Journal of Chemical Neuroanatomy 20 (2000) 93 – 114
www.elsevier.com/locate/jchemneu
Recommendations for straightforward and rigorous methods of
counting neurons based on a computer simulation approach
Christoph Schmitz a,*, Patrick R. Hof b
b
a
Department of Anatomy and Cell Biology, RWTH Uni6ersity of Aachen, Pauwelsstrasse/Wendlingweg 2, 52057 Aachen, Germany
Kastor Neurobiology of Aging Laboratories and Fishberg Research Center for Neurobiology, Mount Sinai School of Medicine, Box 1639,
One Gusta6e L. Le6y Place, New York, NY 10029, USA
Abstract
Any investigation of the total number of neurons in a given brain region must first address the following questions. What is
the best method for estimating the total number of neurons? What are the validity and the expected precision of the obtained
data? What precision must the estimates attain with respect to the scientific question? In the present study, these questions were
addressed using a computer simulation. Virtual brain regions with various spatial distributions of virtual neurons were modeled.
The total numbers of virtual neurons in the modeled brain regions were repeatedly estimated by simulation of modern
design-based stereology, either by using the ‘fractionator’ method or by the established method based on the product of estimated
neuron density and estimated volume of the reference space. We show that estimates of total numbers of neurons obtained using
the fractionator are from a statistical and economical standpoint more efficient than corresponding estimates obtained using the
density/volume procedure. Furthermore, the use of two simple prediction methods (one for homogeneous and the other for
clustered neuron distributions) permits satisfactory predictions about the variation of presumably any estimates of total numbers
of neurons obtained using the fractionator. Finally, we show that assessing the reliability of estimates of mean total neuronal
numbers using the ratio between the mean of the squared coefficients of error of the estimates and the squared coefficient of
variation of the estimated total neuronal numbers, a frequently employed method in stereological studies, is neither useful nor
informative. The present results may constitute a new set of recommendations for the rigorous usage of design-based stereology.
In particular, we strongly recommend counting considerably more neurons than is currently done in the literature when estimating
total neuronal numbers using design-based stereology. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Cell count; Fractionator; Morphometry; Stereology
1. Introduction
The proper assessment of the total number of neurons in a given brain region depends on the accuracy of
the method used as an estimator, on the validity and
the expected precision of the obtained data, and on the
precision that the estimates must attain to address the
scientific question. Unfortunately, none of these issues
has an easy solution.
First, in modern, design-based stereology there are
two methods available for estimating total numbers of
neurons, namely the so-called ‘fractionator’ (Gun* Corresponding author. Tel.: +49-241-8089548; fax: + 49-2418888431.
E-mail address: [email protected] (C.
Schmitz).
dersen, 1986) and the so-called ‘Vref × NV’ method
(West and Gundersen, 1990). Using the fractionator,
neurons in a defined, systematically and randomly sampled part of the entire brain region of interest are
counted, and the total neuronal number is estimated by
multiplying the number of counted neurons by the
reciprocal value of the sampling probability (henceforth
referred to as ‘fractionator estimates’). Using the Vref ×
NV method, an estimate of the total number of neurons
is obtained by multiplying the estimated total volume
of this brain region (6ref) by the estimated mean neuron
density (nV) in a systematically and randomly sampled
part of the entire region (henceforth referred to as
‘Vref × NV estimates’; Vref and NV are real values,
whereas 6ref and nV are estimated values). The choice
between these methods may be based on both statistical
and economical efficiency (the term ‘statistical effi-
0891-0618/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 8 9 1 - 0 6 1 8 ( 0 0 ) 0 0 0 6 6 - 1
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
94
ciency’ refers to the precision of an estimator, which is
more precise as its variation is smaller (Bronshtein and
Semendyayev, 1985). In contrast, the term ‘economical
efficiency’ refers to the amount of work needed to
guarantee a given precision of the estimates (for details
see Schmitz, 1998). However, a systematic comparison
between the fractionator and the Vref ×NV method with
respect to statistical and economical efficiency has not
been performed.
Second, for the Vref ×NV method there are three
different methods available for predicting the precision
of the obtained estimates (Table 1; henceforth referred
to as ‘prediction methods’). Moreover, 16 different
prediction methods were reported in the literature
which have been or might be applied to assess the
precision of fractionator estimates (Table 1; the early,
preliminary method given by Gundersen, 1986; Equation (2.11) is not considered here). Most of these prediction methods (V1, V2, F1 – F4, F6, F8 – F15) are
based on complex theoretical statistics. Others, such as
Table 1
Methods for predicting the precision of estimated total neuronal
numbers obtained using either the Vref×NV method (V1–V3) or the
fractionator (F1–F15), available in the literature
Method
Source
V1
V2
V3
F1
Table 3 in West and Gundersen (1990)
Table 4 in Geinisman et al. (1996)
Appendix A in Simic et al. (1997)
Equation (6) in Gundersen and Jensen (1987); cf.
also Table 5 in West et al. (1991)
Equation 20 in Gundersen and Jensen (1987)
Equation 20 in Cruz-Orive (1990)
Discussion in Thioulouse et al. (1993)
Equation A.2 in Larsen (1998)
Chapter 4 in Scheaffer et al. (1996)
Equation A.4 in Larsen (1998); cf. also Appendix in
Glaser and Wilson (1998); Appendix in Glaser and
Wilson (1999)
Figure 6 [‘P6’] in Schmitz (1998); Equation A.5 in
Glaser and Wilson (1998); Equation A.5 in Glaser
and Wilson (1999); Equation 24 in Nyengaard
(1999)
‘Explicit nugget formula’ (Equation 3.12) with m =0
in Cruz-Orive (1999)
‘Explicit nugget formula’ (Equation 3.12) with m =1
in Cruz-Orive (1999)
‘Implicit nugget formula’ with m= 0 (Subsection 3.4)
in Cruz-Orive (1999)
‘Implicit nugget formula’ with m= 1 (Subsection 3.4)
in Cruz-Orive (1999)
Use of Equation 12; i.e. m= 0 in Gundersen et al.
(1999); Table 2 in West et al. (1996)
Use of Equation 13; i.e. m= 1 in Gundersen et al.
(1999)
Use of Equation 14; i.e. m= 0 in Gundersen et al.
(1999)
Use of Equation 15; i.e. m= 1 in Gundersen et al.
(1999)
F2
F3
F4
F5
F6
F7
F8
F9
F10
F11
F12
F13
F14
F15
F7, have been developed mainly by computer simulation (Schmitz, 1998; Glaser and Wilson, 1998, 1999) or
were applied without presentation of the theoretical
background (V3, F5). In three recent reports some of
these prediction methods have been compared using
computer simulation (Schmitz, 1998; Glaser and
Wilson, 1998, 1999). However, a comprehensive comparison of all these prediction methods is not available
in the literature.
Third, to design properly an experiment, important
quantities to determine are, for example, the minimal
difference one wants to detect between the means of
two populations under comparison on the one hand,
and the biological variation and the precision of the
estimates on the other. A recent attempt to find a
solution to this problem (Geinisman et al., 1996) is
demonstratedly unsatisfactory (Schmitz et al., 1999b).
Another, frequently used approach to demonstrate the
reliability of an estimated mean total number of neurons is to show that the mean of the squared coefficients of error of the estimates as predicted with one of
the prediction methods summarized in Table 1 is less
than half of the squared coefficient of variation of the
estimated total neuronal numbers (henceforth referred
to as ‘CE2/OCV2 approach’; for recent examples see
Geinisman et al., 1996; Begega et al., 1999; Korbo and
West, 2000; among others). This approach is based on
a 20-year-old scheme designed to optimize the sampling
efficiency of stereological studies in biology (Gundersen
and Østerby, 1981) and has often been described in
guidelines how to carry out stereological studies (for
recent examples see West, 1993; Larsen, 1998; Nyengaard, 1999; among others). On the other hand, this
approach has been often criticized (Schmitz, 1997,
2000) based on the relevant statistical literature
(Nicholson, 1978; Searle, 1987). An in-depth evaluation
of the power of the CE2/OCV2 approach is not
available.
This study is aimed to clarify this unsettled situation.
It is not intended to provide a comprehensive analytical
approach for solving the mentioned problems. If at all
possible, an analytical approach would require detailed
theoretical–statistical considerations and would therefore be difficult to understand by neuroscientists not
familiar with the relevant statistics but interested in
quantitative neuroanatomy. Rather, this study is intended to provide simple solutions for the mentioned
problems by analyzing the results of repeated estimates
of the total number of neurons in the same brain
region, or by repeated estimates of mean total neuronal
numbers of populations of individuals. For methodological reasons this cannot be achieved by biological
experiments (Cruz-Orive, 1994; Schmitz, 1998). Therefore, we have addressed this issue by a computer simulation approach. The description of the computer
simulation is presented in parallel to descriptions of real
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
studies involving estimates of total numbers of neurons.
We show that estimates of total neuronal numbers
obtained using the fractionator are from a statistical
and economical standpoint more efficient than corresponding estimates obtained using the Vref × NV
method. Furthermore, the use of two simple prediction
methods (one for homogeneous and the other for clustered neuron distributions) permits satisfactory predictions about the variation of presumably any estimates
of total neuronal numbers obtained using the fractionator. Finally, we show that assessing the reliability of
estimates of mean total neuronal numbers using the
CE2/OCV2 approach is neither useful nor informative.
The presented results may constitute a new set of
recommendations for the rigorous usage of designbased stereology.
We are aware that many scientists interested in quantitative neuroanatomy are not familiar with details of
computer simulations. On the other hand, many explanations are necessary to facilitate repeating the work by
other laboratories. Therefore, Sections 2 and 3 of this
study are presented in the following manner. Readers
not familiar with stereological nomenclature or interested only in a fast overview should only read the text
given in normal fonts. Readers interested in details of
the work should read the entire text in the presented
order.
2. Materials and methods
2.1. Experiment 1
Experiment 1 was intended to investigate the influence of the shape of the reference space, of the spatial
distribution of neurons within this reference space and
of stereological sampling on the variation of estimated
total neuronal numbers. This was achieved by modeling
various virtual brain regions (VBR) with different virtual reference spaces (VRS) and different neuron distributions within these reference spaces, and by modeling
estimates of the total neuronal numbers of these VBR
with different stereological sampling schemes.
2.1.1. Modeling of 6irtual brain regions
Sixteen different VBR were modeled and consisted of
virtual neurons (VN) in VRS. Details of these VBR are
summarized in Table 2; schematic illustrations of the
VRS are shown in Fig. 1. VRSi might be interpreted as
a virtual rat external globus pallidus, and VRSii as a
virtual rat striatum considering only the striosomes.
For VBR1 –VBR4 and VBR9 – VBR12, the VRS was a
sphere with radius r =850 mm and a volume of 6= 2.57
mm3. This volume was similar to the estimated volume
of the rat external globus pallidus (Oorschot, 1996).
