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On a non-local phase-field model for tumour growth with single-well Lennard-Jones potential

Maurizio Grasselli, Luca Melzi, Andrea Signori

TL;DR

The paper addresses a non-local phase-field model for tumour growth that couples a non-local Cahn–Hilliard equation with a parabolic nutrient-diffusion equation, incorporating a viscous relaxation term $\tau\,\partial_t\varphi$ and a singular single-well Lennard–Jones potential to capture cell–cell adhesion. It develops a rigorous analytical framework using Faedo–Galerkin approximations and potential regularization to prove the existence of weak solutions for $\tau>0$, establishes a separation property for the phase variable, and derives a continuous dependence estimate on initial data; it further analyzes the asymptotic limit as $\tau\to0^+$ to obtain a weak solution of the $\tau=0$ problem. The results extend the mathematical understanding of non-local, single-well phase-field models in tumour dynamics, including long-range adhesion and chemotactic coupling, with robust well-posedness, regularity, and stability properties. Overall, the work provides a solid theoretical foundation for biologically plausible tumour-growth models and informs future numerical and analytical explorations in non-local phase-field systems with singular potentials.

Abstract

In the present work, we develop a comprehensive and rigorous analytical framework for a non-local phase-field model that describes tumour growth dynamics. The model is derived by coupling a non-local Cahn-Hilliard equation with a parabolic reaction-diffusion equation, which accounts for both phase segregation and nutrient diffusion. Previous studies have only considered symmetric potentials for similar models. However, in the biological context of cell-to-cell adhesion, single-well potentials, like the so-called Lennard-Jones potential, seem physically more appropriate. The Cahn-Hilliard equation with this kind of potential has already been analysed. Here, we take a step forward and consider a more refined model. First, we analyse the model with a viscous relaxation term in the chemical potential and subject to suitable initial and boundary conditions. We prove the existence of solutions, a separation property for the phase variable, and a continuous dependence estimate with respect to the initial data. Finally, via an asymptotic analysis, we recover the existence of a weak solution to the initial and boundary value problem without viscosity, provided that the chemotactic sensitivity is small enough.

On a non-local phase-field model for tumour growth with single-well Lennard-Jones potential

TL;DR

The paper addresses a non-local phase-field model for tumour growth that couples a non-local Cahn–Hilliard equation with a parabolic nutrient-diffusion equation, incorporating a viscous relaxation term and a singular single-well Lennard–Jones potential to capture cell–cell adhesion. It develops a rigorous analytical framework using Faedo–Galerkin approximations and potential regularization to prove the existence of weak solutions for , establishes a separation property for the phase variable, and derives a continuous dependence estimate on initial data; it further analyzes the asymptotic limit as to obtain a weak solution of the problem. The results extend the mathematical understanding of non-local, single-well phase-field models in tumour dynamics, including long-range adhesion and chemotactic coupling, with robust well-posedness, regularity, and stability properties. Overall, the work provides a solid theoretical foundation for biologically plausible tumour-growth models and informs future numerical and analytical explorations in non-local phase-field systems with singular potentials.

Abstract

In the present work, we develop a comprehensive and rigorous analytical framework for a non-local phase-field model that describes tumour growth dynamics. The model is derived by coupling a non-local Cahn-Hilliard equation with a parabolic reaction-diffusion equation, which accounts for both phase segregation and nutrient diffusion. Previous studies have only considered symmetric potentials for similar models. However, in the biological context of cell-to-cell adhesion, single-well potentials, like the so-called Lennard-Jones potential, seem physically more appropriate. The Cahn-Hilliard equation with this kind of potential has already been analysed. Here, we take a step forward and consider a more refined model. First, we analyse the model with a viscous relaxation term in the chemical potential and subject to suitable initial and boundary conditions. We prove the existence of solutions, a separation property for the phase variable, and a continuous dependence estimate with respect to the initial data. Finally, via an asymptotic analysis, we recover the existence of a weak solution to the initial and boundary value problem without viscosity, provided that the chemotactic sensitivity is small enough.

Paper Structure

This paper contains 13 sections, 5 theorems, 142 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Statements of the results
  4. Proof of Theorem \ref{['thm:existence_weak_tau>0']}
  5. Controlling the spatial average
  6. The approximating problem
  7. A priori estimates
  8. Passage to the limit
  9. Further results in the viscous case
  10. Proof of Theorem \ref{['thm:strong_solutions_tau>0']}.
  11. Proof of Theorem \ref{['thm:separation_property_varphi_tau>0']}
  12. Proof of Theorem \ref{['thm:continuous_dependence_tau>0']}
  13. Proof of Theorem \ref{['thm:asymptotic_tau_to_0']}

Key Result

Theorem 2.2

Let $\tau\in (0,1]$ be fixed. Assume that eq:assumption1-ass:kernel hold. Moreover, let the initial data fulfil Then, there exists a triplet $(\varphi,\mu,\sigma)$ that is a weak solution to eq:varphi-eq:ics in the sense of Definition def:weak_sol. Moreover, it holds that If we also suppose then, $\sigma\in L^\infty(Q)$ with

Figures (1)

  • Figure 1: The blue line represents the Lennard--Jones potential \ref{['def:LJ']}, and the green dashed line its regularisation \ref{['eq:regularized_potential']}-\ref{['eq:regularized_potential2']}, corresponding to the value $\lambda=0.1$. Here $h$ has been selected as $h=0.4$, corresponding to ${\overline{\varphi}}=0.6$.

Theorems & Definitions (9)

  • Definition 2.1: Weak solution
  • Theorem 2.2: Existence of a weak solution
  • Remark 2.3
  • Theorem 2.4: Existence of a strong solution
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7: Strict separation property
  • Theorem 2.8: Continuous dependence on the initial data
  • Theorem 2.9: Asymptotics as $\tau\to0^+$