Improved variance prediction for systematic sampling on ℝ

Improving Cavalieri volume estimation based on non-equidistant planar sections: The trapezoidal estimator

The Cavalieri estimator allows one to infer the volume of an object from area measurements in equidistant planar sections. It is known that applying this estimator in the non-equidistant case may inflate the coefficient of error considerably. We therefore consider a newly introduced variant, the trapezoidal estimator, and make it available to practitioners. Its typical variance behaviour for natural objects is comparable to the equidistant case. We state this unbiased estimator, describe variance estimates and explain how the latter can be simplified under rather general but realistic models for the gaps between sections. Simulations and an application to a synthetic area function based on parietal lobes of 18 monkeys illustrate the new methods.

Basic quantitative morphological methods applied to the central nervous system

Generating numbers has become an almost inevitable task associated with studies of the morphology of the nervous system. Numbers serve a desire for clarity and objectivity in the presentation of results and are a prerequisite for the statistical evaluation of experimental outcomes. Clarity, objectivity, and statistics make demands on the quality of the numbers that are not met by many methods. This review provides a refresher of problems associated with generating numbers that describe the nervous system in terms of the volumes, surfaces, lengths, and numbers of its components. An important aim is to provide comprehensible descriptions of the methods that address these problems. Collectively known as design-based stereology, these methods share two features critical to their application. First, they are firmly based in mathematics and its proofs. Second and critically underemphasized, an understanding of their mathematical background is not necessary for their informed and productive application. Understanding and applying estimators of volume, surface, length or number does not require more of an organizational mastermind than an immunohistochemical protocol. And when it comes to calculations, square roots are the gravest challenges to overcome. Sampling strategies that are combined with stereological probes are efficient and allow a rational assessment if the numbers that have been generated are "good enough." Much may be unfamiliar, but very little is difficult. These methods can no longer be scapegoats for discrepant results but faithfully produce numbers on the material that is assessed. They also faithfully reflect problems that associated with the histological material and the anatomically informed decisions needed to generate numbers that are not only valid in theory. It is within reach to generate practically useful numbers that must integrate with qualitative knowledge to understand the function of neural systems.

Sampling the Mouse Hippocampal Dentate Gyrus

Sampling is a critical step in procedures that generate quantitative morphological data in the neurosciences. Samples need to be representative to allow statistical evaluations, and samples need to deliver a precision that makes statistical evaluations not only possible but also meaningful. Sampling generated variability should, e.g., not be able to hide significant group differences from statistical detection if they are present. Estimators of the coefficient of error (CE) have been developed to provide tentative answers to the question if sampling has been “good enough” to provide meaningful statistical outcomes. We tested the performance of the commonly used Gundersen-Jensen CE estimator, using the layers of the mouse hippocampal dentate gyrus as an example (molecular layer, granule cell layer and hilus). We found that this estimator provided useful estimates of the precision that can be expected from samples of different sizes. For all layers, we found that a smoothness factor (m) of 0 generally provided better estimates than an m of 1. Only for the combined layers, i.e., the entire dentate gyrus, better CE estimates could be obtained using an m of 1. The orientation of the sections impacted on CE sizes. Frontal (coronal) sections are typically most efficient by providing the smallest CEs for a given amount of work. Applying the estimator to 3D-reconstructed layers and using very intense sampling, we observed CE size plots with m = 0 to m = 1 transitions that should also be expected but are not often observed in real section series. The data we present also allows the reader to approximate the sampling intervals in frontal, horizontal or sagittal sections that provide CEs of specified sizes for the layers of the mouse dentate gyrus.

