\(M^5\) mesoscopic and macroscopic models for mesenchymal motion. (English) Zbl 1112.92003

Summary: 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 walks 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.


92C17 Cell movement (chemotaxis, etc.)
47N60 Applications of operator theory in chemistry and life sciences
60G35 Signal detection and filtering (aspects of stochastic processes)
92C37 Cell biology
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[1] Alt W. (1981) Singular perturbation of differential integral equations describing biased random walks. J. Reine Angew. Math. 322, 15–41 · Zbl 0437.60059
[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
[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) · Zbl 1052.92005
[5] Dallon J.C., Sherratt J.A. (2000) A mathematical model for spatially varying extracellular matrix alignment. SIAM J. Appl. Math. 61, 506–527 · Zbl 1012.92019
[6] Dallon J.C., Sherratt J.A., Maini P.K. (2001) Modelling the effects of transforming growth factor-{\(\beta\)} on extracellular alignment in dermal wound repair. Wound Rep. Reg. 9, 278–286
[7] Dickinson R. (2000) A generalized transport model for biased cell migration in an anisotropic environment. J. Math. Biol. 40, 97–135 · Zbl 0998.92005
[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 · Zbl 0912.92002
[9] Dolak Y., Schmeiser C. (2005) Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanics. J. Math. Biol. 51, 595–615 · Zbl 1077.92003
[10] Friedl P., Bröcker E.B. (2000) The biology of cell locomotion within three dimensional extracellular matrix. Cell Motility Life Sci. 57, 41–64
[11] Friedl P., Wolf K. (2003) Tumour-cell invasion and migration: diversity and escape mechanisms. Nat. Rev. 3, 362–374
[12] Hillen T. (2004) On L 2-closure of transport equations: the Cattaneo closure. Discrete Cont. Dyn. Syst. B 4(4): 961–982 · Zbl 1052.92006
[13] Hillen T. (2005) On the L 2-closure of transport equations: the general case. Discrete Cont. Dyn. Syst. B 5(2): 299–318 · Zbl 1077.92004
[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 · Zbl 1002.35120
[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 · Zbl 1099.82018
[16] Levermore C.D. (1996) Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065 · Zbl 1081.82619
[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
[18] Othmer H.G., Dunbar S.R., Alt W. (1988) Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 · Zbl 0713.92018
[19] Othmer H.G., Hillen T. (2001) The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62(4): 1122–1250 · Zbl 1103.35098
[20] Robinson J.C. (2001) Infinite-Dimensional Dynamical Systems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge
[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 · Zbl 0963.60093
[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 · Zbl 1001.92528
[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
[25] Yurchenco P.D., Birk D.E., Mechan R.P. (1994) Extracellular Matrix Assembly. Academic Press, San Diego
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