Bifurcation Law Research Articles

Morphometry of human coronary arterial trees

Background Some groups, including ours, have been generating arterial tree models using constrained constructive optimization (CCO). Arterial trees have been grown to arbitrary resolution without input of anatomical data. We performed this study to learn about the shortcomings that might have resulted from neglecting the anatomical data in CCO models. Methods In a total of 450 segments obtained from 4 human cast hearts, the ratio of bifurcating daughter segment radii (0 < Sbif = r(2)/r(1) < 1) was examined, which corresponds to the split of the total flow of the mother segment. For any complete bifurcation, where the radii of the parent segments and the radii of daughters were known, the area expansion ratio was computed (Aexp = [r(1)2 + r(2)2]/ r(parent)2). Results The bifurcating ratio was found to be distributed in a nonnormal fashion, with a median of 0.76. The average area expansion ratio Aexp, characterizing the change of cross-sectional area of the vasculature from proximal to distal, was 0.93 ± 0.26. The ‘rate of branching’ (d(i)/(d(0)) was defined by the segment diameter relative to the diameter of the root segment. Averaging the rate of branching over segments within each bifurcation level resulted in a decreasing function of bifurcation level. Conclusions This article provides new experimental data on branching geometry of coronary arteries (i.e., the trees evaluated in this study are purely delivering rather than conveying). Based on these facts, we suggest that the analytical bifurcation law in CCO might be replaced by the bifurcation rule obeyed on a stochastic basis only. Anat. Rec. 251:50–59, 1998. © 1998 Wiley-Liss, Inc.

Geometrical and topological optimization and their relation to the bifurcation laws in optimized arterial tree models

In generating models of vascular trees, the method of Constrained Constructive Optimization (CCO) draws on the assumption that optimization criterions can be utilized to derive the geometrical and structural features of the tree models, without using informations from topographic anatomy. The boundary conditions imposed describe pressures and flows at the inlet and outlets of the binary branching model tree as well as the ratios of radii at a bifurcation (power-law for bifurcations). During phylogenesis of species, bifurcation laws, tree geometry and tree topology have most likely been simultaneously optimized. In contrast, a supposed optimum form of bifurcation law is prescribed in the CCO-simulation, and geometry together with structure are optimized within that frame. It is shown that CCO trees optimized under different bifurcation laws initially show slightly different geometries but identical topological structures. At a certain stage of development, however, accumulated geometric deviations lead to changes in topology as a result of optimization. From there on we observe different evolutionary paths of tree development.