THE ANATOMICAL RECORD 207239-106 (1983) Methods for Decreasing the Statistical Variance of Stereological Estimates ROBERT P. BOLENDER Department ofBiologica1 Structure, University of Washington School of Medicine, Seattle, WA 98195 ABSTRACT Three methods are described for decreasing the statistical variance of stereological estimates. Method 1 uses profile boundaries and surface densities of nuclear membranes, measured in thin sections, to estimate the mean diameter, surface area, and numerical density of spherical and nonspherical nuclei. For the guinea pig pancreas (number(m) = 4), the standard deviations (s.d.) as a percent of the mean for the estimates of the diameters of the exocrine, duct, and endothelial cell nuclei were 1.5%, 3.3% and 1.4%. The estimate for the mean diameter of exocrine nuclei (6.4 k 0.1 pm) was based on a spherical model, whereas the estimates for the diameters of the nonspherical (and nonconvex) nuclei of the duct (6.4 f 0.2 pm) and endothelial (6.7 f 0.1 pm) cells were calculated from the numerical density of the exocrine cells and the relative frequencies of the three cell types (determined from serial reconstructions). In a n average cubic centimeter, there were 6.17 x 10' + 0.32 x 10' (s.d. 5.1% of mean) exocrine cells, 1.64 x 10' f 0.18 x 10' (10.9%)duct cells, and 0.803 x 10' 0.13 x 10' (16.6%)endothelial cells. In contrast to method 1,conventional stereological approaches were found to have standard deviations two- to eightfold larger. Method 2 uses a mean nuclear surface area and a ratio of surface densities to estimate the surface area of a membrane compartment in a n average cell. A s.d. equal to 6.5% of the mean was found for the surface area of the outer mitochondria1 membrane in a n average exocrine cell (672 & 43.6 pm2),which represented a n almost fourfold reduction in the s.d. compared with a n earlier estimate (Bolender, 1974).Method 3 relates the surface area of a membrane compartment to a standard number of cells. Referenced to lo6 cells, for example, the surface area of the inner nuclear membrane of endothelial cells had a s.d. equal to 2.9% of the mean, whereas the surface density of the same membrane compartment-referenced to a cm3 of cells-had a s.d. at 19.1% of the mean. In this case, method 3 produced almost a sevenfold reduction in the standard deviation. Similar results were found for exocrine and duct cells. The results of the study indicate that the standard deviation of a stereological estimate can be reduced to a minimum by using a mean nuclear profile boundary to generate a n estimate for a nuclear numerical density, which, in turn, can be combined with a surface density to obtain average cell information. If a convex nucleus were rotated between the tines of a caliper and measured at all possible orientations, the resulting mean value would represent its mean caliper d i ameter (D). In biological stereology, the D characterizing a nuclear population has become one of the most sought after pieces of information because it allows one to estimate the numerical density (Nv) of nuclei, as, for 0 1983 ALAN R. LISS, INC. example, defined by the equation given by Wicksell (1925,1926) and DeHoff and Rhines (1961): Nv = N*/D, Received April 6,1982; accepted June 2,1983. (1) 90 R.P. BOLENDER where NA equals the number of nuclear profiles (N) counted from several random sections having a total test area A. The numerical densities of nuclei are needed to calculate the volumes and frequencies of average cells, as well as average values for cytoplasmic compartments. Earlier studies have shown repeatedly that stereological information can be interpreted more effectively when data are related to average cells and the number thereof (Bolender, 1979a,b, 1981a,b, 1982). A reliable solution to equation (1)depends on collecting two pieces of information, NA and D. The NA represents a variable that is obtained by counting nuclear profiles as they appear in sections. The D, however, is a variable that must be estimated from measurements of profile diameters and a mathematical reconstruction. The methods of reconstruction are expected to work well only when the nuclei can be likened to spheres or slight ellipsoids (see reviews by DeHoff and Rhines ,Underwood [I9701 and Weibel 11979, 19801). When this is not the case, the reliability of a n estimate for a numerical density becomes difficult to assess. The fundamental problem with most of the numerical density equations is that they require information for their solution that cannot be extracted routinely from biological systems. Many stereological equations, for example, ask for information about regular profiles of simple spheroids, but, more often than not, cells that have been fixed and sectioned for electron microscopy can supply only irregular profiles from asymmetric nuclei. This has led to several provisional sohtions to the problem of estimating the D (Ebbeson and Tang, 1965; Greeley et al., 1978; Loud et al., 1978; Cruz-Orive, 19801, but most of these methods require some form of serial sectioning and do not enjoy the attractiveness of a simple stereological solution. The purpose of this paper is to examine ways to improve the interaction between stereological theory and experimental data. To do this, we will approach the problem of numerical densities from a different perspective. Surface densities, which represent the total surface area of a compartment contained within a unit of reference volume (e.g., pm3, cm3), are considered to be one of the best forms of stereological information, because the data needed to obtain these estimates can be collected directly from sections (Weibel, 1974). Earlier (Bolender, 1979a,b,), surface densities were used as a basis for estimating the relative volumes and frequencies of average cells and for evaluating changes in organellar compartments. The strategy of these surface area ratio papers was to find a membrane compartment within the cells that maintained a constant surface area throughout a n experiment and then to