## Abstract

ABSTRACT. The most reliable method for estimation of mean glomerular volume (MGV), the disector/Cavalieri method, is technically demanding and time consuming. Other methods suffer either from a lack of precise correlation with the gold standard or from the need for a large number of glomeruli in the sample. Here, a new method (the 2-profile method) is described; it provides a reliable estimate of MGV by measuring the profile area of glomeruli in two arbitrary parallel sections. MGV was estimated in renal biopsies from 16 diabetic patients and 13 normal subjects using both the Cavalieri and the 2-profile methods. The range of individual glomerular volumes based on the Cavalieri measurements was 0.31 to 4.02 ×10^{6} μm^{3}. There was a high correlation between the two methods for MGV (*r* = 0.97; *P* < 0.0001). However, the 2-profile method systematically overestimated MGV (*P* = 0.0005, paired *t* test). This overestimation was corrected by introducing a multiplication factor of 0.91, after which statistical criteria of interchangeability with the Cavalieri method were met. The optimal distance between two sections was determined as 20 μm with a coefficient of variation of 7.4% in repeated measurements of MGV. On the basis of findings that values for MGV stabilize after ten glomeruli are measured by the disector/Cavalieri method, it was determined that the accuracy of MGV by the 2-profile method obtained by eight glomeruli was less than 7% different from ten in all cases. Thus, the 2-profile method is a practical alternative to the disector/Cavalieri method for estimating MGV, especially in small samples and blocks with limited residual tissue.

Studies of the relationships between glomerular structure and renal function have documented the importance of unbiased quantitative methods (1–6). Glomerular volume may be increased in diabetes (7–9) and, as a key determinant of total glomerular filtration surface, represents an important structural parameter in demonstrating the relationship of filtration surface and GFR in type 1 diabetic patients (1,2). Glomerular hypertrophy has been established as a predictor of glomerulosclerosis in a variety of human and animal glomerular disorders (10–14). Also, a decrease in glomerular volume has been observed consequent to ischemic renal damage (15) and is a concomitant of atubular glomeruli (16). Thus, glomerular volume is an important parameter for many morphometric studies of glomerular structure (17,18).

The ideal method for estimating mean glomerular volume would be based on number weighted sampling, which determines that the likelihood of measuring a given structure, such as a glomerulus, is dependent on its frequency within the sample, rather than there being a greater possibility for larger glomeruli to be sampled (19). Glomeruli vary in shape and deviate from a perfect sphere; therefore, the ideal method should also be independent of shape. Finally, the ideal method should be capable of measuring individual glomerular volumes, thus providing their size distribution. Several methods have been proposed for estimating glomerular volume (19–22). It is generally accepted that estimation based on the Cavalieri principle is the most accurate and unbiased method (19). This method is independent of shape, able to measure individual glomerular volume and provide a size distribution of glomeruli and, if used with a sampling strategy based on the disector principle, is number-weighted (22). However, this method requires careful serial sectioning of the specimen, and it is only applicable to complete glomeruli, eliminating those glomeruli already partially sectioned. Thus, the Cavalieri method may be of limited value when only a small residual of the biopsy specimen is available, which is often the case in retrospective human studies in which tissues have not been serially sectioned by protocol and sectioning for routine pathologic examination has already been performed. Other methods used to estimate mean glomerular volume include maximal profile area (MPA), the Weibel-Gomez method, and a method based on the disector principle (22), each with its own advantages and disadvantages.

The new 2-profile method, introduced here, is similar to MPA in assuming that glomerular shape is spherical. However, the 2-profile method is simpler and at least as accurate as MPA and requires only two arbitrary parallel sections through each glomerulus.

## Materials and Methods

### Cases

Twenty-nine renal biopsies from 16 diabetic patients and 13 normal kidney donors aged 30 ± 9 yr (10 to 62 yr) [mean ± SD (range)] were randomly selected from an extensive collection of biopsies in which serial sectioning had already been done and at least ten glomeruli had been found for mean glomerular volume (MGV) determination by the Cavalieri method. The duration of diabetes was 18 ± 5 yr (5 to 19 yr). All research biopsies were performed with permission of the Committee on the Use of Human Subjects in Research at the University of Minnesota and after informed consent was obtained.

### Tissue Processing

Biopsy specimens were fixed in Zenker solution, embedded in paraffin, serially sectioned at 5 μm onto sequentially numbered slides, and stained with periodic acid-Schiff (PAS). On the basis of the discussion of Weibel (23) regarding systematic errors due to finite section thickness, we compared the average diameter of glomeruli to the section thickness and concluded that the Holmes’ effect would be minimal in 5-μm sections. A precise knowledge of section thickness is necessary to obtain a reliable estimate of GV by the 2-profile method. To minimize thickness variation, the same microtome and technician was used for this study.

