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Method for decreasing the statistical variance of stereological estimates.

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THE ANATOMICAL RECORD 207239-106 (1983)
Methods for Decreasing the Statistical Variance
of Stereological Estimates
ROBERT P. BOLENDER
Department ofBiologica1 Structure, University of Washington School of
Medicine, Seattle, WA 98195
ABSTRACT
Three methods are described for decreasing the statistical
variance of stereological estimates. Method 1 uses profile boundaries and
surface densities of nuclear membranes, measured in thin sections, to estimate
the mean diameter, surface area, and numerical density of spherical and
nonspherical nuclei. For the guinea pig pancreas (number(m) = 4), the standard deviations (s.d.) as a percent of the mean for the estimates of the diameters of the exocrine, duct, and endothelial cell nuclei were 1.5%, 3.3% and
1.4%. The estimate for the mean diameter of exocrine nuclei (6.4 k 0.1 pm)
was based on a spherical model, whereas the estimates for the diameters of the
nonspherical (and nonconvex) nuclei of the duct (6.4 f 0.2 pm) and endothelial
(6.7 f 0.1 pm) cells were calculated from the numerical density of the exocrine
cells and the relative frequencies of the three cell types (determined from serial
reconstructions). In a n average cubic centimeter, there were 6.17 x 10' + 0.32
x 10' (s.d. 5.1% of mean) exocrine cells, 1.64 x 10' f 0.18 x 10' (10.9%)duct
cells, and 0.803 x 10'
0.13 x 10' (16.6%)endothelial cells. In contrast to
method 1,conventional stereological approaches were found to have standard
deviations two- to eightfold larger. Method 2 uses a mean nuclear surface area
and a ratio of surface densities to estimate the surface area of a membrane
compartment in a n average cell. A s.d. equal to 6.5% of the mean was found
for the surface area of the outer mitochondria1 membrane in a n average
exocrine cell (672 & 43.6 pm2),which represented a n almost fourfold reduction
in the s.d. compared with a n earlier estimate (Bolender, 1974).Method 3 relates
the surface area of a membrane compartment to a standard number of cells.
Referenced to lo6 cells, for example, the surface area of the inner nuclear
membrane of endothelial cells had a s.d. equal to 2.9% of the mean, whereas
the surface density of the same membrane compartment-referenced to a cm3
of cells-had a s.d. at 19.1% of the mean. In this case, method 3 produced
almost a sevenfold reduction in the standard deviation. Similar results were
found for exocrine and duct cells.
The results of the study indicate that the standard deviation of a stereological estimate can be reduced to a minimum by using a mean nuclear profile
boundary to generate a n estimate for a nuclear numerical density, which, in
turn, can be combined with a surface density to obtain average cell information.
If a convex nucleus were rotated between
the tines of a caliper and measured at all
possible orientations, the resulting mean
value would represent its mean caliper d i
ameter (D). In biological stereology, the D
characterizing a nuclear population has become one of the most sought after pieces of
information because it allows one to estimate
the numerical density (Nv) of nuclei, as, for
0 1983 ALAN R. LISS, INC.
example, defined by the equation given by
Wicksell (1925,1926) and DeHoff and Rhines
(1961):
Nv
=
N*/D,
Received April 6,1982; accepted June 2,1983.
(1)
90
R.P. BOLENDER
where NA equals the number of nuclear profiles (N) counted from several random sections having a total test area A. The
numerical densities of nuclei are needed to
calculate the volumes and frequencies of average cells, as well as average values for
cytoplasmic compartments. Earlier studies
have shown repeatedly that stereological information can be interpreted more effectively
when data are related to average cells and
the number thereof (Bolender, 1979a,b,
1981a,b, 1982).
A reliable solution to equation (1)depends
on collecting two pieces of information, NA
and D. The NA represents a variable that is
obtained by counting nuclear profiles as they
appear in sections. The D, however, is a variable that must be estimated from measurements of profile diameters and a mathematical reconstruction. The methods of reconstruction are expected to work well only
when the nuclei can be likened to spheres or
slight ellipsoids (see reviews by DeHoff and
Rhines [1968],Underwood [I9701 and Weibel
11979, 19801). When this is not the case, the
reliability of a n estimate for a numerical
density becomes difficult to assess.
The fundamental problem with most of the
numerical density equations is that they require information for their solution that cannot be extracted routinely from biological
systems. Many stereological equations, for
example, ask for information about regular
profiles of simple spheroids, but, more often
than not, cells that have been fixed and sectioned for electron microscopy can supply only
irregular profiles from asymmetric nuclei.
This has led to several provisional sohtions
to the problem of estimating the D (Ebbeson
and Tang, 1965; Greeley et al., 1978; Loud et
al., 1978; Cruz-Orive, 19801, but most of these
methods require some form of serial sectioning and do not enjoy the attractiveness of a
simple stereological solution.
The purpose of this paper is to examine ways
to improve the interaction between stereological theory and experimental data. To do
this, we will approach the problem of numerical densities from a different perspective.
Surface densities, which represent the total
surface area of a compartment contained
within a unit of reference volume (e.g., pm3,
cm3), are considered to be one of the best
forms of stereological information, because
the data needed to obtain these estimates can
be collected directly from sections (Weibel,
1974). Earlier (Bolender, 1979a,b,), surface
densities were used as a basis for estimating
the relative volumes and frequencies of average cells and for evaluating changes in organellar compartments. The strategy of these
surface area ratio papers was to find a membrane compartment within the cells that
maintained a constant surface area throughout a n experiment and then to use it as a
reference for interpreting changes in cytoplasmic compartments. In looking for a way
to test the methodological assumption that
the nuclear membranes of the exocrine cells
of the guinea pig pancreas remained constant
when treated with a secretagogue, a remarkable piece of information was found. The nuclear surface area of both the duct and endothelial cells of the pancreas also remained
constant (Bolender, 1979a). This meant that
when it came to estimating numerical densities and events in average cells, the surface
area ratio methods could deal as effectively
with cells having nonspherical and nonconvex nuclei (duct and endothelial) as with cells
having spherical nuclei (exocrine).
