Search

Professor Hans Jørgen G. Gundersen MD, DMSc (1943–2021) was a pioneering stereologist whose work has inspired and influenced researchers across the world for almost half a century. He was a charismatic character and one of the founding fathers of modern stereology, whose achievements and contributions are fondly remembered below by colleagues and co-workers. It was an enormous pleasure to be in his company and although future generation will miss this opportunity, his work will live on, to inspire and influence future generations of researchers.

The direct superimposition of a standard test grid of congruent quadrats onto an image bearing a population of particles exhibiting perspective artifacts, tends to increase the variance of the population size estimator, because the quadrat contents become unbalanced. If the quadrats are transformed according to the same projection mechanism affecting the particles, however, then the variance is restored into moderate values. Our purpose was to provide exact, easily programmable equations for the relevant transform.

Intersection formulae of Croton type for general geometric probes are well known in integral geometry. For the special case of cylinders with non necessarily convex direktrix, however, no equivalent formula seems to exist in the literature. We derive such formula resorting to motion invariant probability elements associated with test systems, instead of using a traditional approach. Because cylinders are seldom used as probes in stereological practice, however, this note is mainly of a theoretical nature.

A surface area estimator for three-dimensional convex sets, based on the invariator principle of local stereology, has recently motivated its generalization by means of new rotational Crofton-type formulae using Morse theory. We follow a different route to obtain related formulae which are more manageable and valid for submanifolds in constant curvature spaces. As an application, we obtain a simplified version of the mentioned surface area estimator for non-convex sets of smooth boundary. Work was supported by the UJI project P11B2012-24 and the PROMETEOII/2014/062 project.

Exact Cavalieri estimation amounts to zero variance estimation of an integral with systematic observations along a sampling axis. A sufficient condition is given, both in the continuous and the discrete cases, for exact Cavalieri sampling. The conclusions suggest improvements on the current stereological application of fractionator-type sampling.

Design unbiased estimation of population size by stereological methods is an efficient alternative to automatic computer vision methods, which are generally biased. Moreover, stereological methods offer the possibility of predicting the error variance from a single sample. Here we explore the statistical performance of two alternative variance estimators on a dataset of 26 labelled crowd pictures. The empirical mean square errors of the variance predictors are compared by means of Monte Carlo resampling.

A coherent set of explicit approximations is presented for the variance of planar area and volume estimators obtained under systematic geometric sampling. For planar objects (e.g. a land plot, or a tissue section), sampling is considered with test systems of points, lines, segments, stripes, or quadrats. For three dimensional objects analogous probes are considered. For the formulae to apply the design has to be uniform random (which suffices to estimate planar area or volume only) and also isotropic. The formulae are based on G. Matheron’s transitive theory. A synthetic example on the estimation of canopy cover is explained in detail.

It is shown that, for a three dimensional particle (namely an arbitrary compact domain with piecewise smooth boundary in R^3) the mean wedge volume defined on a given pivotal section is equal to the average nucleator estimator of the particle volume defined on that section. Further, if the particle is convex and it contains the pivotal point, then the flower area of a given pivotal section equals the average surfactor estimator defined on that section. These results are intended to throw some light on the standing conjecture that the functional defined on a pivotal section according to the invariator has a unique general expression. As a plus, the former result leads to a computational formula for the mean wedge volume of a convex polygon which is much simpler than the one published recently, and it is valid whether the fixed pivotal point is interior or exterior to the particle.

The pivotal estimators of the surface area and the volume of a generic "particle" are based on a point sampled test line on an isotropic pivotal plane through a fixed point called the pivotal point. The purpose of this paper is to compare the precision of the pivotal estimator of surface area against the surfactor on the one hand, and of the pivotal estimator of volume against the nucleator on the other. For the sake of a tractable but informative model we compute exact variances for a spherical particle with an eccentric nucleolus.

Summary Recently, a new decomposition has been found for the motion invariant density of straight lines in , with applications in stereology. The new principle, called the invariator, leads to new rotational formulae which express the surface area and the volume of a bounded subset (called a ‘particle’) in terms of an observable functional defined in an isotropically oriented section (called a pivotal section) through a fixed point (called the pivotal point). The results have been extended to intrinsic volumes of manifolds in general space forms. The purpose of this paper is to present new results and computational formulae for three-dimensional particles. Explicit estimators are obtained for a convex polyhedral particle with a pivotal point in its interior, in terms of the coordinates of the vertices of the pivotal section. The results are applied to a population of polyhedral grains from a cemented carbide which was studied earlier by alternative methods.

