## Abstract

To understand the dynamics of islet population, especially during conditions with growth of the total islet mass, it is important to have reliable estimators of parameters describing the quantitative appearance of the islet population. We describe a stereological estimator of the volume-weighted mean islet volume based on unbiased assumption-free stereological principles. The volume-weighted mean islet volume is the mean islet volume if the islets are weighted (sampled) proportional to their volume. This method allows simultaneously unbiased estimation of the total islet mass. With use of this method, 22 male Sprague-Dawley rats within the age span of 34–102 days old were investigated. We found a linear correlation (*P* < 0.001) between total islet mass and the volume-weighted mean islet volume. The results support models demonstrating that the physiological growth of the total islet mass in the period studied is totally or mainly caused by proportional growth of existing islets. The functional meaning of the volume-weighted mean islet volume is discussed, and previous methods to study the mean islet volume and islet number are critically evaluated. We propose the volume-weighted mean islet volume to be a biologically useful parameter when describing the mean volume of the pancreatic islets and investigating the differences between experimental groups.

The total mass of pancreatic β-cells is a critical factor in the regulation of glucose homeostasis. The total β-cell mass consists of a dynamic cell population that either expands or declines to adapt to altered physiological conditions. Previous studies of the β-cell mass in the growing pancreas have shown a significant increase in β-cell mass with age (1,2,3,4). The total β-cell mass and the body weight in rats are linearly correlated after the first month of life, thus indicating an adaptive capacity of β-cells (4). Although changes in β-cell mass have been studied for years, questions still remain about the variation in islet number and volume in different physiological and experimental situations. In theory, an increase in total β-cell mass can be due to both an increase in total number of pancreatic islets and an increase in the size of preexisting islets. The optimal approach to describe the relationship between these two parameters in different situations would be to determine the absolute distribution of islets with respect to both number and size. Such a method has not yet been developed. Due to the extreme variation in islet volume, establishing an efficient method to describe the islet-volume distribution is a true stereological challenge. Alternatively, investigators have evaluated islet morphology by means of the two-dimensional (2-D) mean profile area of islets and referred to this as a parameter of “mean islet size” (5,6,7,8). In this study, we challenge such an approach by introducing the volume-weighted mean islet volume and describe how it is estimated in a simple design based on unbiased principles. By definition, the total islet volume is the sum of volumes of all islets. The arithmetric mean islet volume is the total islet volume divided by the total number of islets. The volume-weighted mean islet volume is the mean islet volume if islets are weighted proportional to their volume. The volume-weighted mean volume can be estimated without assumptions about shape of the islets and provides unbiased information of three-dimensional (3-D) volume, in contrast to the commonly used 2-D estimates of mean islet profile area. To illustrate the physiological value of the volume-weighted mean islet volume, we use the growing rat pancreas as an example.

## THEORY

### Arithmetric mean (number-weighted mean) volume.

Considering a group of 3-D objects, the arithmetric (or number-weighted) mean volume (ν_{N}) can be calculated as follows:
where *n* is the number of objects and *V*_{i} is the volume of the *i*th object (i = {1, 2,., *n*}). The term *number-weighted mean volume* is used when every object has the same weight in the equation, regardless of the volume of that object.

### Example 1.

Take three objects of volumes 1, 10, and 50, respectively. The number-weighted mean volume can then be calculated from:
The number-weighted distribution implies that whenever one investigates only a sample of the total population of objects, a uniform random sampling must be performed to get an unbiased estimate of the mean volume in the entire population. This means that every object has one and the same probability to be sampled independent of size, shape, or orientation in space (9). The number-weighted mean volume, ν_{N}, is what we customarily call the mean.

### Volume-weighted mean volume.

The volume-weighted mean volume is the mean volume in the distribution if objects are weighted proportional to their volume (10). The volume-weighted mean volume, ν_{V}, can be expressed as follows:
where *n* is the number of objects and *V*_{i} is the volume of the *i*th object (i = {1, 2,.., *n*}) (10,11).

Hence, given a group of objects of different volumes, a large object in the group contributes more to the calculated mean value compared with a small object in the same group.

