Stereological Corrections Research Articles

Chondrule sizes within the CM carbonaceous chondrites and measurement methodologies

AbstractThe sizes of chondrules are a valuable tool for understanding relationships between meteorite groups and the affinity of ungrouped chondrites, documenting temporal/spatial variability in the solar nebula, and exploring the effects of parent body processing. Many of the recently reported sizes of chondrules within the CM carbonaceous chondrites differ significantly from the established literature average and are more closely comparable to those of chondrules within CO chondrites. Here, we report an updated analysis of chondrule dimensions within the CM group based on data from 1937 chondrules, obtained across a suite of CM lithologies ranging from petrologic subtypes CM2.2 to CM2.7. Our revised average CM chondrule size is 194 μm. Among the samples examined, a relationship was observed between petrologic subtype and chondrule size such that chondrule long‐axis lengths are greater in the more highly aqueously altered lithologies. These findings suggest a greater similarity between the CM and CO chondrites than previously thought and support arguments for a genetic link between the two groups (i.e., the CM‐CO clan). Using the 2‐D and 3‐D data gathered, we also apply numerous stereological corrections to examine their usefulness in correcting 2‐D chondrule measurements within the CM chondrites. Alongside this analysis, we present the details of a standardized methodology for 2‐D chondrule size measurement to facilitate more reliable inter‐study comparisons.

Stereologically corrected particle size distributions for polymer-mounted additive manufacturing powders

Well-defined particle size distributions are required for good flowability and powder packing properties of additive manufacturing powders. Mounting powders within a polymer and using standard metallurgical preparation techniques to cross-section and prepare powder particles for optical analysis allows for simple characterisation processes. However, measured diameters of cross-sectioned particles are typically underestimates of actual particle diameters and hence require stereological correction. The effectiveness of three stereological corrections are investigated in this work, namely the Scheil-Schwartz-Saltykov method, the Goldsmith-Cruz-Orive method and a Finite Difference Method. These methods are investigated against plasma-atomised, gas-atomised and ultrasonically processed Ti-6Al-4V powders. The corrected outputs are compared to laser size diffraction, benchmark data for each powder. Although all three stereological corrections produce improved estimations of the particle size distributions, the Finite Difference Method is recommended producing cumulative mean absolute error values of 2.4%, 3.1% and 7.5% for the plasma-atomised, gas-atomised and ultrasonically processed powders respectively.

Particle size distribution for additive manufacturing powder using stereological corrections

Additive manufacturing powders require well-defined Particle Size Distributions (PSDs) and spherical morphology for good powder flowability. To simplify characterisation processes, powders can be prepared using standard metallurgical preparation techniques, followed by optical imaging to determine the PSD of cross-sectioned particles. However, this measured PSD typically provides underestimates of the true diameters; hence, stereological corrections must be applied. Three stereological correction methods: the Scheil-Schwartz-Saltykov (SSS) method; the Goldsmith and Cruz-Orive (GCO) method and a Finite Difference Method (FDM) are assessed. Laser Size Diffraction (LSD) analysis provided the ground truth data. Particle cross-sections of 2,280 powder particles (Ti-6Al-4V) were analysed in the size range of 9–76 μm. The mean absolute errors were found to be 2.3% for the SSS method, 2.4% for the GCO method, and 1.6% for the FDM method. Hence, whilst all three methods provided an improved estimate of the PSD, FDM was determined the most effective method in this case.

Quantifying mineral liberation by particle grade and surface exposure using X-ray microCT

Liberation is a key driver in all mineral separation processes as it limits the maximum possible grade for a given recovery. In flotation, this is further complicated by the fact that it is surface exposure of the floatable minerals that determines the ultimate performance. Liberation, grade and surface exposure are commonly quantified using Scanning Electron Microscopy coupled to Energy Dispersive X-ray spectroscopy (SEM/EDX) analysis of polished sections. The intrinsically 2D nature of this technique can result in significant sampling errors and stereological effects that can affect the quantification of the ore’s textural characteristics. X-ray microCT (XMT) is an imaging method that can non-invasively and non-destructively delineate ore fragments in 3D, thus providing an alternative method that eliminates the need for stereological corrections and readily provides surface exposure. A methodology and automated algorithm were developed for extracting this information from images of closely packed particles. By dividing these particles into classes based on both their surface exposure and grade, the extent to which there is preferential breakage of the particles can be assessed—an important consideration if sufficient surface liberation for good flotation performance is to be achieved at coarser particle sizes. Using low energy scanning simple 3D mineral maps can be obtained via XMT, allowing for the assessment of liberation and surface exposure for each mineral species. The methodology was tested on low grade porphyry copper ore as this is representative of the most commonly treated ore types for copper production.

