A Diffuse Interface Framework for Modeling the Evolution of Multi-cell Aggregates as a Soft Packing Problem Driven by the Growth and Division of Cells

  • J. Jiang
  • K. GarikipatiEmail author
  • S. Rudraraju
Special Issue: Multiscale Modeling of Tissue Growth and Shape


We present a model for cell growth, division and packing under soft constraints that arise from the deformability of the cells as well as of a membrane that encloses them. Our treatment falls within the framework of diffuse interface methods, under which each cell is represented by a scalar phase field and the zero level set of the phase field represents the cell membrane. One crucial element in the treatment is the definition of a free energy density function that penalizes cell overlap, thus giving rise to a simple model of cell–cell contact. In order to properly represent cell packing and the associated free energy, we include a simplified representation of the anisotropic mechanical response of the underlying cytoskeleton and cell membrane through penalization of the cell shape change. Numerical examples demonstrate the evolution of multi-cell clusters and of the total free energy of the clusters as a consequence of growth, division and packing.


Cell aggregates Embryogenesis 



  1. Allen SM, Cahn JW (1979) A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall 27(6):1085–1095CrossRefGoogle Scholar
  2. Alt S, Ganguly P, Salbreux G (2017) Vertex models: from cell mechanics to tissue morphogenesis. Philos Trans R Soc Lond B Biol Sci 372(1720):20150520CrossRefGoogle Scholar
  3. Bangerth W, Hartmann R, Kanschat G (2007) deal. II—a general purpose object oriented finite element library. ACM Trans Math Softw 33(4):24/1–24/27MathSciNetCrossRefzbMATHGoogle Scholar
  4. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, BerlinCrossRefzbMATHGoogle Scholar
  5. Brodland GW (2004) Computational modeling of cell sorting, tissue engulfment, and related phenomena: a review. Appl Mech Rev 57:47–76CrossRefGoogle Scholar
  6. Cahn JW, Hilliard JE (1958) Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys 28(2):258–267CrossRefGoogle Scholar
  7. Fletcher A, Osterfield M, Baker RE, Shvartsman SY (2014) Vertex models of epithelial morphogenesis. Biophys J 106(11):2291–2304CrossRefGoogle Scholar
  8. Gilbert SF (2000) Developmental biology, 6th edn. Sinauer Associates, SunderlandGoogle Scholar
  9. Glazier JA, Graner F (1993) Simulation of the differential adhesion driven rearrangement of biological cells. Phys Rev E 47:2128–2154CrossRefGoogle Scholar
  10. Goel NS, Rogers G (1978) Computer simulation of engulfment and other movements of embryonic tissues. J Theor Biol 71(1):103–140CrossRefGoogle Scholar
  11. Goel N, Campbell RD, Gordon R, Rosen R, Martinez H, Yaas M (1970) Self-sorting of isotropic cells. J Theor Biol 28(3):423–468CrossRefGoogle Scholar
  12. Graner F (1993) Can surface adhesion drive cell-rearrangement? Part I: biological cell-sorting. J Theor Biol 164(4):455–476CrossRefGoogle Scholar
  13. Graner F, Glazier JA (1992) Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys Rev Lett 69:2013–2016CrossRefGoogle Scholar
  14. Heroux M, Bartlett R, Hoekstra VHR, Hu J, Kolda T, Lehoucq R, Long K, Pawlowski R, Phipps E, Salinger A, Thornquist H, Tuminaro R, Willenbring J, Williams A (2003) An overview of Trilinos. Technical report SAND2003-2927, Sandia National LaboratoriesGoogle Scholar
  15. Honda H (1978) Description of cellular patterns by Dirichlet domains: the two-dimensional case. J Theor Biol 72(3):523–543MathSciNetCrossRefGoogle Scholar