use it as a reference for interpreting changes in cytoplasmic compartments. In looking for a way to test the methodological assumption that the nuclear membranes of the exocrine cells of the guinea pig pancreas remained constant when treated with a secretagogue, a remarkable piece of information was found. The nuclear surface area of both the duct and endothelial cells of the pancreas also remained constant (Bolender, 1979a). This meant that when it came to estimating numerical densities and events in average cells, the surface area ratio methods could deal as effectively with cells having nonspherical and nonconvex nuclei (duct and endothelial) as with cells having spherical nuclei (exocrine). When the mean profile boundaries of these same pancreatic nuclei were measured from several animals, they were found to have a very small population variance. This observation indicated that there was a membrane compartment in the electron micrographs that could be measured with a remarkable reproducibility from one experimental animal to the next. The three methods described in this paper all use this mean profile boundary as their starting point, and, as a result, can generate experimental estimates with relatively small variances. The key step in the development of these methods consists of finding the surface area of a n average nucleus @& which can be used in combination with a nuclear surface density (Svn)to calculate a nuclear numerical density (Nv,): (2) Several examples will be used to show how equation (2) can generate experimental results with standard deviations that are severalfold smaller than usually expected for stereological estimates. MATERIALS AND METHODS Method 1: Estimation ofNumerica1Density of Nuclei (Cells) Models are given for estimating the numerical density of nuclei from surface areas (Figs. 1,2,3).Both models assume that all the nuclei 91 STEREOLOGICAL ESTIMATES MODEL 1 S MEAN PROFILE SPHERE BOUNDARY Fig. 1. Model 1. When a sphere is sectioned either randomly or systematic_ally (case shown), the resulting mean profile boundary CB)is equal to the mean boundary (b) of the original sphere. The B, which can be measured in sections, is used to calculate 1) in equation (8) the D of the original sphere, and 2) in equation (15) the surface area of thcsphere. In a n experimental setting, the surface area (S) of an average spherical exocrine nucleus is used in combination with the nuclear surface density to estimate the nuclear numerical density (equation 16). For details, see Materials and Methods. I R R Fig. 2. Calculation of a mean radius of a sphere from a mean profile radius (f). The example is simplified by using one quadrant of a two-dimensional section that passes through the center of a sphere of radius R. Eighteen equally spaced lines (profile radii) are drawn along the radius of the sphere perpendicular to the x axis and represent the r. of equation 4. The mean profile radius (3is identified with an arrow. If the positions of the profile radii had been chosen randomly along R, then the resulting F would be expected to have the same value as the one calculated from the equally spaced radii-if enough radii were used. of a particular cell type have the same surface of the nuclear profiles seen in thin sections area and shape. The first model assumes a is equal to the mean boundary (b)of a sphere spherical shape and the second assumes that (see Weibel, 1979, p. 37, for a mathematical a nonspherical nucleus can be treated as a proof). In Figures 1 and 2, the mean boundspherical one by introducing a correction fac- ary of a sphere is defined as the average of tor. The effects of relaxing these assumptions all possible boundary lengths that can be are considered in the Discussion. obtained by cutting the sphere randomly (or systematically) where: Model 1: Numerical density of spherical nuclei b = (bl + b2 + . * * + bJm, (3) The method for spherical nuclei is based o n the assumption that the mean boundary (B) and m is the total number of boundaries. R.P. BOLENDER 92 MODEL 2 I NONSPHERE CORRECTION FACTOR (k) 1 SURFACE AREA OF EOUIVALENT SPHERE MEAN PROFILE BOUNDARY OF EQUIVALENT SPHERE Fig. 3. Model 2. When a nonsphere is sectioned randomly, the resulting B is not equal to the b of a sphere having an equivalent surface area. By introducing a correction factor (k),however, the B of the nonsphere can be set equal to that of the equivalent sphere, and the calculations proceed as described for Model 1. See Materials and Methods for details. Each boundary bj defines a circular profile having a radius r;: r; = b;/2a. (4) The mean value of r with respect to x is defined by the following integral: The final equation is written in terms of-a mean diameter (D) and profile boundary (B) because the experimental estimates are based on measurements collected from many nuclei in sections: R (i-)x = 1 B lo (R2 - - xj2)” dx, D (5) where (R2 - xjzs,” equals the lengths of the individual radii and R equals the radius of the sphere. Evaluating the integral we obtain: F = (7d4). R. (6) i = b/2a, (7) 4B/7r2. (9) Note that equation (9) refers to nuclei all of one size. We know from Buffon (1777) and Smith and Guttman (1953)that the length of a boundary (B) can be estimated in sections by counting the number of intersections (I)that occur between the boundary trace and a set of parallel lines separated by a distance d: Since: B the relationship between the mean boundary (b) and the diameter (D) of a sphere can be given as: = = ( ~ / 2 .) I. d. (10) Moreover, a mean boundary (B) can be obtained for the mean number of intersections (I): - B = (d2) . . d. (11) STEREOLOGICAL ESTIMATES When equation (11)is substituted into equation (91, the intersection counting form of equation (9) is obtained for the nuclei (n): where a equals 1when a grid of parallel test lines is used and a equals 2 when a rectangular grid of horizontal and vertical test lines is used. The experimental evaluation of equation (12) indicated that a dense set of test lines is needed to estimate