Linear Correlation Between the Total Islet Mass and the
Volume-Weighted Mean Islet Volume
Maren Skau,1 Bente Pakkenberg,2 Karsten Buschard,1 and Troels Bock1,2
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. Diabetes 50:
1763–1770, 2001
he total mass of pancreatic ␤-cells is a critical
factor in the regulation of glucose homeostasis.
The total ␤-cell mass consists of a dynamic cell
population that either expands or declines to
adapt to altered physiological conditions. Previous studies
of the ␤-cell mass in the growing pancreas have shown a
significant increase in ␤-cell mass with age (1– 4). The total
␤-cell mass and the body weight in rats are linearly
correlated after the first month of life, thus indicating an
adaptive capacity of ␤-cells (4). Although changes in ␤-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
T
From the 1Diabetes Research Laboratory, Bartholin Instituttet, H.S. Kommunehospitalet, Copenhagen, Denmark; and 2Neurological Research Laboratory, Bartholin Instituttet, H.S. Kommunehospitalet, Copenhagen, Denmark.
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@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.
DIABETES, VOL. 50, AUGUST 2001
increase in total ␤-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– 8). In this study, we
challenge such an approach by introducing the volumeweighted 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 (␯N) can be calculated as
follows:
冘V
n
i
␯¯ N ⫽
i⫽1
n
(1)
where n is the number of objects and Vi is the volume of
the ith 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:
1
␯¯ N ⫽ ⫻ 共1 ⫹ 10 ⫹ 50兲 ⫽ 20.3
3
1763
VOLUME-WEIGHTED MEAN ISLET VOLUME
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, ␯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, ␯V, can be expressed
as follows:
冘
n
␯¯ V ⫽
i⫽1
冢
Vi ⫻
Vi
冘
n
i⫽1
Vi
冣
⫽
␯ N2
␯N
(2)
where n is the number of objects and Vi is the volume of
the ith 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:
␯V ⫽
12 ⫹ 102 ⫹ 502
⫽ 42.6
1 ⫹ 10 ⫹ 50
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 ␯V and both ␯N and the
variation in the number-weighted distribution (11):
␯V ⫽ ␯N ⫻ 共CVN共␯兲2 ⫹ 1兲
(3)
where CVN(␯) 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
␯V ⫽ ␯N. In all other cases, ␯V ⬎ ␯N.
Biological usefulness of ␯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 ␯V of pancreatic islets. The method
to obtain an unbiased estimate of ␯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.
1764
FIG. 1. Particle hit by a point (X) with an associated isotropic line in
space. The length of the intercept (i.e., the length of the isotropically
oriented line running from one particle border to the other) is denoted
␲
lo. The volume (V) of the particle can be estimated by: V̂ ⫽
⫻ l03
3
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:
1
V̂ ⫽ ⫻ ␲ ⫻ l03
3
(4)
where l0 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:
冘 冉␲3 ⫻ l 冊
n
3
0, i
␯ˆ V ⫽
i-1
n
⫽
␲
⫻ l0 3
3
(5)
where n is the number of observations and l0,i is the length
of the ith 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 l0,0) 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
DIABETES, VOL. 50, AUGUST 2001
M. SKAU AND ASSOCIATES
␯ˆ V ⫽
FIG. 2. Nonconvex particle hit by a sampling point (X) with an
associated isotropic line in space. The direction of the isotropic line
given causes the generation of two intercepts belonging to the same
sampling point. l0,0 is the length of the intercept containing the
sampling point. l0,iⴙ and l0,i– represent the longest and shortest distances, respectively, between the sampling point and the end points of
the additional ith intercept (i ⴝ 1, since only one additional intercept
is created in this example).
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 ith (i ⫽ { 0,. . .
, m}) intercept not containing the sampling points can then
be determined by the following formula:
l0,i3 ⫽ l0,1⫹3 ⫺ l0,1⫺3
(6)
where l 0,i⫹ and l0,i– represent the longest and shortest
distance, respectively, between the sampling point and the
end points of the ith intercept (Fig. 2).
The sum of all l0,i3 belonging to a given islet is denoted
l0,e3:
冘l
m
l0,e ⫽
3
3
0,i
(7)
i⫽1
An unbiased estimate of the islet volume can now be
obtained:
␲
V̂ ⫽ 共l0,03 ⫹ 2 ⫻ l0,e3兲
3
(8)
Notice that the expression holds true for both globally
convex and nonconvex islets, since l0,e3 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 ␯V:
DIABETES, VOL. 50, AUGUST 2001
␲
⫻ 共l0,03 ⫹ 2 ⫻ l0,e3兲
3
(9)
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 volumeweighted 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 l0 can be estimated
along these direction-indicating grid lines.
