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 cy linders) and for representative neuroanatomical images, that stereologic vo lumetry of three-dimensional images is more efficient when the sampling gri d is placed randomly on each cross-section rather than identically across s ections. 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 gr id position for each method. The slice spacing, grid size, and starting gri d position were randomly varied within practical constraints for 100 000 tr ials in each image. For right cylinders, the random-grid method is always m ore efficient than the fixed-grid method. For the neuroanatomic images test ed, relative variance was up to three times higher for the fixed-grid metho d than for the random-grid method, especially for test grids with few grid intersections ('hits') per section. With the random-grid method, relative v ariance is primarily dependent on the total number of hits rather than on t he distribution of hits per section. Implementation of the random-grid meth od for stereologic volumetry in three-dimensional images should in general improve sampling efficiency. (C) 1999 Elsevier Science Ireland Ltd. All rig hts resented.