Footnote and Annotations

1. I shall refer to the section as a whole without prejudice to the possibility that only part of it, such as a single microscopic field, may be the subject of investigation. [This is a footnote on page 239 of the original article by M. Abercrombie].

Annotation by R. Williams

RW1. As described further in the paper by Abercrombie, estimates of L tend to be too low. Values of P will tend to be slightly high.

RW2. Here Abercrombie acknowledges obliquely that each size category of nucleus needs its own correction factor, L/(L+M). For example, glial cells, interneurons, and principal neurons need individual correction factors.

RW3. I have added this reference to Literature Cited section. The 1946 paper by Abercrombie and Johnson shows an 8-fold increase in cell number in the rabbit sciatic nerve following transection. However, despite the statement above that both methods were used, the published account does not include any explicit or implicit use of the two-section method: Abercrombie's Method 2, see below.

RW4. The average nuclear dimension at right angle to the plane of section is not sufficient to obtain an unbiased estimate. We need to know the distribution of particle heights. If this distribution is not normal then the average may not be a good measure of central tendency. In the case of nuclei of a particular cell type, the error is likely to be minimal.

RW5. Abercrombie's statement that height cannot be determined in the sections used for counting is valid for relatively thin sections. But when section thickness is greater than 10 µ it is possible to estimate mean nuclear height directly. This does require an optical system with a narrow depth of field (<0.25 µ). Nomarski differential interference contrast combined with an objective with a high numerical aperture makes this feasible. Due to Snell's Law (of optical foreshortening) differences in the refractive indices of tissue, glass, and the space between tissue and the objective may need to be considered in estimating nuclear height. Oil-immersion microscopy minimizes this potental problem.

RW6. Abercrombie considers two values of nuclear height L in this paragraph. One is L as measured in sections that are N µ thick; the other is the true height of the nucleus. If the section thickness N is very small (<1 µ) then the error factor 0.21T/(T+N) for cells with a 10 µ diameter amounts to 2.1 and L as measured will be 7.9 µ. If the section thickness is 10 µ, then the error factor is 0.21(10/10+10) and L as measured will be approximately 8.9 µ.

RW7. Proof of Identity when S is made equal to L/M (thanks to Alex Williams):

         2SM - L
P = A ——————-              ( 2 )
       SM (S + 1)

            L
P = A ——————-              (2.1)
       L (L/M + 1)

           1
P = A ——————-              (2.2)
        L/M + M/M

           1
P = A ——————-              (2.3)
       (L + M)/M

          M
P = A ————-                  ( 1 )
        L + M

RW8. Abercrombie is probably referring to differences in tissue compression and expansion in the cutting axis that varies slightly with section thickness when tissue is embedded in paraffin. This should be a minor problem, and one that can be easily controlled. I have not noticed this to be a problem with either celloidin sections or tissue cut frozen.

RW9. Few investigators who have used Abercrombie's Method 1 have actually gone to the trouble to measure nuclear height in a separate set of sections cut at right angles to those used to count. Instead, nuclear height is almost invariably estimated from nuclear diameter in the plane of section. This shortcut may actually give better results, because investigators typically measure a maximized nuclear diameter (most often, the diameter in the plane of section of the nucleolus). Thus L is unlikely to be underestimated using this procedure. Furthermore, there is no problem of differences in shrinkage of tissue processed and cut separately. However, if nuclei are not randomly oriented, L may be a biased estimate of nuclear height.

A major problem with Abercrombie's Method 1 is that it presumes that nuclei (or whatever particle is counted) are cut by the knife blade. However, tissue is often split apart, and cells, nuclei, and nucleoli may be pushed aside into one or the other section. In the extreme, no Method 1 correction factor should be applied. For further discussion of problems with Abercrombie's Method 1 see Williams & Rakic 1988 3D Counting Paper.

RW10. Method 2 is the first technically unbiased method for counting cells in sectioned material of which I am aware. It is arguably superior to our own 3D counting method or to the optical disector for two reasons: 1. it is not affected by differential z-axis shrinkage; 2. it demands no special equipment. In retrospect, Abercrombie's relatively negative assessment of his own superior method is puzzling. The main problem he identifies is that thick and thin sections may have different areas. This is a small matter and easy to rectify. Such differences in area, if present at all, are easy to assess, minimize, and control. Even if areas of the thick and thin section pairs differ, Method 2 can be used to obtain technically unbiased estimates.

The other problem Abercrombie mentions is that Method 2 is more effort to implement. This is not a compelling objection. It is trivial to generate sets of sections cut at two different thicknesses simply by double-advancing the microtome. A trustworthy microtome combined with a skilled microtomist should be able to provide highly reproducible 1x, 2x, 3x, 4x sections. An analysis of the real section thickness is not mandatory. Differences between 1x and 2x sections can be compared to those of 2x and 3x, 3x and 4x, 1x and 4x, 2x and 4x, etc. A comparison of the ratios of differences with the predicted ratios will quickly reveal any problems with the microtome or the counting procedures. In contrast, the analysis of mean particle size required to apply Method 1 is not always trival, particularly when several cell classes are involved. And of course, the technical bias, however small, opens a crack through which many astringent and ill-founded criticisms can seep.

Provided that the staining of both thick and thin sections is thorough, and that good images can be obtained at all focal planes, Abercrobie's method 2 is probably superior to other methods for quantification of immunohistochemically-stained sections that are often subjected to a brutal dehydration process.

Method 2 is also completely insensitive to the problem of non-linear tissue shrinkage in the z-axis after cutting but before counting (van Bartheld, 1999(. Non-uniform z-axis shrinkage is an acute problem using some embedding methods (and a problem that can add substantial undefined bias to supposedly "unbiased' 3-D counting methods. This problem is considered briefly in the original and at more length in the updated web version of Williams & Rakic 1988.

von Bartheld CS (1999) Systematic bias in an "unbiased" neuronal counting technique. Anat Rec 257:119–120.