Table 1. Measures and theoretical values of the scaling exponents a and b.
N denotes the number of measurements incorporated in the average or slope. The 95% CI indicates the range of confidence on the average or slope. For node-level measures, σ indicates the standard deviation of measurements made for individual junctions or parent-child pairs for conservation- and ratio-based measures respectively. For network-level (regression- and distribution-based) measures, R 2 denotes the correlation in the fit. For distribution-based measures, this correlation is between the natural log of bin size and mean vessel dimension (i.e., radius or length) in the bin. For regression-based measures, this correlation is the correlation between the natural log of dimension and the natural log of the number of downstream endpoints. For conservation-based measures, N counts all junctions of three or more well-segmented vessels (see Software and Algorithm) for which Eqs 1a or 1b had a solution. For ratio-based exponents, N counts all parent-child pairs of vessels in which both the parent and child are well-segmented. For distribution-based exponents, N counts the number of well-segmented vessel segments exceeding the minimum-size threshold described in Data Fitting. For regression-based exponents, N counts all well-segmented vessels. The 95% CIs are derived in a manner appropriate to each method: They are 1.96 times the standard error on the mean for conservation- and ratio-based measures and the confidence interval on the SMA regression slope for regression-based measures. For distribution-based measures, they are the range of the middle 95% of slopes derived from alternative binning as described in Data Fitting.
| Radius Exponent Measures | N | a | 95% CI | σ | R 2 |
| Conservation (Node) | 222 | 0.49 | ± 0.03 | 0.23 | – |
| Ratio (Node) | 703 | 0.43 | ± 0.03 | 0.39 | – |
| Distribution (Network) | 657 | 0.30 | ± 0.04 | – | 0.90 |
| Regression (Network) | 1240 | 0.41 | ± 0.02 | – | 0.66 |
| Theory | |||||
| WBE Small Vessels | – | 0.33 | – | ||
| WBE Large Vessels | – | 0.50 | – | ||
| Banavar et al. [14] | – | 0.50 | – | ||
| Huo and Kassab [16] | – | 0.33–0.50 | – | ||
| Murray’s Law | – | 0.33 | – | ||
| Length Exponent Measures | N | b | 95% CI | σ | R 2 |
| Conservation (Node) | 94 | 1.40 | ± 0.20 | 0.98 | – |
| Ratio (Node) | 703 | 0.17 | ± 0.12 | 1.57 | – |
| Distribution (Network) | 518 | 0.73 | ± 0.10 | – | 0.68 |
| Regression (Network) | 1240 | 0.94 | ± 0.05 | – | 0.19 |
| Theory | |||||
| WBE All Vessels | – | 0.33 | – | ||
| Banavar et al. [14] | – | 0.50 | – | ||
| Huo and Kassab [16] | – | 0.33–1.00 | – |