## Abstract

The objective of this study was to investigate the number of glomerular profiles that are required for accurate estimates of mean profile area in a renal biopsy series. Slides from 384 renal biopsies from one center were reviewed. They contained a median of seven glomerular profiles or of four profiles without sclerosis. Profile areas were measured using stereologic point counting. The “true individual mean” for each biopsy was calculated and the “true population mean” for groups of biopsies derived. Individual and population “random sample means” then were calculated from a random sampling of profiles in each biopsy and were compared with true means for the same biopsies. The effect on the true population means of the entire group of biopsies was also assessed, as the minimum number of glomerular profiles that were required for inclusion was changed. In a single biopsy, random sampling of ≥10 profiles without exclusions and of eight profiles or more without sclerosis reliably estimated the true mean areas. In a group of 30 biopsies, random sampling of five or more glomeruli per biopsy reliably estimated the true population mean. In the aggregate series, inclusion of all 384 biopsies produced the most robust true population mean; the reliability of the estimates decreased as the numbers of eligible biopsies diminished with increasing requisite minimum numbers of profiles per biopsy. We conclude that, while ≥10 profiles might be needed for reliable area estimates in a single biopsy, far fewer profiles per biopsy can suffice when groups of biopsies are studied. In analyses of groups of biopsies, all available biopsies should be used without consideration of the number of glomerular profiles in each. Stipulation of a specific minimum number of glomeruli in each biopsy for inclusion reduces the power of analyses because fewer biopsies are available for evaluation.

It is increasingly recognized that the size of glomeruli in the kidney has important clinical and prognostic implications (1–4). Variations are described by body surface area, by birth weight and nephron number, by degree of sclerosis, and by category and stage of renal disease (5–8). It has been proposed that the susceptibility of some high-risk populations to renal disease is marked by glomerulomegaly, which, in turn, might mark nephron underdosing. Enlarged glomeruli seem to be at risk for premature sclerosis, whether by hyperperfusion mechanisms or other processes (9–12).

The gold standard method of assessment of glomerular size is through volume estimates by the Cavalieri method, which involves sectioning at standard intervals through whole glomeruli. This is more readily done on blocks of kidney tissue, such as wedge biopsies or at autopsy (13,14), although it can be applied to core kidney biopsies if adequate numbers of intact glomeruli are present. However, this is a time-intensive and laborious procedure that is unlikely to be incorporated in routine preparation of tissue for diagnostic examination.

Assessment of the areas of glomerular profiles on routinely sectioned and prepared biopsy material is the most feasible approach to estimates of size in disease processes. Much more could be learned from study of glomerular size from the many biopsies that already are archived in most major centers. Area estimates can be translated into volume estimates by use of formulas such as that of Weibel and Gomez. This method uses the formula glomerular volume = profile area^{1.5}× β/κ, where κ is 1.01, a size distribution coefficient that assumes a coefficient of variation for glomerular size within a single specimen of approximately 10%, and β is the shape coefficient of 1.38, the value of a sphere (15). However, such assumptions introduce bias, and it is not clear that they offer any advantage over area estimates in studies of groups of biopsies.

Several factors contribute to variations in glomerular profile size observed in histologic sections. One is the level of the glomerulus sampled by the section: sections that go through or close to the equator of a glomerulus yield large profiles, whereas those at the poles yield small profiles. Another concern is the variability of glomerular size in an individual, even those without renal disease, which, as we recently defined, can vary between two- and eight-fold (16). There is likely to be even more variation in diseased kidneys. In theory, size estimates should be biased in favor of larger glomeruli, whose cross-sectional profiles are more likely to be represented in any random section. The issue of including glomeruli with and without sclerosis should also be considered, given the variations in glomerular size with sclerosis, both by degree and by type of sclerosis (segmental, ischemic) (8,17). From these perspectives, the variability of mean glomerular profile area should be lessened progressively as more profiles are included in the sample.

When the objective is to estimate glomerular size in an individual kidney biopsy, the more glomeruli that are sampled, the more reliable that estimate should be. McLeod *et al.* (18) suggest that at least eight profiles need to be studied in this setting, even with rigorous study of “true” individual glomerular volume by the Cavalieri method. However, when groups of biopsies are being studied to evaluate associations of glomerular size with demographic, morphologic, or diagnostic variables, the minimum number of glomerular profiles that define eligibility of a biopsy for its inclusion is debated. Different reports of glomerular size differences by disease entity or sclerosis have included biopsies with at least four, at least five, and at least seven profiles (2,8,17,19–21), and a recent critique (unpublished) recommended a minimum number of eight glomerular profiles per biopsy.

