Skip to main content

Advertisement

SpringerLink
Log in

M5 mesoscopic and macroscopic models for mesenchymal motion

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alt W. (1981) Singular perturbation of differential integral equations describing biased random walks. J. Reine Angew. Math. 322, 15–41

    MATH  MathSciNet  Google Scholar 

  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, 137–145

    Article  Google Scholar 

  3. Berenschot, G. Vizualization of Diffusion Tensor Imaging. Eindhoven University of Technology, Master Thesis (2003)

  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)

  5. Dallon J.C., Sherratt J.A. (2000) A mathematical model for spatially varying extracellular matrix alignment. SIAM J. Appl. Math. 61, 506–527

    Article  MATH  MathSciNet  Google Scholar 

  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, 278–286

    Article  Google Scholar 

  7. Dickinson R. (2000) A generalized transport model for biased cell migration in an anisotropic environment. J. Math. Biol. 40, 97–135

    Article  MATH  MathSciNet  Google Scholar 

  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. 149–158

    Google Scholar 

  9. Dolak Y., Schmeiser C. (2005) Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanics. J. Math. Biol. 51, 595–615

    Article  MATH  MathSciNet  Google Scholar 

  10. Friedl P., Bröcker E.B. (2000) The biology of cell locomotion within three dimensional extracellular matrix. Cell Motility Life Sci. 57, 41–64

    Article  Google Scholar 

  11. Friedl P., Wolf K. (2003) Tumour-cell invasion and migration: diversity and escape mechanisms. Nat. Rev. 3, 362–374

    Google Scholar 

  12. Hillen T. (2004) On L 2-closure of transport equations: the Cattaneo closure. Discrete Cont. Dyn. Syst. B 4(4): 961–982

    MATH  MathSciNet  Google Scholar 

  13. Hillen T. (2005) On the L 2-closure of transport equations: the general case. Discrete Cont. Dyn. Syst. B 5(2): 299–318

    Article  MATH  MathSciNet  Google Scholar 

  14. Hillen T., Othmer H.G. (2000) The diffusion limit of transport equations derived from velocity jump processes. SIAM J. Appl. Math. 61(3): 751–775

    Article  MATH  MathSciNet  Google Scholar 

  15. Hwang H.J., Kang K., Stevens A. (2005) Global solutions of nonlinear transport equations for chemosensitive movement. SIAM J. Math. Anal. 36, 1177–1199

    Article  MATH  MathSciNet  Google Scholar 

  16. Levermore C.D. (1996) Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065

    Article  MATH  MathSciNet  Google Scholar 

  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, 265–269

    Article  Google Scholar 

  18. Othmer H.G., Dunbar S.R., Alt W. (1988) Models of dispersal in biological systems. J. Math. Biol. 26, 263–298

    Article  MATH  MathSciNet  Google Scholar 

  19. Othmer H.G., Hillen T. (2001) The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62(4): 1122–1250

    MathSciNet  Google Scholar 

  20. Robinson J.C. (2001) Infinite-Dimensional Dynamical Systems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge

    Google Scholar 

  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): 183–212

    Article  MATH  MathSciNet  Google Scholar 

  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. 159–164

    Google Scholar 

  23. Wang, Z., Hillen, T., Li, M. Global existence and travling waves to models for mesenchymal motion in one dimension. (in preparation) 2006

  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, 267–277

    Article  Google Scholar 

  25. Yurchenco P.D., Birk D.E., Mechan R.P. (1994) Extracellular Matrix Assembly. Academic Press, San Diego

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Alberta, Edmonton, AB, Canada, T6G2G1

    Thomas Hillen

Authors
  1. Thomas Hillen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Hillen.

Additional information

Dedicated to K.P. Hadeler, a great scientist, teacher, and friend.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hillen, T. M5 mesoscopic and macroscopic models for mesenchymal motion. J. Math. Biol. 53, 585–616 (2006). https://doi.org/10.1007/s00285-006-0017-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-006-0017-y

Keywords

Search

Navigation