### Example 2.

With use of the same three objects described in example 1, the volume-weighted mean volume of these is calculated from: Furthermore, if we investigate only a sample of the total population and if objects have been sampled proportional to their volume, then the mean volume of these objects equals the volume-weighted mean volume.

Another characteristic of the volume-weighted mean volume is the relationship between ν* _{V}* and both ν

*and the variation in the number-weighted distribution (11): where*

_{N}*CV*

_{N}(ν) is the coefficient of variation (CV) of the number distribution of object volume. From Equation 3, it can be seen that if all objects are of the same volume, then ν

*= ν*

_{V}*. In all other cases, ν*

_{N}*> ν*

_{V}*.*

_{N}### Biological usefulness of ν_{V}.

_{V}

In a situation in which the volume of an object is proportional to the biological function of that object, then the volume-weighted mean volume can also be said to be a function-weighted mean volume. Assume, for instance, that a pancreatic islet of volume 10 has 10 times as much implication on the overall glucose homeostasis than an islet of volume 1. In the volume-weighted mean islet volume, this functional difference will be accounted for since the islet of volume 10 is weighted 10 times more than the islet of volume 1.

### Estimation of the ν_{V} of pancreatic islets.

_{V}

The method to obtain an unbiased estimate of ν* _{V}* is simple and relies on point-sampled intercepts of object profiles on histological sections. Its efficiency has previously been demonstrated when evaluating nuclear enlargements and tumor malignancy (12,13). The following will describe the principle applied to the pancreas in which the islets of Langerhans are the particles of interest embedded in a reference space—the exocrine tissue.

Assume pancreatic islets to be globally convex objects and place a point at a uniform random position within the islet. The estimate of the islet volume can then be calculated from:
where *l*_{0} is the length of a random intercept isotropic in space (i.e., of random 3-D orientation) passing through the point as depicted in Fig. 1 (10,11).

When a set of points is uniformly randomly positioned within a pancreas, some points fall within islets. The volume of these islets can be estimated from Equation 4. Averaging all of these estimates provides an estimate of the mean islet volume of the islets investigated. Since the probability of the randomly chosen points falling within a given islet is proportional to the volume of that islet, then the estimated mean volume will be an estimate of the volume-weighted mean islet volume:
where *n* is the number of observations and *l*_{0,i} is the length of the *i*th intercept (i = {1,… , *n*}).

For the estimate to be valid, the islets must be globally convex. If the islets are nonconvex, then a single sampling point may generate more than one intercept through the same islet, with only one of these (the length of which is denoted *l*_{0,0}) containing this point (Fig. 2). If the formula above is applied in this situation, one will only estimate the volume of the part of the islet, which can be reached with straight unbroken lines from the sampling point. The “hidden” areas with respect to the sampling point are ignored. The total islet volume must therefore be estimated in a slightly modified way in which all of the intercepts belonging to the islet are taken into account (10,11).

Consider an islet of arbitrary size hit by a sampling point. The third power of the length of the *i*th (*i* = { 0,… , *m*}) intercept not containing the sampling points can then be determined by the following formula:
where *l*_{0,i+} and *l*_{0,i–} represent the longest and shortest distance, respectively, between the sampling point and the end points of the *i*th intercept (Fig. 2).

The sum of all *l*_{0,i}^{3} belonging to a given islet is denoted *l*_{0,e}^{3}:
An unbiased estimate of the islet volume can now be obtained:
Notice that the expression holds true for both globally convex and nonconvex islets, since *l*_{0,e}^{3} will be zero in the former. By using an average of all estimated volumes obtained from the point-sampled islet profiles, we obtain an unbiased estimate of ν* _{V}*:
Consequently, if sampling points are chosen randomly and if isotropic requirements are fulfilled, then an unbiased estimator of volume-weighted mean volume can be obtained.

