Precision of circular systematic sampling

In design stereology, many estimators require isotropic orientation of a test probe relative to the object in order to attain unbiasedness. In such cases, systematic sampling of orientations becomes imperative on grounds of efficiency and practical applicability. For instance, the planar nucleator and the vertical rotator imply systematic sampling on the circle, whereas the Buffon–Steinhaus method to estimate curve length in the plane, or the vertical designs to estimate surface area and curve length, imply systematic sampling on the semicircle. This leads to the need for predicting the precision of systematic sampling on the circle and the semicircle from a single sample. There are two main prediction approaches, namely the classical one of G. Matheron for non‐necessarily periodic measurement functions, and a recent approach based on a global symmetric model of the covariogram, more specific for periodic measurement functions. The latter approach seems at least as satisfactory as the former for small sample sizes, and it is developed here incorporating local errors. Detailed examples illustrating common stereological tools are included.     

Estimating the surface area of nonconvex particles from central planar sections

In this paper, we present a new surface area estimator in local stereology. This new estimator is called the ‘Morse‐type surface area estimator’ and is obtained using a two‐stage sampling procedure. First a plane section through a fixed reference point of a three‐dimensional structure is taken. In this section plane, a modification of the area tangent count method is used. The Morse‐type estimator generalizes Cruz‐Orive's pivotal estimator for convex objects to nonconvex objects. The advantages of the Morse‐type estimator over existing local surface area estimators are illustrated in a simulation study. The Morse‐type estimator is well suited for computer‐assisted confocal microscopy and we demonstrate its practicability in a biological application: the surface area estimation of the nuclei of giant‐cell glioblastoma from microscopy images. We also present an interactive software that allows the user to efficiently obtain the estimator.     

Stereological estimation of mean linear intercept length using the vertical sections and trisector methods

A method for estimating the mean linear intercept length of anisotropic microstructures using vertical sections is presented. A test system of cycloids and points is overlaid on the vertical sections, and the mean linear intercept length is estimated from a simple counting procedure — no length measurements are required. The vertical direction is either arbitrarily chosen or chosen perpendicular to most of the surface boundaries of the objects of interest. A design with the latter choice of the vertical direction and three vertical sections — a trisector — will optimize the precision of the estimate from only three vertical sections. The method was applied to two metallic structures, but it may also be used in a biomedical context.     

The isector: a simple and direct method for generating isotropic,uniform random sections from small specimens

The very simple and strong principle of vertical sections devised by Baddeley et al. has been a major advance in stereology when any kind of anisotropy is present in the specimen under study. On the other hand, some important stereological estimators still require isotropic, uniform random sections. This paper deals with a simple technique for embedding specimens in rubber moulds with spherical cavities. After the embedding, any handling of the resulting sphere independent of the specimen will induce isotropy of the final histological sections.     

Estimating the number of complex particles using the ConnEulor principle

An unbiased counting rule for the number of topologically simple objects of any shape, size and distribution in 3D space is a pertinent problem in stereology. Combining the disector principle with the object's 3D Euler number makes possible number estimation, which until now has been obtainable only by exhaustive serial sections. The disector is a set of two sections where the object's profiles in one section are compared with its profiles on the neighbouring section, and the number of new 2D topological events is recorded. In a disector of known volume the sum of topological events is a direct estimate of the disector contribution to the total Euler number, which forms the basis for an ultimate number estimator in 3D, the ConnEulor. The method is illustrated by an electron microscopic study of the number of mitochondria in the exocrine cells of the pancreas.     

Vertical LM sectioning and parallel CT scanning designs for stereology: application to human lung

Practical, unbiased stereological methods are described to estimate lung volume and external surface area, and total volume and surface area of relatively large and anisotropic structures (bronchi and arteries) inside the lung. The volume of each of five lung strata was estimated first by fluid displacement and then by computed tomography (CT) using Cavalieri's method; the reliability of CT was assessed through a calibration procedure, and image thresholding criteria for an accurate volume estimation using CT were established. The parallel, perfectly registered CT section images were also used to estimate the external surface area of each stratum by the spatial grid method. Unbiased estimation of internal surface areas in lung is a long-standing problem: since the structures are large and essentially void, large sections are needed; to facilitate identification, thin sections have to be used for light microscopy, and since such structures are anisotropic, the sections should be vertical. A practical stereological design is demonstrated here on an infant lung, which fulfils all these requirements. This study illustrates the potential of using unbiased stereology to characterize infant pulmonary hypoplasia.     

