A new stereological principle for test lines in three-dimensional space

A new principle is presented to generate isotropic uniform random (IUR) test lines hitting a geometric structure in three-dimensional space (3D). The principle therefore concerns the estimation of surface area, volume, membrane thickness, etc., of arbitrary structures with piecewise smooth boundary. The principle states that a point-sampled test line on an isotropic plane through a fixed point in 3D is effectively an invariant test line in 3D. Particular attention is devoted to the stereology of particles, where an alternative to the surfactor method is obtained to estimate surface area. An interesting case arises when the particle is convex. The methods are illustrated with synthetic examples.     

Stereology of isolated objects with the invariator

The invariator is a new stereological design to generate motion invariant test lines in three dimensions on an isotropic plane through a fixed point. The theory has been published recently. The purpose of this paper is to illustrate the application of the invariator on a group of rat brains to estimate brain volume and external surface area. Each brain was first split into its two hemispheres and then embedded into a ball filled with agar following a configuration named the antithetic isector, with the idea of reducing the error variance. After rolling the ball at random it was scanned by magnetic resonance imaging into a stack of parallel systematic sections: this is the isotropic Cavalieri design which, combined with the antithetic isector idea, proves to be very accurate. The invariator used only an equatorial section of the ball, and in the present case the coefficient of error of the volume and surface area estimators of an individual brain was about 30%. As it is design unbiased, the invariator may prove its strength mainly to estimate population means.     

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.     

Analysis of cardiac function by MRI and stereology

Design-based stereology and phase contrast magnetic resonance imaging (MRI) were combined to monitor changes in the volume of the four chambers of the human heart during the cardiac cycle. The data set consisted of 18 adjacent slices (or 'scanning levels') of 0.5 cm thickness, perpendicular to the long axis of the body, and encompassing the whole heart of a healthy volunteer. At each scanning level, a cardiac gated MR image was obtained at each of 16 equally spaced time frames within the cardiac cycle. Given stationarity with respect to time, absence of image artefacts and appropriate definition of chamber boundaries, for each time frame unbiased estimates of total blood volume in the relevant heart chambers were efficiently obtained using the Cavalieri method and point counting. Combined with a proper MRI acquisition, modern stereological methods constitute an efficient and reliable tool to quantify cardiac function noninvasively.     

Recent applications of the new stereology have thrown fresh light on how the human placenta grows and develops its form

The availability of design-based stereological methods has made it possible to readdress certain key and contentious issues in placental growth and morphogenesis. Three particular questions are: (i) does the population of cytotrophoblast cells decline during gestation?, (ii) is placental growth biphasic or monophasic? and (iii) what are the consequences for intervillous porosity of the elaboration of terminal villi? These questions cannot be answered definitively without recourse to the new stereology. Applying the disector to estimate nuclear number and star volume to assess pore size, recent studies have helped to resolve these issues. Their findings are reviewed. Nuclei were counted in the trophoblastic epithelium, stroma and vascular endothelium of placental villi. It was found that growth is monophasic and proliferative. All types of nuclei increased in number throughout gestation and this included cytotrophoblast. Trophoblast grows by the continuous recruitment of new proliferative units of uniform mean volume. The so-called ‘loss’ of cytotrophoblast cells is a misinterpretation of what is seen on microscopical sections and is attributable to disproportionate growth in villous surface area. Cells simply become more widely dispersed. Elaboration of finer terminal branches on villous trees leads to a decline in the star volumes of villi and intervillous pores. Some of the functional implications of these findings are discussed.     

Vertical LM sectioning and parallel CT scanning designs for stereology: application to human lung

Practical, unbiased stereological methods are described to estimate lung volume and external surface area, and total volume and surface area of relatively large and anisotropic structures (bronchi and arteries) inside the lung. The volume of each of five lung strata was estimated first by fluid displacement and then by computed tomography (CT) using Cavalieri's method; the reliability of CT was assessed through a calibration procedure, and image thresholding criteria for an accurate volume estimation using CT were established. The parallel, perfectly registered CT section images were also used to estimate the external surface area of each stratum by the spatial grid method. Unbiased estimation of internal surface areas in lung is a long-standing problem: since the structures are large and essentially void, large sections are needed; to facilitate identification, thin sections have to be used for light microscopy, and since such structures are anisotropic, the sections should be vertical. A practical stereological design is demonstrated here on an infant lung, which fulfils all these requirements. This study illustrates the potential of using unbiased stereology to characterize infant pulmonary hypoplasia.     

