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Mathematical Model of a Biological Medium with Account for the Active Interactions and Relative Displacements of Cells That Form It

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Abstract

A three-phase continuum model of a biological medium formed by cells, extracellular fluid, and an additional phase responsible for independently controlled active force interaction between the cells is obtained. The model describes the reconstruction of biological tissues with account for the active stresses exerted at intercellular interactions. The constitutive relation for the active stress tensor takes into account different mechanisms of cell interactions, including the chaotic and directed cell activities as the active stresses are created, as well as the anisotropy of their development due to cell density distribution inhomogeneity. On the basis of the model, the problem of forming a cavity within an initially homogeneous cell spheroid due to the loss of stability of the homogeneous state is solved. The constitutive relation for the medium strain rate due to cell rearrangements takes into account two mechanisms of relative cell motion: related to cell adhesion and cellmotility. The participation of differentmechanisms of cell interaction in the self-organization of the biological system that consists of mechanically active cells is investigated.

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References

  1. N. J. Armstrong, K. J. Painter, and J. A. Sherratt, “A Continuum Approach to Modelling Cell-Cell Adhesion,” J. Theor. Biol. 243 (1), 98–113 (2006).

    Article  MathSciNet  Google Scholar 

  2. P. Domschke, D. Trucu, A. Gerisch, and M. Chaplain, “Mathematical Modelling of Cancer Invasion: Implications of Cell Adhesion Variability for Tumour Infiltrative Growth Patterns,” J. Theor. Biol. 361, 41–60 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Gerisch and M. A. J. Chaplain, “Mathematical Modelling of Cancer Cell Invasion of Tissue: Local and Non-localModels and the Effect of Adhesion,” J. Theor. Biol. 250 (4), 684–704 (2008).

    Article  MATH  Google Scholar 

  4. K. J. Painter, N. J. Armstrong, and J. A. Sherratt, “The Impact of Adhesion on Cellular Invasion Processes in Cancer and Development,” J. Theor. Biol. 264 (3), 1057–1067 (2010).

    Article  MathSciNet  Google Scholar 

  5. L. Preziosi and A. Tosin, “Multiphase Modeling of Tumor Growth and Extracellular Matrix Interaction: Mathematical Tools and Applications,” J.Math. Biol. 58, 625–656 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Arduino and L. Preziosi, “A Multiphase Model of Tumour Aggregation in Situ by a Heterogeneous ExtracellularMatrix,” Internat. J. Non-Lin. Mech. 75, 22–30 (2015).

    Article  ADS  Google Scholar 

  7. C. Giverso, M. Scianna, and A. Grillo, “Growing Avascular Tumours as Elasto-Plastic Bodies by the Theory of Evolving Natural Configurations,” Mech. Res. Commun. 68, 31–39 (2015).

    Article  Google Scholar 

  8. T. L. Jackson and H. M. Byrne, “A Mechanical Model of Tumor Encapsulation and Transcapsular Spread,” Mathematical Biosciences 180, 307–328 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Byrne and L. Preziosi, “Modelling Solid Tumour Growth Using the Theory of Mixtures,” Math.Med. Biol. 20, 341–366 (2003).

    Article  MATH  Google Scholar 

  10. J. E. Green, S. L. Waters, K. M. Shakesheff, and H. M. Byrne, “A Mathematical Model of Liver Cell Aggregation in Vitro,” Bull. Math. Biol. 71, 906–930 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Lemon, J. R. King, H. M. Byrne, O. E. Jensen, and K. M. Shakesheff, “Mathematical Modelling of Engineered Tissue Growth Using a Multiphase Porous Flow Mixture Theory,” J. Math. Biol. 52, 571–594 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  12. R. D. O’Dea, S. L. Waters, and H. M. Byrne, “A Multiphase Model for Tissue Construct Growth in a Perfusion Bioreactor,” Math. Med. Biol. 27 (2), 95–127 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  13. G. F. Oster, J. D. Murray, and A. K. Harris, “Mechanical Aspects of Mesenchymal Morphogenesis,” J. Embriol. Exp. Morph. 78, 83–125 (1983).