95
Table 2
Details on runs A–Y of the simulations carried out in Experiment 1a
Run
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
VBR
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
9
10
11
12
13
14
15
16
VRS
i
i
i
i
ii
ii
ii
ii
i
i
i
i
ii
ii
ii
ii
i
i
i
i
ii
ii
ii
ii
c of VN
500
500
500
500
500
500
500
500
50 000
50 000
50 000
50 000
50 000
50 000
50 000
50 000
50 000
50 000
50 000
50 000
50 000
50 000
50 000
50 000
SPP
a
d3
d6
z3
a
d3
d6
z3
a
d3
d6
z3
a
d3
d6
z3
a
d3
d6
z3
a
d3
d6
z3
VSS
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
a
VBR, virtual brain region; VRS, virtual reference space; VN,
virtual neurons; SPP, spatial point process; VSS, virtual sampling
scheme; i, pallidal VRS; ii, striatal VRS (for 3-D sketchs of VRSi and
VRSii see Fig. 1); a, homogeneous SPP; d3, centripetal SPP; d6,
centrifugal SPP; z3, clustered SPP (for schematic representations of
SPPa, SPPd3, SPPd6 and SPPz3 see Fig. 2). Details on VSS1 to VSS3
are given in Table 3.
For VBR5 –VBR8 and VBR13 –VBR16, the VRS consisted of 1000 small spheres with r= 85 mm, which were
arranged as a cube (VRSii). Hence, the volume of VRSii
was also 2.57 mm3.
Within these VRS either a total number of 500 VN
was modeled (VBR1 –VBR8), or a total number of
50 000 VN (VBR9 –VBR16). VN were modeled as
points and were arranged in the VRS according to four
different so-called ‘spatial point processes’ (SPP) as
schematically shown in Fig. 2. SPPa might be interpreted as a homogeneous distribution of VN, SPPd3 as
a centripetal VN distribution, SPPd6 as a centrifugal
VN distribution and SPPz3 as a clustered VN
distribution.
The SPP applied here were the same as described in
detail as SPPa, SPPd3, SPPd6 and SPPz3 in Schmitz
(1998). SPPa was a ‘homogeneous Poisson process’,
which corresponded to complete spatial randomness.
SPPd3 and SPPd6 were ‘inhomogeneous Poisson processes’, in which the point density was allowed to vary
as a function of distance from the center of the reference space (radius). The density function of SPPd3
decreased as a 6th degree polynomial function of radius, whereas the density function of SPPd6 increased as
96
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
Fig. 1. 3-D sketch of VRS used to model VBR. VRSi is a sphere with
radius r= 850 mm and volume 6= 2.57 mm3. VRSii consists of 1000
small spheres with radius r =85 mm, which are arranged as a cube.
The volume of VRSii is also 2.57 mm3. VRSi might be interpreted as
a virtual rat external globus pallidus and VRSii as a virtual rat
striatum considering only the striosomes.
a 6th degree polynomial function of radius. SPPz3 was
a ‘Poisson cluster process’, based on 250 so-called
‘parent points’ (see Schmitz, 1998, for details). Detailed
approaches to SPP as well as techniques for simulating
them may be found in Cox and Isham, 1980; Diggle,
1983; Ripley, 1987; among others).
2.1.2. Modeling of estimates of total neuronal numbers
Estimates of total neuronal numbers were modeled as
recently described in detail (Schmitz, 1998). All steps
carried out in real estimates of total neuronal numbers
of brain regions of interest using the fractionator or the
Fig. 2. Spatial distributions of VN in VRS used to model VBR. VN
are modeled as points and are arranged in the VRS shown in Fig. 1
according to so-called SPP. The figure shows modeled 100 mm thick
sections through the center of VBR with either pallidal VRS [VRSi]
or striatal VRS [VRSii], with either homogeneous [SPPa], centripetal
[SPPd3], centrifugal [SPPd6] or clustered [SPPz3] VN distribution. The
SPP applied here are the same as described as SPPa, SPPd3, SPPd6 and
SPPz3 in Schmitz (1998).
Fig. 3. Schematic summary of modeling estimates of total neuronal
numbers using either the fractionator or the Vref ×NV method. (a)
Schematic representation of S= 13 parallel, systematically and randomly sampled sections of a VBR with pallidal VRS. (b) Rectangular
lattice with side lengths sl, systematically and randomly placed on the
upper surface of a section of the VBR (shown enlarged for better
demonstration of details). This lattice determines the positions of
cubic counting spaces (dark squares) for counting neurons. Such a
lattice is also used to estimate the surface area of this section by
counting the intersections of the lattice situated within the VBR
(arrows), and is used to estimate the boundary length of this section
by counting the intersections of the lattice with the boundaries of the
section (arrowheads). For clarity only one lattice is shown, although
in the computer simulation three different lattices were used for
counting neurons, estimating surface areas and boundary lengths. (c)
Cubic counting spaces with edge e, systematically and randomly
placed in regular intervals sl in the central part of the section
thickness t. VN situated within the counting spaces are counted.
Vref × NV method were modeled, as illustrated in Fig. 3.
Using the algorithm provided by Cruz-Orive (1997),
the VBR were centered on the origin of a Cartesian
coordinate system (V) and dissected to a total number
of S parallel, isotropic uniform random (IUR) sections
with section thickness t and normal vectors parallel to
the z-axis of V (Fig. 3a).
For modeling fractionator estimates, rectangular lattices with uniform side length slN were placed in a
systematic–random manner on the upper surface of the
sections (Fig. 3b). These lattices determined the positions of cubic counting spaces with edge e in the central
part of the section thickness (Fig. 3c). All VN situated
within the counting spaces (Q−; for the formal definition of the mnemonic − in the context of estimates of
total neuronal numbers see Gundersen, 1986) were
counted. Estimates of total numbers of VN (nF) were
calculated as shown in Eq. (1) (Gundersen, 1986):
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
% Q−(slN)2t
nF =
sc=
(1)
e3
For modeling Vref ×NV estimates, the same rectangular lattices for placing cubic counting spaces in the central
part of the section thickness were used as if applying the
fractionator. Also, all VN situated within the counting
spaces (Q−) were counted. From the number of counted
VN ( Q−) and the number of counting spaces used
( F), estimates of the mean VN density (nV) were
calculated as shown in Eq. (2) (West and Gundersen,
1990):
% Q−
(2)
nV =
e %F
3
For modeling estimates of the average surface area of
the sections (A( ), a second set of rectangular lattices with
uniform side length slA was placed in a systematic-random manner on the upper surface of the sections (Fig.
3b). All intersections of the lattices situated within the
VRS were counted (P; arrows in Fig. 3b). Estimates of
A( (i.e. ā) were calculated as shown in Eq. (3) (Gundersen
and Jensen, 1987):
% P(slA)2
a=
(3)
S
Volume estimates (6ref) were calculated according to
the Cavalieri (1635) principle as shown in Eq. (4)
(Gundersen and Jensen, 1987):
6ref =a× t×S
(4)
Estimates of total numbers of VN (nV × N) were calculated as shown in Eq. (5) (West and Gundersen, 1990):
nV × N =6ref × nV
(5)
To obtain estimates of the average total boundary
length of the sections (B( ), a third set of rectangular
lattices with uniform side length slB was placed in a
systematic–random manner on the upper surface of the
sections (Fig. 3b). All intersections of the lattices with the
boundaries of the sections (IS) were counted. Estimates
of B (i.e. b) were calculated according to the Buffon’s
principle (Buffon, 1777) as shown in Eq. (6) (Cruz-Orive,
1997):
% IS ×0.25×p × slB
b=
S
(6)
From the estimated average surface area (a) and the
estimated average total boundary length of the sections
(b), estimates of the shape coefficient (SC=B/
A) were
calculated as shown in Eq. (7) (cf. Roberts et al., 1994):
97
b
(7)
a
The estimates of total VN numbers using the fractionator or the Vref × NV method were carried out using three
different virtual sampling schemes (VSS), resulting in
different numbers of counted neurons and therefore
different variation of the estimates. Details on these VSS
are provided in Table 3.
VSS1 and VSS2 were intended to simulate sampling
approximately (: ) 150 VN with :150 counting spaces
when estimating the total VN numbers of either VBR1 –
VBR8 (VSS1) or of VBR9 –VBR16 (VSS2). By contrast,
VSS3 was intended to simulate sampling : 750 VN with
:750 counting spaces when estimating the total VN
numbers of VBR9 –VBR16. The average numbers of
points counted for estimating the volume of the corresponding reference spaces were 150 (VSS1 and VSS2) or
750 (VSS3). The average numbers of IS counted for
determining the shape coefficient of the corresponding
reference spaces were 24 (VRSi; VSS1 and VSS2), 235
(VRSii; VSS1 and VSS2), 110 (VRSi; VSS3) and 1093
(VRSii; VSS3). Using these VSS, altogether 24 runs of the
computer simulation were performed (A–Y; see Table 2).
Each run consisted of 1000 repetitions of a simulated
stereological procedure, resulting in 1000 fractionator
estimates of the total number of VN of the investigated
VBR as well as in 1000 Vref × NV estimates of the total
VN number of this VBR.
This was equal to that what has been described by
Cruz-Orive (1994) as a ‘rewinding of a video movie of
the splitting process and repeating the sampling procedure again with fresh random numbers to select the
different sampling units’. The position of the investigated VBR in space as well as the positions of the
lattices on the sections were changed for each repetition
of the simulation, whereas the distributional pattern of
the VN in the VRS was changed after each 50 repetitions. This was carried out to prevent dependence of
the obtained results on a single realization of the apTable 3
Details on the VSS useda
VSS1
t (mm)
S
e (mm)
slN (mm)
slA (mm)
slB (mm)
258
7–11
172.67
258
257
1500
VSS2
258
7–11
37.2
258
257
1500
VSS3
151
11–19
37.2
151
150
550
a
t, Section thickness; S, number of sections; e, edge of the cubic
counting spaces; slN, side length of the rectangular lattices for placing
the counting spaces; slA, side length of the rectangular lattices for
estimating surface areas of sections of VBR; slB, side length of the
rectangular lattices for estimating boundary lengths of sections of
VBR.
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
98
plied SPP. Therefore, 20 distributional patterns of VN
were derived from each type of SPP, and a total of
24× 2×1000= 48 000 estimated total numbers of VN
was obtained. For each estimated total number of VN,
the corresponding predicted coefficients of error
[CEpred(n)] were calculated using all methods shown in
Table 1 (a note on the use of method F8 – F11 is given
in Appendix A).
R3 =
CEemp(nV × N)
CEemp(nF)
(14)
The mean of the 1000 squared predicted coefficients
of error of either nV × N or nF, calculated separately for
each of the applied prediction methods (herein, the
mean of these data is called ‘meanE1’):
1000
% ([CEpred(n)]2)Run
2
meanE1{[CEpred(n)] }=
2.1.3. Analysis of the estimates
The results of Experiment 1 were analyzed for the
relationship between the real variation of estimated
total VN numbers and the predictions of this variation
obtained using the prediction methods summarized in
Table 1. This was carried out by calculating ratio R4
shown in Formula (16). R4 =1 indicates that the corresponding prediction method resulted in an exact mean
prediction of the variation of the estimates. R4 \ 1
indicates that the variation of the estimated total VN
numbers was overestimated, whereas R4 B1 indicates
that the variation of the estimated total VN numbers
was underestimated.
