Centred contact density functions for the statistical analysis of random sets. A stereological study on benign and...

Real structures investigated in the material and biological sciences, such as minerals or tissues, can often be reduced to two phases. In a stochastic approach, the components of such binary structures may be considered as the union of grains--random sets implanted with their centres at random points--and their complementary space, which is called the pore space. The simplest stochastic germ-grain model is the Boolean model of random sets, which we use here instrumentally as a null model (reference model) for comparison with our biological material. After a brief review of basic properties of the Boolean model and related statistical methods, we introduce centred contact density functions as a new approach. Empirical contact density functions are estimated from the empirical contact distribution functions with an image analyser by dilation of the grain phase. Theoretical contact density functions are then predicted from a set of image parameters, under the assumption that the Boolean model holds. A centred contact density function is the difference between the estimated and the predicted contact density function. Apart from a random error term, centred contact density functions amount to zero irrespective of the area fraction of the grain phase, when the germ-grain model is Boolean. As a section of a spatial Boolean model is a planar Boolean model, the method is also applicable in stereological studies where digitized images are obtained from sections of a three-dimensional structure. Centred contact density functions were determined for mastopathic tissue as compared to mammary cancer, and for tumour-free prostatic tissue as compared to prostatic cancer. For each category of specimens, twenty cases with 10 images each were analysed. Benign and malignant glandular tissue of the aforementioned types deviates significantly from the Boolean model. Centred contact density functions show that malignant transformation is connected with profound geometric remodelling of the pore space.