A generalized Cahn-Hilliard equation for biological applications

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.