Individual-based and continuum models of growing cell populations: a comparison

  • Helen Byrne
  • Dirk Drasdo


In this paper we compare two alternative theoretical approaches for simulating the growth of cell aggregates in vitro: individual cell (agent)-based models and continuum models. We show by a quantitative analysis of both a biophysical agent-based and a continuum mechanical model that for densely packed aggregates the expansion of the cell population is dominated by cell proliferation controlled by mechanical stress. The biophysical agent-based model introduced earlier (Drasdo and Hoehme in Phys Biol 2:133–147, 2005) approximates each cell as an isotropic, homogeneous, elastic, spherical object parameterised by measurable biophysical and cell-biological quantities and has been shown by comparison to experimental findings to explain the growth patterns of dense monolayers and multicellular spheroids. Both models exhibit the same growth kinetics, with initial exponential growth of the population size and aggregate diameter followed by linear growth of the diameter and power-law growth of the cell population size. Very sparse monolayers can be explained by a very small or absent cell–cell adhesion and large random cell migration. In this case the expansion speed is not controlled by mechanical stress but by random cell migration and can be modelled by the Fisher–Kolmogorov–Petrovskii–Piskounov (FKPP) reaction–diffusion equation. The growth kinetics differs from that of densely packed aggregates in that the initial spread, as quantified by the radius of gyration, is diffusive. Since simulations of the lattice-free agent-based model in the case of very large random migration are too long to be practical, lattice-based cellular automaton (CA) models have to be used for a quantitative analysis of sparse monolayers. Analysis of these dense monolayers leads to the identification of a critical parameter of the CA model so that eventually a hierarchy of three model types (a detailed biophysical lattice-free model, a rule-based cellular automaton and a continuum approach) emerge which yield the same growth pattern for dense and sparse cell aggregates.


Tumour growth Monolayers Agent-based model Individual-based model Continuum approach 

Mathematics Subject Classification (2000)

