Derivation of a Cell Migration Transport Equation from an Underlying Random Walk Model

Abstract

Cell movement in an anisotropic environment has often been modeled by considering the cell velocity, υ(t) ≡ [υ _x (t) υ _y (t)υ _z (t)]^T, to be a Markov stochastic process in time, t; i.e. the future statistics of υ(t) are assumed dependent only on the current value, not on its history. The statistics of the motion are then determined by the joint probability density, p(υ,r,t), where r(t) ≡ [x(t) y(t) z(t)]^T is the cell position. However, because of its direct analogy with cell density, the probability density of r alone, p(r,t), often is of more interest on a longer time scale where the instantaneous value of υ is less important than its long-time stationary distribution. We show here how an approximate differential equation for p(r,t) can be derived if the time scale of interest is much larger than the relaxation time of υ. This analysis adopts the projector operator formalism used in Gardiner (1983, Chapter 6) and in Kubo et al. (1991).