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Within animals, oxygen exchange occurs within networks containing potentiallybillions of microvessels that are distributed throughout the animal's body.Innovative imaging methods now allow for mapping of the architecture and bloodflows within real microvascular networks. However, these data streams have sofar yielded little new understanding of the physical principles that underliethe organization of microvascular networks, which could allow healthy networksto be quantitatively compared with networks that have been damaged, e.g. due todiabetes. A natural mathematical starting point for understanding networkorganization is to construct networks that are optimized accordingly tospecified functions. Here we present a method for deriving transport networksthat optimize general functions involving the fluxes and conductances withinthe network. In our method Kirchoff's laws are imposed via Lagrangemultipliers, creating a large, but sparse system of auxiliary equations. Bytreating network conductances as adiabatic variables, we derive a gradientdescent method in which conductances are iteratively adjusted, and auxiliaryvariables are solved for by two inversions of O(N^2) sized sparse matrices. Inparticular our algorithm allows us to validate the hypothesis thatmicrovascular networks are organized to uniformly partition the flow of redblood cells through vessels. The theoretical framework can also be used toconsider more general sets of objective functions and constraints withintransport networks, including incorporating the non-Newtonian rheology of blood(i.e. the Fahraeus-Lindqvist effect). More generally by forming linearcombinations of objective functions, we can explore tradeoffs between differentoptimization functions, giving more insight into the diversity of biologicaltransport networks seen in nature.

A Gradient Descent Method for Optimization of Model Microvascular Networks