Such stipulations can influence the ability to conduct a study on a given series of biopsies. In view of the variable and restricted number of glomeruli in needle kidney biopsies, the number of biopsies that are available for inclusion in any analytic series will decrease as the minimum number of glomerular profiles that are stipulated for eligibility increases. This can preclude the ability to conduct a study if too few biopsies are eligible for inclusion, especially when subcategory analyses (*e.g.*, by individual disease entity) are proposed. Conversely, the lower the minimum number of necessary profiles stipulated, the more biopsies will be eligible for inclusion in any series, with the potential for the power of increased numbers to offset the potentially less accurate individual means in the participating biopsies.

We investigated the number of glomerular profiles that are required for reasonably accurate estimates of mean glomerular area in a series of renal biopsies that were performed and processed at a single center.

## Materials and Methods

### Renal Biopsies

Slides from renal biopsies from 384 patients with nondiabetic renal disease, performed from January 1, 1994, to December 31, 2001, and archived in the Department of Pathology at the University of Mississippi (Jackson, MS), were reviewed. The mean age of patients was 36.7 ± 17.5 yr. The biopsy tissue had been fixed in 10% buffered formaldehyde, embedded in paraffin, cut in series of 25 sections at 3 μm, and stained with hematoxylin and eosin, periodic acid-Schiff (PAS; the 12th and 24th sections in every block), Masson’s trichome, and periodic acid-methenamine silver stains. The PAS-stained section with the most cortical tissue from each biopsy was projected onto a white surface using an Olympus BH-2 microscope at a final magnification of approximately ×600 for estimates of glomerular profile areas.

### Measurement of Glomerular Profile Area

The tuft area of each individual glomerular profile was measured using a traditional stereologic point-counting method and an orthogonal grid system (each grid square measured 2.5 × 2.5 cm). Glomerular profiles that were closer than one glomerular diameter to the edge of the sections were excluded from the study. All glomerular profiles within the more central parts of the biopsies were measured. A semiquantitative assessment of the extent of sclerosis in every tuft profile was made, ranging from 0 (no sclerosis) to 4 (75 to 100% sclerosis). Only grade 0 profiles were considered to be “nonsclerosed.” The average area estimates of all profiles in each biopsy were calculated—the “true individual mean” for that biopsy. The means of these individual biopsy means were then calculated for groups of biopsies to derive “true population means.” The distribution of these means was always skewed but was normalized by log transformation, and the average was expressed as the geometric mean (gmean) (confidence interval [CI]). All area measurements were expressed as μm^{2} × 10^{3}.

The means of specified numbers of randomly sampled glomeruli in each biopsy were then calculated, the “random sample individual means.” Their average was derived as the “random sample population means,” which were similarly expressed as the gmean (CI). The individual random sample means were compared with the true individual mean within a single biopsy, and the population random sample means were compared with the true population means when groups of biopsies were compared. All analyses were conducted in two stages, first with all glomerular profiles considered without regard to presence and degree of sclerosis, and second, with only profiles without perceptible sclerosis considered.

### Statistical Analyses

All analyses were performed using STATA statistical software, version 8.2, (Stata Corp., College Station, TX). The random selection of a specified number of glomerular profiles within individual biopsies was performed by a specific STATA program (22,23), and the geometric mean of their areas was calculated. This process was repeated as the number of randomly chosen profiles increased from one through 12.

Lin’s concordance correlation coefficient (R_{c}) was calculated to determine the level of agreement between the individual randomly sampled means and the true individual mean for each biopsy in the stable series of 30 biopsies using a downloadable STATA program (24). This technique combines measures of both precision and accuracy to determine whether the observed data deviate significantly from the 45-degree line of perfect concordance (*i.e.*, line of identity). Agreement is generally considered unsatisfactory when R_{c} < 0.6, satisfactory when R_{c} 0.6 to 0.9 and excellent when R_{c} > 0.90.

### Ethics Approvals

This study was approved by the Ethics Committees of the University of Mississippi Medical Center and Monash University School of Medicine.

## Results

Biopsies contained from one to 23 glomerular profiles with a median of seven profiles, when all glomerular profiles were considered, and contained zero to 17 profiles, with a median of four, when only profiles without visible sclerosis on that section were considered. Figure 1 illustrates the distributions of the numbers of glomerular profiles in the biopsies and the substantial reductions in numbers when profiles with sclerosis are excluded. Figure 2 shows groupings of the biopsies according to the minimum number of glomerular profiles that each contained. Without exclusions as a result of sclerosis, 96.1% of the biopsies had four or more glomerular profiles, 49.7% had eight or more glomerular profiles, and only 21.3% had ≥12 profiles. When only profiles without sclerosis were included, the proportions of biopsies with four or more, eight or more, and ≥12 profiles fell to 55.7, 26.2, and 9.3%, respectively. These findings are important in relation to population data, when the issue of sample size or the number of biopsies that contribute to the group data is of great significance.

