## Abstract

Glomerular volume is an important metric reflecting glomerular filtration surface area within the kidney. Glomerular hypertrophy, or increased glomerular volume, may be an important marker for renal stress. Current stereological techniques report the average glomerular volume (AV_{glom}) within the kidney. These techniques cannot assess the spatial or regional heterogeneity common in developing renal pathology. Here, we report a novel “unfolding” technique to measure the actual distribution of individual glomerular volumes in a kidney from the two-dimensional glomerulus profiles observed by optical microscopy. The unfolding technique was first developed and tested for accuracy with simulations and then applied to measure the number of glomeruli (*N*_{glom}), AV_{glom}, and intrarenal distribution of individual glomerular volume (IV_{glom}) in the oligosyndactyl (Os/^{+}) mouse model compared with wild-type (WT) controls. The Os/^{+} mice had fewer and larger glomeruli than WT mice: *N*_{glom} was 12,126 ± 1,658 (glomeruli/kidney) in the WT mice and 5,516 ± 899 in the Os/^{+} mice; AV_{glom} was 2.01 ± 0.28 × 10^{−4} mm^{3} for the WT mice and 3.47 ± 0.35 × 10^{−4} mm^{3} for the Os/^{+} mice. Comparing the glomerular volume distributions in Os/^{+} and WT kidneys, we observed that the Os/^{+} distribution peaked at a higher value of IV_{glom} than the WT distribution peak, and glomeruli with a radius greater than 55 μm were more prevalent in the Os/^{+} mice (3.4 ± 1.6% of total glomeruli vs. 0.6 ± 1.2% in WT). Finally, the largest profiles were more commonly found in the juxtamedullary region. Unfolding is a novel stereological technique that provides a new quantitative view of glomerular volume distribution in the individual kidney.

- kidney stereology
- glomerular volume distribution
- glomerular hypertrophy
- chronic kidney disease
- unfolding algorithm
- nephron number

the kidney has a significant ability to adapt to acute and chronic stress. However, unremitting stimulation of the renal adaptive mechanisms eventually contributes to the progression of disease. Renal adaptive mechanisms are highlighted clinically by the kidney's ability to maintain a stable glomerular filtration rate, with a wide range in the number of functioning nephrons (11). Because human nephron number is determined at birth and decreases with age (14, 19), glomerular hypertrophy is one response the kidney employs to adapt and provide a constant filtration surface area when there are too few nephrons to maintain the homeostatic function of the kidney.

Glomerular hypertrophy is manifested in a number of diseases, such as diabetes, obesity, pregnancy, autosomal dominant polycystic kidney disease, and focal segmental glomerulosclerosis, and following acute kidney injury (13). Although many types of glomerular hyperfiltration lead to glomerular hypertrophy and glomerulosclerosis, the specific mechanisms depend on the pathology (18), and the damage is distributed heterogeneously throughout the kidney (15). Importantly, disease progression is often observed heterogeneously in individual kidneys, particularly at the earliest stages, and focal changes may not present in estimates of average glomerular volume. Early detection of glomerular hypertrophy in patients or preclinical models may enable targeted therapy earlier in disease progression, improving the chances of halting the progression of chronic kidney disease. Thus, an efficient technique to examine the distribution of glomerular volumes is critical.

Stereological techniques are used to estimate the number (*N*_{glom}) and average volume of glomeruli (AV_{glom}) and other structures. Stereology is typically performed by thinly sectioning the kidney and measuring the number and area of glomeruli visible in a fraction of those slices. A mathematical model based on certain simplifying assumptions is used to extrapolate the *N*_{glom} from the profile measurements. One model-based method, outlined by Weibel and Gomez (21) in 1962, assumes that glomeruli are spheres with radii that are normally distributed in the kidney. An alternative method, the unbiased fractioner/dissector, is generally preferred over model-based methods in studies in humans (6, 17, 20). The fractionator/dissector method dispenses with the geometric assumptions of Weibel and Gomez (21) but requires sampling pairs of tissue slices and counting profiles that appear in only one section of the pair. Another new strategy for measuring glomerular endowment relies on injecting a cationic magnetic resonance imaging (MRI) contrast agent that binds to the anionic glomerular basement membrane (1–4, 12). The number of functioning glomeruli can be determined by counting the resulting dark artifacts in an MR image.

