Analysis of a mathematical model for the growth of cancer cells

TL;DR

This paper introduces a two-dimensional mathematical model for multi-layer tumor growth, analyzing the stability of equilibrium states influenced by nutrient diffusion and internal pressure.

Contribution

The paper develops and proves the existence and stability of solutions for a free boundary tumor growth model with periodic boundary conditions.

Findings

Existence of solutions for the tumor growth model

Identification of stationary solutions and their stability

Local asymptotic stability of equilibrium points

Abstract

In this paper, a two-dimensional model for the growth of multi-layer tumors is presented. The model consists of a free boundary problem for the tumor cell membrane and the tumor is supposed to grow or shrink due to cell proliferation or cell dead. The growth process is caused by a diffusing nutrient concentration $\sigma$ and is controlled by an internal cell pressure $p$. We assume that the tumor occupies a strip-like domain with a fixed boundary at $y=0$ and a free boundary $y=\rho(x)$, where $\rho$ is a $2\pi$-periodic function. First, we prove the existence of solutions $(\sigma,p,\rho)$ and that the model allows for peculiar stationary solutions. As a main result we establish that these equilibrium points are locally asymptotically stable under small perturbations.