Theoretical and numerical analysis of a parabolic system with chemoattraction modeling the growth of glioma cells

Abstract

In this paper, we consider a two dimensional PDE model describing the proliferation-invasion structure of hypoxic glial cells in the brain, in which the cell movement is determined not only by natural diffusion but also, by the gradient of the concentration of oxygen. This model is given by a second order nonlinear system involving the density of cancer cells and the oxygen concentration, which are coupled by a chemoattraction term, a source of logistic growth, and a reaction term of Michaelis-Menten type. The contribution of this paper is summarized in the following two main aspects: first, we prove the existence and uniqueness of strong solutions; and second, we propose a nonlinear fully discrete finite element (FE) approximation for the model. We prove its well-posedness, some uniform estimates, the positivity for the discrete oxygen concentration and approximated positivity for the discrete tumour cells, which are required in this biological model. The key point to develop the numerical analysis is to control properly the second order nonlinear chemotaxis term as well as to obtain a discrete energy estimate which, in particular, gives a bounded energy; this energy estimate is derived by using a regularization technique. Finally, we present some numerical simulations which allow us to validate the theoretical results obtained.