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).     

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.     

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.     

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.     

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.     

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.     

The fine structure of fenestrated adrenocortical capillaries revealed by in-lens field-emission scanning electron microscopy and scanning transmission electron microscopy

Cell biologists probing the physiologic movement of macromolecules and solutes across the fenestrated microvascular endothelial cell have used electron microscopy to locate the postulated pore within the fenestrae. Prior to the advent of in-lens field-emission high-resolution scanning electron microscopy (HRSEM) and ultrathin m et al coating technology, quick-freeze, platinum-carbon replica and grazing thin-section transmission electron microscopy (TEM) methods provided two-dimensional or indirect imaging methods. Wedge-shaped octagonal channels composed of fibrils interwoven in a central mesh were depicted as the filtering structures of fenestral diaphragms in images of platinum replicas enhanced by photographic augmentation. However, image accuracy was limited to replication of the cell surface. Subsequent to this, HRSEM technology was developed and provided a high-fidelity, three-dimensional topographic image of the fenestral surface directly from a fixed and dried bulk adrenal specimen coated with a 1 nm chromium film. First described from TEM replicas, the “flower-like” structure comprising the fenestral pores was readily visualized by HRSEM. High-resolution images contained particulate ectodomains on the lumenal surface of the endothelial cell membrane. Particles arranged in a rough octagonal shape formed the fenestral rim. Digital acquisition of analog photographic recordings revealed a filamentous meshwork in the diaphragm, thus confirming and extending observations from replica and grazing section TEM preparations. Endothelial cell pockets, first described in murine renal peritubular capillaries, were observed in rhesus and rabbit adrenocortical capillaries. This report features recent observations of fenestral diaphragms and endothelial pockets fitted with multiple diaphragms utilizing a Schottky field-emission electron microscope. In-lens staging of bulk and thin section specimens allowed tandem imaging in HRSEM and scanning TEM modes at 25 kV.     

Estimation of the directional distribution of spatial fibre processes using stereology and confocal scanning laser microscopy

Fibrous structures like polymers, glass fibres, muscle fibres and capillaries are important components of materials and tissues. A spatial fibre process is the union of smoothly curved or linear one-dimensional features of finite length, arranged in an unbounded three-dimensional reference space according to some random mechanism. Design-based stereology was combined with confocal scanning laser microscopy to study samples of fibre-reinforced composites, which were considered as realizations of not necessarily isotropic fibre processes. The methods enable the unbiased estimation of the intensity and of the directional distribution of spatial fibre processes from arbitrarily directed pairs of registered parallel optical sections a known distance apart. The directions of fibres sampled by a frame of observation on the reference plane are estimated from the coordinates of the intersection points of the fibres with both optical planes using confocal scanning laser microscopy. The true directional distribution of the fibre process is estimated by weighting each measured direction by the reciprocal of its chance of being sampled, which can be inferred from the data. The concept of complete directional randomness for uniformly and independently distributed spatial directions is introduced. The cumulative distribution function of the angular distances between different directions and other exact relations are derived for complete randomness of vectorial and axial directions. A Monte Carlo method is constructed to test spatial fibre processes, whose fibres have negligibly small curvature, for complete directional randomness. Confocal scanning laser microscopy was used to study the angular distribution of glass fibres in a polymer composite which was subjected to increasing hydrostatic extrusion. The hypothesis of complete directional randomness had to be rejected for all samples with 1% probability of error. The directional distribution was of the bipolar type, with the principal axis directed parallel to the axis of extrusion. Progressive stretching of the material increased the degree of anisotropy of the glass fibres. Although presented for an application in polymer physics, the methods are general and may also be applied in biological investigations.     

Spatial distribution of curve length: Concept and estimation

The length of a curvilinear feature, such as a dendrite tree of a neuron, can, in principle, be estimated by the recent, non-invasive method of total vertical projections (TVPs). Curve length is a measure of size, but it reports nothing about curve shape. The shape of a tree-like structure can be described to some extent by the distribution of branch length in properly defined regions of three-dimensional (3-D) space. A definition of curve length distribution in three dimensions is proposed and implemented here on a human neuron. The relevant 3-D regions overlap after projection, and therefore the TVPs method cannot be used directly to estimate the corresponding feature lengths. However, using the ANALYZE^TM software system running on a SUN®SPARC workstation, dendrite subsets sitting in predefined regions of space were rendered in different colours and measured separately by the TVPs method using a cycloid test system. In combination with non-invasive image acquisition and processing techniques, the length distribution concept is likely to be useful in the metrical analysis of either microscopic or macroscopic arborizations in a wide variety of contexts, including living cells and organisms.     

