Explorative statistical analysis of planar point processes in microscopy

Basic methods of explorative statistical analysis for stationary and isotropic planar point processes are briefly and informally reviewed. At the explorative level, planar point patterns may be characterized in terms of the intensity, the K-function and the pair correlation function. These second-order functions enable one to classify a given point process as completely random, clustering or repulsive. The repulsive behaviour may be quantified by an estimate of the hard-core distance. In the exploratory approach, the statistics are essentially free from model assumptions. Second-order spatial functions have been estimated to characterize genuine planar point processes in the macroscopic domain, for example in forestry, geography and epidemiology. For light microscopy and transmission electron microscopy, two situations are distinguished, which may be summarized as the genuine planar case and the stereological case. In the genuine planar case, a direct interpretation of the results of spatial statistics is feasible. Here, monolayers in cell culture, intramembranous particles on freeze fracture specimens and amacrine cells of the retina are mentioned as examples. In the stereological case, point patterns are generated by sections through 3D structures. Here the observed point patterns may arise as the centres of sectional profiles of particles, or as centres of sectional profiles of spatial fibre processes. In both situations, exploratory spatial point process statistics allow a quantitative characterization of sectional images for the purposes of group comparisons and classification. Moreover, for spatial fibre processes it has recently been shown that the observed pair correlation function of the centres of the fibre profiles is an estimate of the reduced pair correlation function of the fibre process in 3D. Hence for fibre processes a stereological interpretation of point process statistics obtained from sections is an additional option.     

Estimation of surface area from vertical sections

‘Vertical’ sections are plane sections longitudinal to a fixed (but arbitrary) axial direction. Examples are sections of a cylinder parallel to the central axis; and sections of a flat slab normal to the plane of the slab. Vertical sections of any object can be generated by placing the object on a table and taking sections perpendicular to the plane of the table. The standard methods of stereology assume isotropic random sections, and are not applicable to this kind of biased sampling. However, by using specially designed test systems, one can obtain an unbiased estimate of surface area. General principles of stereology for vertical sections are outlined. No assumptions are necessary about the shape or orientation distribution of the structure. Vertical section stereology is valid on the same terms as standard stereological methods for isotropic random sections. The range of structural quantities that can be estimated from vertical sections includes V_v, N_v, S_v and the volume-weighted mean particle volume v?_v, but not L_v. There is complete freedom to choose the vertical axis direction, which makes the sampling procedure simple and ‘natural’. Practical sampling procedures for implementation of the ideas are described, and illustrated by examples.     

Stereology of tubular structures

The morphology of a tubular structure can be characterized, in at least some of its important respects, through stereological methods. We study the geometric meaning of standard stereological quantities when applied to tubular structures, with particular regard to their curvature or tortuosity. Measures defined specifically in relation to tubular structures are also introduced for practical use. The ideal smooth bent cylinder, used here, is not realistic but provides general principles from which a more specialized investigation should be developed. The emphasis is placed on simple counting methods of measurement, specifically the tangent counts introduced by DeHoff, and counts of section profiles exhibiting a particular character (such as ‘figure-eights’). These measures convey information about the tortuosity of tubular structures, whereas the standard indices V_v, S_v, M_v give no information about tortuosity. Some data from human testicular tubules are discussed.     

Second-order stereology of spatial fibre systems

This paper describes methods for second‐order stereology of spatial fibre systems. For stationary and isotropic fibre processes it gives practicable estimators of the K‐function and the pair correlation function, which are based on planar sections. The second‐order methods are applied in transmission electron microscopy analysis of blood capillaries in the rat thyroid. They lead to the result that the capillaries show an inhibitory pattern of their spatial arrangement, with a hard‐core distance of about 2.6 µm. There is a close relationship to three‐dimensional size characteristics estimated recently for these elliptical capillaries.     

Stereological length estimation using spherical probes

Lineal structures in biological tissue support a wide variety of physiological functions, including membrane stabilization, vascular perfusion, and cell‐to‐cell communication. In 1953, Smith and Guttman demonstrated a stereological method to estimate the total length density (L_v) of linear objects based on random intersections with a two‐dimensional sampling probe. Several methods have been developed to ensure the required isotropy of object–probe intersections, including isotropic‐uniform‐random (IUR) sections, vertical‐uniform‐random (VUR) slices, and isotropic virtual planes. The disadvantages of these methods are the requirements for inconvenient section orientations (IUR, VUR) or complex counting rules at multiple focal planes (isotropic virtual planes). To overcome these limitations we report a convenient and straightforward approach to estimate L_v and total length, L, for linear objects on tissue sections cut at any arbitrary orientation. The approach presented here uses spherical probes that are inherently isotropic, combined with unbiased fractionator sampling, to demonstrate total L estimation for thin nerve fibres in dorsal hippocampus of the mouse brain.     