In this work, we develop a stereological technique to measure the intrarenal distribution of individual glomerular volume (IV_{glom}) and the standard deviation of intrarenal glomerular volumes, σ_{glom}, in addition to AV_{glom} and *N*_{glom}, based on an iterative “unfolding” algorithm. By making an empirically derived assumption about the shape of the particles (glomeruli) in the model, we calculate the distribution of profiles that would arise from randomly sectioning a particle given its size. Importantly, the largest observed profiles must come from the center of the largest particles. We thus “unfold” the histogram by iteratively counting the largest particles and then subtracting the smaller profiles from the distribution that would result from sectioning those particles. To apply and test the unfolding algorithm, we first simulated a set of glomerulus profiles using a computer and calculated glomerulus radii with the original “true” simulated radii. Then we repeated this comparison using ellipsoidal, rather than spherical, simulated particles. Next, we applied this unfolding technique to examine the distribution of IV_{glom} in oligosyndactyl (Os/^{+}) mice, a model of kidney pathology that exhibits reduced nephron number by nearly 50% and glomerular hypertrophy compared with wild-type (WT) controls (24). By examining a large population of glomerulus profiles, we elucidated differences in numerical and spatial distributions of hypertrophic glomeruli. We hypothesized that unfolding enables accurate measurements of heterogenous glomerular morphology in individual kidneys.

## METHODS

### Animal Model

All animal experiments were approved by the Institutional Animal Care and Use Committee at the University of Virginia and performed according to the National Institutes of Health (NIH) *Guide for the Care and Use of Laboratory Animals*. Five Os/^{+} and five WT mice were used for the study. One Os/^{+} mouse was later removed from the study as an outlier. The mice were bred in-house on a predominately C57Bl/6 background. The Os/^{+} mice were identified by syndactylism of the toes (1).

### Tissue Preparation

Mouse kidneys were fixed by perfusion using 0.9% sodium chloride, followed by 10% neutral buffered formalin, and stored in 2% gluteraldehyde in 0.1 M cacodylate buffer at 4°C. To achieve histology blocks suitable for sectioning, the kidneys were embedded in gelatin (300 Bloom, 15% gelatin in PBS). These were cross-linked overnight with formalin at 4°C. The kidneys were exhaustively sectioned at a nominal thickness of 75 μm using a Leica VT1000S vibrating microtome. Every eighth section, beginning with a random section, was selected for imaging (e.g., beginning with *section no. 3*, *11*, *19*, etc., where *section no. 3* was the first section chosen using a random number generator). Freshly cut floating sections were blocked with 2% BSA and 1% Triton X-100 for 1 h at room temperature with gentle shaking, stained with 1 μg/ml wheat germ agglutinin-Alexa-555 (WGA-555) conjugate in PBS with 0.1% BSA overnight at 4°C, and then washed three times for 15 min with PBS. Wheat germ agglutinin-Alexa-555 binds to sialic acids and *N*-acetylglucosamine on the surface of podocytes (9). It was used to highlight glomeruli at an imaging wavelength where there is little tissue autofluorescence. The sections were wet-mounted with ProLong Diamond (Life Technologies), covered with no. 1 glass coverslips, and sealed after 24 h.

### Confocal Imaging and Measurements

Three-dimensional images were acquired on a Zeiss LSM710 confocal microscope running Zen 2012 software. Z-stacks of tiles with 0.83 × 0.83 × 2.0-μm resolution at 10-μm intervals were stitched into composite z-stacks using built-in functions in Zen. A 561-nm laser line excited the Alexa-555 fluorophore, and Zen selected the emission filter automatically (566–697 nm). To correct for tilted or wrinkled tissue slices, a smoothing operation was developed in Matlab (MathWorks, Natick, MA). The smoothing operation selected the brightest xy region out of all z-slices and then returned a single composite image of the brightest regions acquired. Glomeruli were clearly identifiable compared with the background tissue (Fig. 3). An average of 705 glomeruli per kidney were identified manually to the edge of the capillary tuft. Segmented mask images were generated in FEI Amira 5.6.0 software (Hillsboro, OR) (this segmentation could easily be performed in many other image-processing software packages).

