  Across mammalian species, solute exchange takes place in complex
microvascular networks. In the human placenta, the primary exchange units are
terminal villi that contain disordered networks of fetal capillaries and are
surrounded externally by maternal blood. Here we show how the irregular
internal structure of a terminal villus determines its exchange capacity for a
wide range of solutes. Distilling geometric features into three scalar
parameters, obtained from image analysis and computational fluid dynamics, we
capture archetypal features of the the structure-function relationship of
terminal villi using a simple algebraic approximation, revealing transitions
between flow- and diffusion-limited transport at vessel and network levels. Our
theory accommodates countercurrent effects, incorporates nonlinear blood
rheology and offers an efficient method for testing network robustness. Our
results show how physical estimates of solute transport, based on carefully
defined geometrical statistics, provide a viable method for linking placental
structure and function, and offer a framework for assessing transport in other
microvascular systems.