the length of the profile boundaries (see Fig. 4a). Since the counts are collected with both the horizontal and vertical test lines, a was set equal to 2 in equation (12). The data are collected in the following way. Electron micrographs of exocrine, duct, and endothelial nuclei are projected onto a test screen (final magnification approximately x 12,000) and the intersections between the traces of the inner nuclear membrane (Fig. 4b) and the lines of the test system (Fig. 4a) were counted. The mean number of intersections (I,‘) for the profiles of the nuclei is calculated by dividing the total number of intersections counted by the number of (m) profiles measured: (I1 + Iz + . . + IJm. 93 densities of nonspheroidal nuclei. The first option relies on the numerical density of the spheroidal exocrine nuclei and the relative frequencies of the exocrine, duct, and endothelial cells. The second option uses a correction factor to convert the mean profile boundary of the nonspherical nuclei to that of a sphere of equivalent surface area. To determine the numerical density of the duct and endothelial nuclei, which are nonspherical and nonconvex structures, two pieces of information are used: 1)the numerical density of the exocrine nuclei (from equation 16), and 2) the relative frequencies of the three nuclear types (from serial reconstructions). The numerical density (Nv) of the combined cellular compartment (34, which includes exocrine, duct, and endothelial cells, is obtained by dividing the numerical density of the exocrine nuclei (Nvexcn)by their relative frequency (NexCnlN3J: Option 1. The numerical densities of the duct and endothelial cells are found by multiplying the numerical density of the combined compartment by the relative frequency of each nuclear type (lc): (13) NVlc,cmn3 = NV3c,crn3 . (Nlcn\J3c). (18) The I,’, which come from micrographs having a magnification M, is standardized to a magnification of x 12,000: Equations 17 and 18 represent a solution to the problem of estimating the numerical density of structures that are nonspherical and nonconvex. It is important to note that the numerical density methods (Wicksell, 1925, 1926; Weibel and Gomez, 1962; DeHoff and Rhines, 1961; and Giger and Riedwyl, 1970) used most often by biologists all assume the presence of convex structures. Consequently, these methods can not be expected to supply the best estimates for numerical densities when they are applied to nonconvex structures such as duct and endothelial cell nuclei. Option 2. In model 1, the nuclei of the exocrine cells are spherical and their mean profile boundary (B) is assumed t o b e equal to the mean boundary of a sphere ($1. Now consider the case where we have a population of nuclei with the same surface area and shape, but that the shape is nonspherical (see Fig. 3). Assume that such a situation exists for the duct and endothelial nuclei (see Figs. 4c, d). The nonspherical structure shown in Figure 3 can also supply a mean profile bound- 1,’ = - - I, = I,’ 1 . (12,000 I M). (14) The solution to equation (12) gives the mean diameter (D,) of the spherical nuclei, which, in turn, is used to generate a mean nuclear surface area (SJ: - S, - 2 = a . D, . (15) The numerical density of the nuclei (Nv,! in a cm3 of reference space in found by dividing the surface density of the nuclei (Sv,) by the mean nuclear surface area (S,): Model 2: Numerical density of nonspherical and nonconvex nuclei Model 2 includes two options for estimating the mean surface areas and numerical Fig. 4. The test grid (4a) and inner nuclear membranes (arrows) of exocrine (4h), duct (4~1,and endothelial (4d) cells are illustrated. The exocrine nucleus can be described as a convex structure, whereas the duct and endothelial nuclei are nonconvex because they can be cut more than once by a section plane. STEREOLOGICAL ESTIMATES Figure 4 (continued). 95 96 R.P. BOLENDER ary (B), but, in contrast to the situation in model 1,it is not equal to the mean boundary of a sphere 6)of equivalent surface area. Consequently, the mean profile boundary (B,) of the nonspherical nuclei (measured in thin sections) must be corrected before it can be used to generate the surface area of a sphere with a n equivalent surface area. Since the numerical densities of the duct and endothelial nuclei are already known from equations (17) and (181, a correction factor can be generated that adjusts the mean boundary of the nonspherical nucleus to that of a sphere of equivalent surface area. The correction factor is introduced at the level of equation (12). By rearranging equation (I..), the mean surface area of the nucleus (S,) is obtained: % = Svn,cm3 1Nvn,cm3, (19) where n represents the nuclei of either the duct or endothelial cells. The mean surface area is then used to calculate the mean diameter of? sphere with a n equivalent surface area (D(eqsp)): - D(eqsp)= (ST)”, (20) (21) The correction factor (k,) is obtained for a given nuclear type by dividing the mean boundary of the equivalent sphere (eqsp) by that of the nonspherical nucleus: B(eqspfi(nonspherica1nucleus). (22) Equation (11)is rearranged and the correction factor is introduced: 7, Si,; = (Svi 1SvJ . sn, (24) where the surface density of the nuclear compartment is in the denominator of the membrane ratio. Since both surface densities in equation (24) share the same reference space (Bolender, 1979a), the ratio of surface densities can be simplified to a ratio of intersections: s,,;= (Ii/I,) . 