Abstract
We consider the movement and viability of individual cells in cell colonies. Cell movement is assumed to take place as a result of sensing the strain energy density as a mechanical stimulus. The model is based on tracking the displacement and viability of each individual cell in a cell colony. Several applications are shown, such as the dynamics of filling a gap within a fibroblast colony and the invasion of a cell colony. Though based on simple principles, the model is qualitatively validated by experiments on living fibroblasts on a flat substrate.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Boussinesq J (1885) Application des potentiels á l’ étude de l’ équilibre et du mouvementdes solides élastiques. Gauthier-Villars, Paris
Burmister D (1945) The general theory of stresses and displacements in layered systems I. J Appl Phys 16: 89–94
Burmister D (1945) The general theory of stresses and displacements in layered soil systems II. J Appl Phys 16: 126–127
Burmister D (1945) The general theory of stresses and displacements in layered soil systems III. J Appl Phys 16: 296–302
Califano JP, Reinhart-King CA (2010) Substrate stiffness and cell area predict cellular traction stresses in single cells and cells in contact. Cell Mol Bioeng 3(1): 68–75
Dallon JC, Ehrlich HP (2008) A review of fibroblast populated collagen lattices. Wound Repair Regen 16: 472–479
Dallon JC (2010) Multiscale modeling of cellular systems in biology. Curr Opin Coll Interface Sci 15: 24–31
Gaffney EA, Pugh K, Maini PK (2002) Investigating a simple model for cutaneous wound healing angiogenesis. J Math Biol 45(4): 337–374
Gefen A (2010) Effects of virus size and cell stiffness on forces, work and pressures driving membrane invagination in a receptor-mediated endocytosis. J Biomech Eng 132: 084501–1-084501-5 (To appear)
Graner F, Glazier J (1992) Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys Rev Lett 69: 2013–2016
Haga H, Irahara C, Kobayashi R, Nakagaki T, Kawabata K (2005) Collective movement of epithelial cells on a collagen gel substrate. Biophys J 88(3): 2250–2256
Hoehme S, Drasdo D (2010) A cell-based simulation software for multi-cellular systems. Bioinformatics 26(20): 2641–2642
Javierre E, Vermolen FJ, Vuik C, van der Zwaag S (2009) A mathematical analysis of physiological and morphological aspects of wound closure. J Math Biol 59: 605–630
Johnson KL (1985) Contact mechanics. Cambridge University Press, Cambridge
Lemmon CA, Chen CS, Romer LH (2009) Cell traction forces direct fibronectin matrix assembly. Biophys J 96: 729–738
Lo CM, Wang HB, Dembo M, Wang YL (2000) Cell movement is guided by the rigidity of the substrate. Biophys J 79(1): 144–152
Luding S (2008) Introduction to discrete element methods: basics of contact force models and how to perform the micro-macro transition to continuum theory. Eur J Environ Civil Eng 12(7–8(Special Issue: Alert Course, Aussois)): 785–826
Maggelakis SA (2004) Modeling the role of angiogenesis in epidermal wound healing. Discret. Cont. Syst. 4: 267–273
Merks MH, Koolwijk P (2009) Modeling morphogenesis in silico and in vitro: towards quantitative, predictive, cell-based modeling. Math Model Nat Phenom 4(4): 149–171
Merkel R, Kirchgesner N, Cesa CM, Hoffmann B (2007) Cell force microscopy on elastic layers of finite thickness. Biophys J 93: 3314–3323
Murray JD (2004) Mathematical biology II: spatial models and biomedical applications. Springer, New York
Olsen L, Sherratt JA, Maini PK (1995) A mechanochemical model for adult dermal wound closure and the permanence of the contracted tissue displacement role. J Theor Biol 177: 113–128
Plank MJ, Sleeman BD (2004) Lattice and non-lattice models of tumour angiogenesis. Bull Math Biol 66: 1785–1819
Reinhart-King CA, Dembo M, Hammer DA (2008) Cell-cell mechanical communication through compliant substrates. Biophys J 95: 6044–6051
Sarvestani AS (2010) On the effect of substrate compliance on cellular mobility. J Biochip Tissue Chip 1: 101. doi:10.4172/2153-0777.1000101
Schugart RC, Friedman A, Zhao R, Sen CK (2008) Wound angiogenesis as a function of tissue oxygen tension: a mathematical model. Proc Nat Acad Sci USA 105(7): 2628–2633
Schwarz US, Bischofs IB (2005) Physical determinants of cell organization in soft media. Med Eng Phys 27: 763–772
Sherratt JA, Murray JD (1991) Mathematical analysis of a basic model for epidermal wound healing. J Math Biol 29: 389–404
Vermolen FJ (2009) A simplified finite-element model for tissue regeneration with angiogenesis. ASCE J Eng Mech 135(5): 450–461
Vermolen FJ, Javierre E (2009) On the construction of analytic solutions for a diffusion-reaction equation with a discontinuous switch mechanism. J Comput Appl Math 231: 983–1003
Wang JHC, Lin J-S (2007) Cell traction force and measurement methods. Biomech Model Mechanobiol 6: 361–371
Xue C, Friedman A, Sen CK (2009) A mathematical model of ischemic cutaneous wounds. Proc Nat Acad Sci USA 106(39): 16783–16787
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Vermolen, F.J., Gefen, A. A semi-stochastic cell-based formalism to model the dynamics of migration of cells in colonies. Biomech Model Mechanobiol 11, 183–195 (2012). https://doi.org/10.1007/s10237-011-0302-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10237-011-0302-6