A phase field model of Cahn-Hilliard type for tumour growth with mechanical effects and damage

TL;DR

This paper develops a comprehensive phase field model for tumor growth that incorporates mechanical effects, nutrient dynamics, and reversible tissue damage, providing a mathematically rigorous foundation for understanding complex tumor behaviors.

Contribution

It introduces a novel diffuse interface model with cellular damage governed by a differential inclusion, and proves global existence of weak solutions for this complex PDE system.

Findings

Established global-in-time existence of weak solutions.

Integrated mechanical effects and tissue damage into tumor growth modeling.

Provided a rigorous mathematical framework for complex tumor dynamics.

Abstract

We introduce a new diffuse interface model for tumour growth in the presence of a nutrient, in which we take into account mechanical effects and reversible tissue damage. The highly nonlinear PDEs system mainly consists of a Cahn-Hilliard type equation that describes the phase separation process between healthy and tumour tissue coupled to a parabolic reaction-diffusion equation for the nutrient and a hyperbolic equation for the balance of forces, including inertial and viscous effects. The main novelty of this work is the introduction of cellular damage, whose evolution is ruled by a parabolic differential inclusion. In this paper, we prove a global-in-time existence result for weak solutions by passing to the limit in a time-discretised and regularised version of the system.