3,. (25) Equation (25) represents an attempt to identify a n expression that can supply a n optimal form of average cell information. In this case, an optimal form is defined as a stereological estimate with the smallest standard deviation. - B(eqsp) = (n2 . ~(eqsp))/q- kn = Method 2: Estimation o f Membrane Surface Areas in Average Cells The surface area of a membrane compartment i (Si) in a n average cell (c) can be estimated from the mean surface area of the nucleus (n) and a ratio of surface densities: Method 3: Estimation o Membrane Surface Areal106fCells and mean profile boundary (B(eqsp)): - the numerical densities of duct and endothelial nuclei. = B, . k,/((d2) . d). (23) The 1, of equation (23) represents the expected number of intersections for a sphere with a surface area equal to that of the mean nonspherical and nonconvex nucleus. It can now be used to solve equations (121, (15), and (16) as given in model 1. The advantage of the correction factor in equation (23) is that it allows us to use the low variance counts of nuclear intersections to estimate a mean surface area, which, in turn, is used with a nuclear surface density to obtain estimates for The surface area of a membrane compartment i (Si)/106cells is calculated by multiplying the surface area of the compartment in a n average cell (Si,;)by 10% si/106 cells = Si,; . lo6. (26) Method for Serial Reconstructions Since the details of this method will be given elsewhere, only a brief description is given here. For two of the four pancreases, serial sections 1 pm in thickness were cut from tissue blocks, photographed a t a magnification of x 1,000, and printed at a final magnification of x 17,000. The relative frequencies of exocrine, duct, and endothelial nuclei were determined. Estimating a Mean Diameter From Profile and Numerical Densities Recall from the introduction that equation (1) can be used to estimate the numerical density of convex structures in a unit of reference volume from two pieces of information: l) the number of profiles of the 97 STEREOLOGICAL ESTIMATES structurehnit area of the section plane (NA), and 2) the mean diameter of the structure. Since Nv can be estimated for the exocrine nuclei (convex structures) with equation (16), and NA by counting the number of nuclear profiles seen in the thin sections, the D is obtained by simply rearranging equation (1): (27) The expected diameters of spheres can also be estimated with the methods of Wicksell (1925, 1926) and Giger and Riedwyl (1970). Using measurements of nuclear profiles coming from the same set of sections that supplied the mean profile boundaries for equation (9), the Ds are calculated with these two methods as well. Sampling Procedures Blocks of tissue were collected from the pancreases of four adult male guinea pigs (BFA strain) and fixed for electron microscopy as described previously (Bolender, 1974). Ten blocks were selected from each animal and thin sections were cut in the range of 40-50 nm. Six electron micrographs were collected from each section giving 60 micrographslanimal for each of the three stages of magnification. Since a random systematic method of sampling did not supply enough samples for estimating the mean profile boundaries of the duct and endothelial nuclei, all such profiles that could be found were micrographed. Stage I ( x 6,000) was used to estimate the volume and surface densities of the nuclei, Stage I1 ( X 12,000) to count the nuclear intersections needed to solve equations (12) and (23), and Stage I11 ( X 20,000) to count intersections for the outer mitochondrial and inner nuclear membrane compartments, which were used to solve equation (25). Two test systems were used: Stage I and I11 (see Weibel, 1979, p. 364 [D36]),and Stage I1 (see Fig. 4a). The micrographs were recorded on 35 mm film using a Philips 300 electron microscope, contact printed, and projected onto the test screens of a projector system (Weibel, 1979). A micrograph of a calibration grating having 21,600 lines/cm (E.F. Fullam, Inc., Schenectady, NY) was included on each film strip. PCS-I software (Bolender et al., 1982) and a Tektronix 4051 or 4052 computer (Beaverton, OR) were used to collect data from elec- tron micrographs, solve equations, and run statistical tests. RESULTS Method 1 Model 1: Spherical nuclei The results for the exocrine nuclei estimated with the methods of model 1are given in Table 1.The most remarkable information in this table is the demonstration that the mean nuclear intersections can be estimated with a standard deviation (s.d.) equal to less than 1.5% of the mean. These counts of nuclear intersections were directly responsible for the unusually small statistical variations of the stereological estimates that, in turn, they were used to generate. It can be seen, for example, that the surface area of the exoocrine nuclei had a s.d. of only 3% of the mean and the s.d. of their numerical density was as low as 5.1% of the mean. The mean diameter of equation (271, which was estimated from the nuclear profile and numerical densities (Table 1, footnote lo), was essentially the same as the estimate coming from method 1(Table 1, footnote 3). Table 2 compares the results of model 1 with those of two other methods that were used earlier to estimate diameters for the exocrine nuclei (Bolender, 1974, 1979a,b). All the data in Table 2 came from the same animals and same sets of thin sections. As described in the footnotes, the numerical densities were estimated from 1) the mean surface area and surface density of the nuclei, and 2) the mean volume and volume density of the nuclei. The results show a broad range of estimates: 5.632 x 10' to 10.42 x lo8 nuclei/cm3 for the numerical densities and 5.1% to 21% (as a percentage of the mean) for the standard deviations. It seems likely that the major cause of the differences between the three methods can be traced to the way the original nuclear profiles were measured. The major diameters of the nuclear profiles were used for the Giger and Riedwyl (1970) method (Bolender, 1974) and, in contrast to the results of model 1 (Table 1,footnote 3), they produced a 9% smaller estimate for the mean diameter (Table 2, footnote 4). Both the major and minor profile diameters were used for the Wicksell (1926) method (see Bolender, 1979a,b) and this led to a n 18% smaller value for the mean diameter (Table 2, footnote 4). When the estimates for the urn 133.4 125.3 130.6 125.8 128.8 3.9 (3.0) urn2 (SJ4 9.113 8.931 8.334 9.857 9.058 k 0.630 (6.9) x 1010/crn3 Pm2 (Svn.12 withjnner nuclear membrane (see Figure 4b). 