### Estimating volume-weighted mean volume in practice.

Islets are not randomly distributed within pancreas. By investigating pancreatic sections, one gets the clear impression that the large islets are often located close to ducts and blood vessels whereas the smaller ones seem to be more randomly embedded within the exocrine tissue. This means that a set of uniform random sections has to be used when one performs the estimation of the volume-weighted mean islet volume. Uniform random sampling (URS) of sections means that if the pancreas was exhaustively sectioned, then all sections should have one and the same probability for being sampled for further analysis. The assumption that the islets are isotropically oriented within the pancreas may not be justified. Isotropic orientation means that islets (at their specific position in pancreas) are rotated randomly in all three directions. Therefore, for such a situation to be created, the pancreas must be oriented isotropically when embedded in the block. The sampling points are chosen randomly within the sections and thus within the pancreas by superimposing a point grid uniformly randomly onto the sections. If parallel lines have been drawn through the points of the grid, then the isotropically orientated *l*_{0} can be estimated along these direction-indicating grid lines.

Measuring *l*_{0} with high precision is a time-consuming and often unnecessary procedure, since the parameter itself clearly is subjected to wide variation. Measurements of *l*_{0}^{3} are obtained much faster by using a *l*_{0}^{3}-ruler (Fig. 3), which groups the intercepts into *n* classes. Any class has a width progression by a factor 10^{1/n} on a cubic scale compared with the previous class (14). The *l*_{0}^{3}-ruler is constructed as shown in Table 1. In practice, this means that when a point hits an islet profile, one reads off the class number of the intercept that passes through the point in the randomly chosen direction (Fig. 4). The observation is checkmarked in the *z*th row of a table with *n* rows (Table 1). In advance, the (length)^{3}-median of every class has been calculated and these values are then directly transferred to the intercept (Table 1). Thus, using the ruler one can obtain an estimate of the volume of the object on a linear scale with the necessary precision in the shortest amount of time. By means of Equation 5, the volume-weighted mean islet volume can be calculated. So that the observations can be converted into true biological dimensions, the estimated value of ν_{V} must be corrected with a factor F:
where *Mag* is the final linear magnification under which the sections are studied (Table 1). It is worth noting that the widths of the first and second classes of the ruler on a linear scale are greater than the rest of the classes. The values of *l*_{0}^{3} belonging to the large islet profiles of the sample will therefore be determined with a higher precision than the smallest. This is an advantage, since the volumes of large islets, which are decisive for the estimate in this way, are determined with the greatest precision.

### Nonconvexity.

The global convexity of the islets of Langerhans is obviously not a universal fact, but, nonetheless, only in very rare occasions in rat pancreas did we find more than one intercept belonging to the same sampling point in an islet profile. The potential bias originating from the volume estimation of such islets using the method described can reasonably be assumed to have a negligible impact on the final estimate. Therefore, we estimated the islet volumes under the assumption of islets being globally convex. However, in other research situations this assumption is not justified. For example, some of the islets in GK rats (a model of type 2 diabetes) have been found to have an irregular appearance due to islet fibrosis (15), and in such a situation the formula shown in Equation 9 should be used.

## RESEARCH DESIGN AND METHODS

### Animals.

The morphological changes of pancreatic islets in the developing pancreas were studied in Sprague-Dawley rats. Six Sprague-Dawley breeder pairs were obtained from M&B (Ll. Skensved, Denmark). The animals were housed in cages in the same stable with standard rat diet and water ad libitum. At the day of birth, the litters were reduced to six siblings with as many males as possible. The day of birth was taken as day 0 of age, and weaning was performed at day 20. A total of 22 male rats that were 34–102 days old were killed, and these animals were included. An additional eight male rats were used to determine the shrinkage ratio. The animals were killed by CO_{2} breathing and cervical dislocation.

### Immunohistochemical staining.

The pancreases were removed, weighed, and fixed in acidic formalin as described previously (16). The pancreases were dehydrated and embedded in paraffin at random orientation. Eight to ten sections of ∼5 μm were sampled from each pancreas by systematic uniform random sampling (SURS) (9). Sampling sections according to SURS means that every *N* section is sampled when the pancreas is exhaustively sectioned, and the first section to be sampled is randomly chosen between the first *N* sections from the block.

