A. D. Bratsun, D. A. Bratsun, I. V. Krasnyakov

MICROSCOPIC SIMULATION OF THE CHEMOMECHANICS OF SQUAMOUS CELL TISSUE

The development of computer technologies makes it possible to implement a mathematical model of tissue dynamics, which includes the behavior of individual cells. The paper describes a mathematical model of a quasi-two-dimensional tissue, which consists of cells represented by dynamically changing polygons. The model includes two important processes that mimic the properties of real cells, namely mitotic division and intercalation. An equation of motion based on the elastic potential energy is written for each vertex of the polygonal cell. In the course of evolution, the tissue tends to take a position corresponding to the minimum of potential energy. The model allows a simple extension to the case of the feedback between the biomechanical and chemical properties of the medium and the introduction of several competing tissue types. The results of numerical simulation of heterogeneous carcinoma of the solid type are given as an example. The prospects for the development of this approach to simulation are discussed.

Acknowledgments: The work was financially supported by the Ministry of Science and Higher Education of the Russian Federation (grant FSNM-2020-0026).

References:

1. Bratsun D., Merkuriev D., Zakharov A., Pismen L. Multiscale modeling of tumor growth induced by circadian rhythm disruption in epithelial tissue. J. Biol. Phys., 2016, vol. 42, pp. 107–132. DOI: 10.1007/s10867-015-9395-y.
2. Lesne A., Foray N., Cathala G., Forne T., Wong H., Victor J.-M. Chromatin fiber allostery and the epigenetic code. J. Phys. Condens. Matter, 2015, vol. 27, 064114. DOI: 10.1088/0953-8984/27/6/064114.
3. Kolobov A.V., Polezhaev A.A. Influence of random malignant cell motility on growing tumor front stability. Computer Research and Modeling, 2009, vol. 1, No. 2, pp. 225–232. (In Russian). DOI: 10.20537/2076-7633-2009-1-2-225-232.
4. Stein A.A., Yudina E.N. Mathematical model of a growing plant tissue as a three-phase deformable medium. Russ. J. Biomech., 2011, vol. 15, no. 1, pp. 42–51. (In Russian).
5. Basan M., Elgeti J., Hannezo E., Rappel W.-J., Levine H. Alignment of cellular motility forces with tissue flow as a mechanism for efficient wound healing. Proc. Natl. Acad. Sci., 2013, vol. 110, No. 7, pp. 2452– 2459. DOI: 10.1073/pnas.1219937110.
6. Von Neumann J. Theory of self-reproducing automata. London, University of Illinois Press, 1966, 388 p.
7. Simpson M.J., Landman K.A., Hughes B.D. Distinguishing between directed and undirected cell motility within an invading cell population. Bull. Math. Biol., 2009, vol. 71, pp. 781–799. DOI: 10.1007/s11538-008-9381-7.
8. Chung C.A., Lin T.H., Chen S.D., Huang H.I. Hybrid cellular automaton modeling of nutrient modulated cell growth in tissue engineering constructs. J. Theor. Biol., 2010, vol. 262, no. 2, pp. 267–278. DOI: 10.1016/j.jtbi.2009.09.031.
9. Interian R., Rodriguez-Ramos R., Valdes-Ravelo F., Ramirez-Torrez A., Ribeiro C.C., Conci A. Tumor growth modelling by cellular automata. Mathematics and Mechanics Complex Systems, 2017, vol. 5, no. 3–4, pp. 239–259. DOI: 10.2140/memocs.2017.5.239.
10. Markov M. A., Markov A.V. Computer simulation of the ontogeny of organisms with different types of symmetry. Paleontol. J., 2014, vol. 48, No. 11, pp. 1–9. DOI: 10.1134/S00310301141 10070.
11. Drasdo D., Loeffler M. Individual-based models to growth and folding in one-layered tissues: Intestinal crypts and early development. Nonlinear Anal., 2001, vol. 47, No. 1, pp. 245–256. DOI: 10.1016/S0362-546X(01)00173-0.
12. Viktorinova I., Pismen L., Aigouy B., Dahmann C. Modeling planar polarity of epithelia: the role of signal relay in collective cell polarization. J. R. Soc. Interface., 2011, vol. 8, pp. 1059–1063. DOI: 10.1098/rsif.2011.0117.
13. Salm M., Pismen L.M. Chemical and mechanical signaling in epithelial spreading. Phys. Biol., 2012, vol. 9, No. 2, pp. 026009–026023. DOI: 10.1088/1478-3975/9/2/026009.
14. Bratsun D.A., Krasnyakov I.V., Pismen L.M. Biomechanical modeling of invasive breast carcinoma under a dynamic change in cell phenotype: collective migration of large groups of cells. Biomech. Model. Mechanobiol., 2020, vol. 19, pp. 723–743. DOI: 10.1007/s10237-019-01244-z.
15. Krasnyakov I.V., Bratsun D.A., Pismen L.M. Mathematical modeling of carcinoma growth with a dynamic change in the phenotype of cells. Computer Research and Modeling, 2018, vol. 10, No. 6, pp. 879–902. DOI: 10.20537/2076-7633-2018-10-6-879-902.
16. Krasnyakov I.V., Bratsun D.A., Pismen L.M.  Mathematical modelling of epithelial tissue growth. Russ. J. Biomech., 2020, vol. 24, no. 4, pp. 375–388. DOI: 10.15593/RJBiomech/ 2020.4.03.
17. Denisov E.V., Gerashchenko T.S., Zavyalova M.V., Litviakov N.V., Tsyganov M.M., Kaigorodova E.V., Slonimskaya E.M., Kzhyshkowska J., Cherdyntseva N.V., Perelmuter V.M. Invasive and drug resistant expression profile of different morphological structures of breast tumors. Neoplasma, 2015, vol. 62, no. 3, pp. 405–411. DOI: 10.4149/neo_2015_041.
18. Guillot C., Lecuit T. Mechanics of epithelial tissue homeostasis and morphogenesis. Science, 2013, vol. 340, No. 6137, pp. 1185–1189. DOI: 10.1126/science.1235249.
19. Bratsun D.A., Krasnyakov I.V. Study of architectural forms of invasive carcinoma based on the measurement of pattern complexity. Math. Model. Nat. Phenom, 2022. (In print).

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