arise from two sources, (1) friction between blood and the vessel wall and (2) impedance losses due to reflection of pulsatile waves at a branch junction. In narrow diameter blood vessels, the blood vessel’s high surface area to volume ratio implies that a large fraction of the fluid will be in contact with the vessel walls. Hence, dissipation due to friction is most important in narrow vessels. One may use the method of Lagrange multipliers (Thomas and Finney 1988) to show that power losses in narrow blood vessels are minimized when (9) In this case, total cross-sectional area of the vessels does not remain constant, but increases as increases. Details of the derivation are provided in Appendix A. In contrast, the smaller surface area to volume ratio found in wide blood vessels implies that friction is not the most important source of dissipation for wide blood vessels. Rather, dissipation due to reflection at node junctions is the dominant source of power loss. This power loss may be completely eliminated via the process of impedance matching. Doing so (details are in Appendix B) yields the result (10) For wide-diameter blood vessels (lower values of ), cross-sectional area of the network remains constant. This is consistent with observations that the cross-sectional area of the vascular bed stays constant in both humans and dogs until the vessels reach a transitional size, at which point the area of the bed begins to increase (Caro et al. 2012, p. 244). In the supplementary material to their paper, Savage et al. (2008) provide the outline for the more general case of rigid blood vessels of any radius. Although this derivation neglects blood vessel elasticity, it still gives the exact result for narrow elastic blood vessels and results that are very close to the exact answer for wide elastic blood vessels. Hence, their simplified derivation illustrates the important features of the much more complicated exact solution (Womersley 1955, Shirazi 1972) for elastic blood vessels. The general solution provides radius scaling given by Eq. (9) for very narrow blood vessels, and radius scaling given by Eq. (10) for very wide blood vessels. This optimization of the circulatory system is so obvious that scientists cannot help but use words like “design” when describing the circulatory system. Li (2000, p. 113) states, “The optimal design features of the mammalian cardiovascular system have been marveled at by us Homo sapiens for many decades.” Li goes on to say (2000, pp. 125-126) Invariant pulse transmission features are embedded in the similar pulse pressure and flow waveforms observed at corresponding anatomical sites. The precision of natural design is even more amazing at vascular branching junctions, where branching vessel impedances are practically matched to ensure pulse wave transmission at utmost efficiency with minimal wave reflection and energy losses. That similar cardiovascular transmission features are observed at “corresponding anatomical sites” for different animals seems too unlikely a coincidence to attribute to “convergent evolution”. In the introduction to his detailed derivation of Womersley’s (1955) results, Shirazi (1972, p. 2) states Womersley’s work forms an important link in the continuing chain of understanding [of the cardiovascular system]. We have chosen to present his version not because it is the most sophisticated work in this area but because within its limitations it is a well-developed treatment of several aspects of the arterial problem, and suggests a rational basis for many of the peculiar characteristics observed in the mammalian cardiovascular system. [emphasis mine] Yet Brown et al. (2000, p. 11) instead attribute such features to natural selection: “Natural selection for efficient design of such distribution and support . . . has resulted in the evolution of networks with self-similar, hierarchically scaled architectures.” Yet, isn’t it reasonable to ask if the phrase “natural selection for” doesn’t “smuggle in” the same intelligent intentionality to presumed naturalistic explanations that is attributed to an intelligent Designer by other scientists? After all, “selection” is always rooted in intelligence and volition, while the word “for” in this context indicates purposeful intentions with the definitive target of “efficient design.” If evolutionary biologists who proceed from an interpretive framework of naturalism have not really provided a non-intelligent explanation, but instead injected a substitute intelligence cloaked in selectionist jargon, then isn’t creation by an intelligent Engineer a more plausible explanation? After all, design by an intelligent Engineer explains 1) optimized features, that 2) can be reduced to mathematical formulas, and 3) operate by the same engineering principles as human-engineered fluid-transport systems. D. Deriving Kleiber’s Law The results of the previous two sections suggest that the way to minimize energy losses due to both friction and reflection is to apply the volume-filling constraint of Eq. (8) for all values of , the constraint of Eq. (9) for narrow blood vessels (higher values) and the constraint of Eq. (10) for wide blood vessels (smaller values). This may be done with a “transitional” value of = ̅, as illustrated in Fig. 6. Savage et al. (2008) explain more clearly the derivation of Kleiber’s Law, some details of which are omitted in West et al. (2000). A key step in this derivation is the realization that blood volume V is proportional to the organism’s mass M. We follow their derivation in Appendix C. They obtain (11) Here A0 and A1 are mass-independent constants. Strictly speaking, the relationship above between body mass M and basal metabolic rate B only becomes Kleiber’s Law in the limit of infinite mass (Savage et al. 2008). Hence, deviations from Kleiber’s Law are more likely for smaller organisms, and this may help to explain some of the contrary results mentioned in Section IV. WBE demonstrated quite a few allometric relationships for char1 1 3 k k k r n r − + = = = HEBERT Allometric and metabolic scaling 2023 ICC 211
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