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.     

Precision of circular systematic sampling

In design stereology, many estimators require isotropic orientation of a test probe relative to the object in order to attain unbiasedness. In such cases, systematic sampling of orientations becomes imperative on grounds of efficiency and practical applicability. For instance, the planar nucleator and the vertical rotator imply systematic sampling on the circle, whereas the Buffon–Steinhaus method to estimate curve length in the plane, or the vertical designs to estimate surface area and curve length, imply systematic sampling on the semicircle. This leads to the need for predicting the precision of systematic sampling on the circle and the semicircle from a single sample. There are two main prediction approaches, namely the classical one of G. Matheron for non‐necessarily periodic measurement functions, and a recent approach based on a global symmetric model of the covariogram, more specific for periodic measurement functions. The latter approach seems at least as satisfactory as the former for small sample sizes, and it is developed here incorporating local errors. Detailed examples illustrating common stereological tools are included.     

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.     

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.     

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.     

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.     

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.     

Sampling designs for stereology*

The purpose of this paper is to propose the necessary sampling techniques for estimating a global parameter defined in a solid opaque specimen (e.g. the total volume of mitochondria in a given liver, the total capillary surface area in a given lung, etc.). The geometry of the specimen often suggests a multi-level or cascade sampling design at different magnifications, whereby the object phase at one level becomes the reference phase in the next level. The final parameter is then estimated as the product of the intermediate ratios with the volume of the specimen, which is estimated independently. Each level can be regarded as an independent sampling design; a given stereological project may be planned in terms of one or more of these designs. Our development is a blend of practical experience and recent theoretical advances on sampling for stereology with well-known sampling techniques previously developed with different purposes in mind.     

Stereology of single objects

Systematic stacks of Cavalieri type, spatial grids, vertical sections and projections, etc., are recent sampling tools for the stereology of single objects, namely of isolated objects that can be orientated and scanned at will in a prescribed way. The increasing use of modern noninvasive scanning devices is facing potential users with the challenge of encompassing the necessary knowledge to implement 'good' stereology. The present paper presents a coherent set of recent stereological methods for single objects in a historical perspective, emphasizing the fact that all relevant techniques — old and new — emanate from a common, relatively small set of basic principles.     

Estimation of surface area and length with the orientator

The orientator is a new technique for the estimation of length and surface density and other stereological parameters using isotropic sections. It is an unbiased, design-based approach to the quantitative study of anisotropic structures such as muscle, myocardium, bone and cartilage. A simple method for the practical generation of such isotropic planes in biological specimens is described. No special technical equipment is necessary. Knowledge of an axis of anisotropy can be exploited to optimize the efficiency. To randomize directions in space, points are selected with uniform probability in a square using various combinations of simple random, stratified random, and systematic random sampling. The point patterns thus produced are mapped onto the surface of a hemisphere. The mapped points define directions of sectional planes in space. The mapping algorithm ensures that these planes arc isotropic, hence unbiased estimates of surface and length density can be obtained via the classical stereological formulae. Various implementations of the orientator are outlined: the prototype version, the orientator-gencrated ortrip, two systematic versions, and the smooth version. Orientator sections can be generated without difficulty in large specimens; we investigated human skeletal muscle, myocardium, placenta, and gut tissue. Slight practical modifications extend the applicability of the method to smaller organs like rat hearts. At the ultrastructural level, a correction procedure for the loss of anisotropic mitochondrial membranes due to oblique orientation relative to the electron beam is suggested. Other potential applications of the orientator in anisotropic structures include the estimation of individual particle surface area with isotropic nucleators, the determination of the connectivity of branching networks with isotropic disectors, and generation of isotropic sections for second-order stereology (three-dimensional pattern analysis).     

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.     

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.     

Computer assisted data collection for stereology: rationale and description of point counting stereology (PCS) software.

The paper describes microcomputer software for point counting stereology. Stereology includes a collection of statistical methods that quantify the images of light and transmission electron microscopy. The methods use test grids placed over images to collect raw data, which includes counts of points, intersections, transections, and profiles. In turn, the counts are included in stereological equations that give estimates of compartmental volumes, surfaces, lengths, or numbers. These parameters describe the composition of a structure in three-dimensional space. The PCS (point counting stereology) System Software III serves as a data collection, storage, and management tool. Users set up point counting protocols without programming, enter data by pressing predefined function (MS-DOS) or alphabetic keys (UNIX), store data in files, select files for analysis, and calculate results as stereological densities. The latest version of the PCS software includes a new user interface and is designed as a research "front end" that can feed data either into the calculation tools of a stereology tutorial (Bolender, 1992, this issue) or into the analysis routines of quantitative morphology databases (Bolender and Bluhm, 1992).     

Sampling problems in an heterogeneous organ: Quantitation of relative and total volume of pancreatic islets by light microscopy

In stereological studies analysis of sampling variances is used for optimizing the sampling design. In organs with a heterogeneous distribution of the phase of interest the analysis of sampling variances can be undertaken only if the observed variance between sections is distributed into the fraction which is due to random variation and the fraction which is due to the heterogeneity. In the present example (pancreatic islet volume estimated by light microscopic point-counting) the density of islets showed a linear increase along the axis of the organ. By analysis of sampling variances it was calculated that the most efficient number of sections (cut perpendicular to the organ) was considerably lower when the isolated contribution from the random variation was considered. The total islet volume was obtained by the product of the fractional islet volume and the pancreatic weight. Analysis of sampling variances of the total islet volume was performed by including the variance contribution from the individual pancreatic weights to the variance of the group mean total islet volume. Due to a negative correlation between the fractional volume and organ weight the total islet volume in the group of animals was more precisely estimated than the fractional islet volume. The methods used for dealing with the heterogeneity of the organ and for estimating sampling variances of total structural quantities generalize to a large number of stereological studies in biology.     

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.     

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.     

The smooth fractionator

A modification of the general fractionator sampling technique called the smooth fractionator is presented. It may be used in almost every situation in which sampling is performed from distinct items that are uniquely defined, often they are physically separated items or clusters like pieces, blocks, slabs, sections, etc. To each item is associated a ‘guesstimate’ or an associated variable with a more‐or‐less close – and possibly biased ? relationship to the content of the item. The smooth fractionator is systematic sampling among the items arranged according to the guesstimates in a unique, symmetric sequence with one peak and minimal jumps. The smooth fractionator is both very simple to implement and so efficient that it should probably always be used unless the natural sequence of the sampling items is equally smooth. So far, there is no theory for the prediction of the efficiency of smooth fractionator designs in general, and their properties are therefore illustrated with a range of real and simulated examples. At the cost of a slightly more elaborate sampling scheme, it is, however, always possible to obtain an unbiased estimate of the real precision and of some of the variance components. The only real practical problem for always obtaining a high precision with the smooth fractionator is specimen inhomogeneity, but that is detectable at almost no extra cost. With careful designs and for sample sizes of about 10, the sampling variation for the primary, smooth fractionator sampling step may in practice often be small enough to be ignored.     

Measuring through the microscope: Development and evolution of stereological methods

Obtaining, by means of microscopy, meaningful measurements pertaining to spatial structures requires methods which allow three-dimensional quantitative information to be derived from the reduced information available on the two-dimensional flat sections of the structure. The most powerful methods to that effect are those of stereology which are based on mathematical principles. This paper reviews the early invention of these methods, which sought to solve practical problems, and their further evolution as more rigorous mathematical foundations were developed. It is demonstrated that stereological methods are essentially sampling methods and that newer trends provide new and sound solutions to old and elusive problems, such as anisotropy or particle number and size.