Statistical analysis of reduced pair correlation functions of capillaries in the prostate gland

Abstract

Blood capillaries are thread-like structures that may be considered as an example of a spatial fibre process in three dimensions. At light microscopy, the capillary profiles appear as a planar point process on sections. It has recently been shown that the observed pair correlation function g(r) of the centres of the fibre profiles on two-dimensional sections may be used to estimate the reduced pair correlation function of stationary and isotropic fibre processes in three dimensions. In the present study, we explored how this approach may be extended to statistical analysis of reduced g-functions of capillaries from multiple specimens of different groups and with replicated observations. The methods were applied to normal prostatic tissue compared with prostate cancer. Confidence intervals for the mean reduced g-functions of groups were estimated for fixed r-values parametrically using the t-distribution, and by bootstrap methods. Each estimated reduced g-function was furthermore characterized in terms of its first maximum and minimum. The mean length of capillaries per unit tissue volume was significantly higher in prostate cancer tissue than in normal prostate tissue. Significant differences between the mean reduced g-functions of malignant and benign lesions could be demonstrated for two domains of r-values. In general, bootstrap-based confidence intervals were slightly wider than parametrically estimated confidence intervals. Falsely negative lower bounds of the intervals, which sometimes arose using the parametric approach, could be avoided by the bootstrap method. Testing of group mean values for significant differences by the bootstrap method yielded more conservative results than multiple t-tests. The functional value of the first maximum of the reduced g-function and a global statistical parameter of short-range ordering was significantly reduced in the carcinoma group. Prostate cancer tissue is more densely supplied with capillaries than normal prostate tissue and the three-dimensional arrangement of the vessels differs with respect to interaction at various distance ranges. In the local approach used here, bootstrap methods can be used as a robust statistical tool for the computation of confidence intervals and group comparisons of mean reduced g-functions at specific ranges of interaction.