Probability density functions for axial ratios of sectioning profiles of anisotropically arranged elliptical microvessels

The article theoretically regards probability density functions (PDFs) for axial ratio ( X / Y ) of sectioning profiles of elliptical microvessels (MVs) arranged with anisotropy in a biological tissue volume. A technique for the PDF X / Y calculations in anisotropy of the elliptical MVs is described. The essence of this technique is introducing anisotropy in PDF ( α , φ ) , i.e. the function of the joint distribution of the polar and planar angles α and φ , which define mutual orientation of the elliptical MVs and sectioning planes. With the aid of this technique, the anisotropy cases are studied with PDF( α , φ ) given by pair combinations of the following distributions: (i) a uniform distribution of the angles α and/or φ , (ii) the angle α distribution with PDF α = sin α ( α ∈ [ 0 , π / 2 ] ) , and (iii) Gaussian distributions of the α or φ values. Specifically, PDF X / Y curves are obtained for MVs with the true, or three-dimensional, axial ratio X 0 / Y 0 = 2.0 , and the anisotropy effects on the X / Y expected frequencies are analysed. Conclusions of this analysis, the PDF X / Y calculation technique, and the PDF X / Y curves obtained are useful for stereological reconstruction of anisotropically organised microcirculatory networks, with an ellipticity of their MVs being taken into consideration.