QUICK DETERMINATION OF CRYSTAL SIZE DISTRIBUTIONS OF ROCKS BY MEANS OF A COLOR SCANNER

An acquisition and analysis method based on a commercial, low-cost, high-resolution film scanner is presented. It allows to collect data from standard rock thin sections with a resolution up to 9.4 μm per pixel. Common and general purpose facilities (scanner + PC + image analysis software) may thus be transformed in an appropriate tool for quantitative textural analysis of rocks. The procedure implies the acquisition of four images with crossed polarizers and one parallel light image. Crystal boundaries are extracted from fields in crossed polarizers, while markers for mineral recognition are obtained thresholding the parallel light image. The method is tested for fresh rocks with simple mineralogy (harzburgites and marbles) with no more than three phases, all exhibiting well distinct optical properties. Image processing is performed developing procedures with VISILOG 5.2 package. 2-D size data from binary images are converted to 3-D size data applying stereological corrections. 3-D data are reported in bi-logarithmic diagrams, plotting the crystal number density versus characteristic lengths. The harzburgite samples show a scale invariance of size distributions of olivine while mosaic equant marbles exhibit a different size distribution pattern, without scale invariance and a relative maximum.

Particle texture analysis using polarized light imaging and grey level intercepts

The usual characterization of particulate ore material with image analysis includes modal analysis and liberation analysis. Both methods include stereological corrections based on intercept length distributions within each phase of interest. The first principle of stereology relies on a simple assumption of uniform random sectioning, whereas the liberation models require all particles to have similar textural properties. However, for the sake of adequate prediction of ore behavior in industrial processes it is becoming more and more important to be able to describe particle populations in terms of texture classes. Among the texture analysis techniques, this paper focuses on mean intercept length (MIL). It brings two important improvements: the first one is to improve grain boundary imaging by using images of bireflectant minerals under plane polarized illumination, the second one is to consider intercept analysis on grey level images instead of binary ones. The method is illustrated on a selection of critical hematite textures found within banded iron formations.

Geometric probability based stereological corrections

In mineral processing, a classic problem is that of estimating the particle composition distribution from particle sections. This problem occurs because mineralogical analysis is performed on particle sections rather than the particles. Thus one wishes to infer three-dimensional information (particle composition distribution) from two-dimensional information (particle sections). There are a number of different techniques available for estimating the particle composition distribution and these are called stereological corrections. This paper provides examples of very simple stereological corrections based on geometric probability equations as well as the principle of maximum entropy. The paper focuses on three methods. The first method is a simple entropy based model based on particle section information only. The second method also uses entropy, but the entropy is based on relating particles to particle sections, and is therefore more complex than the first. The third model uses both the particle section distribution and the linear intercept distribution. The most accurate model is that which uses both particle sections and linear intercepts.

Stereological Corrections for Grain Boundary Number Fractions in Three Dimensions

AbstractTwo-dimensional (2D) cross sections through three-dimensional (3D) polycrystalline materials present a biased picture of the statistical properties of grain boundary networks. These properties are essential to many practical applications such as grain boundary engineering. We show a simple correction that will partly correct for the sampling biases by removing the effect of the correlation between grain boundary type and grain boundary area. This correction alters number fraction estimates by as much as ∼60% for σ3 boundaries in the highly-twinned copper samples we consider. We also estimate the bias introduced by the correlation between boundary type and boundary shape, which for many materials represents perhaps a 10% shift in the measured statistics, so that the simple method we propose should correct for the majority of the bias in favorable cases.This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.