### Data Processing

#### General stereological model.

If a volume containing particles (such as glomeruli) is sectioned with a slice thickness much less than the size of the particles, the sections will contain a distribution of two-dimensional profiles cut from those particles. For a collection of *N* spherical particles, each with the radius *R*_{n}, in a cubic volume with side length l, the problem is to estimate the number and size of the particles from slices taken through the volume. By convention, capital *R* is used to indicate particle radius and lower case *r* is used to indicate profile radius. All variables are summarized in Table 1. Taking an infinitely thin slice at a randomly chosen location through the box, the probability that the slice will intersect the *n*^{th} particle is
(1)
where α_{n}, with values between 0 and 1, is the probability of detecting the *n*^{th} particle. If a single sphere with radius *R* is intersected by a slice at a uniformly randomly distributed location, then the probability density of observing a profile with radius *r* in the slice is (Fig. 1*A*) (22)
(2)
*Equation 2* is illustrated in Fig. 1*B*. We adopt the convention that a single slice through the volume can intersect with each particle only once. If the slice does not intersect with the *n*^{th} particle, the observed radius will be 0. These observed profiles with *r* = 0 are represented by a delta function, δ(*r*), that satisfies
The total probability distribution function is the sum of these two functions weighted by the probability of detecting (α_{n}) or not detecting (1 − α_{n}) the *n*^{th} glomerulus. Because the glomeruli are small and sparse, we assumed that the excluded volume effect was negligible. Therefore, the expected distribution of profile radii in a slice containing *N* particles each with a radius of *R*_{n} will be
(2B)
Pr(*r*) is the probability that the *n*^{th} particle will be detected as a profile of radius *r* in a random slice. Because the location of the *n*^{th} particle is unknown, the slice may not intersect the particle. This case yields an observed radius of 0 for that particle, represented by the second term inside the parentheses. If the slice intersects the *n*^{th} particle, with probability α_{n}, the profile radius depends on the particle radius, as in *Eq. 2*. However, because the particle radius is also unknown, the probability distribution function is averaged over all particles in the volume. Each section results in one measurement of the profile (which may be 0). Thus the equation sums to 1. Because the unobserved profiles with *r* = 0 cannot be measured, we must estimate the number of unobserved profiles from those that were observed (*Eq. 2B* formally describes the probability of detecting a profile of a specific radius, but the formula used in the algorithm is given in *Eq. 6*). The total number of profiles per unit area of the slice, *N*_{A}, is related to the number of particles per unit volume, *N*_{V}, by
(3)
where R_{0} is the average radius of the particles (8) when they are spheres or the mean caliper diameter when they are convex but nonspherical. The parameters of interest in the kidney can therefore be estimated by
(4)
and
Thus by estimating either AV_{glom} or *N*_{glom}, one can calculate the other parameter. This model may be modified to reflect a different assumption about the shape of the particles by replacing *Eqs. 2* and *5* with equations appropriate to that geometry. Because the optical slice thickness here was <3% of the diameter of glomeruli, we considered the slice thickness to be negligible. For a complete background, including illustrations, discussion of nonspherical geometries, and consideration of nonnegligible slice thickness, see Weibel (22).

#### Volume distribution by unfolding.

To estimate the distribution of true glomerular radii *R*_{n} from the measured profile radii *r*_{n}, we developed and applied the unfolding algorithm based on the theory described by Wicksell (23) in 1925. We first sorted the measured *r*_{n} into a normalized histogram of 18 bins evenly spaced from 0 to 60 μm. The number of bins should depend on the number of observations and the variance but will generally fall between 16 and 20 for this procedure (22). We assumed that the profiles in the largest bin of this histogram arose from sections near the center of the largest glomeruli. We also assumed that there were no particles larger than the largest observed profile. To find the probability that an observed profile will have a radius between *r*_{m} and *r*_{m} − Δ*r* we discretized *Eq. 2*:
(6)

where *r*_{m} is the upper bound of the *m*^{th} bin and Δ*r* is the bin width. Next, beginning with the bin of largest profiles, we calculated the fraction of glomeruli with the corresponding radius and the expected number of profiles in each smaller profile bin attributed to glomeruli in this size class. We removed these calculated values of *r* from the histogram and repeated the process for the bin of next largest profiles. The bins with negative frequency, after all expected profiles were subtracted, represented profiles from larger glomeruli that were not identified due to experimental challenges. Frequencies in the final histogram were corrected by a factor of *R*_{n}/*R*_{0} to account for the increased likelihood of observing larger glomeruli explained by *Eq. 1*. Finally, we calculated the average radius, *R*_{0}, from the histograms. V_{cortex} was measured directly from the three-dimensional MR images, and *N*_{glom} and AV_{glom} were calculated according to *Eqs. 4* and *5*. For reproducibility, the Matlab code has been included as supplemental material (Supplemental Material for this article is available online at the *AJP-Renal Physiology* web site).