  16. Honda H (1983) Geometrical models for cells in tissues. Int Rev Cytol 81:191–248CrossRefGoogle Scholar
  17. Honda H, Yamanaka H, Eguchi G (1986) Transformation of a polygonal cellular pattern during sexual maturation of the avian oviduct epithelium: computer simulation. Development 98(1):1–19Google Scholar
  18. Itskovitz-Eldor J, Schuldiner M, Karsenti D, Eden A, Yanuka O, Amit M, Soreq H, Benvenisty N (2000) Differentiation of human embryonic stem cells into embryoid bodies compromising the three embryonic germ layers. Mol Med 6(2):88CrossRefGoogle Scholar
  19. Kamrin K, Rycroft CH, Nave JC (2012) Reference map technique for finite-strain elasticity and fluid–solid interaction. J Mech Phys Solids 60(11):1952–1969MathSciNetCrossRefGoogle Scholar
  20. Li XS (2005) An overview of SuperLU: algorithms, implementation, and user interface. ACM Trans Math Softw 31(3):302–325MathSciNetCrossRefzbMATHGoogle Scholar
  21. Mills KL, Kemkemer R, Rudraraju S, Garikipati K (2014) Elastic free energy drives the shape of prevascular solid tumors. PloS ONE 9(7):e103245CrossRefGoogle Scholar
  22. Mirams GR, Arthurs CJ, Bernabeu MO, Bordas R, Cooper J, Corrias A, Davit Y, Dunn S-J, Fletcher AG, Harvey DG et al (2013) Chaste: an open source C++ library for computational physiology and biology. PLoS Comput Biol 9(3):e1002970MathSciNetCrossRefGoogle Scholar
  23. Mochizuki A, Wada N, Ide H, Iwasa Y (1998) Cell-cell adhesion in limb formation, estimated from photographs of cell sorting experiments based on a spatial stochastic model. Dev Dyn 211(3):204–214CrossRefGoogle Scholar
  24. Mosaffa P, Asadipour N, Millán D, Rodríguez-Ferran A, Muñoz JJ (2015) Cell-centred model for the simulation of curved cellular monolayers. Comput Part Mech 2(4):359–370CrossRefGoogle Scholar
  25. Mosaffa P, Rodríguez-Ferran A, Muñoz JJ (2017) Hybrid cell-centred/vertex model for multicellular systems with equilibrium preserving remodelling. Int J Numer Methods Biomed Eng 34(3):e2928MathSciNetCrossRefGoogle Scholar
  26. Nonomura M (2012) Study on multicellular systems using a phase field model. PLOS ONE 7(4):1–904CrossRefGoogle Scholar
  27. Reya T, Morrison SJ, Clarke MF, Weissman IL (2001) Stem cells, cancer, and cancer stem cells. Nature 414(6859):105CrossRefGoogle Scholar
  28. Rudraraju S, Mills KL, Kemkemer R, Garikipati K (2013) Multiphysics modeling of reactions, mass transport and mechanics of tumor growth. In: Holzapfel G, Kuhl E (eds) Computer models in biomechanics. Springer, Dordrecht, pp 293–303CrossRefGoogle Scholar
  29. Schierenberg E (2006) Embryological variation during nematode development (2 Jan 2006), WormBook, ed. The C. elegans Research Community, WormBookGoogle Scholar
  30. Swat MH, Thomas GL, Belmonte JM, Shirinifard A, Hmeljak D, Glazier JA (2012) Multi-scale modeling of tissues using compucell3d. In: Asthagiri AR, Arkin AP (eds) Methods in cell biology, vol 110. Elsevier, Amsterdam, pp 325–366Google Scholar
  31. Vedantam S, Patnaik BSV (2006) Efficient numerical algorithm for multiphase field simulations. Phys Rev E 73(1):016703CrossRefGoogle Scholar
  32. Verner SN, Garikipati K (2018) A computational study of the mechanisms growth-driven folding patterns on shells, with application to the developing brain. Extreme Mech Lett 18:58–69CrossRefGoogle Scholar

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© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Mechanical EngineeringUniversity of MichiganAnn ArborUSA
  2. 2.Mechanical Engineering and MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Mechanical EngineeringUniversity of Wisconsin-MadisonMadisonUSA

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