Estimating the total number of lymphatic valves in infant lungs with the fractionator

The fractionator is illustrated by means of a biomedical example involving the estimation of the number of lymphatic valves in lungs of infants who had died from sudden infant death syndrome (SIDS) and other known causes. The method is unbiased irrespective of tissue deformations and it does not require external information such as section thickness. An upper bound of the coefficient of error of the estimate of the number of valves within one lung was 6.5%, despite the fact that the number of valves counted per lung at the last stage ranged between 11 and 37 only. The upper bound includes the biological variation of the number of valves among infant lungs. Some theoretical remarks are also made on the efficiency of the fractionator. It is suggested, for instance, that the initial sampling stages cause more impact on the precision of the final estimator than the subsequent stages, and that an optimal arrangement of fragments submitted to systematic sampling should have the smallest fragments at the ends, with fragment contents increasing smoothly toward the middle of the series.     

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.     

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.     

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.     

Evaluation of variance models for fractionator sampling of trees

We compared the performance of several models for predicting, from small samples, the precision of estimates of the total number of blossoms on fruit trees obtained using a three‐stage fractionator, in which the sampling units were defined by the tree structure: (1) primary branches and stem (2) secondary branches and shoots and (3) flowering buds. The models considered were the semiempirical models of Cruz‐Orive (1990, 2004 ) (CO), a random sample model (SR), a Poisson model (P), successive differences (D) and repeated systematic sampling (R). Procedures that relied upon a single sample and a model of the variance (SR, P, D) were not able to predict the estimator variance because the considered structures strongly violated model assumptions. The semiempirical CO model performed acceptably in some cases where model assumptions were violated to a moderate degree. The repeated systematic sampling procedure, which does not rely upon a model of the variance, usually provided very good predictions when the resampling terms were distributed appropriately across more than one sampling stage.     

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.     

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.     

Lung development: AT1 and AT2 property

The human respiratory system consists of the upper and the lower respiratory tracts. Anatomically, the lower respiratory tract consists of the trachea, bronchi, bronchioles (terminal bronchioles and respiratory bronchioles), alveolar duct, alveolar duct sacs, and alveoli. Alveoli are composed of two epithelial cell types, cuboidal alveolar type 2 (AT2) cells that secrete surfactant to prevent alveolar collapse and function as stem cells to regenerate alveolar type 1 (AT1) cells during damage repair, and squamous AT1 cells that cover most of the surface area of the alveoli and mediate gas exchange. Previous studies mainly focused on AT2 cells; this review summarizes the current studies on lung development and property of AT1 cells.     

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.     

Prediction of the variance of stereological volume estimates from systematic sections using computer-intensive methods

The Cavalieri method is an unbiased estimator of the total volume of a body from its transectional areas on systematic sections. The coefficient of error (CE) of the Cavalieri estimator was predicted by a computer‐intensive method. The method is based on polynomial regression of area values on section number and simulation of systematic sectioning. The measurement function is modelled as a quadratic polynomial, with an error term superimposed. The relative influence of the trend and the error component is estimated by techniques of analysis of variance. This predictor was compared with two established short‐cut estimators of the CE based on transitive theory. First, all predictors were applied to data sets from six deterministic models with analytically known CE. For these models, the CE was best predicted by the older short‐cut estimator and by the computer‐intensive approach, if the measurement function had finite jumps. The best prediction was provided by the newer short‐cut estimator when the measurement function was continuous. The predictors were also applied to published empirical datasets. The first data set consisted of 10 series of areas of systematically sectioned rat hearts with 10–13 items, the second data set consisted of 13 series of systematically sampled transectional areas of various biological structures with 38–90 items. On the whole, similar mean values for the predicted CE were obtained with the older short‐cut estimator and the computer‐intensive method. These ranged in the same order of magnitude as resampling estimates of the CE from the empirical data sets, which were used as a cross‐check. The mean values according to the newer short‐cut CE estimator ranged distinctly lower than the resampling estimates. However, for individual data sets, it happened that the closest prediction as compared to the cross‐check value could be provided by any of the three methods. This finding is discussed in terms of the statistical variability of the resampling estimate itself.     

