Theory of Scaling Law at Bifurcations

A scaling law is an analytic formulation describing morphometric relationships, often parameterized by the vessel lengths, diameters, and arterial volume between a feeding segment and the perfused subtree (13, 24, 78). Because the effect of a scaling law on wave reflection at bifurcations is of primary interest in this study, we focus on a specific aspect of the scaling laws: the relationship between the mother vessel diameter (d_m) and the diameter of the daughter vessels (d_d1, d_d2) at each bifurcation. This relationship can be written in a generic form in terms of either diameter or cross-sectional area as

where the subscripts m, d1, and d2 denote the mother and daughter vessels, respectively. τ is the scaling parameter that characterizes the different scaling laws (13). The well-known Murray's law was the first scaling law proposed, where τ = 3 was derived by minimizing the combined viscous power dissipation and the metabolic power expenditure across the vascular network (40, 41). Subsequent investigations (27, 76) demonstrated that the main limitation of Murray's law was that it considered each bifurcation in isolation instead of being part of a complete network and proposed successive improvements (based on the minimization of the cost of fluid conduction and fluid metabolism) culminating in the well-established HK law, with an exponent τ = 7/3 (23, 24). In the analysis below, these two laws are considered in detail.