Measuring l0 with high precision is a time-consuming
and often unnecessary procedure, since the parameter
itself clearly is subjected to wide variation. Measurements
of l03 are obtained much faster by using a l03-ruler (Fig. 3),
which groups the intercepts into n classes. Any class has
a width progression by a factor 101/n on a cubic scale
compared with the previous class (14). The l03-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 zth 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 volumeweighted mean islet volume can be calculated. So that the
observations can be converted into true biological dimensions, the estimated value of ␯V must be corrected with a
factor F:
F⫽
冉 冊
1
Mag
3
(10)
FIG. 3. An l03-ruler with 15 classes.
1765
VOLUME-WEIGHTED MEAN ISLET VOLUME
TABLE 1
An example of the estimation of the volume-weighted mean islet volume
A
C
Upper-limit
length
(␮m ⫻ 10⫺4)
D
Class width
length3
(␮m3 ⫻ 10⫺12)
E
F
G
Class
number z
B
Upper-limit
length3
(␮m3 ⫻ 10⫺12)
Class midpoint
(␮m3 ⫻ 10⫺12)
Observed number
per class
E⫻F
(␮m3 ⫻ 10⫺13)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.71
1.55
2.54
3.70
5.07
6.69
8.59
10.8
13.5
16.6
20.3
24.6
29.7
35.8
42.9
0.89
1.16
1.36
1.55
1.72
1.88
2.05
2.21
2.38
2.55
2.73
2.91
3.10
3.29
3.50
0.71
0.84
0.99
1.16
1.37
1.62
1.91
2.25
2.65
3.12
3.68
4.34
5.11
6.03
7.11
0.36
1.13
2.04
3.12
4.38
5.88
7.64
9.72
12.2
15.1
18.5
22.5
27.2
32.8
39.3
58
37
20
18
6
6
6
4
2
4
3
0
7
0
2
173
2.06
4.18
4.08
5.61
2.63
3.53
4.58
3.89
2.43
6.02
5.54
0
19.0
0
7.86
71.4
The general formula of the upper limit of class z(z ⫽ {1,2,. . ., n⫺1, n} on a cubic scale is:
共L 兲3
⫻ 共10z/共n⫺1兲 ⫺ 1兲
10
⫺1
n
n/n⫺1)
for an n-class ruler of real length Ln. When applied to a ruler with Ln ⫽ 35 mm and n ⫽ 15, values are as shown in column B. The
upper limit length of class z on a normal scale (column C) is used when the ruler is drawn. The class widths on the cubic scale
(column D) are retrieved from column B and used to calculate the class mid-point length3 in column E (i.e. the lengths3 from the
starting point of the ruler to the class mid points). The mid-point lengths3 are multiplied by the number of intercepts recorded per
pancreas in each class (column F; data from a pancreas in our study), and the results are shown in column G. Column G provides
the sum 兺 l03, which is used for the calculation of ␯V. Calculation of volume-weighted mean islet volume corrected for observations at
magnification ⫻151:
␯ˆ V ⫽
␲
7.1 ⫻ 1014␮m3
1
⫽ 1.3 ⫻ 106 ␮m3
⫻ l03 ⫻ F ⫽ 1.05 ⫻
⫻
3
173
1513
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 l03 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.
1766
FIG. 4. Using the l03-ruler for determining the class of l03 for an isotropically oriented intercept belonging to a given sampling point (X).
The starting point of the ruler is placed to coincide with the one end
point of the intercept, while the class number of l03 is read off the ruler
at the other intercept end point. In this case, the class number is 7.
DIABETES, VOL. 50, AUGUST 2001
M. SKAU AND ASSOCIATES
(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 ␯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:
Correction factor ⫽
兺 Vi,0
兺 Vi,d
(11)
where 兺Vi,0 is the summed pancreatic volume before fixation and 兺Vi,d 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.
Estimating total islet mass. The estimation of total islet mass, Vtot, 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 ␤-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:
FIG. 5. A schematized image of a uniform random pancreatic tissue
section. Islet profiles (gray) are surrounded by exocrine tissue (white). A
point grid with direction-indicating lines is randomly superimposed
onto the image. The distance between the marginal grid points and the
image border is larger than the largest islet profile. Five points of the
grid are hitting an islet profile, and each one of these generates an intercept parallel to the direction-indicating lines. Because one of the
islet profiles is hit by two points, two intercepts are recorded for that
islet profile. The encircled point of the grid is also used for point counting
the reference space (i.e., the total area of tissue in the sections).