When the mean profile boundaries of these
same pancreatic nuclei were measured from
several animals, they were found to have a
very small population variance. This observation indicated that there was a membrane
compartment in the electron micrographs
that could be measured with a remarkable
reproducibility from one experimental animal to the next. The three methods described
in this paper all use this mean profile boundary as their starting point, and, as a result,
can generate experimental estimates with
relatively small variances. The key step in
the development of these methods consists of
finding the surface area of a n average nucleus @& which can be used in combination
with a nuclear surface density (Svn)to calculate a nuclear numerical density (Nv,):
(2)
Several examples will be used to show how
equation (2) can generate experimental results with standard deviations that are severalfold smaller than usually expected for
stereological estimates.
MATERIALS AND METHODS
Method 1: Estimation ofNumerica1Density
of Nuclei (Cells)
Models are given for estimating the numerical density of nuclei from surface areas (Figs.
1,2,3).Both models assume that all the nuclei
91
STEREOLOGICAL ESTIMATES
MODEL 1
S
MEAN PROFILE
SPHERE
BOUNDARY
Fig. 1. Model 1. When a sphere is sectioned either
randomly or systematic_ally (case shown), the resulting
mean profile boundary CB)is equal to the mean boundary
(b) of the original sphere. The B, which can be measured
in sections, is used to calculate 1) in equation (8) the D
of the original sphere, and 2) in equation (15) the surface
area of thcsphere. In a n experimental setting, the surface area (S) of an average spherical exocrine nucleus is
used in combination with the nuclear surface density to
estimate the nuclear numerical density (equation 16).
For details, see Materials and Methods.
I
R
R
Fig. 2. Calculation of a mean radius of a sphere from
a mean profile radius (f). The example is simplified by
using one quadrant of a two-dimensional section that
passes through the center of a sphere of radius R. Eighteen equally spaced lines (profile radii) are drawn along
the radius of the sphere perpendicular to the x axis and
represent the r. of equation 4. The mean profile radius
(3is identified with an arrow. If the positions of the
profile radii had been chosen randomly along R, then
the resulting F would be expected to have the same value
as the one calculated from the equally spaced radii-if
enough radii were used.
of a particular cell type have the same surface of the nuclear profiles seen in thin sections
area and shape. The first model assumes a is equal to the mean boundary (b)of a sphere
spherical shape and the second assumes that (see Weibel, 1979, p. 37, for a mathematical
a nonspherical nucleus can be treated as a proof). In Figures 1 and 2, the mean boundspherical one by introducing a correction fac- ary of a sphere is defined as the average of
tor. The effects of relaxing these assumptions all possible boundary lengths that can be
are considered in the Discussion.
obtained by cutting the sphere randomly (or
systematically) where:
Model 1: Numerical density of spherical
nuclei
b = (bl + b2 + . * * + bJm,
(3)
The method for spherical nuclei is based o
n
the assumption that the mean boundary (B) and m is the total number of boundaries.
R.P. BOLENDER
92
MODEL 2
I
NONSPHERE
CORRECTION FACTOR (k)
1
SURFACE AREA
OF EOUIVALENT SPHERE
MEAN PROFILE BOUNDARY
OF EQUIVALENT SPHERE
Fig. 3. Model 2. When a nonsphere is sectioned randomly, the resulting B is not equal to the b of a sphere
having an equivalent surface area. By introducing a
correction factor (k),however, the B of the nonsphere can
be set equal to that of the equivalent sphere, and the
calculations proceed as described for Model 1. See Materials and Methods for details.
Each boundary bj defines a circular profile
having a radius r;:
r;
=
b;/2a.
(4)
The mean value of r with respect to x is
defined by the following integral:
The final equation is written in terms of-a
mean diameter (D) and profile boundary (B)
because the experimental estimates are
based on measurements collected from many
nuclei in sections:
R
(i-)x
= 1
B
lo
(R2 -
-
xj2)”
dx,
D
(5)
where (R2 - xjzs,” equals the lengths of the
individual radii and R equals the radius of
the sphere. Evaluating the integral we obtain:
F = (7d4). R.
(6)
i = b/2a,
(7)
4B/7r2.
(9)
Note that equation (9) refers to nuclei all of
one size.
We know from Buffon (1777) and Smith and
Guttman (1953)that the length of a boundary (B) can be estimated in sections by counting the number of intersections (I)that occur
between the boundary trace and a set of parallel lines separated by a distance d:
Since:
B
the relationship between the mean boundary
(b) and the diameter (D) of a sphere can be
given as:
=
=
( ~ / 2 .) I. d.
(10)
Moreover, a mean boundary (B) can be obtained for the mean number of intersections
(I):
-
B
= (d2)
. . d.
(11)
STEREOLOGICAL ESTIMATES
When equation (11)is substituted into equation (91, the intersection counting form of
equation (9) is obtained for the nuclei (n):
where a equals 1when a grid of parallel test
lines is used and a equals 2 when a rectangular grid of horizontal and vertical test lines
is used.
The experimental evaluation of equation
(12) indicated that a dense set of test lines is
needed to estimate the length of the profile
boundaries (see Fig. 4a). Since the counts are
collected with both the horizontal and vertical test lines, a was set equal to 2 in equation
(12). The data are collected in the following
way. Electron micrographs of exocrine, duct,
and endothelial nuclei are projected onto a
test screen (final magnification approximately x 12,000) and the intersections between the traces of the inner nuclear
membrane (Fig. 4b) and the lines of the test
system (Fig. 4a) were counted.
The mean number of intersections (I,‘) for
the profiles of the nuclei is calculated by dividing the total number of intersections
counted by the number of (m) profiles measured:
(I1 + Iz
+ . . + IJm.
