Particle sizes and their distributions estimated from line- and point-sampled intercepts. Including graphical unfolding

Information about particle size is currently obtained almost exclusively by the use of stereological methods which lead to estimates of the number distribution of linear particle size. The main point of this presentation is to stress the freedom to choose more appropriate parameters for size among a host of options, including particle surface area and volume. Moreover, particle size information may often be considered advantageously in terms of particle distributions based on structural characteristics rather than number distributions. Some of these other distribution types are correctly represented in samples of intercept lengths obtained by line- and point-sampling, respectively. The known and quite simple theory of sampling intercepts is summarized and developed further in several different directions, including a derivation of the distribution of intercept length in ellipsoids, graphical unfolding procedures, and mean size estimators. The potential of the approach is illustrated—but not exhausted—by the existence of a general mean size estimator based on minimal assumptions regarding particle shape.     

Application of design‐based stereology for estimation of absolute volume and surface area of the articular and calcified cartilage compartments of undecalcified human femoral heads

Design‐based stereological methods using systematic uniform random sampling, the Cavalieri estimator and vertical sections are used to investigate undecalcified human femoral heads. Ten entire human femoral heads, obtained from normal women and normal men, were systematically sampled and thin undecalcified vertical sections were obtained. Absolute volumes and surface areas of the entire femoral head, the articular cartilage and the calcified cartilage compartments were estimated. In addition, the average thickness of the articular cartilage and the calcified cartilage were calculated. The stereological procedures applied to the human femoral heads resulted in average coefficient of errors, which were 0.03–0.06 for the volume estimates and 0.03–0.04 for the surface area estimates. We conclude that design‐based stereology using the Cavalieri estimator and vertical sections can successfully be used in large undecalcified tissue specimens, like the human femoral head, to estimate the absolute volume and surface area of macroscopic as well as of microscopic tissue compartments. The application of well‐known design‐based stereological methods carries potential advantage for investigating the pathology in inflammatory and degenerative joint diseases.     

Computer-assisted morphometry: point, intersection, and profile counting and three-dimensional reconstruction.

The use of computers in morphometry can involve 1) automated image analysis, semiautomated image analysis and point, intersection, intercept and profile counts of two-dimensional images on tissue sections with mathematical extrapolation to the third dimension, 2) direct measurement of volumes, surfaces, lengths, and curvature using x,y,z coordinates of serial sectioned images, or 3) stereologic techniques and serial sections which is a combination of 1 and 2 above. Automated and semiautomated image analysis are generally restricted to specimens that are characterized by differential contrast such as interalveolar septa in the lung or histochemically stained mucous granules in pulmonary epithelium. Point, intersection, and profile counts using hand-held, notebook PCs, portable PCs, or standard PCs and MS-DOS-based application programs are extremely efficient, precise, affordable, and convenient methods of quantitating average values of a population. When morphometric measurements of individual structures are required, computer-assisted three-dimensional reconstruction using x,y,z coordinates of the surface outline from serial sections is a tedious yet precise method. We describe a computer program that efficiently estimates mean caliper diameter, volume, and surface area with less than five percent error with five sections per structure. We also describe a program that does digital image subtraction on serial sections, superimposes digitally generated test systems on biological images, and accumulates point, intersection, and profile counts using a Macintosh II series computer.     

Unbiased stereological estimation of the number and volume of nuclei and nuclear size variability in invasive ductal breast carcinomas

The application of design-based stereological methods for estimating nuclear features quantitatively in invasive ductal breast cancer is described. Nuclear number, size and size variability are explored in relation to the tumour grade and patient prognosis. The study includes an examination of the efficiency in estimating different nuclear volumes, and two different estimators of the nuclear size variability are contrasted. Forty-two invasive ductal breast carcinomas diagnosed and graded by two pathologists were used. Both 5-μm and 25-μm-thick sections were obtained from paraffin blocks for stereological study. More undifferentiated tumours show significantly larger nuclei than low-grade tumours. The estimates based on the disector method demonstrate a decrease in the number of tumour cell nuclei per unit volume of tissue from grades 1 to 2 and especially from grades 2 to 3. The univariate survival analysis shows a high prognostic value of the nuclear volume estimates. The study shows that an efficient sampling procedure was performed, particularly when estimating volume-weighted mean nuclear volume using the point-sampled intercepts method. This method is more efficient than estimation of the number-weighted mean nuclear volume using the selector method; however, the latter provides paired estimates of volume- and number-weighted mean nuclear volume, as well as an estimate of the coefficient of variation of nuclear volume in the number distribution of the same cells.     

