The rotator

The mean particle volume can be stereologically estimated using the nucleator principle. In the present paper, we discuss another principle for estimating mean particle volume, namely the rotator. The vertical rotator has already been previously described and is supplemented in the present paper by the isotropic rotator. For a collection of particle profiles, simulations show that the variance of the rotator is smaller than that of the nucleator.     

Stereological estimation using vertical sections in a complex tissue

A method designed for stereological estimation in a very complex tissue using vertical sections is presented. In some tissues, the random rotation of the tissue for vertical sections may obscure recognition of the anatomical structures of interest. The present method overcomes this problem by generating sections with both a particular orientation, 'mapping sections', and ordinary random vertical sections usable for the required observations. A map describing the positions of the vertical sections is produced to make the complex reference space recognizable. The method is illustrated by estimating the number and size of neurones in the dorsal raphe nucleus of the human brainstem with its dense packing of roughly 100 nuclei within a volume less than 50 cm³.     

Monitoring age-related changes of collagen content and vascularity in ganglia using unbiased stereological methods

We describe for the first time application of unbiased stereological techniques to estimate total volume and volume fractions of interest in individual dorsal root ganglia (DRG). Volume estimates were obtained using a two-stage sampling design. Sections were systematically sampled following a random start, from DRG which were embedded in methacrylate and exhaustively sectioned. We further examined the efficiency of point counting irregular volume fractions housed in a regular reference volume. We found that the precision of volume estimates was relatively unaffected by exhaustive sampling, and that the magnitude of error was, in large part, determined by object shape.     

The semi-automatic nucleator

The nucleator is a well-established manual stereological method of estimating mean cell volume from observations on random cell transects through reference points of the cells. In this paper, we present an automated version of the nucleator that uses automatic segmentation of the boundaries of the cell transects. An expert supervises the process. If the segmentation is judged to be satisfactory, an estimate of the cell volume is calculated automatically on the basis of the whole cell transect. In the remaining cases, the expert intervenes and uses the classical nucleator. The resulting estimator is called the semi-automatic nucleator. In this paper, we study the statistical properties of the semi-automatic nucleator. Formulae for the bias and mean square error are derived. The semi-automatic nucleator may have a small bias but will still in most cases be more efficient than the classical nucleator. Procedures for estimating bias and mean square error from a pilot study are provided. The application of the semi-automatic nucleator is illustrated in a study of somatostatin positive inhibitory interneurons which were genetically labelled with green fluorescent protein (GFP). The cells were sampled with an optical disector. The centre of mass in a central cell transect was used as reference point. It is found in this study that the number of cells needed for obtaining, for instance, a 5% precision of the estimate of mean cell volume is 150 and 189 for the semi-automatic and the classical nucleator, respectively. Taking into account that the time spent analysing one cell is shorter for the semi-automatic nucleator than for the classical nucleator, the semi-automatic nucleator is superior to the classical nucleator.     

Estimation of subcellular organelle volume from ultrathin sections through centrioles with a discretized version of the vertical rotator

A method for the fast and efficient estimation of the volume (but not surface area) of subcellular organelles is presented. It consists of a rotator/coaxial-section approach based on the Pappus theorem and represents a discretized version of the vertical rotator where, instead of measuring intercept lengths, the points in distance classes are counted. Centrioles serve as a unique reference 'double-point' with constant size allowing unbiased cell selection from the whole population with equal probability and without the disector application. The sandwich-like method of sample preparation allows comparison of control and experimental cases with the same errors caused by overlapping and overprojection. Test experiments demonstrated that the vertical discretized rotator was an efficient and precise tool for the estimation of volume and that a few independent sections of unknown thickness were sufficient for the quantification of one experimental point.     

The total number of neurons in the human neocortex unbiasedly estimated using optical disectors

An efficient method is presented for obtaining, in under 4h, an unbiased estimate of the total number of neurons in the human neocortex, with a coefficient of error on the estimate of ～ 5%. The novel sampling scheme used in this study is unbiased and was designed so that only a small amount of neocortical grey matter had to be removed. Hence, the majority of the cerebral grey matter and all the internal grey matter was left intact for further resampling and analysis. Each cerebral hemisphere was subdivided into the four major neocortical regions, sliced coronally at 7-mm intervals and the volume of the neocortex determined using Cavalieri's principle. Uniform sampling of neocortex was performed in the hemisphere followed by regional subsampling with a varying sampling fraction being taken from each region. Neuronal density estimates were made in thick plastic sections using optical disectors. Shrinkage estimates were made in parallel with the number estimates and found to be negligible. The total number of neocortical neurons in the right hemisphere of five normal 80-year-old men was found to be 13·7 × 10⁹ with an inter-individual coefficient of variation of 12%.     

Estimation of individual feature surface area with the vertical spatial grid

The area of an individual bounded surface (e.g. the boundary of a properly sampled cell) can be estimated from an isotropic uniform random stack of parallel sections, or of non-invasive planar scans, using the well-known spatial grid. A standing problem was to estimate the area of an individual bounded surface with an arbitrary degree of accuracy from a vertical (i.e. not isotropic) stack of sections or scans. A new tool to do this, called the ‘vertical spatial grid’, is presented.     

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.     

Multidimensional long-term time-lapse microscopy of in vitro peripheral nerve regeneration

In order to test the effectiveness of a new advanced time-lapse microscopy imaging and image processing and analysis system, and to do quantitative and qualitative temporal analyses of in vitro peripheral nerve regeneration, long-term time-lapse imaging of cultures of mouse dorsal root ganglia (DRGs) was performed. DRGs were placed in a Petri dish, covered with collagen gel, their attached peripheral nerves were cut in the middle, creating a gap, and the dish was filled with culture medium. Six preparations were kept on the time-lapse imaging system, which provides a suitable incubation environment and enables to capture images from multiple coordinates at x,y,z axes at desired time intervals for 13 days. In general, the time-lapse imaging system proved quite stable and efficient, although some improvements are certainly required. Two main components of peripheral nerve regeneration, outgrowth of axons and activities of resident cells, were examined. Axons started to grow during the first hour of incubation with a 16.5 microm/h rate and showed the slowest rates (0.7 microm/h) on days 8 and 9, after which they resumed higher speeds again. The first cell came out of the proximal end of the cut nerve on the second day and it was a Schwann cell (SC), which was the prominent cell type in the preparations throughout the experiment. SCs were higher in number (83.15% of all cells) but slower in migration (3.4 vs. 7.3 microm/h, P < 0.001) than other cells. Other observed characteristics of axonal outgrowth and cellular activity and interactions between axons and the cells are discussed.     

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.     

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.     

Estimating surface area by the isotropic fakir method from thick slices cut in an arbitrary direction

The proposed fakir method for estimating surface area is based on counting the intersections between the surface lying within a thick slice, and an isotropic spatial grid consisting of a combination of linear probes called fakir probes. An unbiased procedure using a directly randomized spatial grid rather than sections with randomized directions is presented. The method is applicable if perfectly registered serial sections of the surface are available in a thick slice while the direction of the slice can be arbitrary. The efficiency of the fakir method using different arrangements of orthogonal triplets of fakir probes is evaluated and it is shown that mutually shifted probes are superior to non-shifted ones. The application software for interactive counting of intersections between computer-generated fakir probes and the surface within the stack of digitized images is described and demonstrated by two examples: estimation of the surface area of individual tobacco cell chains using a confocal microscope, and estimation of the total area of exposed surface of mesophyll cells in a barley leaf using a wide-field transmission microscope.