93
densities of nonspheroidal nuclei. The first
option relies on the numerical density of the
spheroidal exocrine nuclei and the relative
frequencies of the exocrine, duct, and endothelial cells. The second option uses a correction factor to convert the mean profile
boundary of the nonspherical nuclei to that
of a sphere of equivalent surface area. To
determine the numerical density of the duct
and endothelial nuclei, which are nonspherical and nonconvex structures, two pieces of
information are used: 1)the numerical density of the exocrine nuclei (from equation 16),
and 2) the relative frequencies of the three
nuclear types (from serial reconstructions).
The numerical density (Nv) of the combined
cellular compartment (34, which includes
exocrine, duct, and endothelial cells, is obtained by dividing the numerical density of
the exocrine nuclei (Nvexcn)by their relative
frequency (NexCnlN3J:
Option 1. The numerical densities of the
duct and endothelial cells are found by multiplying the numerical density of the combined compartment by the relative frequency
of each nuclear type (lc):
(13)
NVlc,cmn3 = NV3c,crn3 . (Nlcn\J3c). (18)
The I,’, which come from micrographs having a magnification M, is standardized to a
magnification of x 12,000:
Equations 17 and 18 represent a solution
to the problem of estimating the numerical
density of structures that are nonspherical
and nonconvex. It is important to note that
the numerical density methods (Wicksell,
1925, 1926; Weibel and Gomez, 1962; DeHoff
and Rhines, 1961; and Giger and Riedwyl,
1970) used most often by biologists all assume the presence of convex structures. Consequently, these methods can not be expected
to supply the best estimates for numerical
densities when they are applied to nonconvex structures such as duct and endothelial
cell nuclei.
Option 2. In model 1, the nuclei of the exocrine cells are spherical and their mean profile boundary (B) is assumed t o b e equal to
the mean boundary of a sphere ($1. Now consider the case where we have a population of
nuclei with the same surface area and shape,
but that the shape is nonspherical (see Fig.
3). Assume that such a situation exists for
the duct and endothelial nuclei (see Figs. 4c,
d). The nonspherical structure shown in Figure 3 can also supply a mean profile bound-
1,’
=
-
-
I, = I,’
1
. (12,000 I M).
(14)
The solution to equation (12) gives the
mean diameter (D,) of the spherical nuclei,
which, in turn, is used to generate a mean
nuclear surface area (SJ:
-
S,
- 2
= a . D,
.
(15)
The numerical density of the nuclei (Nv,! in
a cm3 of reference space in found by dividing
the surface density of the nuclei (Sv,) by the
mean nuclear surface area (S,):
Model 2: Numerical density of nonspherical
and nonconvex nuclei
Model 2 includes two options for estimating the mean surface areas and numerical
Fig. 4. The test grid (4a) and inner nuclear membranes (arrows) of exocrine (4h), duct (4~1,and endothelial (4d) cells are illustrated. The exocrine nucleus can
be described as a convex structure, whereas the duct and
endothelial nuclei are nonconvex because they can be
cut more than once by a section plane.
STEREOLOGICAL ESTIMATES
Figure 4 (continued).
95
96
R.P. BOLENDER
ary (B), but, in contrast to the situation in
model 1,it is not equal to the mean boundary
of a sphere 6)of equivalent surface area.
Consequently, the mean profile boundary (B,)
of the nonspherical nuclei (measured in thin
sections) must be corrected before it can be
used to generate the surface area of a sphere
with a n equivalent surface area. Since the
numerical densities of the duct and endothelial nuclei are already known from equations
(17) and (181, a correction factor can be generated that adjusts the mean boundary of the
nonspherical nucleus to that of a sphere of
equivalent surface area.
The correction factor is introduced at the
level of equation (12). By rearranging equation (I..), the mean surface area of the nucleus (S,) is obtained:
% = Svn,cm3 1Nvn,cm3,
(19)
where n represents the nuclei of either the
duct or endothelial cells. The mean surface
area is then used to calculate the mean diameter of? sphere with a n equivalent surface area (D(eqsp)):
-
D(eqsp)= (ST)”,
(20)
(21)
The correction factor (k,) is obtained for a
given nuclear type by dividing the mean
boundary of the equivalent sphere (eqsp) by
that of the nonspherical nucleus:
B(eqspfi(nonspherica1nucleus). (22)
Equation (11)is rearranged and the correction factor is introduced:
7,
Si,; = (Svi 1SvJ
. sn,
(24)
where the surface density of the nuclear compartment is in the denominator of the membrane ratio. Since both surface densities in
equation (24) share the same reference space
(Bolender, 1979a), the ratio of surface densities can be simplified to a ratio of intersections:
s,,;= (Ii/I,)
. 3,.
(25)
Equation (25) represents an attempt to identify a n expression that can supply a n optimal form of average cell information. In this
case, an optimal form is defined as a stereological estimate with the smallest standard
deviation.
-
B(eqsp) = (n2 . ~(eqsp))/q-
kn =
Method 2: Estimation o f Membrane Surface
Areas in Average Cells
The surface area of a membrane compartment i (Si) in a n average cell (c) can be estimated from the mean surface area of the
nucleus (n) and a ratio of surface densities:
Method 3: Estimation o Membrane Surface
Areal106fCells
and mean profile boundary (B(eqsp)):
-
the numerical densities of duct and endothelial nuclei.
=
B, . k,/((d2) . d).
(23)
The 1, of equation (23) represents the expected number of intersections for a sphere
with a surface area equal to that of the mean
nonspherical and nonconvex nucleus. It can
now be used to solve equations (121, (15), and
(16) as given in model 1. The advantage of
the correction factor in equation (23) is that
it allows us to use the low variance counts of
nuclear intersections to estimate a mean surface area, which, in turn, is used with a nuclear surface density to obtain estimates for
The surface area of a membrane compartment i (Si)/106cells is calculated by multiplying the surface area of the compartment in
a n average cell (Si,;)by 10%
si/106 cells = Si,; . lo6.
(26)
Method for Serial Reconstructions
Since the details of this method will be
given elsewhere, only a brief description is
given here. For two of the four pancreases,
serial sections 1 pm in thickness were cut
from tissue blocks, photographed a t a magnification of x 1,000, and printed at a final
magnification of x 17,000. The relative frequencies of exocrine, duct, and endothelial
nuclei were determined.
