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 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.

Mathematical model for simulation of morphological changes associated to crypt fission in the colon

“…The aim of this work has been to present an interesting application of a mathematical model able to describe an ACF formation in the colonic epithelium, its growth, and associated cell dynamics. This model is a novelty in respect to the previous author works [22,23,24], considering a model for ACF dynamics in a fixed domain or an isolated crypt deformation not ending however in a complete fission. Other works [2,18,40] model only a single crypt fission or an epithelial buckling in tissue domain, not containing however a set of crypts distributed in a two dimensional hexagonal compartment.…”

“…Here p = p(x, t) is the cell-cell adhesion pressure exerted between cells located in the crypt walls and related to the mitotic activity [18,42,63]. This partial differential system (3) is similar to that one used in previous works [22,23,24,25] concerning colonic cell dynamics and classified as a transport/mass conservation model on the dynamics of two types of cells. Moreover, it is also used in the context of a general avascular tumour growth, when different types of cells are considered, see [56].…”

“…However, that model, formulated only in terms of transit cell and pressure, allows for crypt deformations with bifurcation, not ending in a complete fission. Differently, in our work the colon is modelled as a heterogeneous two-dimensional no static medium with a periodic hexagonal crypt distribution, similar to that presented in [22,32] where only a static periodic hexagonal domain has been used. In particular, our model considers a dynamic crypt fission model acting in a small set of seven crypts, with deformations that can undergo a complete fission, which generate two new crypts and move neighbouring crypts in the colonic epithelium.…”

“…In fact, the rate β given in ( 5) is activated only at the top and transforms transit cells T into fully differentiated cells F . Note that, these rates are different to those used in previous works [22,23,24,25], since they are scaled on the dimension of the crypt height, allowing to model colonic cell dynamics independently of the crypt size and scale. The parameters γ > 0 and τ > 0 are obtained by using the property that, in the normal case, cell velocity at the orifice x 3 = h corresponds to the rate of 0.85 positions per hour [41,48,51,63].…”

Numerical analysis of multiscale methods for elliptic-parabolic problems with application in the cell dynamics during the formation of colorectal cancer

“…In (FIGUEIREDO et al, 2016) is considered two population of cells as in FRANKS, 2004). Based on tumor growth, their model is presented by the following elliptic-parabolic coupled model…”