#### Unfolding validation.

To test the unfolding algorithm, we simulated the kidney as a cube filled with 10,000 spherical glomeruli uniformly distributed inside. In the simulated samples, the radii were normally distributed (37 ± 3 μm; Fig. 2, *left*) or bimodally distributed (70% 32 ± 3 μm, 30% 45 ± 3 μm; Fig. 2, *right*), which was approximately the range of glomerular radii measured here (The mean glomerular radii of Os/^{+} and WT mice were 44 ± 1.4 and 36 ± 1.7 μm, respectively). Next, we simulated the profile radii that would be expected if seven uniformly spaced, infinitely thin sections were taken through the volume and calculated the particle radii from the distribution of these observed profile radii using the unfolding algorithm. Finally, we computed the correlation coefficient between the original distribution of particle radii and the distribution of particle radii calculated from the unfolding algorithm. We defined this correlation as “fidelity.” To assess this, we performed the same simulation 1,000 times usisng normal and bimodal distributions of particle radii. To test the sensitivity of unfolding to the standard deviation of the particle radius distribution, we performed the simulation for particles with normally distributed radii 1,000 times while varying the standard deviation from 0 to 5 μm, a conservative estimate of σ_{glom} (Fig. 2, *inset*). Because the unfolding algorithm is applied serially, errors might be systematically propagated across histogram bins. To assess the magnitude of this effect, we examined the correlation between the number of profiles in the bins containing the largest profiles and the fidelity of unfolding. To assess the effect of measurement error on the algorithm, we introduced artificial measurement error (0–25%) and repeated the simulations. We chose this range because, in our experience, typical interinvestigator variability is ∼5–10%. Finally, we ran the simulations a number of times using different values of AV_{glom}, σ_{glom}, and distributions of IV_{glom}, estimating the minimum number of profiles that must be sampled for the unfolding algorithm to have an average fidelity of 90%. The results are not shown, but the estimates are given in the discussion as a guideline for using the unfolding algorithm in other species.

To test how the assumption of spherical glomeruli affects the accuracy of unfolding and estimates of *N*_{glom} and AV_{glom}, we simulated profiles cut from a triaxial ellipsoid defined by two axial ratios and a volume. We generated triaxial ellipsoids in Matlab with equivalent volumes to the spheres used in the first set of simulations (2.12 ± 0.51 × 10^{−4} mm^{3}). The ratio of the lengths of the second and third axes, with respect to the shortest axis, was uniformly distributed between 1 and 1.63. To choose the maximum possible axial ratio (1.63), we varied that parameter from 1 to 2 and selected the value that produced an average two-dimensional profile axial ratio of 1.26 (the glomerular axial ratio that was observed in this work). Next, we found the area of the two-dimensional ellipse formed by the intersection of each three-dimensional ellipsoid and a plane passing in uniformly distributed random orientation through the ellipsoid according to the method of Klein (16). While still assuming the particles were spheres, we repeated the unfolding procedure 1,000 times and compared the estimated sphere volumes with the known ellipsoid volumes.

#### Estimation of AV_{glom} and N_{glom} by Weibel and Gomez.

The technique of Weibel and Gomez (21) assumes that glomeruli are spheres in estimating AV_{glom} and *N*_{glom}. The technique is based on *Eq. 1* and uses the constants *k* and *β* to correct for the size distribution and shape of the glomeruli, respectively, and glomerular area density per kidney unit area (A_{A}):

Previously reported values for *k* and *β* are 1.04 and 1.38, respectively (10). *N*_{A} is the number of profiles per unit area of cortex and A_{A} is the glomerular area per unit area of cortex. Here, *N*_{glom} was calculated according to *Eq. 4*. To calculate AV_{glom}, we set the volume fraction of glomeruli, V_{V} = A_{A} (delesse principle) and used the formula:

#### Measurement of AV_{glom} and N_{glom} by MRI.

We compared the AV_{glom} and *N*_{glom} obtained by unfolding with those obtained by MRI of the same kidneys. The MRI techniques are described in previous work (2, 3, 7). Briefly, the mice were given a total of 5.75 mg/kg cationized ferritin (Sigma Aldrich, St. Louis, MO) by retro-orbital injection in two doses spaced 1.5 h apart; 1.5 h after the last injection the mice were euthanized by CO_{2}. Blood was removed by transcardial perfusion with 0.9 M NaCl and stored in 2% gluteraldehyde-0.1 M cacodylate buffer at 4°C. MRI was performed on a Bruker 7T/30 MRI (Billerica, MA); TE/TR = 20/80 ms, and resolution = 50 × 50 × 55 μm.