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.     

Measurement uncertainty limit analysis with the Cramér-Rao bound in case of biased estimators

The Cramér-Rao lower bound has been proven to be a valuable tool for determining the minimal achievable measurement uncertainty and for analyzing the performance of estimators in terms of efficiency. While this is common for unbiased estimators, a bias does often occur in practice. The performance analysis of biased estimators is more difficult, because the bias has to be taken into account additionally. Furthermore, not the behavior of the biased estimator is finally of interest in measurements, but the behavior of its bias-corrected counterpart. In order to simplify the required performance analysis for biased estimators, the relation between the efficiencies of the biased and the bias-corrected estimator is derived. As result, both efficiencies are shown to be (at least asymptotically) identical. Hence, the bias-corrected estimator attains the Cramér-Rao bound if and only if the biased estimator attains its Cramér-Rao bound. Furthermore, the mean square errors become minimal if and only if the estimators reach the Cramér-Rao bound. Consequently, the optimality of the bias-corrected estimator can also be judged by evaluating the mean square error of the biased estimator.     

Confocal laser scanning microscopy for the study of the morphological changes of the postextraction sites

A better understanding of the remodeling process of postextraction sockets is essential in dental treatment planning. The aim of this study was to evaluate whether confocal laser scanning microscopy (CLSM) can be applied to imaging contour changes of postextraction sites, as well as to its quantification with image analysis of obtained three‐dimensional images. This work describes a new application of the CLSM technique. The system used was the OLS3100‐USS, LEXT model (Olympus®). CLSM was used for the surface analysis of the extraction site. The measurements taken with CLSM were: (1) mesio‐distal distance, (2) alveolar ridge thickness, and (3) vestibular and lingual alveolar ridge height. Results of study cast scanning at baseline, 1 and 3 months after tooth extraction, with CLSM are well‐detailed images of postextraction areas. The CLSM technique used in study casts is a valid method to measure the dimensional changes that happen in the edentulous area after tooth extraction. This technique allows the evaluation of changes in mesio‐distal distance, thickness of the alveolar ridge and alveolar ridge height based on the measurements on the alveolar contours. Microsc. Res. Tech., 2011. © 2011 Wiley‐Liss, Inc.     

The use of simple image analysers in lung morphometry.

We have used several commercially available image analysers to measure the inter-alveolar wall distance on histologic sections of animal and human lungs. Inter- and intra-observer variations using these machines are small, and very similar results are obtained using different machines. The recorded result depends on the magnification used since higher resolution produces more intercepts. The plane in which the measurement is made affects the result, representing the difference in shrinkage from fixed tissue to stained slides in the horizontal and vertical planes. Machine-measured inter-alveolar wall distances are smaller than those measured by humans because of the spurious intercepts made by the image analysers. However, the measurements are highly correlated and human-measured inter-alveolar wall distances can be closely predicted from the machine-measured ones. The number of alveoli per unit volume is less well predicted from the average inter-alveolar wall distance. Image analysers may possibly play a useful role in making similar simple measurements in other organs.     

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.     

