Biomechanics and Modeling in Mechanobiology

, Volume 17, Issue 3, pp 727–743 | Cite as

Modeling mechanical inhomogeneities in small populations of proliferating monolayers and spheroids

  • Emma Lejeune
  • Christian LinderEmail author
Original Paper


Understanding the mechanical behavior of multicellular monolayers and spheroids is fundamental to tissue culture, organism development, and the early stages of tumor growth. Proliferating cells in monolayers and spheroids experience mechanical forces as they grow and divide and local inhomogeneities in the mechanical microenvironment can cause individual cells within the multicellular system to grow and divide at different rates. This differential growth, combined with cell division and reorganization, leads to residual stress. Multiple different modeling approaches have been taken to understand and predict the residual stresses that arise in growing multicellular systems, particularly tumor spheroids. Here, we show that by using a mechanically robust agent-based model constructed with the peridynamic framework, we gain a better understanding of residual stresses in multicellular systems as they grow from a single cell. In particular, we focus on small populations of cells (1–100 s) where population behavior is highly stochastic and prior investigation has been limited. We compare the average strain energy density of cells in monolayers and spheroids using different growth and division rules and find that, on average, cells in spheroids have a higher strain energy density than cells in monolayers. We also find that cells in the interior of a growing spheroid are, on average, in compression. Finally, we demonstrate the importance of accounting for stochastic fluctuations in the mechanical environment, particularly when the cellular response to mechanical cues is nonlinear. The results presented here serve as a starting point for both further investigation with agent-based models, and for the incorporation of major findings from agent-based models into continuum scale models when explicit representation of individual cells is not computationally feasible.


Peridynamics Tumor growth Morphogenesis Cell division 

Mathematics Subject Classification

92C10 74L15 



We would like to thank Claudia Vasquez, Andrew Price, Vipul Vachharajani, and Alex Dunn for the stimulating discussions and helpful comments.

Funding This work was supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-114747.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Supplementary material


