On the efficiency of stereologic volumetry as commonly implemented for three-dimensional digital images

The aim of the study was to demonstrate, for certain ideal shapes (right cylinders) and for representative neuroanatomical images, that stereologic volumetry of three-dimensional images is more efficient when the sampling grid is placed randomly on each cross-section rather than identically across sections. As an example, the special case of mathematical right cylinders is presented, and an informal proof is provided. For neuroanatomical images, a custom computer program estimated volume with either the fixed- or random-grid method, using the same cross-sectional slices and first-slice test grid position for each method. The slice spacing, grid size, and starting grid position were randomly varied within practical constraints for 100,000 trials in each image. For right cylinders, the random-grid method is always more efficient than the fixed-grid method. For the neuroanatomic images tested, relative variance was up to three times higher for the fixed-grid method than for the random-grid method, especially for test grids with few grid intersections ('hits') per section. With the random-grid method, relative variance is primarily dependent on the total number of hits rather than on the distribution of hits per section. Implementation of the random-grid method for stereologic volumetry in three-dimensional images should in general improve sampling efficiency.