Estimating the surface area of nonconvex particles from central planar sections

In this paper, we present a new surface area estimator in local stereology. This new estimator is called the ‘Morse‐type surface area estimator’ and is obtained using a two‐stage sampling procedure. First a plane section through a fixed reference point of a three‐dimensional structure is taken. In this section plane, a modification of the area tangent count method is used. The Morse‐type estimator generalizes Cruz‐Orive's pivotal estimator for convex objects to nonconvex objects. The advantages of the Morse‐type estimator over existing local surface area estimators are illustrated in a simulation study. The Morse‐type estimator is well suited for computer‐assisted confocal microscopy and we demonstrate its practicability in a biological application: the surface area estimation of the nuclei of giant‐cell glioblastoma from microscopy images. We also present an interactive software that allows the user to efficiently obtain the estimator.     

Some remarks on the accuracy of surface area estimation using the spatial grid

A set of three line grids in three orthogonal directions is called a spatial grid. This spatial grid can be used for surface area estimation by counting the number of intersection points of a surface with the grid lines. If direction and localization of the spatial grid are suitably randomized, the expectation of this number is proportional to the surface area of interest. The method was especially developed for cases where the surface to be measured is embedded in a medium, which is the usual case in microscopical applications, and where a stack of serial optical sections of the surface is available. The paper presents an improvement of an earlier version of the counting rule for intersection points. Furthermore, if the direction of sectioning is not uniform random, a bias results. This bias is calculated for a disc as a perfectly anisotropic object. A generalization of the estimator is considered by introducing a weighted mean instead of the usual arithmetic mean. The variance due to the randomized direction is investigated depending on the weights, and the minimum of this variance is derived. The relationship between the covariogram and the variance of the surface area estimated with the spatial grid is considered.     

On semiautomatic estimation of surface area

In this paper, we propose a semiautomatic procedure for estimation of particle surface area. It uses automatic segmentation of the boundaries of the particle sections and applies different estimators depending on whether the segmentation was judged by a supervising expert to be satisfactory. If the segmentation is correct the estimate is computed automatically, otherwise the expert performs the necessary measurements manually. In case of convex particles we suggest to base the semiautomatic estimation on the so‐called flower estimator, a new local stereological estimator of particle surface area. For convex particles, the estimator is equal to four times the area of the support set (flower set) of the particle transect. We study the statistical properties of the flower estimator and compare its performance to that of two discretizations of the flower estimator, namely the pivotal estimator and the surfactor. For ellipsoidal particles, it is shown that the flower estimator is equal to the pivotal estimator based on support function measurements along four perpendicular rays. This result makes the pivotal estimator a powerful approximation to the flower estimator. In a simulation study of prolate and oblate ellipsoidal particles, the surfactor also performs well for particles which are not extremely elongated. In particular, the surfactor is not very much affected by the singularity in the surfactor formula or by possible inaccuracies in the necessary angle measurements. We also assess the performance of the semiautomatic procedure in a study of somatostatin positive inhibitory interneurons from mice hippocampi. Only 35% of the cells needed to be analysed manually and an important decrease in workload was obtained by using the semiautomatic approach.     

Stereological estimation of the volume-weighted mean volume of arbitrary particles observed on random sections*

A stereological estimator of the weighted mean volume of particles of arbitrary shape is described. This unbiased estimator is based on simple point-sampling of linear intercept lengths. The complete absence of shape assumptions effectively breaks the long-standing ‘convexity-barrier’: the only requirement here is that individual particles can be unambiguously identified by their profiles on random sections. Practical details of the simple estimation procedure and an example with very irregular particles are reported. Finally, an estimator of the variance of the weighted distribution of particle volume is discussed. This estimator is also valid for particles of arbitrary shape. For any mixture of ellipsoids (spheres, oblates, prolates and triaxial ellipsoids) the estimator is reduced to a simple function of measurements of diameters in the section plane.     