Evaluation of the mean glomerular profile areas proceeded in several stages, and the results for each stage are summarized in Tables 1, 2, and 3.

### Estimating Individual Mean Areas: Sampling Increasing Numbers of Randomly Chosen Glomeruli in a Single Biopsy

This effect was analyzed in a biopsy with 23 glomerular profiles, 14 of which were without discernible sclerosis on that section. Estimates from increasing numbers of randomly sampled glomeruli were compared with the true individual mean for that biopsy, based on assessment of all eligible glomeruli. As shown in Table 1 and Figure 3, the more glomeruli that were included in the random sample, the better the estimates were. The estimates were particularly stable when 10 or more profiles were sampled, without regard to sclerosis, and when eight or more profiles without sclerosis were sampled.

### Estimating Populations Mean Areas: Sampling Increasing Numbers of Glomerular Profiles When the Number of Participating Biopsies Remains Constant

Table 2 and Figure 4 show the effect of sampling a progressively larger number of randomly chosen glomerular profiles in biopsies with abundant glomeruli, for which the number of participating biopsies did not change. This was ascertained on a group of 30 biopsies that each had ≥12 nonsclerosed glomeruli (range 12 to 23). Mean estimates from three or more randomly sampled nonsclerosed glomerular profiles all reliably reflected the “true population mean.” Their stability was particularly good when five or more glomeruli were sampled, and the CI of the estimate were not substantially reduced by sampling greater numbers. Lin’s concordance coefficient was >0.9 when four or more glomeruli without exclusions and three or more nonsclerosed glomeruli were sampled, indicating excellent agreement between the true biopsy means and the random sampled means. There was little extra benefit from sampling eight or 12 glomeruli compared with the sampling of five or more at both the population and the individual biopsy levels.

### Estimating Population Mean Areas: Sampling Increasing Numbers of Glomerular Profiles with Progressively Lower Numbers of Eligible Biopsies

In practice, many biopsies do not have plentiful glomerular profiles. The challenge is to understand whether there needs to be a minimum number of glomerular profiles specified for inclusion of a biopsy in an analyzed series and what this number might be. We used the entire biopsy series to inform this question.

The first approach was to evaluate the true population mean values as the minimum requisite number of glomerular profiles for inclusion of any biopsy was changed (Table 3). The numbers of biopsies that were included in every estimate can be cross-referenced in Figure 2. Figure 5 shows that none of the population mean values differed significantly from one another (the CI always overlapped). However, inclusion of biopsies with minimum numbers of glomerular profiles from one or more to four or more gave the most robust estimates with the least variance (reflected by the narrowest CI). The variance increased as the minimum number of glomeruli profiles for inclusion increased, as a result of lower numbers of participating biopsies, and was especially conspicuous when a minimum of more than seven or eight profiles in each participating biopsy was stipulated.

Figure 5 also shows that population means of glomerular profile areas without sclerosis tended to be higher than the population mean areas of profile areas without exclusions and that the former remained stable as more glomerular profiles were included, whereas the latter tended to fall. These differences presumably reflect inclusion of profiles of small scarred glomeruli in the latter and their greater chance of being sampled as more profiles were evaluated.

Figure 6 shows, in these same groupings of biopsies, that the random sample mean reliably estimated the true population mean when biopsies with five or more glomerular profiles were included, without regard to sclerosis, and when those with four or more glomeruli without sclerosis were included. As increasing minimum numbers of glomerular profiles per biopsy were stipulated, the mean values remained similar to the true populations mean, but there was increasing variation about the mean. This was especially apparent with requirements of sampling of more than eight profiles without exclusion and for biopsies with more than six or seven nonsclerosed glomerular profiles.

### Analysis of Subgroups of Biopsies

The limiting effect of specifying larger minimum numbers of glomeruli for eligibility of biopsies for inclusion is exacerbated when biopsies are divided into groups, for example by diagnostic category or race. Table 4 shows the major diagnostic categories in the biopsy series already cited. Requirement for robust numbers of glomeruli more likely would jeopardize the feasibility of a study on glomerular size in hypertension or membranoproliferative glomerulonephritis than a study on the first three categories of disease. If the outcome of interest is mean area of glomeruli without sclerosis, then the number of qualifying biopsies is limited further, especially in disease states with extensive glomerulosclerosis.

