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.     

Unbiased estimation of curve length in 3-D using vertical slices

An efficient sampling procedure is presented for estimation of total line length per unit volume L_v. It involves the following steps: (1) choose a vertical axis in the specimen, and cut the specimen to obtain VUR vertical slices of constant thickness Δ such that parallel planes of the slices contain the vertical direction; (2) observe the projected image of a vertical slice using transmission microscopy such that beam direction is perpendicular to the slice; (3) count the number of intersections of the projected images of the lineal features of interest with cycloid-shaped test lines whose minor axis is perpendicular to the vertical axis. The expected value of the number of intersections per unit length _prj is related to L_v as follows: Thus, L_v can be estimated from the measurements performed on the projected images of VUR vertical slices.     

Estimation of length and surface of anisotropic capillaries

Surface density (S_V) and length density (L_V) of myocardial capillaries have hitherto been estimated from their profile boundary length (B_A) and their numerical density (Q_A) on transverse sections by the simplifying assumptions of the Krogh model (perfectly anisotropic, straight, unbranched capillaries with constant cross-sectional area). As the capillaries actually are partially anisotropic, curved, branching cylinders with variable cross-sectional area, a geometrical bias arises from the model-reality discrepancies. We have applied and compared two methods to overcome these inconsistencies: (1) estimation of L_V and S_V by a more realistic model (the Dimroth-Watson distribution); (2) estimation of L_V and S_V from isotropic uniform random (IUR) sections. Twelve male Wistar rats were fixed by retrograde vascular perfusion. One pair of longitudinal and transverse sections, and six IUR sections per animal were selected at random from the left ventricular papillary muscles. Ultrathin sections were silver-impregnated and studied by light microscopic morphometry. Nearly identical estimates of L_V and S_V were found by both methods. The model-based estimation provides biologically meaningful anisotropy constants, but it presupposes knowledge of the anisotropy axis. The IUR method provides no measure of anisotropy, but it can be applied in tissues where the anisotropy axis is not known. Both methods are equally efficient and practically unbiased in S_V estimation, but the model-based estimation is far more efficient in L_V estimation.     

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.     

Design-based estimation of surface area in thick tissue sections of arbitrary orientation using virtual cycloids

Surface area is a first‐order stereological parameter with important biological applications, particularly at the intersection of biological phases. To deal with the inherent anisotropy of biological surfaces, state‐of‐the‐art design‐based methods require tissue rotation around at least one axis prior to sectioning. This paper describes the use of virtual cycloids for surface area estimation of objects and regions in thick, transparent tissue sections cut at any arbitrary (convenient) orientation. Based on the vertical section approach of Baddeley et al., the present approach specifies the vertical axis as the direction of sectioning (i.e. the direction perpendicular to the tissue section), and applies computer‐generated cycloids (virtual cycloids) with their minor axis parallel to the vertical axis. The number of surface‐cycloid intersections counted on focal planes scanned through the z‐axis is proportional to the surface area of interest in the tissue, with no further assumptions about size, shape or orientation. Optimal efficiency at each x–y location can be achieved by three virtual cycloids orientated with their major axes (which are parallel to the observation planes) mutually at an angle of 120°. The major practical advantage of the present approach is that estimates of total surface area (S) and surface density (S_V) can be obtained in tissue sections cut at any convenient orientation through the reference space.     

Estimation of length density Lv from vertical slices of unknown thickness

Length density of lineal features, L_v, is an important stereological parameter. The efficient stereological procedure for the estimation of L_v from the counting measurement performed on the projected images of the vertical slices (foils) is modified and improved: L_v can be now estimated from vertical slices of unknown thickness, and the slices need not be of the same thickness. The required assumption-free stereological relationship is is the average number of intersections of straight test lines parallel to the vertical axis with the projected images of the lineal features in the vertical slices, per unit test line length. is the average number of intersections of the lineal features with the parallel planes of the vertical slices, per unit area. Note that there are two parallel planes in a slice, and therefore their total area is twice the area of the observed projected image frame. is the average number of intersections of orientated cycloid-shaped test lines (minor axis perpendicular to vertical axis) with the projected images of the lineal features in the vertical slices, per unit length. For practical applications of this result, it is necessary uniquely to identify the points of intersections of the lineal features with the parallel planes of the slices in the projected images, so that can be estimated unambiguously. However, in practice, this is not a problem in biological, as well as materials microstructures.     

