Sizing Up Allometric Scaling Theory
Figure 9
Influence of the branching ratio.
The scaling exponent α as a function of the number of levels N̅b at which the branching ratio switches from n = 2 to the indicated value of 3 (black circles), 4 (red circles) or 5 (green circles). N̅b varied from 0 (a branching ratio of n = 2 at all levels) to the depth of the entire network (a branching ratio of n = 3, 4, or 5 at all levels). As in Figure 8, each exponent was calculated from networks that spanned eight orders of magnitude in blood volume. In these calculations, network levels with vessel radii ≤1 mm were built according to area-increasing branching, while vessels with radii larger than 1 mm followed area-preserving branching. These curves correspond to a cardiovascular system in which the branching ratio n is smaller near the heart and larger toward the capillaries. In all cases, a change in branching ratio within the network decreases the predicted scaling exponent, bringing it closer to the empirical value of 3/4 without ever touching it.