Estimating a Mean Diameter From Profile
and Numerical Densities
Recall from the introduction that equation
(1) can be used to estimate the numerical
density of convex structures in a unit of reference volume from two pieces of information: l) the number of profiles of the
97
STEREOLOGICAL ESTIMATES
structurehnit area of the section plane (NA),
and 2) the mean diameter of the structure.
Since Nv can be estimated for the exocrine
nuclei (convex structures) with equation (16),
and NA by counting the number of nuclear
profiles seen in the thin sections, the D is
obtained by simply rearranging equation (1):
(27)
The expected diameters of spheres can also
be estimated with the methods of Wicksell
(1925, 1926) and Giger and Riedwyl (1970).
Using measurements of nuclear profiles coming from the same set of sections that supplied the mean profile boundaries for
equation (9), the Ds are calculated with these
two methods as well.
Sampling Procedures
Blocks of tissue were collected from the
pancreases of four adult male guinea pigs
(BFA strain) and fixed for electron microscopy as described previously (Bolender, 1974).
Ten blocks were selected from each animal
and thin sections were cut in the range of
40-50 nm. Six electron micrographs were
collected from each section giving 60 micrographslanimal for each of the three stages of
magnification. Since a random systematic
method of sampling did not supply enough
samples for estimating the mean profile
boundaries of the duct and endothelial nuclei, all such profiles that could be found were
micrographed. Stage I ( x 6,000) was used to
estimate the volume and surface densities of
the nuclei, Stage I1 ( X 12,000) to count the
nuclear intersections needed to solve equations (12) and (23), and Stage I11 ( X 20,000)
to count intersections for the outer mitochondrial and inner nuclear membrane compartments, which were used to solve equation
(25). Two test systems were used: Stage I and
I11 (see Weibel, 1979, p. 364 [D36]),and Stage
I1 (see Fig. 4a).
The micrographs were recorded on 35 mm
film using a Philips 300 electron microscope,
contact printed, and projected onto the test
screens of a projector system (Weibel, 1979).
A micrograph of a calibration grating having
21,600 lines/cm (E.F. Fullam, Inc., Schenectady, NY) was included on each film strip.
PCS-I software (Bolender et al., 1982) and a
Tektronix 4051 or 4052 computer (Beaverton, OR) were used to collect data from elec-
tron micrographs, solve equations, and run
statistical tests.
RESULTS
Method 1
Model 1: Spherical nuclei
The results for the exocrine nuclei estimated with the methods of model 1are given
in Table 1.The most remarkable information
in this table is the demonstration that the
mean nuclear intersections can be estimated
with a standard deviation (s.d.) equal to less
than 1.5% of the mean. These counts of nuclear intersections were directly responsible
for the unusually small statistical variations
of the stereological estimates that, in turn,
they were used to generate. It can be seen,
for example, that the surface area of the exoocrine nuclei had a s.d. of only 3% of the
mean and the s.d. of their numerical density
was as low as 5.1% of the mean. The mean
diameter of equation (271, which was estimated from the nuclear profile and numerical densities (Table 1, footnote lo), was
essentially the same as the estimate coming
from method 1(Table 1, footnote 3).
Table 2 compares the results of model 1
with those of two other methods that were
used earlier to estimate diameters for the
exocrine nuclei (Bolender, 1974, 1979a,b). All
the data in Table 2 came from the same animals and same sets of thin sections. As described in the footnotes, the numerical
densities were estimated from 1) the mean
surface area and surface density of the nuclei, and 2) the mean volume and volume
density of the nuclei. The results show a
broad range of estimates: 5.632 x 10' to 10.42
x lo8 nuclei/cm3 for the numerical densities
and 5.1% to 21% (as a percentage of the mean)
for the standard deviations. It seems likely
that the major cause of the differences between the three methods can be traced to the
way the original nuclear profiles were measured. The major diameters of the nuclear
profiles were used for the Giger and Riedwyl
(1970) method (Bolender, 1974) and, in contrast to the results of model 1 (Table 1,footnote 3), they produced a 9% smaller estimate
for the mean diameter (Table 2, footnote 4).
Both the major and minor profile diameters
were used for the Wicksell (1926) method (see
Bolender, 1979a,b) and this led to a n 18%
smaller value for the mean diameter (Table
2, footnote 4). When the estimates for the
urn
133.4
125.3
130.6
125.8
128.8
3.9
(3.0)
urn2
(SJ4
9.113
8.931
8.334
9.857
9.058
k 0.630
(6.9)
x 1010/crn3
Pm2
(Svn.12
withjnner nuclear membrane (see Figure 4b).
16.08
6.516
15.58
6.315
15.91
6.447
15.62
6.329
15.80
6.402
0.24 i 0.100
(1.5)
(1.5)
urn
number of intersections
45.49
44.09
45.01
44.18
44.69
+ 0.67
(1.5)
6%)'
@",3
(NAn,lJ7
NO. x 105/cm3
4.139
4.424
3.841
4.053
4.114
i 0.240
(5.9)
(Nvn,d
NO. x 1 0 8 / d
6.833
7.128
6.383
7.833
7.044
+ 0.610
(8.6)
=
T
= No. of nuclei n in 1 em3 of reference space.
'Nvn
7NA,'3, = No. of nuclear profiles dcm2 of the combined (3c) exocrine, duct, and endothelial cell profile areas.
8S~,.b,= 2 . InLT(where L,p = 2 . PaC' d; P,, = total points on 3 cells [3c]: exocrine, duct, and endothelial).
'Nv.,~. = S V , ~ , / (No.
~ , of nuclei n [in this case exocrine nuclei] in 1cm3 of reference space consisting of the 3 cells 13~1).
"& = N A ~ . ~ J N see
v ~ equation
, ~ ~ ; (1).
"I,(corr) = 8, ' k,/((a/2) ' d)); see equation (23).
=
' n",e,8p,)i4; see equation (21) and Fig'
'"B,(eqsp)
Footnotes 11-14 are found in the text and appear in Tables 3 and 4
i3~n(eqsp)
= (~,,,,,)/~)~h; see equation (20) and Fig. 3.
