Pattern Formation in Hybrid Models of Cell Populations

Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 15)

Abstract

The paper is devoted to hybrid discrete-continuous models of cell populations dynamics. Cells are considered as individual objects which can divide, die by apoptosis, differentiate and move under external forces. Intra-cellular regulatory networks are described by ordinary differential equations while extracellular species by partial differential equations. We illustrate the application of this approach to some model examples and to the problem of tumor growth. Hybrid models of cell populations present an interesting nonlinear dynamics which is not observed for the conventional continuous models.

References

  1. 1.
    Thompson DW(1992) On growth and form, 2nd edn, Dover reprint of 1942, Dover, New YorkGoogle Scholar
  2. 2.
    Murray JD Mathematical biology, 3rd edn. In: 2 vols: Mathematical biology: I. An introduction (2002), Mathematical biology: II. Spatial models and biomedical applications (2003). Springer, New YorkGoogle Scholar
  3. 3.
    Othmer HG, Painter K, Umulis D, Xue C (2009) The intersection of theory and application in elucidating pattern formation in developmental biology. Math Model Nat Phenom 4(4):3–82MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Alber M, Chen N, Glimm T, Lushnikov P (2007) Two-dimensional multiscale model of cell motion in a chemotactic field. 53–76Google Scholar
  5. 5.
    Dallon JC (2007) Models with lattice-free center-based cells interacting with continuum environment variables. Single Cell Based Model Biol Med 197–219Google Scholar
  6. 6.
    Deutsch A (2007) Lattice-gas cellular automaton modeling of developing cell systems. 29–51Google Scholar
  7. 7.
    Meinhardt H (2008) Models of biological pattern formation: from elementary steps to the organization of embryonic axes. Curr Top Dev Biol 81:1–63CrossRefGoogle Scholar
  8. 8.
    Anderson ARA (2007) A hybrid multiscale model of solid tumour growth and invasion: evolution and the microenvironment 3–28Google Scholar
  9. 9.
    Anderson ARA, Chaplain M, Rejniak KA Eds. (2007) Single cell based models in biology and medicine. Birkhauser, BaselGoogle Scholar
  10. 10.
    Anderson ARA, Rejniak KA, Gerlee P, Quaranta V (2007) Modelling of cancer growth. Evolution and invasion: bridging scales and models. Math Model Nat Phenom 2(3):1–29MathSciNetCrossRefGoogle Scholar
  11. 11.
    Drasdo D (2007) Center-based single-cell models: an approach to multi-cellular organization based on a conceptual analogy to colloidal particles. 171–196Google Scholar
  12. 12.
    Karttunen M, Vattulainen I, Lukkarinen A (2004) A novel methods in soft matter simulations. Springer, BerlinCrossRefGoogle Scholar
  13. 13.
    Bessonov N, Pujo-Menjouet L, Volpert V (2006) Cell modelling of hematopoiesis. Math Model Nat Phenom 1(2):81–103MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bessonov N, Demin I, Pujo-Menjouet L, Volpert V (2009) A multi-agent model describing self-renewal or differentiation effect of blood cell population. Math Comput Model 49:2116–2127MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Bessonov N, Kurbatova P, Volpert V (2010) Particle dynamics modelling of cell populations. Math Model Nat Phenom 5(7):42–47MathSciNetCrossRefGoogle Scholar
  16. 16.
    Swat A, Dolado I, Rojas JM, Nebreda AR (2009) Cell density-dependent inhibition of epidermal growth factor receptor signaling by p38 alpha mitogen-activated protein kinase via Sprouty2 downregulation. Mol Cell Biol 29(12):3332–3343CrossRefGoogle Scholar
  17. 17.
    Sark S, Jeanjean R, Zhang CC, Arcondeguy T (2006) Inhibition of cell division suppresses heterocyst development in Anabaena sp. Strain PCC 7120. J Bacteriol 188(4):1396–1404CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute of Mechanical Engineering ProblemsSaint PetersburgRussia
  2. 2.Institut Camille Jordan, University Lyon 1, UMR 5208 CNRSVilleurbanneFrance

Personalised recommendations