92C05 92-08 92B05 


  1. 1.
    Adam J, Belomo N (1997) A survey of models for tumor-immune system dynamics. Birkhäuser, BostonzbMATHGoogle Scholar
  2. 2.
    Alarcon T, Byrne H, Maini P (2004) A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells. J Theor Biol 229: 395–411CrossRefMathSciNetGoogle Scholar
  3. 3.
    Alber MS, Kiskowski MA, Glazier JA, Jiang Y (2002) On cellular automaton approaches to modeling biological cells. In: Rosenthal J, Gilliam DS (eds) Mathematical systems theory in biology, communication, and finance, IMA 142. Springer, New York, pp 1–40Google Scholar
  4. 4.
    Alcaraz J, Buscemi L, Grabulosa M, Trepat X, Fabry B, Farre R, Navajas D (2003) Microrheology of human lung epithelial cells measured by atomic force microscopy. Biophys J 84: 2071–2079CrossRefGoogle Scholar
  5. 5.
    Allen M, Tildersley D (1987) Computer Simulation of Liquids. Oxford Science Publications, OxfordzbMATHGoogle Scholar
  6. 6.
    Ambrosi D, Mollica F (2002) On the mechanics of a growing tumor. Int J Eng Sci 40(12): 1297–1316CrossRefMathSciNetGoogle Scholar
  7. 7.
    Ambrosi D, Mollica F (2004) The role of stress in the growth of a multicell spheroid. J Math Biol 48(5): 477–479zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Anderson A, Chaplain MAJ, Rejniak K (2007) Single-cell-based models in biology and medicine. Birkhäuser, BaselCrossRefGoogle Scholar
  9. 9.
    Anderson A, Chaplain MAJ, Newman E, Steele R, Thompson A (2000) Mathematical modeling of tumor invasion and metastasis. J Theor Med 2: 129–154zbMATHGoogle Scholar
  10. 10.
    Araujo R, McElwain D (2004) A history of the study of solid tumour growth: the contribution of mathematical models. Bull Math Biol 66: 1039–1091CrossRefMathSciNetGoogle Scholar
  11. 11.
    Beysens D, Forgacs G, Glazier J (2000) Cell sorting is analogous to phase ordering in fluids. Proc Natl Acad Sci USA 97(17): 9467–9471CrossRefGoogle Scholar
  12. 12.
    Block M, Schoell E, Drasdo D (2007) Classifying the expansion kinetics and critical surface dynamics of growing cell populations. Phys Rev Lett 99: 248,101–248,104CrossRefGoogle Scholar
  13. 13.
    Breward C, Byrne H, Lewis C (2002) The role of cell–cell interactions in a two-phase model for avasular tumour growth. J Math Biol 45: 125–152zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bru A, Albertos S, Subiza J, Garcia-Arsenio J, Bru I (2003) The universal dynamics of tumor growth. Biophys J 85: 2948–2961CrossRefGoogle Scholar
  15. 15.
    Bru A, Pastor J, Fernaud I, Bru I, Melle S, Berenguer C (1998) Super-rough dynamics of tumor growth. Phys Rev Lett 81(18): 4008–4011CrossRefGoogle Scholar
  16. 16.
    Byrne H (1997) The importance of intercellular adhesion in the development of carcinomas. IMA J Math Appl Med Biol 14: 305–323zbMATHCrossRefGoogle Scholar
  17. 17.
    Byrne H, Chaplain J (1996) Modelling the role of cell–cell adhesion in the growth and development of carcinomas. Math Comput Model 12: 1–17CrossRefGoogle Scholar
  18. 18.
    Byrne H, Chaplain M (1997) Free boundary value problem associated with the growth and development of multicellular spheroids. Eur J Appl Math 8: 639–658zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Byrne HM, King JR, McElwain DLS, Preziosi L (2003) A two-phase model of solid tumor growth. Appl Math Lett 16(4): 567–573zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Byrne H, Preziosi L (2003) Modelling solid tumour growth using the theory of mixtures. Math Med Biol 20: 341–366zbMATHCrossRefGoogle Scholar
  21. 21.
    Carpick R, Ogletree DF, Salmeron M (1999) A gerneral equation for fitting contact area and friction vs load measurements. J Colloid Interface Sci 211: 395–400CrossRefGoogle Scholar
  22. 22.
    Chaplain M, Graziano L, Preziosi L (2006) Mathematical modelling of the loss of of tissue compression responsiveness and its role in solid tumour development. Math Med Biol 23(3): 197–229zbMATHCrossRefGoogle Scholar
  23. 23.
    Chen C, Byrne H, King J (2001) The influence of growth-induced stress from the surrounding medium on the development of multicell spheroids. J Math Biol 43: 191–220zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Chesla S, Selvaraj P, Zhu C (1998) Measuring two-dimensional receptor-ligand binding kinetics by micropipette. Biophys J 75: 1553–1557CrossRefGoogle Scholar
  25. 25.
    Chu YS et al (2005) Johnson–Kendall–Roberts theory applied to living cells. Phys Rev Lett 94: 028,102CrossRefGoogle Scholar
