On a phase field model of Cahn-Hilliard type for tumour growth with mechanical effects

TL;DR

This paper develops a coupled phase field model incorporating mechanical effects for tumor growth, combining Cahn-Hilliard, elasticity, and nutrient diffusion equations, and establishes mathematical well-posedness results.

Contribution

It introduces a novel macroscopic model coupling Cahn-Hilliard with elasticity for tumor growth, addressing non-linear challenges beyond gradient flow frameworks.

Findings

Proved existence, uniqueness, and regularity of solutions.

Established continuous dependence results for weak solutions.

Provided mathematical foundations for future optimal control studies.

Abstract

Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn-Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell-cell adhesion effects are taken into account with the help of a Ginzburg--Landau type energy. In the overall model an equation of Cahn-Hilliard type is coupled to the system of linear elasticity and a reaction-diffusion equation for a nutrient concentration. The highly non-linear coupling between a fourth-order Cahn-Hilliard equation and the quasi-static elasticity system lead to new challenges which cannot be dealt within a gradient flow setting which was the method of choice for other elastic Cahn-Hilliard systems. We show existence, uniqueness and regularity results. In addition, several continuous dependence results with respect to different topologies are shown. Some of these results give uniqueness for weak solutions and other results will be helpful for optimal control problems.