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

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Abstract

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.

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References

  1. Adam J, Belomo N (1997) A survey of models for tumor-immune system dynamics. Birkhäuser, Boston

    MATH  Google Scholar 

  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: 395–411

    Article  MathSciNet  Google Scholar 

  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 1–40

    Google Scholar 

  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: 2071–2079

    Article  Google Scholar 

  5. Allen M, Tildersley D (1987) Computer Simulation of Liquids. Oxford Science Publications, Oxford

    MATH  Google Scholar 

  6. Ambrosi D, Mollica F (2002) On the mechanics of a growing tumor. Int J Eng Sci 40(12): 1297–1316

    Article  MathSciNet  Google Scholar 

  7. Ambrosi D, Mollica F (2004) The role of stress in the growth of a multicell spheroid. J Math Biol 48(5): 477–479

    Article  MATH  MathSciNet  Google Scholar 

  8. Anderson A, Chaplain MAJ, Rejniak K (2007) Single-cell-based models in biology and medicine. Birkhäuser, Basel

    Book  Google Scholar 

  9. Anderson A, Chaplain MAJ, Newman E, Steele R, Thompson A (2000) Mathematical modeling of tumor invasion and metastasis. J Theor Med 2: 129–154

    MATH  Google Scholar 

  10. Araujo R, McElwain D (2004) A history of the study of solid tumour growth: the contribution of mathematical models. Bull Math Biol 66: 1039–1091

    Article  MathSciNet  Google Scholar 

  11. Beysens D, Forgacs G, Glazier J (2000) Cell sorting is analogous to phase ordering in fluids. Proc Natl Acad Sci USA 97(17): 9467–9471

    Article  Google Scholar 

  12. Block M, Schoell E, Drasdo D (2007) Classifying the expansion kinetics and critical surface dynamics of growing cell populations. Phys Rev Lett 99: 248,101–248,104

    Article  Google Scholar 

  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: 125–152

    Article  MATH  MathSciNet  Google Scholar 

  14. Bru A, Albertos S, Subiza J, Garcia-Arsenio J, Bru I (2003) The universal dynamics of tumor growth. Biophys J 85: 2948–2961

    Article  Google Scholar 

  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): 4008–4011

    Article  Google Scholar 

  16. Byrne H (1997) The importance of intercellular adhesion in the development of carcinomas. IMA J Math Appl Med Biol 14: 305–323

    Article  MATH  Google Scholar 

  17. Byrne H, Chaplain J (1996) Modelling the role of cell–cell adhesion in the growth and development of carcinomas. Math Comput Model 12: 1–17

    Article  Google Scholar 

  18. Byrne H, Chaplain M (1997) Free boundary value problem associated with the growth and development of multicellular spheroids. Eur J Appl Math 8: 639–658

    Article  MATH  MathSciNet  Google Scholar 

  19. Byrne HM, King JR, McElwain DLS, Preziosi L (2003) A two-phase model of solid tumor growth. Appl Math Lett 16(4): 567–573

    Article  MATH  MathSciNet  Google Scholar 

  20. Byrne H, Preziosi L (2003) Modelling solid tumour growth using the theory of mixtures. Math Med Biol 20: 341–366

    Article  MATH  Google Scholar 

  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: 395–400

    Article  Google Scholar 

  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): 197–229

    Article  MATH  Google Scholar 

  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: 191–220

    Article  MATH  MathSciNet  Google Scholar 

  24. Chesla S, Selvaraj P, Zhu C (1998) Measuring two-dimensional receptor-ligand binding kinetics by micropipette. Biophys J 75: 1553–1557

    Article  Google Scholar 

  25. Chu YS et al (2005) Johnson–Kendall–Roberts theory applied to living cells. Phys Rev Lett 94: 028,102

    Article  Google Scholar 

  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): 273–288

    Article  Google Scholar 

  27. Please CP, Pettet G, McElwain D (1998) A new approach to modelling the formation of necrotic regions in tumours. Appl Math Lett 11: 89–94

    Article  MATH  MathSciNet  Google Scholar 

  28. Cristini V, Lowengrub J, Nie Q (2003) Nonlinear simulations of tumor growth. J Math Biol 46: 191–224

    Article  MATH  MathSciNet  Google Scholar 

  29. Dallon J, Othmer H (2004) How cellular movement determines the collective force generated by the dictyostelium discoideum slug. J Theor Biol 231: 203–222

    Article  MathSciNet  Google Scholar 

  30. DeMasi A, Luckhaus S, Presutti E (2005) Two-scale hydrodynamic limit for a model of malignant tumor cells. MPI-MIS (Preprint) 2: 1–47

    Google Scholar 

  31. Dormann S, Deutsch A (2002) Modeling of self-organized avascular tumor growth with a hybrid cellular automaton. In Silico Biol 2: 0035

    Google Scholar 

  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 281–291

    Google Scholar 

  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, Basel

    Google Scholar 

  34. Drasdo D (2005) Coarse graining in simulated cell populations. Adv Complex Syst 8(2 & 3): 319–363

    Article  MATH  MathSciNet  Google Scholar 

  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)

  36. Drasdo D, Forgacs G (2000) Modelling the interplay of generic and genetic mechanisms in cleavage, blastulation and gastrulation. Dev Dyn 219: 182–191

    Article  Google Scholar 

  37. Drasdo D, Hoehme S (2005) A single-cell based model to tumor growth in-vitro: monolayers and spheroids. Phys Biol 2: 133–147