First moment formulae for the intersection between a fixed geometric object and one moving according to the kinematic density constitute the basis of stereology and they are essentially well known. However, the corresponding second moment formulae are less known; some of them are scattered in the literature and their proofs are often absent. Such formulae may be useful to compute exact variances when the objects involved have a known geometric shape on the one hand, and to obtain practical predictors of the estimation variance on the other. The purpose of this paper is to present a partly new, coherent set of second moment formulae for a fixed compact set intersected by mobile flats, bounded probes, or test systems endowed with the corresponding invariant measures, or by a stationary manifold process. We thereby contribute to a better understanding of the so-called variance ‘paradoxes’, arising when a probe performs better than a higher-dimensional one. For completeness, we include the necessary tools in ...

Integral section formulae for totally geodesic submanifolds (planes) intersecting a compact submanifold in a space form are available from appropriate representations of the motion invariant density (measure) of these planes. Here we present a new decomposition of the invariant density of planes in space forms. We apply the new decomposition to rewrite Santaló's sectioning formula and thereby to obtain new mean values for lines meeting a convex body. In particular we extend to space forms a recently published stereological formula valid for isotropic plane sections through a fixed point of a convex body in R3. © 2008 Elsevier B.V. All rights reserved.

Summary We present a collection of variance models for estimators obtained by geometric systematic sampling with test points, quadrats, and n-boxes in general, on a bounded domain in n-dimensional Euclidean space ℝn, n = 1, 2, ... , and for systematic rays and sectors on the circle. The approach adopted − termed the filtering approach − is new and different from the current transitive approach. This report is only preliminary, however, because it includes only variance models in terms of the covariogram of the measurement function. The estimation step is in preparation.

A general variance predictor is presented for a Cavalieri design with slices of an arbitrary thickness t >or= 0. So far, prediction formulae have been available either for measurement functions with smoothness constant q = 0, 1, ... , and t >or= 0, or for fractional q in [0, 1] with t = 0. Because the possibility of using a fractional q adds flexibility to the variance prediction, we have extended the latter for any q in [0, 1] and t >or= 0. Empirical checks with previously published human brain data suggest an improved performance of the new prediction formula with respect to the hitherto available ones.

Sampling designs dictated by stereology have proven very useful in recent years to estimate in situ the total number of deposited particles, or of macrophages, in different lung compartments at the light microscopical level. The sampling methods are based on parallel slabs which are subsequently subsampled by disectors. The resulting number estimators are unbiased irrespective of tissue shrinkage or swelling, and they are readily applicable in other contexts (notably in neuroscience). Several variants of the design are available, however, and, although they all yield the same number estimates, their precision, and the mathematical prediction of it, vary among the different estimators and are subjected to theoretical improvement. The present paper constitutes a detailed survey of the latest advances, and it illustrates methods and formulae alike by way of a real example stemming from an earlier study on particle retention and clearance.

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 estimator Qˆ 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(Qˆ) 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. *Current address: Centre for Medical Statist...

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.

The purpose of this study was to compare magnetic resonance imaging (MRI) against physical sectioning techniques to estimate the volume of human cerebral hemisphere compartments (cortex, subcortex, and their union, called "total"). The volume of these compartments was estimated postmortem for six human subjects from MRI virtual sections and from physical sections using the Cavalieri design with point counting. Cursory paired t tests revealed no significant differences between the two methods for any of the three compartments considered, although P = 0.06 for the subcortex. A sharper analysis incorporating recent error prediction formulae revealed a significant discrepancy between the two methods in the estimation of subcortex and total volume for three of the specimens. Yet, none of these analyses is adequate to detect possible biases. The incorporation of an explanatory variable, namely hemisphere weight, and the adoption of a specific gravity rho = 1.04 g/cm(3) for the material, enabled us to carry out an allometric analysis for the total compartment which revealed a significant bias of the MRI data. The new error prediction formulae are illustrated by way of example, and their accuracy is checked by a resampling experiment on a data set of 274 MRI sections.