### Glomerular Volume Estimation Based on the Cavalieri Principle

Measuring the areas of glomerular profiles in serial parallel sections a known distance apart provides an estimate of its volume (Figure 1). The disector principle of particle sampling proposed by Sterio (22) is an unbiased strategy in which the probability that a particle may be sampled is independent of its volume (Figure 2). A detailed description of this principle is provided elsewhere. Together, the Cavalieri method and the disector principle provide a number weighted estimate of MGV. PAS-stained slides were projected onto a white surface with an Olympus BH-2 microscope at ×150, and glomeruli were selected for study by the Cavalieri and the 2-profile method (see below) using the disector principle. The profile area of every fourth section (20 μm apart) was measured by point counting using a grid with 9 points/4 μm (2). The outline of the glomerular tuft as a complex polygon was used to define the profile area (24).

Results of point counting in the ten randomly selected glomeruli were used to generate glomerular volume estimates by summing the areas of profiles and multiplying by the distance between them (19):

where V_{GCAV} is the estimated glomerular volume based on the Cavalieri principle, T is the distance between sections, and A_{i} is the area of the i^{th} profile.

### Glomerular Volume Estimation by the 2-Profile Method

Given the geometry of spheres, it is possible to fit one and only one sphere through two parallel circles located on a common axis (Figure 3). Reducing the three-dimensional shape of a sphere to a two-dimensional circle, these two circles become two parallel lines. The equation describing these relationships is derived as follows:

where BD and AE are the radii of two parallel profiles of the sphere and OE is the radius of the sphere itself.

Replacing OE with R (glomerular radius), BD with r_{1} (the radius of the smaller profile), AE with r_{2} (the radius of the larger profile), and AB with h (the distance between two sections, then:

and given the formula for the volume of a sphere:

where V_{GTP} is GV measured by the 2-profile method.

These equations are written for the situation in which both sections are on the same side of the sphere’s equator. However, when one section is on the other side of the equator, the negative sign for AB (h) is squared and equation 4 does not change. Thus, knowing the areas of any two parallel random profiles of a sphere and the distance between them is sufficient to calculate its volume. The radius of a circle with the same area as the profile was calculated in the two randomly selected profiles in each glomerulus. Glomerular volume was estimated using equation 4.

### Development of a Correction Factor for the 2-Profile Method

To evaluate the interchangeability of the new 2-profile method with the Cavalieri method, MGV estimated from paired sections at random distances and the Cavalieri method were compared for relationship and agreement (25,26). The correction of the overestimated values obtained by the 2-profile method was based on that of Bland and Altman (25). However, in contrast to the Bland and Altman approach, the Cavalieri method was accepted as the gold standard. To reduce the range of within-subject variation and, consequently, to expand the range within which the two methods could be considered interchangeable after correction, a logarithmic transformation was applied to the difference of MGV values obtained by two methods. Mean difference after logarithmic transformation was considered as the bias and subtracted from the MGV estimated by the 2-profile method:

where V_{GTPi} is the volume of the glomerulus estimated by the 2-profile method before correction.

And after back-transformation:

And after introducing this correction into equation 4:

### Repeatability of the 2-Profile Method and Derivation of the Optimal Distance between Two Sections

The optimal distance between two sections was defined as the minimum distance providing less than 10% coefficient of variation (CV) between repeated measurements by the same observer. Ten random cases were selected to check for repeatability of the new method. Four paired random locations on each glomerulus at distances of 5, 10, 15, and 20 μm (a total of 16 measurements per glomerulus) were selected in ten randomly selected glomeruli per case. Repeatability of the 2-profile method was evaluated separately for each distance according to the British Standards Institution criteria (25,27).

### The Minimum Number of Glomeruli Required

Preliminary studies indicated that the CV of MGV estimated by the Cavalieri method was almost always below 30% after ten individual glomeruli were measured at 20-μm intervals (unpublished data). This meets published recommendations for the reliability of an estimate (28). Given that these two methods using ten glomeruli per biopsy were essentially interchangeable (see Results), we considered the confidence interval (CI) of 95% of MGV obtained by ten glomeruli as a measure of accuracy for MGV obtained by numbers of glomeruli sampled below ten (MGV_{i}). This was done by calculating type 1 and type 2 errors based on t-distribution for degrees of freedom from 1 to 9 in each case. The order of selection of glomeruli for MGV estimation was random.

