Figure 1.
Schematic vessel architecture and branching.
A vessel at level k branches into two daughter vessels at level k+1. The branching ratio is thus n = 2. The radii, rk+1, and lengths, lk+1, of the two daughter vessels are identical by Assumption 3. The ratios of the radii and lengths at level k+1 to those at level k are defined as γ, β> and β< in Equations 2 and 3. The choice of β> for the radial ratio corresponds to area-preserving branching and of β< to area-increasing branching. In the WBE model, the cardiovascular system is composed of successive generations of these vascular branchings, from level 0 (the heart) to level N (the capillaries).
Figure 2.
Schematic scaling relation for finite-size corrections in networks with only area-preserving branching.
The dashed line schematically depicts the 3/4 power law that relates the number of capillaries, Ncap, to the blood volume Vblood. This scaling relationship is a straight line in logarithmic space (ln Ncap versus ln Vblood) and represents the leading-order behavior in the limit of infinite blood volume and organism size. The solid line dramatizes the curvature for the scaling relation for finite-size networks obtained when vessel radii are determined solely by area-preserving branching. The dotted line illustrates the consequences of a linear regression on the curve for finite-size organisms (solid line). Since the solid line depicts the predicted curvilinear relationship that deviates above and away from the infinite-size asymptote, Equation 16, the WBE model predicts that fits to data for organisms whose vascular networks are built only with area-preserving branching will yield scaling exponents smaller than 3/4.
Figure 3.
Finite-size corrections for networks with only area-preserving branching.
(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.
Figure 4.
Finite-size corrections for networks with only area-increasing branching.
As in Figure 3, but networks are now constructed with area-increasing branching only. The abscissa reflects the absolute size range, NS+1/NL+1, within each group used to determine the scaling exponent. NS and NL are the number of levels in the smallest and largest networks, respectively. Note that NS+1/NL+1 is always smaller than 1 and the scaling exponent α has an accumulation point at 1, the infinite size limit. Black circles: numerical data. Red line: analytical approximation, Equation 20.
Figure 5.
Schematic scaling relation for finite-size corrections in networks with both area-preserving and area-increasing branching.
The dashed line schematically depicts the 3/4 power law of ln Ncap versus ln Vblood in the infinite network limit. The solid line dramatizes the curvature for the scaling relation that is obtained when the network has a transition point above which it has area-preserving branching and below which it has area-increasing branching. The dotted line illustrates the consequences of a linear regression on what is a curvilinear relationship that deviates below and away from the infinite-size limit, Equation 22. As a result, the WBE model predicts that fits to data for organisms whose vascular networks are built in mixed mode will yield scaling exponents that are larger than 3/4.
Figure 6.
Finite-size corrections for networks with both area-preserving and area-increasing branching.
(A) As in Figure 3B, we numerically determine the scaling exponent α by OLS regression within a group of artificial networks spanning roughly 8 orders of magnitude in body mass (blood volume). The exponent obtained from a group is plotted against the size of the smallest network in that group (as measured by the number of capillaries, Ncap,S). Many groups are built by systematically increasing the size of the smallest network, resulting in the depicted graph. In all cases the branching ratio was n = 2. Black circles: numerical values. Red curve: analytical approximation, Equation 23. Green curve: Best fit to the shape of Equation 23, . (B) As in (A), except that each exponent is plotted against the number of levels NS of the smallest network in the group from which it was determined. We display results obtained for a branching ratio n = 2 (black circles) and n = 3 (green circles). The red circles mark the predictions of the WBE model, since NS = 25 for the smallest network (a shrew, meaning N̅ = 24 plus 1 level for pulsatile flow) in the case of n = 2, and NS = 16 for n = 3.
Figure 7.
Influence of the location of the transition between area-preserving and area-increasing branching.
The scaling exponent α is plotted against the number of levels N̅ with area-increasing branching in the network. For each N̅ the exponent was determined from a group of artificial networks that start from a smallest organism of fixed size and span eight orders of magnitude in blood volume to the largest organism, as described in section “Finite-size corrections to 3/4 allometric scaling”. N̅ is varied from 0 (pure area-preserving branching) to the entire network (pure area-increasing). Black circles: Networks with branching ratio n = 2 and a smallest organism size of N = 25 levels. Green circles: Networks with branching ratio n = 3 and a smallest organism size of N = 16 levels. These graphs capture both finite-size effects and the effects of varying the extent of the network that is built with area-increasing branching. The exponent α changes from 3/4 to 1 as N̅ grows, which is suggested by considering a composite of Figures 3B and 4. The red circles mark the prediction of the finite-size corrected WBE model (N̅ = 24 for n = 2 and N̅ = 15 for n = 3).
Figure 8.
Influence of an extended transition region.
The three curves are analogous to those in Figure 6B. (In fact, the black curves are identical.) The figure shows how a transition over 12 (red circles) and 24 levels (green circles) shifts the curve relative to the WBE assumption of a transition over a single level (black circles). The more extended the transition region, the fewer the levels built with area-preserving branching. The scaling exponent increases as a consequence. Inset: The linear interpolation of rk+1/rk as a function of network level used in generating the red curve. The transition occurs from β = 2−1/2 = 0.707 to β = 2−1/3 = 0.794 over 12 levels centered at the WBE transition level N̅ = 24.
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.
Figure 10.
Dependency of the scaling exponent on body mass range as determined by ordinary least squares regression on empirical data.
The data are binned in orders of magnitude for body mass as described in the text. (A) Cumulative binning starting with smallest mammals. (B) Cumulative binning starting from largest mammals. (C) Exponents from individual order-of-magnitude bins. The exponents computed from these aggregations of empirical data vary both above and below 3/4. Note, however, that in all cases the allometric exponents tend to increase with increasing body mass. The error bars represent the 95% confidence intervals. When data is scarce, the confidence intervals become so large that the exponents cannot be trusted. (The full range of some error bars is cut off by the scale of the plots.)