The entire analysis of the results of Experiment 1
comprised calculation of the following variables:
Mean, S.D. and empirically estimated coefficient of
error of the 1000 estimates of Vref and of the 1000
estimates of NV obtained using the Vref ×NV method:
S.D.(6ref)
mean(6ref)
(8)
S.D.(nV)
mean(nV)
(9)
CEemp(6ref)=
CEemp(nV)=
The ratio between CEemp(6ref) and CEemp(nV):
R1 =
CEemp(6ref)
CEemp(nV)
(10)
Mean, S.D. and empirically estimated coefficient of
error of the 1000 estimated total numbers of VN obtained using the Vref ×NV method:
CEemp(nV × N)=
S.D.(nV × N)
mean(nV × N)
(11)
The ratio between [CEemp(6ref) + CEemp(nV)] and
CEemp(nV × N):
R2 =
[CEemp(6ref) + CEemp(nV)]
CEemp(nV × N)
(12)
Mean, S.D. and empirically estimated coefficient of
error of the 1000 estimated total numbers of VN obtained using the fractionator:
CEemp(nF)=
S.D.(nF)
mean(nF)
The ratio between CEemp(nV × N) and CEemp(nF):
(13)
Run = 1
1000
(15)
The ratio between meanE1{[CEpred(n)]2} and
[CEemp(n)]2, also calculated separately for each of the
applied prediction methods:
R4 =
meanE1{[CEpred(n)]2}
[CEemp(n)]2
(16)
2.2. Experiment 2
Experiment 2 was intended to investigate the influence of both stereological sampling and biological variability (as well as the relationship between these
variables) on the observed interindividual variation of
estimated total numbers of neurons of a sample of
individuals.
2.2.1. Modeling of populations of 6irtual brain regions
Five different virtual populations (P1 –P5) of 15 000
VBR each were modeled, differing in the VN distributions within the VRS and the frequency distributions of
the total VN numbers of the 15 000 VBR each (for
illustration see Table 4 and Fig. 4). The mean total
number of VN per VBR was approximately 50 000 for
each population.
VBR in populations P1 –P4 consisted of homogeneously distributed VN in a spherical VRS with radius
r=850 mm (according to VRSi − SPPa in Fig. 2),
whereas VBR in population P5 consisted of clustered
VN in a spherical VRS with the same radius (according
to VRSi − SPPz3 in Fig. 2).
The total VN numbers of the 15 000 VBR each in
P1 –P5 (i.e. the frequency distributions of the total VN
number of P1 –P5; FDP1 –FDP5) were obtained by using
the integer values of 15 000 pseudorandom numbers
each generated with the pseudorandom number generator implemented in MS Excel for Windows 95, version
7.0. This pseudorandom number generator allows the
generation of a preselected number of pseudorandom
numbers (here, 1000, 14 000 or 15 000 as given in detail
in Table 4) according to a preselected distribution
(here, ‘standard’), a preselected mean (here, each time
50 000) and a preselected S.D. (here, 50, 2000, 6000,
7500 or 23 500 as given in detail in Table 4). FDP1 and
FDP3 were generated by one realization of this pseudorandom number generator, and FDP2 and FDP4 by
two realizations each of this pseudorandom number
generator. FDP5 was identical to FDP1.
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
99
Table 4
Details on the virtual populations [P1–P5] of VBR modeled in Experiment 2a
P1
c of VBR
VRS
SPP
c of PRN (first realization)
Preselected mean (first realization)
Preselected S.D. (first realization)
c of PRN (second realization)
Preselected mean (second realization)
Preselected S.D. (second realization)
c of VN-minimum
c of VN-maximum
c of VN-mean
c of VN-S.D.
c of VN-CV
15 000
i
a
15 000
50 000
2000
–
–
–
42 487
57 888
49 973
2012
0.040
P2
P3
15 000
i
a
14 000
50 000
50
1000
50 000
7500
28 314
79 843
50 023
1977
0.040
P4
15 000
i
a
15 000
50 000
6,000
–
–
–
28 179
75 146
50 020
5999
0.120
P5
15 000
i
a
14 000
50 000
50
1000
50 000
23 500
1000
149 912
50 012
5981
0.120
15 000
i
z3
†
†
†
–
–
–
42 487
57 888
49 973
2012
0.040
a
VBR, virtual brain region; VRS, virtual reference space; SPP, spatial point process; PRN, pseudorandom numbers generated with the
pseudorandom number generator implemented in MS Excel for Windows 95, version 7.0. Preselected mean, preselected mean of PRN when
generating PRN according to a ‘standard’ distribution with this pseudorandom number generator. Preselected S.D., preselected standard deviation
of PRN when generating PRN according to a ‘standard’ distribution with this pseudorandom number generator. †, No use of the pseudorandom
number generator, since the frequency distribution of the total VN number of P5 was identical to the frequency distribution of the total VN
number of P1. VN, virtual neurons; i, pallidal VRS (for a 3-D sketch of VRSi see Fig. 1); a, homogeneous SPP; z3, clustered SPP (for schematic
f
representations of SPPa and SPPz3 see Fig. 2).
% ([CEpred(n)]2)VBR
2.2.2. Modeling of estimates of mean total neuronal
numbers
All steps carried out in real estimates of the mean
total neuronal number of a sample of individuals selected from a population were modeled. A number of
VBR was selected from the investigated population,
and the total VN numbers of the selected VBR were
estimated using the fractionator. With respect to the
investigated populations, the numbers of selected VBR
and the VSS applied, 18 different virtual stereological
studies (VSTST) were carried out as summarized in
Table 5 [c 1 to c18].
Each VSTST consisted of the following steps. First,
either f =6 or f =12 VBR were uniformly and randomly selected from the investigated population. ‘Uniformly and randomly selected’ means that each VBR
had the same chance to be selected. From the real
numbers of VN of the selected VBR, mean, S.D. and
coefficient of variation were calculated. According to
the literature, the square of this coefficient of variation
was named ‘real inherent biological variance of the
individuals’ (West, 1993; ICV2). Second, the total numbers of VN of the selected VBR were estimated once
using the fractionator as explained above, using either
VSS2 or VSS3 (Table 3). From the estimated total
numbers of VN of the selected VBR, mean, S.D. and
coefficient of variation were calculated. According to
the literature, the square of this coefficient of variation
was named ‘observed relative variance of group’ (West,
1993; OCV2). Predicted coefficients of error of the
estimated total numbers of VN were calculated using
method F7 (Table 1). Herein, the mean of these data is
called ‘meanE2’:
2
meanE2{[CEpred(n)] }=
VBR = 1
(17)
f
Each VSTST was carried out 1000 times, starting
with selecting VBR from the investigated population.
2.2.3. Analysis of the estimates
The results of Experiment 2 were analyzed for the
relationship between the interindividual variation of the
estimated total VN numbers of the selected VBR on the
one hand (OCV2), and the sum of the interindividual
variation of the true total VN numbers of the selected
VBR and the predicted mean variation of the estimates
on the other hand (SICV − CE). This was carried out by
calculating ratio R5 shown in Formula (19). R5 =0
indicates that there was no difference between OCV2
and SICV − CE. R5 \ 0 indicates that SICV − CE was
greater than OCV2, whereas R5 B 0 indicates that
SICV − CE was smaller than OCV2.
Furthermore, the relationship between the predicted
mean variation of the estimates and the interindividual
variation of the estimated total VN numbers of the
selected VBR was analyzed. This was carried out by
calculating ratio R6 shown in Formula (20).
The entire analysis of the results of Experiment 2
comprised calculation of the following variables for
each run of the VSTST.
The sum of ICV2 and [meanE2{[CEpred(n)]2}]:
SICV − CE = ICV2 + meanE2{[CEpred(n)]2}
(18)
2
The difference between SICV − CE and OCV :
R5 =
SICV − CE − OCV2
SICV − CE + OCV2
(19)
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
100
The ratio between [meanE2{[CEpred(n)]2}] and OCV2:
R6 =
meanE2{[CEpred(n)]2}
OCV2
(20)
The mean of the 1000 values of OCV2:
1000
% (OCV2)Run
2
mean(OCV )=
Run = 1
(21)
1000
The mean of the 1000 values of SICV − CE:
2.3. Source of randomness
A pseudorandom number generator provided by
L’Ecuyer (1988) (Figure 3 in this study; PRNG1) was
used as source of randomness (details of PRNG1 are
given in Appendix C). Repeating the entire simulation
with another pseudorandom number generator provided by L’Ecuyer (1988) (Figure 4 in this study;
PRNG2) led to almost identical results (details of
PRNG2 are also given in Appendix C). Therefore, only
results obtained using PRNG1 are presented.
1000
% (SICV − CE)Run
mean(SICV − CE)=
Run = 1
The difference
mean(OCV2):
(22)
1000
between
mean(SICV − CE)
and
2
R7 =
mean(SICV − CE) − mean(OCV )
mean(SICV − CE) + mean(OCV2)
(23)
Further analysis of the data is presented in Appendix
B.
3. Results
3.1. Results of Experiment 1
To describe the results of Experiment 1, it is necessary to compare results from different runs of the
computer simulations. For the sake of clarity, these
comparisons will be presented in an abbreviated format. For example, a comparison between runs A–D of
Experiment 1 are abbreviated as ‘runs A–B–C–D’.
Fig. 4. Frequency distributions of the total number of VN of five modeled populations [P1 – P5] of VBR. Each population consists of 15 000 VBR.
VBR in populations P1 – P4 are modeled as homogeneous VN distribution in a pallidal VRS (according to VRSi −SPPa in Fig. 2), whereas VBR
in population P5 are modeled as clustered VN distribution in pallidal VRS (according to VRSi −SPPz3 in Fig. 2). For P1 and P5, total VN number
is 49 97392012 (mean9 S.D.), for P2 it is 50 0239 1977, for P3 50 020 9 5999, and for P4 50 012 9 5981. The frequency distributions of total
VN number of P1, P3 and P5 approximate a Gaussian distribution, whereas the frequency distributions of P2 and P4 do not. Notation of the data
in brackets means that the smaller value was not included in the corresponding class.
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
Table 5
Details on the VSTST modeled in Experiment 2a
VSTST
c1
c2
c3
c4
c5
c6
c7
c8
c9
c10
c11
c12
c13
c14
c15
c16
c17
c18
f
P
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
5
5
6
6
6
6
12
12
12
12
6
6
6
6
12
12
12
12
6
6
VSS
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
2
3
a
P, investigated population; f, number of selected individuals;
VSS, applied sampling scheme. Details on VSS2 and VSS3 are given
in Table 3.
For all 24 runs of the computer simulation, the mean
of the 1000 estimated total numbers of VN obtained
using either the fractionator or the Vref ×NV method
Fig. 5.
101
was approximately 500 [VBR1 –VBR8] or approximately 50 000 [VBR9 –VBR16].
Fig. 5a shows the empirically estimated coefficients
of error of 6ref obtained using the Vref × NV method.
