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/YX/Y) of sectioning profiles of elliptical microvessels (MVs) arranged with anisotropy in a biological tissue volume. A technique for the PDFX/YPDFX/Y calculations in anisotropy of the elliptical MVs is described. The essence of this technique is introducing anisotropy in PDF(α,φ)PDF(α,φ), i.e. the function of the joint distribution of the polar and planar angles αα and φφ, 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,φα,φ) given by pair combinations of the following distributions: (i) a uniform distribution of the angles αα and/or φφ, (ii) the angle αα distribution with PDFα=sinα(α∈[0,π/2]), and (iii) Gaussian distributions of the αα or φφ values. Specifically, PDFX/YPDFX/Y curves are obtained for MVs with the true, or three-dimensional, axial ratio X0/Y0=2.0X0/Y0=2.0, and the anisotropy effects on the X/YX/Y expected frequencies are analysed. Conclusions of this analysis, the PDFX/YPDFX/Y calculation technique, and the PDFX/YPDFX/Y curves obtained are useful for stereological reconstruction of anisotropically organised microcirculatory networks, with an ellipticity of their MVs being taken into consideration.