Geometric sampling, and local stereology in particular, often require observations at isotropic random directions on the sphere, and some sort of systematic design on the sphere becomes necessary on grounds of efficiency and practical applicability. Typically, the relevant probes are of nucleator type, in which several rays may be contained in a sectioning plane through a fixed point (e.g. through a nucleolus within a biological cell). The latter requirement considerably reduces the choice of design in practice; in this paper, we concentrate on a nucleator design based on splitting the sphere into regions of equal area, but not of identical shape; this design is pseudosystematic rather than systematic in a strict sense. Firstly, we obtain useful exact representations of the variance of an estimator under pseudosystematic sampling on the sphere. Then we adopt a suitable covariogram model to obtain a variance predictor from a single sample of arbitrary size, and finally we examine the prediction accuracy by way of simulation on a synthetic particle model.

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.

The theory of Cavalieri sampling - or systematic sampling along an axis - has received a recent impetus. The error variance may be represented by the sum of three components, namely the extension term, the 'Zitterbewegung', and higher order terms. The extension term can be estimated from the data, and it constitutes the standard variance approximation used so far. The Zitterbewegung oscillates about zero, and neither this nor higher order terms have hitherto been considered to predict the variance. The extension term is always a good approximation of the variance when the number of observations is very large, but not necessarily when this number is small. In this paper we propose a more general representation of the variance, and from it we construct a flexible extension term which approximates the variance satisfactorily for an arbitrary number of observations. Furthermore, we generalize the current connection between the smoothness properties of the measurement function (e.g. the section area function of an object when the target is the volume) and the corresponding properties of its covariogram to facilitate the computation of the new variance approximations; this enables us to interpret the behaviour of the variance from the 'overall shape' of the measurement function. Our approach applies mainly to measurement functions whose form is known analytically, but it helps also to understand the behaviour of the variance when the measurement function is known at sufficiently many points; in fact, we illustrate the concepts with both synthetic and real data.

Useful approximations have been developed along the years to predict the precision of systematic sampling for measurable functions of a bounded support in ℝd. Recently, the theory of systematic sampling on ℝ has received a thrust. In geometric sampling, design based unbiased estimators exist, however, which imply systematic sampling on the circle (𝕊1) and the semicircle (ℍ1); the planimeter estimator of an area, or the Buffon-Steinhaus estimator of curve length in the plane constitute popular examples. Over the last two decades, many other estimators of geometric measures have been obtained which imply systematic sampling also on the sphere (𝕊2). In this paper we adapt the theory available for non-periodic functions of bounded support on ℝ to periodic functions, and thereby to 𝕊1and ℍ1, and we obtain new estimators of the corresponding variance approximations. Further we consider - we believe for the first time - the problem of predicting the precision of systematic sampling in 𝕊2. The paper starts with a historical perspective, and ends with suggestions for further research.

Recent theory has been developed to estimate volume from a systematic sample of tissue slices of a given thickness and to predict the corresponding error. Our goal was to check the error prediction formulas by resampling and to determine the minimum number of MR slices required to estimate the volumes of the cerebrum and of the compartments of gray matter (GM) and white matter (WM) with prescribed errors.Our working data set comprised the GM and WM segmentations obtained from a paradigmatic high signal-to-noise ratio 3D spoiled GRASS MR volume data set for a single healthy human subject. The data were classified using a fuzzy clustering minimum distance algorithm. We thereby obtained a stack of 183 serial coronal slices of 1 mm thickness encompassing the whole cerebrum. Empirical resampling was carried out using the corresponding data vectors, and the theoretical error predictors were thereby checked for slice thicknesses of 1, 3, 9, and 27 mm, with a distance of 45 mm between slice midplanes.Irrespective of slice thickness, a minimum of 3, 5, and 10 slices provided estimates of the true total volume of GM and WM in the cerebrum with coefficients of error (CEs) of 10, 5, and 3%, respectively, where CE(V)% = 100 x SE(V)/V. For the cerebrum, a minimum of two, three, and four slices were required for CEs of the same precision.In combination with high signal-to-noise ratio and enhanced tissue contrast, Cavalieri slices are the most appropriate for MRI, they supply unbiased and highly efficient volume estimates of brain compartments. For a given number of slices, CE(V) decreases rapidly when the slices are thicker than the gaps between them; when the slices are thinner than the gaps, then CE(V) is similar to that in the situation when the slice thickness is zero.

