A new fractionator principle with varying sampling fractions: exemplified by estimation of synapse number using electron microscopy

The quantification of ultrastructure has been permanently improved by the application of new stereological principles. Both precision and efficiency have been enhanced. Here we report for the first time a fractionator method that can be applied at the electron microscopy level. This new design incorporates a varying sampling fraction paradigm. The method allows for systematic random sampling from blocks of variable slab thickness, thereby eliminating the need for exhaustive serial sectioning through an entire containing space. This novel approach acknowledges the inaccuracy inherent in estimating the total object number using section sampling fractions based on the average thickness of sections of variable thicknesses. As an alternative, this approach estimates the correct particle section sampling probability based on an estimator of the Horvitz–Thompson type, resulting in a theoretically more satisfying and accurate estimate of the expected number of particles for the defined containing space.     

Assessment of particle retention and clearance in the intrapulmonary conducting airways of hamster lungs with the fractionator1

A modified version of the fractionator was used to estimate the total number of polystyrene microspheres retained in the airways of hamster lungs at two different time points after inhalation. A systematic three-stage subsampling procedure with known sampling fractions was adopted. First, each lung was cut into slices, from which primary disectors were sampled systematically with a known sampling fraction. From each primary disector, smaller sub-disectors were subsampled, and the corresponding sampling fraction was estimated by point counting. Finally, a few particles were counted at the microscopic level in the sub-disectors, and the final estimate of total particle number (which is unbiased irrespective of any tissue deformations) was easily computed as a product of the counted number times the reciprocal of the successive sampling fractions. The error variance of each estimate was assessed from the data using a new estimator. An average of 6% of the deposited particles were retained on the epithelial surface of the intrapulmonary conducting airways shortly after the inhalation, from which at least one-third was already phagocytosed by macrophages. After 24 h, an average of 87% of the particles retained shortly after the inhalation had been cleared. The proportion of particles ingested by macrophages had increased to at least 87%, in three out of four animals studied.     

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.     

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.     

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.     

A simplification of the ‘step’ method for estimating mean section thickness

An inexpensive and efficient method for estimating mean section thickness based on an estimate of the sectioning depth is presented. This value is required for the estimation of particle number using the disector.     

Automatic sampling for unbiased and efficient stereological estimation using the proportionator in biological studies

Quantification of tissue properties is improved using the general proportionator sampling and estimation procedure: automatic image analysis and non-uniform sampling with probability proportional to size (PPS). The complete region of interest is partitioned into fields of view, and every field of view is given a weight (the size) proportional to the total amount of requested image analysis features in it. The fields of view sampled with known probabilities proportional to individual weight are the only ones seen by the observer who provides the correct count. Even though the image analysis and feature detection is clearly biased, the estimator is strictly unbiased. The proportionator is compared to the commonly applied sampling technique (systematic uniform random sampling in 2D space or so-called meander sampling) using three biological examples: estimating total number of granule cells in rat cerebellum, total number of orexin positive neurons in transgenic mice brain, and estimating the absolute area and the areal fraction of β islet cells in dog pancreas. The proportionator was at least eight times more efficient (precision and time combined) than traditional computer controlled sampling.     

On the empirical variance of a fractionator estimate

The fractionator consists of several sampling stages with systematic sampling at each stage; data are collected only at the last stage. Therefore, predicting the error variance of a fractionator estimator is a non-trivial problem, because the observations are correlated in a complicated, unknown way. Gundersen proposed to split the material sampled at the first stage into two subsets, and to compute the variance of the pooled estimate empirically using the corresponding pair of observations made in these two subsets. The idea is very effective, but the estimator thus proposed needed some corrections. The purpose of this paper is to present an improved estimator of the coefficient of error of a fractionator estimator using Gundersen's design.     

Letter to the Editor

A double-disector method for obtaining estimates of the number of particles inside other particles, illustrated by an estimation of the average cell number in a glomerulus, has been developed. The method is suitable for paraffin-embedded tissue because it does not require knowledge about the section thickness. Most importantly, the estimate is absolutely unaffected by tissue shrinkage. The average number of cells in a human glomerulus is 2850 whereas in rats it is 580.     

