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Reliable Stereological Method for Estimating the Number of Microscopic Hepatocellular Foci from Their Transections¹

Departments of Pathology [T. D. P., J. H. K., H. K., Y. H., S. G.], Mathematics [J. C.], and Statistics [D. N., G. W.], University of Wisconsin, Madison, Wisconsin 53706

We have derived a mathematical equation ?? where ?? is the estimated number of foci per cu cm with radii larger than , and is the lower limit for radii of foci or profiles (x_i) that can be reliably observed in tissue sections with area (A), which provides an unbiased estimate of the number of microscopic, hepatic foci from their profiles in tissue sections. The significant feature of the formula is the recognition of the experimenter's inability to reliably identify profiles below a certain size limit. In order to test the formula on a real data set, we used serial sections to reconstruct all 716 complete and incomplete foci in blocks of liver from 12 mice and compared the actual number of foci with the number estimated by application of the formula. The foci were identified after 20 and 28 weeks in male C57BL/6 x C3H F₁ mice that had been given i.p. injections of diethylnitrosamine (5 µg/g body weight) when they were 15 days old. An additional mathematical method was used to confirm the spheroidal shape of the foci and to measure the partially cut foci. Since a study of known spherical objects (agarose beads) demonstrated that the paraffin sections stretched by 16% during their preparation, it was also necessary to correct for this distortion before applying the formula to the profile data. For the 12 blocks of liver, the mean number of foci estimated by the new method was 91% of the number found in the reconstructions. The mean estimate was not significantly different, at the 95% level, from the actual number of foci. In contrast, when 2 other currently used methods were applied to this same data set, the mean estimates were significantly biased, underestimating the number of foci by 26 and 42%.

Experience with this real data set has helped to develop guidelines for application of the new mathematical-stereological method. Before applying the formula, it is necessary to confirm the generally spheroidal shape of the foci, to correct for distortions in the tissue sections, and to decide on a lower limit for reliable identification of profiles. Since the new method also provides a basis for estimating the size distribution of foci, it should eventually have general application for the study of in vivo growth kinetics of early preneoplastic and neoplastic cell populations.

¹ Supported by NIH Grants CA15664 and CA25522 and ARO Grant DAAG2980-K-0042. Presented in part at the 1982 meeting of the American Association for Cancer Research in St. Louis, Mo. (11).

² Recipient of a fellowship award from NIH Training Grant 5-T32-ES 0701507. Present address: Department of Industrial Engineering, Northwestern University, Evanston, Ill. 60201.

³ Present address: McArdle Laboratory for Cancer Research, University of Wisconsin-Madison, Madison, Wis. 53706.

⁴ To whom requests for reprints should be addressed.

Received 4/30/82. Accepted 12/ 8/82.

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