Measuring error and sampling variation in stereology: comparison of the efficiency of various methods for planar image analysis

An evaluation is made of the relative efficiency (precision of the final estimate per unit time of measurement on a given set of sections) of different methods for planar analysis aimed at estimating aggregate, overall stereological parameters (such as V_v, S_v). The methods tested are point-counting with different densities of test points (4 ≤ P_T ≤ 900 per picture), semiautomatic computer image analysis with MOP and automatic image analysis with Quantimet, for obtaining V_v and S_v estimates. One biological sample as well as three synthetic model structures with known coefficients of variation between sections are used. The standard error of an estimate is mainly determined by the coefficient of variation between sampling units (= sections in the present paper) so that measuring each sample unit with a very high precision is not necessary. Automatic image analysis and point-counting with a 100-point grid were the most efficient methods for reducing the relative standard errors of the V_v and S_v estimates to equivalent levels in the synthetic models. Using a 64-point grid was as precise, and about 11 times faster than using a tracing device for obtaining the estimate of V_v in the biological sample.     

Centred contact density functions for the statistical analysis of random sets A stereological study on benign and malignant glandular tissue using image analysis

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.     

Estimating volumes from common carp hepatocytes using design‐based stereology and examining correlations with profile areas: Revisiting a nutritional assay and unveiling guidelines to microscopists

Assessing fish liver status is common in aquaculture nutrition assays. This often implies determining hepatocytes profile areas in routine thin (5–7 μm) histological sections. However, there are theoretical problems using planar morphometry in thin sections: inherent sampling cells biases, too small numbers of sampled cells, under/overestimation of size, measuring size as areas when cells are three‐dimensional (3D) entities. The gold standard for assessing/validate cell size is stereology using thick sections (20–40 μm). Here, we estimated the volume of hepatocytes and their nuclei by the nucleator and optical disector stereological probes (in thick sections), and, innovatively, in thin sections too (using single‐section disectors). The liver of common carp eating feed containing either low or high level of lipids was targeted. Results were compared with prior profile areas from planar morphometry using thin sections, and with profile areas estimated here with the two‐dimensional (2D) nucleator. Ratios between nucleus and cell/cytoplasm (N/C) areas and volumes were calculated and compared. There was high positive correlation between volumes in thin and thick sections (r = .85 to .89; p < .001), empirically validating the single‐section disector. Strong correlations existed between profile‐derived versus 2D‐nucleator areas (r = .74 to .83; p < .001). There was systematic underestimation of cells and nucleus size using planar morphometry. The N/C ratios derived from the 2D‐nucleator data were higher than those from planar morphometry. Despite theoretical premises for using simple planar morphometry in thin sections are flawed, our results support that such morphometry on carp/fish hepatocytes may offer some valid biological conclusions. Anyway, we advanced guidelines for implementing proper methods.     

Systematic and random errors in least squares estimation for circular contours

It is assumed that stochastically independent measurements are made of the distance from a reference centre to n points spaced at equi-angular intervals along an arc of length θ on the circular contour of a part. The least squares estimates of the radius of the circular contour and the coordinates of the true centre relative to the reference centre are derived assuming that the distance between the true and reference centres is small, relative to the radius of the part. These estimates are shown to be expressible as linear combinations of the measured distances with coefficients that are explicit functions of n and θ.

The estimates are shown to be systematically in error (biased) unless the true and reference centres coincide. Formulas are given for the calculation of the bias error in any situation. The random error of the estimated quantities is expressed in terms of the random error of an individual measurement and the values of n and θ. The correlation coefficient between the estimated quantities is also found. The results enable the determination, in advance of making measurements, of whether the likely random and systematic errors will be tolerable for a given choice of n and θ and a given centring accuracy     

Improved estimation of the pair correlation function of random sets

The texture of binary spatial structures can be characterized by second-order methods of spatial statistics. The pair correlation function, which describes the structure in terms of spatial correlation as a function of distance, is of central importance in this context. Conventionally, the pair correlation function of stationary and isotropic random sets is estimated as the ratio of the covariance to the square of volume fraction of the phase of interest. In the present paper, an improved estimator of the pair correlation function is presented, where the covariance is divided by the square of a distance-adapted estimator of volume fraction. The new estimator is explained mathematically and applied to simulated images of the Boolean model and to microscopic images from neoplastic and non-neoplastic human glandular tissues. It leads to a considerable reduction of bias and variance of estimated pair correlation functions, in particular for large distances.     

