Imaging platforms for generating highly multiplexed histological images are being continually developed and improved. Significant improvements have also been made in the accuracy of methods for automated cell segmentation and classification. However, less attention has focused on the quantification and analysis of the resulting point clouds, which describe the spatial coordinates of individual cells. We focus here on a particular spatial statistical method, the cross-pair correlation function (cross-PCF), which can identify positive and negative spatial correlation between cells across a range of length scales. However, limitations of the cross-PCF hinder its widespread application to multiplexed histology. For example, it can only consider relations between pairs of cells, and cells must be classified using discrete categorical labels (rather than labeling continuous labels such as stain intensity). In this paper, we present three extensions to the cross-PCF which address these limitations and permit more detailed analysis of multiplex images: topographical correlation maps can visualize local clustering and exclusion between cells; neighbourhood correlation functions can identify colocalization of two or more cell types; and weighted-PCFs describe spatial correlation between points with continuous (rather than discrete) labels. We apply the extended PCFs to synthetic and biological datasets in order to demonstrate the insight that they can generate.

Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ0,! such that there exists a point process η where Φ=Φ0,!∪η in distribution and Φ0,!∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n→∞. We also prove that for some DPPs, there exists an R∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √nR∗. This R∗ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high-dimensional Boolean models is given.

We consider a Markov chain of point processes such that each state is a superposition of an independent cluster process with the previous state as its centre process together with some independent noise process and a thinned version of the previous state. The model extends earlier work by Felsenstein (1975) and Shimatani (2010) describing a reproducing population. We discuss when closed-form expressions of the first- and second-order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.

ORAI1 proteins are ion channel subunits and the essential pore-forming units of the calcium release-activated calcium channel complex essential for T-cell activation and many other cellular processes. In this study, we used environmental scanning electron microscopy (ESEM) with scanning transmission electron microscopy (STEM) detection to image plasma membrane expressed ORAI1 proteins in whole Jurkat T cells in the liquid state. Utilizing a stably transfected Jurkat T cell clone expressing human ORAI1 with an extracellular human influenza hemagglutinin (HA) tag we investigated if liquid-phase STEM can be applied to detect recombinant surface expressed protein. Streptavidin coated quantum dots were coupled in a one-to-one stoichiometry to ORAI1 proteins detected by biotinylated anti-HA fragmented antibody fragments. High-resolution electron microscopic images revealed the individual label locations from which protein pair distances were determined. These data were analyzed using the pair correlation function and, in addition, an analysis of cluster size and frequency was performed. ORAI1 was found to be present in hexamers in a small fraction only, and ORAI1 resided mostly in monomers and dimers.

The random surface measure of a stationary Boolean model with grains from the convex ring is considered. A sufficient condition and a necessary condition for the existence of the density of the second-order moment measure of are given and a representation of this density is derived. As applications, the surface pair correlation functions of a Boolean model with spheres and a Boolean model with randomly oriented right circular cylinders in ℝ3 are determined.

The set covariance of a dead leaves model, constructed from hard spheres of constant diameter, is calculated analytically. The calculation is based on the covariance of a single sphere and on the pair correlation function of the centres of the spheres. There exist applications in the field of random sequential adsorption and in the interpretation of small-angle scattering experiments.

Previous quantitative descriptions of cellular ultrastructure have focused on spatial content (volume, surface area and number of organelles and membrane domains). It is possible to complement such descriptions by also quantifying spatial arrangements. Hitherto, applications of stereological methods for achieving this (notably, estimation of covariance and pair correlation functions) have been confined to organ and tissue levels. This study explores 3-dimensional subcellular arrangements of key organelles within acinar cells of rabbit parotid salivary glands, highly polarised epithelial cells specialised for exocrine secretion of α-amylase. It focuses on spatial arrangements of secretion product stores (zymogen granules), rough endoplasmic reticulum (RER) and mitochondria. Systematic random samples of electron microscopical fields of view from 3 rabbits were analysed using test grids bearing linear dipole probes of different sizes. Unbiased estimates of organelle volume densities were obtained by point counting and estimates of covariance and pair correlation functions by dipole counting. Plots of pair correlation functions against dipole length identified spatial arrangement differences between organelle types. Volumes within RER and mitochondrial compartments were positively correlated with themselves at distances below 4 μm and 2 μm respectively but were essentially randomly arranged at longer distances. In sharp contrast, zymogen granules were not randomly arranged. They were clustered at distances below 6–7 μm and more widely scattered at greater distances. These findings provide quantitative confirmation of the polarised arrangement of zymogen granules within acinar cells and further support for the relative invariance of biological organisation between subjects.

Formulas for anisotropic stereology of fibre and surface processes are presented. They concern the relation between second-order quantities of the original process and its projections and sections. Various mathematical tools for handling these formulas are presented, including stochastic optimization. Finally applications in stereology are discussed, relating to intensity estimators using anisotropic sampling designs. Variances of these estimators are expressed and evaluated for processes with the Poisson property.