To identify islets profiles, the sections were immunostained for insulin using guinea pig anti–swine insulin (Dako, Denmark), followed by horseradish peroxidase–conjugated rabbit anti–guinea pig (Dako). Diaminobenzidine was used as chromogen. Finally, counterstaining was performed with both Mayer’s hematoxylin and eosin (H-E). The commonly used counterstaining with hematoxylin only can make it difficult to clearly identify the border of islets. The immunostaining enabled us to define even those islets located adjacent to or within the connective tissue surrounding ducts, as these can be difficult to identify using H-E staining alone.

### Estimating ν_{V}.

_{V}

In our study, every section was investigated using an Olympus BH-2 light microscope (final magnification 151) with a projecting arm to project the image onto the table. A grid with 396 points and a set of parallel lines of random orientation were superimposed randomly onto the image (Fig. 5). The distance between the image border and the marginal points was larger than the largest islet profile. The ruler used was a 15-class ruler with a total length of 35 mm. Data were entered in a result sheet and ν* _{V}* was estimated (Table 1). The sections were investigated systematically with an adjusted step-length to give a total of ∼300 points hitting islet profiles per pancreas.

### Correction of estimated values of ν_{V}.

_{V}

Processing fresh pancreases to histological sections involves steps (primarily dehydration) that cause tissue shrinkage. Assuming that the islets shrink by the same factor as the rest of the pancreas, we corrected the volumes obtained from measurements on the histological sections for the tissue shrinkage. The correction factor was determined by measuring the total pancreatic volume according to the principle of Scherle (17) before fixation and after dehydration. The correction factor (i.e., shrinkage ratio) could then be estimated:

where ∑*V _{i,0}* is the summed pancreatic volume before fixation and ∑

*V*is the summed volume after dehydration. A total of eight pancreases were used, four of which were obtained from day-21 rats and four from day-73 rats.

_{i,d}### Estimating total islet mass.

The estimation of total islet mass, *V*_{tot}, was performed at the same time as the estimation of volume-weighted mean islet volume. This was done by volume-fraction estimation using the principle of Delesse (18). Estimating the total islet or β-cell mass according to the Delesse principle has been performed by many previous studies using either point counting (4) or linear scanning (2). The total number of points hitting the islet profiles per pancreas equals the total number of recorded islet intercepts. The point counting of the reference space was done by choosing one point of the grid and only recording it if this point hit the reference space. If a point grid containing—e.g., 396 points—is used, then the total islet mass can be calculated from the following formula:

with islet points and reference points being the number of sampling points hitting the islet profiles and the section profiles, respectively, of a given pancreas.

### Statistics.

Because the aim of the study was to illustrate the functional character of the volume-weighted mean islet volume, the paired data of total islet mass and the volume-weighted mean islet volume in the rats were investigated in a linear regression model based on the least-squares fit on log_{10}-transformed data. Correlation analysis was based on Pearson’s *r*.

## RESULTS

There was a positive correlation between age and the total islet mass (*r* = 0.76, *P* < 0.00005) and between age and the volume-weighted mean islet volume (*r* = 0.64, *P* < 0.005) in the period investigated (Fig. 6). The figures show that, of the 22 rats investigated, 6 were 34 days of age and 6 were 69 days of age, allowing a reasonable estimation of the total variance (the sum of the biological variation and the estimator-induced variation) of the parameters investigated. At day 34, the CV, the standard deviation divided by the mean, was 24% between the estimates of total islet mass and 33% between the estimates of the volume-weighted mean islet volume. At day 69, the CV was 9% for the total islet mass and 39% for the volume-weighted mean islet volume. In rats, an increase with age of both total islet mass and volume-weighted mean islet volume was seen. The paired log_{10}-transformed data for the volume-weighted mean islet volume and total islet mass were investigated in a linear regression model (Fig. 7). The equation for the regression line shown is:
where *b* = 0.76 (0.13; [0.51–1.0]; *P* < 0.001 vs. *H*_{0}: *b* = 0) and *a* = 6.2 (0.10; [6.0–6.4]); (SE; 95% CI). All estimated data of ν* _{V}* were corrected with a factor equal to the shrinkage ratio caused by fixation and dehydration. This factor was determined to be 1.4.