Volume estimation from projections

We describe a new estimator of the volume of axially convex objects from total vertical projections with known position of the vertical axis. The estimator combines the Cavalieri method with the known formula for area in terms of the support function of a convex body. We examine the accuracy of the proposed estimator for ellipsoidal objects having exactly known support function and volume. In addition, we illustrate practical problems of accuracy by implementing the method for some biological products.     

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.     

A simple and efficient alternative to implementing systematic random sampling in stereological designs without a motorized microscope stage

When properly applied, stereology is a very robust and efficient method to quantify a variety of parameters from biological material. A common sampling strategy in stereology is systematic random sampling, which involves choosing a random sampling relevant objects start point outside the structure of interest, and sampling at sites that are placed at pre‐determined, equidistant intervals. This has proven to be a very efficient sampling strategy, and is used widely in stereological designs. At the microscopic level, this is most often achieved through the use of a motorized stage that facilitates the systematic random stepping across the structure of interest. Here, we report a simple, precise and cost‐effective software‐based alternative to accomplishing systematic random sampling under the microscope. We believe that this approach will facilitate the use of stereological designs that employ systematic random sampling in laboratories that lack the resources to acquire costly, fully automated systems.     

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.     

Unbiased estimation of particle density in the tandem scanning reflected light microscope

The tandem scanning reflected light microscope has the property of being able to obtain information from ‘inside’ solid objects by taking a thin optical section at the focal plane of the objective lens. This plane can be focused up and down through the specimen. We describe an unbiased 3-D counting rule for the TSRLM, which is applied to the estimation of osteocyte lacunar density in whole bone. This is shown to be an extremely efficient way of making such an estimate. Further possibilities for the application of the microscope in the field of stereology are discussed.     

Non‐uniform systematic sampling in stereology*

Non‐uniform systematic sampling designs in stereology are studied. Various methods of constructing non‐uniform systematic sampling points from prior knowledge of the measurement function are presented. As an example, we consider area estimation from lengths of linear intercepts. The efficiency of two area estimators, based on non‐uniform sampling of parallel lines, is compared to that of the classical 2D Cavalieri estimator, based on uniform sampling, in a sample of planar profiles from transverse sections of 41 small myelinated axons. The comparison is based on simulations. It is concluded that for profiles of this type one of the non‐uniform sampling schemes is more efficient than the traditional uniform sampling scheme. Other examples where non‐uniform systematic sampling may be used are in area estimation from lines emanating from a fixed point, area estimation from concentric circles or spirals and curve length estimation from sweeping lines. It is shown that proportional‐to‐size sampling is a special case of non‐uniform systematic sampling. Finally, the effect of noise in the observations is discussed.     

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.     

Global spatial sampling with isotropic virtual planes: estimators of length density and total length in thick, arbitrarily orientated sections

Existing design-based direct length estimators require random rotation around at least one axis of the tissue specimen prior to sectioning to ensure isotropy of test probes. In some tissue it is, however, difficult or even impossible to define the region of interest, unless the tissue is sectioned in a specific, nonrandom orientation. Spatial uniform sampling with isotropic virtual planes circumvents the use of physically isotropic or vertical sections. The structure that is contained in a thick physical section is investigated with software-randomized isotropic virtual planes in volume probes in systematically sampled microscope fields using computer-assisted stereological analysis. A fixed volume of 3D space in each uniformly sampled field is probed with systematic random, isotropic virtual planes by a line that moves across the computer screen showing live video images of the microscope field when the test volume is scanned with a focal plane. The intersections between the linear structure and the virtual probes are counted with columns of two dimensional disectors.
Global spatial sampling with sets of isotropic uniform random virtual planes provides a basis for length density estimates from a set of parallel physical sections of any orientation preferred by the investigator, i.e. the simplest sampling scheme in stereology. Additional virtues include optimal conditions for reducing the estimator variance, the possibility to estimate total length directly using a fractionator design and the potential to estimate efficiently the distribution of directions from a set of parallel physical sections with arbitrary orientation.
Other implementations of the basic idea, systematic uniform sampling using probes that have total 3D × 4π freedom inside the section, and therefore independent of the position and the orientation of the physical section, are briefly discussed.     