Advantages and pitfalls of using free‐hand sections of frozen needles for three‐dimensional analysis of mesophyll by stereology and confocal microscopy

The anatomical structure of mesophyll tissue in the leaf is tightly connected with many physiological processes in plants. One of the most important mesophyll parameters related to photosynthesis is the internal leaf surface area, i.e. the surface area of mesophyll cell walls exposed to intercellular spaces. An efficient design‐based stereological method can be applied for estimation of this parameter, using software‐randomized virtual fakir test probes in stacks of optical sections acquired by a confocal microscope within thick physical free‐hand sections (i.e. acquired using a hand microtome), as we have shown in the case of fresh Norway spruce needles recently. However, for wider practical use in plant ecophysiology, a suitable form of sample storage and other possible technical constraints of this methodology need to be checked. We tested the effect of freezing conifer needles on their anatomical structure as well as the effect of possible deformations due to the cutting of unembedded material by a hand microtome, which can result in distortions of cutting surfaces. In the present study we found a higher proportion of intercellular spaces in mesophyll in regions near to the surface of a physical section, which means that the measurements should be restricted only to the middle region of the optical section series. On the other hand, the proportion of intercellular spaces in mesophyll as well as the internal needle surface density in mesophyll did not show significant difference between fresh and frozen needles; therefore, we conclude that freezing represents a suitable form of storage of sampled material for proposed stereological evaluation.     

Sampling theory and automated simulations for vertical sections,applied to human brain

In recent years, there have been substantial developments in both magnetic resonance imaging techniques and automatic image analysis software. The purpose of this paper is to develop stereological image sampling theory (i.e. unbiased sampling rules) that can be used by image analysts for estimating geometric quantities such as surface area and volume, and to illustrate its implementation. The methods will ideally be applied automatically on segmented, properly sampled 2D images – although convenient manual application is always an option – and they are of wide applicability in many disciplines. In particular, the vertical sections design to estimate surface area is described in detail and applied to estimate the area of the pial surface and of the boundary between cortex and underlying white matter (i.e. subcortical surface area). For completeness, cortical volume and mean cortical thickness are also estimated. The aforementioned surfaces were triangulated in 3D with the aid of FreeSurfer software, which provided accurate surface area measures that served as gold standards. Furthermore, a software was developed to produce digitized trace curves of the triangulated target surfaces automatically from virtual sections. From such traces, a new method (called the ‘lambda method’) is presented to estimate surface area automatically. In addition, with the new software, intersections could be counted automatically between the relevant surface traces and a cycloid test grid for the classical design. This capability, together with the aforementioned gold standard, enabled us to thoroughly check the performance and the variability of the different estimators by Monte Carlo simulations for studying the human brain. In particular, new methods are offered to split the total error variance into the orientations, sectioning and cycloid components. The latter prediction was hitherto unavailable – one is proposed here and checked by way of simulations on a given set of digitized vertical sections with automatically superimposed cycloid grids of three different sizes. Concrete and detailed recommendations are given to implement the methods.     

Precision of Cavalieri sections and slices with local errors

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.     

The saucor, a new stereological tool for analysing the spatial distributions of cells, exemplified by human neocortical neurons and glial cells

The 3D spatial arrangement of particles or cells, for example glial cells, with respect to other particles or cells, for example neurons, can be characterized by the radial number density function, which expresses the number density of so-called 'secondary' particles as a function of their distance to a 'primary' particle. The present paper introduces a new stereological method, the saucor, for estimating the radial number density using thick isotropic uniform random or vertical uniform random sections. In the first estimation step, primary particles are registered in a disector. Subsequently, smaller counting windows are drawn with random orientation around every primary particle, and the positions of all secondary particles within the windows are recorded. The shape of the counting windows is designed such that a large portion of the volume close to the primary particle is examined and a smaller portion of the volume as the distance to the primary object increases. The experimenter can determine the relation between these volumina as a function of the distance by adjusting the parameters of the window graph, and thus reach a good balance between workload and obtained information. Estimation formulae based on the Horvitz-Thompson theorem are derived for both isotropic uniform random and vertical uniform random designs. The method is illustrated with an example where the radial number density of neurons and glial cells around neurons in the human neocortex is estimated using thick vertical sections for light microscopy. The results indicate that the glial cells are clustered around the neurons and the neurons have a tendency towards repulsion from each other.