Search

The estimation of object volume from rotationally randomised sections relies on an ancient geometrical principle due to the Alexandrian mathematician Pappus. In this paper we describe two studies that make estimates of eye volume using a stereological version of the Pappus principle. The first study uses a design-based version of the Pappus theorem to make estimates of eye volume in Sprague–Dawley rats and the second uses a model-based implementation for estimating eye volume in juvenile Dover sole. In the sole study we compare the estimates of volume with estimates made using the Cavalieri method and show that the Pappus method gives identical volume estimates with a markedly lower coefficient of variation (2.5%) than the Cavalieri method (13.5%).

Classical stereology is capable of quantifying the total amount or ‘density’ of a geometrical feature from sampled information, but gives no information about the local spatial arrangement of the feature. However, stereological methods also exist for quantifying the ‘local’ spatial architecture of a 3D microstructure from sampled information. These methods are capable of quantifying, in a statistical manner, the spatial interaction in a structure over a range of distances. One of the key quantities used in a second-order analysis of a volumetric feature is the set covariance. Previous applications of covariance analysis have been ‘model-based’ and relied upon computerized image analysis. In this paper we describe a new ‘design-based’ manual method, known as linear dipole probes, that is suitable for estimating covariance from microscopic images. The approach is illustrated in practice on vertically sectioned lung tissue. We find that only relatively sparse sampling per animal is required to obtain estimates of covariance that have low inter-animal variability.

The estimation of ‘size’ for an interconnected nonconvex phase is an ongoing problem in materials science and biomedicine. Examples of this type of phase include the pore space in a porous rock such as sandstone or the marrow space found in cancellous bone. A reasonable definition of size for this type of phase is provided by the mean volume-weighted star volume. This is defined as the average volume of the phase of interest seen unobscured from a typical point within the phase. One practical advantage of this definition of size is that it is easy to estimate stereologically. In this paper we extend the concept of star volume to consider the mean surface-weighted star volume. The mean surface-weighted star volume is the volume of a continuous phase seen unobscured from a typical point of the interface. We expect that this parameter could be useful in biological problems involving mass transfer across an interface. The stereological estimation of mean surface-weighted star volume is described and the method is illustrated on piglet lung tissue.