Inspection of Fig. 2 shows that, in addition to being the number of new pipes at each junction node (or branching point), n is also the ratio of the number of pipes in level + 1 compared to the number of pipes in level : (7) Eqs. (6) and (7) gives us the “volume filling” or “space filling” constraint that must be met by the network: (8) Because the branching ratio n is assumed to be the same for all values of , γ equals γ, a constant. C. Minimizing power losses in the cardiovascular system Biologist Cecil D. Murray (1926) suggested that the human cardiovascular system was constructed in such a way as to minimize the work required to oxygenate the body, and he provided simple calculations to demonstrate that construction. J. R. Womersley (1955) published a mathematical solution for velocity, flow rate, and viscous drag in arteries and noted a phase-lag between pressure gradient and flow, similar to the voltage-current phase lag that can exist in an alternating electrical circuit. Womersley did not explicitly state all the details of his derivation, but Shirazi (1972) “fills in” the details. Womersley’s solutions strongly suggest that the cardiovascular system is designed to minimize energy losses as oxygenated blood is transported throughout the body. Within a mammalian or avian cardiovascular system, energy losses Figure 5. An organism contains NN service volumes, each with individual volume N ∝ 3N. (a) Because an organism’s entire volume must be supplied with nutrients, V NN N. (b) But one may treat the terminal branches themselves as being part of each service volume. In that case, there are only NN− service volumes, each with a volume proportional to 3N−. Because the organism’s entire volume V must still be supplied we have V NN− N−. In general, V N ∝ N 3 , for all . Note: the radius of N and N are not to scale. Figure 4. Because the radius of a terminal branch N within the network (i.e., a capillary in the mammalian cardiovascular system) is much smaller than its length N, it is reasonable to assume that the “service volume” N of biological material supplied by each terminal branch is proportional to 3N. 1 1 3 1 1 3 k k k k k l N n l N − − + + = = = HEBERT Allometric and metabolic scaling 2023 ICC 210
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