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 CO2
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 ␮m 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 ␯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 ␯V was estimated
DIABETES, VOL. 50, AUGUST 2001
Total islet mass ⫽
islet points
times pancreas weight
396 times reference points
(12)
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
log10-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 volumeweighted 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 log10-transformed data for the volumeweighted mean islet volume and total islet mass were
investigated in a linear regression model (Fig. 7). The
equation for the regression line shown is:
log vv ⫽ a ⫹ (b ⫻ log Vlot)
(13)
where b ⫽ 0.76 (0.13; [0.51–1.0]; P ⬍ 0.001 vs. H0: b ⫽ 0)
and a ⫽ 6.2 (0.10; [6.0 – 6.4]); (SE; 95% CI). All estimated
data of ␯V were corrected with a factor equal to the
shrinkage ratio caused by fixation and dehydration. This
factor was determined to be 1.4.
1767
VOLUME-WEIGHTED MEAN ISLET VOLUME
FIG. 6. Age versus total islet mass (A) and volume-weighted mean islet
volume (B).
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 ␮m due to the size of the smaller islets.
In addition, the set of sections must proceed at least 600
␮m (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 glyce-
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 volumeweighted 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 ␤-cell mass from a rat pancreas can be performed in less than 1.5 h with use of a
microscope.
1768
FIG. 7. The volume-weighted mean islet volume as a function of total
islet mass, linear regression with log10-transformed axes. The two
reference lines illustrate the 95% CI for the regression line. The insert
shows a scatter plot with linear axes of the same data. There was a
positive correlation between the total islet mass and the volumeweighted mean islet volume in both the log10-transformed data (r ⴝ
0.80, P < 0.00001) and in the nontransformed data (r ⴝ 0.70, P <
0.001).
DIABETES, VOL. 50, AUGUST 2001
M. SKAU AND ASSOCIATES
mic 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– 8). In the following description this
parameter is denoted aisl. The geometrical meaning of aisl
can be described using the formula of Abercrombie (22)
with the assumption that the sections used are sampled by
URS:
M
P⫽A⫻
(14)
L⫹M
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 NV is the number density of islets in the pancreas and
NA is the profile density of islets in the section, then:
P ⫽ NV ⫻ Vtissue ⫽ NV ⫻ Atissue ⫻ t
(15)
A ⫽ NA ⫻ Atissue
(16)
and:
where Vtissue is the volume of the tissue in the section,
Atissue 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:
NA ⫽ NV ⫻ 共h ⫹ t兲
(17)
where h (⫽L) is the mean height of the islets measured
perpendicular to the cutting surface. Because:
NV ⫽
with f ⫽
⫽
f
f
⵩ NA ⫽
VN
Aisl
(18)
total islet profile area
total area of sections
total islet volume
(17), Eq. 18 can be rewritten
total tissue volume
as a stereological expression of aisl:
aisl ⫽
vN
h⫹t
(19)
From Equation 19 it is seen that aisl 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 aisl is not direct or indirect
evidence of an increased mean islet volume, since the ratio
between the arithmetric mean islet volume and the mean
DIABETES, VOL. 50, AUGUST 2001
FIG. 8. Proposed models of the expansion of total islet mass. Model A
illustrates the formation of new islets while keeping the relative
volume distribution of islets unchanged. Model B illustrates the expansion of preexisting islets with the same factor without the addition of
new islets.
islet height is influenced by both the shape of the islets as
well as the volume distribution of islets. Thus, aisl 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 aisl.
The number of islet profiles per section area, NA, has
also been used as a parameter to compare the number of
islets in pancreas. Equation 17 can be rewritten as:
NA ⫽
nislets ⫻ 共h ⫹ t兲
Vpancreas
(20)
where nislets is the total number of islets in pancreas and
Vpancreas is the total volume of the pancreas. Thus, NA is
not a direct or an indirect estimator of the total islet
number, because NA 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,
1769
VOLUME-WEIGHTED MEAN ISLET VOLUME
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 volumeweighted 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.
REFERENCES
1. McEvoy RC: Changes in the volumes of the A-, B-, and D-cell populations
in the pancreatic islets during the postnatal development of the rat.
Diabetes 30:813– 817, 1981
2. Kaung HL: Growth dynamics of pancreatic islet cell populations during
fetal and neonatal development of the rat. Dev Dyn 200:163–175, 1994
3. Finegood DT, Scaglia L, Bonner-Weir S: Dynamics of ␤-cell mass in the
growing rat pancreas: estimation with a simple mathematical model.
