  We analyse mathematical models in order to understand how microstructural
features of vascular networks may affect blood-flow dynamics, and to identify
particular characteristics that promote the onset of self-sustained
oscillations. By focusing on a simple three-node motif, we predict that network
"redundancy", in the form of a redundant vessel connecting two main
flow-branches, together with differences in haemodynamic resistance in the
branches, can promote the emergence of oscillatory dynamics. We use existing
mathematical descriptions for blood rheology and haematocrit splitting at
vessel branch-points to construct our flow model; we combine numerical
simulations and stability analysis to study the dynamics of the three-node
network and its relation to the system's multiple steady-state solutions.
While, for the case of equal inlet-pressure conditions, a "trivial" equilibrium
solution with no flow in the redundant vessel always exists, we find that it is
not stable when other, stable, steady-state attractors exist. In turn, these
"nontrivial" steady-state solutions may undergo a Hopf bifurcation into an
oscillatory state. We use the branch diameter ratio, together with the inlet
haematocrit rate, to construct a two-parameter stability diagram that
delineates regimes in which such oscillatory dynamics exist. We show that flow
oscillations in this network geometry are only possible when the branch
diameters are sufficiently different to allow for a sufficiently large flow in
the redundant vessel, which acts as the driving force of the oscillations.
These microstructural properties, which were found to promote oscillatory
dynamics, could be used to explore sources of flow instability in biological
microvascular networks.
