Bridging the gap between individual-based and continuum models of growing cell populations

  • Mark A. J. Chaplain
  • Tommaso Lorenzi
  • Fiona R. MacfarlaneEmail author


Continuum models for the spatial dynamics of growing cell populations have been widely used to investigate the mechanisms underpinning tissue development and tumour invasion. These models consist of nonlinear partial differential equations that describe the evolution of cellular densities in response to pressure gradients generated by population growth. Little prior work has explored the relation between such continuum models and related single-cell-based models. We present here a simple stochastic individual-based model for the spatial dynamics of multicellular systems whereby cells undergo pressure-driven movement and pressure-dependent proliferation. We show that nonlinear partial differential equations commonly used to model the spatial dynamics of growing cell populations can be formally derived from the branching random walk that underlies our discrete model. Moreover, we carry out a systematic comparison between the individual-based model and its continuum counterparts, both in the case of one single cell population and in the case of multiple cell populations with different biophysical properties. The outcomes of our comparative study demonstrate that the results of computational simulations of the individual-based model faithfully mirror the qualitative and quantitative properties of the solutions to the corresponding nonlinear partial differential equations. Ultimately, these results illustrate how the simple rules governing the dynamics of single cells in our individual-based model can lead to the emergence of complex spatial patterns of population growth observed in continuum models.


Growing cell populations Pressure-driven cell movement Pressure-limited growth Individual-based models Continuum models 

Mathematics Subject Classification

92C17 35Q92 92-08 92B05 35C07 



FRM is funded by the Engineering and Physical Sciences Research Council (EPSRC) (Grant No. EP/N014642/1). TL and FRM gratefully acknowledge Dirk Drasdo and Luís Neves de Almeida for insightful discussions.

