Existence of weak solutions for a volume-filling model of cell invasion into extracellular matrix

TL;DR

This paper proves the existence of weak solutions for a complex cell invasion model involving coupled PDEs and ODEs, highlighting the role of the ECM and demonstrating traveling wave solutions through simulations.

Contribution

It establishes the existence of weak solutions for a coupled PDE-ODE model of cell invasion with volume-filling effects, using a partial gradient flow approach.

Findings

Existence of weak solutions for the model.

Presence of traveling wave solutions.

Asymptotic behavior as ECM degradation rate increases.

Abstract

We study the existence of weak solutions for a model of cell invasion into the extracellular matrix (ECM), which consists of a non-linear partial differential equation for the density of cells, coupled with an ordinary differential equation (ODE) describing the ECM density. The model contains cross-species density-dependent diffusion and proliferation terms that capture the role of the ECM in providing structural support for the cells during invasion while also preventing growth via volume-filling effects. Furthermore, the model includes ECM degradation by the cells. We present an existence result for weak solutions which is based on carefully exploiting the partial gradient flow structure of the problem which allows us to overcome the non-regularising nature of the ODE involved. In addition, we present simulations based on a finite difference scheme that illustrate that the system exhibits travelling wave solutions, and we investigate numerically the asymptotic behaviour as the ECM degradation rate tends to infinity.