  1. Akanuma T, Chen C, Sato T, Merks R, Sato T (2016) Memory of cell shape biases stochastic fate decision-making despite mitotic rounding. Nat Commun 7:11963CrossRefGoogle Scholar
  2. Ambrosi D, Mollica F (2002) On the mechanics of a growing tumor. Int J Eng Sci 40:1297–1316MathSciNetCrossRefzbMATHGoogle Scholar
  3. Ambrosi D, Pezzuto S (2012) Active stress versus active strain in mechanobiology: constitutive issues. J Elast 107(2):199–212CrossRefzbMATHGoogle Scholar
  4. Ambrosi D, Preziosi L (2008) Cell adhesion mechanisms and stress relaxation in the mechanics of tumours. Biomech Model Mechanobiol 8(5):397–413CrossRefGoogle Scholar
  5. Ambrosi D, Ateshian G, Arruda E, Cowin S, Dumais J, Goriely A, Holzapfel G, Humphrey J, Kemkemer R, Kuhl E et al (2011) Perspectives on biological growth and remodeling. J Mech Phys Solids 59(4):863–883MathSciNetCrossRefzbMATHGoogle Scholar
  6. Ambrosi D, Preziosi L, Vitale G (2012) The interplay between stress and growth in solid tumors. Mech Res Commun 42:87–91CrossRefGoogle Scholar
  7. Ambrosi D, Pezzuto S, Riccobelli D, Stylianopoulos T, Ciarletta P (2016) Solid tumors are poroelastic solids with a chemo mechanical feedback on growth. J Elast 129:1–18MathSciNetzbMATHGoogle Scholar
  8. Araujo RP, McElwain DLS (2004) A linear-elastic model of anisotropic tumour growth. Eur J Appl Math 15(3):365–384MathSciNetCrossRefzbMATHGoogle Scholar
  9. Bosveld F, Markova O, Guirao B, Martin C, Wang Z, Pierre A, Balakireva M, Gaugue I, Ainslie A, Christophorou N, Lubensky D (2016) Epithelial tricellular junctions act as interphase cell shape sensors to orient mitosis. Nature 530(7591):495CrossRefGoogle Scholar
  10. Breslin S, O’Driscoll L (2013) Three-dimensional cell culture: the missing link in drug discovery. Drug Discov Today 18(5):240–249CrossRefGoogle Scholar
  11. Byrne H, Drasdo D (2008) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58(4):657–687MathSciNetzbMATHGoogle Scholar
  12. Byrne H, Preziosi L (2003) Modelling solid tumour growth using the theory of mixtures. Math Med Biol 20(4):341–366CrossRefzbMATHGoogle Scholar
  13. Cheng G, Tse J, Jain RK, Munn L (2009) Micro-environmental mechanical stress controls tumor spheroid size and morphology by suppressing proliferation and inducing apoptosis in cancer cells. PLoS ONE 4(2):e4632CrossRefGoogle Scholar
  14. Deisboeck TS, Wang Z, Macklin P, Cristini V (2011) Multiscale cancer modeling. Annu Rev Biomed Eng 13:127–55CrossRefGoogle Scholar
  15. Dortdivanlioglu B, Javili A, Linder C (2017) Computational aspects of morphological instabilities using isogeometric analysis. Comput Methods Appl Mech Eng 316:261–279MathSciNetCrossRefGoogle Scholar
  16. Drasdo D, Höhme S (2005) A single-cell-based model of tumor growth in vitro: monolayers and spheroids. Phys Biol 2(3):133–147CrossRefGoogle Scholar
  17. Drasdo D, Loeffler M (2001) Individual-based models to growth and folding in one-layered tissues: intestinal crypts and early development. Nonlinear Anal Theory Methods Appl 47(1):245–256MathSciNetCrossRefzbMATHGoogle Scholar
  18. Drasdo D, Höhme S, Block M (2007) On the role of physics in the growth and pattern formation of multi-cellular systems: what can we learn from individual-cell based models? J Stat Phys 128(1–2):287–345MathSciNetCrossRefzbMATHGoogle Scholar
  19. Folkman J, Hochberg M (1973) Self-regulation of growth in three dimensions. J Exp Med 138(4):745–753CrossRefGoogle Scholar
  20. Galle J, Aus G, Schaller G, Beyer T, Drasdo D (2006) Individual cell based models of the spatial temporal organization of multicellular systems-achievements and limitations. Cytometry A 69(7):704–710CrossRefGoogle Scholar
  21. Galle J, Preziosi L, Tosin A (2009) Contact inhibition of growth described using a multiphase model and an individual cell based model. Appl Math Lett 22(10):1483–90MathSciNetCrossRefzbMATHGoogle Scholar
  22. Gibson W, Veldhuis J, Rubinstein B, Cartwright H, Perrimon N, Brodland G, Nagpal R, Gibson M (2011) Control of the mitotic cleavage plane by local epithelial topology. Cell 144(3):427–438CrossRefGoogle Scholar
  23. Gillies TE, Cabernard C (2011) Cell division orientation in animals. Curr Biol 21(15):R599–R609CrossRefGoogle Scholar
  24. Giverso C, Ciarletta P (2016) On the morphological stability of multicellular tumour spheroids growing in porous media. Eur Phys J E Soft Matter Biol Phys 39(10):92CrossRefGoogle Scholar
