Journal of Mathematical Biology

, Volume 76, Issue 1–2, pp 205–234 | Cite as

From short-range repulsion to Hele-Shaw problem in a model of tumor growth

  • Sebastien Motsch
  • Diane Peurichard


We investigate the large time behavior of an agent based model describing tumor growth. The microscopic model combines short-range repulsion and cell division. As the number of cells increases exponentially in time, the microscopic model is challenging in terms of computational time. To overcome this problem, we aim at deriving the associated macroscopic dynamics leading here to a porous media type equation. As we are interested in the long time behavior of the dynamics, the macroscopic equation obtained through usual derivation method fails at providing the correct qualitative behavior (e.g. stationary states differ from the microscopic dynamics). We propose a modified version of the macroscopic equation introducing a density threshold for the repulsion. We numerically validate the new formulation by comparing the solutions of the micro- and macro- dynamics. Moreover, we study the asymptotic behavior of the dynamics as the repulsion between cells becomes singular (leading to non-overlapping constraints in the microscopic model). We manage to show formally that such asymptotic limit leads to a Hele-Shaw type problem for the macroscopic dynamics. We compare the micro- and macro- dynamics in this asymptotic limit using explicit solutions of the Hele-Shaw problem (e.g. radially symmetric configuration). The numerical simulations reveal an excellent agreement between the two descriptions, validating the formal derivation of the macroscopic model. The macroscopic model derived in this paper therefore enables to overcome the problem of large computational time raised by the microscopic model, but stays closely linked to the microscopic dynamics.


