The mathematical science of quantitative stereology has established relationships for the quantitation of elements in three-dimensional space from observations on two-dimensional planes. This report describes the utilization and importance of such mathematical relationships for the quantitative analysis of focal hepatic lesions in terms relative to the volume of the liver.

Three examples are utilized to demonstrate the utility of such calculations in the three-dimensional quantitation of hepatic focal lesions. The first is that of a computer-simulated experiment based on defined hypothetical situations. The simulations demonstrate the applicability of the computations described in this report to the evaluation of two-dimensional data from typical animal experiments. The other two examples are taken from actual experiments and involve the transplantation of hepatic cell populations into the liver of suitably prepared hosts and the quantitation of altered foci produced by initiation with diethylnitrosamine-partial hepatectomy followed by promotion with phenobarbital. The quantitation of altered foci by means of a two-dimensional analysis (simple enumeration of focal intersections/area of tissue section) is proportional to the quantitation of foci per volume of liver provided that the mean diameter of the foci for each treatment is sufficiently uniform, as exemplified in the text by the transplantation experiment. When such mean diameters are unequal as in the diethylnitrosamine-phenobarbital experiment described herein, quantitation from three-dimensional analysis gives significantly different results as compared with enumeration of focal intersections on two-dimensional areas. These studies clearly demonstrate that the frequency and size of foci intersections viewed on two-dimensional tissue sections do not necessarily reflect the number or size of foci in the three-dimensional tissue. Only by quantitating the number and size of the foci in relation to the three-dimensional volume of the tissue can one determine the validity of the proportionality of data from two-dimensional measurements to the total number of foci per volume of tissue. Such a conclusion has important implications for quantitative studies on hepatocarcinogenesis as well as for the enumeration of premalignant lesions which occur during the natural history of carcinogenesis in any solid tissue.

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The studies reported herein were supported in part by CA-07175, CA-22484, CA-24818, and CA-17334 from the National Cancer Institute.

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