In design stereology, and in the context of geometric sampling in general, the problem often arises of estimating the integral of a bounded non-random function over a bounded manifold D ⊂ ℝ n by systematic sampling with geometric probes. Variance predictors, often based on Matheron's theory of regionalized variables, are available when the relevant function is sampled at the points of a grid intersecting D, but not when the dimension of the probes is greater than zero. For instance, the volume of a bounded object may be estimated using parallel systematic planes, which amounts to sampling on ℝ1 with systematic points, or using parallel systematic slabs of thickness t > 0, which amounts to sampling on ℝ1 with non-overlapping systematic segments of length t > 0. Useful variance predictors exist for the former case, but not for the latter. In this paper we set out a general scheme to predict estimation variances when the dimension of either D, or of the probes, is n. We make some progress when both dimensions are equal to n, and obtain explicit results for n = 1 (e.g. for systematic slice sampling). We check and illustrate our results for the volume estimators of ellipsoids and of rat lung.

In design-based stereology, fixed parameters (such as volume, surface area, curve length, feature number, connectivity) of a non-random geometrical object are estimated by intersecting the object with randomly located and oriented geometrical probes (e.g. test slabs, planes, lines, points). Estimation accuracy may in principle be increased by increasing the number of probes, which are usually laid in a systematic pattern. An important prerequisite to increase accuracy, however, is that the relevant estimators are unbiased and consistent. The purpose of this paper is therefore to give sufficient conditions for the unbiasedness and strong consistency of design-based stereological estimators obtained by systematic sampling. Relevant mechanisms to increase sample size, compatible with stereological practice, are considered.

A version of the Rao–Blackwell theorem is shown to apply to most, but not all, stereological sampling designs. Estimators based on random test grids typically have larger variance than quadrat estimators; random s-dimensional samples are worse than random r-dimensional samples for s < r. Furthermore, the standard stereological ratio estimators of different dimensions are canonically related to each other by the Rao–Blackwell process. However, there are realistic cases where sampling with a lower-dimensional probe increases efficiency. For example, estimators based on (conditionally) non-randomised test point grids may have smaller variance than quadrat estimators. Relative efficiency is related to issues in geostatistics and the theory of wide-sense stationary random fields. A uniform minimum variance unbiased estimator typically does not exist in our context.

Summary The classical methods for estimating the volume of human body compartments in vivo (e.g. skin-fold thickness for fat, radioisotope counting for different compartments, etc.) are generally indirect and rely on essentially empirical relationships — hence they are biased to unknown degrees. The advent of modern non-invasive scanning techniques, such as X-ray computed tomography (CT) and magnetic resonance imaging (MRI) is now widening the scope of volume quantification, especially in combination with stereological methods. Apart from its superior soft tissue contrast, MRI enjoys the distinct advantage of not using ionizing radiations. By a proper landmarking and control of the scanner couch, an adult male volunteer was scanned exhaustively into parallel systematic MR ‘sections’. Four compartments were defined, namely bone, muscle, organs and fat (which included the skin), and their corresponding volumes were easily and efficiently estimated by the Cavalieri method: the total section area of a compartment times the section interval estimates the volume of the compartment without bias. Formulae and nomograms are given to predict the errors and to optimize the design. To estimate an individual's muscle volume with a 5% coefficient of error, 10 sections and less than 10min point counting (to estimate the relevant section areas) are required. Bone and fat require about twice as much work. To estimate the mean muscle volume of a population with the same error contribution, from a random sample of six subjects, the workload per subject can be divided by √6, namely 4 min per subject. For a given number of sections planimetry would be as accurate but far more time consuming than point counting.

A new rotational version of Crofton's formula is derived for the intrinsic volumes of a domain Y in a space form. More precisely, a functional is defined on the intersection between Y and a totally geodesic submanifold (plane) through a fixed point, such that the rotational average of this functional is equal to the intrinsic volumes of Y. Particular cases of interest in stereology are considered for the Euclidean case. © 2009 Elsevier Inc. All rights reserved.

The recently developed method of total vertical projections is illustrated to estimate the total dendritic length of a human Substantia Nigra neuron. Next, the length of the different orders of dendritic branches, and the mean segment length for each order--commonly regarded as important parameters in neuron physiology--are also estimated. Finally, it is shown how to estimate the mean dendritic length in a population of neurons from vertical slices of arbitrary and unknown thickness. Being unbiased and highly efficient, the proposed methods offer interesting alternatives to current procedures used for the metric analysis of neuron arborizations.