A novel method for mean cell volume estimation

A novel method is described for the estimation of the mean cell volume of cell populations grown in suspension. The cells are filtered onto a nitrocellulose filter to form a cylindrical pellet which is embedded in epoxy resin. Using estimates of pellet height and radius, the number of cells in the pellet and of the volume density of the cells in the pellet, it is possible to produce an unbiased estimate of the mean cell volume. This method is compared, using cell suspensions of the blood parasite Trypanosoma brucei, with mean cell volume estimation using a Coulter channellizer. A Coulter channellizer was also used to compare the mean cell volume of living trypanosomes with that of aldehyde-fixed populations, and the values obtained were compared with those obtained using the new method. The estimated mean cell volume of a T. brucei clone was used to derive values from volume densities obtained by point and intersection counts for the absolute volumes of the flagellar pocket, the nucleus, and endocytic organelles containing internalized horseradish peroxidase and transferrin-gold after 30-min incubations at 310 K. Estimated values for the surface area of the flagellar pocket and the surface area of a cell were also obtained. From known data on the total amount of variant surface glycoprotein molecules per cell and the known packing density of membrane proteins, it was estimated that approximately 80% of the molecules must reside in intracellular compartments. It was estimated that the equivalent of 5% of the surface membrane may be internalized per minute, an amount which is almost the size of the entire flagellar pocket membrane.     

Prediction of the variance of stereological volume estimates from systematic sections using computer-intensive methods

The Cavalieri method is an unbiased estimator of the total volume of a body from its transectional areas on systematic sections. The coefficient of error (CE) of the Cavalieri estimator was predicted by a computer‐intensive method. The method is based on polynomial regression of area values on section number and simulation of systematic sectioning. The measurement function is modelled as a quadratic polynomial, with an error term superimposed. The relative influence of the trend and the error component is estimated by techniques of analysis of variance. This predictor was compared with two established short‐cut estimators of the CE based on transitive theory. First, all predictors were applied to data sets from six deterministic models with analytically known CE. For these models, the CE was best predicted by the older short‐cut estimator and by the computer‐intensive approach, if the measurement function had finite jumps. The best prediction was provided by the newer short‐cut estimator when the measurement function was continuous. The predictors were also applied to published empirical datasets. The first data set consisted of 10 series of areas of systematically sectioned rat hearts with 10–13 items, the second data set consisted of 13 series of systematically sampled transectional areas of various biological structures with 38–90 items. On the whole, similar mean values for the predicted CE were obtained with the older short‐cut estimator and the computer‐intensive method. These ranged in the same order of magnitude as resampling estimates of the CE from the empirical data sets, which were used as a cross‐check. The mean values according to the newer short‐cut CE estimator ranged distinctly lower than the resampling estimates. However, for individual data sets, it happened that the closest prediction as compared to the cross‐check value could be provided by any of the three methods. This finding is discussed in terms of the statistical variability of the resampling estimate itself.     

Stereology for anisotropic cells: Application to growth cartilage*

A number of either new or recently available stereological methods are described for estimating volume, surface area and number of anisotropic cells. The methods are illustrated with direct reference to the epiphyseal growth plate. Different estimates of a given quantity are obtained by applying alternative methods to the same set of sections, in order to compare the relative merits of the methods. For instance, the surface area of the cells is estimated via the Dimroth–Watson model (which gives a measure of the degree of anisotropy in addition to the surface area estimate) and from vertical sections using cycloid test systems. Cell number is estimated by traditional unfolding methods and by the new disector method. Also, volume-weighted mean cell volume is estimated from vertical sections via point-sampled intercepts using two different kinds of rulers to classify intercept lengths. Finally, nested design statistics is applied to a set of data from twelve animals in order to compare the relative impacts of biological and stereological (sampling) variations on the observed coefficient of error of a group mean estimate. The preferred methods are listed in the final section.     

