On a Reaction-Cross-Diffusion System Modelling the Growth of Glioblastoma

TL;DR

This paper analyzes a complex cross-diffusion model for glioblastoma growth, combining mathematical analysis, existence proofs, traveling wave solutions, and numerical simulations in 1D and 2D, considering anisotropic effects.

Contribution

It provides the first complete existence analysis of the model in one dimension and explores traveling waves and numerical simulations including tissue anisotropy.

Findings

Existence of solutions in 1D established.

Traveling wave solutions related to tumor growth dynamics.

Numerical simulations demonstrate effects of anisotropic diffusion.

Abstract

We investigate a recently proposed cross-diffusion system modelling the growth of gliobastoma taking into account size exclusion both in the migration and proliferation process. In addition to degenerate nonlinear cross-diffusion the model includes reaction terms as in the Fisher-Kolmogorov equation and linear ones modelling transition between states of proliferation and migration. We discuss the mathematical structure of the system and provide a complete existence analysis in spatial dimension one. The proof is based on exploiting partial entropy dissipation techniques and fully implicit time discretisations. In order to prove existence of the latter appropriate variational and fixed-point techniques are used, together with suitable a-priori estimates. Moreover, we review the existence of travelling waves and their relation to potential growth or decay of glioblastoma. Finally we provide extensive numerical studies in one and two spatial dimensions, including the effect of anisotropic diffusions as found in neural tissues.