Modeling and Computer Simulation of Viscoelastic Crypt Deformation

E. P. Oliveira, G. Romanazzi


Colorectal cancer morphogenesis begins at the cellular level from cell mutations in the intestinal epithelium cavities called crypts. These mutations lead to a pressure difference in the epithelium crypt walls, which causes deformation and generates visible abnormalities in the epithelium. The geometrical modeling of these crypts and the mathematical modeling of the physical process that cause the deformations can be simulated by using a Finite Element Method. The method solves numerically the system of PDE equations that governs this phenomenon and permits to estimate the deformations of the crypt walls. We simulate in this work the crypt deformation when the cell mutations appear in several regions of the crypt epithelium.


Colorectal cancer; Crypts; Finite Element Method; Computer simulation

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I. M. M. Van Leeuwen, H. M. Byrne, O. E. Jensen, and J. R. King, "Crypt

dynamics and colorectal cancer: advances in mathematical modelling", Cell

proliferation, vol. 39, no. 3, pp. 157-181, 2006.

K. Drasdo and M. Loeffler, "Individual-based models to growth and folding in one-layered tissues: intestinal crypts and early development", Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 1, pp. 245-256, 2001.

I. N. Figueiredo, C. Leal, G. Romanazzi, and B. Engquist, "Biomathematical

modle for simulating abnormal orifice patterns in colonic crypts", Mathematical biosciences, vol. 315, p. 108221, 2019.

P. Hogeweg, "Evolving Mechanisms of Morphogenesis: on the Interplay between Differential Adhesion and Cell Differentiation, "Journal of Theoretical Biology", vol. 203, no. 4, pp. 317-333, 2000.

S. Y. Wong, K. H. Chiam, C. T. Lim, and P. Matsudaira, "Computational

model of cell positioning: directed and collective migration in the intestinal

crypt epithelium", Journal of The Royal Society Interface, vol. 7, no. 3,

pp. S351-S363, 2010.

A. A. Almet, B. D. Hughes, K. A. Landman, I. S. Näthke, and J. M. Osborne, "A Multicellular Model of Intestinal Crypt Buckling and Fission", Bulletin of Mathematical Biology, vol. 80, no. 2, pp. 335-359, 2018.

P. Buske, J. Galle, N. Barker, G. Aust, H. Clevers, and M. Loeffler, "A Comprehensive Model of the Spatio-Temporal Stem Cell and Tissue Organisation in the Intestinal Crypt", PLoS Computational Biology, vol. 7, no. 1, p. e1001045, 2011.

J. Galle, M. Homann, and G. Aust, "From single cells to tissue architecture - a bottom-up approach to modelling the spatio-temporal organisation of complex multi-cellular systems", Journal of Mathematical Biology, vol. 58, no. 1, pp. 261-283, 2009.

I. N. Figueiredo, C. Leal, G. Romanazzi, and B. Engquist, "Homogenization model for aberrant crypt foci", SIAM Journal on Applied Mathematics, vol. 76, no. 3, pp. 1152-1177, 2016.

A. A. Almet, P. K. Maini, D. E. Moulton, and H. M. Byrne, "Modeling perspectives on the intestinal crypt, a canonical system for growth, mechanics, and remodeling", Current Opinion in Biomedical Engineering, vol. 15, pp. 32-39, 2020.

G. De Matteis, A. Graudenzi, and M. Antoniotti, "A review of spatial computational models for multi-cellular systems, with regard to intestinal crypts and colorectal cancer development", Journal of Mathematical Biology, vol. 66, no. 7, pp. 1409-1462, 2013.

S. K. Kershaw, H. M. Byrne, D. J. Gavaghan, and J. M. Osborne, "Colorectal cancer through simulation and experiment", IET Systems Biology, vol. 7, no. 3, pp. 57-73, 2013.

C. M. Edwards and S. J. Chapman, "Biomechanical Modelling of Colorectal Crypt Budding and Fission", Bulletin of Mathematical Biology, vol. 69, no. 6, pp. 1927-1942, 2007.

M. R. Nelson, J. R. King, and O. E. Jensen, "Buckling of a growing tissue and the emergence of two-dimensional patterns", Mathematical Biosciences, vol. 246, no. 2, pp. 229-241, 2013.

M. Bjerknes, "Expansion of Mutant Stem Cell Populations in the Human Colon", Journal of Theoretical Biology, vol. 178, no. 4, pp. 81-385, 1996.

B. M. Boman, J. Z. Fields, O. Bonham-Carter, and O. A. Runquist, "Computer Modeling Implicates Stem Cell Overproduction in Colon Cancer Initiation", Cancer Research, vol. 61, no. 23, pp. 8408-8411, 2001.

A. Di Garbo, M. D. Johnston, S. J. Chapman, and P. K. Maini, "Variable

renewal rate and growth properties of cell populations in colon crypts", Physical Review E, vol. 81, no. 6, p. 061909, 2010.

A. d'Onofrio and I. P. M. Tomlinson, "A nonlinear mathematical model of cell turnover, differentiation and tumorigenesis in the intestinal crypt", Journal of Theoretical Biology, vol. 244, no. 3, pp. 367-374, 2007.

I. N. Figueiredo, C. Leal, G. Romanazzi, B. Engquist, and P. N. Figueiredo, "A convection-diffusion-shape model for aberrant colonic crypt morphogenesis", Computing and Visualization in Science, vol. 14, no. 4, pp. 157-166, 2011.

J. Salençon, Handbook of Continuum Mechanics: General Concepts, Thermoelasticy. Springer Science & Business Media, 2012.

N. K. Kyslstad, Simulating the viscoelastic response of the spinal cord. PhD thesis, Faculty of Mathematics and Natural Sciences, University of Oslo, Oslo, 2014.

F. Hecht, O. Pironneau, A. Le Hyaric, and K. Ohtsuka, "Freefem++ manual", 2005.

D. V. Guebel and N. V. Torres, "A computer model of oxygen dynamics in human colon mucosa: implications in normal physiology and early tumor development", Journal of Theoretical Biology, vol. 250, no. 3, pp. 389-409, 2008.


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