To measure *N*_{glom} and AV_{glom}, we developed an image-processing algorithm in Matlab. Local minima in the images were identified using a watershed algorithm, and individual glomeruli were counted if their component pixels were connected within a specified radius of 26 pixels. Next, we isolated a line profile of 11 pixels in each direction (*x*-axis) of the center of the glomerulus and measured the width of the artifact at 55% of the maximum depth. This width was chosen because it produced the same AV_{glom} as the stereology of Weibel and Gomez (21).

### Location of the Largest Glomeruli

We noted in manual inspection of the image slices that larger glomerulus profiles were more frequently present in the juxtamedullary region than in the cortex. Because the sagittal tissue sections were randomly selected, the locations of glomeruli could not be compared directly between samples. Therefore, to determine whether larger glomeruli were juxtamedullary, we collected coronal sections from the center of WT (*n* = 3) and Os/^{+} (*n* = 3) kidneys (minimum of 74 profiles per kidney). Next, we calculated the distance between each glomerular profile center and the nearest edge of the kidney section. We compared the average distance for the largest 10% of profiles to the average distance of all profiles in that kidney. Results shown are the average plus or minus the intermouse standard deviation.

### Statistics

Student's *t*-test was used to compare pairs of samples between groups. For multiple comparisons, one-way-ANOVA tests were performed in Matlab using the “multcompare” function. A *P* value of <0.05 was considered significant.

## RESULTS

### Validation of the Unfolding Algorithm

To test the accuracy of the Unfolding algorithm, we used a simulated data set consisting of random populations of spherical particles (as a model for glomeruli), and the corresponding circular profiles of these particles was observed by randomly sectioning the three-dimensional volume. The intrarenal distribution of IV_{glom} was calculated from profile radii for particles of normally distributed size, as shown in Fig. 2. The correlation coefficient between the calculated and true radii was 0.96 for normally distributed radii and 0.95 for bimodally distributed radii. Increasing the standard deviation of particle radii in the simulation to 5 μm decreased the correlation coefficient slightly, to 0.95 (Fig. 2, *inset*).

To determine the importance of error caused by assuming that glomeruli are spherical, we simulated the case in which glomeruli are triaxial ellipsoids. We based the shape on our experimental findings, where the mean ratio of the major and minor axes of the measured glomerular profiles was 1.26 ± 0.03 in Os/^{+} and WT mice (with no difference between genotypes). In the model, this corresponded to a set of ellipsoids in which the length of shortest axis was a, and the other two axes were b × a and c × a, where b and c were uniformly distributed values between 1 and 1.63. When we used the unfolding algorithm to estimate the volumes of these triaxial ellipsoids from random profiles, the correlation coefficient between the histograms of the calculated and true volumes was 0.88, and AV_{glom} was underestimated by 3.6% on average. To test the fidelity of unfolding in the presence of measurement error, we repeated simulations with artificial error of 0–25% of the profile radius. Measurement error below ±15% decreased the fidelity by ∼2–3%; ±25% measurement error decreased the fidelity by ∼10%. We estimate that typical interinvestigator variability is ∼5–10%. To test the robustness of the unfolding algorithm to sampling error, we examined the simulations with an unusually large or small frequency of profiles in the bins containing the largest profiles. We found no correlation (*r*^{2} ∼10^{−4}) between the frequency of the largest profiles and the fidelity of the algorithm.

