  Can three-dimensional, microvasculature networks still ensure blood supply if
individual links fail? We address this question in the sinusoidal network, a
plexus-like microvasculature network, which transports nutrient-rich blood to
every hepatocyte in liver tissue, by building on recent advances in
high-resolution imaging and digital reconstruction of adult mice liver tissue.
We find that the topology of the three-dimensional sinusoidal network reflects
its two design requirements of a space-filling network that connects all
hepatocytes, while using shortest transport routes: sinusoidal networks are
sub-graphs of the Delaunay graph of their set of branching points, and also
contain the corresponding minimum spanning tree, both to good approximation. To
overcome the spatial limitations of experimental samples and generate
arbitrarily-sized networks, we developed a network generation algorithm that
reproduces the statistical features of 0.3-mm-sized samples of sinusoidal
networks, using multi-objective optimization for node degree and edge length
distribution. Nematic order in these simulated networks implies anisotropic
transport properties, characterized by an empirical linear relation between a
nematic order parameter and the anisotropy of the permeability tensor. Under
the assumption that all sinusoid tubes have a constant and equal flow
resistance, we predict that the distribution of currents in the network is very
inhomogeneous, with a small number of edges carrying a substantial part of the
flow. We quantify network resilience in terms of a permeability-at-risk, i.e.\
permeability as function of the fraction of removed edges. We find that
sinusoidal networks are resilient to random removal of edges, but vulnerable to
the removal of high-current edges. Our findings suggest the existence of a
mechanism counteracting flow inhomogeneity to balance metabolic load on the
liver.