Efficient and Unbiased Estimation of Population Size

Population sizing from still aerial pictures is of wide applicability in ecological and social sciences. The problem is long standing because current automatic detection and counting algorithms are known to fail in most cases, and exhaustive manual counting is tedious, slow, difficult to verify and unfeasible for large populations. An alternative is to multiply population density with some reference area but, unfortunately, sampling details, handling of edge effects, etc., are seldom described. For the first time we address the problem using principles of geometric sampling. These principles are old and solid, but largely unknown outside the areas of three dimensional microscopy and stereology. Here we adapt them to estimate the size of any population of individuals lying on an essentially planar area, e.g. people, animals, trees on a savanna, etc. The proposed design is unbiased irrespective of population size, pattern, perspective artifacts, etc. The implementation is very simple—it is based on the random superimposition of coarse quadrat grids. Also, an objective error assessment is often lacking. For the latter purpose the quadrat counts are often assumed to be independent. We demonstrate that this approach can perform very poorly, and we propose (and check via Monte Carlo resampling) a new theoretical error prediction formula. As far as efficiency, counting about 50 (100) individuals in 20 quadrats, can yield relative standard errors of about 8% (5%) in typical cases. This fact effectively breaks the barrier hitherto imposed by the current lack of automatic face detection algorithms, because semiautomatic sampling and manual counting becomes an attractive option.

Sampling theory and automated simulations for vertical sections, applied to human brain

In recent years, there have been substantial developments in both magnetic resonance imaging techniques and automatic image analysis software. The purpose of this paper is to develop stereological image sampling theory (i.e. unbiased sampling rules) that can be used by image analysts for estimating geometric quantities such as surface area and volume, and to illustrate its implementation. The methods will ideally be applied automatically on segmented, properly sampled 2D images - although convenient manual application is always an option - and they are of wide applicability in many disciplines. In particular, the vertical sections design to estimate surface area is described in detail and applied to estimate the area of the pial surface and of the boundary between cortex and underlying white matter (i.e. subcortical surface area). For completeness, cortical volume and mean cortical thickness are also estimated. The aforementioned surfaces were triangulated in 3D with the aid of FreeSurfer software, which provided accurate surface area measures that served as gold standards. Furthermore, a software was developed to produce digitized trace curves of the triangulated target surfaces automatically from virtual sections. From such traces, a new method (called the 'lambda method') is presented to estimate surface area automatically. In addition, with the new software, intersections could be counted automatically between the relevant surface traces and a cycloid test grid for the classical design. This capability, together with the aforementioned gold standard, enabled us to thoroughly check the performance and the variability of the different estimators by Monte Carlo simulations for studying the human brain. In particular, new methods are offered to split the total error variance into the orientations, sectioning and cycloid components. The latter prediction was hitherto unavailable--one is proposed here and checked by way of simulations on a given set of digitized vertical sections with automatically superimposed cycloid grids of three different sizes. Concrete and detailed recommendations are given to implement the methods.

The precision of estimates in stereological analyses

When quantifying the total volume, number, length, or surface of morphological features in complex structures such as the brain, it is neither desirable nor feasible to make absolute determinations of these parameters. It would take a lifetime to count all of the neurons or measure the total length of dendrites in many parts of the brain. Estimates based on a relatively small number of samples of even the most complex structures will suffice, if the estimates are unbiased and have a sufficient amount of precision. It is important to have an understanding of what the precision of an estimate actually means and how it can be calculated and altered, to optimize studies with regard to the number of individuals, sections, and probes used in an analysis. An optimal sampling scheme is one that involves a minimum of sampling but still provides enough precision to draw a conclusion with a predetermined level of confidence.

Orbital prefrontal cortex volume predicts social network size: an imaging study of individual differences in humans

The social brain hypothesis, an explanation for the unusually large brains of primates, posits that the size of social group typical of a species is directly related to the volume of its neocortex. To test whether this hypothesis also applies at the within-species level, we applied the Cavalieri method of stereology in conjunction with point counting on magnetic resonance images to determine the volume of prefrontal cortex (PFC) subfields, including dorsal and orbital regions. Path analysis in a sample of 40 healthy adult humans revealed a significant linear relationship between orbital (but not dorsal) PFC volume and the size of subjects' social networks that was mediated by individual intentionality (mentalizing) competences. The results support the social brain hypothesis by indicating a relationship between PFC volume and social network size that applies within species, and, more importantly, indicates that the relationship is mediated by social cognitive skills.