### Statistical Analyses

The relationship between the two methods was evaluated using the Pearson correlation. Interchangeability was assessed by the criteria of Lee *et al*. (29), except for individual-subject agreement for MGV, which was assessed by the Bland and Altman method (25,26). The criterion of repeatability was that 95% of differences be less than 2 within subject SD and the coefficient of repeatability was calculated as 2.83ς_{r}, where ς_{r} is the within subject SD obtained by one-way ANOVA as defined by the British Standards Institution and described by Bland and Altman (25–27). The definition for accuracy was: 1 − (type I error + type II error). *P* < 0.05 was considered significant. Except where noted, data are presented as mean ± SD.

## Results

The volume of individual glomeruli based on Cavalieri measurements ranged from 0.31 to 4.02 × 10^{6} μm^{3}. The correlation between individual glomerular volumes measured by the Cavalieri and the 2-profile methods was *r* = 0.87, *P* < 0.001 (Figure 4). MGV estimated by the Cavalieri and the 2-profile method were 1.76 ± 0.73 × 10^{6} μm^{3} and 1.92 ± 0.67 × 10^{6} μm^{3}, respectively (*P* < 0.0005). The correlation between the MGV obtained by the two methods was *r* = 0.97, *P* < 0.0001 (Figure 5). The mean difference of MGV obtained by the two methods after logarithmic transformation was 0.041. Substituting this value in equation 5 and applying back transformation in equation 6, the corrected equation to estimate VG is:

After applying this correction factor to the 2-profile method, the difference between MGV estimated by this method and the Cavalieri method was calculated. The limits of agreement defined as d̄_{log} (corrected) ± 1.96 SD were 0.0047 ± 0.068 and, after back transformation, 0.86 and 1.18 for lower and upper limits, respectively (Figure 6, A and B). Ninety-six percent of MGV estimated by the new method were between 0.86 and 1.18 times the Cavalieri values. The same approach was used to evaluate the interchangeability of these two methods for estimating individual glomerular volumes after applying the correction factor. Ninety-five percent of individual differences were located in d̄ ± 1.96 SD; with 757023 and −773235 μm^{3} as the upper and lower limits of agreement (Figure 7).

The coefficients of repeatability, within-group SD and coefficients of variation of repeated measurements between the 2-profile method, improved by increasing the distance between the two sections (Table 1).

Increasing the number of glomeruli in estimating MGV resulted in greater accuracy and lower SEM values (Figure 8, A and B).

## Discussion

Glomerular volume measurements have been widely used to study structural changes in various renal diseases, renal physiology, and normal kidney growth and development (10–15,8,30,31). Measurement of glomerular volume isneeded for estimating the absolute values of various structural parameters, such as filtration surface area, capillary length, or number of cells per glomerulus (32,33). MGV estimates have prognostic value in idiopathic nephrotic syndrome (11). The Cavalieri method may be considered the gold standard method for estimating MGV. This estimate is independent of the shape of the glomerulus (design-based) and, in contrast to model-based methods, requires no assumptions regarding glomerular shape. However, it requires sufficient tissue for an adequate number of complete glomeruli for a reliable estimate of MGV. This problem may be particularly important when dealing with limited human biopsy material. This is the usual case with archived tissues, which have not been serially sectioned, or with limited numbers of glomeruli in the original biopsy specimen. Given the interchangeability of the Cavalieri and 2-profile methods, it would be possible, with limited number of glomeruli, to determine GV by the 2-profile method on both complete and incomplete glomeruli. McLeod *et al*. (34) suggested that increasing section thickness to approximately 20 μm and decreasing the number of glomeruli to five would not increase the SD of the mean values obtained by the Cavalieri by more than 3%. As noted above, our unpublished data showed similar results regarding section thickness. However, the statistical reasoning of McLeod *et al*. is based on the estimated changes in SD of the between-patient mean values, while reducing the number of glomeruli studied. This analysis ignores glomerular variations within a biopsy specimen, which may be important where there are small and large glomeruli in the same tissue, as one would expect in tissues with atubular glomeruli (16).

Several other methods have been proposed to estimate MGV or the volume of individual glomeruli. This laboratory previously compared the Cavalieri, the maximal profile area (MPA), and the Weibel-Gomez and disector methods (35). The MPA method was highly correlated with the Cavalieri method (before: *r* = 0.93, *P* < 0.01; after: *r* = 0.91, *P* < 0.001). However, the MPA method requires at least five serial sections, the area of the first two and the last two sections to be smaller than the middle one, and at least one of these smaller profiles to be 2700 μm^{2} less than the middle profile (35). In addition, for incomplete glomeruli to be usable, it is necessary that the equator and a distance including two profiles after that should be located within the available tissue. These conditions make this method more difficult to use and less likely to find adequate numbers of glomeruli in small biopsy specimens than the 2-profile method.

