Is there section deformation resulting in differential change of nuclear numerical densities along the z axis of thick methacrylate or paraffin sections?

The optical disector, a three‐dimensional counting frame or probe in stereology, is often positioned in the middle (depth) of a thick section for unbiased nuclear counting. Using 30–40 μm thick methacrylate or paraffin sections for nuclear counting of neurons with the optical disector, however, some studies showed markedly higher nuclear densities at 10% of the section thickness near the top or bottom surface of the section, suggestive of deformation of section along its z axis and thus affecting the number estimation. To verify the findings, this study obtained two sets of 12–14 methacrylate sections (average thicknesses 21.7 and 29.4 μm) and two sets of 12 paraffin sections (average thicknesses 13.8 and 29.2 μm) from mature rat testes. Each section was used to count round spermatid nuclei in the seminiferous epithelium densely packed with the cells, using 3–4 consecutive disectors placed vertically (along the z axis of the section) from the top surface of the section, through the whole section thickness (two sets of methacrylate and paraffin sections) or in 80–83% of the thickness (other sections). The results demonstrated that, overall, there were no considerable nonuniform changes of the nuclear densities along the z axis of the 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.     

Estimation of variance components of local stereological volume estimators: a pilot study

Local stereological techniques can be used for particle volume estimation based on information collected on a section plane through a reference point of the particle. We present methods for variability estimation of the local stereological volume estimators. This variability arises during the stereological estimation procedure and in the particle population. Both of these components can be estimated separately from planar sections. Our aim is to give a preliminary analysis of the possibility to include the particle structure interaction into the estimation procedure. For this reason, not only the section profiles, but also their locations, have to be recorded. The methods are applied for the sectional data obtained from neurons in the hippocampal brain region subiculum of four 3-month-old male Wistar rats. The proposed procedure enables one to obtain information about particle volume distribution.     

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.     

Unbiased stereological estimation of the number and volume of nuclei and nuclear size variability in invasive ductal breast carcinomas

The application of design-based stereological methods for estimating nuclear features quantitatively in invasive ductal breast cancer is described. Nuclear number, size and size variability are explored in relation to the tumour grade and patient prognosis. The study includes an examination of the efficiency in estimating different nuclear volumes, and two different estimators of the nuclear size variability are contrasted. Forty-two invasive ductal breast carcinomas diagnosed and graded by two pathologists were used. Both 5-μm and 25-μm-thick sections were obtained from paraffin blocks for stereological study. More undifferentiated tumours show significantly larger nuclei than low-grade tumours. The estimates based on the disector method demonstrate a decrease in the number of tumour cell nuclei per unit volume of tissue from grades 1 to 2 and especially from grades 2 to 3. The univariate survival analysis shows a high prognostic value of the nuclear volume estimates. The study shows that an efficient sampling procedure was performed, particularly when estimating volume-weighted mean nuclear volume using the point-sampled intercepts method. This method is more efficient than estimation of the number-weighted mean nuclear volume using the selector method; however, the latter provides paired estimates of volume- and number-weighted mean nuclear volume, as well as an estimate of the coefficient of variation of nuclear volume in the number distribution of the same cells.     

Pronounced loss of cell nuclei and anisotropic deformation of thick sections

The local deformation and variations in section thickness are studied in 100-μm thick vibratome sections of well-fixed human brain tissue. During processing, including drying on glass slides, the section thickness is reduced to less than half, but close to the edges there is less shrinkage of the section thickness. Close to both surfaces there is a pronounced reduction in the number of neuronal nucleoli. At the scale of the original section, the upper 15 μm and the lower 10 μm are depleted. The loss is most pronounced at the upper surface, which is unprotected during processing. In the central 70% of the section height, where one would ordinarily use an optical disector for sampling, there is no indication of non-uniform shrinkage. The simplest explanation for the observed loss of nucleoli is that all cells opened by the knife may lose their nuclei across an unprotected section surface. The observations do not generalize to other tissues and other preparation techniques, but illustrate the magnitude of some of the problems for uniform sampling and unbiased estimation in very thick sections. The uniform optical disector sampling of nucleoli in thick sections, as opposed to that of cell nuclei, raises a special problem, which is discussed briefly.     

Estimating volumes from common carp hepatocytes using design‐based stereology and examining correlations with profile areas: Revisiting a nutritional assay and unveiling guidelines to microscopists

Assessing fish liver status is common in aquaculture nutrition assays. This often implies determining hepatocytes profile areas in routine thin (5–7 μm) histological sections. However, there are theoretical problems using planar morphometry in thin sections: inherent sampling cells biases, too small numbers of sampled cells, under/overestimation of size, measuring size as areas when cells are three‐dimensional (3D) entities. The gold standard for assessing/validate cell size is stereology using thick sections (20–40 μm). Here, we estimated the volume of hepatocytes and their nuclei by the nucleator and optical disector stereological probes (in thick sections), and, innovatively, in thin sections too (using single‐section disectors). The liver of common carp eating feed containing either low or high level of lipids was targeted. Results were compared with prior profile areas from planar morphometry using thin sections, and with profile areas estimated here with the two‐dimensional (2D) nucleator. Ratios between nucleus and cell/cytoplasm (N/C) areas and volumes were calculated and compared. There was high positive correlation between volumes in thin and thick sections (r = .85 to .89; p < .001), empirically validating the single‐section disector. Strong correlations existed between profile‐derived versus 2D‐nucleator areas (r = .74 to .83; p < .001). There was systematic underestimation of cells and nucleus size using planar morphometry. The N/C ratios derived from the 2D‐nucleator data were higher than those from planar morphometry. Despite theoretical premises for using simple planar morphometry in thin sections are flawed, our results support that such morphometry on carp/fish hepatocytes may offer some valid biological conclusions. Anyway, we advanced guidelines for implementing proper methods.     

