  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.
