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.     

A simple and efficient alternative to implementing systematic random sampling in stereological designs without a motorized microscope stage

When properly applied, stereology is a very robust and efficient method to quantify a variety of parameters from biological material. A common sampling strategy in stereology is systematic random sampling, which involves choosing a random sampling relevant objects start point outside the structure of interest, and sampling at sites that are placed at pre‐determined, equidistant intervals. This has proven to be a very efficient sampling strategy, and is used widely in stereological designs. At the microscopic level, this is most often achieved through the use of a motorized stage that facilitates the systematic random stepping across the structure of interest. Here, we report a simple, precise and cost‐effective software‐based alternative to accomplishing systematic random sampling under the microscope. We believe that this approach will facilitate the use of stereological designs that employ systematic random sampling in laboratories that lack the resources to acquire costly, fully automated systems.     

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.     

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.     

Estimation of the number of stellate cells in a liver with the smooth fractionator

To better evaluate the activation and proliferative response of hepatic stellate cells (HSC) in hepatic fibrosis, it is essential to have sound quantitative data in non‐pathological conditions. Our aim was to obtain the first precise and unbiased estimate of the total number of HSC in the adult rat, by combining the optical fractionator, in a smooth sampling design, with immunocytochemistry against glial fibrillary acidic protein. Moreover, we wanted to verify whether there was sufficiently relevant specimen inhomogeneity that could jeopardize the high expected estimate precision when using the smooth fractionator design for HSC. Finally, we wanted to address the question of what sampling scheme would be advisable a priori for future studies. Microscopical observations and quantitative data provided no evidence for inhomogeneity of tissue distribution of HSC. Under this scenario, we implemented a baseline sampling strategy estimating the number (N?) of HSC as 207E⁰⁶ (CV = 0.17). The coefficient of error [CE(N?)] was 0.04, as calculated by two formerly proposed approaches. The biological difference among animals contributed ? 95% to the observed variability, whereas methodological variance comprised the remaining 5%. We then carried out a half reduction of sampling effort, at the level of both sections and fields. In either occasion, the CE(N?) values were low (? 0.05) and the biological variance continued to be far more important than methodological variance. We concluded that our baseline sampling (counting 650–1000 cells/rat) would be appropriate to assess the lobular distribution and the N? of HSC. However, if the latter is the only parameter to be estimated, around half of our baseline sampling (counting 250–600 cells/rat) would still generate precise estimates [CE(N?) < 0.1], being in this case more efficient to reduce the number of sections than to reduce the sampled fields.     

Precision of the fractionator from Cavalieri designs

A popular procedure to predict the variance of the fractionator consists in splitting the initial collection of fragments into two subsets, in order to use the corresponding particle counts (or any other pertinent measure), in the calculation. The current formula does not account for local or ‘nugget’ errors inherent in the estimation of fragment contents, however. Moreover, it does not account for the fact that the contribution of the variability between fragments or slices should rapidly decrease as the sampling fraction increases. For these reasons, an update to the formula is overdue. It should be stressed, however, that the formula applies to Cavalieri slices designs – its application for arbitrary partition designs is therefore not warranted.     

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.     

A simple technique to measure the movements of the microscope stage along the x and y axes for stereological methods

Measurements of microscope stage movements in the x and y directions are of importance for some stereological methods such as the optical disector and optical fractionator. The length of stage movements can be measured with great precision and accuracy using a suitable motorized stage, which is generally a computer-assisted instrument. This type of equipment is generally too expensive for and not readily available in many laboratories. This paper describes a simple method to measure the movements of the microscope stage along the x and y directions, which can be used for purposes such as systematic uniform random sampling. It needs a microscope attachment consisting of two dial indicators; one of them is used to measure the amount of stage movement along the x -axis and the other measures the amount of movement along the y -axis. Movements of the stage on the micrometre-scale can be measured easily using this device.     

Characterization of nickel silicides using EELS-based methods

We present an unbiased estimator of the total number of alveolar structures distal to the transition from a bronchiole to an alveolar duct system (‘ventilatory units’, VUs). In species without respiratory bronchioles, including mice, the number of VUs is equivalent to the number of acini. The acinus is a functional unit of gas exchange, defined as a parenchymal unit distal to a terminal bronchiole in which all airways contain alveoli and thus participate in gas exchange. The estimator combines two different estimators of the number of VUs: (1) an estimator derived from the Euler number of all the openings of the bronchial tree and (2) an estimator derived from direct counts of topological changes occurring at bronchiole‐alveolar duct junctions. Combining the two estimators eliminates the requirement to be able to identify even vanishingly small pieces of bronchial tissue in physical disectors. We implemented the fractionator estimator in five adult mice lungs using physical fractionators with varying but known sampling fractions (Horvitz–Thompson estimator). We obtained total values of about 4200 VUs (CV = 0.05) in 21‐day‐old and 4480 (CV = 0.06) in 69‐day‐old animals. Being fractionator estimates, these total numbers are independent of shrinkage. The densities of VUs per unit volume of tissue (values corrected for tissue shrinkage) were similar in left and right lungs.     

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.     

Optimizing sampling efficiency of stereological studies in biology: or ‘Do more less well!‘

The aim of the sampling design for stereology is to obtain the maximal amount of quantitative structural information at a given total cost or effort. Principles of such optimal designs are discussed and methods for generating them are illustrated by a biological example. In general, the variation between different individuals—the biological variation—is the major determinant of overall efficiency, whereas the variation between single microscopic features is unimportant. It follows that the expenditure of time and/or money in order to increase the precision of the individual measurements is irrational in almost all studies where the emphasis is on the biological results.     

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.     

Stereological estimation of integral mixed curvature with application

This article addresses a bounded system of compact smooth surfaces in three-dimensional space. Recently, an unbiased estimator of the integral mixed curvature of the given system based on a vertical sampling design has been proposed. The aims of the present paper are: (i) to show that the proposed estimator may have an infinite variance; (ii) to suggest a modification that has better statistical properties; (iii) to extend the point estimator to a function that monitors the curvature along the gradient microstructures; (iv) to present an application of the method to real microscopic images.     

A handheld support system to facilitate stereological measurements and mapping of branching structures

BranchSampler is a system for computer-assisted manual stereology written for handheld devices running Windows CE. The system has been designed specifically to streamline data collection and optimize sampling of tree-like branching structures, with particular aims of reducing user errors, saving time, and saving data in formats suited for further analysis in other software, for example, a spreadsheet. The system can be applied in a wide range of applications, from biomedical science to agriculture and horticulture. It can be applied for sampling nested generations of lung bronchioles and renal arterioles or for collection and optimizing sampling of crops for precision agriculture. Although the system has been designed specifically for sampling branching structures, it is sufficiently flexible to be used for other applications involving nested stereological designs. We describe the system specifications, software and Graphical User Interface development, functionality and application of the handheld system using four examples: (a) sampling monkey lung bronchioles for estimation of diameter and wall thickness (b) sampling rat kidney for estimating number of arteries and arterioles in a specific generation (c) mapping fruit (apple) tree yield in an orchard and (d) estimating the total leaf surface area of chrysanthemum plants in a greenhouse.     

用非均匀采样实现谐波的高精度分析

信号的谐波分析一直是信号处理的基本方法之一。通过对国内外一些提高谐波分析精度的有效方法的研究,从傅里叶算法的基本原理出发,提出了一种新型的谐波分析方法,可以在不增加硬件设备及采样数据的前提下,利用双速率采样的非均匀序列,简单有效地提高谐波分析的精度。计算机仿真证实了该方法的有效性。     

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.