A simple tool for stereological assessment of digital images: the STEPanizer

STEPanizer is an easy-to-use computer-based software tool for the stereological assessment of digitally captured images from all kinds of microscopical (LM, TEM, LSM) and macroscopical (radiology, tomography) imaging modalities. The program design focuses on providing the user a defined workflow adapted to most basic stereological tasks. The software is compact, that is user friendly without being bulky. STEPanizer comprises the creation of test systems, the appropriate display of digital images with superimposed test systems, a scaling facility, a counting module and an export function for the transfer of results to spreadsheet programs. Here we describe the major workflow of the tool illustrating the application on two examples from transmission electron microscopy and light microscopy, respectively.     

Estimation of average particle size from vertical projections

A new stereological relationship is derived for the estimation of average size (average width) of a collection of convex particles in a 3D microstructure. The average size is estimated from measurements performed on projected images of the microstructure generated by total vertical projections. The stereological relationship is as follows: D = Ī _C /(2 N ₀β). D is the average width, ¯ I _C is the average absolute number of intersections between the specifically oriented and regularly spaced cycloid shape test lines and particle boundaries observed in the total vertical projections, N ₀ is the total number of particles observed in the total vertical projection and the parameter β is a characteristic of the measurement grid; it has units of reciprocal of length. The result is applicable to any arbitrary collection of convex particles; the particle orientations need not be isotropic. Only 'intersection counts' are required; it is not necessary to measure sizes of the particles in the projected images.     

Estimation of structural anisotropy based on volume orientation. A new concept

The quantification of anisotropy—its main direction and the degree of dispersion around it—is desirable in numerous research fields dealing with physical structures. Conventional methods are based on the orientation of interface elements. The results of these methods do not always agree with perceived anisotropy, and anisotropic structures do not necessarily turn out to be ‘anisotropic’ using these methods. In the present paper, we propose an alternative to curve and surface orientation, namely volume orientation. Using trabecular bone as an example of a two-phase anisotropic structure, the new concept is studied in some detail. In particular, a parametric method of estimating volume orientation from sections is presented and discussed.     

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.     

Study of the structure of the air and blood capillaries of the gas exchange tissue of the avian lung by serial section three-dimensional reconstruction

We have previously reconstructed the gas exchange tissue of the adult muscovy duck, Cairina moschata using a method of manually aligning sections and tracing the contours of the components of the gas exchange tissue. This reconstruction method demonstrated that the air capillaries are comprised of an expanded globular part interconnected by narrow air channels. The blood capillaries completely surround the air capillaries forming an anastomosing meshwork of short segments. However, the resulting reconstruction was limited in scope because of the laborious process of tracing the profiles of each component through the sequence of micrographs. We have now reconstructed a larger proportion of the exchange tissue by using a cross-correlation based alignment strategy and have demonstrated that the staining intensity of each of the exchange tissue components is sufficiently different to allow them to be identified by simple filtering and thresholding. The resulting reconstructions sample a much larger proportion of the exchange tissue and demonstrate the heterogeneity of structures from different locations in the parabronchus. We have shown that a sheet-flow-type arrangement of blood capillaries surrounds the infundibulum; this represents an unexpected functional convergence with the arrangement of blood capillaries surrounding the mammalian alveoli. It is feasible, using this reconstruction strategy, to analyse the exchange tissue of a large number of avian species in order to determine structural correlates of function. The resulting reconstructions could be analysed in order to determine the basis of the functional efficiency and rigidity of the avian lung.     

Hydroforming of anisotropic aluminum tubes: Part II analysis

Part I presented an experimental investigation of hydroforming of Al-6260-T4 tubes and a simple two-dimensional model of the process. Relatively long, extruded circular tubes were formed against a square die with rounded corners, with simultaneous application of axial feeding. Localized wall thinning was reported to occur at mid-span which, accentuated by friction, led to burst. Part II presents fully 3D models of the process that include friction as well as more advanced constitutive models shown in previous studies to be essential for simulation of burst in free hydroforming of aluminum alloy tubes. The models are used to simulate several of the experiments of Part I, emphasizing the prediction of all aspects of the forming process, including wall thinning and its localization that lead to rupture. A shell element model is shown to capture the majority of the structural features of the process very successfully. However, even with the implementation of advanced constitutive models, it fails to reproduce correctly the localization of wall thinning. It is demonstrated that switching to solid elements coupled to non-quadratic yield functions results in accurate predictions of all aspects of the problem, including the onset of rupture. Apparently, slow growing depressions that develop at the interface between the flattened part of the cross section that is in contact with the die and the rounded part that is not, have a complex three dimensional stress state requiring accurate modeling offered by solid elements. Furthermore, the evolution of these depressions is only reproduced with accuracy when in addition non-quadratic yield functions are adopted.