#### AV_{glom}, N_{glom}, and IV_{glom} distribution.

Glomeruli were traced manually from the confocal images. A sample image is shown in Fig. 3. Sample histograms, fits of measured glomerular profiles, and calculated glomerular volumes are shown in Fig. 4. AV_{glom} and σ_{glom} for each mouse were calculated from the distributions of IV_{glom} according to *Eq. 5* (Table 2). AV_{glom} was 2.01 ± 0.28 × 10^{−4} mm^{3} for the WT mice (*n* = 5) and 3.47 ± 0.35 × 10^{−4} mm^{3} for the Os/^{+} mice (*n* = 4). The intrasample volume standard deviation σ_{glom} was significantly larger for Os/^{+} mice (1.4 ± 0.19 × 10^{−4} mm^{3}) than for WT mice (0.85 ± 0.28 × 10^{−4} mm^{3}). *N*_{glom} was estimated according to *Eq. 2*. *N*_{glom} was 12,126 ± 1,658 (glomeruli/kidney) in the WT mice and 5,516 ± 899 in the Os/^{+} mice. In summary, the Os/^{+} mice had 54% fewer (*P* < 0.001) and 73% larger (*P* < 0.001) glomeruli than the WT mice. Os/^{+} −3 was clearly an outlier; *N*_{glom} and AV_{glom} were similar to the WT mice and were omitted from the population comparisons.

We compared the glomerulus volume distributions of the WT and Os/^{+} mice, as shown in Fig. 5. The largest Os/^{+} peak (3.0 × 10^{−4} mm^{3}) was larger than the largest WT peak (1.8 × 10^{−4} mm^{3}). The Os/^{+} mice also had a larger standard deviation of intrarenal volume; σ_{glom} was 1.80 ± 0.23 (×10^{−4} mm^{3}) in Os/^{+} mice and 1.06 ± 0.35 (×10^{−4} mm^{3}) in WT mice. Glomeruli in the largest size bin (*r* > 55μm) were observed significantly more frequently (*P* < 0.05) in the Os/^{+} mice (3.4 ± 1.6%) than in the WT mice (0.6 ± 1.2%). Thus Os/^{+} mice had fewer and larger glomeruli with a broader intrarenal standard deviation.

Finally, we examined the spatial distribution of glomeruli based on their profile sizes, as shown in Fig. 6. For both WT and Os/^{+} mice, there were apparently three populations of glomeruli. The superficial glomeruli near the cortical surface were the smallest, the juxtamedullary glomeruli were visibly larger, and the glomeruli near major vessels were the largest. To verify this across all sections, we calculated the distance to the edge of the kidney section for every glomerulus (Fig. 6*B*). In the WT kidneys, the largest 10% of profiles were 1.12 ± 0.26 mm from the edge, whereas all profiles were 0.59 ± 0.12 mm from the edge. In the Os/^{+} kidneys, the largest 10% of profiles was 0.82 ± 0.14 mm from the edge, whereas all profiles were 0.46 ± 0.14 mm from the edge. The same trend was observed in sagittal sections of the kidney but was less pronounced. We did not compare between genotypes because WT kidneys are larger in general and the glomeruli should be farther from the edge of the kidney in WT mice. We conclude that larger glomeruli were on average located deeper in the cortex.

### Comparison With Other Methods

To validate the unfolding algorithm as a predictor of *N*_{glom} and AV_{glom}, we compared our results for the WT and Os/^{+} glomerular endowment to estimates made with other stereological techniques (Table 2). There were no statistically significant differences in *N*_{glom} estimated by unfolding or MRI. Others have found that the Weibel and Gomez method overestimated AV_{glom} by 23% compared with the dissector method (5). In this work, the Weibel Gomez estimate of AV_{glom} was 20 and 22% (for WT and Os/^{+} mice) larger than that estimated by unfolding, even when both numbers were calculated using the exact same data. This is because the unfolding algorithm implicitly considers the undetected small profiles when determining AV_{glom}.

## DISCUSSION

In this work, we present a model-based stereological approach to measure intrarenal distribution of IV_{glom}. Although other stereological techniques exist for estimating *N*_{glom} and AV_{glom}, they cannot be used to estimate the distribution of IV_{glom}. We tested the ability of the unfolding technique to recover the distribution of particles from a set of profile measurements by simulating spherical profiles with a known volume distribution; the correlation between the known and calculated distribution was 96%. Next we tested the algorithm using simulated profiles cut from random triaxial ellipsoids rather than spheres. The correlation between the true and calculated volume distribution was 88%, and AV_{glom} was underestimated by 4%. When we applied the unfolding algorithm to WT and Os/^{+} mouse models, we found that the Os/^{+} had 54% fewer glomeruli and 73% larger AV_{glom} than the WT mice. This is consistent with previous observations of the Os/^{+} mouse (24). Finally, we compared the estimate of *N*_{glom} by unfolding with the estimates by MRI and Weibel and Gomez (21) and found no significant difference between the different techniques. The measurements required for unfolding are the same as those required for the dissector technique. Thus, the unfolding approach could be used as a supplement to a design-based technique, calculating *N*_{glom} and AV_{glom} from the dissector method and distribution of IV_{glom} from unfolding. If the geometry of the glomeruli is known, unfolding may be used as a stand-alone approach.

