Viscoelastic Cahn--Hilliard models for tumour growth

TL;DR

A new phase field model for tumour growth where viscoelastic effects are taken into account and stability and existence of solutions for the fully-discrete finite element scheme are shown and positive definiteness of the discrete Cauchy--Green tensor is proved.

Abstract

We introduce a new phase field model for tumour growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn--Hilliard equation with source terms for the tumour cells and a convected reaction-diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behaviour of tumours, are taken into account. The model is completed by a viscoelastic system constisting of the Navier--Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy--Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumour growth in two space dimensions $d=2$ by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. $Δt\leq Ch^2$, is required. Moreover, in arbitrary dimensions $d\in\{2,3\}$, we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy--Green tensor is proved with a regularization technique that was first introduced by J. W. Barrett, S. Boyaval ("Existence and approximation of a (regularized) Oldroyd-B model". In: M3AS. 21.9 (2011), pp. 1783--1837). After that, we improve the regularity results in arbitrary dimensions $d\in\{2,3\}$ and in two dimensions $d=2$, where a CFL condition is required. Then, for $d=2$, we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results.

Figures (14)

• Figure 1: Configurations of a viscoelastic cell-mixture within a fixed domain, described by a virtual decomposition of the total deformation. Adapted from malek_prusa_2018.
• Figure 2: The elastic energy densities for the Oldroyd-B model \ref{['eq:energy_oldroyd']} with $\kappa=\kappa_0=1$ (black), the FENE-P model \ref{['eq:energy_fene-p']} with $b=3.5$ (blue) and the generalized Peterlin model \ref{['eq:energy_gen_peterlin']} (red) in the one dimensional case.
• Figure 3: The functions $G$ (left) and $\beta$ (right) and their regularizations.
• Figure 4: Initial tumour (left), initial nutrient (center) and initial mesh (right).
• Figure 5: Comparison of the fully viscous model (first row) to the viscoelastic model (second row) at time $t=2$. In the first three columns, $\varphi$, $\sigma$ and $\left\lvert{\pmb{\mathrm{v}}}\right\rvert$ are visualized. In the last column, the final mesh and $\left\lvert{\mathbb{T}_{\mathrm{el}}(\mathbb{B})}\right\rvert$ are shown.

Theorems & Definitions (21)

• Definition 3.2: Weak solution
• Theorem 3.3: Existence of weak solutions
• Remark 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 4.1
• Remark 4.4
• Remark 4.5
• Lemma 4.6
• Lemma 4.7: Stability