The model has three main assumptions: Within the = N level of the network, the pipe sizes are “size invariant.” In other words, the length N and radius N of the very smallest pipes (e.g., capillaries in mammals or birds) do not depend on the mass M of the organism. These characteristic sizes are determined by basic physical principles and limitations, not by the overall size specifics of the organism itself. Capillaries in different creatures should be the same size, regardless of differences in their masses: a capillary in a dinosaur should be the same size as a capillary in a mouse. Organisms minimize the energy needed to transport materials through the network. This is prima facie evidence that organisms were intelligently engineered, although evolutionists attribute this optimization to evolution and natural selection (Brown et al. 2000). However, the WBE model itself makes no evolution-based assumptions. If not for the perfunctory (and apparently obligatory!) assertion that such optimization is achieved by natural selection, the WBE theory could easily be viewed as a design-based theory. The hierarchical branching network is “volume-filling” in order to ensure that nutrients are supplied to the entire organism’s volume. Because this requirement is often poorly explained in the technical literature, I elaborate on it below. B. Volume-filling One can imagine that each terminal pipe (capillary) in the network provides nutrients to a group of cells having a “service volume” N . Each service volume may be thought of as a biological “black box”. We do not necessarily know the precise shape of each service volume. However, because these nutrient-carrying vessels are narrow, we know N that the radius of each vessel is much smaller than its length. It is therefore reasonable to assume that N ∝ 3 (Fig. 4). In other words, the service volume is proportional to the length of the capillary which supplies nutrients to it. Because the network must supply nutrients to the organism’s entire volume V, the organism’s total volume V must equal the sum of the N N service volumes (Fig. 5a): (4) Note that the number of service volumes NN is also equal to the number of capillaries. However, what is counted as a service volume is somewhat arbitrary. One could, without loss of generality, treat the capillaries themselves as belonging to the service volumes (Fig. 5b). In that case, the service volumes are a little larger. Their volumes are proportional to, not the cube of the pipe lengths in level N, but to the cube of the pipe lengths in level N − . But the number of service volumes has now decreased to N N − , rather than NN . However, the organism’s entire volume must still be supplied with nutrients: (5) West et al. implicitly assume that the proportionality constant is the same in both cases, although they acknowledged that this “volume-filling” assumption breaks down somewhat for small values of or N (West et al. 2000). Ignoring this complication, we have for every level (every value of ) within the network: (6) Figure 3. Relationships between vessel radii and lengths in two adjacent levels of the network. HEBERT Allometric and metabolic scaling 2023 ICC 209
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