We extended our computer simulations to estimate the number of profiles per kidney that must be sampled for the unfolding algorithm to be accurate (defined by at least a 90% correlation with the true distribution). In general, the number of samples depends on the expected distribution of glomerular (particle) radii. Regardless of the number of profiles needed for accurate Unfolding, uniform random sampling with good spatial coverage is required for an accurate estimate of *N*_{glom}. The following guidelines are based on calculations in our laboratory (data not shown); for normally distributed glomerular radii with *R* = 100 ± 5 μm, 300 profiles are sufficient. If the standard deviation is increased to 20 μm, then 700 profiles are required. If a second population of larger glomeruli is present, e.g., a bimodal distribution where 80% of radii are 100 ± 5 μm and 20% are 125 ± 5 μm, 500 profiles are necessary. If the second population is smaller than the median, e.g., a bimodal distribution where 80% of radii are 100 ± 5 μm and 20% are 75 ± 5 μm, only 400 profiles are necessary. This is because larger glomeruli are more likely to be sampled, so a second population of large glomeruli biases the distribution more than a second population of small ones. For a trimodal distribution, e.g., 75 ± 5, 100 ± 5, and 125 ± 5 μm radii in a 10:80:10 ratio, ∼500 profiles are needed. For reference, ∼75 profiles were observed in a single axial section of a mouse kidney. Regardless of the required total number of profiles, uniform random sampling procedures must be observed. These estimates are meant to serve as a guideline rather than a precise number, and preliminary data should be used to gauge the expected distribution.

Several sources of error and bias arise in measuring glomeruli from thin sections of kidneys. These include assumptions about size variance, shape, finite section thickness, ambiguity in identifying glomerular profiles, tissue preparation artifacts, and random sampling error. Bias due to variance in glomerular size arises from the fact that larger glomeruli are more likely to be sampled. This bias scales with (σ_{glom}/*R*_{0})^{2}, which is up to 4% for mouse glomeruli. This source of bias is easily corrected if σ_{glom} is determined. Bias from finite section thickness is due to the projection of a curved three-dimensional profile onto a two-dimensional imaging plane and scales with (1 + 3t/2D)^{−1}; this was an ∼4% overestimation of radius in this work (section thickness *t* = 2 μm). The nonspherical shape of glomeruli affects the estimations in two ways; it changes the probability of detecting a glomerulus (*Eq. 1*) and causes underestimation of glomerular volume. If the profiles are considered ellipses rather than circles, then the term *R*_{n} in *Eq. 1* must be converted to the mean caliper radius rather than the radius of a circle of equivalent area (measured here). The ratio between these is given by
where q is the axial ratio of the ellipse and results in ∼1.3% underestimation of *N*_{glom} for the axial ratio of 1.26 measured for glomeruli in this work (22). The effect of assuming shape on the unfolding algorithm is more complex. Here, we underestimated the volume of nonspherical simulated data by 3.6%. However, more detailed studies of glomerular shape and shape distribution will improve the application of Unfolding. Ambiguity due to sampling error is particularly complex, because it depends on stain quality, imaging quality, tissue preparation, and researcher skill. In this work we identified almost no profiles with radius <20 μm^{2}, which likely constitute up to 20% of possible profiles from the glomeruli. This is a systematic error in all stereological techniques. One can “correct” the observed number of profiles to include this estimate of lost small profiles. However, because these small profiles are only assumed to exist, we chose not to make this correction here. Tissue shrinkage, another source of error, depends highly on the method of embedding and sectioning. Estimates range from 5 to 15% shrinkage for plastic embedding to 50% for paraffin embedding. Here, we used a vibrating microtome to section fixed tissue (6). This alleviated the need to infiltrate the tissue with a hypertonic solvent (such as benzyl alcohol/paraffin, which causes tissue shrinkage) but required a confocal microscope be used to image the thick sections. Finally, based on the comparison between the MRI and stereology estimates of *N*_{glom}, we estimate ± 5–10% sampling error when the number of observations is ∼700. We conclude that systematic error and bias arising from tissue preparation, profile identification, and sampling may be much greater than the bias from size variance and assumptions about glomerulus shape.

