Signed line intercept count

Elements of surfaces that bound a phase (β) in a two phase mixture (α+β) may be classified as: (a) convex (++) if both principal curvatures are positive; (b) concave (? ?) if both are negative; and (c) saddle (+ ?) if one is positive and the other negative. This classification excludes the limiting cases for which one or both of the principal curvatures is zero. The traces of these surfaces that form the boundaries of the β areas on a representative two dimensional section may also be: (a) convex (+) if the local curvature is positive; or (b) concave (?) if it is negative. Line intercepts may be tabulated separately for intersections with convex (+) and concave (?) segments of boundary. This paper presents a derivation of fundamental stereological formulae that relate these counting measurements to three-dimensional geometric properties of the structure they sample.     

Measuring through the microscope: Development and evolution of stereological methods

Obtaining, by means of microscopy, meaningful measurements pertaining to spatial structures requires methods which allow three-dimensional quantitative information to be derived from the reduced information available on the two-dimensional flat sections of the structure. The most powerful methods to that effect are those of stereology which are based on mathematical principles. This paper reviews the early invention of these methods, which sought to solve practical problems, and their further evolution as more rigorous mathematical foundations were developed. It is demonstrated that stereological methods are essentially sampling methods and that newer trends provide new and sound solutions to old and elusive problems, such as anisotropy or particle number and size.     

Image analysis applied to engineering geology, a literature review

Image analysis is a technique used in a number of subjects, but to date not extensively used in the geological sciences. The paper describes the way image analysis is used in the various disciplines and presents an overview of the methods and its potential uses in geology and particularly engineering geology. Emphasis is placed on the way image analysis can be applied to rock structures and textures. It is hoped that this overview will provide an introduction to the literature, which is published in a variety of related fields. Electronic Publication     

Foreword

The volume of thin foil specimens, which contain precipitate or other particles, viewed in the TEM is needed to determine particle density and spacing. It can be determined from the locations of the particles, measured using stereo pairs. A calculation that determines the volume between planes (not necessarily parallel or horizontal) that enclose the points is described. These planes will systematically underestimate the actual surface spacing and hence the volume, but a simple correction factor based on the number of points used in the fit can be used to estimate the actual volume. The method is tested, and its accuracy is evaluated using simulated data and applied to precipitate particles in creep-tested silicon carbide.     

Unbiased linear property estimation for spheres,from sections exhibiting overprojection and truncation

In the biological sciences, stereological techniques are frequently used to infer changes in structural parameters (volume fraction, for example) between samples from different populations or subject to differing treatment regimes. Non-homogeneity of these parameters is virtually guaranteed, both between experimental animals and within the organ under consideration. A two-stage strategy is then desirable, the first stage involving unbiased estimation of the required parameter, separately for each experimental unit, the latter being defined as a subset of the organ for which homogeneity can reasonably be assumed. In the second stage, these point estimates are used as data inputs to a hierarchical analysis of variance, to distinguish treatment effects from variability between animals, for example. Techniques are therefore required for unbiased estimation of parameters from potentially small numbers of sample profiles. This paper derives unbiased estimates of linear properties in one special case—the sampling of spherical particles by transmission microscopy, when the section thickness is not negligible and the resulting circular profiles are subject to lower truncation. The derivation uses the general integral equation formulation of Nicholson (1970); the resulting formulae are simplified, algebraically, and their efficient computation discussed. Bias arising from variability in slice thickness is shown to be negligible in typical cases. The strategy is illustrated for data examining the effects, on the secondary lysosomes in the digestive cells, of exposure of the common mussel to hydrocarbons. Prolonged exposure, at 30 μg 1^?1 total oil-derived hydrocarbons, is seen to increase the average volume of a lysosome, and the volume fraction that lysosomes occupy, but to reduce their number.     

Stereological determination of curvature distributions of spatial fibre systems

Under the assumption of homogeneity and isotropy, this paper solves the problem of determining spatial curvature distributions of fibre processes from curvatures of projection curves from thin sections. As an example, a sample of dislocation lines in a hot-rolled Fe-Ni alloy is discussed.     

Stereological equations for phases within particles

Using data from sections, stereological equations can be used to estimate the volume-weighted averages of both the volume and the volume-squared of a set of particles. In mineral processing, the volume distribution of particles can be readily estimated using sieving, so it is uncommon to use stereological methods to determine the volume distribution of particles. A far more important application of stereology for mineral processing is to determine the volumetric grade distribution of minerals within particles. Different minerals or compounds within the particles are called phases. In this paper, stereological equations for estimating various volume-weighted averages are modified so that they can be applied to phases within particles. From these equations, various volume-weighted moments of the mineral grade distribution can be estimated. The equations are also modified so that the volume-weighted covariance of different phases can also be estimated. The equations derived are studied numerically by simulating sections though a double-capped sphere.     

PCS System I: point counting stereology programs for cell biology

PCS System I (PCS) is a set of four software modules designed to simplify the application of stereology to problems in cell biology. It is written in BASIC for the Tektronix 4052A microcomputer (Beaverton, OR). A Counting Module collects raw data counts in either a Density Mode (points, intersections, transections, profiles) or a Boundary Mode (intersections with complete nuclear profiles). This information is stored on tape or disk data files and can also be printed. Three analysis modules use data files created with the Counting Module. The Density Module uses Density Mode data files to calculate volume, surface, length, and numerical densities. The B Numerical Density Module uses both Boundary Mode and Density Mode data to calculate the means for the boundary, diameter, and surface area of a nuclear compartment. The mean nuclear surface area is then used with the nuclear surface density to estimate the nuclear numerical density, which, in turn, is used to calculate surface areas of membrane compartments in average cells and in 10(6) cells. The Format Module reformats raw data files for analysis with Tektronix statistical software.     

Spatial orientation of arterial sections determined from aligned vascular smooth muscle

The alignment of the arterial axis can be used as the required reference for three-dimensional (3-D) measurements of structures within the artery wall. Our hypothesis is that this alignment reference may be derived mathematically if one uses medial smooth muscle, a tissue component we have found to have a circumferential organization in human cerebral and coronary arteries. We tested this hypothesis for angles of sectioning up to 60°, using arteries fixed at normal distending pressure. These segments of artery were sectioned at precisely measured angles, using a specially designed mitre box, and the sections were stained to enhance birefringence of the smooth muscle. Arteries from eleven autopsies were obtained from the heart and brain, and measurements of 3-D orientation on medial smooth muscle were made with a polarizing light microscope equipped with a four-axis Universal stage. By comparing the cutting angle, derived from our mathematically obtained values, to the predetermined cutting angle, we deduce that we can determine the obliqueness of cut relative to the arterial axis to within 4.6°. The application that is important to us, and possibly to other laboratories, is that the necessary direction reference for an artery is completely contained within a single tissue section.     

The efficient and unbiased estimation of nuclear size variability using the ‘selector’

The selector was used to make an unbiased estimation of nuclear size variability in one benign naevocellular skin tumour and one cutaneous malignant melanoma. The results showed that the estimates obtained using the selector were comparable to those obtained using the more time consuming Cavalieri-disector approach. Employing ‘optical sections’, the selector was found to be between five and ten times more efficient than the Cavalieri-disector method when using physical sections.