Agent-based models Tumor growth Porous media equation Hele-Shaw problem 

Mathematics Subject Classification

35K55 35B25 76D27 82C22 92C15 

Supplementary material


  1. Aoki I (1982) A simulation study on the schooling mechanism in fish. Bull Jpn Soc Sci Fish 48(8):1081–1088 (Japan)Google Scholar
  2. Balagué D, Carrillo JA, Laurent T, Raoul G (2013) Nonlocal interactions by repulsive-attractive potentials: radial ins/stability. Phys. D 260:5–25MathSciNetCrossRefzbMATHGoogle Scholar
  3. Ballerini M, Cabibbo N, Candelier R, Cavagna A, Cisbani E, Giardina I, Orlandi A, Parisi G, Procaccini A, Viale M (2008) Empirical investigation of starling flocks: a benchmark study in collective animal behaviour. Anim Behav 76(1):201–215CrossRefGoogle Scholar
  4. Bellomo N, Preziosi L (2000) Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math Comput Modell 32(3):413–452MathSciNetCrossRefzbMATHGoogle Scholar
  5. Berthelin F (2002) Existence and weak stability for a pressureless model with unilateral constraint. Math. Models Methods Appl. Sci. 12(02):249–272MathSciNetCrossRefzbMATHGoogle Scholar
  6. Berthelin F, Broizat D (2012) A model for the evolution of traffic jams in multi-lane. Kinet Relat Models 5(4):697–728MathSciNetCrossRefzbMATHGoogle Scholar
  7. Berthelin F, Degond P, Delitala M, Rascle M (2008) A model for the formation and evolution of traffic jams. Arch Ration Mech Anal 187(2):185–220MathSciNetCrossRefzbMATHGoogle Scholar
  8. Berthelin F, Degond P, Le Blanc V, Moutari S, Rascle M, Royer J (2008) A traffic-flow model with constraints for the modeling of traffic jams. Math Models Methods Appl Sci 18(supp01):1269–1298MathSciNetCrossRefzbMATHGoogle Scholar
  9. Bouchut F, Brenier Y, Cortes J, Ripoll J-F (2000) A hierarchy of models for two-phase flows. J Nonlinear Sci 10(6):639–660MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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):2321–2344MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bresch D, Perrin C, Zatorska E (2014) Singular limit of a Navier–Stokes system leading to a free/congested zones two-phase model. CR Math 352(9):685–690MathSciNetzbMATHGoogle Scholar
  12. Bruna M, Chapman S (2012) Excluded-volume effects in the diffusion of hard spheres. Phys Rev E 85(1):011103CrossRefGoogle Scholar
  13. Bruna M, Chapman S (2014) Diffusion of finite-size particles in confined geometries. Bull Math Biol 76(4):947–982MathSciNetCrossRefzbMATHGoogle Scholar
  14. Burger M, Capasso V, Morale D (2007) On an aggregation model with long and short range interactions. Nonlinear Anal Real World Appl 8(3):939–958MathSciNetCrossRefzbMATHGoogle Scholar
  15. Burger M, Di Francesco M, Pietschmann J-F, Schlake B (2010) Nonlinear cross-diffusion with size exclusion. SIAM J Math Anal 42(6):2842–2871MathSciNetCrossRefzbMATHGoogle Scholar
  16. Burger M, Fetecau R, Huang Y (2014) Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion. SIAM J Appl Dyn Syst 13(1):397–424MathSciNetCrossRefzbMATHGoogle Scholar
  17. Byrne H, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58(4–5):657–687MathSciNetCrossRefzbMATHGoogle Scholar
  18. Byrne HM, Chaplain M (1997) Free boundary value problems associated with the growth and development of multicellular spheroids. Eur J Appl Math 8(06):639–658MathSciNetCrossRefzbMATHGoogle Scholar
  19. Carrillo J, Chertock A, Huang Y (2015) A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. Commun Comput Phys 17(01):233–258MathSciNetCrossRefzbMATHGoogle Scholar
  20. Cheng CHA, Coutand D, Shkoller S (2014) Global existence and decay for solutions of the Hele-Shaw flow with injection. Interfaces Free Bound 16(3):297–338MathSciNetCrossRefzbMATHGoogle Scholar
  21. Colli P, Gilardi G, Hilhorst D (2014) On a Cahn–Hilliard type phase field system related to tumor growth, arXiv preprint arXiv:1401.5943
  22. Couzin ID, Krause J, James R, Ruxton GD, Franks NR (2002) Collective memory and spatial sorting in animal groups. J Theor Biol 218(1):1–11MathSciNetCrossRefGoogle Scholar
  23. Deaconu M, Herrmann S, Maire S (2015) The walk on moving spheres: a new tool for simulating Brownian motion’s exit time from a domain. Math Comput Simul 28–38Google Scholar
  24. Degond G, Hua J, Navoret L (2011) Numerical simulations of the Euler system with congestion constraint. J Comput Phys 230(22):8057–8088MathSciNetCrossRefzbMATHGoogle Scholar
  25. Degond P, Dimarco G, Mac T, Wang N (2014) Macroscopic models of collective motion with repulsion, arXiv preprint arXiv:1404.4886
  26. Degond P, Hua J (2013) Self-organized hydrodynamics with congestion and path formation in crowds. J Comput Phys 237:299–319MathSciNetCrossRefzbMATHGoogle Scholar
  27. Degond P, Navoret L, Bon R, Sanchez D (2010) Congestion in a macroscopic model of self-driven particles modeling gregariousness. J Stat Phys 138(1–3):85–125CrossRefzbMATHGoogle Scholar
  28. Egly H, Despres B, Sentis R (2011) Ablative Hele-Shaw model for ICF flows modeling and numerical simulation. Math Models Methods Appl Sci 21(07):1571–1600MathSciNetCrossRefzbMATHGoogle Scholar