'"%(eqsp) = s V ~ ( ~ ~ ~ ~ J N V ~ ( ~ ~ ~ ~ )
"S
. D2;
e.g., (animal 1):gn = K ' (6.516 am? = 133.4 am2.
5 S c , , ~ , = 2 . I,LT((where LT = 2 . PI, . d); l c [cell] indicates that the reference space is filled by only one cell type).
n
%, = (~112). I, ' d; e.g., (animal 1): B, = (d2). 45.49 . 0.225 pm = 16.08 pm.
% = 4 . sn/a2;
e.g., (animal 1): = (4 . 16.08 pm)/li2 = 6.517 pm.
'Mean
s.d.
(s.d. as %
of mean)
2
3
4
Mean
1
Animal
(EnY
~~
8.351
.
8.244
7.683
7.512
7.947
f 0.410
(5.2)
P.m2
x 101°/cm3
(svn,3J8
urn
6.611
6.732
6.528
6.789
6.663
2 0.120
(1.7)
6.261
6.580
5.884
5.970
6.174
? 0.320
(5.1)
(DnP
No. x108/crn3
(~vn.3~'
TABLE 1. Estimates for mean intersections Cj,,), boundaries @,J, diameters @,J, surface areas (g,,),and for surface (SvJ and numerical (Nv,,)
densities for the exocrine cell nuclei
m
l'd
F
00
W
STEREOLOGICAL ESTIMATES
99
mean diameters are used to calculate the
surface areas and volumes of mean nuclei,
and combined with either the surface or volume densities to estimate the nuclear numerical densities, the other methods gave
mean values (Table 2, footnotes 2 and 3) that
are substantially different than those of
method 1: they were 169% and 178% larger
for the Wicksell method and 28% and 35%
larger for the Giger and Riedwyl method.
The choice of either a nuclear volume or
surface area for estimating numerical densities also influences the results (see footnotes
2 and 3 in Table 2). The mean surface areas
and surface densities of nuclei generate the
smallest standard deviations for the method
of model 1, whereas the mean volume and
volume density gave the best results for the
other methods. Notice in model 1 that the
s.d. of the numerical density is more than
twice as large for the volume based estimate
(Table 2, footnote 3) as it is for the one based
on surface areas (Table 2, footnote 2).
+I
Model 2: Nonspherical and nonconvex
nuclei
Estimates for the duct and endothelial nuclei are given in Tables 3 and 4. The data
"Uncorrected for shape" were generated with
model 1and those "Corrected for shape" with
model 2. Table 3 contains a surprising result.
When the nonspherical and nonconvex nuclei of the duct cells are treated as spheres
(model l), the mean number of profile intersections is essentially the same as when they
are corrected for shape (model 2). The difference between the estimates of the two models
is less than 1%.Although these results may
be fortuitous, the possibility exists that model
1 may also be applicable to certain classes of
nonspherical nuclei. Notice that the duct cell
nucleus had a mean surface area (corrected)
of 128.4 pm' (Table 3, footnote 14) and a
standard deviation of 6.5% of the mean. Its
numerical density was 1.640 x 1
0' nuclei/
cm3 (Table 3, footnote 9). The similarity between the surface area of a duct cell nucleus
(128.4 pm') and that of the exocrine cell
(128.8 pm2) is also striking. Here we have a
situation where two cells with the same embryonic origin have practically the same nuclear surface area.
Table 4 summarizes the results for the endothelial nuclei. The pattern of a similarity
between the nuclear intersections of the duct
cells for models 1 and 2 did not occur for the
endothelial nuclei; the results of the two
15.90
f 0.52
(3.3)
44.99
& 1.48
(3.3)
16.06
15.16
16.01
16.38
(3.3)
f 0.21
6.444
6.507
6.143
6.490
6.637
130.6
f 8.4
(6.5)
133.0
118.5
132.3
138.4
39.74
k 1.41
(3.5)
39.48
41.46
38.05
39.95
(8.1)
k 0.17
2.098
2.330
2.078
2.063
1.920
(3.3)
k 1.46
44.62
45.05
42.53
44.93
45.95
47.81
46.21
46.68
46.93
46.91
k 0.67
(1.4)
1
2
3
4
Mean
s.d.
(s.d. a s %
Pm
6.849
6.619
6.686
6.722
6.719
f 0.10
(1.4)
Irm
16.90
16.33
16.50
16.59
16.58
i- 0.24
(1.4)
(En)3
147.4
137.6
140.4
142.0
141.8
f 4.1
(2.9)
pm2
(SJ4
(SVn . 3 2
clm2
x1010/cm3
1.659
1.210
1.120
1.324
1.328
f 0.24
(17.8)
x10'0~cm3
82.19
54.25
57.80
65.86
65.03
f 12.43
(19.1)
m2
(sVn,d5
6.391
k 0.21
(3.3)
6.454
6.092
6.436
6.582
(6.5)
f 8.3
128.4
130.8
116.6
130.1
136.1
(10.9)
1.640
k 0.179
31.25
f 2.96
(9.5)
1.781
1.782
1.585
1.411
30.18
35.56
29.25
29.35
7.321
7.075
7.148
7.186
17.73
50.14
f 0.10
(1.4)
f 0.25
(1.4)
+ 0.72
(1.4)
7.183
w
Irm
18.06
17.46
17.67
17.73
(Gn)'3
51.11
49.40
49.90
50.17
(Bn)'2
162.1
(2.9)
f 4.7
(16.1)
f 6.43
39.98
(15.0)
i- 0.1223
0.8172
0.9855
0.7692
0.6980
0.8160
48.81
34.49
36.01
40.60
168.4
157.3
160.5
162.2
(Nvn 3c)'
NO.
x 1o8/~m3
(Nvn,lc)'
No.
x 108/cm3
flm2
(Sn)'4
Model 2
Corrected for nonspherical shape
(7,)"
densities for the endothelial cell nuclei
For footnotes, see Table 1. The correction factor for the endothelial cell nuclei (ken,,,) was 1.0558;see equation (23)
(Td1
Animal
(E,Y
(3.3)
f 0.52
15.77
15.92
15.03
15.88
16.24
(id,boundaries (ZJ,diameters (FJ,surface areas (SJ, and for surface (SvJ and numerical (NvJ
Model 1
Uncorrected for nonspherical shape
TABLE 4. Estimates for mean intersections
For footnotes, see Table 1.The correction factor for the duct cell nuclei (kdJ was 0.9795; see equation (23)
Mean
s.d.
