Stereological length estimation using spherical probes

Lineal structures in biological tissue support a wide variety of physiological functions, including membrane stabilization, vascular perfusion, and cell‐to‐cell communication. In 1953, Smith and Guttman demonstrated a stereological method to estimate the total length density (L_v) of linear objects based on random intersections with a two‐dimensional sampling probe. Several methods have been developed to ensure the required isotropy of object–probe intersections, including isotropic‐uniform‐random (IUR) sections, vertical‐uniform‐random (VUR) slices, and isotropic virtual planes. The disadvantages of these methods are the requirements for inconvenient section orientations (IUR, VUR) or complex counting rules at multiple focal planes (isotropic virtual planes). To overcome these limitations we report a convenient and straightforward approach to estimate L_v and total length, L, for linear objects on tissue sections cut at any arbitrary orientation. The approach presented here uses spherical probes that are inherently isotropic, combined with unbiased fractionator sampling, to demonstrate total L estimation for thin nerve fibres in dorsal hippocampus of the mouse brain.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Autoradiographic studies on the mode of growth in jejunal crypt cells of the mouse

The structures of primary concern in Part A are varieties—the generalization in higher dimensions of smooth curves and surfaces. Fixed orientation random s-sections (s-dimensional flat sections) of a fixed t-variety (t-dimensional variety) in R^d (d-dimensional space) are considered in Section 2. The t-variety induces a density-cum-orientation distribution which is related to corresponding sectional quantities. The by now classical basic formula; of stereology are all special cases of the simple multi-dimensional formula (2.16). Next (Section 3), statistics of the variety of intersection of several statistically homogeneous random varieties are related to the corresponding statistics of the parent varieties. Part B is concerned with the analysis by random s-sections of fixed aggregates of t-dimensional opaque particles (i.e. varieties for 1 ≤ t ≤ d — 1, domains for t = d) embedded within an opaque specimen. Fixed orientation random sections are considered in Section 5, and isotropically oriented ones in Sections 6–10. It is shown in Section 7 that the mean particle ‘caliper diameter’, a key quantity, may in theory be estimated by isotropic slice sectioning. The theory is particularly rich when the particles are convex, witness the arrays of useful formula; in Sections 8–10. Crofton's remarkable ‘second theorem’ comes into its own in Section 9, permitting simple estimates from isotropic test lines of mean area squared to mean perimeter (when d = 2) and mean volume squared to mean surface area (when d = 3); in fact, the formulæ of Sections 7–9 suggest estimates for both mean and variance of the areas (when d = 2) and volumes (when d = 3) of aggregates of embedded convex particles. Further results for aggregates of convex polytopes are given in Section 10. Certain general considerations and principles relating to estimation from random sections in the practical cases, i.e. d ≤ 3, are included in the final Section 11. Equations which are the specializations of multi-dimensional formulæ to the practical cases are asterisked. A feature is the attention given to the proper probabilistic specification of random sections through the specimen, a matter that seems largely to have been ignored.     

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).     

Precision of circular systematic sampling

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.     

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.     

Second-order stereology of benign and malignant alterations of the human mammary gland

The purpose of the present study was a quantitative characterization of the three-dimensional arrangement of the epithelial component of benign and malignant alterations of the female breast by combining stereology with stochastic geometry. Twenty cases of fibrous mastopathy and 20 cases of invasive ductal mammary cancer were studied at the light microscopic level. Segmentation of the epithelial tissue component was performed with an image analyser. From the resulting binary images, unbiased estimates of the covariance C(r) and the intensity V_v of the epithelial volume component were obtained automatically by computer. From these data, estimates of the correlation function k(r), of the pair correlation function g(r), of the radial distribution function RDF(r) and of the reduced second moment function K(r) of epithelial volume were determined. The estimates of C(r) and RDF(r) differed between groups, but these functions depend on spatial pattern and V_v. As carcinomas showed a significantly higher epithelial volume density V_v than mastopathies, estimation of C(r) and RDF(r) alone did not permit a safe distinction between possibly different types of spatial arrangement of epithelium in the benign and malignant lesions. Analysis of the estimates of k(r), g(r) and K(r), which are not influenced by V_v, showed definite interaction between epithelial volume elements, with clustering at short distances and repulsion at long distances. In both groups, the null hypothesis of purely random arrangement of epithelium had to be rejected. The clearest distinction between groups was obtained by estimation of g(r), which showed that short-range, tubular pattern as well as long-range, lobular architecture are better preserved in benign than in malignant lesions. The low interindividual scatter of k(r), g(r) and K(r) indicates a high biological significance of spatial pattern, which is presumably under strict genetic control. Potential uses of the method are: (i) the identification of biomathematical models which could contribute to a better understanding of the growth processes involved, (ii) conditional simulation of the underlying three-dimensional structures by computer, and (iii) supporting the diagnosis of mammary lesions with borderline histopathological appearance.     

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.     

Shear-torsion of large curvature beams-Part II. Applications to thin bodies

The shear-torsion state of stress in a curved beam, whose cross sections is a thin rectangle with sides not parallel to the plane of the beam, are determined in closed form. The maximum value of the shear stress is attained at the concave boundary of the beam. The shear-torsion moment of inertia J_st, in multi-connected thin-walled cross sections is evaluated. Several examples of cross sections are discussed.     

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.     

Some stereological and statistical consequences derived from Cartier's formula

Let X and Y be two random variables with distribution functions F_x and F_Y, respectively. If Cartier's formula E[Y|X]=X holds, then F_x is necessarily less dispersed than F_Y. In this paper, the main consequences of Cartier's formula are derived. Some examples where such a formula holds are also given, namely in stereology (comparison of various estimators of Minkowsky's functionals) and in geostatistics (the change of support problem).