Let’s start by looking at an ‘extreme’ case for it’s instructive value, even though it may be biologically unlikely or impossible. We do the area-fraction-fractionator point-counting and find that 100% of the tissue is cell bodies. Let’s also suppose that we used point-sampled-intercepts or the nucleator to estimate a mean soma volume of 5,000 cubic microns.

percent of tissue that is somas (VV) = 100 percent

mean soma volume (vN) = 5,000 cubic microns

numerical density (Nv) = VV/vN = 1/5,000 cubic microns = 1 cell body per five thousand cubic microns

Now let’s consider a situation where half of the volume of the tissue is taken up by cell bodies. In other words, we do area-fraction-fractionator point-counting and find that half of the points fall on somas and the other half on neuropil. We will keep the cell body volume the same, as determined by the nucleator or point-sampled-intercepts, 5,000 cubic microns.

percent of tissue that is somas (VV) = 50 percent

mean soma volume (vN) = 5,000 cubic microns

numerical density (Nv) = VV/vN = 1/10,000 cubic microns = 1 cell body per ten thousand cubic microns

These examples are easy to understand. If there is nothing but cell bodies and they are X cubic microns in size, there has to be one cell body per X cubic microns. If the tissue’s make-up is one-half cell bodies by volume, and the cells are X cubic microns in size, there has to be one cell body per 2X cubic microns; half of any volume unit is somas, the other half is not somas. Here is a more complicated example:

percent of tissue that is somas (VV) = 63.5 percent

mean soma volume (vN) = 8,181 cubic microns

numerical density (Nv) = VV/vN = 0.635/8,181 cubic microns = 1/12,880 cubic microns = 1 cell body per twelve thousand eight-hundred and eighty cubic microns

Cruz-Orive, L.M. (1987) Particle Number can be Estimated Using a Disector of Unknown Thickness: the Selector. J. of Microscopy, 145, 121-142