  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
consensual 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 an 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 an heterogeneous
media, 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 proven, applying a fixed
point theorem. Numerical results showing the convergence of the periodic model
to the homogenized model are presented.
