The basic stereological formulas for estimating volume (Vv) and surface (Sv) densities are strictly valid only for true infinitely thin sections; the use of "ultrathin" sections of finite thickness T introduces systematic errors, mostly in the sense of overestimation of the parameters. These errors depend on the size and shape of the structural elements and on T. Correction factors for this effect of T are derived by considering model structures that simulate the shape and arrangement of subcellular organelles: (a) spherical vesicles, (b) disks as models for rough endoplasmic reticulum (RER) cisternae, (c) cylindrical tubules as models for smooth endoplasmic reticulum (SER) tubules, microvilli, etc. For vesicles, a model of discrete convex spherical particles is assumed; the correction factors consider loss of caps due to grazing sections and size distribution of the vesicles. The disk and tubule models are used in connection with the new integral geometric formulas of R. E. Miles which consider random aggregates of "interpenetrating" particles so that the resultant structure is non-convex and thus approximates in nature the networks characteristic of endoplasmic reticulum (ER). Some practical examples relative to liver cells show that the errors due to section thickness may be of the order of 20-40% or more. Computation formulas as well as graphs are given for the determination of the correction factors for Vv and Sv.KEY WORDS stereology 9 section thickness correction factors -organelle volume density membrane surface density Stereological methods have hitherto been used in cell biology mostly in the framework of studies where control and experimental preparations could be considered geometrically similar. Although it was known that the measurements were fraught with serious and systematic methodological errors, these could be disregarded on the assumption that all preparations were similarly affected by them.In the attempt to correlate the membrane areas collected in subcellular fractions with those originally present in the intact tissue, as described in the companion paper (1), such assumptions were not warranted. Among the errors to be considered, that introduced by section thickness was of particular concern: it depends on the shape of the 584 J. CELL BIOLOGY 9 The Rockefeller University Press 9

SUMMARYEstimating surface and volume density of subcellular membrane systems at different magnifications yield different results. As the magnification is increased from × 18,000 to × 130,000 the estimates of surface density of endoplasmic reticulum and inner mitochondrial membranes increase by a factor of 3, whereas that for outer mitochondrial membranes increases only by 20%. The estimate of volume density of endoplasmic reticulum also increases by a factor of 3. No further increase is observed at magnifications above × 130,000 which is therefore called critical magnification. The findings are interpreted on the basis of the concept of fractals proposed by Mandelbrot, and the fractal dimensions of the membrane systems considered are estimated. This can lead to the derivation of resolution correction factors which permit measurements obtained at any magnification to be converted to estimates at critical magnification. These findings may explain, at least in part, the large discrepancy in the estimates of the surface of cytomembranes found in the literature.

Previous attempts to relate the structure and function of hepatocytic membranes have compared biochemical data of fractions to morphological data derived from either intact tissue or fractions. The effects of the original homogenization aside, biochemical recoveries comparing membrane marker enzymes of the homogenate to subsequent fractions suggest a general conservation of activity. A stereological study was undertaken to estimate membrane surface areas in the intact tissue, homogenate, and fractions of the same livers and then to test the comparability of these data with membrane marker enzymes by calculating both morphological and biochemical recoveries. The stereological data were corrected for errors due to section thickness and compression.

The purpose of the study was to consider quantitatively the relationships between the surface area of the endoplasmic reticulum (ER) and constituent marker enzyme activities, as they occur in fractions collected from rat liver homogenates .The ER surface area was estimated in five membrane-containing fractions by use of a combined cytochemical-stereological technique (5), while, at the same time, ER marker enzymes were assayed biochemically . Fraction/homogenate recoveries for the ER enzymes averaged 100%, total membrane surface area 98%, and ER surface area 96% . Relative specific activities, which compare the relative amounts of ER marker enzyme activities to the relative ER surface area in the membranecontaining fractions, indicate variable distributions for glucose-6-phosphatase and NADPH cytochrome c reductase, but not for esterase .The endoplasmic reticulum (ER) of rat liver hepatocytes has been studied extensively by use of marker enzymes to identify the presence of these membranes in fractions (17,14,15,3,1,23,10,11,21) . In recent years, the question as to whether a given amount of marker enzyme activity is associated with a similar amount of ER membrane has become one of increasing importance (15,25,13,44,32,8) . Several papers have suggested that (a) marker enzymes are heterogeneously distributed on the ER and that (b) "ER marker enzymes" may also be attached to membranes other than those of the ER (15,46,2,18,22,29,24,14,41,39,4,38,30) . The existence of such marker enzyme heterogeneities would be expected to have important consequences when one is interpreting bio-J. CELL BIOLOGY © The Rockefeller University Press " 0021-9525/80/06/0577/10 $1 .00 Volume 85 June 1980 577-586 chemical data extrapolated from one fraction to the entire liver, or when one is making corrections for contaminations, particularly if the fractionation procedure leads to a sorting of the heterogeneous ER membranes .In an earlier study (9), the surface areas of the hepatic membranes were determined in each of several fractions collected by differential centrifugation . An average 96% fraction-homogenate recovery for the membrane surface areas, accompanied by an average 95% recovery for several membrane marker enzymes, suggested that, for the most part, both membrane surface area and enzyme activity were being similarly conserved during the fractionation procedure . The study showed the extent to which homogenization forms a pool