cardiovascular system). WEB claim that (1) metabolism should scale proportionally to nutrient flow (or, equivalently, total fluid flow) in this circulatory system and (2) in order to minimize the energy dissipated in transport, the volume of fluid used to transport nutrients (i.e., blood volume) is a fixed fraction of body mass. They then proceed by analyzing the consequences of these two claims at the level of the smallest circulatory tubules (capillaries, alveoli, etc.). Experimentally, the volume contained in those smallest tubules is constant across a wide range of masses. Because fluid flow through a tubule is determined by the volume thereof, the total fluid flow is proportional to the total number of smallest tubules. Thus, if denotes the basal metabolic rate, the total fluid flow, and the number of minimal tubules, Circulatory systems do not grow by simply scaling proportionally larger; they become more deeply nested. The depth of nesting depends on the self-similarity exponents of the tubule dimensions, and the effects of that depth depend on how many "child" tubules each branching produces. Connecting these values to macroscopic quantities depends (very loosely) on a precise model of tubules. WEB show that, if the tubules are well-approximated by rigid cylinders, then, in order to prevent the fluid from "getting clogged" in small cylinders, the total fluid volume satisfies (Despite conceptual similarities, this condition is inconsistent with Murray's law.) Because blood volume is a fixed fraction of body mass, Non-power-law scaling Closer analysis suggests that Kleiber's law does not hold over a wide variety of scales. Metabolic rates for smaller animals (birds under , or insects) typically fit to much better than ; for larger animals, the reverse holds. As a result, log-log plots of metabolic rate versus body mass appear to "curve" upward, and fit better to quadratic models. In all cases, local fits exhibit exponents in the range. Modified circulatory models Adjustments to the WBE model that retain assumptions of network shape predict larger scaling exponents, worsening the discrepancy with observed data. But one can retain a similar theory by relaxing WBE's assumption of a nutrient transport network that is both fractal and circulatory. Different networks are less efficient, in that they exhibit a lower scaling exponent, but a metabolic rate determined by nutrient transport will always exhibit scaling between and . (WBE argued that fractal circulatory networks would necessarily evolve to minimize energy used for transport, but other