The influence of tissue processing on quantitative histopathology in breast cancer

Objective grading of breast cancer by morphometry has been suggested for improving the precision of the prognostic prediction. However, the tissue components evaluated might be influenced by variations in the processing, reducing the clinical value. In the present study, the impact of the period of fixation, of the acidity of the fixative and of the embedding medium was investigated by allocating tissue samples from 27 surgical breast cancer specimens systematically randomly to different modes of processing. The volume-weighted mean volume of cancer cell nuclei, v?_V(nuc), was estimated using the method of point-sampled intercepts on vertical sections. In addition, estimates of the mean nuclear profile area, ā_H(nuc), the nuclear volume fraction, V_V(nuc), the nuclear profile density, ND, and the mitotic profile frequency, MF, were obtained. The quantitative histopathological estimates were stable with respect to the investigated variables of the tissue processing. No significant differences were found when comparing the estimates obtained in samples from five tumours fixed in formalin at pH 5·0, 6·0, 7·0, 7·4 and 8·0, respectively. Similarly, no significant correlations between the estimates and the period of formalin fixation (24 h, 3 days and 3 months) were found in samples from five other tumours. However, the v?_V(nuc) was 13% larger (2p = 0·004) and the mean ND 17% smaller (2p = 0·04) in hydroxyethyl-methacrylate-embedded samples from 17 tumours as compared to paraffin-embedded samples. Thus, the shrinkage observed in paraffin seems to affect nuclei less than tissue.     

Classification of spatial textures in benign and cancerous glandular tissues by stereology and stochastic geometry using artificial neural networks

Stereology and stochastic geometry can be used as auxiliary tools for diagnostic purposes in tumour pathology. Whether first-order parameters or stochastic-geometric functions are more important for the classification of the texture of biological tissues is not known. In the present study, volume and surface area per unit reference volume, the pair correlation function and the centred quadratic contact density function of epithelium were estimated in three case series of benign and malignant lesions of glandular tissues. The information provided by the latter functions was summarized by the total absolute areas between the estimated curves and their horizontal reference lines. These areas are considered as indicators of deviation of the tissue texture from a completely uncorrelated volume process and from the Boolean model with convex grains, respectively. We used both areas and the first-order parameters for the classification of cases using artificial neural networks (ANNs). Learning vector quantization and multilayer feedforward networks with backpropagation were applied as neural paradigms. Applications included distinction between mastopathy and mammary cancer (40 cases), between benign prostatic hyperplasia and prostatic cancer (70 cases) and between chronic pancreatitis and pancreatic cancer (60 cases). The same data sets were also classified with linear discriminant analysis. The stereological estimates in combination with ANNs or discriminant analysis provided high accuracy in the classification of individual cases. The question of which category of estimator is the most informative cannot be answered globally, but must be explored empirically for each specific data set. Using learning vector quantization, better results could often be obtained than by multilayer feedforward networks with backpropagation.     

The optical rotator

The optical rotator is an unbiased, local stereological principle for estimation of cell volume and cell surface area in thick, transparent slabs. The underlying principle was first described in 1993 by Kiêu &38; Jensen ( J. Microsc170, 45–51) who also derived an estimator of length. In this study we further discuss the methods derived from this principle and present two new local volume estimators.
The optical rotator benefits from information obtained in all three dimensions in thick sections but avoids over-/underprojection problems at the extremes of the cell. Using computer-assisted microscopes the extra measurements demand minimal extra effort and make this estimator even more efficient when it comes to estimation of individual cell size than many of the previous local estimators. We demonstrate the principle of the optical rotator in an example (the cells in the dorsal root ganglion of the rat), evaluate its efficiency and compare it with other unbiased, local stereological principles available for estimation of cell volume and surface area.     