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

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.     

Estimating aggregate and overall characteristics from thick sections by transmission microscopy

The random spatial structure considered is the union X of an aggregate of random opaque convex particles in transparent space, the particle centroids being sited at the points of a homogeneous Poisson process. The model is thus the union of an aggregate of non-necessarily-convex disjoint regions. Expressions are derived for the fundamental stereological quantities V_V, S_V, K_V (integrated mean curvature density) and G_V (integrated gaussian curvature density) for X; and for A_A, L_A and C_A (integrated curvature density) for the projection of a slice of thickness t of this model onto a plane parallel to the slice, where t may take any non-negative value. The estimates relevant to transmission microscopy suggested by this theory are presented.     

Second-order stereology of benign and malignant alterations of the human mammary gland

The purpose of the present study was a quantitative characterization of the three-dimensional arrangement of the epithelial component of benign and malignant alterations of the female breast by combining stereology with stochastic geometry. Twenty cases of fibrous mastopathy and 20 cases of invasive ductal mammary cancer were studied at the light microscopic level. Segmentation of the epithelial tissue component was performed with an image analyser. From the resulting binary images, unbiased estimates of the covariance C(r) and the intensity V_v of the epithelial volume component were obtained automatically by computer. From these data, estimates of the correlation function k(r), of the pair correlation function g(r), of the radial distribution function RDF(r) and of the reduced second moment function K(r) of epithelial volume were determined. The estimates of C(r) and RDF(r) differed between groups, but these functions depend on spatial pattern and V_v. As carcinomas showed a significantly higher epithelial volume density V_v than mastopathies, estimation of C(r) and RDF(r) alone did not permit a safe distinction between possibly different types of spatial arrangement of epithelium in the benign and malignant lesions. Analysis of the estimates of k(r), g(r) and K(r), which are not influenced by V_v, showed definite interaction between epithelial volume elements, with clustering at short distances and repulsion at long distances. In both groups, the null hypothesis of purely random arrangement of epithelium had to be rejected. The clearest distinction between groups was obtained by estimation of g(r), which showed that short-range, tubular pattern as well as long-range, lobular architecture are better preserved in benign than in malignant lesions. The low interindividual scatter of k(r), g(r) and K(r) indicates a high biological significance of spatial pattern, which is presumably under strict genetic control. Potential uses of the method are: (i) the identification of biomathematical models which could contribute to a better understanding of the growth processes involved, (ii) conditional simulation of the underlying three-dimensional structures by computer, and (iii) supporting the diagnosis of mammary lesions with borderline histopathological appearance.     

Characterisation of lubricants on ball bearings by FT‐IR using an integrating sphere

Fourier transform‐infrared reflectance microspectroscopy has been used extensively for the examination of coatings on non‐planar surfaces such as ball bearings. While this technique offers considerable advantages, practical application has many drawbacks, some of which are easily overcome by the use of integrating sphere technology. This paper described the use of an integrating sphere for the quantification of thin layers of lubricant on the surface of ball bearings and the parameters that require optimisation in order to obtain reliable data. Several applications of the technique were discussed including determination of lubricant load on 12.7mm steel ball bearings and the examination of degraded lubricant on post‐mortem specimens. Copyright © 2007 John Wiley & Sons, Ltd.     

Estimation of surface area and length with the orientator

The orientator is a new technique for the estimation of length and surface density and other stereological parameters using isotropic sections. It is an unbiased, design-based approach to the quantitative study of anisotropic structures such as muscle, myocardium, bone and cartilage. A simple method for the practical generation of such isotropic planes in biological specimens is described. No special technical equipment is necessary. Knowledge of an axis of anisotropy can be exploited to optimize the efficiency. To randomize directions in space, points are selected with uniform probability in a square using various combinations of simple random, stratified random, and systematic random sampling. The point patterns thus produced are mapped onto the surface of a hemisphere. The mapped points define directions of sectional planes in space. The mapping algorithm ensures that these planes arc isotropic, hence unbiased estimates of surface and length density can be obtained via the classical stereological formulae. Various implementations of the orientator are outlined: the prototype version, the orientator-gencrated ortrip, two systematic versions, and the smooth version. Orientator sections can be generated without difficulty in large specimens; we investigated human skeletal muscle, myocardium, placenta, and gut tissue. Slight practical modifications extend the applicability of the method to smaller organs like rat hearts. At the ultrastructural level, a correction procedure for the loss of anisotropic mitochondrial membranes due to oblique orientation relative to the electron beam is suggested. Other potential applications of the orientator in anisotropic structures include the estimation of individual particle surface area with isotropic nucleators, the determination of the connectivity of branching networks with isotropic disectors, and generation of isotropic sections for second-order stereology (three-dimensional pattern analysis).     

