SUMMARY ‘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 Vv, Nv, Sv and the volume‐weighted mean particle volume v̄v, but not Lv. 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.

With the advent of many new tools over the last five years, stereology has become simpler, assumption-free, and more efficient but, at the same time, new terms and concepts have proliferated, which risk overwhelming potential users. The present review is intended to meet the urgent need for a structured classification and evaluation of the newest stereological methods. Being fairly comprehensive, the exposition is necessarily succinct: the reader is referred to selected references for the necessary details and examples.

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K E Y W O R D S . Cavalieri estimator, cycloid test system, fluid displacement, lung volume, pleural surface area, point counting, rabbit lung, stereology, stratification, systematic sampling, vertical sections. S U M M A R YA practical methodology is proposed for the stereological analysis of lung and other organs using recently developed unbiased procedures. This study concentrates on the unbiased estimation of lung volume using Cavalieri's principle compared with the fluid displacement method and measurement of pleural surface area using vertical sections. Furthermore, the proposed design, in addition to the sampling of extensive slices for the initial steps, also allows sampling of vertical sections for light and electron-microscopical stereology.The procedures are described in detail by reference to biological data from the right lungs of four rabbits. We found excellent agreement between estimates of lung volume measured with Cavalieri's principle and those measured by fluid displacement. Pertinent details of the statistical analysis of the sources of variation (namely biological, systematic sectioning and point-counting variation) are given in an appendix.

SUMMARY 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.

Many problems, in stereology and elsewhere (geometric sampling, calculus, etc.) reduce to estimating the integral Q of a non-random measurement function f over a bounded support on R. The unbiased estimatorQ based on systematic sampling of period T > 0 (such as the popular Cavalieri estimator) is usually convenient and highly precise. The purpose of this paper is twofold. First, to obtain a new, general representation of var(Q) in terms of the smoothness properties of f . We extend the current theory, which holds for smoothness constant q ∈ N, to any q ≥ 0; to this end we develop a new version of the Euler-MacLaurin summation formula, making use of fractional calculus. Our second purpose is to apply the mentioned representation to obtain a new variance estimator for any q ≥ 0; we concentrate on the useful case q ∈ [0, 1]. By means of synthetic data, and real data from a human brain, we show that the new estimator performs better than its current alternatives.

A number of either new or recently available stereological methods are described for estimating volume, surface area and number of anisotropic cells. The methods are illustrated with direct reference to the epiphyseal growth plate. Different estimates of a given quantity are obtained by applying alternative methods to the same set of sections, in order to compare the relative merits of the methods. For instance, the surface area of the cells is estimated via the Dimroth-Watson model (which gives a measure of the degree of anisotropy in addition to the surface area estimate) and from vertical sections using cycloid test systems. Cell number is estimated by traditional unfolding methods and by the new disector method. Also, volume-weighted mean cell volume is estimated from vertical sections via point-sampled intercepts using two different kinds of rulers to classify intercept lengths. Finally, nested design statistics is applied to a set of data from twelve animals in order to compare the relative impacts of biological and stereological (sampling) variations on the observed coefficient of error of a group mean estimate. The preferred methods are listed in the final section.

SUMMARYAn evaluation is made of the relative efficiency (precision of the final estimate per unit time of measurement on a given set of sections) of different methods for planar analysis aimed at estimating aggregate, overall stereological parameters (such as Vv, Sv). The methods tested are point‐counting with different densities of test points (4 ≤ PT ≤ 900 per picture), semiautomatic computer image analysis with MOP and automatic image analysis with Quantimet, for obtaining Vv and Sv estimates. One biological sample as well as three synthetic model structures with known coefficients of variation between sections are used. The standard error of an estimate is mainly determined by the coefficient of variation between sampling units (= sections in the present paper) so that measuring each sample unit with a very high precision is not necessary. Automatic image analysis and point‐counting with a 100‐point grid were the most efficient methods for reducing the relative standard errors of the Vv and Sv estimates to equivalent levels in the synthetic models. Using a 64‐point grid was as precise, and about 11 times faster than using a tracing device for obtaining the estimate of Vv in the biological sample.