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Approximation and existence of a viscoelastic phase-field model for tumour growth in two and three dimensions

Harald Garcke, Dennis Trautwein

TL;DR

The paper develops a diffuse-interface, viscoelastic phase‑field model for tumor growth that couples a Cahn–Hilliard type evolution of the order parameter $\varphi$ to a quasi‑static nutrient equation, a viscoelastic Oldroyd‑B framework for the elastic tensor $\mathbb{B}$, and a momentum balance with nonzero divergence. It proves global‑in‑time existence of weak solutions in $d=2,3$ under stress diffusion, using a fully discrete finite element scheme with energy estimates, regularizations, and limit passages to recover the continuous problem. The authors also establish convergence of discrete solutions to a global weak solution and validate the approach with numerical simulations in two and three dimensions that illustrate invasion patterns, mechanotaxis, and stress effects. The work provides a rigorous, practically implementable computational framework for analyzing viscoelastic tumor growth with diffuse interfaces and growth‑induced stresses, offering a tool for exploring mechanochemical tumor dynamics.

Abstract

In this work, we present a phase-field model for tumour growth, where a diffuse interface separates a tumour from the surrounding host tissue. In our model, we consider transport processes by an internal, non-solenoidal velocity field. We include viscoelastic effects with the help of a general Oldroyd-B type description with relaxation and possible stress generation by growth. The elastic energy density is coupled to the phase-field variable which allows to model invasive growth towards areas with less mechanical resistance. The main analytical result is the existence of weak solutions in two and three space dimensions in the case of additional stress diffusion. The idea behind the proof is to use a numerical approximation with a fully-practical, stable and (subsequence) converging finite element scheme. The physical properties of the model are preserved with the help of a regularization technique, uniform estimates and a limit passage on the fully-discrete level. Finally, we illustrate the practicability of the discrete scheme with the help of numerical simulations in two and three dimensions.

Approximation and existence of a viscoelastic phase-field model for tumour growth in two and three dimensions

TL;DR

The paper develops a diffuse-interface, viscoelastic phase‑field model for tumor growth that couples a Cahn–Hilliard type evolution of the order parameter to a quasi‑static nutrient equation, a viscoelastic Oldroyd‑B framework for the elastic tensor , and a momentum balance with nonzero divergence. It proves global‑in‑time existence of weak solutions in under stress diffusion, using a fully discrete finite element scheme with energy estimates, regularizations, and limit passages to recover the continuous problem. The authors also establish convergence of discrete solutions to a global weak solution and validate the approach with numerical simulations in two and three dimensions that illustrate invasion patterns, mechanotaxis, and stress effects. The work provides a rigorous, practically implementable computational framework for analyzing viscoelastic tumor growth with diffuse interfaces and growth‑induced stresses, offering a tool for exploring mechanochemical tumor dynamics.

Abstract

In this work, we present a phase-field model for tumour growth, where a diffuse interface separates a tumour from the surrounding host tissue. In our model, we consider transport processes by an internal, non-solenoidal velocity field. We include viscoelastic effects with the help of a general Oldroyd-B type description with relaxation and possible stress generation by growth. The elastic energy density is coupled to the phase-field variable which allows to model invasive growth towards areas with less mechanical resistance. The main analytical result is the existence of weak solutions in two and three space dimensions in the case of additional stress diffusion. The idea behind the proof is to use a numerical approximation with a fully-practical, stable and (subsequence) converging finite element scheme. The physical properties of the model are preserved with the help of a regularization technique, uniform estimates and a limit passage on the fully-discrete level. Finally, we illustrate the practicability of the discrete scheme with the help of numerical simulations in two and three dimensions.
Paper Structure (26 sections, 10 theorems, 191 equations, 11 figures)

This paper contains 26 sections, 10 theorems, 191 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The full mathematical model
  3. Main results and key strategy
  4. Notation
  5. Approximation with a fully-practical numerical scheme
  6. Discrete setting
  7. Discrete function spaces
  8. Results concerning interpolation and projections
  9. The fully-practical numerical scheme
  10. Technical preliminaries and regularizations
  11. Some regularizations
  12. Approximation of the convective term
  13. Another discrete chain rule
  14. Proof of the existence and stability theorem
  15. Regularized discrete scheme with cut-offs
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $T>0$ and $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$, be a bounded Lipschitz domain with polygonal (or polyhedral, respectively) boundary $\partial\Omega$. Moreover, we assume that $\partial_\mathrm{D}\Omega \subset \partial\Omega$ is closed and of positive surface measure, and we set $\partial_ and with constants $C, C(T, c_\infty)>0$, where $C(T, c_\infty)$ depends exponentially on $T$ and

Figures (11)

  • Figure 1: Typical setting for phase-field models, where a smooth interface with width related to $\varepsilon>0$ separates the pure phases.
  • Figure 2: Framework of multiple configurations.
  • Figure 3: Snapshots of the first example with $\kappa_t=0$. First row: $\varphi_h^n$ at $t=0, 5, 10, 14$, where $\varphi_h^n=1$ (red) in the tumour tissue and $\varphi_h^n=-1$ (blue) in the host tissue. Second row: the nutrient $\sigma_h^n$, $\left\lvert{\mathbf{v}_h^n}\right\rvert$ (with the velocity field $\mathbf{v}_h^n$), and both eigenvalues of $\mathbb{B}_h^n$ at $t=14$.
  • Figure 4: Snapshots of the first example with $\kappa_t=0.5$. First row: $\varphi_h^n$ at $t=0, 5, 10, 14$, where $\varphi_h^n=1$ (red) in the tumour tissue and $\varphi_h^n=-1$ (blue) in the host tissue. Second row: the nutrient $\sigma_h^n$, $\left\lvert{\mathbf{v}_h^n}\right\rvert$ (with the velocity field $\mathbf{v}_h^n$), and both eigenvalues of $\mathbb{B}_h^n$ at $t=14$.
  • Figure 5: Snapshots of the first example with $\kappa_t=-0.5$. First row: $\varphi_h^n$ at $t=0, 5, 10, 14$, where $\varphi_h^n=1$ (red) in the tumour tissue and $\varphi_h^n=-1$ (blue) in the host tissue. Second row: the nutrient $\sigma_h^n$, $\left\lvert{\mathbf{v}_h^n}\right\rvert$ (with the velocity field $\mathbf{v}_h^n$), and both eigenvalues of $\mathbb{B}_h^n$ at $t=14$.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Definition 1.1: Weak solution
  • Theorem 1.2: Existence of weak solutions
  • Remark 2.1
  • Theorem 2.2: Well-posedness of the numerical scheme
  • Remark 2.3
  • Remark 2.4: Approximation of the initial and boundary data
  • Lemma 2.5
  • Lemma 2.6
  • Remark 2.7: $\delta\to 0$
  • Lemma 2.8
  • ...and 6 more