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.     

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.     

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.     

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.     

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.     

Estimating surface area by the isotropic fakir method from thick slices cut in an arbitrary direction

The proposed fakir method for estimating surface area is based on counting the intersections between the surface lying within a thick slice, and an isotropic spatial grid consisting of a combination of linear probes called fakir probes. An unbiased procedure using a directly randomized spatial grid rather than sections with randomized directions is presented. The method is applicable if perfectly registered serial sections of the surface are available in a thick slice while the direction of the slice can be arbitrary. The efficiency of the fakir method using different arrangements of orthogonal triplets of fakir probes is evaluated and it is shown that mutually shifted probes are superior to non-shifted ones. The application software for interactive counting of intersections between computer-generated fakir probes and the surface within the stack of digitized images is described and demonstrated by two examples: estimation of the surface area of individual tobacco cell chains using a confocal microscope, and estimation of the total area of exposed surface of mesophyll cells in a barley leaf using a wide-field transmission microscope.     

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.     

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.     

The optical rotator

The optical rotator is an unbiased, local stereological principle for estimation of cell volume and cell surface area in thick, transparent slabs. The underlying principle was first described in 1993 by Kiêu &38; Jensen ( J. Microsc170, 45–51) who also derived an estimator of length. In this study we further discuss the methods derived from this principle and present two new local volume estimators.
The optical rotator benefits from information obtained in all three dimensions in thick sections but avoids over-/underprojection problems at the extremes of the cell. Using computer-assisted microscopes the extra measurements demand minimal extra effort and make this estimator even more efficient when it comes to estimation of individual cell size than many of the previous local estimators. We demonstrate the principle of the optical rotator in an example (the cells in the dorsal root ganglion of the rat), evaluate its efficiency and compare it with other unbiased, local stereological principles available for estimation of cell volume and surface area.     

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.     

Tissue shrinkage and unbiased stereological estimation of particle number and size*

This paper is a review of the stereological problems related to the unbiased estimation of particle number and size when tissue deformation is present. The deformation may occur during the histological processing of the tissue. It is especially noted that the widely used optical disector may be biased by dimensional changes in the z‐axis, i.e. the direction perpendicular to the section plane. This is often the case when frozen sections or vibratome sections are used for the stereological measurements. The present paper introduces new estimators to be used in optical fractionator and optical disector designs; the first is, as usual, the simplest and most robust. Finally, it is stated that when tissue deformation only occurs in the z‐direction, unbiased estimation of particle size with several estimators is possible.     

Estimation of the directional distribution of spatial fibre processes using stereology and confocal scanning laser microscopy

Fibrous structures like polymers, glass fibres, muscle fibres and capillaries are important components of materials and tissues. A spatial fibre process is the union of smoothly curved or linear one-dimensional features of finite length, arranged in an unbounded three-dimensional reference space according to some random mechanism. Design-based stereology was combined with confocal scanning laser microscopy to study samples of fibre-reinforced composites, which were considered as realizations of not necessarily isotropic fibre processes. The methods enable the unbiased estimation of the intensity and of the directional distribution of spatial fibre processes from arbitrarily directed pairs of registered parallel optical sections a known distance apart. The directions of fibres sampled by a frame of observation on the reference plane are estimated from the coordinates of the intersection points of the fibres with both optical planes using confocal scanning laser microscopy. The true directional distribution of the fibre process is estimated by weighting each measured direction by the reciprocal of its chance of being sampled, which can be inferred from the data. The concept of complete directional randomness for uniformly and independently distributed spatial directions is introduced. The cumulative distribution function of the angular distances between different directions and other exact relations are derived for complete randomness of vectorial and axial directions. A Monte Carlo method is constructed to test spatial fibre processes, whose fibres have negligibly small curvature, for complete directional randomness. Confocal scanning laser microscopy was used to study the angular distribution of glass fibres in a polymer composite which was subjected to increasing hydrostatic extrusion. The hypothesis of complete directional randomness had to be rejected for all samples with 1% probability of error. The directional distribution was of the bipolar type, with the principal axis directed parallel to the axis of extrusion. Progressive stretching of the material increased the degree of anisotropy of the glass fibres. Although presented for an application in polymer physics, the methods are general and may also be applied in biological investigations.     

