  Understanding vascular adaptation, namely what drives veins to shrink or
grow, is key for the self-organization of flow networks and their optimization.
From the top-down principle of minimizing flow dissipation at a fixed metabolic
cost within flow networks, flow shear rate resulting from the flows pervading
veins is hypothesized to drive vein adaptation. Yet, there is no bottom-up
derivation of how flow forces impact vein dynamics. From the physical principle
of force balance, shear rate acts parallel to vein walls, and hence, naively
shear rate could only stretch veins and not dilate or shrink them. We, here,
resolve this paradox by theoretically investigating force balance on a vein
wall in the context of the vascular network of the model organism Physarum
polycephalum. We propose, based on previous mechanical studies of cross-linked
gels, that shear induces a nonlinear, anisotropic response of the actomyosin
gel, making up vein walls that can indeed drive vein dilatation. Furthermore,
our force balance approach allows us to identify that shear feedback occurs
with a typical timescale and with a typical target shear rate that are not
universal properties of the material but instead depend smoothly on the
location of the vein within the network. In particular, the target shear rate
is related to the vein's hydrostatic pressure, which highlights the role of
pressure in vascular adaptation. Finally, since our derivation is based on
force balance and fluid mechanics, we believe our approach can be extended to
vascular adaptation in other organisms.