16.08 6.516 15.58 6.315 15.91 6.447 15.62 6.329 15.80 6.402 0.24 i 0.100 (1.5) (1.5) urn number of intersections 45.49 44.09 45.01 44.18 44.69 + 0.67 (1.5) 6%)' @",3 (NAn,lJ7 NO. x 105/cm3 4.139 4.424 3.841 4.053 4.114 i 0.240 (5.9) (Nvn,d NO. x 1 0 8 / d 6.833 7.128 6.383 7.833 7.044 + 0.610 (8.6) = T = No. of nuclei n in 1 em3 of reference space. 'Nvn 7NA,'3, = No. of nuclear profiles dcm2 of the combined (3c) exocrine, duct, and endothelial cell profile areas. 8S~,.b,= 2 . InLT(where L,p = 2 . PaC' d; P,, = total points on 3 cells [3c]: exocrine, duct, and endothelial). 'Nv.,~. = S V , ~ , / (No. ~ , of nuclei n [in this case exocrine nuclei] in 1cm3 of reference space consisting of the 3 cells 13~1). "& = N A ~ . ~ J N see v ~ equation , ~ ~ ; (1). "I,(corr) = 8, ' k,/((a/2) ' d)); see equation (23). = ' n",e,8p,)i4; see equation (21) and Fig' '"B,(eqsp) Footnotes 11-14 are found in the text and appear in Tables 3 and 4 i3~n(eqsp) = (~,,,,,)/~)~h; see equation (20) and Fig. 3. '"%(eqsp) = s V ~ ( ~ ~ ~ ~ J N V ~ ( ~ ~ ~ ~ ) "S . D2; e.g., (animal 1):gn = K ' (6.516 am? = 133.4 am2. 5 S c , , ~ , = 2 . I,LT((where LT = 2 . PI, . d); l c [cell] indicates that the reference space is filled by only one cell type). n %, = (~112). I, ' d; e.g., (animal 1): B, = (d2). 45.49 . 0.225 pm = 16.08 pm. % = 4 . sn/a2; e.g., (animal 1): = (4 . 16.08 pm)/li2 = 6.517 pm. 'Mean s.d. (s.d. as % of mean) 2 3 4 Mean 1 Animal (EnY ~~ 8.351 . 8.244 7.683 7.512 7.947 f 0.410 (5.2) P.m2 x 101°/cm3 (svn,3J8 urn 6.611 6.732 6.528 6.789 6.663 2 0.120 (1.7) 6.261 6.580 5.884 5.970 6.174 ? 0.320 (5.1) (DnP No. x108/crn3 (~vn.3~' TABLE 1. Estimates for mean intersections Cj,,), boundaries @,J, diameters @,J, surface areas (g,,),and for surface (SvJ and numerical (Nv,,) densities for the exocrine cell nuclei m l'd F 00 W STEREOLOGICAL ESTIMATES 99 mean diameters are used to calculate the surface areas and volumes of mean nuclei, and combined with either the surface or volume densities to estimate the nuclear numerical densities, the other methods gave mean values (Table 2, footnotes 2 and 3) that are substantially different than those of method 1: they were 169% and 178% larger for the Wicksell method and 28% and 35% larger for the Giger and Riedwyl method. The choice of either a nuclear volume or surface area for estimating numerical densities also influences the results (see footnotes 2 and 3 in Table 2). The mean surface areas and surface densities of nuclei generate the smallest standard deviations for the method of model 1, whereas the mean volume and volume density gave the best results for the other methods. Notice in model 1 that the s.d. of the numerical density is more than twice as large for the volume based estimate (Table 2, footnote 3) as it is for the one based on surface areas (Table 2, footnote 2). +I Model 2: Nonspherical and nonconvex nuclei Estimates for the duct and endothelial nuclei are given in Tables 3 and 4. The data "Uncorrected for shape" were generated with model 1and those "Corrected for shape" with model 2. Table 3 contains a surprising result. When the nonspherical and nonconvex nuclei of the duct cells are treated as spheres (model l), the mean number of profile intersections is essentially the same as when they are corrected for shape (model 2). The difference between the estimates of the two models is less than 1%.Although these results may be fortuitous, the possibility exists that model 1 may also be applicable to certain classes of nonspherical nuclei. Notice that the duct cell nucleus had a mean surface area (corrected) of 128.4 pm' (Table 3, footnote 14) and a standard deviation of 6.5% of the mean. Its numerical density was 1.640 x 1 0' nuclei/ cm3 (Table 3, footnote 9). The similarity between the surface area of a duct cell nucleus (128.4 pm') and that of the exocrine cell (128.8 pm2) is also striking. Here we have a situation where two cells with the same embryonic origin have practically the same nuclear surface area. Table 4 summarizes the results for the endothelial nuclei. The pattern of a similarity between the nuclear intersections of the duct cells for models 1 and 2 did not occur for the endothelial nuclei; the results of the two 15.90 f 0.52 (3.3) 44.99 & 1.48 (3.3) 16.06 15.16 16.01 16.38 (3.3) f 0.21 6.444 6.507 6.143 6.490 6.637 130.6 f 8.4 (6.5) 133.0 118.5 132.3 138.4 39.74 k 1.41 (3.5) 39.48 41.46 38.05 39.95 (8.1) k 0.17 2.098 2.330 2.078 2.063 1.920 (3.3) k 1.46 44.62 45.05 42.53 44.93 45.95 47.81 46.21 46.68 46.93 46.91 k 0.67 (1.4) 1 2 3 4 Mean s.d. (s.d. a s % Pm 6.849 6.619 6.686 6.722 6.719 f 0.10 (1.4) Irm 16.90 16.33 16.50 16.59 16.58 i- 0.24 (1.4) (En)3 147.4 137.6 140.4 142.0 141.8 f 4.1 (2.9) pm2 (SJ4 (SVn . 3 2 clm2 x1010/cm3 1.659 1.210 1.120 1.324 1.328 f 0.24 (17.8) x10'0~cm3 82.19 54.25 57.80 65.86 65.03 f 12.43 (19.1) m2 (sVn,d5 6.391 k 0.21 (3.3) 6.454 6.092 6.436 6.582 (6.5) f 8.3 128.4 130.8 116.6 130.1 136.1 (10.9) 1.640 k 0.179 31.25 f 2.96 (9.5) 1.781 1.782 1.585 1.411 30.18 35.56 29.25 29.35 7.321 7.075 7.148 7.186 17.73 50.14 f 0.10 (1.4) f 0.25 (1.4) + 0.72 (1.4) 7.183 w Irm 18.06 17.46 17.67 17.73 (Gn)'3 51.11 49.40 49.90 50.17 (Bn)'2 162.1 (2.9) f 4.7 (16.1) f 6.43 39.98 (15.0) i- 0.1223 0.8172 0.9855 0.7692 0.6980 0.8160 48.81 34.49 36.01 40.60 168.4 157.3 160.5 162.2 (Nvn 3c)' NO. x 1o8/~m3 (Nvn,lc)' No. x 108/cm3 flm2 (Sn)'4 Model 2 Corrected for nonspherical shape (7,)" densities for the endothelial cell nuclei For footnotes, see Table 1. The correction factor for the endothelial cell nuclei (ken,,,) was 1.0558;see equation (23) (Td1 Animal (E,Y (3.3) f 0.52 15.77 15.92 15.03 15.88 16.24 (id,boundaries (ZJ,diameters (FJ,surface areas (SJ, and for surface (SvJ and numerical (NvJ Model 1 Uncorrected for nonspherical shape TABLE 4. Estimates for mean intersections For footnotes, see Table 1.The correction factor for the duct cell nuclei (kdJ was 0.9795; see equation (23) Mean s.d. (s.d. as % of mean) 3 4 45.43 42.89 45.31 46.33 5 'd F 101 STEREOLOGICAL ESTIMATES models differed by 7%. The