Precise and general conditions for the validity of a comprehensive set of stereological fundamental formulae

The usual derivations of the well-known fundamental formulae of stereology are unsatisfactory. Precise and general conditions for the validity of a comprehensive system of such formulae are given. They take the form of unbiased ratio estimators of stereological fractions. These estimators are defined for various types of ‘weighted’ random plane and line sections of the specimen. Thanks to certain probabilistic equivalences, these sections may in practice be implemented fairly easily. The system includes new formulae which allow the estimation of the mean lineal and areal projections of embedded features. The mean square errors of several alternative estimators of the same feature characteristics are compared.     

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.     

Estimation of length and surface of anisotropic capillaries

Surface density (S_V) and length density (L_V) of myocardial capillaries have hitherto been estimated from their profile boundary length (B_A) and their numerical density (Q_A) on transverse sections by the simplifying assumptions of the Krogh model (perfectly anisotropic, straight, unbranched capillaries with constant cross-sectional area). As the capillaries actually are partially anisotropic, curved, branching cylinders with variable cross-sectional area, a geometrical bias arises from the model-reality discrepancies. We have applied and compared two methods to overcome these inconsistencies: (1) estimation of L_V and S_V by a more realistic model (the Dimroth-Watson distribution); (2) estimation of L_V and S_V from isotropic uniform random (IUR) sections. Twelve male Wistar rats were fixed by retrograde vascular perfusion. One pair of longitudinal and transverse sections, and six IUR sections per animal were selected at random from the left ventricular papillary muscles. Ultrathin sections were silver-impregnated and studied by light microscopic morphometry. Nearly identical estimates of L_V and S_V were found by both methods. The model-based estimation provides biologically meaningful anisotropy constants, but it presupposes knowledge of the anisotropy axis. The IUR method provides no measure of anisotropy, but it can be applied in tissues where the anisotropy axis is not known. Both methods are equally efficient and practically unbiased in S_V estimation, but the model-based estimation is far more efficient in L_V estimation.     

A stereological method for estimating volume and surface of sarcoplasmic reticulum

The stereological analysis of volume and surface of sarcoplasmic reticulum tubules in muscle cells meets with particular problems because these structures are (a) highly oriented in the longitudinal direction of the muscle cells, (b) lined up in parallel, (c) periodically arranged in relation to sarcomeres and (d) internally symmetrical in analogy to sarcomere symmetry. On the whole this system may be characterized as a parallel linear array of units with translational-reflexive symmetry. On the basis of a simple model it is demonstrated that the problems of sampling and stereological measurement can be overcome if a combination of oblique and transverse sections are used. The procedure can be simplified to operate exclusively with oblique sections only slightly inclined to the transverse plane if a small systematic error can be accepted.     

Some remarks on the accuracy of surface area estimation using the spatial grid

A set of three line grids in three orthogonal directions is called a spatial grid. This spatial grid can be used for surface area estimation by counting the number of intersection points of a surface with the grid lines. If direction and localization of the spatial grid are suitably randomized, the expectation of this number is proportional to the surface area of interest. The method was especially developed for cases where the surface to be measured is embedded in a medium, which is the usual case in microscopical applications, and where a stack of serial optical sections of the surface is available. The paper presents an improvement of an earlier version of the counting rule for intersection points. Furthermore, if the direction of sectioning is not uniform random, a bias results. This bias is calculated for a disc as a perfectly anisotropic object. A generalization of the estimator is considered by introducing a weighted mean instead of the usual arithmetic mean. The variance due to the randomized direction is investigated depending on the weights, and the minimum of this variance is derived. The relationship between the covariogram and the variance of the surface area estimated with the spatial grid is considered.     

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.     

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.