A Cahn--Hilliard--Darcy model for tumour growth with chemotaxis and active transport

TL;DR

This paper derives a thermodynamically consistent Cahn--Hilliard--Darcy model for tumour growth that incorporates nutrient diffusion, chemotaxis, active transport, and other biological processes, and analyzes its stability and interface behaviors.

Contribution

It introduces a new tumour growth model including active transport mechanisms ensuring thermodynamic consistency and develops novel sharp interface models with jump conditions.

Findings

Active transport significantly influences tumour growth dynamics.

New sharp interface models with nutrient density jumps are proposed.

Active transport affects stability and growth scenarios.

Abstract

Using basic thermodynamic principles we derive a Cahn--Hilliard--Darcy model for tumour growth including nutrient diffusion, chemotaxis, active transport, adhesion, apoptosis and proliferation. The model generalises earlier models and in particular includes active transport mechanisms which ensure thermodynamic consistency. We perform a formally matched asymptotic expansion and develop several sharp interface models. Some of them are classical and some are new which for example include a jump in the nutrient density at the interface. A linear stability analysis for a growing nucleus is performed and in particular the role of the new active transport term is analysed. Numerical computations are performed to study the influence of the active transport term for specific growth scenarios.