Practical implementation of the planar and spatial rotator in a complex tissue: the brain.

In neuroscience, application of widely used stereological local volume estimators, including the planar rotator, is challenged by the combination of a complex tissue organisation and an estimator requirement of either isotropic or vertical sections, i.e. randomly oriented tissue. The spatial rotator is applicable with any tissue orientation but is sensitive to projection artefacts. The challenge is thus to select the most appropriate method for individual analyses. In this study, agreement between estimates of mean cell volume acquired with the vertical planar and the spatial rotator is assessed for two brain regions with different types of cytoarchitecture (motor cortex and hippocampal cornu ammonis 1). The possibility of using the planar rotator in tissues cut in an arbitrary direction is explored and requirements for a theoretically unbiased result as well as histological considerations are provided. LAY DESCRIPTION: Cells may change volume both during disease and with advancing age. Assessment of the volume of individual cells can therefore serve as a useful indicator of general tissue state. Most available methods to estimate cell volume in tissue sections, however, require that the tissue analysed has random orientation. Particularly for complex tissues such as the brain this is a challenge as identification, delineation and subdivision of many brain areas rely heavily on the use of anatomical atlases where illustrations depict the tissue in a few well-known orientations. In this study, the practical application of two different methods for estimating mean cell volumes in tissues cut in a preferred orientation is evaluated. Requirements for the feasibility of cell volume estimation without random tissue orientation as well as histological considerations are provided.

A note on the stereological implications of irregular spacing of sections.

Stereological methods for serial sections traditionally assume that the sections are exactly equally spaced. In reality, the spacing and thickness of sections can be quite irregular. This may affect the validity and accuracy of stereological techniques, especially the Cavalieri estimator of volume. We present a new formula for the accuracy of the Cavalieri estimator that includes the effect of random variability in section spacing. A modest amount of variability in section spacing can cause a substantial increase in estimator variance.

On error prediction in circular systematic sampling.

An extended covariogram model is discussed for estimating the precision of circular systematic sampling. The extension is motivated by recent developments in shape analysis of featureless planar objects. Preliminary simulation results indicate that it is important to consider the extended covariogram model.

Volume estimation from projections.

We describe a new estimator of the volume of axially convex objects from total vertical projections with known position of the vertical axis. The estimator combines the Cavalieri method with the known formula for area in terms of the support function of a convex body. We examine the accuracy of the proposed estimator for ellipsoidal objects having exactly known support function and volume. In addition, we illustrate practical problems of accuracy by implementing the method for some biological products.

Shape modelling of spatial particles from planar central sections -- a case study.

In this paper we develop statistical tools for shape modelling of spatial particles from central sections through the particles. The particles are assumed to be star-shaped with respect to a reference point inside the particles and are modelled as stochastic deformations of spheres centred at the reference points. The resulting particles are rotation invariant with respect to the reference point. As an illustration, the model is applied to study shape differences between neurons in the Granular and CA1 layer in the human hippocampus.