  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 kinetic description for the cell-velocity
probability density function, and model the flow-dependent stimuli as forcing
terms which bias 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 also predicts that, because the tensotactic stimulus
depends strongly on the cell volume fraction, upstream, tensotaxis-dominated
migration occurs only transiently when the cells are initially seeded, and
transitions to downstream, chemotaxis-dominated migration occur at later times
due to the dispersive effect of cell diffusion.
