A coupled convection-diffusion level set model for tracking epithelial cells in colonic crypts

Abstract

Colorectal cancer is initiated in colonic crypts as a consequence of alterations leading to the disruption of the normal colonic cellular process. We propose a new model, which couples a convection-di usion type equation with a level set equation, for tracking the time evolution of an epithelial cell set, inside a colonic crypt, until it reaches the top of the crypt. The convection-di usion equation describes the evolution of the density of the cells in the epithelial cell set. The parameters of this equation regulate the geometric and temporal cellular mechanism, and di erent parameter choices lead to distinct cell behavior. The level set equation tracks the location and shape of the epithelial cell set, inside the crypt, as well as its interface, separating the cell set from the others cells, which reside within the crypt. The interfacial velocity of the epithelial cell set is obtained from the convection-di usion type equation. Some in silico experiments are described. They are performed in a relative small time, with respect to the real biological evolution.