Cities, wealth, and earthquakes follow continuous power-law probability distributions such as the Pareto distribution, which are canonically associated with scale-free behavior and self-similarity. However, many self-similar processes manifest as discrete steps that do not produce a continuous scale-free distribution. We construct a discrete power-law distribution that arises naturally from a simple model of hierarchical self-similar processes such as turbulence and vasculature, and we derive the maximum-likelihood estimate (MLE) for its exponent. Our distribution is self-similar, in contrast to previously studied discrete power laws such as the Zipf distribution. We show that the widely used MLE derived from the Pareto distribution leads to inaccurate estimates in systems that lack continuous scale invariance such as branching networks and data subject to logarithmic binning. We apply our MLE to data from bronchial tubes, blood vessels, and earthquakes to produce new estimates of scaling exponents and resolve contradictions among previous studies.

Modern models that derive allometric relationships between metabolic rate and body mass are based on the architectural design of the cardiovascular system and presume sibling vessels are symmetric in terms of radius, length, flow rate, and pressure. Here, we study the cardiovascular structure of the human head and torso and of a mouse lung based on three-dimensional images processed via our software Angicart. In contrast to modern allometric theories, we find systematic patterns of asymmetry in vascular branching, potentially explaining previously documented mismatches between predictions (power-law or concave curvature) and observed empirical data (convex curvature) for the allometric scaling of metabolic rate. To examine why these systematic asymmetries in vascular branching might arise, we construct a mathematical framework to derive predictions based on local, junction-level optimality principles that have been proposed to be favored in the course of natural selection and development. The two most commonly used principles are material-cost optimizations (construction materials or blood volume) and optimization of efficient flow via minimization of power loss. We show that material-cost optimization solutions match with distributions for asymmetric branching across the whole network but do not match well for individual junctions. Consequently, we also explore random branching that is constrained at scales that range from local (junction-level) to global (whole network). We find that material-cost optimizations are the strongest predictor of vascular branching in the human head and torso, whereas locally or intermediately constrained random branching is comparable to material-cost optimizations for the mouse lung. These differences could be attributable to developmentally-programmed local branching for larger vessels and constrained random branching for smaller vessels.

Scientists have long sought to understand how vascular networks supply blood and oxygen to cells throughout the body. Recent work focuses on principles that constrain how vessel size changes through branching generations from the aorta to capillaries and uses scaling exponents to quantify these changes. Prominent scaling theories predict that combinations of these exponents explain how metabolic, growth, and other biological rates vary with body size. Nevertheless, direct measurements of individual vessel segments have been limited because existing techniques for measuring vasculature are invasive, time consuming, and technically difficult. We developed software that extracts the length, radius, and connectivity of in vivo vessels from contrast-enhanced 3D Magnetic Resonance Angiography. Using data from 20 human subjects, we calculated scaling exponents by four methods-two derived from local properties of branching junctions and two from whole-network properties. Although these methods are often used interchangeably in the literature, we do not find general agreement between these methods, particularly for vessel lengths. Measurements for length of vessels also diverge from theoretical values, but those for radius show stronger agreement. Our results demonstrate that vascular network models cannot ignore certain complexities of real vascular systems and indicate the need to discover new principles regarding vessel lengths.