# A tractable mathematical model for tissue growth

## TL;DR

This paper develops a simplified mathematical model for tissue growth using free boundary problems and mean curvature flow, providing analytical and numerical insights into tumor dynamics.

## Contribution

It introduces a tractable free boundary model for tissue growth incorporating a PDE-driven forcing term and offers linear stability analysis and diffuse-interface approximations.

## Key findings

- Model captures key features of tissue growth and death.
- Finite-element simulations validate the diffuse-interface approximation.
- Stability analysis provides insights into interface evolution.

## Abstract

Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow (together with a `kinetic under-cooling' regularisation) where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We perform linear stability analysis and derive a diffuse-interface approximation of the model. Finite-element discretisations of two closely related models are presented, together with computational results comparing the approximate solutions.