A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

Abstract

We derive a Cahn-Hilliard-Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions. Via a coupling of the Cahn-Hilliard-Darcy equations to a system of reaction-diffusion equations a multitude of phenomena such as nutrient diffusion and consumption, angiogenesis, hypoxia, blood vessel growth, and inhibition by toxic agents, which are released for example by the necrotic cells, can be included. A new feature of the modelling approach is that a volume-averaged velocity is used, which dramatically simplifies the resulting equations. With the help of formally matched asymptotic analysis we develop new sharp interface models. Finite element numerical computations are performed and in particular the effects of necrosis on tumour growth is investigated numerically.

Optimal Control of a Phase Field System Modelling Tumor Growth with Chemotaxis and Singular Potentials

“…The capital letters A, B, D, P, χ denote positive coefficients that stand for the apoptosis rate, nutrient supply rate, nutrient consumption rate, proliferation rate, and chemotaxis coefficient, in this order. In addition, let us point out that the contributions χ σ and χ ∆ϕ model pure chemotaxis, namely, the movement of tumor cells towards regions of high nutrients, and the active transport that describes the movement of the nutrient towards the tumor cells (see [17,18,20] for more details). Furthermore, the function h has originally been introduced as an interpolation function between −1 and 1 in order to have h(−1) = 0 and h(1) = 1, so that the mechanisms modelled by the terms (P σ − A − u)h(ϕ) and Dσh(ϕ) are switched off in the healthy case, which corresponds to ϕ = −1, and are fully active in the tumorous case ϕ = 1.…”

The phase field method for geometric moving interfaces and their numerical approximations

“…Another example of broad applications of phase field modeling and simulations in biomedical field is on the cancer and tumor growth models. This is a subject that have been studied by a number of authors [222,411,310], see [316] for a review on the models and numerical methods. 7.5.…”

Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach

“…Furthermore, by interpreting the tumor and the healthy cells as inertia-less fluids, the contribution of the velocity field can be included in the investigation by assuming a Darcy law or a Stokes-Brinkman equation. In this regards, let us refer to [16,19,21,[23][24][25][26]28], where further mechanisms such as chemotaxis and active transport are also taken into account. Furthermore, we also point out the paper [22], where a non-local model is proposed.…”