  25. Guz N, Dokukin M, Kalaparthi V, Sokolov I (2014) If cell mechanics can be described by elastic modulus: study of different models and probes used in indentation experiments. Biophys J 107(3):564–575CrossRefGoogle Scholar
  26. Helmlinger G, Netti PA, Lichtenbeld HC, Melder RJ, Jain RK (1997) Solid stress inhibits the growth of multicellular tumor spheroids. Nat Biotechnol 15:778–783CrossRefGoogle Scholar
  27. Hertwig O (1884) Investigations on the morphology and physiology of the cell: the problem of fertilization and isotropy of the egg, a theory of heredity, vol 3. FischerGoogle Scholar
  28. Jagiella N, Müller B, Müller M, Vignon-Clementel I, Drasdo D (2016) Inferring growth control mechanisms in growing multi-cellular spheroids of nsclc cells from spatial-temporal image data. PLoS Comput Biol 12(2):1004,412Google Scholar
  29. Janet M, Cheng G, Tyrrell J, Wilcox-Adelman S, Boucher Y, Jain R, Munn L (2012) Mechanical compression drives cancer cells toward invasive phenotype. Proc Natl Acad Sci 109(3):911–916CrossRefGoogle Scholar
  30. Jones A, Byrne H, Gibson J, Dold J (2000) A mathematical model of the stress induced during avascular tumour growth. J Math Biol 40(6):473–499MathSciNetCrossRefzbMATHGoogle Scholar
  31. Keyomarsi K, Sandoval L, Band V, Pardee A (1991) Synchronization of tumor and normal cells from g1 to multiple cell cycles by lovastatin. Cancer Res 51(13):3602–3609Google Scholar
  32. Kilic B, Madenci E (2010) An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theor Appl Fract Mech 53(3):194–204CrossRefGoogle Scholar
  33. Kim Y, Stolarska MA, Othmer HG (2007) A hybrid model for tumor spheroid growth in vitro I: theoretical development and early results. Math Models Methods Appl Sci 17:1773–1798MathSciNetCrossRefzbMATHGoogle Scholar
  34. Lai V, Nedrelow D, Lake S, Kim B, Weiss E, Tranquillo R, Barocas V (2016) Swelling of collagen-hyaluronic acid co-gels: an in vitro residual stress model. Ann Biomed Eng 44(10):2984–2993CrossRefGoogle Scholar
  35. Lamb B, Luo W, Nagdas S, Yousaf M (2014) Cell division orientation on biospecific peptide gradients. ACS Appl Mater Interfaces 6(14):11523–11528CrossRefGoogle Scholar
  36. Lejeune E, Linder C (2017a) Modeling tumor growth with peridynamics. Biomech Model Mechanobiol 1–17Google Scholar
  37. Lejeune E, Linder C (2017b) Quantifying the relationship between cell division angle and morphogenesis through computational modeling. J Theor Biol 418:1–7MathSciNetCrossRefzbMATHGoogle Scholar
  38. Lejeune E, Javili A, Weickenmeier JE, Kuhl Linder C (2016) Tri-layer wrinkling as a mechanism for anchoring center initiation in the developing cerebellum. Soft Matter 12:5613–5620CrossRefGoogle Scholar
  39. Lima E, Almeida R, Oden J (2015) Analysis and numerical solution of stochastic phase field models of tumor growth. Numer Methods Partial Differ Equ 31(2):552–574MathSciNetCrossRefzbMATHGoogle Scholar
  40. Littlewood D (2015) Roadmap for peridynamic software implementation. SAND Report, Sandia National Laboratories, Albuquerque, NM and Livermore, CAGoogle Scholar
  41. Lorenzo G, Scott M, Tew K, Hughes T, Zhang Y, Liu L, Vilanova G, Gomez H (2016) Tissue-scale, personalized modeling and simulation of prostate cancer growth. Proc Natl Acad Sci 113(48):E7663–E7671CrossRefGoogle Scholar
  42. Madenci E, Oterkus E (2014) Peridynamic theory and its applications. Springer, New YorkCrossRefzbMATHGoogle Scholar
  43. Mascheroni P, Boso D, Preziosi L, Schrefler B (2017) Evaluating the influence of mechanical stress on anticancer treatments through a multiphase porous media model. J Theor Biol 421:179–188MathSciNetCrossRefzbMATHGoogle Scholar
  44. Minc N, Piel M (2012) Predicting division plane position and orientation. Trends Cell Biol 22(4):193–200CrossRefGoogle Scholar
  45. Minc N, Burgess D, Chang F (2011) Influence of cell geometry on division-plane positioning. Cell 144(3):414–426CrossRefGoogle Scholar
  46. Mpekris F, Angeli S, Pirentis A, Stylianopoulos T (2015) Stress-mediated progression of solid tumors: effect of mechanical stress on tissue oxygenation, cancer cell proliferation, and drug delivery. Biomech Model Mechanobiol 14(6):1391–1402CrossRefGoogle Scholar
  47. Nestor-Bergmann A, Goddard G, Woolner S (2014) Force and the spindle: mechanical cues in mitotic spindle orientation. Semin Cell Dev Biol 34:133–139CrossRefGoogle Scholar
  48. Oterkus S (2015) Peridynamics for the solution of multiphysics problems. PhD thesis, The University of ArizonaGoogle Scholar