(s.d. as %
of mean)
3
4
45.43
42.89
45.31
46.33
5
'd
F
101
STEREOLOGICAL ESTIMATES
models differed by 7%. The mean surface area
of the endothelial nuclei was 162.1 pm2 (Table 4, footnote 14), and the s.d. was 2.9% of
the mean. There were 0.8172 x lo8 endothelial nuclei/cm3 (Table 4, footnote 9).
Method 2
A mean nuclear surface area can be used
to estimate 1)the numerical densities of nuclei (and cells), and 2) the surface areas of
membrane compartments in average cells.
Thus far, we have seen that model 1 can
generate the most reproducible results for
the numerical densities of cells. However, can
model 1continue to give estimates with small
standard deviations when it is used to estimate organellar compartments in average
cells?
The surface area of the outer mitochondrial
membranes in a n average exocrine cellbased on the method of model 1-is given on
the left of Table 5. Notice that the mitochondrial to nuclear intersection ratio (Table 5 ,
footnote 3) varies only slightly from animal
to animal; the s.d. is only 3.7% of the mean.
The surface area of the outer mitochondrial
membrane was 674 pm2/average cell and the
s.d. was 6.5% of the mean (Table 5 , footnote
5). Usually, estimates for the surface area of
a compartment in a n average cell are calculated from the volume of a n average cell and
the surface density of the compartment
(Loud, 1968; Weibel et al., 1969; Bolender,
1974; Crapo et al., 1980; Mori and Christensen, 1980; Haies et al., 1981; Crapo et al.,
1982). Table 5 also contains two sets of results based on the volumes of average cells.
In contrast to the results of model 1(Table 5 ,
footnote 5 ) , the other two methods gave estimates (Table 5, footnote 8) for the outer mitochondrial membranes in a n average
exocrine cell that were 51%and 13% smaller,
but these mean values had 86% and 168%
larger standard deviations.
While a n estimate for the mitochondrial
membranes (Table 5 ) with a s.d. of 6.5% of
the mean seems attractive, it is more than
twice as large as the s.d. for the nuclear
membranes (Table 1, footnote 4). This was a
surprising result because there are 5.2 times
more mitochondrial than nuclear membranes in a n average cell, and, given the
similarity in sample size (i.e., equal numbers
of animals and micrographs), a smaller s.d.
was expected).
Can the 6.5% s.d. be explained by the
method of sampling? Tissue blocks had been
+I
-
0000
l nw
w
mm
mw
m
sp.
m
102
R.P. BOLENDER
taken from five regions of the pancreas (A to
E), spaced equally along its length from head
to tail (Bolender, 1974).To look for a n answer
to this question, 60 micrographs were collected from each of the 5 regions (for one
animal), and the surface area of the mitochondrial membranes in each region was estimated according to footnote 5 in Table 5. A
statistical analysis of the results indicated
that a n average exocrine cell in regions A, B,
and C of the pancreas had about 30% more
outer mitochondrial membranes than those
in regions D and E (P < 0.05). This tells us
that the exocrine cells can have different
amounts of mitochondria according to their
location in the pancreas, an observation reported earlier by Loud (1968)and Schmucker
et al. (1978) for hepatocytes in the liver lobule. It also suggests that the method of regional sampling may have been responsible
for part of the 6.5% s.d. The new areaweighted method of sampling (see Weibel,
1979) is designed specifically to deal with
such problems of organ heterogeneity, and it
seems likely that had this method been used
smaller statistical variations for the outer
mitochondrial membranes would have been
found.
Method 3
The surface area of the nuclear membranes/1O6 cells is compared to the surface
densities of the nuclear membranes in Table
6. The results show that the s.d. (expressed
as a percentage of the mean) can be reduced
from 5.2 to 3.0%for exocrine cells, 8.1to 6.5%
for duct cells, and 17.8 to 2.9% for endothelial
cells. Therefore, by relating the nuclear surface area to lo6 cells-instead of to a cm3 of
reference volume-the s.d. of the estimate (as
a percentage of the mean) is decreased by
73%for exocrine cells, 25% for duct cells, and
514% for endothelial cells.
Serial Reconstructions
The relative frequencies of the exocrine,
duct, and endothelial cells coming from the
serial reconstructions are given in Table 7.
The results are based on 649 reconstructed
nuclei for animal 1and 582 nuclei for animal
2. This estimate of nuclear frequencies was
used in option 2 of model 2 to calculate a
correction factor for the duct and endothelial
nuclei that were applied to the measurements of mean profile boundaries (see Fig.
3). The results of these calculations are given
TABLE 6. Comvarison of nuclear surface areasll0' cells and nuclear surface densities
Exocrine cells
Sn/106cells
pm2
Animal
x 108/106cells
1
2
3
4
Mean
s.d.
(s.d. as %
Duct cells
Endothelial cells
SVn,Sc
1*m2
x1 0 ~ ~ i c m ~
s,/106
SVn,Bc
Pm2
x 108/106cells
x 101°/cm3
1*m2
1.334
1.253
1.306
1.258
8.351
8.244
7.683
7.512
1.308
1.166
1.301
1.361
1.288
*(3.0)
0.039
7.947
f 0.410
(5.2)
1.284
2.098
1.621
1.328
k 0.083
k 0.170
f 0.047
k 0.240
(6.5)
(8.1)
(2.9)
(17.8)
1.684
1.573
1.605
1.622
2.330
2.078
2.063
1.920
1.659
1.210
1.120
1.324
of mean)
TABLE Z Comparison o f nuclear frequencies determined from serial reconstructions and models
1 and 2: Relative frequencies ofpancreatic nuclei
Serial reconstructions
% Duct
% Endothelial
cell nuclei
cell nuclei
cell nuclei
% Exocrine
Animal
68.90
73.19
Mean
s.d.