Diabetes 44:249 –256, 1995
4. Montanya E, Nacher V, Biarnes M, Soler J: Linear correlation between
␤-cell mass and body weight throughout the lifespan in Lewis rats: role of
␤-cell hyperplasia and hypertrophy. Diabetes 49:1341–1346, 2000
5. Tse EO, Gregoire FM, Reusens B, Remacle C, Hoet JJ, Johnson PR, Stern
1770
JS: Changes of islet size and islet size distribution resulting from proteinmalnutrition in lean (Fa/Fa) and obese (fa/fa) Zucker rats. Obes Res
5:563–571, 1997
6. Elayat AA, el Naggar MM, Tahir M: An immunocytochemical and morphometric study of the rat pancreatic islets. J Anat 186:629 – 637, 1995
7. Tomita T, Doull V, Pollock HG, Krizsan D: Pancreatic islets of obese
hyperglycemic mice (ob/ob). Pancreas 7:367–375, 1992
8. Parsons JA, Bartke A, Sorenson RL: Number and size of islets of Langerhans in pregnant, human growth hormone– expressing transgenic, and
pituitary dwarf mice: effect of lactogenic hormones. Endocrinology 136:
2013–2021, 1995
9. Gundersen HJ, Jensen EB: The efficiency of systematic sampling in
stereology and its prediction. J Microsc 147:229 –263, 1987
10. Gundersen HJ: Stereology of arbitrary particles: a review of unbiased
number and size estimators and the presentation of some new ones, in
memory of William R. Thompson. J Microsc 143:3– 45, 1986
11. Gundersen HJ, Jensen EB: Stereological estimation of the volumeweighted mean volume of arbitrary particles observed on random sections.
J Microsc 138:127–142, 1985
12. Nielsen K, Colstrup H, Nilsson T, Gundersen HJ: Stereological estimates of
nuclear volume correlated with histopathological grading and prognosis of
bladder tumor. Virchows Arch B Cell Pathol 52:41–54, 1986
13. Ladekarl M: Objective malignancy grading: a review emphasizing unbiased
stereology applied to breast tumors. APMIS Suppl 79:1–34, 1998
14. Sorensen FB: Stereological estimation of the mean and variance of nuclear
volume from vertical sections. J Microsc 162:203–229, 1991
15. Movassat J, Saulnier C, Serradas P, Portha B: Impaired development of
pancreatic beta-cell mass is a primary event during the progression to
diabetes in the GK rat. Diabetologia 40:916 –925, 1997
16. Bock T, Svenstrup K, Pakkenberg B, Buschard K: Unbiased estimation of
total beta-cell number and mean beta-cell volume in rodent pancreas.
APMIS 107:791–799, 1999
17. Scherle W: A simple method for volumetry of organs in quantitative
stereology. Mikroskopie 26:57– 60, 1970
18. Delesse MA: Procédé mécanique pour déterminer la composition des
roches. C R Acad Sci Paris 25:544 –545, 1847
19. Lifson N, Lassa CV, Dixit PK: Relation between blood flow and morphology
in islet organ of rat pancreas. Am J Physiol 249:E43–E48, 1985
20. Hellman B: The numerical distribution of the islets of Langerhans at
different ages of the rat. Acta Endocrinol 32:63–77, 1959
21. Hellman B: The volumetric distribution of the pancreatic islet tissue in
young and old rats. Acta Endocrinol 31:91–106, 1959
22. Abercrombie M: Estimation of nuclear population from microtomic sections. Anat Rec 94:239 –247, 1946
23. Wicksell SD: The corpuscle problem: a mathematical study of a biometric
problem. Biometrika 17:84 –99, 1925
24. Wicksell SD: The corpuscle problem: second memoir. case of ellipsoidal
corpuscles. Biometrika 18:151–172, 1926
25. Cruz-Orive LM: Distribution-free estimation of sphere size distributions
from slabs showing overprojection and truncation, with a review of
previous methods. J Microsc 131:265–290, 1983
26. Gundersen HJ, Jensen EB: Particle sizes and their distributions estimated
from line- and point-sampled intercepts. Including graphical unfolding.
J Microsc 131:291–310, 1983
27. Garofano A, Czernichow P, Breant B: In utero undernutrition impairs rat
beta-cell development. Diabetologia 40:1231–1234, 1997
28. Scaglia L, Cahill CJ, Finegood DT, Bonner-Weir S: Apoptosis participates in
the remodeling of the endocrine pancreas in the neonatal rat. Endocrinology 138:1736 –1741, 1997
29. Bonner-Weir S, Deery D, Leahy JL, Weir GC: Compensatory growth of
pancreatic beta-cells in adult rats after short-term glucose infusion. Diabetes 38:49 –53, 1989
DIABETES, VOL. 50, AUGUST 2001