  49. Preziosi L, Tosin A (2008) Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J Math Biol 58(4):625–656MathSciNetzbMATHGoogle Scholar
  50. Price A, Huang E, Sebastiano V, Dunn A (2017) A semi-interpenetrating network of polyacrylamide and recombinant basement membrane allows pluripotent cell culture in a soft, ligand-rich microenvironment. Biomaterials 121:179–192CrossRefGoogle Scholar
  51. Rahman M, Feng Y, Yankeelov T, Oden J (2017) A fully coupled space-time multiscale modeling framework for predicting tumor growth. Comput Methods Appl Mech Eng 320:261–286MathSciNetCrossRefGoogle Scholar
  52. Ren H, Zhuang X, Cai Y, Rabczuk T (2016) Dual-horizon peridynamics. Int J Numer Meth Eng 108(12):1451–1476MathSciNetCrossRefGoogle Scholar
  53. Roose T, Netti P, Munn L, Boucher Y, Jain R (2003) Solid stress generated by spheroid growth estimated using a linear poroelasticity model. Microvasc Res 66(3):204–212CrossRefGoogle Scholar
  54. Silling S, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17):1526–1535CrossRefGoogle Scholar
  55. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48:175–209MathSciNetCrossRefzbMATHGoogle Scholar
  56. Silling SA, Lehoucq RB (2010) Peridynamic theory of solid mechanics. Adv Appl Mech 44:73–168CrossRefGoogle Scholar
  57. Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88(2):151–184MathSciNetCrossRefzbMATHGoogle Scholar
  58. Stylianopoulos T (2017) The solid mechanics of cancer and strategies for improved therapy. J Biomech Eng 139(2):021,004Google Scholar
  59. Stylianopoulos T, Martin J, Chauhan V, Jain S, Diop-Frimpong B, Bardeesy N, Smith B, Ferrone C, Hornicek F, Boucher Y, Munn L (2012) Causes, consequences, and remedies for growth-induced solid stress in murine and human tumors. Proc Natl Acad Sci 109(38):15101–15108.Google Scholar
  60. Stylianopoulos T, Martin J, Snuderl M, Mpekris F, Jain S, Jain R (2013) Coevolution of solid stress and interstitial fluid pressure in tumors during progression: implications for vascular collapse. Cancer Res 73(13):3833–3841CrossRefGoogle Scholar
  61. Su Y, Chiang P, Cheng L, Lee C, Swami N, Chou C (2015) High aspect ratio nanoimprinted grooves of poly (lactic-co-glycolic acid) control the length and direction of retraction fibers during fibroblast cell division. Biointerphases 10(4):041,008Google Scholar
  62. Sugimura K, Lenne P, Graner F (2016) Measuring forces and stresses in situ in living tissues. Development 143(2):186–196CrossRefGoogle Scholar
  63. Sutherland RM (1988) Cell and environment interactions in tumor microregions: the multicell spheroid model. Science 240(4849):177–184CrossRefGoogle Scholar
  64. Taber L (1995) Biomechanics of growth, remodeling, and morphogenesis. Appl Mech Rev 48(8):487–545CrossRefGoogle Scholar
  65. Tepole AB, Kuhl E (2016) Computational modeling of chemo-bio-mechanical coupling: a systems-biology approach toward wound healing. Comput Methods Biomech Biomed Eng 19(1):13–30CrossRefGoogle Scholar
  66. Théry M, Jiménez-Dalmaroni A, Racine V, Bornens M, Jülicher F (2007) Experimental and theoretical study of mitotic spindle orientation. Nature 447(7143):493–496CrossRefGoogle Scholar
  67. Vavourakis V, Wijeratne P, Shipley R, Loizidou M, Stylianopoulos T, Hawkes D (2017) A validated multiscale in-silico model for mechano-sensitive tumour angiogenesis and growth. PLoS Comput Biol 13(1):e1005,259Google Scholar
  68. Voutouri C, Polydorou C, Papageorgis P, Gkretsi V, Stylianopoulos T (2016) Hyaluronan-derived swelling of solid tumors, the contribution of collagen and cancer cells, and implications for cancer therapy. Neoplasia 18(12):732–741CrossRefGoogle Scholar
  69. Wang Z, Butner J, Kerketta R, Cristini V, Deisboeck TS (2015) Simulating cancer growth with multiscale agent-based modeling. Semin Cancer Biol 30:70–78CrossRefGoogle Scholar
  70. Wyatt T, Harris A, Lam M, Cheng Q, Bellis J, Dimitracopoulos A, Kabla A, Charras G, Baum B (2015) Emergence of homeostatic epithelial packing and stress dissipation through divisions oriented along the long cell axis. Proc Natl Acad Sci 112(18):5726–5731CrossRefGoogle Scholar
  71. Xue S, Li B, Feng X, Gao H (2016) Biochemomechanical poroelastic theory of avascular tumor growth. J Mech Phys Solids 94:409–432MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringStanford UniversityStanfordUSA

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