(s.d. as %
of mean
71.04
f 3.03
(4.27)
21.21
17.26
9.794
9.553
Model 1
% Exocrine
cell nuclei
69.35
72.52
72.03
72.83
19.24
9.673
71.68
f 2.80
f 0.170
f 1.59
(14.5)
(1.759)
Model 2
% Endothelial
cell nuclei
cell nuclei
% Duct
(2.2)
19.73
19.63
19.42
17.21
+
19.00
1.20
(6.2)
10.92
7.85
8.55
9.95
9.32
f 1.38
(14.8)
103
STEREOLOGICAL ESTIMATES
in Table 7 (expressed as relative frequencies)
in order that comparisons can be made with
the serial reconstructions. Agreement is
good. The mean values for the two sets of
estimates differ by 0.9% for exocrine nuclei,
1.3%for duct nuclei, and 3.8%for endothelial
nuclei.
DISCUSSION
Numerical Densities and Average Cell
Information From Surface Areas
The results for exocrine cells (see Tables 1,2,
5 ) demonstrate that estimates for numerical
densities and compartmental areas in average cells can be obtained with a smaller s.d.
when they are based on surface areas (equations 16 and 24). Mayhew (1981) has also reported that surface area references, for a given
size of a sample, offer a smaller coefficient of
variation than those based on volumes.
The surface area approach is successful in
lowering the statistical variance, because it
is based on the mean nuclear profile boundary that can be estimated with a s.d. as small
as 1.4% of the mean (Table 4, footnote 2).
When this nuclear boundary is used to estimate the surface area of the nucleus, the s.d.
increases to as little a s 2.9% of the mean
(Table 4,footnote 4). In contrast, the methods
of Wicksell (1925, 1926) and Giger and Riedwyl(1970) rely 1)on measurements of profile
diameters that, in a n experimental setting,
often depart from the assumed ellipsoid or circular shapes, and 2) on mean nuclear diameters that must be cubed to calculate the nuclear volumes. Consequently,
these estimates for numerical densities had
statistical variations that were often several
times larger than those based on mean nuclear surface areas.
Assumptions o f Models 1 and 2
Model 1 assumes that all the exocrine nuclei are spheres with equal surface areas, i.e.,
that they are a homogeneous population. Although this may be true in the living cells, a
brief survey of electron micrographs reveals
nuclear profiles ranging from circles and ellipses to moderately distorted shapes. In the
thin sections, therefore, the assumed homogeneous nuclei are sometimes being represented 1)by noncircular profiles that are the
result of section compression, and 2) by nuclei that have had their shape altered by the
fixation and embedding procedures.
When a section is compressed, the diameter of a circular profile is thought to remain
intact as the major axis of a n ellipse and the
ratio of the major to minor axes of the ellipse
serve as a quantitative measure of the
compression (Loud, 1968; Weibel, 1979).
When a circular profile is compressed into a n
ellipse, however, the length of the original
boundary is reduced (see Weibel, 1979, p. 150)
as well as the surface area of a mean sphere
that would be generated by model 1. When
the percentage of section compression (x) is
plotted against the percentage of underestimate of surface area (y),the following expression is obtained:
y = d(1.0
+ O.O05x),
(28)
which indicates that the loss of surface area
is roughly equal to the degree of compression. Since the profiles in the study had a
mean compression of about lo%, a 10% overestimate for the numerical densities of nuclei
in the blocks would be expected.
Nuclei that have their spherical shapes altered by the preparative procedures are expected to have a smaller mean diameter and
a smaller probability of being included in a
section. The model in Figure 5 considers the
consequences of not conforming to the assumption that all the nuclei have the same
likelihood of being sectioned. Heterogeneity
is introduced into the reference space (the
cubes) by decreasing the diameters of the
alpha spheres (black) while the beta spheres
(white) are held constant. The results of introducing this error are expressed as a ratio
of observed and predicted surface areas. As
the alpha spheres become smaller (cubes B
to F), their likelihood of being sectioned decreases, which accounts for the progressive
increase in the error. Notice that a moderate
amount of heterogeneity in the sizes of the
spheres produces only a small error. Cube D,
where the ratio of the mean diameters of the
alpha and beta spheres is 0.75, for example,
introduces only a 3% underestimate for the
mean surface area of the six spheres. Consequently, slight alterations in the shapes of
the nuclei are not expected to be a major
source of error.
What if all the nuclei do not have the same
surface area? In Figure 5 , we know that a
mean profile boundary of the six spheres in
a given cube will be influenced by the separate diameters of the alpha and beta spheres
and their individual frequencies. In cube C,
for example, the combined s heres have a
surface density of 15.46 cm /cube, and ex-
zp
104
R.P. BOLENDER
e
A
2
100
09
10
D,,
08
07
6,,(u = 1 O l o O 5 c m . ~=
06
05
1 Ocm)
Fig. 5. The effects of differences in the diameters of
the spheres on the estimates for the mean caliper diameter (D). The model includes six cubes containing two
subpopulations of spheres, alpha (black) and beta (white).