Sulcal morphology and volume of Broca's area linked to handedness and sex

We investigated the effect of handedness and sex on: (i) sulcal contours defining PO and PTR and (ii) volume estimates of PO and PTR subfields in 40 left- and 42 right-handers. Results show an effect of handedness on discontinuity of the inferior frontal sulcus (IFS: P<0.01). Discontinuity of IFS was observed in: 43% left- and 62% right hemispheres in right-handers and in 65% left- and 48% right-hemispheres in left-handers. PO volume asymmetry was rightward in left-handed males (P=0.007) and females (P=0.02), showed a leftward trend in right-handed males (P=0.06), and was non-asymmetrical in right-handed females (P=0.96, i.e. left- and right-hemisphere PO volumes did not differ significantly). PO volume asymmetry in males differed significantly between handedness groups (P=0.001). Findings indicate a high degree of variability in the sulcal contours of PO and PTR and volume asymmetry of PO: the factors sex and handedness can explain some of this variability.

Comparison of three methods for the estimation of pineal gland volume using magnetic resonance imaging

Pineal gland is a very important neuroendocrine organ with many physiological functions such as regulating circadian rhythm. Radiologically, the pineal gland volume is clinically important because it is usually difficult to distinguish small pineal tumors via magnetic resonance imaging (MRI). Although many studies have estimated the pineal gland volume using different techniques, to the best of our knowledge, there has so far been no stereological work done on this subject. The objective of the current paper was to determine the pineal gland volume using stereological methods and by the region of interest (ROI) on MRI. In this paper, the pineal gland volumes were calculated in a total of 62 subjects (36 females, 26 males) who were free of any pineal lesions or tumors. The mean ± SD pineal gland volumes of the point-counting, planimetry, and ROI groups were 99.55 ± 51.34, 102.69 ± 40.39, and 104.33 ± 40.45 mm³, respectively. No significant difference was found among the methods of calculating pineal gland volume (P > 0.05). From these results, it can be concluded that each technique is an unbiased, efficient, and reliable method, ideally suitable for in vivo examination of MRI data for pineal gland volume estimation.

Comparison of three methods for the estimation of the pituitary gland volume using magnetic resonance imaging: a stereological study

Stereological techniques using point counting and planimetry have been used to estimate pituitary gland volume. However, many studies have estimated pituitary gland volume by the mathematical approach the elliptic formula. The objective of the current study was to determine pituitary gland volume using stereological methods and elliptic formula on magnetic resonance imaging (MRI). In this study, pituitary gland volumes were estimated in a total of 28 subjects (22 females, 6 males,) who were free of any pituitary or neurological symptoms and signs. The mean ± SD pituitary gland volumes for the point counting, planimetry and elliptic formulae groups were 582.14 ± 140.16 mm³, 610.08 ± 133.17 mm³, and 432.82 ± 147.38 mm³, respectively. The mean CE for the pituitary gland volume estimates derived from the point counting technique was 8.07%. No significant difference was found between point counting and planimetric methods for the pituitary gland volume (P > 0.05). In addition, there was a 26.14 and 29.71% underestimation of pituitary volume as measured by the elliptic formula compared to the point counting and planimetric techniques, respectively. From these results, it can be concluded that stereological techniques are unbiased, efficient and reliable methods and are ideally suitable for in vivo examination of MRI data for pituitary gland volume estimation. Hence, we suggest that estimating pituitary gland volume using MRI study and stereology may be clinically relevant for pituitary surgeons for the investigation of the structure and function of the pituitary gland.

Nondestructive, stereological estimation of canopy surface area

We describe a stereological procedure to estimate the total leaf surface area of a plant canopy in vivo, and address the problem of how to predict the variance of the corresponding estimator. The procedure involves three nested systematic uniform random sampling stages: (i) selection of plants from a canopy using the smooth fractionator, (ii) sampling of leaves from the selected plants using the fractionator, and (iii) area estimation of the sampled leaves using point counting. We apply this procedure to estimate the total area of a chrysanthemum (Chrysanthemum morifolium L.) canopy and evaluate both the time required and the precision of the estimator. Furthermore, we compare the precision of point counting for three different grid intensities with that of several standard leaf area measurement techniques. Results showed that the precision of the plant leaf area estimator based on point counting is high. Using a grid intensity of 1.76 cm(2)/point we estimated plant and canopy surface areas with accuracies similar to or better than those obtained using image analysis and a commercial leaf area meter. For canopy surface areas of approximately 1 m(2) (10 plants), the fractionator leaf approach with sampling fraction equal to 1/9 followed by point counting using a 4.3 cm(2)/point grid produced a coefficient of error of less than 7%. The smooth fractionator can be used to ensure that the additional contribution to the estimator variance due to between-plant variability is small.

