A General Scaling Law of Vascular Tree: Optimal Principle of Bifurcations in Pulsatile Flow

Murray’s law, as the best-known optimal relationship between bifurcation calibers, is obtained based on the assumption of steady-state Poiseuille blood flow and is mostly accurate in small vessels. In middle sized and large vessels such as the aorta and coronary arteries, the pulsatile nature of the flow is dominant and deviations from Murray law have been observed. In the present study, a general scaling law is proposed, which describes the optimum relationship between the characteristics of bifurcations and pulsatile flow. This scaling law takes into account the deviations from Murray law in large vessels, and proposes optimal flow (i.e. less flow resistance) for the full range of the vascular system, from the small vessels to large ones such aorta. As a general scaling law, it covers both symmetrical and asymmetrical bifurcations. One of the merits of this scaling law is that bifurcation characteristics solely depend on the Womersley number of parent vessels. The diameter ratios suggested by this scaling law are in acceptable agreement with available clinical morphometric data such as those reported for coronary arteries and aortoiliac bifurcations. A numerical simulation of pulsatile flow for several Womersley numbers in bifurcation models according to the proposed scaling law and Murray law has been performed, which suggests that the general scaling law provides less flow resistance and more efficiency than Murray law in pulsatile flow.