Journal of Mathematical Biology

, Volume 53, Issue 4, pp 585–616 | Cite as

M5 mesoscopic and macroscopic models for mesenchymal motion

  • Thomas HillenEmail author


In this paper mesoscopic (individual based) and macroscopic (population based) models for mesenchymal motion of cells in fibre networks are developed. Mesenchymal motion is a form of cellular movement that occurs in three-dimensions through tissues formed from fibre networks, for example the invasion of tumor metastases through collagen networks. The movement of cells is guided by the directionality of the network and in addition, the network is degraded by proteases. The main results of this paper are derivations of mesoscopic and macroscopic models for mesenchymal motion in a timely varying network tissue. The mesoscopic model is based on a transport equation for correlated random walk and the macroscopic model has the form of a drift-diffusion equation where the mean drift velocity is given by the mean orientation of the tissue and the diffusion tensor is given by the variance-covariance matrix of the tissue orientations. The transport equation as well as the drift-diffusion limit are coupled to a differential equation that describes the tissue changes explicitly, where we distinguish the cases of directed and undirected tissues. As a result the drift velocity and the diffusion tensor are timely varying. We discuss relations to existing models and possible applications.


Transport Equation Drift Velocity Diffusion Tensor Macroscopic Model Moment Closure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alt W. (1981) Singular perturbation of differential integral equations describing biased random walks. J. Reine Angew. Math. 322, 15–41zbMATHMathSciNetGoogle Scholar
  2. 2.
    Barocas V.H., Tranquillo R.T. (1997) An anisotropic biphasic theory of tissue-equivalent mechanics: The interplay among cell traction, fibrillar network deformation, fibril alignment and cell contact guidance. J. Biomech. Eng. 119, 137–145CrossRefGoogle Scholar
  3. 3.
    Berenschot, G. Vizualization of Diffusion Tensor Imaging. Eindhoven University of Technology, Master Thesis (2003)Google Scholar
  4. 4.
    Chalub, F.A.C.C., Markovich, P.A., Perthame, B., Schmeiser, C. Kinetic models for chemotaxis and their drift-diffusion limits. Monatsh. Math. 123–141 (2004)Google Scholar
  5. 5.
    Dallon J.C., Sherratt J.A. (2000) A mathematical model for spatially varying extracellular matrix alignment. SIAM J. Appl. Math. 61, 506–527zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dallon J.C., Sherratt J.A., Maini P.K. (2001) Modelling the effects of transforming growth factor-β on extracellular alignment in dermal wound repair. Wound Rep. Reg. 9, 278–286CrossRefGoogle Scholar
  7. 7.
    Dickinson R. (2000) A generalized transport model for biased cell migration in an anisotropic environment. J. Math. Biol. 40, 97–135zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dickinson R.B. (1997). A model for cell migration by contact guidance. In: Alt W., Deutsch A., Dunn G. (eds). Dynamics of Cell and Tissue Motion. Birkhauser, Basel, pp. 149–158Google Scholar
  9. 9.
    Dolak Y., Schmeiser C. (2005) Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanics. J. Math. Biol. 51, 595–615zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Friedl P., Bröcker E.B. (2000) The biology of cell locomotion within three dimensional extracellular matrix. Cell Motility Life Sci. 57, 41–64CrossRefGoogle Scholar
  11. 11.
    Friedl P., Wolf K. (2003) Tumour-cell invasion and migration: diversity and escape mechanisms. Nat. Rev. 3, 362–374Google Scholar
  12. 12.
    Hillen T. (2004) On L 2-closure of transport equations: the Cattaneo closure. Discrete Cont. Dyn. Syst. B 4(4): 961–982zbMATHMathSciNetGoogle Scholar
  13. 13.
    Hillen T. (2005) On the L 2-closure of transport equations: the general case. Discrete Cont. Dyn. Syst. B 5(2): 299–318zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Hillen T., Othmer H.G. (2000) The diffusion limit of transport equations derived from velocity jump processes. SIAM J. Appl. Math. 61(3): 751–775zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hwang H.J., Kang K., Stevens A. (2005) Global solutions of nonlinear transport equations for chemosensitive movement. SIAM J. Math. Anal. 36, 1177–1199zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Levermore C.D. (1996) Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Mori S., Crain B.J., Chacko V.P., Zijl P.C.M. (1999) Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging. Ann. Neurol. 45, 265–269CrossRefGoogle Scholar
  18. 18.
    Othmer H.G., Dunbar S.R., Alt W. (1988) Models of dispersal in biological systems. J. Math. Biol. 26, 263–298zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Othmer H.G., Hillen T. (2001) The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62(4): 1122–1250MathSciNetGoogle Scholar
  20. 20.
    Robinson J.C. (2001) Infinite-Dimensional Dynamical Systems. Cambridge Texts in Applied Mathematics. Cambridge University Press, CambridgeGoogle Scholar
  21. 21.
    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
  22. 22.
    Tranquillo R.T., Barocas V.H. (1997). A continuum model for the role of fibroblast contact guidance in wound contraction. In: Alt W., Deutsch A., Dunn G. (eds). Dynamics of Cell and Tissue Motion. Birkhauser, Basel, pp. 159–164Google Scholar
  23. 23.
    Wang, Z., Hillen, T., Li, M. Global existence and travling waves to models for mesenchymal motion in one dimension. (in preparation) 2006Google Scholar
  24. 24.
    Wolf K., Mazo I., Leung H., Engelke K., von Andria U., Deryngina E.I., Strongin A.Y., Bröcker E.B., Friedl P. (2003) Compensation mechanism in tumor cell migration mesenchymal-amoeboid transition after blocking of pericellular proteolysis. J. Cell Biol. 160, 267–277CrossRefGoogle Scholar
  25. 25.
    Yurchenco P.D., Birk D.E., Mechan R.P. (1994) Extracellular Matrix Assembly. Academic Press, San DiegoCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.University of AlbertaEdmontonCanada

Personalised recommendations