Diameter-defined Strahler system and connectivity matrix of the pulmonary arterial tree

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Abstract
For modeling of a vascular tree for hemodynamic analysis, the well-known Weibel, Horsfield, and Strahler systems have three shortcomings: vessels of the same order are all treated as in parallel, despite the fact that some are connected in series; histograms of the diameters of vessels in the successive orders have wide overlaps; and the “small-twigs-on-large-trunks” phenomenon is not given a quantitative expression. To improve the accuracy of the hemodynamic circuit model, we made a distinction between vessel segments and vessel elements: a segment is a vessel between two successive nodes of bifurcation; an element is a union of a group of segments of the same order that are connected in series. In an equivalent circuit, all elements of the same order are considered as arranged in parallel. Then, we follow the ordering method of Horsfield and Strahler, with introduction of an additional rule for the assignment of order numbers. If Dn and SDn denote the mean and standard deviation of the diameters of vessels of order n, then our rule divides the gap between Dn--SDn and Dn--1 + SDn--1 evenly between orders n and n--1. Finally, we introduced a connectivity matrix with a component in the mth row and the nth column that is the average number of vessels of order m that grow out of the vessels of order n. This method was applied to the rat. We found that the rat pulmonary arterial tree has 11 orders of vessels and that the geometry is fractal within these orders. The ratios of diameters, lengths, and numbers of elements in successive orders are 1.58, 1.60, and 2.76, respectively. The connectivity matrix reveals interesting features beyond the fractal concept. New features are found in the variation of the total cross-sectional area of elements with order numbers.

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