Measuring error and sampling variation in stereology: comparison of the efficiency of various methods for planar image analysis

An 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 V_v, S_v). The methods tested are point-counting with different densities of test points (4 ≤ P_T ≤ 900 per picture), semiautomatic computer image analysis with MOP and automatic image analysis with Quantimet, for obtaining V_v and S_v 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 V_v and S_v 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 V_v in the biological sample.     

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).     

A simple algorithm to measure the volume-weighted and number-weighted mean volume of particles

An algorithm is presented which offers an alternative approach for measuring volume- and number-weighted mean volume and standard deviation of particles. Using a computer-assisted manual method the following intermediate steps are performed automatically: generation of linear probes emanating from the sampling point of the object and intersecting the profile periphery, measurement of their lengths, and measurement of the area of the transect required for estimating the standard deviation of the volume-weighted mean volume. By first tracing manually the outline of the periphery of the object with a cursor, on a magnetic tablet or on an image acquired into the computer with a video camera, the location of all pixels of the periphery is registered and the area of the transect is measured concurrently. The computer is informed of the coordinates of the selection point in the uniform random (UR) sampling grid by clicking the cursor. All ensuing operations are automatic. In the case of isotropic UR (IUR) sections the algorithm traces a series of uniform systematic random linear probes between the sampling point and the object profile periphery emanating from this selection point, radiating at angular intervals of 29–30° to the periphery. In the case of vertical sections, similar lines are generated at intervals where the sine of the angle changes by a value of 0·33. The volume-weighted mean volume of the object is estimated from the average of all the products , where l represents the length of each individual random linear probe. As the periphery is traced, the algorithm can automatically determine the area of the cross-section of the object, from which the standard deviation of the volume-weighted mean volume can be calculated. Some elements of the above algorithm are also used for the measurement of the number-weighted mean volume. The latter procedure is facilitated using an acoustic vertical depth monitor attached to the microscope. The impact of truncation (‘lost caps’) on the precision of the measurements is discussed. The algorithm is of particular use in light microscopy for measuring cell nuclei by direct visual inspection of the microscopic field using a side-arm mirror assembly interfaced with a magnetic tablet.     

On the estimation of particle number

Two methods are proposed for estimating the number of separated particles within a solid structure per unit volume of structure, N_v. Apart from being arranged with independence of any size parameter, no special assumptions upon the size, shape and orientation of the particles are made. The first method is based on the identity N_V = (N_A)_u ? μ_u^?1, where (N_A)_u is the mean number of particle sections per unit area of a plane probe T_u which is uniform random within the structure and perpendicular to a given direction u, whereas μ_u is the mean particle caliper length along u. The second method uses N_V = A_A?V^?1, where A_A is the mean areal fraction of the particles per unit area of section, whereas v is the mean particle volume. The estimation of (N_A)_u, μ_u, and v requires the examination of parallel serial sections above and below T_u. Particle model reconstructions are not needed, however. Previous approaches to the problem are discussed.     

Estimating length density and quantifying anisotropy in skeletal muscle capillaries

The accurate estimation of stereological parameters defined on anisotropic structures is a long-standing problem. In this paper we seek to estimate the capillary length density J_v in skeletal muscle tissue. A well-known model for directional anisotropy in space, namely the ‘spherical normal’ or ‘Fisher axial distribution’ model, is found to fit the relevant data satisfactorily. Based on this model, a short-cut estimation method is proposed and illustrated with a numerical example. This method essentially consists in taking the ratio of mean capillary profile counts, as obtained from transversal and longitudinal sections of the muscle tissue, and making use of a table or a graph given in the paper to estimate J_v. The conditions under which the methods are applicable and practicable are discussed in detail. Apart from an accurate estimation of J_v, an important feature of our method is the possibility of quantifying the degree of anisotropy by a coefficient K (called the concentration parameter of the Fisher axial distribution), which enjoys both a biological significance and a sound statistical basis.     