Predicting the permeability of sandstone from image analysis of pore structure

A model is developed that allows accurate prediction of the permeability of a core sample of sedimentary rock, based on two-dimensional image analysis of its pore structure. The pore structure is idealized as consisting of a cubic network of pore tubes, with the tubes having an arbitrary distribution of cross-sectional areas and shapes. The areas and perimeters of the individual pores are estimated from image analysis of scanning electron micrographs of thin sections, with appropriate stereological corrections introduced to account for the angle between axis of the pore tube and plane of the thin section. The individual conductances of each tube are estimated from the measured areas and perimeters, using the hydraulic radius approximation. Variations in the pore diameter along the length of the tube are accounted for with a “constriction factor” whose derivation is based on laminar flow through an irregular tube. Effective-medium theory is used to find the effective single-tube conductance, based on the measured distribution of individual conductances. This procedure is applied to several consolidated North Sea reservoir sandstones, and some outcrop sandstones, with permeabilities ranging from 20 to 1400 mD. The predicted permeabilities are typically within a factor of 2 of the measured values, with an average error in logk of only 0.168.

Power law olivine crystal size distributions in lithospheric mantle xenoliths

Olivine crystal size distributions (CSDs) have been measured in three suites of spinel- and garnet-bearing harzburgites and lherzolites found as xenoliths in alkaline basalts from Canary Islands, Africa; Victoria Land, Antarctica; and Pali Aike, South America. The xenoliths derive from lithospheric mantle, from depths ranging from 80 to 20 km. Their textures vary from coarse to porphyroclastic and mosaic–porphyroclastic up to cataclastic. Data have been collected by processing digital images acquired optically from standard petrographic thin sections. The acquisition method is based on a high-resolution colour scanner that allows image capturing of a whole thin section. Image processing was performed using the VISILOG 5.2 package, resolving crystals larger than about 150 μm and applying stereological corrections based on the Schwartz–Saltykov algorithm. Taking account of truncation effects due to resolution limits and thin section size, all samples show scale invariance of crystal size distributions over almost three orders of magnitude (0.2–25 mm). Power law relations show fractal dimensions varying between 2.4 and 3.8, a range of values observed for distributions of fragment sizes in a variety of other geological contexts. A fragmentation model can reproduce the fractal dimensions around 2.6, which correspond to well-equilibrated granoblastic textures. Fractal dimensions >3 are typical of porphyroclastic and cataclastic samples. Slight bends in some linear arrays suggest selective tectonic crushing of crystals with size larger than 1 mm. The scale invariance shown by lithospheric mantle xenoliths in a variety of tectonic settings forms distant geographic regions, which indicate that this is a common characteristic of the upper mantle and should be taken into account in rheological models and evaluation of metasomatic models.

Comparative morphometric study of immunohistochemical versus conventional staining for the evaluation of megakaryocytopoiesis in normal and pathological bone marrow biopsies

Identification of megakaryocytes by immunohistochemistry may be superior to hematoxylin-eosin (HE) stain method for assessing megakaryocyte size and number in clinical specimens; however, a side-by-side comparison of the two methods has not been reported. In the present study, comparative morphometry using both methods was performed on marrow biopsies of normal individuals, and of patients with myelodysplastic syndrome, chronic myeloid leukemia and immune thrombocytopenia. Morphometric results in the present study showed that precise megakaryocyte size can be calculated in normal and pathologic bone marrow sections by using HE stain if one employs stereological corrections. In contrast, megakaryocyte numbers can be more precisely detected by immunohistochemistry than by HE stain, particularly in myelodysplastic syndrome and chronic myeloid leukemia. Differentiation disturbances and ineffective megakaryocytopoiesis in myelodysplastic syndrome were demonstrated by immunomorphometric analyses.

Reduction in spine density associated with long-term potentiation in the dentate gyrus suggests a spine fusion-and-branching model of potentiation.

Approximately 2,700 dendritic spines in Golgi-impregnated hippocampal granule cells were quantified via image analysis 24 h after the unilateral induction of long-term potentiation in seven rats. Stereological corrections were made using a tilting disector and analytical unfolding technique. In the potentiated hemisphere the mean spine density along dendrites was reduced by approximately 20%. The relative frequency of shorter, thicker spines was increased in potentiated tissue. Physiological consequences of two morphological changes leading to a reduction in spine density (retraction or fusion of spines) were examined using a compartmental model of a simplified granule cell. The model was constructed in the NEURON modeling environment and included a realistic population of 60 dendritic spines (with dual-component synapses and active Ca(2+)-dependent mechanisms). Simulations demonstrated that potentiation of postsynaptic responses was compatible with fusion (with branching) of a proportion of spines with their neighbors but was not compatible with retraction of spines. This result held over wide variations of model parameters as long as dendritic membranes were assumed to be excitable.