Maternal dexamethasone treatment reduces ovarian follicle number in neonatal rat offspring

Elevated glucocorticoid levels in the gravid female circulation affect a number of endocrine functions in the fetuses and neonates. The aim of this study was to examine the effects of maternal dexamethasone (Dx) administration during late pregnancy on the ovaries of neonatal offspring. On the 16th day of pregnancy, experimental dams received subcutaneously 1.0 mg Dx/kg b.w., followed by 0.5 mg Dx/kg b.w./day on the 17th and 18th days of gestation. The control gravid females received the same volume of saline vehicle. Left ovaries from 5‐day‐old female pups were stereologically analyzed. The ovary volumes were estimated using Cavalieri's principle. The number of healthy and atretic primordial and primary follicles was estimated using a fractionator–physical disector method. The number of secondary follicles was determined by exact counts of every fourth section encompassing whole cross‐sections of the ovary. The ovary volume was significantly decreased (by 44.4%; P < 0.05) in the group of female pups from Dx‐treated mothers comparing to the controls. The numbers of healthy primordial and atretic follicles were 38.8% (P < 0.05) and 50.9% (P < 0.05), respectively, reduced in the ovaries of pups from the Dx‐treated mothers, when compared with the control values. There were 53.4% (P < 0.05) fewer healthy primary and 41.8% (P < 0.05) fewer healthy secondary follicles as well. The numbers of atretic primary and secondary follicles were reduced by 60.0% (P < 0.05) and 61.7% (P < 0.05), respectively. It can be concluded that fetal exposure to glucocorticoids decreased the pool of non‐growing follicles in the neonatal ovary, whereas the processes of folliculogenesis and atresia remained unaffected.     

Unbiased and efficient estimation of the total number of terminal bronchiolar duct endings in lung: a modified physical disector

A novel modification of the physical disector is described which was used to estimate the total number of terminal bronchiolar duct endings (TBDEs) in human infant lung. TBDEs are closed three-dimensional space curves of complex shape that are inherently difficult to count from histological sections. However, careful consideration of the microanatomy of the terminal duct endings provides us with the opportunity to define a very simple and unbiased counting rule. To apply the rule in practice we also need to determine a suitable disector height. Owing to the complex shape of the TBDE we had no prior knowledge of what disector height would be suitable for counting the TBDE structures. Exhaustive serial sectioning of complete TBDE structures was carried out and showed that any disector height under 90 μm would give unbiased counts. A further empirical study was then undertaken to determine the most efficient disector height. This was found to be 50 μm.
The total number of TBDEs in the upper lobe of the right lung of six human infants aged between 13 and 25 weeks was also estimated. The estimates of numerical density obtained with our modification of the physical disector were multiplied by estimates of lung lobe volume obtained using Cavalieri's Principle. The total number of TBDEs in the lobes ranged from 15 323 to 57 768, with a mean of 40 306. The average coefficient of error of the number estimates was 19%, which was deemed precise enough given the biological coefficient of variation between TBDE number of 36%.     

The rotator

The mean particle volume can be stereologically estimated using the nucleator principle. In the present paper, we discuss another principle for estimating mean particle volume, namely the rotator. The vertical rotator has already been previously described and is supplemented in the present paper by the isotropic rotator. For a collection of particle profiles, simulations show that the variance of the rotator is smaller than that of the nucleator.     

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.     

Unbiased stereological estimation of the rat fetal pituitary volume and of the total number and volume of TSH cells after maternal dexamethasone application

Glucocorticoids have an inhibitory influence on proliferation activity of the pituitary cells while stimulating apoptosis. Therefore, it was hypothesized that the synthetic glucocorticoid, dexamethasone (DX), has an inhibitory influence on the number of thyroid‐stimulating hormone (TSH) cells during fetal development. The effects of maternal administration of DX on stereological parameters of TSH cells, and TSH serum concentration were investigated in 21‐day‐old rat fetuses. On day 16 of pregnancy, the experimental dams received 1.0 mg DX/kg b.w. subcutaneously, followed by 0.5 mg DX/kg b.w./day on days 17 and 18 of gestation. The control gravid females received the same volume of saline vehicle. TSH cells were stained immunocytochemically by the peroxidase–antiperoxidase (PAP) method. The fetal pituitary volumes were estimated using Cavalieri's principle. A physical disector counting technique in combination with the fractionator sampling method was used for estimation of pituitary TSH cell number. Cell and nuclear volumes were measured with a planar rotator. Maternal DX application was found to cause a significant decrease of pituitary volume and number of TSH cells per pituitary in 21‐day‐old fetuses in comparison with the control fetuses. TSH cell number expressed per body weight unit declined significantly after maternal DX administration. These results indicate an inhibitory DX influence on proliferative activity of precursors and likely differentiated TSH cells and increased apoptotic prevalence. The histological appearance, volume of TSH cells and TSH serum concentration suggest intensive synthetic activity in TSH cells of DX exposed fetuses. Microsc. Res. Tech. 73:1077–1085, 2010. © 2010 Wiley‐Liss, Inc.