CEemp(6ref) varied as a function of the VRS [runs A–E,
B–F, etc.] and as a function of the VSS used [runs I–R,
K–S, etc.]. The highest values of CEemp(6ref) were obtained if estimating Vref of VBR with striatal VRS by
Fig. 5. Results of Experiment 1, shown as a function of the runs of
the simulation [A – Y; details of these runs are given in Table 2]. Each
run consists of 1000 modeled estimates of the total number of VN of
one of the modeled VBR shown in Fig. 2 using the Vref ×NV method,
as well as of 1000 modeled estimates of the total VN number of the
same VBR using the fractionator. The ordinates of all graphs are
truncated at the shown values. (a) Empirically estimated coefficient of
error of the volume estimates (6ref) of the investigated VBR obtained
using the Vref ×NV method [CEemp(6ref)]. The highest values of
CEemp(6ref) are obtained for VBR with striatal VRS by using the
virtual sampling schemes VSS1 or VSS2 (runs E, F, G, H and N, O,
P, Q; for VSS1 and VSS2 see Table 3), and the smallest values for
VBR with pallidal VRS by using VSS3 (runs R, S, T, U; for VSS3 see
also Table 3). (b) Empirically estimated coefficient of error of the VN
density estimates (nV) of the investigated VBR obtained using the
Vref ×NV method [CEemp(nV)]. The highest values of CEemp(nV) are
obtained for VBR with clustered VN distribution (runs D, H, M, Q,
U and Y), and the smallest values for VBR with homogeneous VN
distribution (runs A, E, I, N, R and V). (c) Results obtained for ratio
R1 [Eq. (10); i.e. ratio between CEemp(6ref) and CEemp(nV)]. The
highest values of ratio R1 are obtained for VBR with striatal VRS
and homogeneous VN distribution (runs E, N and V), and the
smallest values for VBR with pallidal VRS and clustered VN distribution (runs D, M and U). (d) Empirically estimated coefficient of error
of estimated total VN numbers [nV × N] obtained using the Vref × NV
method [CEemp(nV × N)]. The highest values of CEemp(nV × N) are
obtained for VBR with clustered VN distribution (runs D, H, M, Q,
U and Y), and the smallest values for VBR with homogeneous VN
distribution (runs A, E, I, N, R and V). Note that the values obtained
for CEemp(nV × N) are similar to the values obtained for CEemp(nV,
shown in b). (e) Results obtained for ratio R2 [Eq. (12); i.e. ratio
between [CEemp(6ref) +CEemp(nV)] and CEemp(nV × N)]. This ratio is
always greater than 1. (f) Empirically estimated coefficient of error of
estimated total VN numbers [nF] obtained using the fractionator
[CEemp(nF)]. The highest values of CEemp(nF) are obtained for VBR
with clustered VN distribution (runs D, H, M, Q, U and Y), and the
smallest values for VBR with homogeneous VN distribution (runs A,
E, I, N, R and V). Note that the values obtained for CEemp(nF) are
similar to the values obtained for CEemp(nV × N) (shown in d). (g)
Results obtained for ratio R3 [Eq. (14); i.e. ratio between CEemp(nV ×
N) and CEemp(nF)]. Ratio R3 is greater than 1, except for VBR with
centrifugal VN distribution of 50 000 VN in pallidal VRS (runs L
and T). There are three essential findings of Experiment 1 shown in
this figure. First, CEemp(nV × N) and CEemp(nF) vary as a function of
the spatial VN distribution in the VRS (compare run A with run B,
C, D; E– F– G – H, etc.), as a function of the VRS (runs A –E, B–F;
etc.), as a function of the total number of VN (runs A – I, B–K; etc.)
and as a function of the VSS used (runs I – R, K – S; etc.). Second, the
variation of Vref ×NV estimates is not simply the sum of the variation
of the number of counted neurons and the variation of the volume
estimates (e). Rather, the covariance between these variables has to be
considered if calculating the variation of Vref ×NV estimates (Gundersen and Jensen, 1987). Third, except for VBR with centrifugal VN
distribution of 50 000 VN in pallidal VRS (runs L and T), the
fractionator estimates have a greater precision than the corresponding
Vref ×NV estimates.
102
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
using VSS1 or VSS2 [runs E, F, G, H and N, O, P, Q],
and the smallest values if estimating Vref of VBR with
pallidal VRS by using VSS3 [runs R, S, T, U].
Fig. 5b displays the empirically estimated coefficients
of error of nV obtained using the Vref ×NV method.
CEemp(nV) varied as a function of the applied SPP [runs
A– B–C –D, E–F– G – H, etc.], as a function of the
VRS [runs A–E, B– F, etc.], as a function of the total
number of VN [runs A – I, B – K, etc.] and as a function
of the VSS used [runs I – R, K – S, etc.]. The highest
values of CEemp(nV) were obtained for VBR with clustered VN distribution [runs D, H, M, Q, U and Y], and
the smallest values for VBR with homogeneous VN
distribution [runs A, E, I, N, R and V].
Fig. 5c shows the results obtained for ratio R1. This
ratio varied as a function of the applied SPP [runs
A – B –C –D, E–F– G – H, etc.], as a function of the
VRS [runs A–E, B– F, etc.], as a function of the total
number of VN [runs A – I, B – K, etc.] and as a function
of the VSS used [runs I – R, K – S, etc.]. The highest
values of R1 were obtained for VBR with striatal VRS
and homogeneous VN distribution [runs E, N and V],
and the smallest values for VBR with pallidal VRS and
clustered VN distribution [runs D, M and U].
Fig. 5d displays the empirically estimated coefficients
of error of nV × N obtained using the Vref ×NV method.
Like CEemp(nV), CEemp(nV × N) varied as a function of
the applied SPP [runs A – B – C – D, E – F – G – H, etc.], as
a function of the VRS [runs A – E, B – F, etc.], as a
function of the total number of VN [runs A – I, B–K,
etc.] and as a function of the VSS used [runs I – R, K–S,
etc.]. The highest values of CEemp(nV × N) were obtained
for VBR with clustered VN distribution [runs D, H, M,
Q, U and Y], and the smallest values for VBR with
homogeneous VN distribution [runs A, E, I, N, R and
V].
Fig. 5e shows the results obtained for ratio R2. This
ratio was always greater than 1.
Fig. 5f shows the empirically estimated coefficients of
error of nF obtained using the fractionator. Like
CEemp(nV × N), CEemp(nF) varied as a function of the
applied SPP [runs A– B – C – D, E – F – G – H, etc.], as a
function of the VRS [runs A – E, B – F, etc.], as a
function of the total number of VN [runs A – I, B–K,
etc.] and as a function of the VSS used [runs I – R, K–S,
etc.]. The highest values of CEemp(nF) were obtained for
VBR with clustered VN distribution [runs D, H, M, Q,
U and Y], and the smallest values for VBR with
homogeneous VN distribution [runs A, E, I, N, R and
V].
Fig. 5g shows the results obtained for ratio R3. This
ratio was greater than 1, except for VBR with centrifugal VN distribution of 50 000 VN in pallidal VRS [runs
L and T].
Fig. 6a–c displays the results obtained for ratio R4
for all runs of the simulation as a function of the
applied method for predicting the precision of nV × N or
nF (additional data are given in Appendix A). The
results can be summarized as follows. If defining a
range of 0.75 B R4 B 1.25 as satisfactory mean prediction of the precision of nV × N or nF, no method led to
satisfactory predictions in any runs of the simulation.
For VBR with a total number of 500 VN, application
of all prediction methods resulted either in underestimation or overestimation of CEemp(nV × N) or CEemp(nF)
[runs A–H]. Note in particular that F5 resulted in
underestimations of CEemp(nF). The methods V1 and
V3 considerably underestimated CEemp(nV × N), except
in the case of VBR with centripetal VN distribution in
‘pallidal’ VRS [runs B, K, S] and of VBR6 [run F].
Also, methods F1 and F3 considerably underestimated
CEemp(nF), except in cases such as V1 and V3. Satisfactory mean predictions of the precision of nF were
obtained using F7 for investigating VBR9 –VBR11 and
VBR13 –VBR15 [runs R–T and V–X; in these runs VN
distributions were modeled according to homogeneous
or inhomogeneous Poisson processes]. Satisfactory
mean predictions were also obtained using F6 for investigating VBR12 and VBR16 [runs U and Y; in these runs
VN distributions were modeled according to Poisson
cluster processes]. Furthermore, satisfactory mean predictions of the precision of nF were also be obtained
using F9 or F13 for investigating VBR9 –VBR11 and
VBR13 –VBR15 [runs R–T and V–X], and using F11 or
F15 for investigating VBR12 and VBR16 [runs U and
Y].
3.2. Results of Experiment 2
For all 18 VSTST, the mean of the 1000 estimated
mean total numbers of VN was approximately 50 000.
Fig. 7a displays the frequency distributions of the 1000
values of ratio R5 each, and Fig. 7b of ratio R6. It was
found that both ratios varied in a broad range. For
ratio R5 the largest range was found for c 18 (−0.963
to + 0.763) and the smallest for c 15 ( − 0.272 to
0.353). The frequency distributions of R6 were composed of values between 0 and over 100% except c15
and c 17. The largest range was found for c 12 (0.5–
4814%), and the smallest for c 15 (3.1–53.6%).
Fig. 8a shows the results obtained for mean(OCV2),
and Fig. 8b the results obtained for mean(SICV − CE).
Except c 17 and c18, nearly identical results were
found for mean(OCV2) and mean(SICV − CE). Both variables depend on the interindividual variation of the
number of VN among the VBR of the investigated
population [compare c1 with c 3; c2 vs. c 4; etc.],
and on the applied VSS [c1 vs. c 9; c2 vs. c10;
etc.]. Fig. 8c displays the results obtained for the difference between mean(SICV − CE) and mean(OCV2). Except
for c 17 and c18, there was virtually no difference
between these variables.
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
103
Fig. 6.
4. Discussion
4.1. Validity of the results
Modeling of stereological estimates involves various
random elements (actual spatial VN distributions of
VBR, planes of section, thickness of the first sections,
positions of the lattices onto the sections). Therefore, it
represents a so-called ‘stochastic simulation’ (see Ripley, 1987). For each stochastic simulation a source of
randomness is required. Here, a pseudorandom number
generator (L’Ecuyer, 1988; PRNG1) was applied as
104
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
source of randomness (details on PRNG1 are given in
Appendix C). Formally, pseudorandom numbers are
generated by a computer using a simple numerical
algorithm. Consequently, pseudorandom numbers are
not truly random. Rather, any given sequence of pseudorandom numbers is supposed to appear random to
someone who does not know the algorithm. Furthermore, pseudorandom numbers are considered ‘random’
if a sequence of pseudorandom numbers has the same
probability of passing certain statistical tests as truly
random numbers would have (Knuth, 1981). L’Ecuyer
(1988) demonstrated the ‘randomness’ of PRNG1 by
using 21 different tests of randomness, details of which
may be found in Knuth (1981) or Marsaglia (1985).
Based on L’Ecuyer’s (1988) evaluation, James (1990)
has recommended the use of PRNG1 for stochastic
simulations. However, despite the demonstration of the
‘randomness’ of the applied pseudorandom number
generator, one cannot rule out the possibility that other
results would have been obtained if another pseudorandom number generator would have been used (see
Ripley, 1987). Aside from using only such generators
which have been exhaustively tested (Knuth, 1981;
Marsaglia, 1985), it is therefore recommended to carry
out any stochastic simulation with different pseudorandom number generators (Ripley, 1987). This was
achieved here by repeating the entire simulation using
another pseudorandom number generator developed
and tested by L’Ecuyer (1988); PRNG2; see Appendix
C for details). PRNG2 yielded nearly identical results as
PRNG1.
4.2. Rele6ance of the results to quantitati6e
neuroanatomy
Brain regions as modeled in these simulations do not
occur naturally. Nonetheless, these simulations have
their biological relevance if the following points are
considered. First, for methodological reasons, it is vir-
tually impossible to determine the exact 3-D distribution pattern of neurons in a brain region of interest
(Reed and Howard, 1997). Second, detailed information on the frequency distributions of total neuronal
numbers is currently not available from the literature.
