pipes, the total resistance of the network is the sum of the resistances of each level of the network, just as the equivalent resistance of electrical resistors in series is equal to the sum of the individual resistances: (A3) Each level has N identical pipes of length and radius . Hence, each level of the network has hydrodynamic resistances in parallel with one another. Eq. (A3) becomes (A4) If a total current Q ̇ 0 flows through the network, the total power dissipated is (A5) The method of Lagrange multipliers (Thomas and Finney 1988) is used to find the parameters that minimize this power loss, assuming that the organism’s mass M, blood volume V , and products N 3∝ V are known quantities. The method requires the construction of an ‘auxiliary function’ F′ that is a function of the variables allowed to vary in order to minimize the power loss, i.e., , , and , as well as the undetermined multipliers λ′ , λ′ M , and the λ′ : (A6) Note that the volume filling constraint has been applied N + 1 times, because it must hold for all N + 1 levels of the network. The equation for the volume of a cylinder was used to obtain an expression for the total volume of blood V within the network. As noted in the Supplementary Materials section of Savage et al (2008), this expression may be simplified considerably since μ, π, and Q ̇ 2 0 are constants: (A7) An optimizing expression for is obtained by taking the partial derivatives of F with respect to and setting that partial derivative equal to 0: (A8) Since must be independent of , one may be tempted to solve for directly: (A9) But it is much easier to observe that Eq. (A8) is also satisfied if (A10) and one then imposes the constraint that be independent of : (A11) This implies that (A12) where we have introduced the symbol β> to show that this constraint applies for narrow blood vessels, denoted by values greater than some particular = ̅. Taking the partial derivative of F with respect to and setting the derivative equal to 0 leads to the “volume filling” requirement obtained in Section IIIB: (A13) One may then show that the first and fourth terms (which are themselves sums) on the right-hand-side of Eq. (A7 add to zero. This results in a simplified expression for F: (A14) Since F does not depend on mass M, diferentiating Eq. (A14) with respect to M yields (A15) Integrating Eq. (A 5 and imposing the requirement that V 0 when M 0 implies that (A16) Eq. (A16) plays an important role in deriving Kleiber’s Law, as shown in Appendix C. APPENDIX B: OPTIMIZATION FOR WIDE BLOOD VESSELS The smaller surface area to volume ratio found in thicker blood vessels implies that dissipation due to wave reflection at node junctions is a much greater source of power loss than friction. This power loss may (in theory) be completely eliminated via the process of impedance matching. This discussion follows the methodology of Savage et al. (2008) and Caro et al. (2012). Pressure in the arterial system consists of both steady-state and oscillatory components (Caro et al. 2012). Since the steady-state components do not change, it is sufficient to consider just the oscillatory incident, reflected, and transmitted pressure and current waveforms: HEBERT Allometric and metabolic scaling 2023 ICC 224
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