The colonic epithelium is filled in its internal walls by small cavities called crypts. These crypts are small tubular glands, containing proliferative cells, that obey to a programmed cell mechanism. The aberrant crypts occur as a consequence of an abnormal behavior of this programmed cell mechanism, inside the crypts, and are thought to be the precursors of colon cancer [1]. Normal crypts have roundish orifices grouped in a regular pattern. However in aberrant crypts, the shape and pattern of the correspondent orifices have different forms, as for example, star-like, elliptical, tubular [2], that can evolve up to have a complete fission of the crypts. The different patterns of colonic pits are important to be modelled and simulated since there is a correlation between histopathology and distinct pit patterns. The goal of this talk is to describe biomechanical differential models representing the crypt pit pattern and a crypt fission in time when a disruption of the healthy functioning of the cell mechanism is active. We use two-dimensional crypt geometries similar to that presented in [3], which enable a better representation of a three-dimensional crypt in such a way that it emulates a top view of the crypt and its orifice, as in a clinician’s 2D visualisation during a colonoscopy exam. The model couples the cell dynamics occurring inside the crypt with the mechanical visco-elastic behaviour of the material surrounding the crypt orifice, and its numerical implementation is based on a finite element method on moving meshes. We present also some numerical approximation of the differential problem of the proliferation in a fixed domain. By using numerical simulations, we show finally how a disturbance of the programmed mechanism of the cells, yield to abnormal colonic pit patterns and crypt fission. References [1] P. Figueiredo, M. Donato, M. Urbano, H. Goulão, H. Gouveia, C. Sofia, M. Leitão, and Diniz Freitas. Aberrant Crypt Foci: Endoscopic Assessment and Cell Kinetics Characterization. International Journal of Colorectal Disease, 24(4):441–450, 2009. [2] Huang, Qiyang, et al. Interobserver and Intra-observer Consistency in the Endoscopic Assessment of Colonic Pit Patterns. Gastrointestinal Endoscopy, 60(4): 520-526, 2004. [3] Isabel N. Figueiredo, Carlos Leal, Giuseppe Romanazzi, Bjorn Engquist. Homogenization Model for Aberrant Crypt Foci. SIAM Journal on Applied Mathematics, 76(3):1152–1177, 2016.
|