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 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).     

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.     

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.     

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.     

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.     

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.     

Application of stereologic methods to stratified gingival epithelia

Stereologic point-counting procedures were applied to estimate quantitative tissue and cytoplasm parameters of the oral and the junctional epithelium of the human gingiva. Three gingival biopsies of female children, obtained and processed under standardized conditions, served (1) to determine the minimal sample size of cytoplasmic area required for estimating representative volumetric parameters in different strata of both epithelia, and (2) to compare average volume and surface density data derived from standardized samples in epithelial cross sections cut at two planes orientated perpendicular to each other. Sampling of electron micrographs, performed at two levels of magnifications according to a systematic stratified random sampling procedure consisted of recording field strips parallel to the epithelial or tooth surface within each of the various strata. The optimal sample size required varied for different organelles and epithelial strata. Minimal sample size of cytoplasmic area per biopsy were 300–440 μm² for basal, and 450 μm² for stratum spinosum cells of oral epithelium, and 130–185 μm² for basal cells of the junctional epithelium. Average morphometric parameters for gross tissue components (cells, nuclei, cytoplasm, intercellular space) which resulted from analysing 4900 μm² tissue area in each of the epithelia in each biopsy, were almost identical when determined in two different section planes. Volume and surface density data of cytoplasmic organelles (mitochondria, rough and smooth endoplasmic reticulum, cytoplasmic filaments) resulting from a sample of 280 μm² cytoplasmic area per stratum and biopsy, revealed differences of varying magnitude, when determined in the two section planes. These differences were markedly smaller than those between comparable strata of both epithelia. Discussion of sampling procedures concluded that the type of sampling used for the present study is suitable for comparative morphometric studies.     

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.     

Stereological estimation of surface area and barrier thickness of fish gills in vertical sections

Previous morphometric methods for estimation of the volume of components, surface area and thickness of the diffusion barrier in fish gills have taken advantage of the highly ordered structure of these organs for sampling and surface area estimations, whereas the thickness of the diffusion barrier has been measured orthogonally on perpendicularly sectioned material at subjectively selected sites. Although intuitively logical, these procedures do not have a demonstrated mathematical basis, do not involve random sampling and measurement techniques, and are not applicable to the gills of all fish. The present stereological methods apply the principles of surface area estimation in vertical uniform random sections to the gills of the Brazilian teleost Arapaima gigas. The tissue was taken from the entire gill apparatus of the right‐hand or left‐hand side (selected at random) of the fish by systematic random sampling and embedded in glycol methacrylate for light microscopy. Arches from the other side were embedded in Epoxy resin. Reference volume was estimated by the Cavalieri method in the same vertical sections that were used for surface density and volume density measurements. The harmonic mean barrier thickness of the water‐blood diffusion barrier was calculated from measurements taken along randomly selected orientation lines that were sine‐weighted relative to the vertical axis. The values thus obtained for the anatomical diffusion factor (surface area divided by barrier thickness) compare favourably with those obtained for other sluggish fish using existing methods.     

Stereology of isolated objects with the invariator

The invariator is a new stereological design to generate motion invariant test lines in three dimensions on an isotropic plane through a fixed point. The theory has been published recently. The purpose of this paper is to illustrate the application of the invariator on a group of rat brains to estimate brain volume and external surface area. Each brain was first split into its two hemispheres and then embedded into a ball filled with agar following a configuration named the antithetic isector, with the idea of reducing the error variance. After rolling the ball at random it was scanned by magnetic resonance imaging into a stack of parallel systematic sections: this is the isotropic Cavalieri design which, combined with the antithetic isector idea, proves to be very accurate. The invariator used only an equatorial section of the ball, and in the present case the coefficient of error of the volume and surface area estimators of an individual brain was about 30%. As it is design unbiased, the invariator may prove its strength mainly to estimate population means.     

The saucor, a new stereological tool for analysing the spatial distributions of cells, exemplified by human neocortical neurons and glial cells

The 3D spatial arrangement of particles or cells, for example glial cells, with respect to other particles or cells, for example neurons, can be characterized by the radial number density function, which expresses the number density of so-called 'secondary' particles as a function of their distance to a 'primary' particle. The present paper introduces a new stereological method, the saucor, for estimating the radial number density using thick isotropic uniform random or vertical uniform random sections. In the first estimation step, primary particles are registered in a disector. Subsequently, smaller counting windows are drawn with random orientation around every primary particle, and the positions of all secondary particles within the windows are recorded. The shape of the counting windows is designed such that a large portion of the volume close to the primary particle is examined and a smaller portion of the volume as the distance to the primary object increases. The experimenter can determine the relation between these volumina as a function of the distance by adjusting the parameters of the window graph, and thus reach a good balance between workload and obtained information. Estimation formulae based on the Horvitz-Thompson theorem are derived for both isotropic uniform random and vertical uniform random designs. The method is illustrated with an example where the radial number density of neurons and glial cells around neurons in the human neocortex is estimated using thick vertical sections for light microscopy. The results indicate that the glial cells are clustered around the neurons and the neurons have a tendency towards repulsion from each other.     

