  Within animals, oxygen exchange occurs within networks containing potentially
billions of microvessels that are distributed throughout the animal's body.
Innovative imaging methods now allow for mapping of the architecture and blood
flows within real microvascular networks. However, these data streams have so
far yielded little new understanding of the physical principles that underlie
the organization of microvascular networks, which could allow healthy networks
to be quantitatively compared with networks that have been damaged, e.g. due to
diabetes. A natural mathematical starting point for understanding network
organization is to construct networks that are optimized accordingly to
specified functions. Here we present a method for deriving transport networks
that optimize general functions involving the fluxes and conductances within
the network. In our method Kirchoff's laws are imposed via Lagrange
multipliers, creating a large, but sparse system of auxiliary equations. By
treating network conductances as adiabatic variables, we derive a gradient
descent method in which conductances are iteratively adjusted, and auxiliary
variables are solved for by two inversions of O(N^2) sized sparse matrices. In
particular our algorithm allows us to validate the hypothesis that
microvascular networks are organized to uniformly partition the flow of red
blood cells through vessels. The theoretical framework can also be used to
consider more general sets of objective functions and constraints within
transport networks, including incorporating the non-Newtonian rheology of blood
(i.e. the Fahraeus-Lindqvist effect). More generally by forming linear
combinations of objective functions, we can explore tradeoffs between different
optimization functions, giving more insight into the diversity of biological
transport networks seen in nature.