Estimation of individual feature surface area with the vertical spatial grid

The area of an individual bounded surface (e.g. the boundary of a properly sampled cell) can be estimated from an isotropic uniform random stack of parallel sections, or of non-invasive planar scans, using the well-known spatial grid. A standing problem was to estimate the area of an individual bounded surface with an arbitrary degree of accuracy from a vertical (i.e. not isotropic) stack of sections or scans. A new tool to do this, called the ‘vertical spatial grid’, is presented.     

Quantitative serial sectioning analysis: preview

Quantitative serial sectioning analysis provides all of the geometric information that is available from ordinary stereological analysis. In addition, it circumvents the weaknesses inherent in geometric models that are used to estimate distribution functions and topological properties. It further makes accessible some geometric aspects of microstructures that are not available stereologically, even with model assumptions. At the present state of the art it requires a prodigious effort to prepare and analyse a single sample from a single specimen. However, the current generation of image analysing computers already has the capability for automating data collection and analysis, and there is reasonable potential for automating the sample preparation procedure. Quantitative serial sectioning analysis would render obsolete all of the elegant geometrical probability arguments of stereology. Is it the wave of the future?     

Design-based estimation of surface area in thick tissue sections of arbitrary orientation using virtual cycloids

Surface area is a first‐order stereological parameter with important biological applications, particularly at the intersection of biological phases. To deal with the inherent anisotropy of biological surfaces, state‐of‐the‐art design‐based methods require tissue rotation around at least one axis prior to sectioning. This paper describes the use of virtual cycloids for surface area estimation of objects and regions in thick, transparent tissue sections cut at any arbitrary (convenient) orientation. Based on the vertical section approach of Baddeley et al., the present approach specifies the vertical axis as the direction of sectioning (i.e. the direction perpendicular to the tissue section), and applies computer‐generated cycloids (virtual cycloids) with their minor axis parallel to the vertical axis. The number of surface‐cycloid intersections counted on focal planes scanned through the z‐axis is proportional to the surface area of interest in the tissue, with no further assumptions about size, shape or orientation. Optimal efficiency at each x–y location can be achieved by three virtual cycloids orientated with their major axes (which are parallel to the observation planes) mutually at an angle of 120°. The major practical advantage of the present approach is that estimates of total surface area (S) and surface density (S_V) can be obtained in tissue sections cut at any convenient orientation through the reference space.     

Applied stereology in materials science and engineering

A general view of the role of stereology in materials science and engineering (MSE), is followed by a discussion of applied stereology for one-, two- and multi-phase structures, and by examples of stereology applications to MSE, such as the impact of stereology on the creation and evolution of modern high–strength, low-alloy steels. The present state and prospects of applied stereology in MSE are informally discussed. It is demonstrated that, without stereology, materials science cannot evolve into a truly quantitative science, and that the needs of both materials scientists and materials engineers should be met by the continuing cooperation of those who develop stereological methods and those who apply them. To solve such problems as quantifying the interaction between features of similar and different dimensions in two- or multi-phase structures, a general parameter for volumetrical contiguity of second-phase particles, a contiguity parameter for second-phase particle profiles in grain-boundary surfaces, and techniques termed ‘reversed stereology’ are proposed and discussed.     

Stereological estimation of covariance using linear dipole probes

Classical stereology is capable of quantifying the total amount or 'density' of a geometrical feature from sampled information, but gives no information about the local spatial arrangement of the feature. However, stereological methods also exist for quantifying the 'local' spatial architecture of a 3D microstructure from sampled information. These methods are capable of quantifying, in a statistical manner, the spatial interaction in a structure over a range of distances. One of the key quantities used in a second-order analysis of a volumetric feature is the set covariance. Previous applications of covariance analysis have been 'model-based' and relied upon computerized image analysis. In this paper we describe a new 'design-based' manual method, known as linear dipole probes, that is suitable for estimating covariance from microscopic images. The approach is illustrated in practice on vertically sectioned lung tissue. We find that only relatively sparse sampling per animal is required to obtain estimates of covariance that have low inter-animal variability.     

Efficient embedding technique for preparing small specimens for stereological volume estimation: zebrafish larvae

Stereological sampling regimes, in particular volume and number estimation, often require systematic uniformly random sections throughout a specimen. A method has been developed to increase the efficiency of preparing fish larvae for sectioning prior to histological or stereological analysis. Embedding a group of larvae in a resin block using this technique greatly reduces the quantity of sections produced and allows easy assessment of sample groups. Saving time in this way therefore makes stereology a more viable research tool.