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

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.

References in corpus (3)

• Viscoelastic Cahn--Hilliard models for tumour growth
• Numerical analysis for a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport
• On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities

Cited by in corpus (4)

• A phase field model of Cahn-Hilliard type for tumour growth with mechanical effects and damage
• On a diffuse interface model for incompressible viscoelastic two-phase flows
• Structure-preserving approximation of the non-isothermal Cahn-Hilliard system based on the entropy equation
• Parametric finite element approximation of two-phase Navier--Stokes flow with viscoelasticity