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Mathematical Medicine and Biology Advance Access originally published online on October 17, 2007
Mathematical Medicine and Biology 2007 24(4):379-400; doi:10.1093/imammb/dqm007
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Shock formation and non-linear dispersion in a microvascular capillary network

S. R. Pop, G. Richardson, S. L. Waters and O. E. Jensen{dagger}

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

{dagger} Email: oliver.jensen{at}nottingham.ac.uk

Received on May 14, 2007. Revised on August 1, 2007. Accepted on August 3, 2007.

Temporal and spatial fluctuations are a common feature of blood flow in microvascular networks. Among many possible causes, previous authors have suggested that the non-linear rheological properties of capillary blood flow (notably the Fåhræus effect, the Fåhræus–Lindqvist effect and the phase-separation effect at bifurcations) may be sufficient to generate temporal fluctuations even in very simple networks. We have simulated blood flow driven by a fixed pressure drop through a simple arcade network using coupled hyperbolic partial differential equations (PDEs) that incorporate well-established empirical descriptions of these rheological effects, accounting in particular for spatially varying haematocrit distributions; we solved the PDE system using a characteristic-based method. Our computations indicate that, under physiologically realistic conditions, there is a unique steady flow in an arcade network which is linearly stable and that plasma skimming suppresses the oscillatory decay of perturbations. In addition, we find that non-linear perturbations to haematocrit distributions can develop shocks via the Fåhræus effect, providing a novel mechanism for non-linear dispersion in microvascular networks.

Keywords: microcirculation; capillary network; haematocrit; blood rheology


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