SUMMARY A new method is proposed to estimate the total length of a bounded, isolated linear feature in three dimensions from ‘total vertical projections’, obtained by rotating the curve about a fixed axis (arbitrarily called ‘vertical’) and projecting it onto a fixed vertical plane. No sections are required. Properly stained and embedded neuron dendrites, mycelial trees, fluorescent cytoskeletal filaments within a cell, etc., are candidate specimens for the method, especially in combination with the new devices for non-invasive three-dimensional microscopy. It is necessary that the specimen curve is rigid (i.e. of constant shape), that its length density is not too high (so that overlapping effects are not important) and that the embedding medium is fairly transparent. Given these requirements, the method can be very accurate and convenient to use, as is exemplified here.

A new stereological method has recently been developed to estimate the total length of a bounded curve in 3D from a sample of projections about a vertical axis. Unlike other methods based on serial section reconstructions, the new method is unbiased (i.e., it has zero systematic error). A basic requirement, not difficult to fulfill in many cases, is that the masking of one structure by another is not appreciable. The application of the new method to real curvilinear structures using a clinical magnetic resonance (MR) imager is illustrated. The first structure measured was a twisted water-filled glass tube of known length. The accuracy of the method was assessed: With six vertical projections, the tube length was measured to within 2% of the true value. The second example was a living bonsai tree, and the third was a clinical application of MR angiography. The possibility of applying the method to other scientific disciplines, for example, the monitoring of plant root growth, is discussed.

The invariator is a method to generate a test line within an isotropically oriented plane through a fixed point, in such a way that the test line is effectively motion invariant in three dimensional space. Generalizations exist for non Euclidean spaces. The invariator design is convenient to estimate surface area and volume simultaneously. In recent years a number of new results have appeared which call for an updated survey. We include two new estimators, namely the a posteriori weighting estimator for surface area and volume, and the peak-and-valley formula for surface area.

Summary Cavalieri sections — and more recently Cavalieri slices, especially in combination with non-invasive scanning — are widely used to estimate volumes. Physical Cavalieri slices are also increasingly used to estimate neuron numbers via the optical fractionator. In either case, the prediction of the error variance is important to assess optimal sample sizes. The error variance consists of two components, one due to the variation among the true contents of sections or slices, and the other due to local or ‘nugget’ errors. The latter may arise for instance when estimating section areas by point counting, or when counting discrete particles in slices or disectors. In this paper, a fairly comprehensive set of prediction formulae is presented to separate both variance components.

Cavalieri sampling is often used to estimate the volume of an object with systematic sections a constant distance T apart. The variance of the corresponding estimator can be expressed as the sum of the extension term (which gives the overall trend of the variance and is used to estimate it), the 'Zitterbewegung' (which oscillates about zero) and higher order terms. The extension term is of order T2m+2 for small T, where m is the order of the first non-continuous derivative of the measurement function f, (namely of the area function if the target is the volume). A key condition is that the jumps of the mth derivative f (m) of f are finite. When this is not the case, then the variance exhibits a fractional trend, and the current theory fails. Indeed, in practice the mentioned trend is often of order T2q+2, typically with 0

In design-based stereology, fixed parameters (such as volume, surface area, curve length, feature number, connectivity) of a non-random geometrical object are estimated by intersecting the object with randomly located and oriented geometrical probes (e.g. test slabs, planes, lines, points), [4], [5], [8], [11], [12]. Estimation accuracy may in principle be increased by increasing the number of probes, which are usually laid in a systematic pattern, [1], [2], [3], [7], [9], [10]. An important prerequisite to increase accuracy, however, is that the relevant estimators are unbiased and consistent. Our purpose is therefore to give sufficient conditions for the unbiasedness and strong consistency of design-based stereological estimators obtained by systematic sampling.

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.

A 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.

A theory for volume estimation from independent, unbounded plane probes is available. In practice, however, probes cannot be unbounded and independent at the same time, hence the interest of systematic sectioning. Previous empirical studies on real specimens showed that the mean square error of a volume ratio estimator obtained from m systematic sections behaved roughly as m–3 for small m. This drastic increase in efficiency with respect to independent sectioning prompted us to develop some theoretical models in this paper. We study specimens consisting of two ellipsoids in Rn. In virtue of an invariance result, an ellipsoid–ellipsoid model can be studied via a simple model consisting of two concentric n-balls. Explicit results are obtained for n = 1 and n = 3. In the latter case the bias and the mean square error of the relevant volume ratio estimator are both shown to be of O (m –4).