Both the alpha and beta spheres in cube A have the
same diameters and therefore show the same probability
of being sectioned (the probability is expressed as a ratio
of the diameters). In cubes B to F, however, the alpha
spheres become less likely to be sectioned as their diameters decrease. In cube F, for example, where the alpha
diameter is 50%that of the beta, the alpha spheres will
be expected to appear in a section only half as often as
the beta spheres. The error produced by this sampling
probability is expressed as a ratio of the observed and
expected values. It is less than 5% for four of the six
cubes. The mean caliper diameters of the alpha and beta
spheres in each cube reflect the skes and frequencies of
the spheres (D = [C D,, +-CD,]/Gj:D(A) = l.0 cm; D(B) =
0.95g-n; D(C) = 0.9 cm; D(D) = 0.89 cm; D(Ej = 0.8 cm;
and D(F) = 0.75 cm.
pected values for the mean profile boundary
of 1.48 cm, a mean diameter of 0.9 cm, and a
mean surface area of 2.577 cm2. Dividing the
surface density by the mean surface area
gives the predicted value of six spheres in
the cube. By introducing the error produced
by the smaller probability of sectioning the
alpha spheres, the observed value would be
6.06 spherestcube, which represents a n error
of only 1%. Given the examples developed
with Figure 5, it appears likely that a mean
profile boundary taken from a slightly heterogeneous and distorted population of nuclei would still lead to a reasonably accurate
estimate for a nuclear numerical density. The
largest source of error seems to be the section
compression, which can be estimated and
corrected (see Weibel, 1979, p. 150).
Model 2 included two options for estimating the mean surface area and numerical
density of nonspherical nuclei. Option 1used
the numerical density of the spheroidal exocrine nuclei and the relative frequencies of
the exocrine, duct, and endothelial cells. Option 2 used a correction factor to convert the
mean profile boundary of the nonspherical
nuclei to that of a sphere of equivalent surface area. The mean profile boundaries from
the duct cell nuclei provided a most unexpected result (Table 3, footnotes 2 and 12).
The uncorrected and corrected estimates for
the mean diameter differed by less than 1%.
While a n explanation for such a result can
not yet be given, it is still possible to speculate that equation (12) may be appropriate
for both spheres and certain classes of nonspherical nuclei-without a correction factor.
Sources of Statistical Variance in
Stereological Estimates
The standard deviations in Tables 1,3, and
4 are relatively small for the mean profile
boundaries and surface areas of nuclei
(range: 1.4 to 6.5% of the mean), but several
times larger for the surface and numerical
densities (range: 3.5 to 19.1% of the mean).
The surface densities, which are used with
the mean nuclear surface areas to estimate
the numerical densities, are responsible for
the larger standard deviations. Why are the
surface densities so variable?
A network of stereological information is
illustrated in Figure 6, which extends from
the organellar contents of a n average exocrine cell to the total number of cells in the
pancreas. Reference spaces are identified for
each set of information along with the equations that were used to make the calculations. Data are included from four animals
along with means and standard deviations
(expressed a s a percentage of the mean).
The important thing to notice in this figure
is that for each of the four animals, the
cubic centimeter of reference volume always contains a different number of cells.
I n a cubic centimeter of exocrine cells, for
105
STEREOLOGICALESTIMATES
Total
pancreatic
volume
Nn
,Nn
=
Nvn ' "pan
ID8
pancreas1
5 340
8 724
6 676
6 694
crn3 of
pancreas
Nv,
=
INn .
cm3 of
exocrine
cells
Sv, i S,
NV"
CmJl
IN,
10'
5 621
5 778
5 384
6 375
=
'
svn S"
10' cm'/
6 833
7 128
6 383
7 833
Average
exocrine
cell
S" =
T(D.)*
I V 2 l
133 4
125.3
130.6
125 8
6.859
5.789
7.044
128.8
( % 20.35%)
( 2 7.299%)
( 2 8.639%)
( ? 3.029%)
Fig. 6. Numerical densities of exocrine cells. The frequencies of the cells are shown at several reference points
along a network of information that extends from an
average cell to the total pancreas. Estimates are included for four animals and standard deviations are expressed as a percentage of the mean. The point to be
made by the figure is that except for the average exocrine cell estimate, which uses a single cell, all the
remaining estimates for the mean values of the numerical densities are based on unequal numbers of cells. If a
standard number of cells is used (see Table 61, then the
standard deviation for the surface density (or numerical
density) estimate falls from 7.3% (column 2) or 8.6%
(column 3) of the mean to 3.0%.Notice that this is the
same statistical advantage gained by using the average
cell. For details, see text.
example, the surface areas of the nuclear
membranes came from as many a s 7.833 x
10' cells (animal 4) to a s few as 6.383 x 10'
(animal 3)-a difference of 23% or 144.7 million cells. This tells us that a t least part of
the experimental error associated with the
estimates for surface densities can be attributed to a .sampling procedure that accepts data coming from a different number
of cells for each animal. Method 3 modifies
this sampling procedure by always collecting information from the same number of
cells (lo6 cells/animal), and the result is a
substantial decrease in the variance (see
Table 6). This method of relating results to
lo6 cells is widely used in biochemical studies, and, if extended to stereological data,
can be expected to facilitate the task of correlating cellular function with structure
(Bolender, 1981b, 1982).
One can certainly expect that "biological
variation" contributes to the differences we
find in the number of cell/cm3 from one
animal to the next, but it seems likely that
many of the variations we measure actually
come from artifacts, which are produced by
the methods t h a t must be used to gain access to the information contained in thin
sections. In attempting to find correlations
of structure to function, we have found it
especially helpful to remember that thin
sections are not always representative of
the blocks (section thickness and compression), t h a t blocks are not always representative of the living tissue (shrinkage and
swelling), and that blocks of living tissue
are not always representative of the nonhomogeneous tissues and organs.
CONCLUDING REMARKS
The goal of this study was to identify methods that can decrease the variance of stereological estimates. The strategy of the
approach consisted of looking for ways to optimize the interactions between stereological
equations and experimental data. By defining the best form of experimental data as the
one that has the smallest population variance, and then writing stereological equa-
106
R.P. BOLENDER
tions that could take advantage of this
optimal starting point, stereological estimates for biological structures could be obtained with standard deviations as small as
1.4%of the mean.
ACKNOWLEDGMENTS
The assistance of S. Anderson, S. Allen,
and D. Ringer is gratefully acknowledged. A
referee, Dr. J.F. Bertram, and M.P. Larsen
provided a number of constructive comments
and suggestions, and they are sincerely
thanked. This work was supported by
USPHS grants GM-22757 and GM-29853
from the National Institutes of Health.
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