All stereological techniques are biased from tissue preparation artifacts, small pieces of glomeruli that are lost during sectioning and finite section thickness. The dissector technique makes no assumptions about the size or shape of the glomeruli; however, it can be subject to the tissue-processing artifacts and assumes unambiguously identified profiles. The Weibel and Gomez (21) technique assumes the size variance and shape of glomeruli but can be performed extremely quickly. Neither the dissector nor the Weibel and Gomez (21) techniques alone can be used to measure the distribution of IV_{glom}. Unfolding assumes that glomeruli are spherical and is the only microscopy technique to determine the distribution of IV_{glom}. In this work, we used very thick (75 μm) physical sections and confocal microscopy to obtain very thin optical sections (2 μm), but standard light microscopy with hematoxylin and eosin staining would also be sufficient if thin (<5 μm) sections were used. Although the unfolding technique introduces some bias, this bias is small compared with the error associated with tissue preparation, profile identification, and sampling error. Furthermore, because large numbers of profiles are quickly identified and measured, the bias can easily be quantified and corrected.

In summary, unfolding is extremely efficient, requires minimal equipment, produces results comparable with other stereological methods, and can be used to measure σ_{glom} and the intrarenal distribution of IV_{glom} (in addition to AV_{glom} and *N*_{glom}). In principle, unfolding could be applied to measure intraorgan distributions of any number of structures. In the kidney, this could include vessel, podocyte, and tubule diameters. Unfolding is thus a flexible technique that provides a new, quantitative view of morphology in the individual organ.

## CONCLUSION

This work describes a novel stereological technique to measure intrarenal glomerular volume, which is applied to detect glomerular hypertrophy in the Os/^{+} mouse model of low nephron endowment. The technique is based on an unfolding algorithm applied to recover the original distribution of glomerular radii from a sample of profile measurements. In simulated data, the correlation between true and calculated particle radii was 96% accurate for Gaussian distributions and 95% accurate for bimodal distributions of glomerular radii. When we made an alternate assumption, that particles are triaxial ellipsoids, the algorithm was 88% accurate and underestimated AV_{glom} by 3.6%. Comparing the volume distributions in Os/^{+} and WT kidneys, we observed that the largest peak in the Os/^{+} mice was shifted from the WT peak and that glomeruli with a radius greater than 55 μm were more prevalent. The Os/^{+} mice had significantly fewer and larger glomeruli than the WT mice. Comparing several different means of estimation (unfolding, Weibel and Gomez, or MRI), we found no significant difference in estimates of *N*_{glom}. We conclude that the unfolding algorithm is an efficient technique to measure the intrarenal glomerular volume distribution. This work may enable detailed studies of common kidney diseases that heterogeneously affect glomerular and nephron morphology.

## GRANTS

This work was funded by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-091722 (K. M. Bennett) and The Hartwell Foundation (J. R. Charlton).

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## AUTHOR CONTRIBUTIONS

B.D.H., E.J.B., J.R.C., and K.M.B. conception and design of research; B.D.H., E.J.B., and J.R.C. performed experiments; B.D.H. and E.J.B. analyzed data; B.D.H., J.R.C., and K.M.B. interpreted results of experiments; B.D.H. prepared figures; B.D.H. drafted manuscript; B.D.H., J.R.C., and K.M.B. edited and revised manuscript; B.D.H., E.J.B., J.R.C., and K.M.B. approved final version of manuscript.

## ACKNOWLEDGMENTS

We gratefully acknowledge A. Eggers and the laboratory of R. Gates at the Hawaii Institute for Marine Biology, T. Carvalho and M. Dunlap at the Biological Electron Microscope Facility, and M. Bellinger at the Histology and Imaging Facility at the John A. Burns School of Medicine of the University of Hawaii for expertise in confocal imaging. We also thank Q. Liu and G. Turner at the Barrow Neurogical Institute-Arizona State University Center for Pre-Clinical Imaging. Finally, we gratefully acknowledge D. Takagi at the University of Hawaii Department of Mathematics for reviewing the mathematics.

- Copyright © 2016 the American Physiological Society