    Article  Google Scholar 

  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): 319–363

    MathSciNet  Google Scholar 

  39. Drasdo D, Höhme S (2003) Individual-based approaches to birth and death in avascular tumors. Math Comput Model 37: 1163–1175

    Article  MATH  Google Scholar 

  40. Drasdo D, Kree R, McCaskill J (1995) Monte-carlo approach to tissue-cell populations. Phys Rev E 52(6): 6635–6657

    Article  Google Scholar 

  41. Friedman A (2007) Mathematical analysis and challenges arrising from models of tumor growth. Math Model Methods Appl Sci 17: 1751–1772

    Article  MATH  Google Scholar 

  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)

  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: 62–75

    Article  Google Scholar 

  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): 1802–1811

    Article  Google Scholar 

  45. Graner F, Glazier J (1992) Simulation of biological cell sorting using a two-dimensional extended potts model. Phys Rev Lett 69(13): 2013–2016

    Article  Google Scholar 

  46. Greenspan H (1976) On the growth and stability of cell cultures and solid tumors. J Theor Biol 56(1): 229–242

    Article  MathSciNet  Google Scholar 

  47. Hogeweg P (2000) Evolving mechanisms of morphogenesis: on the interplay between differential adhesion and cell differentiation. J Theor Biol 203: 317–333

    Article  Google Scholar 

  48. Johnson K, Kendall K, Roberts A (1971) Surface energy and the contact of elastic solids. Proc R Soc A 324: 301–313

    Article  Google Scholar 

  49. King J, Franks S (2004) Mathematical analysis of some multidimensional tissue-growth models. Eur J Appl Math 15(3): 273–295

    Article  MATH  MathSciNet  Google Scholar 

  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: 191–220

    Article  MathSciNet  Google Scholar 

  51. Preziosi L (2003) Cancer modelling and simulation. Chapman & Hall/CRC Press, London/West Palm Beach

    MATH  Google Scholar 

  52. MacArthur B, Please C (2004) Residual stress generation and necrosis formation in multicell tumour spheroids. J Math Biol 49: 537–552

    Article  MATH  MathSciNet  Google Scholar 

  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: 880–883

    Article  Google Scholar 

  54. Merks R, Glazier J (2005) A cell-centered approach to developmental biology. Physica A 352: 113–130

    Article  Google Scholar 

  55. Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21: 1087–1092

    Article  Google Scholar 

  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): 312–316

    Article  Google Scholar 

  57. Mombach J, Glazier J (1996) Single cell motion in aggregates of embryonic cells. Phys Rev Lett 76(16): 3032–3035

    Article  Google Scholar 

  58. Moreira J, Deutsch A (2002) Cellular automata models of tumour development—a critical review. Adv Complex Syst 5(1): 247–267

    Article  MATH  MathSciNet  Google Scholar 

  59. Nelson CM, Jean RP, Tan JL, Liu WF, Sniadecki NJ, Spector AA, Chen CS (2005) Proc Natl Acad Sci USA 102: 11594–11599

    Article  Google Scholar 

  60. Newman T (2005) Modeling multi-cellular systems using sub-cellular elements. Math Biosci Eng 2(3): 613–624

    MATH  MathSciNet  Google Scholar 

  61. Palsson E, Othmer H (2000) A model for individual and collective cell movement in dictyostelium discoideum. Proc Natl Acad Sci USA 12(18): 10,448–10,453

    Google Scholar 

  62. Piper J, Swerlick R, Zhu C (1998) Determining force dependence of two-dimensional receptor-ligand binding affinity by centrifugation. Biophys J 74: 492–513

    Article  Google Scholar 

  63. Preziosi L, Tosin A, Multiphase modeling of tumor growth and extracellular matrix interaction: mathematical tools and applications. J Math Biol

  64. Ramis-Conde I, Chaplain MAJ, Anderson A (2008) Mathematical modelling of cancer cell invasion of tissue. Math Comput Model (in press)

  65. Roose T, Chapman S, Maini P (2007) Mathematical models of avascular tumour growth: a review. SIAM Rev 49(2): 179–208

    Article  MATH  MathSciNet  Google Scholar 

  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: 204–212

    Article  Google Scholar 

  67. Schaller G, Meyer-Hermann M (2005) Multicellular tumor spheroid in an off-lattice voronoi-delaunay cell model. Phys Rev E 71: 051,910-1–051,910-16

    Article  MathSciNet  Google Scholar 

  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): 5462–5471

    Article  Google Scholar 

  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: 7221–7231

    Google Scholar 

  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): 183–212

    Article  MATH  MathSciNet  Google Scholar 

  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: 183–198

    Article  Google Scholar 

  72. Ward J, King J (1997) Mathematical modelling of avascular-tumor growth. IMA J Math Appl Med Biol 14: 39–69

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK

    Helen Byrne

  2. INRIA, Domaine Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Dirk Drasdo

  3. IZBI, University of Leipzig, Leipzig, Germany

    Dirk Drasdo

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  1. Helen Byrne
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  2. Dirk Drasdo
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Correspondence to Dirk Drasdo.

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Byrne, H., Drasdo, D. Individual-based and continuum models of growing cell populations: a comparison. J. Math. Biol. 58, 657–687 (2009). https://doi.org/10.1007/s00285-008-0212-0

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  • DOI: https://doi.org/10.1007/s00285-008-0212-0

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