(B1) Note that the additional minus sign in the expressions in the second row of Eq. (B1) take into account the fact that the reflected waves propagate in the direction opposite to the incident waves. Also, the incident and reflected wave numbers are both denoted by κ1, since wavenumber is a property of the artery and not the wave itself. Here, the currrent amplitudes are positive real numbers, but the pressure amplitudes are complex, with the complex parts of each current incorporated into each (complex) pressure amplitude. This allows for possible phase differences between waveforms. At the node junction (x = 0) and at all times t, the sum of the incident and reflected waves must equal that of the transmitted wave: (B2) If this were not the case, then any existing pressure difference would quickly drive blood toward the region of lower pressure, removing the pressure difference. Impedance Z is defined as the ratio of applied oscillatory pressure to resulting oscillatory fluid flow. It is a property of the blood vessel and not the wave per se. Hence, the expressions relating the incident and reflected pressure waveforms to their corresponding currents will both be expressed in terms of the same impedance Z , and the transmitted wave will be expressed in terms of the impedance Z . The pressure and current amplitudes are related by (B3) Since the pressure amplitudes may generally be complex, the impedances may be complex, as well. At all times, the net inbound current at the junction must equal the outbound current, so (B4) Substituting the expressions from Eq. (B3) into Eq. (B4) yields (B5) Adding together and then subtracting Eqs. (B2) and (B5 results in an expression for the reflected pressure amplitude in terms of the incident pressure amplitude: (B6) To minimize power losses, the reflected amplitude should be zero. This condition is met if 1 1 1 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) i x t i x t i i i x t i x t r r i x t i x t t t pe Qe p e Qe pe Q e − − − − − − − − (B7) For the special case of inviscid (negligible viscosity) fluid flow, the hydrodynamic impedance is (Caro et al. 2012): (B8) where ρ is blood density and 0 is the Korteweg-Moens velocity (Moens 1878; Korteweg 1878), the velocity at which a blood pressure pulse propagates through the arterial system when viscosity is negligible. Eqs. (B7 and (B8) imply that, for large blood vessels, impedance matching is achieved if (B9) where we have introduced the symbol β< to show that this constraint applies for wide blood vessels, denoted by values less than some particular = ̅. Appendix C: Derivation of Kleiber’s Law Obtaining a general expression for blood vessels of intermediate length is extraordinarily difficult (see Shirazi 1972), and we omit this discussion here. However, from the results of the previous two appendices, it is apparent that efficiency of the cardiovascular system is increased by using the volume-filling condition expressed in Eq. (A13), as well as Eq. (A12) for narrow blood vessels and Eq. (B9) for wide blood vessels, as shown in Figure 6. Here we derive Kleiber’s Law, as illustrated by Savage et al. (2008). The total volume of fluid (here assumed to be blood) in the organism is equal to the total volume of the network: (C1) This sum may be partitioned into two parts, where = ̅ marks the transition between wide and narrow blood vessels: (C2) Since the volume-filling constraint holds for all values of , we have (C3) and our generalized expression for the radius of each vessel is (C4) where we have defined (C5) 1 2 1 3 for for k n k k B n k k − − = = HEBERT Allometric and metabolic scaling 2023 ICC 225
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