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.     

Estimation error of Leydig cell numbers in atrophied rat testes due to the assumption of spherical nuclei

In this study, Leydig cell numbers in control and atrophied testes (induced via subcutaneous implants of testosterone plus 17β estradiol for 16 weeks; TE-implanted) of rats, estimated via the fractionator method (independent of any assumptions) were compared to those estimated via the disector (unbiased, but dependent on shrinkage) and Floderus (assumes spherical particles, dependent on shrinkage) methods. Estimates of Leydig cell numbers in control rats produced by all three stereological methods were similar. In rats with atrophied testes, both the fractionator and the disector methods produced significantly lower (P < 0.01; 47% and 41 % with fractionator and disector, respectively) Leydig cell number estimates per testis than in the controls. By contrast, the estimates of Leydig cell number in atrophied testes derived via the Floderus equation were not significantly different from those of controls, but larger than those obtained via the fractionator and the disector methods. These results suggested that the assumptions of the Floderus method were violated in the atrophied rat testes. Why was the Floderus method of estimating Leydig cell number applicable to control rats but not to the TE-implanted rats? In an attempt to answer this question the diameter measurement together with its correction factor used in the Floderus equation (i.e. D + t ? 2H) was also derived from the data collected for the disector method. The values for D + t ? 2H used in the Floderus method and also calculated via the disector method were found to be identical in controls, but for the TE-implanted rats a 32% lower value was obtained with the Floderus equation when compared to the disector. These findings suggested that this estimation error caused an overestimation of Leydig cell numbers in the TE-implanted rat testes.     

Stereological ratio estimation based on counts from integral test systems

The stereological practice of using integral test systems in the estimation of the fundamental stereological ratios is studied in the light of recent theoretical developments in sampling. The estimation of a ratio is based on counts only, obtained from two, in general different, test sets constituting the integral test system. The ordinary ratio-of-sums estimator based on counts from uniformly positioned integral test systems is compared with two estimators based on non-uniform, weighted sampling. It is shown that the estimators based on weighted sampling are not, in general, unbiased. Furthermore, it is pointed out that the mean-of-ratios estimator based on replicated weighted sampling needs not have smaller MSE than the ordinary ratio-of-sums estimator based on replicated uniform sampling. The fact that the estimation is based on counts as opposed to complete 2-d observations does not necessarily mean a reduction in information. For certain types of stereological ratios, the ordinary ratio-of-sums estimator based on complete observation is shown, counter to intuition, to be less accurate than that based on simple and fast counting.     

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.     

The efficiency of systematic sampling in stereology and its prediction*

The superior efficiency of systematic sampling at all levels in stereological studies is emphasized and various commonly used ways of implementing it are briefly described. Summarizing recent theoretical and experimental studies a set of very simple estimators of efficiency are presented and illustrated with a variety of biological examples. In particular, a nomogram for predicting the necessary number of points when performing point counting is provided. The very efficient and simple unbiased estimator of the volume of an arbitrary object based on Cavalieri's principle is dealt with in some detail. The efficiency of the systematic fractionating of an object is also illustrated.     

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.     

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.     

Light microscopic identification and semiquantification of polyethylene particles in methylmethacrylate and paraffin-embedded experimental bone implant specimens

The aim of this study was to evaluate the identification of polyethylene (PE) particles in relatively thick methylmethacrylate (MMA) sections widely used in bone implant research. The sensitivity and specificity were compared between decalcified paraffin-embedded oil red O (ORO) stained and MMA-embedded sections using polarized light. Furthermore, we introduced a grading system to semiquantify the level of PE particles in peri-implant tissue. Paraffin-embedded and MMA-embedded sections were compared concerning intra-observer agreement of the grading system. Moreover, the semiquantitative assessment of particle level was compared between the two section types. We found a sensitivity and specificity of polarized light of 100% for both paraffin ORO-stained and MMA sections. The intra-observer agreement on both types was comparable and acceptable. The ratings of differently processed blocks (MMA- and paraffin-embedded) originating from the same bone implant specimen showed good correlation. Our study showed that relatively thick MMA sections were just as suitable as ORO-stained paraffin sections concerning peri-implant PE particle migration analysis. MMA sections do not allow analysis at cellular level, but unbiased estimation of bone ingrowth into the implant surface based on stereological principles is possible.     

Unbiased volume estimation with coaxial sections: an application to the human bladder

A problem of current medical interest is how to estimate the volume of an object from coaxial sections. Such images can be produced in ultrasound scanning, when the planar scan sections all emanate from a common axis. Two unbiased volume estimators for this sampling regime, previously published without derivation, are derived and presented here in some detail. The estimators are based on an ancient theorem of Pappus of Alexandria (c. ad 320). One of the estimators was used to estimate the volume of urine in the bladder of six human volunteers. The case example was attractive because the corresponding volumes could be voided and directly measured after the scanning session, thus confirming the unbiasedness of the estimator used. Sample size guidelines are also given.     

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.