Resilience of three-dimensional sinusoidal networks in liver tissue

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—a feature known for hierarchical networks, but unexpected for plexus-like networks. 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.
Author summary The liver is the largest metabolic organ of the human body and pivotal for blood detoxification and the uptake of many medically relevant drugs. Inside liver tissue, a dense meshwork of blood micro-vessels, the sinusoidal network, contacts every hepatocyte cell. The architecture of these networks, and especially how their function responds to local damage is not well understood. Previous theoretical work addressed network resilience in simple two-dimensional networks such as vein networks of leaves but not in three-dimensional networks, partly due to the considerable difficulties of imaging three-dimensional tissues. Here, we build on unprecedented advances in imaging mouse liver tissue. By simulating arbitrarily-sized networks that faithfully reproduce the statistical features of spatially restricted experimental samples, we computationally characterize the relation between a weak alignment of the network along the flow direction and enhanced transport along this direction. A simple transport model predicts an inhomogeneous distribution of flow in the sinusoidal network. Concomitantly, these networks are resilient against the random removal of edges, but vulnerable against removal of ‘highway’ edges that carried a high flow in the unperturbed state. We speculate that yet unknown adaptive mechanisms balance the distribution of flows and thereby increase the resilience of this physiologically important network.