Determination of microstructural parameters of random spatial surface fractals by measuring chord length distributions

An experimental procedure for determining the micro-structural parameters of random spatial surface fractals is proposed. The procedure is based on chord length distribution measurements of planar sections of the investigated material at different linear resolutions, and comparison of the experimental results with the corresponding parameters of a well-defined structure model. The variability of the model allows microstructures with arbitrary volume fraction and fractal dimension to be considered. Furthermore, both discrete and continuous scaling behaviour have been studied.     

Quantification of microvessels in canine lymph nodes

Quantification of microvessels in tumors is mostly based on counts of vessel profiles in tumor hot spots. Drawbacks of this method include low reproducibility and large interobserver variance, mainly as a result of individual differences in sampling of image fields for analysis. Our aim was to test an unbiased method for quantifying microvessels in healthy and tumorous lymph nodes of dogs. The endothelium of blood vessels was detected in paraffin sections by a combination of immunohistochemistry (von Willebrand factor) and lectin histochemistry (wheat germ agglutinin) in comparison with detection of basal laminae by laminin immunohistochemistry or silver impregnation. Systematic uniform random sampling of 50 image fields was performed during photo-documentation. An unbiased counting frame (area 113,600 microm(2)) was applied to each micrograph. The total area sampled from each node was 5.68 mm(2). Vessel profiles were counted according to stereological counting rules. Inter- and intraobserver variabilities were tested. The application of systematic uniform random sampling was compared with the counting of vessel profiles in hot spots. The unbiased estimate of the number of vessel profiles per unit area ranged from 100.5 +/- 44.0/mm(2) to 442.6 +/- 102.5/mm(2) in contrast to 264 +/- 72.2/mm(2) to 771.0 +/- 108.2/mm(2) in hot spots. The advantage of using systematic uniform random sampling is its reproducibility, with reasonable interobserver and low intraobserver variance. This method also allows for the possibility of using archival material, because staining quality is not limiting as it is for image analysis, and artifacts can easily be excluded. However, this method is comparatively time-consuming.     

The new stereological tools in metallography: estimation of pore size and number in aluminium

A number of simple, unbiased and efficient methods are now available in stereology for estimating the number and size of arbitrary particles or voids in a material, with the only assumption that particles must be identifiable on serial sections or confocal planes through the material. In recent years, these methods have been developed and applied mainly in a biomedical context: this paper reviews and illustrates them with the aid of a metallographic example, namely the pore population of a sand-cast aluminium alloy. Our goal is to convey the fact that stereology is sampling in three dimensions, and therefore its principles remain valid and applicable in no matter what context. The disector, the selector, and an indirect method to estimate the distance between two parallel planes of polish are thereby illustrated. It is also shown how to split the error variance of the estimator of the pore volume fraction (‘porosity’) into the three components due to blocks, sections within blocks and systematic point counting on sections.     

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.     

Unbiased estimation of capillary length from vertical slices*

Previous stereological approaches to estimate feature length include isotropic sections, which tend to be inefficient for highly anisotropic structures such as skeletal muscle capillaries, and semiparametric model-based methods, which require transverse and longitudinal sections only, but are biased to a variable, unknown degree. The recent method of vertical slices combines the advantages of both approaches, namely it is unbiased, efficient and convenient. This study illustrates for the first time how to apply the vertical slices method in biology by direct light microscopy and intersection counting with a properly orientated cycloid test system. Neither image processing nor confocal microscopy are used. The purpose of the study was to estimate capillary length in the left ventricle of rat heart. Beyond this, a novel histochemical method enables the staining of the venular capillary region in red and the arteriolar capillary region in blue, and hence estimates their separate lengths. The vertical slices method to estimate feature length seems to be a promising approach for biology.     

Anisotropic properties of bovine nasal cartilage

To investigate the structural anisotropy in bovine septal cartilage, quantitative procedures in microscopic magnetic resonance imaging (μMRI), polarized light microscopy (PLM), and mechanical indentation were used to measure the tissue in three orthogonal planes: vertical, medial, and caudocephalic. The quantitative T2 imaging experiments in μMRI found strong anisotropy in the images of both vertical and caudocephalic planes but little anisotropy in the images from the medial plane. The PLM birefringent experiments found that the retardation values in the medial section were only about 10% of these in the vertical and caudocephalic sections and that the angle values in all three sections followed the rotation of the tissue section in the microscope stage. The stress relaxation experiments in mechanical indentation showed reduced stiffness in the medial plane compared to stiffness in either the vertical or caudocephalic planes. Collectively, the results in this project coherently indicate a marked structural anisotropy in cartilage from the nasal septum, where the long axis of the collagen fibrils is oriented in parallel with the medial axis.