mean surface area of the endothelial nuclei was 162.1 pm2 (Table 4, footnote 14), and the s.d. was 2.9% of the mean. There were 0.8172 x lo8 endothelial nuclei/cm3 (Table 4, footnote 9). Method 2 A mean nuclear surface area can be used to estimate 1)the numerical densities of nuclei (and cells), and 2) the surface areas of membrane compartments in average cells. Thus far, we have seen that model 1 can generate the most reproducible results for the numerical densities of cells. However, can model 1continue to give estimates with small standard deviations when it is used to estimate organellar compartments in average cells? The surface area of the outer mitochondrial membranes in a n average exocrine cellbased on the method of model 1-is given on the left of Table 5. Notice that the mitochondrial to nuclear intersection ratio (Table 5 , footnote 3) varies only slightly from animal to animal; the s.d. is only 3.7% of the mean. The surface area of the outer mitochondrial membrane was 674 pm2/average cell and the s.d. was 6.5% of the mean (Table 5 , footnote 5). Usually, estimates for the surface area of a compartment in a n average cell are calculated from the volume of a n average cell and the surface density of the compartment (Loud, 1968; Weibel et al., 1969; Bolender, 1974; Crapo et al., 1980; Mori and Christensen, 1980; Haies et al., 1981; Crapo et al., 1982). Table 5 also contains two sets of results based on the volumes of average cells. In contrast to the results of model 1(Table 5 , footnote 5 ) , the other two methods gave estimates (Table 5, footnote 8) for the outer mitochondrial membranes in a n average exocrine cell that were 51%and 13% smaller, but these mean values had 86% and 168% larger standard deviations. While a n estimate for the mitochondrial membranes (Table 5 ) with a s.d. of 6.5% of the mean seems attractive, it is more than twice as large as the s.d. for the nuclear membranes (Table 1, footnote 4). This was a surprising result because there are 5.2 times more mitochondrial than nuclear membranes in a n average cell, and, given the similarity in sample size (i.e., equal numbers of animals and micrographs), a smaller s.d. was expected). Can the 6.5% s.d. be explained by the method of sampling? Tissue blocks had been +I - 0000 l nw w mm mw m sp. m 102 R.P. BOLENDER taken from five regions of the pancreas (A to E), spaced equally along its length from head to tail (Bolender, 1974).To look for a n answer to this question, 60 micrographs were collected from each of the 5 regions (for one animal), and the surface area of the mitochondrial membranes in each region was estimated according to footnote 5 in Table 5. A statistical analysis of the results indicated that a n average exocrine cell in regions A, B, and C of the pancreas had about 30% more outer mitochondrial membranes than those in regions D and E (P < 0.05). This tells us that the exocrine cells can have different amounts of mitochondria according to their location in the pancreas, an observation reported earlier by Loud (1968)and Schmucker et al. (1978) for hepatocytes in the liver lobule. It also suggests that the method of regional sampling may have been responsible for part of the 6.5% s.d. The new areaweighted method of sampling (see Weibel, 1979) is designed specifically to deal with such problems of organ heterogeneity, and it seems likely that had this method been used smaller statistical variations for the outer mitochondrial membranes would have been found. Method 3 The surface area of the nuclear membranes/1O6 cells is compared to the surface densities of the nuclear membranes in Table 6. The results show that the s.d. (expressed as a percentage of the mean) can be reduced from 5.2 to 3.0%for exocrine cells, 8.1to 6.5% for duct cells, and 17.8 to 2.9% for endothelial cells. Therefore, by relating the nuclear surface area to lo6 cells-instead of to a cm3 of reference volume-the s.d. of the estimate (as a percentage of the mean) is decreased by 73%for exocrine cells, 25% for duct cells, and 514% for endothelial cells. Serial Reconstructions The relative frequencies of the exocrine, duct, and endothelial cells coming from the serial reconstructions are given in Table 7. The results are based on 649 reconstructed nuclei for animal 1and 582 nuclei for animal 2. This estimate of nuclear frequencies was used in option 2 of model 2 to calculate a correction factor for the duct and endothelial nuclei that were applied to the measurements of mean profile boundaries (see Fig. 3). The results of these calculations are given TABLE 6. Comvarison of nuclear surface areasll0' cells and nuclear surface densities Exocrine cells Sn/106cells pm2 Animal x 108/106cells 1 2 3 4 Mean s.d. (s.d. as % Duct cells Endothelial cells SVn,Sc 1*m2 x1 0 ~ ~ i c m ~ s,/106 SVn,Bc Pm2 x 108/106cells x 101°/cm3 1*m2 1.334 1.253 1.306 1.258 8.351 8.244 7.683 7.512 1.308 1.166 1.301 1.361 1.288 *(3.0) 0.039 7.947 f 0.410 (5.2) 1.284 2.098 1.621 1.328 k 0.083 k 0.170 f 0.047 k 0.240 (6.5) (8.1) (2.9) (17.8) 1.684 1.573 1.605 1.622 2.330 2.078 2.063 1.920 1.659 1.210 1.120 1.324 of mean) TABLE Z Comparison o f nuclear frequencies determined from serial reconstructions and models 1 and 2: Relative frequencies ofpancreatic nuclei Serial reconstructions % Duct % Endothelial cell nuclei cell nuclei cell nuclei % Exocrine Animal 68.90 73.19 Mean s.d. (s.d. as % of mean 71.04 f 3.03 (4.27) 21.21 17.26 9.794 9.553 Model 1 % Exocrine cell nuclei 69.35 72.52 72.03 72.83 19.24 9.673 71.68 f 2.80 f 0.170 f 1.59 (14.5) (1.759) Model 2 % Endothelial cell nuclei cell nuclei % Duct (2.2) 19.73 19.63 19.42 17.21 + 19.00 1.20 (6.2) 10.92 7.85 8.55 9.95 9.32 f 1.38 (14.8) 103 STEREOLOGICAL ESTIMATES in Table 7 (expressed as relative frequencies) in order that comparisons can be made with the serial reconstructions. Agreement is good. The mean values for the two sets of estimates differ by 0.9% for exocrine nuclei, 1.3%for duct nuclei, and 3.8%for endothelial nuclei. DISCUSSION Numerical Densities and Average Cell