A generalized Cahn-Hilliard equation for biological applications

Professor Leonard M. Sander
Physics Dept., University of Michigan, Ann Arbor


We study fronts of cells such as those invading a wound or in a growing tumor. First we look at a discrete stochastic model in which cells can move, proliferate, and experience cell-cell adhesion. We compare this with a coarse-grained, continuum description of this phenomenon by means of a generalized Cahn-Hilliard equation (GCH) with a proliferation term.

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. The results of continuum and discrete models are in a good agreement with each other for the different regimes we analyzed.