Sizing Up Allometric Scaling Theory
(A) The logarithm of the number of capillaries is regressed with ordinary least squares (OLS) on the logarithm of blood volume for a set of artificial networks, spanning 8 orders of magnitude, built with only area-preserving branching. In this particular example the scaling exponent is determined to be 0.743, very close to 3/4. Black circles: numerical values. Red curve: power-law regression. (B) A scaling exponent α is determined by OLS regression for each group of artificial networks spanning roughly 8 orders of magnitude in body mass (blood volume). Exponents so-determined are paired with the size of the smallest network (as measured by the number of capillaries, Ncap,S) in the corresponding group. Groups are built by systematically increasing the size of the smallest network, while always maintaining a range of 8 orders of magnitude in body volume (mass), resulting in the depicted graph. In all cases the branching ratio was n = 2. Black circles: numerical values. Red curve: analytical approximation, Equation 17.