Optimal Use of Sampled Tissue Sections for Estimating the Number of Hepatocellular Foci

Statistical techniques were applied to computer-simulated data to evaluate the importance of various factors that may affect the estimation of the number of hepatocellular foci from foci transections. The simulations were modeled after mouse foci for which three-dimensional size distributions and densities were determined from serial section reconstructions. The foci had been induced in male C57BL/6 × C3H F1 mice at 20 and 28 weeks after a single injection, in infancy, of diethylnitrosamine. In an earlier report, we had shown that the number of foci per cu cm could be accurately estimated from profiles using a conditional estimator when the investigator is unable to identify profiles smaller than a certain size (e). In the present study, emphasis was placed on assessing the value of step serial sections in order to make optimal use of the small tissue samples in mouse liver. Since all mathematical estimators of N 3 are based on measurement of sampled profiles (n2e) and ultimately derive from the fundamental relationship N 3 , = N 2 , / 2 μ , (where N 3e is the number of foci per cu cm, N 2e is the number of profiles per sq cm, and µe is the average adjusted focus radius), the relative importance of N 2e and µe on the conditional estimator was evaluated. This was accomplished by comparing the errors resulting from use of the conditional estimator with those resulting from use of two other estimators. The latter two estimators consisted of a “sampled focus estimator,” which used sampled intact foci to estimate N 3, and a “reference estimator,” which used profiles from foci with a µe that was known. Additionally, in order to provide a stable variance for the conditional estimator, we adopted a simple smoothing procedure. As expected, none of the estimators showed any significant bias. However, somewhat surprising was the finding that the standard deviations from use of all three estimators were almost identical. Consequently, it appears that the variance resulting from application of the smoothed conditional estimator to large sets of profile data is not due to difficulty in estimating µe. In these instances, the faulty estimates of N 2e resulted almost entirely from the variability in sampling foci from tissue blocks which constituted only about 1/25 of the liver volume. In addition, the ability to reduce the error in estimating the number of foci by increasing the number of profiles was also evaluated. We simulated a sectioning protocol for the 28-week mouse livers in which the distances between the 5-µm-thick step sections in a 1-mm block of liver were progressively decreased from 1000 to 50 µm. For the particular size distribution and density of simulated foci, the error in estimating the number of foci per cu cm from profiles in 11-step sections was only slightly greater than that obtained by estimating the numerical density of foci from the actual three-dimensional sample. We conclude that the application of an appropriate estimator to profiles of foci in step sections should be of great value for estimating the number of spheroidal neoplastic and preneoplastic foci. Furthermore, computer simulations should enable the investigator to determine the advantages and limitations of particular sectioning schemes prior to embarking on time-consuming projects.