Allometric scaling law and ergodicity breaking in the vascular system

TL;DR

This paper explores how fractal vascular networks exhibit ergodicity breaking, affecting biophysical scaling laws and biomedical applications, by generalizing the West-Brown-Enquist model and analyzing fluid flow dynamics.

Contribution

It introduces a generalized fractal branching model and investigates ergodicity breaking in vascular systems, highlighting its significance in biophysical scaling and biomedical contexts.

Findings

Fluid flow in fractal vascular systems is generally non-ergodic.

Fractal structure contributes to ergodicity breaking beyond aging and crowding.

Accounting for non-ergodicity impacts biomedical and microfluidic applications.

Abstract

Allometry or the quantitative study of the relationship of body size to living organism physiology is an important area of biophysical scaling research. The West-Brown-Enquist (WBE) model of fractal branching in a vascular network explains the empirical allometric Kleiber law (the 3/4 scaling exponent for metabolic rates as a function of animal's mass). The WBE model raises a number of new questions, such as how to account for capillary phenomena more accurately and what are more realistic dependencies for blood flow velocity on the size of a capillary. We suggest a generalized formulation of the branching model and investigate the ergodicity in the fractal vascular system. In general, the fluid flow in such a system is not ergodic, and ergodicity breaking is attributed to the fractal structure of the network. Consequently, the fractal branching may be viewed as a source of ergodicity breaking in biophysical systems, in addition to such mechanisms as aging and macromolecular crowding. Accounting for non-ergodicity is important for a wide range of biomedical applications where long observations of time series are impractical. The relevance to microfluidics applications is also discussed.