Methods for Digital Particle Image Sizing (DPIS): Comparisons and improvements

This paper explores the accuracy of particle image sizing using direct processing of digitally recorded images. Traditional methods for particle image sizing were considered and, four new algorithms were developed to deliver improved accuracy and robustness. Statistical error analysis was performed using Monte Carlo simulations in order to quantify the dependence of these methods on noise, discretization, and particle size distribution. The performance of these methods were compared against Phase Doppler Analyzer measurements of spray atomization.We introduce a novel two-dimensional four-point Gaussian estimator and an alternative Gaussian estimator based on a local least squares (LLS) fit. These methods were further advanced to account for pixel discretization effects using integral formulations (continuous methods). All new methods were compared against conventional pixel counting and the established three-point Gaussian estimator. The new methods significantly reduced the total error in the diameter estimation compared to the three-point Gaussian estimator and pixel counting. The least squares Gaussian estimator and its continuous version demonstrated almost identical results and superior performance for diameters over 4 pixels. For smaller diameters, the continuous four-point Gaussian estimator delivered the highest accuracy. For uniform particle size distribution between 2–14 pixels image diameter, the least squares estimators delivered error less than 5% with respect to the true diameter for 80% of the particles. The remaining methods demonstrated error of 5% (or better) for less than 60% of the particles. Validation in an experiment of high-pressure spray atomization showed that the Gaussian local least squares methods and the continuous four-point method delivered similar particle size distribution compared to PDA. The particle mean diameter estimated by the two methods differed only by 3% and 6% respectively with respect to the PDA measurements.The novel particle image sizing schemes developed here can deliver accurate, robust, and computationally efficient apparent diameter measurement, thus providing a viable, simple and inexpensive solution for performing sizing on conventional particle image velocimetry images. This capability enables simultaneous measurements of both velocity and particle size for a wide range of multi-phase flows.     

Unbiased estimation of human body composition by the Cavalieri method using magnetic resonance imaging

The classical methods for estimating the volume of human body compartments in vivo (e.g. skin-fold thickness for fat, radioisotope counting for different compartments, etc.) are generally indirect and rely on essentially empirical relationships — hence they are biased to unknown degrees. The advent of modern non-invasive scanning techniques, such as X-ray computed tomography (CT) and magnetic resonance imaging (MRI) is now widening the scope of volume quantification, especially in combination with stereological methods. Apart from its superior soft tissue contrast, MRI enjoys the distinct advantage of not using ionizing radiations. By a proper landmarking and control of the scanner couch, an adult male volunteer was scanned exhaustively into parallel systematic MR ‘sections’. Four compartments were defined, namely bone, muscle, organs and fat (which included the skin), and their corresponding volumes were easily and efficiently estimated by the Cavalieri method: the total section area of a compartment times the section interval estimates the volume of the compartment without bias. Formulae and nomograms are given to predict the errors and to optimize the design. To estimate an individual's muscle volume with a 5% coefficient of error, 10 sections and less than 10min point counting (to estimate the relevant section areas) are required. Bone and fat require about twice as much work. To estimate the mean muscle volume of a population with the same error contribution, from a random sample of six subjects, the workload per subject can be divided by √6, namely 4 min per subject. For a given number of sections planimetry would be as accurate but far more time consuming than point counting.     

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.     

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.     

Rapid Prototyping Applications in Medicine. Part 2: STL File Generation and Case Studies

Rapid prototyping (RP) technology has extended traditional manufacturing applications in areas other than product engineering. Using RP to fabricate custom implants and prosthesis for surgical planning and education is now an important area of research. Although, in theory, RP is capable of producing objects of any complexity, designing freeform shapes is difficult using current CAD systems. These CAD systems are geared toward the design of parts manufactured by traditional methods; they do not help designers exploit the expanded opportunities offered by RP technology. Medical data cannot be input into these CAD systems directly for further modification and manipulation. The purpose of this project is to explore a new approach for modelling and prototyping biomedical objects. The work extends from volume modelling to RP and medicine. In Part 1 of two papers, a new approach to modelling complex objects, NURBS-based volume modelling, is proposed. A NURBS representation of volumes is developed to represent not only the surface boundary but also the interior of a 3D object. NURBS-based volume modelling inherits advantages from both NURBS modelling and voxel-based modelling. The key idea of the NURBS-based volume modelling is to exploit the flexibility of NURBS modelling and use the voxelised NURBS volumes as components for constructing complex objects. This paper, Part 2, deals mainly with issues of interfacing volume models to RP systems. A new approach to generate STL files through volume modelling and iso-surface extraction is proposed. This approach guarantees the validity of the final STL file inherently. Software development and case studies are also given.