On variables that affect estimates of the true sizes and densities of radioactively labeled cell nuclei

Tritiated thymidine has been widely used as a nuclear marker of cell birth. The true diameters and packing densities (nuclei/microns 3) of such radioactively labeled nuclei cannot be measured directly from tissue sections. Here we show that existing stereological corrections cannot be applied to data from radioactively labeled nuclei. We empirically measured the number of silver grains exposed by nuclei containing tritiated thymidine. The nuclei were separated from the photographic emulsion by known thicknesses of fixed, embedded avian telencephalon. The results of this experiment were used to develop an equation that estimates the number of silver grains exposed by a cell nucleus of any given diameter, containing a given amount of radioactive label, and located at any given distance from the photographic emulsion. The equation also allows one to calculate the probability that a label-containing nucleus will be correctly classified as labeled. Simulations of the equation revealed that not all label-containing nuclei are correctly classified by using commonly employed identification procedures and that larger nuclei are less likely to be correctly classified than smaller nuclei, given the same amount of label. The equation can be used to modify one class of existing stereological equations so as to be applicable to measurements of radioactively labeled nuclei. Finally, we discuss the assumptions and limitations of this modification.

Changes in the numerical density of synaptic contacts with long-term potentiation in the hippocampal dentate gyrus.

Long-term potentiation (LTP) in the rat dentate gyrus is a multifaceted phenomenon, including synaptic potentiation; simultaneous synaptic depression at neighboring, unconditioned synapses; and a change in the amount of cell firing produced by a specified amount of synaptic current (see Levy and Desmond: In G. Buzsaki and C. Vanderwolf (eds): Electrical Activity of The Archicortex. Budapest: Akademiai Kiado, pp. 359-373, '85b). This study presents long-term anatomical modifications that seem related to excitatory synaptic modification. These anatomical alterations appear early and persist for at least 60 minutes following conditioning stimulation. Each animal received test pulse stimulation delivered alternately to the angular bundles before and after brief, unilateral high-frequency conditioning stimulation that is typical of many LTP paradigms. Anatomical preparation followed standard procedures. Double-blind scoring procedures quantified the number of asymmetric synapses in the dentate molecular layer. These counts were converted to the number of synapses per unit volume using stereological corrections that combined geometrically derived theory and modest serial sectioning. Multivariate analysis of variance evaluated the statistical significance of changes in synapse density. Across all three groups of animals, conditioning stimulation does not significantly change the density of synaptic contacts across the entire molecular layer. There is a trend for a decreased density of synaptic contacts in the middle molecular layer, the region activated by the conditioning stimulation. Here the density of concave spine profiles increases significantly in all three groups of animals with conditioning stimulation. This increase accompanies significant decreases in the density of nonconcave, simple and ellipsoid, spine profiles. No significant changes in the density of shaft synapses occur with LTP-inducing conditioning stimulation. These data suggest that the concave spine profiles are a correlate of LTP-inducing stimulation and may be the potentiated synapses. We hypothesize that with synaptic potentiation there occurs an interconversion of spine synapses such that some nonconcave spine profiles become concave spine profiles. Such an interconversion apparently begins shortly after the conditioning stimulation and persists for at least 60 minutes.

Quantification of branched neuronal projections labelled by retrograde fluorescent tracing. A study of olivo-cerebellar projections

A method for estimating the number of branched and unbranched neurones projecting from a nucleus to two target sites is presented, based on the retrograde transport of fluorescent tracers. The method initially involves stereological corrections for the size of cytoplasm and nuclei respectively labelled by the two tracers. A second correction is applied to account for doubly labelled cells whose cytoplasm, but not nuclei, are in the plane of section. Finally, the detection rates of the two tracers are determined and appropriate corrections are applied. The projection from the medial accessory olive to the cerebellar vermis was studied using true blue and diamidino yellow to illustrate the method. Application of the method increased the number of branched neurones detected by 18.5%. Of the total increase, 48.4% was due to the correction for size, 9.2% to the correction for doubly labelled cells with nuclei outside the plane of the section and 42.4% to the correction for detection rates. There was no significant masking of one tracer by another, but true blue enhanced the fluorescence of diamidino yellow.