## DISCUSSION

In this article, we describe the volume-weighted mean islet volume. This parameter is not equal to the arithmetric (number-weighted) mean islet volume, since the volume-weighted mean islet volume is the mean volume in the distribution if islets are weighted (sampled) proportional to their volume. If it is true that the impact of an islet on the overall glucose homeostasis is proportional to the volume of the islet, then sampling of islets proportional to volume can be seen as a sampling proportional to function, making the volume-weighted mean islet volume a function-weighted mean volume. The volume-weighted mean islet volume can be estimated by an estimator based on unbiased principles in a simple design without need for computer-assisted image-analysis systems, even though commercial stereological software packages include the possibility for estimating the volume-weighted mean islet volume by the point-sampled intercepts method. Efficient estimation of the total islet mass, volume-weighted mean islet volume, and the total β-cell mass from a rat pancreas can be performed in less than 1.5 h with use of a microscope.

When describing and evaluating differences in the islet volumes under physiological and pathological conditions, the most optimal raw data to obtain would be data on the true volume distribution of the pancreatic islets. Such data can be obtained using design-based stereological principles, but since islets in such cases must be sampled by URS, the method would require a disector-based design, because this is the only known way to sample 3-D objects in a uniformly random fashion when investigating 2-D sections. The sections used for sampling islets according to URS by the disector principle must have a maximum thickness of ∼7–8 μm due to the size of the smaller islets. In addition, the set of sections must proceed at least 600 μm (corresponding to ∼85 sections) further into the block to be able to obtain unbiased volume estimates of the biggest islets sampled in the disector. It seems realistic to assume that at least six to eight of such sets of sections have to be investigated to obtain reasonable accuracy of the volume distribution of islets within one pancreas. Thus, to describe the volume distribution by the available stereological techniques would be a considerably labor-intensive task.

Alternatively, another approach to describe the between-group variation in islet volume would be to estimate the arithmetric mean volume of the islets. The estimation can be performed by estimating the total number of islets using the disector principle combined with an estimation of the total islet volume using—for example—the Delesse principle. It would be considerably more labor-intensive than the estimation of the volume-weighted mean islet volume but nonetheless much less labor-intensive than the estimation of the islet-volume distribution. However, the question is whether the arithmetric mean volume itself is of primary interest. The population of islets of Langerhans is very heterogeneous, with a large number of very small islets and a small number of very large islets (19,20). Although it is the medium-sized and large islets that mainly contribute to the total islet mass (21) and thus the glycemic control, it is the small islets that have the major effect on the level of the arithmetric mean volume and presumably only a small influence on the glycemic control. Thus, in a functional perspective, it could be argued that the arithmetric mean is not very relevant because of the impact on this parameter of the many small and functionally less important islets. This favors the view of the mean islet volume in the volume-weighted distribution as a functionally relevant parameter to describe the islet population.

In previous articles giving data on “mean islet size,” an often used method has been to count the number of islet profiles in a histological section and then divide this number into the total cross-sectional area occupied by islets in the section (5,6,7,8). In the following description this parameter is denoted *a*_{isl}. The geometrical meaning of *a*_{isl} can be described using the formula of Abercrombie (22) with the assumption that the sections used are sampled by URS:
where *P* is the number of “center points” (i.e., any predefined geometrical point of the same relative position in all islets) in a section, *A* is the number of islets seen in the section, *L* is the average length of islets measured perpendicular to the section, and *M* is the section thickness.

If *N*_{V} is the number density of islets in the pancreas and *N*_{A} is the profile density of islets in the section, then:
and:
where *V*_{tissue} is the volume of the tissue in the section, *A*_{tissue} is the cross-sectional area of the tissue section, and *t* (=*M*) is the thickness of the section.

