Analysis of a mathematical model for tumor growth with Gibbs-Thomson relation

TL;DR

This paper investigates a mathematical model for nonnecrotic tumor growth incorporating a Gibbs-Thomson relation, revealing multiple stationary solutions and analyzing their stability based on model parameters.

Contribution

It introduces a novel free boundary model with Gibbs-Thomson relation for tumor growth and analyzes the existence, uniqueness, and stability of solutions.

Findings

Existence and uniqueness of solutions are established.

Two stationary solutions are identified depending on parameters.

The larger stationary solution is asymptotically stable for small diffusion-to-growth ratio.

Abstract

In this paper we study a mathematical model for the growth of nonnecrotic solid tumor. The tumor is assumed to be radially symmetric and its radius R(t) is an unknown function of time t as tumor growth, and the model is in the form of a free boundary problem. The feature of the model is that a Gibbs-Thomson relation is taken into account, which resulting an interesting phenomenon that there exist two stationary solutions (depending on the model parameters). The global existence and uniqueness of solution are established. By denoting c the ratio of the diffusion time scale to the tumor doubling time scale, we prove that for sufficiently small c>0, the stationary solution with the larger radius is asymptotically stable, and the other smaller one is unstable.