It has been shown that the MPA method overestimates MGV (35,36), as does the 2-profile method before applying the correction factor. Both the MPA and 2-profile methods follow almost the same assumption regarding shape; therefore, it may be concluded that this assumption leads to overestimation of MGV. The Weibel-Gomez method, based on assumptions for glomerular shape and size, has the advantage of simplicity (32). Usually a few sections are enough, but a correction factor is needed to compensate for overestimation (35). However, volume-weighted sampling, the shape coefficient and the size distribution coefficient may be sources of imprecision in this method (36,37). Weibel-Gomez correlates with the Cavalieri method, albeit not as precisely as MPA method (36,37). The method based on the disector principle is both independent of glomerular shape and represents number-weighted sampling (21) but, at least in small samples, does not correlate well with the Cavalieri method (36), whereas increasing sample size eliminates its advantage, which is time-saving.

The data in the present study showed that, in most instances, individual glomerular volumes are predictable by the 2-profile method and that these data met the criteria of Bland and Altman for interchangeability with the Cavalieri method (25). However, considering the wide limit of agreement, this method cannot be recommended for estimating individual glomerular volumes. However, the shape aberrations reflected in these wide limits resulted in both overestimation and underestimation, thus providing a good estimate of MGV. Given large samples, the 2-profile method might provide a reasonable approximation of glomerular size distribution, but, for this parameter, the Cavalieri method is probably more appropriate. Paraffin embedding causes a marked shrinkage in glomeruli compared with plastic-embedded tissues (38). Moreover, removal of perfusion pressure and compression by the biopsy needle probably results in immediate decrease in glomerular volume compared with the *in vivo* condition. Thus, glomerular volume, especially in paraffin-embedded tissue, should be considered only in relative and not in absolute terms. It has been suggested that MPA could be an alternative to the Cavalieri method for measuring individual glomerular volume. Yet, no appropriate statistical proof of this is available (35,36,37). Neither the Weibel-Gomez nor the disector methods can estimate individual glomerular volume (21,22).

As might have been predicted, as the distance between two sections is reduced, the negative effect of errors, imprecision and deviation from spherical shape on the estimate of mean glomerular volume, increases. On the other hand, increasing the distance between sections might increase the probability that smaller glomeruli could be missed. Inherently, the 2-profile method provides a volume-weighted estimate of MGV because there is a greater probability that larger glomeruli will be sectioned twice in two parallel sections. Increasing the section distance would probably add to this bias. This study tried to find the least possible distance to preserve repeatability and, thus, did not go beyond section distances of 20 μm. This provided a satisfactory coefficient of variation of 7.4%. It should be noted that the nature of this CV value is different from those previously reported (36,37). Although the methodologies used to calculate CV were not fully explored, it can be surmised that the CV of MGV was calculated from repeated measurements in different samples (glomeruli). This provides a mixed measure of variation originating from the method itself and from variations in sample size and glomerular size and shape variations in the population of glomeruli under study. The CV in the present report, in contrast, simply estimates the repeatability of the method. The 2-profile method can be used with unbiased sampling (the disector method) for a number-weighted estimate of MGV.

The minimum number of glomeruli required to obtain a reliable estimate of MGV requires establishment for each method. We used a mixed sample of type 1 diabetic patients and normal controls with ages ranging from 10 to 62 yr and duration of diabetes from 5 to 19 yr. Consequently, this study may be applicable to this wide range of human situations. This study illustrated the different variances obtained from different numbers of glomeruli on the accuracy (reliability) of the resultant mean. As suggested, these values are not true values of accuracy, because it was assumed that the CI of MGV_{10} represents the correct 95% limits for MGV. However, these data provide a reasonable comparison for MGV obtained with fewer than ten glomeruli. This study cannot recommend any specific number, as each study will need to balance the importance of accuracy against the limitation of resources. Nevertheless, the accuracy of MGV obtained by eight glomeruli is 93% and by five glomeruli is more than 70% of that obtained by ten glomeruli. This is in accordance with findings by other investigators (34,35).

The correction factor derived here may not apply to age groups, renal disorders, or fixatives that are different from those of the current study. However, the use of a correction factor may not be important as long as the study conditions are similar within the groups to be compared.

In summary, the 2-profile method is interchangeable with the Cavalieri method for estimating MGV, is easier than the Cavalieri method, and is probably more accurate than methods other than the Cavalieri gold standard. The 2-profile method makes it more likely that adequate numbers of glomeruli will be available from small tissue samples or from tissues that previously have been extensively sectioned. Thus this technique is especially useful when limited tissue is available.

## Acknowledgments

This work was supported by grants from the National Institutes of Health (DK-13083 and DK-54638) and the National Center for Research Resources (M01-RR00400). We thank Ms. Patricia L. Erickson for her secretarial assistance.

- © 2002 American Society of Nephrology