Therefore, in investigations comparable to those presented here, the common usage is to apply exactly
defined point distribution patterns for modeling any
individuals (here, VBR) containing any type of particles
(here, VN; König et al., 1991; McShane and Palmatier,
1994; Schmitz, 1998; Glaser and Wilson, 1998, 1999).
Third, the volume of the reference spaces and the mean
total numbers of VN were selected to be similar to
estimates of the mean volume and the mean total
neuronal number of the rat external globus pallidus as
reported by Oorschot (1996). Fourth, the number of
VBR investigated in Experiment 2 (i.e. 6 or 12), the
average number of counted VN (i.e. 150 or 750), and
the values of OCV obtained in Experiment 2 cover the
ranges of these variables reported in most stereological
studies published to date. In summary, the computer
simulations presented here may serve as a useful substitute for quantitative neuroanatomical studies.
4.3. Statistical and economical efficiency of the
Vref × NV method and the fractionator in estimating
total neuronal numbers
Estimates of total neuronal numbers using the fractionator require only counting of neurons in a part of
the brain region of interest. In contrast, estimates of
total neuronal numbers using the Vref × NV method
require counting of neurons in a part of the brain
region of interest and estimating the total volume of
this brain region. Therefore, already from a theoretical
point of view the Vref × NV method has the smaller
economical efficiency (West, 1993).
Fig. 6. Results of all runs of the simulation in Experiment 1 (A – Y; details of these runs are given in Table 2). Each run consists of 1000 modeled
estimates of the total number of VN of one of the modeled VBR shown in Fig. 2 using the Vref ×NV method, as well as of 1000 modeled estimates
of the total VN number of the same VBR using the fractionator. The figure shows the results obtained for ratio R4 as a function of the applied
prediction method [Eq. (16); i.e. ratio between the mean of the 1000 squared predicted coefficients of error [meanE1{[CEpred(n)]2}]] and the
empirically estimated squared coefficient of error [[CEemp(n)]2] after estimating the total VN number of the investigated VBR 1000 times and
predicting the variation of the estimates with one of the prediction methods shown in Table 1. The ordinates of all graphs are limited to
0.45 R4 5 1.6. (a) Results obtained for runs A–H, that is, investigating VBR with 500 VN by using the virtual sampling scheme VSS1 (Table 3)
resulting in counting of approximately 150 VN per estimate. (b) Results obtained for runs I – Q, that is, investigating VBR with 50 000 VN by
using VSS2 (Table 3) resulting in counting of approximately 150 VN per estimate. (c) Results obtained for runs R – Y, that is, investigating VBR
with 50 000 VN by using VSS3 (Table 3) resulting in counting of approximately 750 VN per estimate. There are three essential findings of
Experiment 1 shown in this figure. First, if defining a range of 0.75 BR4 B1.25 as satisfactory mean prediction of the precision of the estimated
total VN numbers, no prediction method leads to satisfactory predictions in any runs of the simulation. Second, for runs A – H, application of
all prediction methods results either in underestimation or overestimation of CEemp(nV × N) or CEemp(nF). Third, for runs I – Y, the use of three
pairs of prediction methods (F7–F6, F9–F11 and F13–F15; each time the first prediction method used when investigating VBR with VN
distributions according to homogeneous or inhomogeneous Poisson processes and the second prediction method used when investigating VBR
with VN distributions according to Poisson cluster processes) results in satisfactory predictions of the variation of most of the modeled
fractionator estimates.
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
105
Fig. 7. Results of all VSTST carried out in Experiment 2 (c 1 to c18; details of these VSTST are given in Table 5). Each VSTST consists of
1000 repetitions of selecting either 6 or 12 VBR of one of the modeled populations P1 – P5 of 15 000 VBR each (for P1 – P5 see Table 4), estimating
the total number of VN of each selected VBR once using the fractionator and either virtual sampling scheme VSS2 or VSS3 (for VSS2 and VSS3
see Table 3), and predicting the variation of the estimates using prediction method F7 (for F7 see Table 1). For each repetition, the squared
coefficient of variation of the real total VN numbers of the selected VBR is calculated [ICV2], the mean squared predicted coefficient of error of
the estimated total VN numbers [meanE2{[CEpred(n)]2}], and the squared coefficient of variation of the estimated total VN numbers [OCV2]. The
ordinates of all graphs are truncated at the shown values. (a) Frequency distributions of the results obtained for ratio R5 (Eq. (19)), describing
the difference between [ICV2 + meanE2{[CEpred(n)]2}] and OCV2. (b) Frequency distributions of the results obtained for ratio R6 [Eq. (20); that
is, ratio between meanE2{[CEpred(n)]2} and OCV2]. There are two essential findings of Experiment 2 shown in this figure. First, for each VSTST,
ratio R5 varies in a broad range. The largest range is found for c 18 ( − 0.963 to + 0.763) and the smallest for c 15 ( −0.272 to + 0.353).
Second, ratio R6 varies also in a broad range and was composed of values between 0 and over 100% except c 15 and c17. The largest range
is found for c 12 (0.5 – 4814%), and the smallest for c15 (3.1 – 53.6%).
The results of Experiment 1 show that the statistical
efficiency of both fractionator estimates and Vref × NV
estimates depends on the shape of a brain region of
interest, on the spatial distribution of neurons within
this brain region, and on the sampling scheme used for
estimating total neuronal numbers. This confirms results of previous studies (Schmitz, 1998; Glaser and
Wilson, 1998, 1999). Furthermore, the results of Experiment 1 demonstrate that the statistical efficiency of
both fractionator estimates and Vref × NV estimates depends on the ratio between the mean number of
counted neurons and the total number of neurons in
the brain region of interest (ratio R8). Interestingly, for
neuron distributions corresponding to homogeneous or
106
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
estimates depends on the variation of the number of
counted neurons and on the variation of the volume
estimates. However, the variation of Vref × NV estimates
is not simply the sum of the variation of the number of
counted neurons and the variation of the volume estimates, as demonstrated in Fig. 5e. Rather the covariance between these variables has to be considered if
calculating the variation of Vref × NV estimates (Gundersen and Jensen, 1987). If volume estimates are obtained by using the point counting method and the
Cavalieri’s principle, the variation of the estimates depends on the number of counted points and on the
so-called ‘average shape coefficient’ (SC) of the sections
of the brain region of interest (Gundersen and Jensen,
1987; Roberts et al., 1994). This average shape coefficient is defined as the ratio between the average total
inhomogeneous Poisson processes, there was only a
small difference in the statistical efficiency of both
fractionator estimates and Vref ×NV estimates between
R8 = 150/500 = 0.3 and R8 =150/50 000 =0.003 (Fig.
5d and f; compare run A, B and C with run J, K and
L). In contrast, for neuron distributions corresponding
to Poisson cluster processes there was a high difference
in the statistical efficiency of both fractionator estimates and Vref × NV estimates between R8 =0.3 and
R8 = 0.003 (Fig. 5d and f; compare run D with run M).
Moreover, the results of Experiment 1 show that
fractionator estimates have a greater statistical efficiency than Vref × NV estimates (Fig. 5g). This is due to
the fact that the variation of fractionator estimates
depends solely on the variation of the number of
counted neurons, whereas the variation of Vref × NV
Fig. 7.
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
107
mated total numbers of neurons using the Vref ×NV
method. This was demonstrated in Experiment 1 [Fig.
5, runs E, F, G, H, N, P, V and X].
In summary, fractionator estimates have both the
greater economical efficiency as well as the greater
statistical efficiency. We therefore recommend to estimate total neuronal numbers preferably using the
fractionator.
4.4. Prediction of the 6ariation of estimated total
numbers of neurons obtained using the Vref × NV
method or the fractionator
Fig. 8. Results of all VSTST carried out in Experiment 2 ( c 1 to
c18; details of these VSTST are given in Table 5). Each VSTST
consists of 1000 repetitions of selecting either 6 or 12 VBR of one of
the modeled populations P1 –P5 of 15 000 VBR each (for P1 – P5 see
Table 4), estimating the total number of VN of each selected VBR
once using the fractionator and either virtual sampling scheme VSS2
or VSS3 (for VSS2 and VSS3 see Table 3), and predicting the variation
of the estimates using prediction method F7 (for F7 see Table 1).
Fractionator estimates are modeled using either the virtual sampling
scheme VSS2 (VSTST c 1 to c8 and c 17) or using VSS3 (VSTST
c 9 to c 16 and c 18; for VSS2 and VSS3 see Table 3). For each
repetition, the squared coefficient of variation of the real total VN
numbers of the selected VBR is calculated [ICV2], the mean squared
predicted coefficient of error of the estimated total VN numbers
[meanE2{[CEpred(n)]2}], and the squared coefficient of variation of the
estimated total VN numbers [OCV2]. The ordinates of all graphs are
truncated at the shown values. (a) Mean of the 1000 values obtained
for OCV2 [mean(OCV2)]. This variable depends on the interindividual
variation of the number of VN among the VBR of the investigated
population (compare c 1 with c 3; c 2 vs. c 4; etc.), and on the
VSS used ( c 1 vs. c 9; c 2 vs. c10; etc.). (b) Mean of the 1000
values obtained for the sum of ICV2 and meanE2{[CEpred(n)]2}
[mean(SICV − CE)]. Like mean(OCV2), mean(SICV − CE) depends on the
interindividual variation of the number of VN among the VBR of the
investigated population (compare c 1 with c 3; c 2 vs. c 4; etc.),
and on the VSS used (c1 vs. c9; c 2 vs. c 10; etc.). (c) Mean of
the 1000 values obtained for ratio R7 (Eq. (23)), describing the
difference between mean(SICV − CE) and mean(OCV2). There is one
essential finding of Experiment 2 shown in this figure. Except for
c 17 and c 18, there is virtually no difference between mean(OCV2)
and mean(SICV − CE).
boundary length of the sections and the square root of
the average surface area of the sections of the brain
region of interest (see above; Eq. (7)). The greater this
average shape coefficient is, the greater is the variation
of volume estimates by using the point counting
method and the Cavalieri’s principle (Gundersen and
Jensen, 1987). SC was approximately 3.4 for the pallidal VRS [VRSi; VBR 1 – 4 and 9 – 12], and was approximately 28.5 for the striatal VRS [VRSii; VBR 5 – 8 and
13 – 16]. Therefore, the variation of the Vref ×NV estimates was greater if investigating VBR with striatal
VRS than investigating VBR with pallidal VRS [Fig.
5d, runs A–E, B–F, etc.]. The difference between the
statistical efficiency of fractionator estimates and Vref ×
NV estimates is a function of the contribution of the
variation of volume estimates to the variation of esti-
The results of Experiment 1 show complex interrelations between the shape of a brain region of interest,
the number and the spatial distribution of neurons
within this brain region, the variation of estimates of
total neuronal numbers, and the precision of predictions of this variation (henceforth abbreviated as ‘predictions’) using the various prediction methods listed in
Table 1. It is beyond the scope of this study to provide
a complete analysis of these interrelations. Rather,
some aspects relevant for the use of the fractionator or
the Vref × NV method in quantitative neuroanatomy will
be briefly discussed in the following.
