Homogenization Model for Aberrant Crypt Foci

TL;DR

This paper develops a homogenization model to simulate and predict the spread of Aberrant Crypt Foci in the colon epithelium, aiding understanding of early colorectal cancer development through in silico analysis.

Contribution

It introduces a homogenization approach to model cellular dynamics of ACF, simplifying complex tissue heterogeneity into an averaged, computationally manageable model.

Findings

Homogenized model accurately predicts ACF evolution.

Numerical results confirm convergence of periodic to homogenized model.

Theoretical proof of solution existence for the homogenized model.

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 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.