Fractals and fractal dimension of systems of blood vessels: An analogy between artery trees, river networks, and urban hierarchies

TL;DR

This paper explores the fractal structure and hierarchical scaling laws of blood vessel networks, drawing analogies with river and urban systems, and reveals new allometric and power law relationships in biological and natural networks.

Contribution

It introduces exponential and linear scaling laws for blood vessel hierarchies, extending river network laws to biological systems and comparing human and animal vascular structures.

Findings

Human blood vessel data aligns more with natural fractal rules than dogs.

Derived power laws include Zipf's law for vessel length distribution.

Hierarchies of blood vessels, rivers, and urban systems show deep structural similarities.

Abstract

An analogy between the fractal nature of networks of arteries and that of systems of rivers has been drawn in the previous works. However, the deep structure of the hierarchy of blood vessels has not yet been revealed. This paper is devoted to researching the fractals, allometric scaling, and hierarchy of blood vessels. By analogy with Horton-Strahler's laws of river composition, three exponential laws have been put forward. These exponential laws can be reconstructed and transformed into three linear scaling laws, which can be named composition laws of blood vessels network. From these linear scaling laws it follows a set of power laws, including the three-parameter Zipf's law on the rank-size distribution of blood vessel length and the allometric scaling law on the length-diameter relationship of blood vessels in different orders. The models are applied to the observed data on human beings and animals early given by other researchers, and an interesting finding is that human bodies more conform to natural rules than dog's bodies. An analogy between the hierarchy of blood vessels, river networks, and urban systems are further drawn, and interdisciplinary studies of hierarchies will probably provide new revealing examples for the science of complexity.