Application of design‐based stereology for estimation of absolute volume and surface area of the articular and calcified cartilage compartments of undecalcified human femoral heads

Design‐based stereological methods using systematic uniform random sampling, the Cavalieri estimator and vertical sections are used to investigate undecalcified human femoral heads. Ten entire human femoral heads, obtained from normal women and normal men, were systematically sampled and thin undecalcified vertical sections were obtained. Absolute volumes and surface areas of the entire femoral head, the articular cartilage and the calcified cartilage compartments were estimated. In addition, the average thickness of the articular cartilage and the calcified cartilage were calculated. The stereological procedures applied to the human femoral heads resulted in average coefficient of errors, which were 0.03–0.06 for the volume estimates and 0.03–0.04 for the surface area estimates. We conclude that design‐based stereology using the Cavalieri estimator and vertical sections can successfully be used in large undecalcified tissue specimens, like the human femoral head, to estimate the absolute volume and surface area of macroscopic as well as of microscopic tissue compartments. The application of well‐known design‐based stereological methods carries potential advantage for investigating the pathology in inflammatory and degenerative joint diseases.     

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.     

Sampling theory and automated simulations for vertical sections,applied to human brain

In recent years, there have been substantial developments in both magnetic resonance imaging techniques and automatic image analysis software. The purpose of this paper is to develop stereological image sampling theory (i.e. unbiased sampling rules) that can be used by image analysts for estimating geometric quantities such as surface area and volume, and to illustrate its implementation. The methods will ideally be applied automatically on segmented, properly sampled 2D images – although convenient manual application is always an option – and they are of wide applicability in many disciplines. In particular, the vertical sections design to estimate surface area is described in detail and applied to estimate the area of the pial surface and of the boundary between cortex and underlying white matter (i.e. subcortical surface area). For completeness, cortical volume and mean cortical thickness are also estimated. The aforementioned surfaces were triangulated in 3D with the aid of FreeSurfer software, which provided accurate surface area measures that served as gold standards. Furthermore, a software was developed to produce digitized trace curves of the triangulated target surfaces automatically from virtual sections. From such traces, a new method (called the ‘lambda method’) is presented to estimate surface area automatically. In addition, with the new software, intersections could be counted automatically between the relevant surface traces and a cycloid test grid for the classical design. This capability, together with the aforementioned gold standard, enabled us to thoroughly check the performance and the variability of the different estimators by Monte Carlo simulations for studying the human brain. In particular, new methods are offered to split the total error variance into the orientations, sectioning and cycloid components. The latter prediction was hitherto unavailable – one is proposed here and checked by way of simulations on a given set of digitized vertical sections with automatically superimposed cycloid grids of three different sizes. Concrete and detailed recommendations are given to implement the methods.     

Stereology for anisotropic cells: Application to growth cartilage*

A number of either new or recently available stereological methods are described for estimating volume, surface area and number of anisotropic cells. The methods are illustrated with direct reference to the epiphyseal growth plate. Different estimates of a given quantity are obtained by applying alternative methods to the same set of sections, in order to compare the relative merits of the methods. For instance, the surface area of the cells is estimated via the Dimroth–Watson model (which gives a measure of the degree of anisotropy in addition to the surface area estimate) and from vertical sections using cycloid test systems. Cell number is estimated by traditional unfolding methods and by the new disector method. Also, volume-weighted mean cell volume is estimated from vertical sections via point-sampled intercepts using two different kinds of rulers to classify intercept lengths. Finally, nested design statistics is applied to a set of data from twelve animals in order to compare the relative impacts of biological and stereological (sampling) variations on the observed coefficient of error of a group mean estimate. The preferred methods are listed in the final section.     

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.     

Estimation of subcellular organelle volume from ultrathin sections through centrioles with a discretized version of the vertical rotator

A method for the fast and efficient estimation of the volume (but not surface area) of subcellular organelles is presented. It consists of a rotator/coaxial-section approach based on the Pappus theorem and represents a discretized version of the vertical rotator where, instead of measuring intercept lengths, the points in distance classes are counted. Centrioles serve as a unique reference 'double-point' with constant size allowing unbiased cell selection from the whole population with equal probability and without the disector application. The sandwich-like method of sample preparation allows comparison of control and experimental cases with the same errors caused by overlapping and overprojection. Test experiments demonstrated that the vertical discretized rotator was an efficient and precise tool for the estimation of volume and that a few independent sections of unknown thickness were sufficient for the quantification of one experimental point.     