  26. 26.
    Cickovski T, Huang C, Chaturvedi R, Glimm T, Hentschel H, Alber M, Glazier JA, Newman SA, Izaguirre JA (2005) A framework for three-dimensional simulation of morphogenesis. IEEE/ACM Trans Comput Biol Bioinformatics 2(3): 273–288CrossRefGoogle Scholar
  27. 27.
    Please CP, Pettet G, McElwain D (1998) A new approach to modelling the formation of necrotic regions in tumours. Appl Math Lett 11: 89–94zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Cristini V, Lowengrub J, Nie Q (2003) Nonlinear simulations of tumor growth. J Math Biol 46: 191–224zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Dallon J, Othmer H (2004) How cellular movement determines the collective force generated by the dictyostelium discoideum slug. J Theor Biol 231: 203–222CrossRefMathSciNetGoogle Scholar
  30. 30.
    DeMasi A, Luckhaus S, Presutti E (2005) Two-scale hydrodynamic limit for a model of malignant tumor cells. MPI-MIS (Preprint) 2: 1–47Google Scholar
  31. 31.
    Dormann S, Deutsch A (2002) Modeling of self-organized avascular tumor growth with a hybrid cellular automaton. In Silico Biol 2: 0035Google Scholar
  32. 32.
    Drasdo D (1996) Different growth regimes found in a monte-carlo model of growing tissue cell populations. In: Schweitzer F (eds) Self organization of complex structures: from individual to collective dynamics. Gordon & Breach, New York, pp 281–291Google Scholar
  33. 33.
    Drasdo D (2003) On selected individual-based approaches to the dynamics of multicellular systems. In: Alt W, Chaplain M, Griebel M (eds) Multiscale modeling. Birkhäuser, BaselGoogle Scholar
  34. 34.
    Drasdo D (2005) Coarse graining in simulated cell populations. Adv Complex Syst 8(2 & 3): 319–363zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Drasdo D (2008) Center-based single-cell models: an approach to multi-cellular organization based on a conceptual analogy to colloidal particles. In: Anderson A, Chaplain M, Rejniak K (eds) Single-cell-based models in biology and medicine. Birkhäuser, Basel (in press)Google Scholar
  36. 36.
    Drasdo D, Forgacs G (2000) Modelling the interplay of generic and genetic mechanisms in cleavage, blastulation and gastrulation. Dev Dyn 219: 182–191CrossRefGoogle Scholar
  37. 37.
    Drasdo D, Hoehme S (2005) A single-cell based model to tumor growth in-vitro: monolayers and spheroids. Phys Biol 2: 133–147CrossRefGoogle Scholar
  38. 38.
    Drasdo D, Hoehme 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): 319–363MathSciNetGoogle Scholar
  39. 39.
    Drasdo D, Höhme S (2003) Individual-based approaches to birth and death in avascular tumors. Math Comput Model 37: 1163–1175zbMATHCrossRefGoogle Scholar
  40. 40.
    Drasdo D, Kree R, McCaskill J (1995) Monte-carlo approach to tissue-cell populations. Phys Rev E 52(6): 6635–6657CrossRefGoogle Scholar
  41. 41.
    Friedman A (2007) Mathematical analysis and challenges arrising from models of tumor growth. Math Model Methods Appl Sci 17: 1751–1772zbMATHCrossRefGoogle Scholar
  42. 42.
    Galle J, Aust G, Schaller G, Beyer T, Drasdo D (2006) Single-cell based mathematical models to the spatio-temporal pattern formation in multi-cellular systems. Cytometry A (in press)Google Scholar
  43. 43.
    Galle J, Loeffler M, Drasdo D (2005) Modelling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro. Biophys J 88: 62–75CrossRefGoogle Scholar
  44. 44.
    Galle J, Sittig D, Hanisch I, Wobus M, Wandel E, Loeffler M, Aust G (2006) Individual cell-based models of tumor-environment interactions: multiple effects of cd97 on tumor invasion. J Am Path 169(5): 1802–1811CrossRefGoogle Scholar
  45. 45.
    Graner F, Glazier J (1992) Simulation of biological cell sorting using a two-dimensional extended potts model. Phys Rev Lett 69(13): 2013–2016CrossRefGoogle Scholar
  46. 46.
    Greenspan H (1976) On the growth and stability of cell cultures and solid tumors. J Theor Biol 56(1): 229–242CrossRefMathSciNetGoogle Scholar
  47. 47.
    Hogeweg P (2000) Evolving mechanisms of morphogenesis: on the interplay between differential adhesion and cell differentiation. J Theor Biol 203: 317–333CrossRefGoogle Scholar
  48. 48.
    Johnson K, Kendall K, Roberts A (1971) Surface energy and the contact of elastic solids. Proc R Soc A 324: 301–313CrossRefGoogle Scholar
  49. 49.