Information From Surface Areas The results for exocrine cells (see Tables 1,2, 5 ) demonstrate that estimates for numerical densities and compartmental areas in average cells can be obtained with a smaller s.d. when they are based on surface areas (equations 16 and 24). Mayhew (1981) has also reported that surface area references, for a given size of a sample, offer a smaller coefficient of variation than those based on volumes. The surface area approach is successful in lowering the statistical variance, because it is based on the mean nuclear profile boundary that can be estimated with a s.d. as small as 1.4% of the mean (Table 4, footnote 2). When this nuclear boundary is used to estimate the surface area of the nucleus, the s.d. increases to as little a s 2.9% of the mean (Table 4,footnote 4). In contrast, the methods of Wicksell (1925, 1926) and Giger and Riedwyl(1970) rely 1)on measurements of profile diameters that, in a n experimental setting, often depart from the assumed ellipsoid or circular shapes, and 2) on mean nuclear diameters that must be cubed to calculate the nuclear volumes. Consequently, these estimates for numerical densities had statistical variations that were often several times larger than those based on mean nuclear surface areas. Assumptions o f Models 1 and 2 Model 1 assumes that all the exocrine nuclei are spheres with equal surface areas, i.e., that they are a homogeneous population. Although this may be true in the living cells, a brief survey of electron micrographs reveals nuclear profiles ranging from circles and ellipses to moderately distorted shapes. In the thin sections, therefore, the assumed homogeneous nuclei are sometimes being represented 1)by noncircular profiles that are the result of section compression, and 2) by nuclei that have had their shape altered by the fixation and embedding procedures. When a section is compressed, the diameter of a circular profile is thought to remain intact as the major axis of a n ellipse and the ratio of the major to minor axes of the ellipse serve as a quantitative measure of the compression (Loud, 1968; Weibel, 1979). When a circular profile is compressed into a n ellipse, however, the length of the original boundary is reduced (see Weibel, 1979, p. 150) as well as the surface area of a mean sphere that would be generated by model 1. When the percentage of section compression (x) is plotted against the percentage of underestimate of surface area (y),the following expression is obtained: y = d(1.0 + O.O05x), (28) which indicates that the loss of surface area is roughly equal to the degree of compression. Since the profiles in the study had a mean compression of about lo%, a 10% overestimate for the numerical densities of nuclei in the blocks would be expected. Nuclei that have their spherical shapes altered by the preparative procedures are expected to have a smaller mean diameter and a smaller probability of being included in a section. The model in Figure 5 considers the consequences of not conforming to the assumption that all the nuclei have the same likelihood of being sectioned. Heterogeneity is introduced into the reference space (the cubes) by decreasing the diameters of the alpha spheres (black) while the beta spheres (white) are held constant. The results of introducing this error are expressed as a ratio of observed and predicted surface areas. As the alpha spheres become smaller (cubes B to F), their likelihood of being sectioned decreases, which accounts for the progressive increase in the error. Notice that a moderate amount of heterogeneity in the sizes of the spheres produces only a small error. Cube D, where the ratio of the mean diameters of the alpha and beta spheres is 0.75, for example, introduces only a 3% underestimate for the mean surface area of the six spheres. Consequently, slight alterations in the shapes of the nuclei are not expected to be a major source of error. What if all the nuclei do not have the same surface area? In Figure 5 , we know that a mean profile boundary of the six spheres in a given cube will be influenced by the separate diameters of the alpha and beta spheres and their individual frequencies. In cube C, for example, the combined s heres have a surface density of 15.46 cm /cube, and ex- zp 104 R.P. BOLENDER e A 2 100 09 10 D,, 08 07 6,,(u = 1 O l o O 5 c m . ~= 06 05 1 Ocm) Fig. 5. The effects of differences in the diameters of the spheres on the estimates for the mean caliper diameter (D). The model includes six cubes containing two subpopulations of spheres, alpha (black) and beta (white). Both the alpha and beta spheres in cube A have the same diameters and therefore show the same probability of being sectioned (the probability is expressed as a ratio of the diameters). In cubes B to F, however, the alpha spheres become less likely to be sectioned as their diameters decrease. In cube F, for example, where the alpha diameter is 50%that of the beta, the alpha spheres will be expected to appear in a section only half as often as the beta spheres. The error produced by this sampling probability is expressed as a ratio of the observed and expected values. It is less than 5% for four of the six cubes. The mean caliper diameters of the alpha and beta spheres in each cube reflect the skes and frequencies of the spheres (D = [C D,, +-CD,]/Gj:D(A) = l.0 cm; D(B) = 0.95g-n; D(C) = 0.9 cm; D(D) = 0.89 cm; D(Ej = 0.8 cm; and D(F) = 0.75 cm. pected values for the mean profile boundary of 1.48 cm, a mean diameter of 0.9 cm, and a mean surface area of 2.577 cm2. Dividing the surface density by the mean surface area gives the predicted value of six spheres in the cube. By introducing the error produced by the smaller probability of sectioning the alpha spheres, the observed value would be 6.06 spherestcube, which represents a n error of only 1%. Given the examples developed with Figure 5, it appears likely that a mean profile boundary taken from a slightly heterogeneous and distorted population of nuclei would still lead to a reasonably