Figure 7 shows the average area of nonsclerosed glomeruli by disease categories, as the minimum required number of nonsclerosed glomeruli was increased from two to eight. They are arranged in order of glomerular size, and membranoproliferative glomerulonephritis is excluded. The destabilizing effects on the mean estimates imposed by smaller numbers of participating biopsies is especially evident with mesangial proliferative glomerulonephritis and with lupus nephritis class V (SLE V).

Table 5 shows how the ability to detect a difference of significance between two categories of disease can be lost as the minimum glomeruli required for inclusion of a biopsy in a series is increased. The size difference in nonsclerosed glomeruli between lupus nephritis other than class V (SLE O) and SLE V, suggested in Figure 6, is clearly significant when biopsies with one or more, two or more, or four or more nonsclerosed glomerular profiles are included but becomes marginal when six or more glomeruli are required, and the significance is lost among biopsies with eight or more nonsclerosed glomeruli.

## Discussion

In an individual biopsy, good estimates of mean glomerular profile area were obtained by measuring as few as eight nonsclerosed glomerular profiles. In an analysis in which the number of biopsies was held constant, sampling of as few as five glomerular profiles in each biopsy gave a good representation of the population mean profile area. Inclusion of glomeruli with sclerosis introduced more variation and increased the requisite number of profiles. In a series in which the numbers of participating biopsies decreased as the minimum number of glomerular profiles required for inclusion rose, the best estimates of the population mean profile areas were supplied by including as many biopsies as possible; there were no grounds for exclusion of any biopsy on the basis of minimum numbers of profiles.

One of the arguments for recommending evaluation of a generous minimum number of glomerular profiles in every biopsy in any particular series is based on the purported potential for the biopsy mean profile area to fall as increasing numbers of profiles are sampled, as a result of minimization of the effect of preferential sampling of larger glomeruli. It could be argued, however, that larger glomeruli will always be overrepresented in systematically sampled series of sections. In any case, the phenomenon was not apparent in our analyses when glomeruli with sclerosis were excluded from consideration.

In a single biopsy, the more glomeruli that are available for measurement, the more valid the final estimate of glomerular profile area should be. In a group of biopsies with plentiful glomeruli, the same should apply, although our data showed minimal benefit in sampling more than six or seven profiles. However, the situation is different with grouped data, for which a compromise is needed between the desired stability of the glomerular size estimates and the diminishing number of eligible biopsies, as the minimum number of glomerular profiles required for inclusion is increased. The potentially weakened accuracy of the profile area estimates if biopsies with relatively few glomeruli are included in any series is more than balanced by the increase in power contributed by inclusion of the larger numbers of “eligible” biopsies. In this analysis, inclusion of biopsies with four or more nonsclerosed glomeruli permitted inspection of 2.5 times as many biopsies than did inclusion of biopsies with a minimum of eight nonsclerosed glomeruli, but gave a similar mean and narrower CI. When no biopsies were excluded, four times as many biopsies could be evaluated, and the mean area estimates were equally or even more robust.

In practice, biopsies contain variable and limited numbers of glomeruli, and all qualifying glomerular profiles of biopsies with “sufficient” profiles are measured to achieve an individual biopsy mean. If the minimum specified number of profiles is too high, then the sample size to analyze any phenomenon or differences by groupings might well be too small. This study shows that it is not reasonable to limit ability to discriminate a difference by restricting, in advance, the number of biopsies that are available for inclusion by specifying a larger requisite minimal number of glomerular profiles. Rather, the associations of glomerular tuft volume with features of interest can be explored within any biopsy series. If variation operates randomly, then the inclusion of data from biopsies with lower numbers of glomeruli should not bias the findings in any particular direction, and enrichment of numbers of contributing biopsies adds power. If the conclusions and associations are statistically significant and biologically plausible, then they are likely to be correct. This approach is no less valid than exploring a body of epidemiologic data for associations of interest, even when those data were not originally collected for that specific purpose. In this instance, it allows large bodies of archived material to be put to great use in exploring matters of great interest.

The concepts outlined here should be generalizable across biopsy series, although the specific quantitative findings are likely to vary somewhat, according to the number of available biopsies, biopsy practices (numbers of glomeruli included in biopsies), subgroupings of interest, the degrees of glomerulosclerosis, and other factors.

## Footnotes

Published online ahead of print. Publication date available at www.jasn.org.

- © 2006 American Society of Nephrology