  We derive the main properties of adaptive Hagen-Poiseuille flows in elastic
microchannel networks akin to biological veins in organisms. We show that
adaptive Hagen-Poiseuille flows successfully simulate key features of
\textit{Physarum polycephalum} networks, replicating physiological
out-of-equilibrium phenomena like peristalsis and shuttle streaming, associated
with the mechanism of nutrient transport in \textit{Physarum}. A new
topological steady state has been identified for asynchronous adaptation,
supporting out-of-equilibrium laminar fluxes. Adaptive Hagen-Poiseuille flows
show saturation effects on the fluxes in contractile veins, as observed in
animal and artificial contractile veins.
