Global existence of classical solutions and numerical simulations of a cancer invasion model

TL;DR

This paper proves the global existence of classical solutions for a complex cancer invasion PDE model and supports the theory with numerical simulations in 2D and 3D.

Contribution

It establishes the first global existence results for this nonlinear cancer invasion model and develops a numerical scheme for its simulation.

Findings

Global classical solutions exist in 2D and 3D domains.

Numerical simulations validate the theoretical results.

The scheme effectively models cancer invasion dynamics.

Abstract

In this paper, we study a cancer invasion model both theoretically and numerically. The model is a nonstationary, nonlinear system of three coupled partial differential equations modeling the motion of cancer cells, degradation of the extracellular matrix, and certain enzymes. We first establish existence of global classical solutions in both two- and three-dimensional bounded domains, despite the lack of diffusion of the matrix-degrading enzymes and corresponding regularizing effects in the analytical treatment. Next, we give a weak formulation and apply finite differences in time and a Galerkin finite element scheme for spatial discretization. The overall algorithm is based on a fixed-point iteration scheme. In order to substantiate our theory and numerical framework, several numerical simulations are carried out in two and three spatial dimensions.