Combining Equations 14, 15, and 16 gives:
where *h* (=*L*) is the mean height of the islets measured perpendicular to the cutting surface. Because:
Eq. 18 can be rewritten as a stereological expression of *a*_{isl}:
From Equation 19 it is seen that *a*_{isl} is influenced by the mean islet volume, the mean height of the islets, and the section thickness. Even if the contribution from the latter is ignored, an increase in *a*_{isl} is not direct or indirect evidence of an increased mean islet volume, since the ratio between the arithmetric mean islet volume and the mean islet height is influenced by both the shape of the islets as well as the volume distribution of islets. Thus, *a*_{isl} is not an unbiased estimator of “mean volume,” nor are we able to conclude whether the mean islet volume is different in two experimental situations by investigating *a*_{isl}.

The number of islet profiles per section area, *N*_{A}, has also been used as a parameter to compare the number of islets in pancreas. Equation 17 can be rewritten as:
where *n*_{islets} is the total number of islets in pancreas and *V*_{pancreas} is the total volume of the pancreas. Thus, *N*_{A} is not a direct or an indirect estimator of the total islet number, because *N*_{A} is influenced by the true number of islets, the mean height of the islets, the section thickness, and the total volume of the pancreas.

In our study, we found a linear correlation between total islet mass and volume-weighted mean islet volume in the developing rat pancreas under physiological conditions. Knowledge of the volume-weighted mean islet volume (or the arithmetric mean islet volume) per se does not allow distinct conclusions of the real volume distribution of islets. Nonetheless, the finding of a linear relation gives some information on the way the islet population changes when the total islet volume is expanding during physiological growth. As examples, consider two models (A and B) of the change in the volume distribution of islets during physiological growth (Fig. 8). In model A, an increase in the total islet volume is reached by the formation of new islets while keeping the relative islet volume distribution unchanged, whereas in model B an increase in the total islet volume is accomplished by all islets expanding their volume by the same factor without adding new islets. Model A can be rejected from our data, since it would lead to a constant value for the volume-weighted mean islet volume. Model B would imply that the slope (*b*) in Equation 13 equals 1, and because the 95% CI for *a* included the value 1, this model interestingly cannot be rejected. Models A and B are extreme models in the spectrum of possible models for physiological islet growth. In fact, model A is not defined on a continuous scale, since all islets at some point in time will originate from a small volume (one cell or one cluster of cells, depending on the definition of an islet). On the other hand, model B cannot describe the entire life span for an islet population, because obviously at some time new islets must have been generated. Hellman (20,21) previously investigated the islet population at different ages in Wistar rats using the methods described by Wicksell (23,24). Even though these methods generally have been abandoned in stereology due to a number of weaknesses when applied to biological organs (25,26), the distribution curves for islet volumes given by Hellman (20,21) probably hold some truth. The results, however, should be interpreted with some caution. For example, the values for the total volume of the islet tissue calculated by these methods differ approximately three- to fourfold from the values later obtained in the investigation of rats of similar strain and age by other groups using volume-fraction–based stereological techniques (15,27). Nevertheless, the data given by Hellman are not in opposition to the theory that physiological growth of the total islet volume is chiefly caused by increasing volumes of preexisting islets. Changes in islet volume can be reached in different ways. The number of endocrine cells might change due to apoptosis or replication of islet cells (28), and the mean volume of the islet cells are also prone to alterations depending on factors such as blood glucose concentration (29). It should be noted that the method described here does not provide any information on how changes in islet volumes are reached.

In conclusion, we describe how to estimate the volume-weighted mean islet volume, which we propose to be an important parameter when describing the islet population. Our data strongly suggest that during physiological growth in young rats, the increase in total islet mass chiefly originates from increasing the volumes of existing islets proportional to their volume.

## Acknowledgments

This study was supported by the University Hospital of Copenhagen and the Danish Research Council.

We thank Pernille Albrechtsen for technical assistance.

## Footnotes

Address correspondence and reprint requests to Troels Bock, MD, PhD, Bartholin Instituttet, University Hospital of Copenhagen, H.S. Kommunehospitalet, DK-1399 Copenhagen K, Denmark. E-mail: tbock{at}post12.tele.dk.

Received for publication 28 November 2000 and accepted in revised form 26 April 2001.

2-D, two-dimensional; 3-D, three-dimensional; CV, coefficient of variation; H-E, hematoxylin and eosin; SURS, systematic uniform random sampling; URS, uniform random sampling.