Generalized Cahn-Hilliard equation for biological applications.
Recently we considered a stochastic discrete model which describes fronts of cells invading a wound [E. Khain, L. M. Sander, and C. M. Schneider-Mizell, J. Stat. Phys. 128, 209 (2007)]. 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, similar to those of the Fisher-Kolmogorov equation. The problem of front velocity selection is examined, and our theoretical predictions are in the 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 good agreement with each other for the different regimes we analyzed.
Khain
Evgeniy
E
Department of Physics, Oakland University, Rochester, Michigan 48309, USA.
Sander
Leonard M
LM
D000465:Algorithms; D002448:Cell Adhesion / Q000502:physiology*; D002449:Cell Aggregation / Q000502:physiology*; D002450:Cell Communication / Q000502:physiology*; D003198:Computer Simulation; D008954:Models, Biological; D014945:Wound Healing / Q000502:physiology*