    King J, Franks S (2004) Mathematical analysis of some multidimensional tissue-growth models. Eur J Appl Math 15(3): 273–295zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Landman K, Please C (2001) The influence of growth-induced stress from the surrounding medium on the development of multicell spheroids. J Math Biol 43: 191–220CrossRefMathSciNetGoogle Scholar
  51. 51.
    Preziosi L (2003) Cancer modelling and simulation. Chapman & Hall/CRC Press, London/West Palm BeachzbMATHGoogle Scholar
  52. 52.
    MacArthur B, Please C (2004) Residual stress generation and necrosis formation in multicell tumour spheroids. J Math Biol 49: 537–552zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Mahaffy R, Shih C, McKintosh F, Kaes J (2000) Scanning probe-based frequency-dependent microrheology of polymer gels and biological cells. Phys Rev Lett 85: 880–883CrossRefGoogle Scholar
  54. 54.
    Merks R, Glazier J (2005) A cell-centered approach to developmental biology. Physica A 352: 113–130CrossRefGoogle Scholar
  55. 55.
    Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21: 1087–1092CrossRefGoogle Scholar
  56. 56.
    Lekka M, Laidler P, Gil D, Lekki J, Stachura Z, Hrynkiewicz AZ (1999) Elasticity of normal and cancerous human bladder cells studied by scanning force microscopy. Eur Biophys J 28(4): 312–316CrossRefGoogle Scholar
  57. 57.
    Mombach J, Glazier J (1996) Single cell motion in aggregates of embryonic cells. Phys Rev Lett 76(16): 3032–3035CrossRefGoogle Scholar
  58. 58.
    Moreira J, Deutsch A (2002) Cellular automata models of tumour development—a critical review. Adv Complex Syst 5(1): 247–267zbMATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Nelson CM, Jean RP, Tan JL, Liu WF, Sniadecki NJ, Spector AA, Chen CS (2005) Proc Natl Acad Sci USA 102: 11594–11599CrossRefGoogle Scholar
  60. 60.
    Newman T (2005) Modeling multi-cellular systems using sub-cellular elements. Math Biosci Eng 2(3): 613–624zbMATHMathSciNetGoogle Scholar
  61. 61.
    Palsson E, Othmer H (2000) A model for individual and collective cell movement in dictyostelium discoideum. Proc Natl Acad Sci USA 12(18): 10,448–10,453Google Scholar
  62. 62.
    Piper J, Swerlick R, Zhu C (1998) Determining force dependence of two-dimensional receptor-ligand binding affinity by centrifugation. Biophys J 74: 492–513CrossRefGoogle Scholar
  63. 63.
    Preziosi L, Tosin A, Multiphase modeling of tumor growth and extracellular matrix interaction: mathematical tools and applications. J Math BiolGoogle Scholar
  64. 64.
    Ramis-Conde I, Chaplain MAJ, Anderson A (2008) Mathematical modelling of cancer cell invasion of tissue. Math Comput Model (in press)Google Scholar
  65. 65.
    Roose T, Chapman S, Maini P (2007) Mathematical models of avascular tumour growth: a review. SIAM Rev 49(2): 179–208zbMATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    Roose T, Netti PA, Munn L, Boucher Y, Jain R (2003) Solid stress generated by spheroid growth estimated using a linear poroelasticity model. Microvasc Res 66: 204–212CrossRefGoogle Scholar
  67. 67.
    Schaller G, Meyer-Hermann M (2005) Multicellular tumor spheroid in an off-lattice voronoi-delaunay cell model. Phys Rev E 71: 051,910-1–051,910-16CrossRefMathSciNetGoogle Scholar
  68. 68.
    Schienbein M, Franke K, Gruler H (1994) Random walk and directed movement: comparison between inert particles and self-organized molecular machines. Phys Rev E 49(6): 5462–5471CrossRefGoogle Scholar
  69. 69.
    Schiffer I, Gebhard S, Heimerdinger C, Heling A, Hast J, Wollscheid U, Seliger B, Tanner B, Gilbert S, Beckers T, Baasner S, Brenner W, Spangenberg C, Prawitt D, Trost T, Schreiber W, Zabel B, Thelen M, Lehr H, Oesch F, Hengstler J (2003) Switching off her-2/neu in a tetracycline-controlled mouse tumor model leads to apoptosis and tumor-size-dependent remission. Cancer Res 63: 7221–7231Google Scholar
  70. 70.
    Stevens A (2000) The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J Appl Math 61(1): 183–212zbMATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Stott E, Britton N, Glazier J, Zajac M (1999) Stochastic simulation of benign avascular tumor growth using the potts model. Math Comput Model 30: 183–198CrossRefGoogle Scholar
  72. 72.
    Ward J, King J (1997) Mathematical modelling of avascular-tumor growth. IMA J Math Appl Med Biol 14: 39–69zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  2. 2.INRIADomaine RocquencourtLe Chesnay CedexFrance
  3. 3.IZBIUniversity of LeipzigLeipzigGermany

Personalised recommendations