Using a probabilistic approach to derive a two-phase model of flow-induced cell migration

Interstitial fluid flow is a feature of many solid tumours. In vitro experiments have shown that such fluid flow can direct tumour cell movement upstream or downstream depending on the balance between the competing mechanisms of tensotaxis and autologous chemotaxis. In this work we develop a probabilistic-continuum, two-phase model for cell migration in response to interstitial flow. We use a Fokker-Planck type equation for the cell-velocity probability density function, and model the flow-dependent mechanochemical stimulus as a forcing term which biases cell migration upstream and downstream. Using velocity-space averaging, we reformulate the model as a system of continuum equations for the spatio-temporal evolution of the cell volume fraction and flux, in response to forcing terms which depend on the local direction and magnitude of the mechanochemical cues. We specialise our model to describe a one-dimensional cell layer subject to fluid flow. Using a combination of numerical simulations and asymptotic analysis, we delineate the parameter regime where transitions from downstream to upstream cell migration occur. As has been observed experimentally, the model predicts downstream-oriented, chemotactic migration at low cell volume fractions, and upstream-oriented, tensotactic migration at larger volume fractions. We show that the locus of the critical volume fraction, at which the system transitions from downstream to upstream migration, is dominated by the ratio of the rate of chemokine secretion and advection. Our model predicts that, because the tensotactic stimulus depends strongly on the cell volume fraction, upstream migration occurs only transiently when the cells are initially seeded, and transitions to downstream migration occur at later times due to the dispersive effect of cell diffusion.