    MATH  Google Scholar 

  14. R. J. Dyson, J.E.F. Green, J. P. Whiteley, and H.M. Byrne, “An Investigation of the Influence of Extracellular Matrix Anisotropy and Cell-Matrix Interactions on Tissue Architecture,” J. Math. Biol. 72 (7), 1775–1809 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  15. L. A. Davidson, S. D. Joshi, H. Y. Kim, M. Dassow, L. Zhang, and J. Zhou, “Emergent Morphogenesis: ElasticMechanics of a Self-deforming Tissue,” J. Biomech. 43 (1), 63–70 (2010).

    Article  Google Scholar 

  16. L. V. Beloussov, S. A. Logvenkov, and A. A. Stein, “Mathematical Model of an Active Biological Continuous Medium with Account for the Deformations and Rearrangements of the Cells,” Fluid Dynamics 47 (1), 1–9 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  17. S. A. Logvenkov and A. A. Stein, “MathematicalModel of Spatial Self-organization in aMechanically Active CellularMedium,” Biophysics 62 (6), 926–934 (2017).

    Article  Google Scholar 

  18. N. N. Kizilova, S. A. Logvenkov, and A. A. Stein, “Mathematical Modeling of Transport-Growth Processes in Multiphase Biological Continua,” Fluid Dynamics 47 (1), 1–9 (2012).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. P. Tracqui, “Biophysical Models of Tumour Growth,” Rep. Prog. Phys. 72 (5), 056701 (2009).

    Article  ADS  Google Scholar 

  20. I. Vlahinic, H.M. Jennings, J. E. Andrade, and J. J. Thomas, “A Novel and General Form of Effective Stress in a Partially Saturated PorousMaterial: The Influence ofMicrostructure,” Mech.Mater. 43, 25–35 (2011).

    Article  Google Scholar 

  21. R. I. Nigmatulin, Fundamentals of Mechanics of Heterogeneous Media (Nauka, Moscow, 1978) [in Russian].

    Google Scholar 

  22. D. A. Drew and L. A. Segel, “Averaged Equations for Two-Phase Flows,” Stud. Appl.Math. 50 (3), 205–231 (1971).

    Article  MATH  Google Scholar 

  23. A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1977) [in Russian].

    MATH  Google Scholar 

  24. A. A. Samarskii and P. N. Vabishchevich, “Difference Schemes for the Transfer Equations,” Dif. Equations 34 (12), 1675–1685.

  25. J. C. Gerhart, “Mechanisms Regulating Pattern Formation in the Amphibian Egg and Early Embryo,” in Biological Regulation and Development, Ed. by R. Goldberger (Plenum Press, New York, 1980), Vol. 2, pp. 133–316.

    Article  Google Scholar 

  26. M. D. White, J. Zenker, S. Bissiere, and N. Plachta, “How Cells Change Shape and Position in the Early Mammalian Embryo,” Curr. Opin. Cell Biol. 44, 7–13 (2017).

    Article  Google Scholar 

  27. J. C. Fierro-Gonzalez, M. D. White, J. C. Silval, and N. Plachta, “Cadherin-Dependent Filopodia Control Preimplantation Embryo compaction,” Nat. Cell. Biol. 15 (12), 1424–1433 (2013).

    Article  Google Scholar 

  28. T. P. Fleming, E. Butler, J. Collins, B. Sheth, and A. E. Wild, “Cell Polarity andMouse Early Development,” Adv. Mol. and Cell Biol. 26, 67–94 (1998).

    Article  Google Scholar 

  29. S. F. Gilbert, Developmental Biology, 6th ed. (Sinauer Associates, Sunderland, Mass., 2000).

    Google Scholar 

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Authors and Affiliations

  1. National Research University Higher School of Economics, ul. Myasnitskaya 20, Moscow, 101000, Russia

    S. A. Logvenkov

  2. Institute of Mechanics, Moscow State University, Michurinskii pr. 1, Moscow, 119192, Russia

    S. A. Logvenkov

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  1. S. A. Logvenkov
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Correspondence to S. A. Logvenkov.

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Original Russian Text © S.A. Logvenkov, 2018, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2018, No. 5, pp. 3–16.

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Logvenkov, S.A. Mathematical Model of a Biological Medium with Account for the Active Interactions and Relative Displacements of Cells That Form It. Fluid Dyn 53, 583–595 (2018). https://doi.org/10.1134/S0015462818050129

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  • DOI: https://doi.org/10.1134/S0015462818050129

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