Design-based estimation of surface area in thick tissue sections of arbitrary orientation using virtual cycloids.

Surface area is a first-order stereological parameter with important biological applications, particularly at the intersection of biological phases. To deal with the inherent anisotropy of biological surfaces, state-of-the-art design-based methods require tissue rotation around at least one axis prior to sectioning. This paper describes the use of virtual cycloids for surface area estimation of objects and regions in thick, transparent tissue sections cut at any arbitrary (convenient) orientation. Based on the vertical section approach of Baddeley et al., the present approach specifies the vertical axis as the direction of sectioning (i.e. the direction perpendicular to the tissue section), and applies computer-generated cycloids (virtual cycloids) with their minor axis parallel to the vertical axis. The number of surface-cycloid intersections counted on focal planes scanned through the z-axis is proportional to the surface area of interest in the tissue, with no further assumptions about size, shape or orientation. Optimal efficiency at each x-y location can be achieved by three virtual cycloids orientated with their major axes (which are parallel to the observation planes) mutually at an angle of 120 degrees. The major practical advantage of the present approach is that estimates of total surface area (S) and surface density (SV) can be obtained in tissue sections cut at any convenient orientation through the reference space.

Stereological length estimation using spherical probes.

Lineal structures in biological tissue support a wide variety of physiological functions, including membrane stabilization, vascular perfusion, and cell-to-cell communication. In 1953, Smith and Guttman demonstrated a stereological method to estimate the total length density (Lv) of linear objects based on random intersections with a two-dimensional sampling probe. Several methods have been developed to ensure the required isotropy of object-probe intersections, including isotropic-uniform-random (IUR) sections, vertical-uniform-random (VUR) slices, and isotropic virtual planes. The disadvantages of these methods are the requirements for inconvenient section orientations (IUR, VUR) or complex counting rules at multiple focal planes (isotropic virtual planes). To overcome these limitations we report a convenient and straightforward approach to estimate Lv and total length, L, for linear objects on tissue sections cut at any arbitrary orientation. The approach presented here uses spherical probes that are inherently isotropic, combined with unbiased fractionator sampling, to demonstrate total L estimation for thin nerve fibres in dorsal hippocampus of the mouse brain.

Bias of a length density estimator based on vertical projections.

Attention is paid to the stereological estimator of the length density of lineal features in three-dimensional space. In Gokhale (J. Microsc. (1990) 159, 133-141), the estimator based on measurements performed on a projection of the content of vertical slices with a given thickness Delta was derived. The aim of this paper is to show that using five vertical slices with systematically chosen orientations yields a reliable result, i.e. the bias of the estimator is smaller than 5%. In the case of a choice of the vertical axis such that most lineal features are not perpendicular or nearly perpendicular to the vertical axis, a reliable result can be obtained using only three systematic orientations of vertical slices.

Efficient estimation of number density in opaque material microstructures: the large-area disector.

Unbiased and efficient estimation of number density of features in opaque material microstructures has been quite difficult. In this contribution a montage-based efficient serial sectioning technique is presented as a practical solution for efficient estimation of number density in such microstructures. The new technique utilizes a combination of digital image analysis and unbiased disector sampling procedures.