Some properties of the Voronoi polygon associated with a moving particle in a regular tessellation are studied. In particular, the area of the variable polygon has played a relevant role in previous studies on plant competition, animal predation, adjustment of the sex ratio in bisexual populations and the gregarious behaviour of animal species.

SUMMARY A distribution-free method is presented for unfolding the size distribution of an aggregate of disjoint opaque spherical particles in a translucent medium, from the circular profiles observed in the orthogonal projection of sections of a certain thickness through the medium. This situation pertains especially to light and transmission electron microscopy. The model accounts for overprojection (or ‘Holmes effect’) due to section thickness, and for two forms of truncation (or ‘missing profiles’ mechanisms), simultaneously. The profile data may be grouped into an arbitrary class system, chosen by the user, and the procedure can be programmed in a medium-size desk calculator. In order to incorporate the above features, however, we had to choose a finite-difference unfolding algorithm, the advantages and disadvantages of which are discussed early in the paper in the context of the main alternative unfolding procedures hitherto available. The performance of the method is tested against three sets of real data previously analysed by a maximum likelihood method (Keidinget al., 1972).

SUMMARY The development of stereological methods for the study of dilute phases of particles, voids or organelles embedded in a matrix, from measurements made on plane or linear intercepts through the aggregate, has deserved a great deal of effort. With almost no exception, the problem of describing the particulate phase is reduced to that of identifying the statistical distribution—histogram in practice—of a relevant size parameter, with the previous assumption that the particles are modelled by geometrical objects of a constant shape (e.g. spheres). Therefore, particles exhibiting a random variation about a given type of shape as well as a random variation in size, escape previous analyses. Such is the case of unequiaxed particles modelled by triaxial ellipsoids of variable size and eccentricity parameters. It has been conjectured (Moran, 1972) that this problem is indetermined in its generality (i.e. the elliptical sections do not furnish a sufficient information which permits a complete description of the ellipsoids). A proof of this conjecture is given in the Appendix. When the ellipsoids are biaxial (spheroids) and of the same type (prolate or oblate), the problem is identifiable. Previous attempts to solve it assume statistical independence between size and shape. A complete, theoretical solution of the spheroids problem—with the independence condition relaxed—is presented. A number of exact relationships—some of them of a striking simplicity—linking particle properties (e.g. mean-mean caliper length, mean axial ratio, correlation coefficient between principal diameters, etc.) on the one hand, with the major and minor dimensions of the ellipses of section on the other, emerge, and natural, consistent estimators of the mentioned properties are made easily accessible for practical computation. Finally, the scope and limitations of the mathematical model are discussed.

SUMMARY At present a model-free, design-based theory of unbiased estimation, and a model-based one of linear unbiased estimation of minimum variance, are available for stereology. The main developments rest upon the nested scheme {section (quadrat)}, whence the raw data are expressible in terms of area, length and number. The main aim of this paper is to complete the available model-based theory by introducing the step in which sections are analysed by point-counting via coherent test systems (CTSs). Using this development, the stereologist should be able to handle raw point and intersection counts optimally, in order to find the best estimator of a ratio R in a given specimen in a wide range of circumstances. The latter include, for instance, the use of different CTSs on different sections and of double CTSs on each section, as well as the case—(not uncommon in electron microscopy)—in which different sections from the same sample are observed at slightly different magnifications but analysed by quadrats (via automatic or semi-automatic image analysers, for instance), or CTSs of fixed sizes. The main conclusion pertaining to the latter case is that the estimators obtained via section-wise magnification corrections are in general superior to those corrected via a global, average magnification. In order to illustrate the methodology, a synthetic numerical example, and a real one, are given.

Certain assumptions on the movement and signalling of a species, as well as on the dimensionality of the space, give rise to a variety of “spatial models”, and yield some conclusions relative to the gregarious behaviour of that species and the adjustment of the sex ratio in bisexual populations. In the former context, Hamilton's (1971) arguments are affected. In the latter, Kalmus & Smith's (1960) treatment is discussed and elaborated to include pertinent new factors. This is done in a quantitative manner by applying a number of occupancy models. It appears that the sex ratio which maximizes the expected number of fertilized females will be biased towards the mobile sex, subject to the relative fecundity of the males. It is argued that the optimal sex ratio constitutes a socially advantageous but individually disadvantageous character, and may survive under certain conditions.