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.     

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.     

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.     

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.     

Stereological estimation using vertical sections in a complex tissue

A method designed for stereological estimation in a very complex tissue using vertical sections is presented. In some tissues, the random rotation of the tissue for vertical sections may obscure recognition of the anatomical structures of interest. The present method overcomes this problem by generating sections with both a particular orientation, 'mapping sections', and ordinary random vertical sections usable for the required observations. A map describing the positions of the vertical sections is produced to make the complex reference space recognizable. The method is illustrated by estimating the number and size of neurones in the dorsal raphe nucleus of the human brainstem with its dense packing of roughly 100 nuclei within a volume less than 50 cm³.     

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).     

MicroCT examination of human bone specimens: effects of polymethylmethacrylate embedding on structural parameters

X‐ray microtomography permits the nondestructive investigation of trabecular and cortical bone specimens without special preparation of the sample. To do a quantitative characterization, the cross‐section images have to be binarized, separating bone from nonbone. For this purpose, a widely used method is uniform thresholding. However, for commonly available microtomography scanners which use a polychromatic X‐ray source, it is unclear what effect the surrounding medium (e.g. air, saline solution, polymethylmethacrylate) has on the threshold value used for the binarization. In the literature an easy procedure to find the optimal uniform threshold value for a given acquisition condition is reported. By applying this procedure, the present work investigated whether a microtomography scan of trabecular bone samples in air or embedded in polymethylmethacrylate gave the same results in terms of structural parameters. The gold standard, that is, histological sections, was used as a reference. Two fixed threshold values were found, one for the microtomography scans performed in air and one for the scans with the same samples embedded in polymethylmethacrylate. These were applied on the correspondent microtomography images for the estimation of structural parameters, such as bone volume fraction, direct trabecular thickness, direct trabecular separation and structure model index. Paired comparisons were made in bone volume fraction between histological sections and microtomography cross‐sections for the same bone samples scanned first in air and then embedded in polymethylmethacrylate, by which no significant differences were found. Paired comparisons were also made in bone volume fraction, direct trabecular thickness, direct trabecular separation and structure model index for the same samples over volumes of interest of 4 × 4 × 4 mm³ between microtomography scans in air and scans with the samples embedded in polymethylmethacrylate. Neither these comparisons showed significant differences. This leads to the conclusion that structural parameters estimated by microtomography for human trabecular bone samples scanned either in air or embedded in polymethylmethacrylate are not affected by the surrounding medium (i.e. presence or absence of polymethylmethacrylate), provided that the corresponding optimal threshold value is applied for each acquisition condition.     

Stereological analysis and modelling of gradient structures

Gradient structures are inhomogeneous along a particular gradient direction but homogeneous perpendicular to that direction. Consequently, structural parameters such as volume fraction or surface area density are local characteristics which depend on the 'vertical' coordinate with respect to the 'vertical' gradient axis.
Analogously, models for gradient structures have model parameters depending on the vertical coordinate z . For example, a Voronoi tessellation with a gradient is generated by a gradient point process with a local intensity which is a function of z . Similarly, a gradient germ grain model is obtained from a gradient point process where the grain size distribution may also depend on z . For a gradient Boolean model, local volume fraction V_V ( z ) and local surface area density S_V ( z ) can be calculated from the model parameters.
Stereological methods for gradient structures are based on vertical sections parallel to the gradient direction. Estimation of V_V ( z ), S_V ( z ) and local length density L_V ( z ) is done by lineal analysis using horizontal test lines with vertical coordinate z . Similarly, lineal analysis is used to estimate local mean cell volume of gradient tessellations. For the estimation of local particle number density and size in the spirit of the Wicksell problem the use of kernel methods and distributional assumptions is required.     

A new stereological principle for test lines in three-dimensional space

A new principle is presented to generate isotropic uniform random (IUR) test lines hitting a geometric structure in three-dimensional space (3D). The principle therefore concerns the estimation of surface area, volume, membrane thickness, etc., of arbitrary structures with piecewise smooth boundary. The principle states that a point-sampled test line on an isotropic plane through a fixed point in 3D is effectively an invariant test line in 3D. Particular attention is devoted to the stereology of particles, where an alternative to the surfactor method is obtained to estimate surface area. An interesting case arises when the particle is convex. The methods are illustrated with synthetic examples.     

