Modelling solute transport in the brain microcirculation: is it really well mixed inside the blood vessels?
Résumé
Most network models describing solute transport in the brain microcirculation use
the well-mixed hypothesis and assume that radial gradients inside the blood vessels
are negligible. Recent experimental data suggest that these gradients, which may
result from heterogeneities in the velocity field or consumption in the tissue, may
in fact be important. Here, we study the validity of the well-mixed hypothesis in
network models of solute transport using theoretical and computational approaches.
We focus on regimes of weak coupling where the transport problem inside the
vasculature is independent of the concentration field in the tissue. In these regimes,
the boundary condition between vessels and tissue can be modelled by a Robin
boundary condition. For this boundary condition and for a single cylindrical capillary,
we derive a one-dimensional cross-section average transport problem with dispersion
and exchange coefficients capturing the effects of radial gradients. We then extend
this model to a network of connected tubes and solve the problem in a complex
anatomical network. By comparing with results based on the well-mixed hypothesis,
we find that dispersive effects are a fundamental component of transport in transient
situations with relatively rapid injections, i.e. frequencies above one Hertz. For slowly
varying signals and steady states, radial gradients also significantly impact the spatial
distribution of vessel/tissue exchange for molecules that easily cross the blood brain
barrier. This suggests that radial gradients cannot be systematically neglected and that
there is a crucial need to determine the impact of spatio-temporal heterogeneities on
transport in the brain microcirculation.
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