Voronoi polyhedra analysis of optimized arterial tree models

Topological and metric properties of Voronoi polyhedra (VP) generated by the distal end points of terminal segments in arterial tree models grown by the method of constrained constructive optimization (CCO) are analyzed with the aim to characterize the spatial distribution of their supply sites relative to randomly distributed points as a reference model. The distributions of the number Nf of Voronoi cell faces, cell volume V, surface area S, area A of individual cell faces, and asphericity parameter alpha of the CCO models are all significantly different from the ones of random points, whereas the distributions of V, S, and alpha are also significantly different among CCO models optimized for minimum intravascular volume and minimum segment length (p < 0.0001). The distributions of Nf, V, and S of the CCO models are reasonably well approximated by two-parameter gamma distributions. We study scaling of intravascular blood volume and arterial cross-sectional area with the volume of supplied tissue, the latter being represented by the VP of the respective terminal segments. We observe scaling exponents from 1.20 +/- 0.007 to 1.08 +/- 0.005 for intravascular blood volume and 0.77 +/- 0.01 for arterial cross-sectional area. Setting terminal flows proportional to the associated VP volumes during tree construction yields a relative dispersion of terminal flows of 37% and a coefficient of skewness of 1.12.