Supplementary material

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  1. Ambrosi D, Mollica F (2002) On the mechanics of a growing tumor. Int J Eng Sci 40(12):1297–1316MathSciNetzbMATHGoogle Scholar
  2. Ambrosi D, Preziosi L (2002) On the closure of mass balance models for tumor growth. Math Models Methods Appl Sci 12(05):737–754MathSciNetzbMATHGoogle Scholar
  3. Araujo RP, McElwain DS (2004) A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull Math Biol 66(5):1039–1091MathSciNetzbMATHGoogle Scholar
  4. Basan M, Risler T, Joanny JF, Sastre-Garau X, Prost J (2009) Homeostatic competition drives tumor growth and metastasis nucleation. HFSP J 3(4):265–272Google Scholar
  5. Binder BJ, Landman KA (2009) Exclusion processes on a growing domain. J Theor Biol 259(3):541–551MathSciNetzbMATHGoogle Scholar
  6. Bresch D, Colin T, Grenier E, Ribba B, Saut O (2010) Computational modeling of solid tumor growth: the avascular stage. SIAM J Sci Comput 32(4):2321–2344MathSciNetzbMATHGoogle Scholar
  7. Brú A, Albertos S, Subiza JL, García-Asenjo JL, Brú I (2003) The universal dynamics of tumor growth. Biophys J 85(5):2948–2961Google Scholar
  8. Buttenschoen A, Hillen T, Gerisch A, Painter KJ (2018) A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis. J Math Biol 76(1–2):429–456MathSciNetzbMATHGoogle Scholar
  9. Byrne HM (2010) Dissecting cancer through mathematics: from the cell to the animal model. Nat Rev Cancer 10(3):221Google Scholar
  10. Byrne HM, Chaplain MAJ (1995) Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math Biosci 130(2):151–181zbMATHGoogle Scholar
  11. Byrne HM, Chaplain MAJ (1996) Growth of necrotic tumors in the presence and absence of inhibitors. Math Biosci 135(2):187–216zbMATHGoogle Scholar
  12. Byrne HM, Chaplain MAJ (1997) Free boundary value problems associated with the growth and development of multicellular spheroids. Eur J Appl Math 8(6):639–658MathSciNetzbMATHGoogle Scholar
  13. Byrne HM, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58(4–5):657MathSciNetzbMATHGoogle Scholar
  14. Byrne HM, Preziosi L (2003) Modelling solid tumour growth using the theory of mixtures. Math Med Biol 20(4):341–366zbMATHGoogle Scholar
  15. Byrne HM, King JR, McElwain DS, Preziosi L (2003) A two-phase model of solid tumour growth. Appl Math Lett 16(4):567–573MathSciNetzbMATHGoogle Scholar
  16. Champagnat N, Méléard S (2007) Invasion and adaptive evolution for individual-based spatially structured populations. J Math Biol 55(2):147MathSciNetzbMATHGoogle Scholar
  17. Chaplain MAJ, Graziano L, Preziosi L (2006) Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development. Math Med Biol 23(3):197–229zbMATHGoogle Scholar
  18. 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(3):191–220MathSciNetzbMATHGoogle Scholar
  19. Ciarletta P, Foret L, Amar MB (2011) The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis. J R Soc Interface 8(56):345–368Google Scholar
  20. Dancer EN, Hilhorst D, Mimura M, Peletier LA (1999) Spatial segregation limit of a competition-diffusion system. Eur J Appl Math 10(2):97–115MathSciNetzbMATHGoogle Scholar
  21. Deroulers C, Aubert M, Badoual M, Grammaticos B (2009) Modeling tumor cell migration: from microscopic to macroscopic models. Phys Rev E 79(3):031917MathSciNetGoogle Scholar
  22. Drasdo D (2005) Coarse graining in simulated cell populations. Adv Complex Syst 8(02–03):319–363MathSciNetzbMATHGoogle Scholar
  23. Drasdo D, Hoehme S (2012) Modeling the impact of granular embedding media, and pulling versus pushing cells on growing cell clones. New J Phys 14(5):055025Google Scholar
  24. Dyson L, Maini PK, Baker RE (2012) Macroscopic limits of individual-based models for motile cell populations with volume exclusion. Phys Rev E 86(3):031903Google Scholar
  25. Engblom S, Wilson DB, Baker RE (2018) Scalable population-level modelling of biological cells incorporating mechanics and kinetics in continuous time. R Soc Open Sci 5(8):180379Google Scholar
  26. Fernando AE, Landman KA, Simpson MJ (2010) Nonlinear diffusion and exclusion processes with contact interactions. Phys Rev E 81(1):011903Google Scholar
  27. Greenspan H (1976) On the growth and stability of cell cultures and solid tumors. J Theor Biol 56(1):229–242MathSciNetGoogle Scholar
  28. Hillen T, Othmer HG (2000) The diffusion limit of transport equations derived from velocity-jump processes. SIAM J Appl Math 61(3):751–775MathSciNetzbMATHGoogle Scholar
  29. Hillen T, Painter KJ (2009) A user’s guide to PDE models for chemotaxis. J Math Biol 58(1–2):183MathSciNetzbMATHGoogle Scholar
  30. Inoue M et al (1991) Derivation of a porous medium equation from many markovian particles and the propagation of chaos. Hiroshima Math J 21(1):85–110MathSciNetzbMATHGoogle Scholar
  31. Johnston ST, Simpson MJ, Baker RE (2012) Mean-field descriptions of collective migration with strong adhesion. Phys Rev E 85(5):051922Google Scholar
  32. Johnston ST, Baker RE, McElwain DS, Simpson MJ (2017) Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves. Sci Rep 7:42134Google Scholar
  33. Kim I, Požár N (2018) Porous medium equation to Hele-Shaw flow with general initial density. Trans Am Math Soc 370(2):873–909MathSciNetzbMATHGoogle Scholar
  34. Kim IC, Perthame B, Souganidis PE (2016) Free boundary problems for tumor growth: a viscosity solutions approach. Nonlinear Anal 138:207–228MathSciNetzbMATHGoogle Scholar
  35. Landman KA, Fernando AE (2011) Myopic random walkers and exclusion processes: single and multispecies. Phys A Stat Mech Appl 390(21–22):3742–3753Google Scholar
