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An Allen-Cahn tumor growth model with temperature

Stefania Gatti, Erica Ipocoana, Alain Miranville

TL;DR

The paper addresses incorporating temperature effects into a phase-field tumor-growth model by deriving a thermodynamically consistent non-isothermal Allen-Cahn system with temperature and nutrient coupling. The authors develop a three-equation PDE framework for the order parameter $\varphi$, temperature $\theta$, and nutrient $\sigma$ using microforces and Gibbs free energy, yielding an order-parameter equation, a temperature equation with nonconstant conductivity, and a nutrient diffusion-reaction equation. They prove local-in-time existence and uniqueness via a decoupled fixed-point approach and establish global-in-time existence through a priori estimates ensuring positivity and boundedness of the variables. The results provide a mathematically rigorous basis for studying thermally influenced diffuse-interface tumor growth and pave the way for analyses of therapies involving temperature variations and nutrient dynamics.

Abstract

In this paper, we propose a new non-isothermal Allen-Cahn (Ginzburg-Landau) model for tumor growth. After deriving it using a microforces approach, we study its well-posedness. In particular, we are able to prove the existence and uniqueness of a local and global-in-time solution to our PDE system.

An Allen-Cahn tumor growth model with temperature

TL;DR

The paper addresses incorporating temperature effects into a phase-field tumor-growth model by deriving a thermodynamically consistent non-isothermal Allen-Cahn system with temperature and nutrient coupling. The authors develop a three-equation PDE framework for the order parameter , temperature , and nutrient using microforces and Gibbs free energy, yielding an order-parameter equation, a temperature equation with nonconstant conductivity, and a nutrient diffusion-reaction equation. They prove local-in-time existence and uniqueness via a decoupled fixed-point approach and establish global-in-time existence through a priori estimates ensuring positivity and boundedness of the variables. The results provide a mathematically rigorous basis for studying thermally influenced diffuse-interface tumor growth and pave the way for analyses of therapies involving temperature variations and nutrient dynamics.

Abstract

In this paper, we propose a new non-isothermal Allen-Cahn (Ginzburg-Landau) model for tumor growth. After deriving it using a microforces approach, we study its well-posedness. In particular, we are able to prove the existence and uniqueness of a local and global-in-time solution to our PDE system.
Paper Structure (9 sections, 4 theorems, 148 equations)

This paper contains 9 sections, 4 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Derivation of the model
  3. Order parameter equation
  4. Temperature equation
  5. Nutrient equation
  6. Well-posedness
  7. Assumptions and notation
  8. Local in time existence of a solution
  9. Global in time existence of the solution

Key Result

Theorem 3.1

For any $R>0$ there exists $T=T(R)>0$ such that if $(\theta_0,{\varphi}_0,\sigma_0)$ satisfies assumptions initialtemp--initialnutr and then there exists a unique local solution $(\theta,{\varphi},\sigma)$ to system system in $(0,T)$, namely, Besides,

Theorems & Definitions (10)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['localthm']}
  • Remark 3.5
  • Theorem 3.6
  • proof