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

Hillen, Thomas

J. Math. Biol. 53, No. 4, 585-616 (2006).

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.

MSC:

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

Keywords:

moment closure; parabolic scaling; hydrodynamic scaling; protease dynamics

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