Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Giulia Cavalleri

TL;DR

The paper develops a diffuse-interface phase-field model for tumor growth that couples a Cahn–Hilliard equation for tissue phase with nutrient diffusion, a hyperbolic balance for mechanics, and a novel damage variable governed by a parabolic differential inclusion. The main contribution is a rigorous global-in-time existence proof for weak solutions, achieved by a time-discretised, regularised scheme that employs a convex–concave energy split and Moreau–Yosida regularisation, accompanied by discrete energy estimates and compactness arguments to pass to the limit. The analysis handles strong nonlinearity and coupling, including a $p$-Laplacian damage term and a nonconvex elastic energy, while ensuring nutrient bounds and adherence to a comprehensive weak formulation. This work provides a mathematically rigorous foundation for biologically relevant CH-type models with mechanical effects and reversible tissue damage, enabling reliable simulations of tumor evolution under nutrition and mechanical loading.

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.

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

TL;DR

The paper develops a diffuse-interface phase-field model for tumor growth that couples a Cahn–Hilliard equation for tissue phase with nutrient diffusion, a hyperbolic balance for mechanics, and a novel damage variable governed by a parabolic differential inclusion. The main contribution is a rigorous global-in-time existence proof for weak solutions, achieved by a time-discretised, regularised scheme that employs a convex–concave energy split and Moreau–Yosida regularisation, accompanied by discrete energy estimates and compactness arguments to pass to the limit. The analysis handles strong nonlinearity and coupling, including a -Laplacian damage term and a nonconvex elastic energy, while ensuring nutrient bounds and adherence to a comprehensive weak formulation. This work provides a mathematically rigorous foundation for biologically relevant CH-type models with mechanical effects and reversible tissue damage, enabling reliable simulations of tumor evolution under nutrition and mechanical loading.

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.
Paper Structure (12 sections, 11 theorems, 180 equations)

This paper contains 12 sections, 11 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Derivation of the model
  3. Aim of the paper
  4. Main result
  5. Notation and preliminaries
  6. Hypotheses
  7. Weak formulation and existence result
  8. Proof of the existence theorem
  9. Time discretisation
  10. A priori estimates for the time-discrete system
  11. Compactness assertions
  12. Passage to the limit in the discrete system

Key Result

Theorem 2.1

Let $\Omega \subseteq \mathbb{R}^d$ be a Lipschitz bounded domain. Given there exists a constant $C$ such as for every $v \in H^s$, the following inequality holds true:

Theorems & Definitions (27)

  • Theorem 2.1: Gagliardo--Nirenberg inequality
  • Theorem 2.2: Ehrling's lemma
  • Lemma 2.3: Discrete Gronwall inequality
  • Lemma 2.4
  • Lemma 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 17 more