We defined a range of 0.75 B R4 B 1.25 as satisfactory mean prediction of the precision of estimated total
neuronal numbers (for R4 see Eq. (16)). Using this
definition, we found that no prediction method led to
satisfactory predictions in any runs of the simulation
(Fig. 6). Rather, for each prediction method, situations
could be modeled in which the variation of the corresponding fractionator estimates or Vref × NV estimates
was considerably overestimated. As well, except for
method F6, F11 and F15, situations could be modeled
in which the variation of the corresponding neuron
number estimates was considerably underestimated.
Therefore, no prediction method can be regarded perfect. This is in line with recent theoretical work concerned with the variation of stereological estimates
obtained using the fractionator or the Vref × NV method
(Cruz-Orive, 1999; Gundersen et al., 1999). On the
other hand, there was no modeled situation for which it
was impossible to obtain any satisfactory prediction.
Therefore, we looked for pairs of prediction methods,
the use of which resulted in satisfactory predictions of
the variation of neuron number estimates of as many
modeled situations as possible. For the Vref ×NV
method, it was not possible to find such a pair of
prediction methods. In contrast, for fractionator estimates with R8 = 150/50 000 = 0.003 (Fig. 6b) or R8 =
750/50 000 = 0.015 (Fig. 6c), three pairs of prediction
methods were found leading to this goal. These pairs
were F7–F6, F9–F11 and F13–F15. Each time, use of
the first method (F7, F9 and F13) resulted in satisfactory mean predictions if the spatial neuron distribution
108
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
was modeled according to homogeneous or inhomogeneous Poisson processes, whereas use of the second
method (F6, F11 and F15) resulted in satisfactory mean
predictions if the spatial neuron distribution was modeled according to Poisson cluster processes. The computer simulation did not show any advantage of one of
the mentioned pairs of prediction methods over the
other pairs. However, a clear advantage of F7 – F6 over
both F9–F11 and F13 – F15 is the fact that predictions
of the variation of fractionator estimates are many
times easier to calculate using F7 – F6 than using F9–
F11 or F13–F15 (details are shown in Appendix D). It
should be mentioned that the use of F9 and F13
resulted in satisfactory predictions even if modeling
fractionator estimates of VBR with centripetal VN
distribution (Fig. 6b and c, runs K and S). F9 and F13
(as well as V1–V3, F1 – F4, F8, F10 – F12, F14 and
F15) are based on complex theoretical statistics, namely
on Matheron’s ‘theory of regionalized variables’
(Matheron, 1965, 1971). In a former study, which was
published before F9 and F13 had been reported in the
literature, it was found that the use of all prediction
methods based on Matheron’s (1965, 1971) theory resulted in considerable overestimation of the precision of
fractionator estimates if modeling VBR with centripetal
VN distribution (Schmitz, 1998; see also Fig. 6b and c
here, runs K and S). This finding was interpreted as
indicating ‘that Matheron’s (1965, 1971) theory can in
principle not serve as the optimum basis for predicting
the precision of fractionator estimates, independent of
the manner of how it is adapted to the fractionator’
(Schmitz, 1998). The development of F9 and F13 has
shown that this interpretation can no longer be
maintained.
For the fractionator estimates with R8 =150/500 =
0.3, we found that there was no prediction method the
use of which led to satisfactory predictions of the
variation of the estimates (Fig. 6a). Particularly, F5
resulted in underestimations of this variation. F5 is a
slight modification of F7, considering the sampling
fraction of fractionator estimates (i.e. if the space occupied by the counting spaces is, say, the 1000th part of
the reference space of the brain region of interest, the
sampling fraction sf is 1/1000). At first glance it seems
useful to consider sf if R8 is small. This may be seen
from the borderline case if sf=1 and thus, R8 = 1. In
this case there is no variation of fractionator estimates,
but the use of all prediction methods except F5 results
in mean predictions of this variation greater than 0.
However, already with R8 =150/500 = 0.3, F5 resulted
in underestimations of the variation of fractionator
estimates, whereas F7 resulted in overestimations of
this variation (Fig. 6a). With R8 =150/50 000 =0.003
(Fig. 6b) or R8 =750/50 000 = 0.015 (Fig. 6c), F5 led to
almost identical predictions as F7 did. Therefore, there
is no advantage of using F5 rather than F7.
In summary, using F7 when investigating VBR with
VN distributions according to homogeneous or inhomogeneous Poisson processes and using F6 when investigating VBR with VN distributions according to
Poisson cluster processes facilitated satisfactory predictions of the variation of the modeled fractionator estimates. We recommend to predict the variation of
fractionator estimates in quantitative neuroanatomy always using these simple prediction methods. Most
likely, inspection of the investigated sections on the
microscope will be sufficient to decide whether the
neuron distribution is homogeneous (warranting the
use of F7) or clustered (warranting the use of F6).
Otherwise, there are methods available in the literature
for investigating the distribution pattern of neurons in a
brain region of interest (for example, see Duyckaerts
and Godefroy, 2000). In any case, however, it appears
necessary to interpret predictions of the variation of
estimated total neuronal numbers carefully.
4.5. Optimization of stereological sampling schemes
A frequently used approach to demonstrate the reliability of an estimated mean total number of neurons is
to show that the mean of the squared predicted coefficients of error of estimated total neuronal numbers (i.e.
meanE2{[CEpred(n)]2}) is less than half of the squaredE2
coefficient of variation of these estimated total neuronal
numbers (i.e. OCV2; for recent examples see Geinisman
et al., 1996; Begega et al., 1999; Korbo and West, 2000;
among others). For example, in West (1993) this approach was explained as follows. If for a number of
individuals the mean total number of neurons in a
given brain region is estimated using the fractionator or
the Vref × NV method, the squared coefficient of variation of the estimated total numbers of neurons (OCV2)
is affected not only by the real inherent biological
variance of the individuals (ICV2), but also by the
variance of the estimates (CE2), which is related to the
amount of sampling performed in each individual. According to West (1993) the relation between OCV2,
ICV2 and CE2 can be calculated as shown in Eq. (24):
OCV2 = ICV2 + CE2
(24)
2
2
If the major contributor to OCV is CE (in this case
is CE2 greater than ICV2, and ratio R6 is greater than
50%), the most efficient way to reduce OCV2 would be
to reduce CE2 by increasing the precision of the estimates (West, 1993). If the major contribution to OCV2
is ICV2 (in this case is CE2 smaller than ICV2, and ratio
R6 is smaller than 50%), the most efficient way to
reduce OCV2 would be to reduce ICV2 by increasing
the number of investigated individuals (West, 1993).
The results of Experiment 2 reveal however that this
recommendation be considered critically. Both OCV2
and (ICV2 + CE2) are random variables, and the differ-
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
ence between OCV2 and (ICV2 +CE2) may vary considerably if the same stereological study is carried out
repeatedly (Fig. 7a). In consequence, the ratio between
CE2 and OCV2 is a random variable as well, which may
also vary considerably (Fig. 7b). The actual relation
between OCV2 and (ICV2 +CE2) is shown in Eq. (25),
as already given by Gundersen (1986):
OCV2( · )= ICV2( · )+ CE2
(25)
OCV2( · ) is a stereological estimate of mean(OCV2)
[and ICV2( · ) a stereological estimate of mean (ICV2)],
which would be obtained if the same stereological study
would be repeated unlimited times. Provided that the
predictions of the variation of the estimated total neuronal numbers equal the real variation of the estimates,
the mean of the observed values of OCV2 would equal
the mean of the observed values of (ICV2 +CE2) (Fig.
8c). In this case, the ratio between the mean of CE2 and
the mean of OCV2 would indeed provide a basis for
evaluating the reliability of estimated mean total neuronal numbers (details are given in Appendix B). However, this is obviously not the case in quantitative
neuroanatomy. Without often repeating the same stereological study, the CE2/OCV2 approach is neither useful nor informative.
However, this CE2/OCV2 approach served as the
basis for statements such as ‘there are no prior reasons
at all for expecting that the counting of more than 50
or 100 items (i.e. neurons) per individual or organ is
necessary’ (Gundersen, 1986), or ‘an advantage of systematic random sampling is that one need count only
about 100 cells or synapses to get sampling variances to
be negligible in comparison with interanimal variances’
(Coggeshall and Lekan, 1996). As a consequence,
counting of no more than 100 – 200 cells per individual
has become a general recommendation in design-based
stereology (Gundersen et al., 1988; West, 1993; Mayhew and Gundersen, 1996; among others) and has been
applied in many stereological studies published in the
literature. The results of Experiment 2 show the need to
handle this recommendation very carefully. At present,
there is no method available to perform valid comparisons between sampling variances and interanimal variances in quantitative neuroanatomy.
In summary, assessing the reliability of estimates of
mean total neuronal numbers using the CE2/OCV2
approach is neither useful nor informative. We feel that
this approach is not optimal. Finally, there is urgent
need to find a new, analytical solution of this major
problem in design-based stereology. Based on our own
experience, we have decided to increase the number of
counted neurons to at least 700 – 1000 per individual
whenever possible (Schmitz et al., 1999a; Heinsen et al.,
1999, 2000). For example, for brain regions with a
homogeneous neuron distribution, counting of approximately 900 neurons results in a predicted coefficient of
109
error of 0.033. Accordingly, one may expect the true
total number of neurons in the investigated brain region with a probability of approximately 95% in a
range of approximately 9 7% about the estimated total
neuronal number. The amount of time necessary to
carry out these estimates is about 1 day per brain
region of interest, which appears justified as a reasonable compromise between the amount of time dedicated
to the analysis and the precision of the obtained estimates. We are aware that this can be accepted only as
one possible empirical solution to the mentioned problem, and that an analytical solution is still lacking. We
expect however that our study will initiate new discussions and, hopefully, will stimulate new approaches to
solve
this
important
issue
of
quantitative
neuroanatomy.
Acknowledgements
We thank our many colleagues who prompted us to
carry out this study. We gratefully acknowledge Hubert
Korr and Helmut Heinsen for their constructive and
helpful comments. This study was supported by the
START-program of the Faculty of Medicine at the
RWTH University of Aachen, Germany (C.S.), and by
NIH grants AG02219, AG05138, and MH58911
(P.R.H.).
Appendix A. Note on the use of the prediction methods
F8–F11
For F8–F11, a term t must be calculated (CruzOrive, 1999). In this study, the use of F8–F11 was
demonstrated by means of an example, i.e. by application of F8–F11 on data presented by West et al. (1996).
In the latter study, which was concerned with the
number of somatostatin neurons in the striatum of rats,
neurons were counted on every tenth section throughout the striatum. Section thickness was 20 mm, and the
height of the counting spaces was 15 mm. Therefore, the
so-called ‘section sampling fraction’ (ssf) was 0.1, and
the so-called ‘thickness sampling fraction’ (tsf) was
0.75. Accordingly, in the example given by Cruz-Orive
(1999), t was 0.1. In the computer simulations presented here, every section was analyzed, and ssf was
therefore 1. Tsf was 0.669 for VSS1, 0.144 for VSS2,
and 0.246 for VSS3. According to Cruz-Orive (1999), t
would also be 1 here. In this case, however, F8 and F9
equal F7. Therefore, we decided in the present study to
calculate t= tsf, yielding t =0.669 for VSS1, t=0.144
for VSS2, and t= 0.246 for VSS3. The results obtained
this way are presented in Fig. 6. However, results
obtained using t= 1 was very similar, as shown in Fig.
