Allometry and Ergodicity breaking in a fractal capillary network

Allometry is a type of scaling in biology, which explains how various parameters (such as the metabolism rate, speed, and even the lifespan of an animal) depend on the average body mass. Ergodicity is a concept in the theory of dynamical systems, which implies that the time average as a parameter can be substituted by the ensemble (or phase-space) average. The flow of biological liquids is often non-ergodic.

In this paper, we discuss the famous West-Brown-Enquist (WBE) model of fractal branching in a vascular network. The WBE model explains, for example, why the average lifespan of an animal is proportional to power four root of its average body mass. We suggest an enhancement of the model and also suggest a possibility that fractal branching can lead to ergodicity breaking.

We consider the dependency of the time fraction on the volume fraction. The volume fraction, however, is not a fraction of the vascular circulatory system volume, but a fraction of organism tissue volume. When experimental observations are performed to trace a particle, the parameter of interest is the physical volume of the tissue, not the volume of liquid. So, the relevant parameter it is not just the volume of the fluid in capillaries, but the volume of the entire tissue, which is served by capillaries. The capillaries in this model provide equal access to all tissues. Consequently, the ensemble-averaged probability to find a blood particle in a certain volume of a tissue is just proportional to the volume.

Contrary to that, due to the fractal geometry, the time fraction is not necessarily proportional to the volume of the tissue. This is because a particle spends more time in smaller regions. The time-averaged probability is the fraction of time, which a particle spends in a given region to the total time of the circulation. Ergodicity depends on whether the time average is equal to the ensemble average.

Of course, in reality there is a limit how small capillaries can be, so in a physical system there is always a limit of fractal behavior, and a blood particle does not spend infinite time in smallest capillaries. However, fractal models imply that mathematically capillaries may be as small as one wants, which means the total time of circulation is unbounded (approaches infinity), however, time ratios are still finite.

This is, to my knowledge, a new result (nobody formulated it in this way), and our work is an extension of the WBE model.

M. Nosonovsky, P. Roy. “Allometric scaling law and ergodicity breaking in the vascular system”. Microfluidics and Nanofluidics (2020) 24:53
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