A Glioblastoma PDE-ODE model including chemotaxis and vasculature

TL;DR

This paper develops a PDE-ODE model for Glioblastoma that includes chemotaxis and vasculature, providing analytical estimates, a stable numerical scheme, and insights into tumor morphology influenced by key parameters.

Contribution

It introduces a novel PDE-ODE model for Glioblastoma with chemotaxis, offers a priori estimates, a finite element scheme, and analyzes parameter effects on tumor structure.

Findings

Model does not develop finite-time blow-up under certain conditions

Finite element scheme preserves key estimates of the continuous model

Main parameters influence tumor ring width and surface regularity

Abstract

In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes chemotaxis term directed to vasculature. First, we obtain some a priori estimates for the (possible) solutions of the model. In particular, under some conditions on the parameters, we obtain that the system does not develop blow-up at finite time. In addition, we design a fully discrete finite element scheme for the model which preserves some pointwise estimates of the continuous problem. Later, we make an adimensional study in order to reduce the number of parameters. Finally, we detect the main parameters determining different width of the ring formed by proliferative and necrotic cells and different regular/irregular behaviour of the tumor surface.