  We build an evolutionary scenario that explains how some crucial
physiological constraints in the arterial network of mammals - i.e. hematocrit,
vessels diameters and arterial pressure drops - could have been selected by
evolution. We propose that the arterial network evolved while being constrained
by its function as an organ. To support this hypothesis, we focus our study on
one of the main function of blood network: oxygen supply to the organs. We
consider an idealized organ with a given oxygen need and we optimize blood
network geometry and hematocrit with the constraint that it must fulfill the
organ oxygen need. Our model accounts for the non-Newtonian behavior of blood,
its maintenance cost and F\aa hr\ae us effects (decrease in average
concentration of red blood cells as the vessel diameters decrease). We show
that the mean shear rates (relative velocities of fluid layers) in the tree
vessels follow a scaling law related to the multi-scale property of the tree
network, and we show that this scaling law drives the behavior of the optimal
hematocrit in the tree. We apply our scenario to physiological data and reach
results fully compatible with the physiology: we found an optimal hematocrit of
0.43 and an optimal ratio for diameter decrease of about 0.79. Moreover our
results show that pressure drops in the arterial network should be regulated in
order for oxygen supply to remain optimal, suggesting that the amplitude of the
arterial pressure drop may have co-evolved with oxygen needs.
