Resilience of three-dimensional sinusoidal networks in liver tissue.

Jens Karschau,André Scholich,Marino Zerial,Jonathan Wise,Benjamin M Friedrich,Hernán Morales-Navarrete,Yannis Kalaidzidis

doi:10.1371/journal.pcbi.1007965

Abstract

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.

Highlights

Leaf venation [1], fungal mycelium [2, 3], and animal trails networks [4], river deltas [5], and even force networks in granular materials [6], each represent natural transport networks formed by self-organization
We develop a network generation algorithm that reproduces statistical features of the sinusoidal network, enabling us to simulate arbitrarily sized networks from spatially restricted biological samples and, to explore in silico a design space of three-dimensional networks
We developed a Monte-Carlo algorithm to generate synthetic networks that faithfully reproduce the statistical features of spatially restricted samples of hepatic sinusoidal networks, using multi-objective optimization of both node degree and edge length distribution, see Fig 2A

Summary

Introduction

Leaf venation [1], fungal mycelium [2, 3], and animal trails networks [4], river deltas [5], and even force networks in granular materials [6], each represent natural transport networks formed by self-organization. Past research addressed resilience properties almost exclusively in two-dimensional biological transport networks. It was shown that self-organization by local feedback rules can generate hierarchical networks resembling those of leaf networks, which optimize flow resistance upon removal of a single link [1]. Work on two-dimensional networks addressed the balance between the cost of network formation and network resilience to random failure [9], the cost of repair after perturbations [10], or adaptation to fluctuations in load [11]

Methods

Results

Conclusion