## Introduction

Capillaries are defined as blood vessels whose wall consists only of endothelial cells and a basement membrane. The density of capillarization is an important factor for the oxygen supply of a tissue. The growth of new vessels in general is denoted as angiogenesis. Capillary angiogenesis in tumours is a topic of central importance in tumour biology. In the exploration of mechanisms of angiogenesis, the basic structural background remains the capillary network itself, which can be visualized by microscopy. To obtain objective findings in such investigations, it is obligatory to quantify the capillarization. For this purpose methods of quantitative stereology are relevant (Mattfeldt & Mall, 1984; Mattfeldt *et al*., 2004a,b). These methods are rooted in the mathematical domain of stochastic geometry, where capillaries may be considered as an example of a three-dimensional (3D) fibre process.

Fibre processes are random geometrical models for fibrous structures. They are used in applications in biology, medicine and material science, for example (Mattfeldt *et al*., 1994; Stoyan *et al*., 1995; Krasnoperov & Stoyan, 2004). Fibres may be intuitively defined as thread-like structures, i.e. filamentous or thin tubular structures whose length greatly exceeds their width. After cutting, such fibres appear on a microscopic section as dots, e.g. small ellipses when the true fibres are circular or elliptical cylinders. In the case of isotropic and stationary capillary networks, three simple first-order parameters can be estimated by using information from sections of arbitrary location and orientation by using elementary stereological equations

*V*=

_{V}*A*(1)

_{A}*S*= (4/π)

_{V}*B*(2)

_{A}*L*= 2

_{V}*Q*(3)

_{A}(see, e.g. Weibel, 1979; Mattfeldt *et al*., 1990, 2004a,b; Howard & Reed, 2005). Here the stereological shorthand denotes: *V*_{V}, the volume of capillaries per unit reference volume; *A*_{A}, the area of capillary profiles per unit reference area on sections; *S*_{V}, the surface area of capillaries per unit reference volume; *B*_{A}, the boundary length of capillary profiles per unit reference area on sections; *L*_{V}, the length of capillaries per unit reference volume, i.e. the intensity of the 3D fibre process and *Q*_{A}, the number of capillary profiles per unit reference area on sections. The three parameters on the left side of Eqs (1)–(3) express the density of capillary supply in 3D space in different terms.

Even the combination of all three model parameters, *V*_{V}, *S*_{V} and *L*_{V}, does not provide a complete geometrical characterization of a capillary network. The first-order parameters tell nothing about the geometrical architecture (pattern) of the blood vessels, i.e. their spatial arrangement relative to each other. To describe arrangements of random sets in space, a well-established approach consists of methods of second-order stereology. Such techniques have hitherto been used mostly for random sets with positive volume fraction (volume processes) (Cruz-Orive, 1989; Mattfeldt *et al*., 1993, 1996, 2000; Mattfeldt & Stoyan, 2000; Mattfeldt, 2003). In principle, however, they may also be used for the second-order characterization of surface processes and fibre processes in 3D space (Mattfeldt *et al*., 1994; Stoyan *et al*., 1995). A recent study showed how second-order stereological inference on isotropic spatial fibre processes may be performed on the basis of observations on two-dimensional sections (Krasnoperov & Stoyan, 2004). The ordinary planar pair correlation function *g*(*r*) of the sectional profiles of the fibres can be used to estimate the reduced pair correlation function *g*_{3}(*r*) of the 3D fibre process (Krasnoperov & Stoyan, 2004). In this methodological study, the emphasis was put on point estimation of the reduced pair correlation function. However, in an experimental or clinical research project with more data, it is desired to provide confidence intervals of the function for groups of cases and to test for significant differences between groups. Such an attempt at statistical inference was made in the present study.