Erratum and addendum to "Three-dimensional counting: An accurate and direct method to estimate numbers of cells in sections material"

Figure 1. A The counting frame and its 8 neighboring frames. Cells that cross into shaded neighbors are not counted (cells 1, 8, 9, and 10). Cell 7 was excluded in figure 2 of our original paper, but here it is clear that this cell must be counted. Note that the frame has been rotated 90 degrees in comparison to figure 2 of the original paper. B The counting box as seen in perspective, as if from the lower right corner of figure 1A. The central counting box is surrounded by 26 similar boxes, although only 9 of these are actually illustrated. Cells that intrude into half of these 26 neighbors (e.g., cells 1, 8, 9, and 10: solid black transects) are excluded. The other four cells—3, 5, 7, and 13—only intrude into permitted neighbors and are counted. Surfaces that separate forbidden and permitted compartments are shaded. These forbidden surfaces extend outward without change in orientation as indicated by the arrows. Cells in figure 1A that are entirely inside the box (e.g., 2, 11, 12, and 14), which are always counted, are not shown in figure 1B.

The problem is that the bold line of the counting frame in figure 2 (page 346) should not extend straight beyond point C, but must be bent 90 degrees as shown in figure 1A above (Gundersen, 1977). Making this change eliminates all bias because cells that cross over into half of the 8 neighboring frames—in this case, the 4 shaded neighbors—are excluded from the count. As a counterbalance, cells that only cross into the 4 unshaded neighbors are included. The bold line in figure 1A separates forbidden and permitted half-planes. It is simple to extend this unbiased rule to three-dimensions. A counting box can be though of as the central element of a 3x3x3 array of equivalent boxes (only part of the array is shown in figure 1B). As is the case for a counting frame, half of the neighbors of the counting box—13 of 26— are forbidden. The "forbidden 13" include all 9 bottom neighbors and 4 neighbors on the same level as the counting box. In contrast, all 9 top neighbors and the 4 other neighbors that are on the same level as the counting box are permitted (note that none of these permitted neighbors have been illustrated in figure 1B). According to these simple rules, the forbidden left and front surfaces of the counting box illustrated in figure 3 of our paper do not extend straight up to the surface of the tissue section, but must be bent back 90 degrees and run parallel to the top side of the box as does the upper, shaded surface in figure 1B, above. This definition of the counting box and forbidden surfaces gives unbiased counts, no matter what the size or shape of the cells. This proper procedure also simplifies three-dimensional counting because fewer cells need to be examined carefully to decide whether they cross over into forbidden regions.

We thank Dr. H.J.G. Gundersen (University of Aarhus), Dr. M.L. Schwartz (Yale University), and Dr. V. Howard (University of Liverpool) for their improvements and helpful suggestions.

Gundersen, H.J.G. (1977) Notes on the estimation of the numerical density of arbitrary profiles. The edge effect. J. Microscopy 111:219-222.
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