Estimating multivariate linear profiles change point with a monotonic change in the mean of response variables

In this paper, a maximum likelihood estimator (MLE) is developed to estimate change point when monotonic change occurs in the mean of response variables in multivariate linear profiles in Phase II. Performance of the proposed estimator is compared to the performance of step change and linear drift estimators under different shift types. To conduct comparisons, accuracy and precision of the estimators are considered as performance measures. Simulation results show that the average change point estimate of the proposed estimator is less biased than the one for the step and drift estimators in small shifts, because \( {\overline{\widehat{\tau}}}_{\mathrm{monotonic}} \) is closer to the actual change point of 25 in small shifts. Also, the precision of the proposed estimator is approximately better than that of the step and drift estimators, because its precision values are higher. Hence, the proposed estimator has better performance in terms of both accuracy and precision in small shifts under any kinds of increasing changes. In single step and linear drift changes when the magnitude of shifts increases, the accuracy and precision of their corresponding estimators become better than the accuracy and precision of the proposed estimator. However, the proposed estimator has an advantage that it does not require assumptions about the change type, and its only assumption is that the mean of the response variables changes in an increasing manner. Additional evaluations on the effect of smoothing constant show that with smaller values of the smoothing constant, the proposed change point estimator has less biased estimates and smaller values of mean square error in small shifts rather than the step and drift estimators, leading to a better performance. Also, the larger values of smoothing constant lead to the better performance of the monotonic estimator in large shifts. Finally, the application of the proposed estimator is shown through a real case in the calibration process in the automotive industry.     

Edge-corrected estimators of the nearest-neighbour distance distribution function for three-dimensional point patterns

In multiphase systems consisting of ‘particles’ embedded in a matrix the three-dimensional spatial distribution of the particles may represent important structural information. In systems where the matrix is transparent or translucent recent developments in microscopy allow the three-dimensional location of particles to be recorded. Using these data a spatial statistical, or second-order stereological, analysis can be carried out. In second-order stereology functions of interparticle distances are used as summary statistics of the spatial distributions. These statistics show whether the particles are randomly arranged or, more commonly, either clustered together or inhibited from close approach to each other. This paper focuses on the estimation of one of these spatial statistics, the nearest-neighbour distance distribution function or G-function. In practice, estimation of the G-function is plagued by an ‘edge-effect’ bias introduced by the sampling process itself. There exist a number of G-function estimators that tackle this edge effect problem; for single sample ‘bricks’ it can be shown that these estimators become increasingly accurate as the brick size increases, i.e. they are consistent. However, in many practical cases the size of a sampling brick is fixed by experimental constraints and in these circumstances the only way to increase sample size is to take replicated sampling regions. In this paper we review a number of existing G-function estimators and propose a new estimator. These estimators are compared using the criterion of how well they overcome the edge-effect when they are applied to replicated samples of a fixed size of brick. These comparisons were made using Monte-Carlo simulation methods; the results show that three existing estimators are clearly unsuitable for estimating the G-function from replicated sample bricks. Of the other estimators the recommended estimator depends upon the number of replicates taken; however, we conclude that if a total of more than about 800 points are analysed then the bias in the pooled estimate of the G-function can be reduced to tolerable levels.     

Using biased image analysis for improving unbiased stereological number estimation – a pilot simulation study of the smooth fractionator

The smooth fractionator was introduced in 2002. The combination of a smoothing protocol with a computer‐aided stereology tool provides better precision and a lighter workload. This study uses simulation to compare fractionator sampling based on the smooth design, the commonly used systematic uniformly random sampling design and the ordinary simple random sampling design. The smooth protocol is performed using biased information from crude (but fully automatic) image analysis of the fields of view. The different design paradigms are compared using simulation in three different cell distributions with reference to sample size, noise and counting frame position. Regardless of clustering, sample size or noise, the fractionator based on a smooth design is more efficient than the fractionator based on a systematic uniform random design, which is more efficient than a fractionator based on simple random design. The fractionator based on a smooth design is up to four times more efficient than a simple random design.     

基于模糊径向基函数神经网络的航空煤油干点估计

研究以模糊聚类和径向基函数网络结合的模糊径向基函数网络ＦＲＢＦＮ,并用主元分析对高维输入变量进行预处理,降低了模型的输入变量维数,进而构造基于ＰＣＡ－ＦＲＢＦＮ的估计模型,这一方法通过对加氢裂化装置分馏塔航空煤油干占估计得到验证。