  29. Fetecau R, Huang Y, Kolokolnikov T (2011) Swarm dynamics and equilibria for a nonlocal aggregation model. Nonlinearity 24(10):2681MathSciNetCrossRefzbMATHGoogle Scholar
  30. Gauss CF (1831) Besprechung des Buchs von LA Seeber: Intersuchungen über die Eigenschaften der positiven ternären quadratischen Formen usw. Göttingsche Gelehrte Anzeigen 2:188–196Google Scholar
  31. Greenspan HP (1972) Models for the growth of a solid tumor by diffusion. Stud Appl Math 51(4):317–340CrossRefzbMATHGoogle Scholar
  32. Harpold H, Alvord E, Swanson K (2007) The evolution of mathematical modeling of glioma proliferation and invasion. J Neuropathol Exp Neurol 66(1):1–9CrossRefGoogle Scholar
  33. Hawkins-Daarud A, Prudhomme S, van der Zee K, Oden T (2013) Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth. J Math Biol 67(6–7):1457–1485MathSciNetCrossRefzbMATHGoogle Scholar
  34. Helbing D, Molnar P (1985) Social force model for pedestrian dynamics. Math Comput Simul Phys Rev E 51:4282Google Scholar
  35. Kipnis C, Olla S, Varadhan SRS (1989) Hydrodynamics and large deviation for simple exclusion processes. Commun Pure Appl Math 42(2):115–137MathSciNetCrossRefzbMATHGoogle Scholar
  36. Labbé S, Maitre E (2013) A free boundary model for Korteweg fluids as a limit of barotropic compressible Navier–Stokes equations. Methods Appl Anal 20(2):165–178MathSciNetzbMATHGoogle Scholar
  37. Leroy Lerêtre M (2014) Etude de la croissance tumorale via la modélisation agent-centré du comportement collectif des cellules au sein d’une population cellulaire, PhD thesis, Univ. Paul Sabatier, ToulouseGoogle Scholar
  38. Lowengrub J, Frieboes H, Jin F, Chuang Y-L, Li X, Macklin P, Wise S, Cristini V (2010) Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23(1):R1–R91MathSciNetCrossRefzbMATHGoogle Scholar
  39. Maury B (2006) A time-stepping scheme for inelastic collisions. Numer Math 102(4):649–679MathSciNetCrossRefzbMATHGoogle Scholar
  40. Maury B, Roudneff-Chupin A, Santambrogio F (2010) A macroscopic crowd motion model of gradient flow type. Math Models Methods Appl Sci 20(10):1787–1821MathSciNetCrossRefzbMATHGoogle Scholar
  41. Maury B, Venel J (2011) A discrete contact model for crowd motion. ESAIM Math Model Numer Anal 45(01):145–168MathSciNetCrossRefzbMATHGoogle Scholar
  42. Mellet A, Perthame B, Quiros F (2015) A Hele-Shaw problem for tumor growth, arXiv:1512.06995 [math]
  43. Morale D, Capasso V, Oelschläger K (2005) An interacting particle system modelling aggregation behavior: from individuals to populations. J Math Biol 50(1):49–66MathSciNetCrossRefzbMATHGoogle Scholar
  44. Moussaïd M, Guillot E, Moreau M, Fehrenbach J, Chabiron O, Lemercier S, Pettré J, Appert-Rolland C, Degond P, Theraulaz G (2012) Traffic instabilities in self-organized pedestrian crowds. PLoS Comput Biol 8(3):e1002442CrossRefGoogle Scholar
  45. Muller M (1956) Some continuous Monte Carlo methods for the Dirichlet problem. Ann Math Stat 27(3):569–589Google Scholar
  46. Oden T, Hawkins A, Prudhomme S (2010) General diffuse-interface theories and an approach to predictive tumor growth modeling. Math Models Methods Appl Sci 20(03):477–517MathSciNetCrossRefzbMATHGoogle Scholar
  47. Oelschläger K (1990) Large systems of interacting particles and the porous medium equation. J Differ Equ 88(2):294–346MathSciNetCrossRefzbMATHGoogle Scholar
  48. Oksendal B (1992) Stochastic differential equations: an introduction with applications. Springer, New YorkCrossRefzbMATHGoogle Scholar
  49. Perrin C, Zatorska E (2015) Free/congested twophase model from weak solutions to multi-dimensional compressible Navier–Stokes equations. Comm Partial Differential Equations 40(8):1558–1589Google Scholar
  50. Perthame B, Quirós F, Tang M, Vauchelet N (2014) Derivation of a Hele-Shaw type system from a cell model with active motion. Interfaces Free Bound 16:489–508MathSciNetCrossRefzbMATHGoogle Scholar
  51. Perthame B, Quirós F, Vázquez J (2014) The Hele-Shaw asymptotics for mechanical models of tumor growth. Arch Ration Mech Anal 212(1):93–127MathSciNetCrossRefzbMATHGoogle Scholar
  52. Perthame B, Tang M, Vauchelet N (2014) Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient. Math Models Methods Appl Sci 24(13):2601–2626MathSciNetCrossRefzbMATHGoogle Scholar
  53. Reynolds C W (1987) Flocks, herds and schools: a distributed behavioral model. ACM SIGGRAPH Comput Gr 21:25–34CrossRefGoogle Scholar
  54. Roose T, Chapman J, Maini P (2007) Mathematical models of avascular tumor growth. SIAM Rev 49(2):179–208MathSciNetCrossRefzbMATHGoogle Scholar
  55. Swanson K, Bridge C, Murray JD, Alvord E Jr (2003) Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J Neurol Sci 216(1):1–10CrossRefGoogle Scholar
  56. Wilson D, King J, Byrne H (2007) Modelling scaffold occupation by a growing, nutrient-rich tissue. Math Models Methods Appl Sci 17(supp01):1721–1750MathSciNetCrossRefzbMATHGoogle Scholar
  57. Zhao H (2005) A fast sweeping method for eikonal equations. Math Comput 74(250):603–627MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.Faculty of MathematicsWien UniversityViennaAustria

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