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.     

Controlled serial grinding for high-resolution three-dimensional reconstruction

This work reports an improved preparation procedure for acquiring SEM images for three‐dimensional (3D) reconstruction. The images are acquired consecutively after serial grinding and polishing. Ugelstad beads are used as landmarks for registration purposes and for determining the thickness of the abraded sections. An estimation of the bead sizes necessary for suitable thickness quantification at the corresponding spatial resolution and uncertainty is given. Shape‐based interpolation is used for filling the gaps between the cross‐sections. An indication of the distance between cross‐sections necessary for good interpolation is also given. An example of a 3D reconstructed paper volume is presented. The method is suitable for preparation of fibre and paper as well as other materials.     

Confocal microscopy and stereology: estimating volume, number, surface area and length by virtual test probes applied to three-dimensional images

The opportunities of confocal microscopy applied to morphometry of microscopical structures are presented and demonstrated on stereological methods based on evaluation of optical sections within a thick slice and using computer-generated virtual test probes. Such methods, allowing arbitrary orientation of the thick slice, can be used for estimating volume, number, surface area, and length. The methods using spatial grid of points, disector, fakir, and slicer probes are described and illustrated by different examples using our freeware 3DTOOLS software and their variance and applicability are discussed. It is shown that shifted triple or quadruple spatial grids of lines are very efficient for the surface area and volume estimation by the fakir method.     

A simple algorithm to measure the volume-weighted and number-weighted mean volume of particles

An algorithm is presented which offers an alternative approach for measuring volume- and number-weighted mean volume and standard deviation of particles. Using a computer-assisted manual method the following intermediate steps are performed automatically: generation of linear probes emanating from the sampling point of the object and intersecting the profile periphery, measurement of their lengths, and measurement of the area of the transect required for estimating the standard deviation of the volume-weighted mean volume. By first tracing manually the outline of the periphery of the object with a cursor, on a magnetic tablet or on an image acquired into the computer with a video camera, the location of all pixels of the periphery is registered and the area of the transect is measured concurrently. The computer is informed of the coordinates of the selection point in the uniform random (UR) sampling grid by clicking the cursor. All ensuing operations are automatic. In the case of isotropic UR (IUR) sections the algorithm traces a series of uniform systematic random linear probes between the sampling point and the object profile periphery emanating from this selection point, radiating at angular intervals of 29–30° to the periphery. In the case of vertical sections, similar lines are generated at intervals where the sine of the angle changes by a value of 0·33. The volume-weighted mean volume of the object is estimated from the average of all the products , where l represents the length of each individual random linear probe. As the periphery is traced, the algorithm can automatically determine the area of the cross-section of the object, from which the standard deviation of the volume-weighted mean volume can be calculated. Some elements of the above algorithm are also used for the measurement of the number-weighted mean volume. The latter procedure is facilitated using an acoustic vertical depth monitor attached to the microscope. The impact of truncation (‘lost caps’) on the precision of the measurements is discussed. The algorithm is of particular use in light microscopy for measuring cell nuclei by direct visual inspection of the microscopic field using a side-arm mirror assembly interfaced with a magnetic tablet.     

Sampling problems in an heterogeneous organ: Quantitation of relative and total volume of pancreatic islets by light microscopy

In stereological studies analysis of sampling variances is used for optimizing the sampling design. In organs with a heterogeneous distribution of the phase of interest the analysis of sampling variances can be undertaken only if the observed variance between sections is distributed into the fraction which is due to random variation and the fraction which is due to the heterogeneity. In the present example (pancreatic islet volume estimated by light microscopic point-counting) the density of islets showed a linear increase along the axis of the organ. By analysis of sampling variances it was calculated that the most efficient number of sections (cut perpendicular to the organ) was considerably lower when the isolated contribution from the random variation was considered. The total islet volume was obtained by the product of the fractional islet volume and the pancreatic weight. Analysis of sampling variances of the total islet volume was performed by including the variance contribution from the individual pancreatic weights to the variance of the group mean total islet volume. Due to a negative correlation between the fractional volume and organ weight the total islet volume in the group of animals was more precisely estimated than the fractional islet volume. The methods used for dealing with the heterogeneity of the organ and for estimating sampling variances of total structural quantities generalize to a large number of stereological studies in biology.