Stochastic modelling of biased cell migration and collagen matrix modification

Matrix dynamics plays a crucial role in several physiological and pathological processes. In this paper we develop a model framework, which describes the temporal fibre network evolution depending on the influence of migrating fibroblasts. The cells are regarded as discrete objects in the plane, whose velocities are determined by a generalised Langevin equation. For its solution we verify existence and uniqueness. The courses of the trajectories are affected by two external impulses, chemotaxis and contact guidance, respectively. The extracellular matrix is described by a continuous vector field which contains both information on density and orientation of the fibrous material. Modelling dynamic interaction between the discrete and the continuum variables is an essential point of this paper. In particular, the smoothing of the fluctuating paths plays a key role. Besides a detailed description of the formulated equations, we also supply the condensed pseudo code of the algorithm. We investigate several examples and present results both from artificial and real data.