9.
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
110
Appendix B. Bias in predicting the relative contribution
of stereological sampling to the variation of an
estimated mean total neuronal number
Suppose a population of F VBR containing VN. To
evaluate the mean total number of VN in this population, select a small, random sample of f VBR and
estimate the total numbers of VN of the selected VBR
using the fractionator. Since the estimates are unbiased,
their mean (ĝ) is an unbiased estimator of the true
mean total number of VN of the selected VBR (G. ) as
well as of the true mean total number of VN in the
population (G; notations of ĝ, G. and G as introduced
by Nicholson, 1978). This is shown in Eqs. (B.1) and
(B.2))
E[ĝ] =E[G. ] =G
(B.1)
with
n
f
n
ĝ = % j ;
f
j=1
f
n
N
G. = % j ;
j=1 f
F
N
G= % j
j=1 F
n
(B.2)
where E[…] is the expected value, n is the estimated
total number of VN, and N is the ‘true total number of
VN.
The variation of ĝ depends on interindividual variability and on stereological sampling, as shown in Eq.
(B.3) (Nicholson, 1978):
s 2ĝ = s 2G. +E[s 2ĝG. ]=
s 2N
+E[s 2ĝG. ]
f
(B.3)
where s 2ĝ is the variance of the distribution of ĝ about
G; s 2G. the variance of the distribution of G. about G; s 2N
the variance of N among the VBR in the population;
and E[s 2ĝG. ] is the expected value of the variance of ĝ
about G. for all possible independent random samples
of f VBR from the population and all possible estimates of G. of the selected VBR. If s 2G. is greater than
E[s 2ĝG. ], the ratio R9 shown in Eq. (B.4)
R9 =
(E[s 2ĝG. ] )
s 2ĝ
(B.4)
is smaller than 50%. In this case the variation of ĝ is
mainly due to interindividual variability. If the ratio R9
is greater than 50%, the variation of ĝ is mainly due to
stereological sampling. The relative contribution of
stereological sampling to the variation of ĝ can be
calculated as shown in Eq. (B.5):
Contrel = 1−
s 2G.
s 2ĝ
(B.5)
In real experiments using the fractionator, usually
only one random sample of f individuals is selected
from a population. Therefore, s 2ĝ, s 2G. , s 2N and E[s 2ĝG. ]
are unknown, and R9 or Contrel cannot be calculated.
Rather ratio R6 (see above, Section 2.2.3) is frequently
used to predict the relative contribution of stereological
sampling to the variation of an estimated mean total
number of neurons (for recent examples see Geinisman
et al., 1996; Begega et al., 1999; Korbo and West,
2000).
As explained in the main text, ratio R6 shows considerable variation if the same stereological study is repeated (Fig. 7b). Moreover, ratio R6 is a biased
estimator of Contrel, as may be deduced from the
literature (Searle, 1987). This bias depends on the frequency distribution of the number of VN of the corresponding population of VBR, on the number of
Fig. 9. Results of all runs of the simulation in Experiment 1 (A – Y). The graphs show results obtained for ratio R4 (Eq. (16)) as a function of
the applied method for predicting the variation of the fractionator estimates. Black bars, ratio R4 calculated with t =tsf. Grey bars, ratio R4
calculated with t =1. t and tsf are explained in Appendix A. The ordinates of all graphs are limited to 0.45 R4 51.6. Note that the results
obtained using t = tsf are very similar to the corresponding results using t =1.
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
Est[Contrel]= 1−
s 2G.
s 2ĝ
111
(B.6)
This estimate of Contrel was compared with the mean
of the 1000 values of the ratio R6 each as shown in Eq.
(B.7):
R10 =
mean[R6]− Est[Contrel]
mean[R6]+ Est[Contrel]
(B.7)
Furthermore, the ratio between the mean of the 1000
values of [meanE2{[CEpred(n)]2}] (see above, Section
2.2.3) and the mean of the 1000 values of OCV2 was
calculated, as shown in Eq. (B.8):
1000
% (meanE2{[CEpred(n)]2})Run
R11 =
Run = 1
(B.8)
1000
% (OCV )Run
2
Run = 1
Like R6, R11 was compared with the estimate of
Contrel, as shown in Eq. (B.9):
R12 =
Fig. 10. Results of all VSTST carried out in Experiment 2 ( c 1 to
c18). The graphs display the results obtained for s 2G. (a), s 2ĝ (b),
Est[Contrel] (Eq. (B.5); c), the mean of the 1000 values of ratio R6
each (Eq. (20); d), ratio R10 (Eq. (B.7); e), ratio R11 (Eq. (B.8); f) and
ratio R12 (Eq. (B.9); g). The mentioned variables are explained in
Appendix B. The ordinates of all graphs are truncated at the shown
values. Note that (except for c 17 and c18) the results obtained for
Est[Contrel] are very similar to the corresponding results obtained for
ratio R11. Accordingly, except for c 17 and c 18, there is virtually
no difference between ratio R11 and Est[Contrel], as shown by calculating ratio R12.
selected VBR, the stereological sampling scheme used,
and the accuracy of meanE2{[CEpred(n)]2} in predicting
the mean squared coefficient of error of the estimated
total numbers of VN.
To the best of our knowledge, the bias of ratio R6 as
an estimator of Contrel has not been investigated. This
was achieved here by using the results of Experiment 2.
After selecting either f = 6 or f =12 VBR from the
investigated population, the mean of the (known) total
numbers of VN of the selected VBR was calculated (G. ),
as well as the mean of the estimated total numbers of
VN (ĝ). After carrying out the simulation 1000 times,
the variance of the 1000 values of G. (s 2G. ) was calculated
as the empirical estimator of s 2G. , as well as the variance
of the 1000 values of ĝ (s 2G. ) as the empirical estimator
of s 2ĝ. From these data an estimate of Contrel could be
calculated as shown in Eq. (B.6):
R11 − Est[Contrel]
R11 + Est[Contrel]
(B.9)
Fig. 10 shows the obtained results. As expected, for
each VSTST, s 2G. (Fig. 10a) was smaller than s 2ĝ (Fig.
10b). Both s 2G. and s 2ĝ depend on the interindividual
variation of the number of VN among the VBR of the
investigated population (compare c 1 with c3; c2
vs. c4; etc.), and on the number of selected VBR
(compare c 1 with c 5; c2 vs. c 6; etc.). For the
VSTST c3, c 4 and c7 to c 16, Est[Contrel] was
smaller than 50% and therefore the variation of ĝ was
mainly due to interindividual variability (Fig. 10c). For
c 1, c 2, c 5, c6, c 17 and c 18, Est[Contrel] was
greater than 50% and therefore the variation of ĝ was
mainly due to stereological sampling (Fig. 10c). As
expected, ratio R6 was a biased estimator of Est[Contrel]
(the mean of the 1000 values of R6 each is shown in
Fig. 10d; Fig. 10e shows ratio R10). This bias was
related to the number of selected VN (compare c1
with c 5; c 2 vs. c6; etc.), on the frequency distribution of N of the investigated population (c 3 vs. c4;
c 7 vs. c8; etc.) and on the accuracy of CEpred(n) in
predicting the coefficient of error of estimated total
numbers of VN (c 1 vs. c 17; c 9 vs. c 18). In
contrast to this, ratio R11 was an unbiased estimator of
Est[Contrel], except for c 17 and c18 in which the
variation of the fractionator estimates was considerably
underestimated (R11 is shown in Fig. 10f; Fig. 10g
shows ratio R12). Therefore, in principle it is quite
possible to use predictions of the variation of fractionator estimates and variations among estimated total
numbers of neurons for drawing conclusions on the
contribution of stereological sampling to the total variation of estimated mean total numbers of neurons.
However, this is only possible if the same stereological
study would be carried out repeatedly.
112
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
The results of c17 and c18 illustrate the fatal
consequences of underestimating the variation of estimated total numbers of VN if evaluating the appropriateness of sampling schemes in modern, design-based
stereology with ratio R6. Although the relative contribution of stereological sampling to the variation of ĝ
was approximately 99% in c17 (94% in c18), the
mean of the 1000 values of ratio R6 was merely 8.6% in
c 17 (8.7% in c18).
Finally, about 20 years ago a so-called ‘concisely
coined rule of thumb’ was established for a broad class
of biological experiments employing stereological
counting techniques, ‘one should look at more individuals […], rather than measure them more precisely (and
in a more time-consuming way)’ (Gundersen and
Østerby, 1981). However, for the VSTST involving P1
and P2, s 2ĝ was smaller when investigating f = 6 VBR
with VSS3 compared with the investigation of f =12
VBR with VSS2 (c 5 and c6 vs. c9 and c 10).
Consequently, there should be some caution with the
application of this ‘rule of thumb’. Its use should be
limited to investigations of mean total numbers of
neurons for which the variation of ĝ is mainly due to
interindividual variability (as shown with c7 vs. c 11
or c 8 vs. c 12). Unfortunately at present, there is no
available method to evaluate this in real stereological
studies.
Appendix C. Pseudorandom number generators used in
the present study
The pseudorandom number generator PRNG1
(L’Ecuyer, 1988) consists of the combination of two
so-called ‘multiplicative congruential linear pseudorandom number generators’ (MLCG), the basic form of
which is shown in Eq. (C.1):
f(l) =(a ×l)MODm;
l
g(l) = ;
m
l= 1, 2, …
represent all integers in the range (− 231 + 85, 231 −85),
can be found in Figure 3 in L’Ecuyer (1988).
What is referred to as PRNG2 here consists of the
combination of three MLCG as shown in Eq. (C.1).
For the first MLCG, m= 32 363 and a= 157. For the
second MLCG, m=31 727 and a=146, and for the
third MLCG, m= 31 657 and a=142. The initial seeds
were l= 123 for the first MLCG, l =456 for the second
MLCG, and l =789 for the third MLCG. The combined generator takes values evenly spread in [0, 1] and
has a period of approximately 8.12544× 1012. A
portable implementation of PRNG2, which works as
long as the computer can represent all integers in the
range (− 32 363, + 32 363), can be found in Figure 4
in L’Ecuyer (1988). Following a recommendation by
Ripley (1987), the complete simulation was based on
one single sequence of PRNG1, and the repetition was
based on one single sequence of PRNG2.
Appendix D. Calculations necessary to predict the
variation of fractionator estimates using the prediction
methods F6, F7, F9, F11, F13 and F15
Suppose that a fractionator estimate is carried out by
counting neurons in R counting spaces, which are distributed over S sections of a brain region of interest.
