M5 mesoscopic and macroscopic models for mesenchymal motion
- 196 Downloads
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.
KeywordsTransport Equation Drift Velocity Diffusion Tensor Macroscopic Model Moment Closure
Unable to display preview. Download preview PDF.
- 3.Berenschot, G. Vizualization of Diffusion Tensor Imaging. Eindhoven University of Technology, Master Thesis (2003)Google Scholar
- 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
- 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
- 11.Friedl P., Wolf K. (2003) Tumour-cell invasion and migration: diversity and escape mechanisms. Nat. Rev. 3, 362–374Google Scholar
- 20.Robinson J.C. (2001) Infinite-Dimensional Dynamical Systems. Cambridge Texts in Applied Mathematics. Cambridge University Press, CambridgeGoogle 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. 159–164Google Scholar
- 23.Wang, Z., Hillen, T., Li, M. Global existence and travling waves to models for mesenchymal motion in one dimension. (in preparation) 2006Google Scholar