Edge-corrected estimators of the nearest-neighbour distance distribution function for three-dimensional point patterns

In multiphase systems consisting of ‘particles’ embedded in a matrix the three-dimensional spatial distribution of the particles may represent important structural information. In systems where the matrix is transparent or translucent recent developments in microscopy allow the three-dimensional location of particles to be recorded. Using these data a spatial statistical, or second-order stereological, analysis can be carried out. In second-order stereology functions of interparticle distances are used as summary statistics of the spatial distributions. These statistics show whether the particles are randomly arranged or, more commonly, either clustered together or inhibited from close approach to each other. This paper focuses on the estimation of one of these spatial statistics, the nearest-neighbour distance distribution function or G-function. In practice, estimation of the G-function is plagued by an ‘edge-effect’ bias introduced by the sampling process itself. There exist a number of G-function estimators that tackle this edge effect problem; for single sample ‘bricks’ it can be shown that these estimators become increasingly accurate as the brick size increases, i.e. they are consistent. However, in many practical cases the size of a sampling brick is fixed by experimental constraints and in these circumstances the only way to increase sample size is to take replicated sampling regions. In this paper we review a number of existing G-function estimators and propose a new estimator. These estimators are compared using the criterion of how well they overcome the edge-effect when they are applied to replicated samples of a fixed size of brick. These comparisons were made using Monte-Carlo simulation methods; the results show that three existing estimators are clearly unsuitable for estimating the G-function from replicated sample bricks. Of the other estimators the recommended estimator depends upon the number of replicates taken; however, we conclude that if a total of more than about 800 points are analysed then the bias in the pooled estimate of the G-function can be reduced to tolerable levels.     

The saucor, a new stereological tool for analysing the spatial distributions of cells, exemplified by human neocortical neurons and glial cells

The 3D spatial arrangement of particles or cells, for example glial cells, with respect to other particles or cells, for example neurons, can be characterized by the radial number density function, which expresses the number density of so-called 'secondary' particles as a function of their distance to a 'primary' particle. The present paper introduces a new stereological method, the saucor, for estimating the radial number density using thick isotropic uniform random or vertical uniform random sections. In the first estimation step, primary particles are registered in a disector. Subsequently, smaller counting windows are drawn with random orientation around every primary particle, and the positions of all secondary particles within the windows are recorded. The shape of the counting windows is designed such that a large portion of the volume close to the primary particle is examined and a smaller portion of the volume as the distance to the primary object increases. The experimenter can determine the relation between these volumina as a function of the distance by adjusting the parameters of the window graph, and thus reach a good balance between workload and obtained information. Estimation formulae based on the Horvitz-Thompson theorem are derived for both isotropic uniform random and vertical uniform random designs. The method is illustrated with an example where the radial number density of neurons and glial cells around neurons in the human neocortex is estimated using thick vertical sections for light microscopy. The results indicate that the glial cells are clustered around the neurons and the neurons have a tendency towards repulsion from each other.