The S sections are a systematic and random sample of
every uth section of an entire series of sections through
this brain region with unique section thickness t. Q−
r is
the number of neurons counted in the rth counting
space, and Q−
s the number of neurons counted in the
sth section. Furthermore, Q−
r is the mean number of
counted neurons per counting space, and Var(Q−
r ) is
the variance of the number of counted neurons among
the counting spaces. Predictions of the variation of this
fractionator estimate are then obtained as follows:
F6:
CEpred(nF)=
F7:
CEpred(nF)=
(C.1)
For the first MLCG, m =2 147 483 563 and a=
40 014. For the second MLCG, m = 2 147 483 399 and
a= 40 692. Before the first call, the integer l must be
initialized. In the present study, the initial seeds were
l = 12 345 for the first MLCG, and l= 67 890 for the
second MLCG. The combined generator takes values
evenly spread in [0, 1] and has a so-called ‘period’ of
approximately 2.30584×1018, which is far more than
the period of 108 as stated by Ripley (1987) as minimum period of a good pseudorandom number generator (the period of a pseudorandom number generator is
the sequence of pseudorandom numbers after which the
generator begins to generate the same sequence of
numbers over again). A portable implementation of
PRNG1, which works as long as the computer can
'
Var(Q−
r )
'
2
R(Q−
r )
1
R
% Q
r=1
−
r
=
(D.1)
'
1
(D.2)
S
% Q
−
s
s=1
For F9, F11, F13 and F15, first of all the following
equations must be calculated:
t=
t
1
=
u× t u
(D.3)
a=
1 [1+ 2t − 2(t 2)][1− t]2
6 40−10(t 2)+ 3(t 3)
(D.4)
S
N= % Q−
s
(D.5)
s=1
S
−
A= % (Q−
s ×Qs )
s=1
(D.6)
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
S−1
−
B = % (Q−
s × Q s + 1)
(D.7)
s=1
S−2
−
C = % (Q−
s × Q s + 2)
(D.8)
s=1
S−3
−
D = % (Q−
s × Q s + 3)
(D.9)
s=1
Using these equations, predictions of the variation of
fractionator estimates are obtained as follows:
CEpred(nF)=
F9
P=
F11
(431−32t 2 +2t 3 +t 4)B
12
(D.11)
(D.12)
(79−16t 2 +2t 3 +t 4)D
20
(D.13)
CEpred(nF)=
F13
a[3(A − N) −4B + C] 1
+
N2
N
(D.10)
(268−37t 2 + 4t 3 +2t 4)C
15
U=
V=
'
A −[1/22 −t 2(P − U+ V)]
N
CEpred(nF)=
(D.14)
(3[A − N] −4B +C)/240 +N
N
(D.15)
F15
CEpred(nF)=
(1320A −2155B + 1072C −237D)/1320
N
(D.16)
References
Begega, A., Cuesta, M., Santin, L.J., Rubio, S., Astudillo, A., Arias,
J.L., 1999. Unbiased estimation of the total number of nervous
cells and volume of medial mammillary nucleus in humans. Exp.
Gerontol. 34, 771 – 782.
Bronshtein, I.N., Semendyayev, K.A., 1985. Handbook of Mathematics. Springer, Berlin.
Buffon, G.L.L.C, 1777. In: Essai d’Arithmétique Morale. Supplément
à l’Histoire Naturelle, vol. 4, Imprimerie Royale, Paris.
Cavalieri, B., 1635. Geometria indivisibilibus continuorum. Typis
Clementis Ferronij, Bononiae (reprinted 1966 as Geometria Degli
Indivisibili. Unione Tipografico-Editrice Torinese, Torino).
Coggeshall, R.E., Lekan, H.A., 1996. Methods for determining numbers of cells and synapses: a case for more uniform standards of
review. J. Comp. Neurol. 364, 6–15.
Cox, D.R., Isham, V., 1980. Point Processes. Chapman & Hall,
London.
Cruz-Orive, L.M., 1990. On the empirical variance of a fractionator
estimate. J. Microsc. 160, 89–95.
Cruz-Orive, L.M., 1994. Toward a more objective biology. Neurobiol. Aging 15, 377 – 378.
Cruz-Orive, L.M., 1997. Stereology of single objects. J. Microsc. 186,
93– 107.
Cruz-Orive, L.M., 1999. Precision of Cavalieri sections and slices with
local errors. J. Microsc. 193, 182–198.
113
Diggle, P.J., 1983. Statistical Analysis of Spatial Point Patterns.
Academic Press, New York.
Duyckaerts, C., Godefroy, G., 2000. Voronoi tessellation to study the
numerical density and the spatial distribution of neurones. J.
Chem. Neuroanat. 20, 83 – 92.
Geinisman, Y., Gundersen, H.J.G., van der Zee, E., West, M.J.,
1996. Unbiased stereological estimation of the total number of
synapses in a brain region. J. Neurocytol. 25, 805 – 819.
Glaser, E.M., Wilson, P.D., 1998. The coefficient of error of optical
fractionator population size estimates: a computer simulation
comparing three estimators. J. Microsc. 192, 163 – 171.
Glaser, E.M., Wilson, P.D., 1999. A comparison of three estimators
of the coefficient of error of optical fractionator cell count estimates. Acta Stereol. 18, 15 – 25.
Gundersen, H.J.G., 1986. Stereology of arbitrary particles. J. Microsc. 143, 3 – 45.
Gundersen, H.J.G., Østerby, R., 1981. Optimizing sampling efficiency
of stereological studies in biology: or ‘do more less well!’. J.
Microsc. 121, 65 – 73.
Gundersen, H.J.G., Jensen, E.B., 1987. The efficiency of systematic
sampling and its prediction. J. Microsc. 147, 229 – 263.
Gundersen, H.J.G., Bagger, A., Bendtsen, T.F., Evans, S.M., Korbo,
L., Marcussen, N., Møller, A., Nielsen, K., Nyengaard, J.R.,
Pakkenberg, B., Sørensen, F.B., Vesterby, A., West, M.J., 1988.
The new stereological tools: disector, fractionator, nucleator and
point sampled intercepts and their use in pathological research
and diagnosis. Acta Pathol. Microbiol. Immunol. Scand. 96,
857 – 881.
Gundersen, H.J.G., Jensen, E.B.V., Kieû, K., Nielsen, J., 1999. The
efficiency of systematic sampling in stereology-reconsidered. J.
Microsc. 193, 199 – 211.
Heinsen, H., Rüb, U., Bauer, M., Ulmar, G., Bethke, B., Schüler, M.,
Böcker, F., Eisenmenger, W., Götz, M., Korr, H., Schmitz, C.,
1999. Nerve cell loss in the thalamic mediodorsal nucleus in
Huntington’s disease. Acta Neuropathol. 97, 613 – 622.
Heinsen, C., Arzberger, T., Schmitz, C., 2000. Celloidin embedding of
human brains — a new, simple and reliable method for producing serial sections of high section thickness. J. Chem. Neuroanat.
20, 49 – 59.
James, F., 1990. A review of pseudorandom number generators.
Comput. Phys. Commun. 60, 329 – 344.
Knuth, D.E., 1981. The Art of Computer Programming (Seminumerical Algorithms), vol. 2. Addison-Wesley, Reading, MA.
König, D., Carvajal-Gonzales, S., Downs, A.M., Vassy, J., Rigaut,
J.P., 1991. Modelling and analysis of 3-D arrangements of particles by point processes with examples of application to biological
data obtained by confocal scanning light microscopy. J. Microsc.
161, 405 – 433.
Korbo, L., West, M., 2000. No loss of hippocampal neurons in AIDS
patients. Acta Neuropathol. 99, 529 – 533.
Larsen, J.O., 1998. Stereology of nerve cross sections. J. Neurosci.
Methods 85, 107 – 118.
L’Ecuyer, P., 1988. Efficient and portable combined random number
generators. Commun. A.C.M. 31, 742 – 749 and 774.
Marsaglia, G., 1985. A current view of random number generators.
In: Billard, G. (Ed.), Computer Science and Statistics: The Interface. Elsevier, Amsterdam, pp. 3 – 10.
Matheron, G., 1965. Les Variables Régionalisées et leur Estimation.
Thesis, Masson et Cie, Paris.
Matheron, G., 1971. The Theory of Regionalized Variables and Its
Applications. Cahiers du Centre de Morphologie Mathématique
de Fontainebleau, no. 5. École Nationale Supérieure des Mines de
Paris.
Mayhew, T.M., Gundersen, H.J.G., 1996. If you assume, you can
make an ass out of u and me: a decade of the disectorfor
sterological counting of particles in 3D space. J. Anat. 188, 1–15.
114
C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114
McShane, L.M., Palmatier, M., 1994. Spatial distribution of neurons in tissue culture wells: implications for sampling methods
to estimate population size. Stat. Med. 13, 523–540.
Nicholson, W.L., 1978. Application of statistical methods in quantitative microscopy. J. Microsc. 113, 223–239.
Nyengaard, J.R., 1999. Stereologic methods and their application
in kidney research. J. Am. Soc. Nephrol. 10, 1100–1123.
Oorschot, D.E., 1996. Total number of neurons in the neostriatal,
pallidal, subthalamic, and substantia nigral nuclei of the rat
basal ganglia: a stereological study using the cavalieri and optical disector methods. J. Comp. Neurol. 366, 580–599.
Reed, M.G., Howard, C.V., 1997. Edge-corrected estimators of the
nearest-neighbour distance distribution function for three-dimensional point patterns. J. Microsc. 186, 177–184.
Ripley, B.D., 1987. Stochastic Simulation. Wiley, New York.
Roberts, N., Garden, A.S., Cruz-Orive, L.M., Whitehouse, G.H.,
Edwards, R.H.T., 1994. Estimation of fetal volume by magnetic
resonance imaging and stereology. Br. J. Radiol. 67, 1067 –
1077.
Scheaffer, R.L., Mendenhall, W., Ott, L., 1996. Elementary Survey
Sampling, fifth ed. PWS-Kent, Boston.
Schmitz, C., 1997. Towards more readily comprehensible procedures in disector stereology. J. Neurocytol. 26, 707–710.
Schmitz, C., 1998. Variation of fractionator estimates and its prediction. Anat. Embryol. 198, 371–397.
Schmitz, C., 2000. Towards the use of state-of-the-art stereology in
experimental gerontology. Exp. Gerontol. 35, 429–431.
Schmitz, C., Bültmann, E., Gube, M., Korr, H., 1999a. Neuron
loss in the mouse hippocampus following prenatal injection of
.
tritiated thymidine or saline. Int. J. Dev. Neurosci. 17, 185–
190.
Schmitz, C., Rüb, U., Korr, H., Heinsen, H., 1999b. Nerve cell
loss in the thalamic mediodorsal nucleus in Huntington’s disease. II. Optimization of a stereological estimation procedure.
Acta Neuropathol. 97, 623 – 628.
Searle, S.R., 1987. Linear Models for Unbalanced Data. Wiley,
New York.
Simic, G., Kostovic, I., Winblad, B., Bogdanovic, N., 1997. Volume and number of neurons of the human hippocampal formation in normal aging and Alzheimer’s disease. J. Comp. Neurol.
379, 482 – 494.
Thioulouse, J., Royet, J.P., Ploye, H., Houllier, F., 1993. Evaluation of the precision of systematic sampling: nugget effect and
covariogram modelling. J. Microsc. 172, 249 – 256.
West, M.J., 1993. New stereological methods for counting neurons.
Neurobiol. Aging 14, 275 – 285.
West, M.J., Gundersen, H.J.G., 1990. Unbiased stereological estimation of the number of neurons in the human hippocampus.
J. Comp. Neurol. 296, 1 – 22.
West, M.J., Slomianka, L., Gundersen, H.J.G., 1991. Unbiased
stereological estimation of the total number of neurons in the
subdivisions of the rat hippocampus using the optical fractionator. Anat. Rec. 231, 482 – 497.
West, M.J., Østergaard, K., Andreassen, O.A., Finsen, B., 1996.
Estimation of the number of somatostatin neurons in the striatum: an in situ hybridization study using the optical fractionator method. J. Comp. Neurol. 370, 11 – 22.