Homogenization Model for Aberrant Crypt Foci

Abstract

Several explanations can be found in the literature about the origin of colorectal cancer. There is, however, some agreement on the fact that the carcinogenic process is a result of several genetic mutations of normal cells. The colon epithelium is characterized by millions of invaginations, very small cavities, called crypts, where most of the cellular activity occurs. It is agreed in the medical community that a potential first manifestation of the carcinogenic process, observed in conventional colonoscopy images, is the appearance of aberrant crypt foci (ACF). These are clusters of abnormal crypts, morphologically characterized by an atypical behavior of the cells that populate the crypts. In this work a homogenization model is proposed for representing the cellular dynamics in the colon epithelium. The goal is to simulate and predict, in silico, the spread and evolution of ACF, as it can be observed in colonoscopy images. By assuming that the colon is a heterogeneous medium, exhibiting a periodic distribution of crypts, we start this work by describing a periodic model that represents the ACF cell-dynamics in a two-dimensional setting. Then, homogenization techniques are applied to this periodic model to find a simpler model, whose solution symbolizes the averaged behavior of ACF at the tissue level. Some theoretical results concerning the existence of solution of the homogenized model are proved, applying a fixed point theorem. Numerical results showing the convergence of the periodic model to the homogenized model are presented.