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 ¾ 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.

异体测量法或人体大小与生物机体生理之间关系的定量研究是生物物理定标研究的重要领域。血管网络中分形分支的West-Brown-Enquist（WBE）模型解释了经验异速Kleiber定律（代谢率随动物质量变化的¾标度指数）。WBE模型提出了许多新问题，例如如何更准确地解释毛细现象，以及血流速度对毛细血管大小的更现实的依赖性。我们建议分支模型的广义公式，并研究在分形血管系统中的遍历性。通常，这种系统中的流体流动不是遍历的，遍历性的破坏归因于网络的分形结构。所以，除了衰老和大分子拥挤等机制外，分形分支还可以看作是生物物理系统中遍历性破坏的来源。对于无法长期观察时间序列的广泛生物医学应用，非遍历性的计算很重要。还讨论了与微流控应用的相关性。