  36. LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, CambridgeGoogle Scholar
  37. Lorenzi T, Lorz A, Perthame B (2017) On interfaces between cell populations with different mobilities. Kinet Relat Models 10(1):299–311MathSciNetzbMATHGoogle Scholar
  38. Lowengrub JS, Frieboes HB, Jin F, Chuang YL, Li X, Macklin P, Wise SM, Cristini V (2009) Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23(1):R1MathSciNetzbMATHGoogle Scholar
  39. Lushnikov PM, Chen N, Alber M (2008) Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact. Phys Rev E 78(6):061904Google Scholar
  40. Mellet A, Perthame B, Quiros F (2017) A Hele-Shaw problem for tumor growth. J Funct Anal 273(10):3061–3093MathSciNetzbMATHGoogle Scholar
  41. Mimura M, Sakaguchi H, Matsushita M (2000) Reaction-diffusion modelling of bacterial colony patterns. Phys A Stat Mech Appl 282(1–2):283–303Google Scholar
  42. Motsch S, Peurichard D (2018) From short-range repulsion to Hele-Shaw problem in a model of tumor growth. J Math Biol 76(1–2):205–234MathSciNetzbMATHGoogle Scholar
  43. Murray PJ, Edwards CM, Tindall MJ, Maini PK (2009) From a discrete to a continuum model of cell dynamics in one dimension. Phys Rev E 80(3):031912Google Scholar
  44. Murray PJ, Edwards CM, Tindall MJ, Maini PK (2012) Classifying general nonlinear force laws in cell-based models via the continuum limit. Phys Rev E 85(2):021921Google Scholar
  45. Oelschläger K (1989) On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes. Probab Theory Relat Fields 82(4):565–586MathSciNetzbMATHGoogle Scholar
  46. Oelschläger K (1990) Large systems of interacting particles and the porous medium equation. J Differ Equ 88(2):294–346MathSciNetzbMATHGoogle Scholar
  47. Othmer HG, Hillen T (2002) The diffusion limit of transport equations II: chemotaxis equations. SIAM J Appl Math 62(4):1222–1250MathSciNetzbMATHGoogle Scholar
  48. Othmer HG, Dunbar SR, Alt W (1988) Models of dispersal in biological systems. J Math Biol 26(3):263–298MathSciNetzbMATHGoogle Scholar
  49. Painter KJ, Sherratt JA (2003) Modelling the movement of interacting cell populations. J Theor Biol 225(3):327–339MathSciNetGoogle Scholar
  50. Penington CJ, Hughes BD, Landman KA (2011) Building macroscale models from microscale probabilistic models: a general probabilistic approach for nonlinear diffusion and multispecies phenomena. Phys Rev E 84(4):041120Google Scholar
  51. Penington CJ, Hughes BD, Landman KA (2014) Interacting motile agents: taking a mean-field approach beyond monomers and nearest-neighbor steps. Phys Rev E 89(3):032714Google Scholar
  52. Perthame B (2014) Some mathematical aspects of tumor growth and therapy. In: International congress of mathematicians (ICM)Google Scholar
  53. Perthame B, Quirós F, Tang M, Vauchelet N (2014) Derivation of a Hele-Shaw type system from a cell model with active motion. Interface Free Bound 16(4):489–508MathSciNetzbMATHGoogle Scholar
  54. Perthame B, Quirós F, Vázquez JL (2014) The Hele-Shaw asymptotics for mechanical models of tumor growth. Arch Ration Mech Anal 212(1):93–127MathSciNetzbMATHGoogle Scholar
  55. Preziosi L (2003) Cancer modelling and simulation. CRC Press, Boca RotonzbMATHGoogle Scholar
  56. Ranft J, Basan M, Elgeti J, Joanny JF, Prost J, Jülicher F (2010) Fluidization of tissues by cell division and apoptosis. Proc Nat Acad Sci USA 107(49):20863–20868Google Scholar
  57. Roose T, Chapman SJ, Maini PK (2007) Mathematical models of avascular tumor growth. SIAM Rev 49(2):179–208MathSciNetzbMATHGoogle Scholar
  58. Sherratt JA, Chaplain MAJ (2001) A new mathematical model for avascular tumour growth. J Math Biol 43(4):291–312MathSciNetzbMATHGoogle Scholar
  59. Simpson MJ, Merrifield A, Landman KA, Hughes BD (2007) Simulating invasion with cellular automata: connecting cell-scale and population-scale properties. Phys Rev E 76(2):021918Google Scholar
  60. Simpson MJ, Landman KA, Hughes BD (2010) Cell invasion with proliferation mechanisms motivated by time-lapse data. Phys A Stat Mech Appl 389(18):3779–3790Google Scholar
  61. Stevens A (2000) The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J Appl Math 61(1):183–212MathSciNetzbMATHGoogle Scholar
  62. Stevens A, Othmer HG (1997) Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J Appl Math 57(4):1044–1081MathSciNetzbMATHGoogle Scholar
  63. Tang M, Vauchelet N, Cheddadi I, Vignon-Clementel I, Drasdo D, Perthame B (2014) Composite waves for a cell population system modeling tumor growth and invasion. In: Partial differential equations: theory, control and approximation. Springer, pp 401–429Google Scholar
  64. Van Liedekerke P, Palm M, Jagiella N, Drasdo D (2015) Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results. Comput Part Mech 2(4):401–444Google Scholar
  65. Wang YY, Lehuédé C, Laurent V, Dirat B, Dauvillier S, Bochet L, Le Gonidec S, Escourrou G, Valet P, Muller C (2012) Adipose tissue and breast epithelial cells: a dangerous dynamic duo in breast cancer. Cancer Lett 324(2):142–151Google Scholar
  66. Ward JP, King J (1997) Mathematical modelling of avascular-tumour growth. Math Med Biol 14(1):39–69zbMATHGoogle Scholar
  67. Ward J, King J (1999) Mathematical modelling of avascular-tumour growth II: modelling growth saturation. Math Med Biol 16(2):171–211zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of St AndrewsSt AndrewsScotland

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