A generalized Cahn-Hilliard equation for biological applications

TL;DR

This paper introduces a generalized Cahn-Hilliard equation with proliferation to model cell invasion, analyzing front propagation, velocity selection, and transient behaviors, with results aligning well with discrete models.

Contribution

It develops a continuum GCH model incorporating proliferation for biological invasion, extending previous models and analyzing different adhesion regimes.

Findings

Propagating fronts in subcritical adhesion match Fisher-Kolmogorov dynamics.

Transient secondary peaks occur in supercritical adhesion regimes.

Self-similar relaxation dynamics are observed in the non-proliferation case.

Abstract

Recently we considered a stochastic discrete model which describes fronts of cells invading a wound \cite{KSS}. In the model cells can move, proliferate, and experience cell-cell adhesion. In this work we focus on a continuum description of this phenomenon by means of a generalized Cahn-Hilliard equation (GCH) with a proliferation term. As in the discrete model, there are two interesting regimes. For subcritical adhesion, there are propagating "pulled" fronts, similarly to those of Fisher-Kolmogorov equation. The problem of front velocity selection is examined, and our theoretical predictions are in a good agreement with a numerical solution of the GCH equation. For supercritical adhesion, there is a nontrivial transient behavior, where density profile exhibits a secondary peak. To analyze this regime, we investigated relaxation dynamics for the Cahn-Hilliard equation without proliferation. We found that the relaxation process exhibits self-similar behavior. The results of continuum and discrete models are in a good agreement with each other for the different regimes we analyzed.