# A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis

## TL;DR

This paper develops a comprehensive multiphase tumour growth model combining Cahn-Hilliard-Darcy equations with reaction-diffusion systems, capturing complex biological phenomena including necrosis, nutrient dynamics, and angiogenesis, and simplifies the equations using volume-averaged velocity.

## Contribution

It introduces a novel multiphase tumour model with volume-averaged velocity and derives sharp interface models through asymptotic analysis.

## Key findings

- Numerical simulations reveal the impact of necrosis on tumour growth.
- The model effectively captures nutrient diffusion, angiogenesis, and hypoxia.
- Finite element computations demonstrate the model's applicability.

## 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.