Second-order stereology and ultrastructural examination of the spatial arrangements of tissue compartments within glomeruli of normal and diabetic kidneys

The present study explores 3D spatial arrangements of compartments within the rat renal glomerulus and tests for differences after chemically-induced diabetes. In particular, the arrangements of capillaries, podocytes, mesangium and urinary space are quantified and compared between (a) kidneys within groups and (b) kidneys from streptozotocin-diabetic rats and age-matched controls. The stereological tool employed is the pair correlation function which is estimated by counting linear dipole probes of different sizes superimposed on ultrathin sections so as to be random in position and orientation. Unbiased estimates of the volume density of each glomerular component were estimated by point counting. Thereafter, estimates of the covariance and pair correlation function were determined from corresponding dipole counts. Plots of covariance and pair correlation functions against dipole length were almost identical in control and diabetic groups, indicating that diabetes did not disturb the normal spatial arrangements within glomeruli. However, differences were detected between compartments within groups. Whilst volume elements within all compartments were clustered at distances below about 8 μm (the approximate size of the basic cellular or other structural unit), the cluster size varied between compartments. The pattern was one of progressively smaller clusters in the sequence capillaries, podocytes, urinary space, mesangium. Beyond a distance of 8 μm, all glomerular components (in both control and diabetic groups) were arranged as expected for a 'random' (meaning neither clustered nor repulsed) volume process. These studies re-emphasize the relative invariance of biological organization and the value and limitations of covariance analysis for quantifying different levels of organization in different tissues and experimental groups.     

Estimating length density and quantifying anisotropy in skeletal muscle capillaries

The accurate estimation of stereological parameters defined on anisotropic structures is a long-standing problem. In this paper we seek to estimate the capillary length density J_v in skeletal muscle tissue. A well-known model for directional anisotropy in space, namely the ‘spherical normal’ or ‘Fisher axial distribution’ model, is found to fit the relevant data satisfactorily. Based on this model, a short-cut estimation method is proposed and illustrated with a numerical example. This method essentially consists in taking the ratio of mean capillary profile counts, as obtained from transversal and longitudinal sections of the muscle tissue, and making use of a table or a graph given in the paper to estimate J_v. The conditions under which the methods are applicable and practicable are discussed in detail. Apart from an accurate estimation of J_v, an important feature of our method is the possibility of quantifying the degree of anisotropy by a coefficient K (called the concentration parameter of the Fisher axial distribution), which enjoys both a biological significance and a sound statistical basis.     

Beam heating in an asymmetrically illuminated thin circular film

The method of images is used to calculate the temperature increase produced in a circular film when the film is illuminated at any point by an electron beam of circular cross-section and constant current density over its cross-section. Formulae are given in dimensionless form for (i) the temperature increase anywhere along the line joining the centres of the beam and of the film, (ii) the temperature increase at the centre of the beam (v_c), and (iii) the position and value of the maximum temperature increase in the film v_m. The relative sizes of v_c and v_m are discussed and it is shown that (v_m – v_c)/v_m ≤ 0.079. The theory applies best to metallic films where power loss by means of thermal conductivity is dominant. A brief discussion is given of the cases where thermal radiation is important, sample calculations are given for carbon and copper films, and the effect of focusing the electron beam at constant current is considered.     

Practical approach to the estimation of the overall mean caliper diameter of a population of spheres and its application to data where small profiles are missed

In the morphometry laboratory a practical, accurate, and computationally simple procedure is needed when the estimation of the number of spherical particles per unit volume (N_v) is pursued. In addition, this procedure should be able to deal with the very real problem of profiles too small to be counted. Two computationally simple methods, the size class analysis method and the mean profile diameter method, were examined in detail. Computer-generated random profiles from various size-distributions of spheres were analysed by both methods. The percents error to be expected with various sphere distributions with a relatively large number of missed small profiles were determined for both methods. The accuracy of the size class analysis method was poor when a significant number of small profiles were missed. In this situation the accuracy of this method could be greatly improved by applying a simple modification to the procedure. A size class was identified which was larger than the largest missed profile but smaller than the diameter of the smallest sphere in the population. All contributions of this size class and all smaller size classes were deleted from the computation of N_v. The mean profile diameter method was found to be difficult to apply to distributions containing missed small profiles. Missed small profiles were handled more predictably by the modified size class analysis method.