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.     

Measurement of geometrical parameters in cocontinuous polymer blends: 3D versus 2D image analysis

Formulae of stereology are used to estimate 3D geometrical parameters of cocontinuous structures measured from 2D micrographs of polymer blends. 3D images of symmetric and nonsymmetric polymer blends made of fluorescently labelled polystyrene and styrene‐ran‐acrylonitrile copolymer were obtained with laser scanning confocal microscopy. Geometrical parameters of the blend interface, specifically volume fraction, surface area per unit volume (S _V ) and average of local mean curvature were measured directly from the 3D images and compared to the values estimated from analysis of a number of 2D slices combined with stereological relations. When the total length of phase boundary considered in the analysis of the 2D slices (L_Tot ) was at least 6000 times bigger than the characteristic length of the microstructure (S^?1_V ), the standard deviation for all the parameters measured became negligible. However, considerable discrepancies between the average values computed from 3D and 2D images were observed for any value of L_Tot . The mean curvature distribution was also measured from both the 3D images and the 2D slices. The distribution was estimated from the 2D slices but with a width about 2.4 times that of the true value obtained from the 3D images.     

The use of a hybrid diffraction unit in the measurement of dislocation densities by electron microscopy

The expression used in the electron microscopical measurement of dislocation densities is ?=2N/Lt, where N is the number of intersections between the dislocation lines in a TEM image, L the total length of a set of sampling lines superimposed upon that image (corrected for magnification) and t the thickness of the thin foil containing the dislocation structure. During the estimation of ? it is therefore important to reduce errors in the measurements of all three variables. It is shown how calibration of the microscope magnification, careful consideration of diffraction conditions and use of the hollow cone and rocking beam facilities on the hybrid diffraction unit fitted to the Philips EM 400T can assist in increasing the speed and accuracy of the measurement procedure.     

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.     

Characterization of coals by automated optical image analysis 1. Vitrinite reflectance

An increasing awareness of the importance of petrographic characterization of coals to efficient selection of coals for pulverized coal combustion and metallurgical coking, has highlighted the need to improve slow and subjective optical microscopic procedures. Automated image-analysis procedures to measure vitrinite random reflectance are examined here in some detail, giving particular attention to inertinite-rich coals. It is shown by consideration of intra- and inter-particle reflectance variance that a given accuracy for vitrinite mean random reflectance (R_v) can be achieved by selection of an appropriate surface sampling procedure. The repeatability of R_v for single coals is similar to that for manual microscopy, but the reproducibility, as established by an international interchange exercise, is not yet good enough to specify a standard procedure. In some coals even vitrinite sub-macerals can be distinguished; however, caution is required when extending this method to vitrinite reflectance distributions of blends containing different rank coals or of heat-altered coals.     

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.     

Study of changes in L32 EELS ionization edges upon formation of Ni-based intermetallic compounds

EELS L₃₂ ionization edges in several Ni‐based intermetallic compounds have been studied and interpreted in terms of the distribution of electrons in the valence d‐bands. It is demonstrated that the integral EELS cross‐sections change only slightly upon the formation of intermetallic compounds and therefore the charge transfer between atoms is negligible. On the other hand, the changes in the fine energy‐loss near‐edge structure (ELNES) of the Ni L₃ edge can be readily detected indicating an important redistribution of d‐electrons at the Ni site with alloying. These features are well reproduced by ab‐initio calculations with a FLAPW method in its WIEN97 implementation. In contrast to the drastic effect of chemical environment, structural transformations in the investigated intermetallics result in smaller ELNES changes, which can be detected by only exceptional instruments with a higher energy resolution.