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Interstitial fluid flow is a feature of many solid tumours. In vitroexperiments have shown that such fluid flow can direct tumour cell movementupstream or downstream depending on the balance between the competingmechanisms of tensotaxis and autologous chemotaxis. In this work we develop aprobabilistic-continuum, two-phase model for cell migration in response tointerstitial flow. We use a Fokker-Planck type equation for the cell-velocityprobability density function, and model the flow-dependent mechanochemicalstimulus as a forcing term which biases cell migration upstream and downstream.Using velocity-space averaging, we reformulate the model as a system ofcontinuum equations for the spatio-temporal evolution of the cell volumefraction and flux, in response to forcing terms which depend on the localdirection and magnitude of the mechanochemical cues. We specialise our model todescribe a one-dimensional cell layer subject to fluid flow. Using acombination of numerical simulations and asymptotic analysis, we delineate theparameter regime where transitions from downstream to upstream cell migrationoccur. As has been observed experimentally, the model predictsdownstream-oriented, chemotactic migration at low cell volume fractions, andupstream-oriented, tensotactic migration at larger volume fractions. We showthat the locus of the critical volume fraction, at which the system transitionsfrom downstream to upstream migration, is dominated by the ratio of the rate ofchemokine secretion and advection. Our model predicts that, because thetensotactic stimulus depends strongly on the cell volume fraction, upstreammigration occurs only transiently when the cells are initially seeded, andtransitions to downstream migration occur at later times due to the dispersiveeffect of cell diffusion.

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