion models are widely used to describe how its cells spread and proliferate. We derive the classical Fisher-KPP equation from mass conservation, Fickian diffusion, and logistic growth, and extend it to fractionated radiotherapy through a linear-quadratic survival update applied to the tumor-density field at each fraction. The Fisher-KPP equation is solved by finite differences on a three-dimensional voxel grid with 0.5-mm spacing; Geant4 is used only for Monte Carlo photon transport and dose calculation. The schedule of 60~Gy in 30 fractions from the clinical case reported by Roda et al. is reproduced, and the simulated evolution is compared qualitatively with the clinical course as a demonstration only, without quantitative validation against patient-specific imaging data. The case report gives lesion dimensions at three time points but no serial pretreatment imaging, so the diffusion and proliferation parameters cannot be estimated for this patient and are taken from the literature. The results are a qualitative demonstration of the coupled model and do not constitute patient-specific validation.