accurate estimate for a nuclear numerical density. The largest source of error seems to be the section compression, which can be estimated and corrected (see Weibel, 1979, p. 150). Model 2 included two options for estimating the mean surface area and numerical density of nonspherical nuclei. Option 1used the numerical density of the spheroidal exocrine nuclei and the relative frequencies of the exocrine, duct, and endothelial cells. Option 2 used a correction factor to convert the mean profile boundary of the nonspherical nuclei to that of a sphere of equivalent surface area. The mean profile boundaries from the duct cell nuclei provided a most unexpected result (Table 3, footnotes 2 and 12). The uncorrected and corrected estimates for the mean diameter differed by less than 1%. While a n explanation for such a result can not yet be given, it is still possible to speculate that equation (12) may be appropriate for both spheres and certain classes of nonspherical nuclei-without a correction factor. Sources of Statistical Variance in Stereological Estimates The standard deviations in Tables 1,3, and 4 are relatively small for the mean profile boundaries and surface areas of nuclei (range: 1.4 to 6.5% of the mean), but several times larger for the surface and numerical densities (range: 3.5 to 19.1% of the mean). The surface densities, which are used with the mean nuclear surface areas to estimate the numerical densities, are responsible for the larger standard deviations. Why are the surface densities so variable? A network of stereological information is illustrated in Figure 6, which extends from the organellar contents of a n average exocrine cell to the total number of cells in the pancreas. Reference spaces are identified for each set of information along with the equations that were used to make the calculations. Data are included from four animals along with means and standard deviations (expressed a s a percentage of the mean). The important thing to notice in this figure is that for each of the four animals, the cubic centimeter of reference volume always contains a different number of cells. I n a cubic centimeter of exocrine cells, for 105 STEREOLOGICALESTIMATES Total pancreatic volume Nn ,Nn = Nvn ' "pan ID8 pancreas1 5 340 8 724 6 676 6 694 crn3 of pancreas Nv, = INn . cm3 of exocrine cells Sv, i S, NV" CmJl IN, 10' 5 621 5 778 5 384 6 375 = ' svn S" 10' cm'/ 6 833 7 128 6 383 7 833 Average exocrine cell S" = T(D.)* I V 2 l 133 4 125.3 130.6 125 8 6.859 5.789 7.044 128.8 ( % 20.35%) ( 2 7.299%) ( 2 8.639%) ( ? 3.029%) Fig. 6. Numerical densities of exocrine cells. The frequencies of the cells are shown at several reference points along a network of information that extends from an average cell to the total pancreas. Estimates are included for four animals and standard deviations are expressed as a percentage of the mean. The point to be made by the figure is that except for the average exocrine cell estimate, which uses a single cell, all the remaining estimates for the mean values of the numerical densities are based on unequal numbers of cells. If a standard number of cells is used (see Table 61, then the standard deviation for the surface density (or numerical density) estimate falls from 7.3% (column 2) or 8.6% (column 3) of the mean to 3.0%.Notice that this is the same statistical advantage gained by using the average cell. For details, see text. example, the surface areas of the nuclear membranes came from as many a s 7.833 x 10' cells (animal 4) to a s few as 6.383 x 10' (animal 3)-a difference of 23% or 144.7 million cells. This tells us that a t least part of the experimental error associated with the estimates for surface densities can be attributed to a .sampling procedure that accepts data coming from a different number of cells for each animal. Method 3 modifies this sampling procedure by always collecting information from the same number of cells (lo6 cells/animal), and the result is a substantial decrease in the variance (see Table 6). This method of relating results to lo6 cells is widely used in biochemical studies, and, if extended to stereological data, can be expected to facilitate the task of correlating cellular function with structure (Bolender, 1981b, 1982). One can certainly expect that "biological variation" contributes to the differences we find in the number of cell/cm3 from one animal to the next, but it seems likely that many of the variations we measure actually come from artifacts, which are produced by the methods t h a t must be used to gain access to the information contained in thin sections. In attempting to find correlations of structure to function, we have found it especially helpful to remember that thin sections are not always representative of the blocks (section thickness and compression), t h a t blocks are not always representative of the living tissue (shrinkage and swelling), and that blocks of living tissue are not always representative of the nonhomogeneous tissues and organs. CONCLUDING REMARKS The goal of this study was to identify methods that can decrease the variance of stereological estimates. The strategy of the approach consisted of looking for ways to optimize the interactions between stereological equations and experimental data. By defining the best form of experimental data as the one that has the smallest population variance, and then writing stereological equa- 106 R.P. BOLENDER tions that could take advantage of this optimal starting point, stereological estimates for biological structures could be obtained with standard deviations as small as 1.4%of the mean. ACKNOWLEDGMENTS The assistance of S. Anderson, S. Allen, and D. Ringer is gratefully acknowledged. A referee, Dr. J.F. Bertram, and M.P. Larsen provided a number of constructive comments and suggestions, and they are sincerely thanked. This work was supported by USPHS grants GM-22757 and GM-29853 from the National Institutes of Health. LITERATURE CITED Bolender, R.P. 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