Confidence intervals for the volume of brain structures in Cavalieri sampling with local errors

The identification and quantification of morphological alterations that occur in the brain due to neurological disease, development and ageing, is of special interest in brain research. Design-based stereological methods have been widely applied in combination with magnetic resonance (MR) imaging to estimate the volume of brain structures. In the Cavalieri method, the volume V is directly estimated from equidistant and parallel MR images of the brain with a uniform random starting position. A second level of sampling is usually required to estimate the section area from each image (e.g., by applying point counting). The mathematical justification and implementation of the methodology is simple and it can be applied to structures of arbitrary shape. However, due to the spatial dependence in the data sample, how to predict the precision of the estimator of V becomes a difficult task. In this paper, we develop a method to construct a confidence interval for V by analysing the variability produced by each level of sampling (i.e., between and within sections' variability). Simulation from two known analytical measurement functions is carried out to show its validity. Finally, the new approach is applied to investigate the significance of hippocampus volume change in a patient with epilepsy after 2 years of disease.

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.

A general variance predictor for Cavalieri slices

A general variance predictor is presented for a Cavalieri design with slices of an arbitrary thickness t >or= 0. So far, prediction formulae have been available either for measurement functions with smoothness constant q = 0, 1, ... , and t >or= 0, or for fractional q in [0, 1] with t = 0. Because the possibility of using a fractional q adds flexibility to the variance prediction, we have extended the latter for any q in [0, 1] and t >or= 0. Empirical checks with previously published human brain data suggest an improved performance of the new prediction formula with respect to the hitherto available ones.

Confidence intervals in Cavalieri sampling

The aim of this study is to derive a formula that allows the prediction, from a Cavalieri data sample, of an appropriate confidence interval for a parameter Q. Two different approaches are used to address the problem. The first approach is to investigate whether the asymptotic distribution of the Cavalieri estimator exists when the sampling period T tends to zero. In particular, the distribution of the standardized version of the Cavalieri estimator z(T) is analysed for a measurement function f whose smoothness constant is an integer number m. The analysis reveals that when the first noncontinuous derivative of f, f((m)), exhibits a unique discontinuity, the asymptotic distribution of z(T) exists and it has a bounded support. An analytical expression of the distribution is derived for the cases m = 0 and 1. However, when f((m)) has two or more discontinuities, the asymptotic distribution of z(T) does not exist and its support may be unbounded. In the second approach, a generalized version of the refined Euler Mac-Laurin summation formula, valid for measurement functions with a fractional, rather than just an integer, smoothness constant, is applied to the Cavalieri estimator. As a result, a formula that predicts a lower and upper bound for the true parameter is derived for small T. This bound prediction formula is applied to Cavalieri data samples of human cerebral cortex. In particular, for sample sizes n = 8, 12 and 16, the true volume of cerebral cortex is bounded by relative distances 8%, 4% and 2% of the Cavalieri estimate, respectively.

New variance expressions for systematic sampling: the filtering approach

We present a collection of variance models for estimators obtained by geometric systematic sampling with test points, quadrats, and n-boxes in general, on a bounded domain in n-dimensional Euclidean space R(n), n = 1, 2, ... , and for systematic rays and sectors on the circle. The approach adopted - termed the filtering approach - is new and different from the current transitive approach. This report is only preliminary, however, because it includes only variance models in terms of the covariogram of the measurement function. The estimation step is in preparation.

No postnatal doubling of number of neurons in human Broca’s areas (Brodmann areas 44 and 45)? A stereological study

In this study we explored whether a postnatal doubling of the total number of neurons occurs in the human Brodmann areas 44 and 45 (Broca's area). We describe the most recent error prediction formulae and their application for the modern stereological estimators for volume and number of neurons. We estimated the number of neurons in 3D optical disector probes systematically random sampled throughout the entire Brodmann areas (BA) 44 and 45 in developing and young adult cases. In the relatively small number of male and female cases studied no substantial postnatal increase in total number of neurons occurred in areas 44 and 45; the volume of these areas reached adult values around 7 years. In addition, we did find indications that a shift from a right-over-left to a left-over-right asymmetry may occur in the volume of BA 45 during postnatal development. No major asymmetry in total number of neurons in BA 44 and 45 was detected.