Recommendations for straightforward and …research.mssm.edu/cnic/pdfs/SchmitzHof.pdfJournal of...

Journal of Chemical Neuroanatomy 20 (2000) 93–114

Recommendations for straightforward and rigorous methods ofcounting neurons based on a computer simulation approach

Christoph Schmitz a,*, Patrick R. Hof b

a Department of Anatomy and Cell Biology, RWTH Uni6ersity of Aachen, Pauwelsstrasse/Wendlingweg 2, 52057 Aachen, Germanyb 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 isthe best method for estimating the total number of neurons? What are the validity and the expected precision of the obtaineddata? What precision must the estimates attain with respect to the scientific question? In the present study, these questions wereaddressed 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 moderndesign-based stereology, either by using the ‘fractionator’ method or by the established method based on the product of estimatedneuron density and estimated volume of the reference space. We show that estimates of total numbers of neurons obtained usingthe fractionator are from a statistical and economical standpoint more efficient than corresponding estimates obtained using thedensity/volume procedure. Furthermore, the use of two simple prediction methods (one for homogeneous and the other forclustered neuron distributions) permits satisfactory predictions about the variation of presumably any estimates of total numbersof neurons obtained using the fractionator. Finally, we show that assessing the reliability of estimates of mean total neuronalnumbers using the ratio between the mean of the squared coefficients of error of the estimates and the squared coefficient ofvariation of the estimated total neuronal numbers, a frequently employed method in stereological studies, is neither useful norinformative. 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 estimatingtotal neuronal numbers using design-based stereology. © 2000 Elsevier Science B.V. All rights reserved.

Keywords: Cell count; Fractionator; Morphometry; Stereology

www.elsevier.com/locate/jchemneu

1. Introduction

The proper assessment of the total number of neu-rons in a given brain region depends on the accuracy ofthe method used as an estimator, on the validity andthe expected precision of the obtained data, and on theprecision that the estimates must attain to address thescientific question. Unfortunately, none of these issueshas an easy solution.

First, in modern, design-based stereology there aretwo methods available for estimating total numbers ofneurons, namely the so-called ‘fractionator’ (Gun-

dersen, 1986) and the so-called ‘Vref×NV’ method(West and Gundersen, 1990). Using the fractionator,neurons in a defined, systematically and randomly sam-pled part of the entire brain region of interest arecounted, and the total neuronal number is estimated bymultiplying the number of counted neurons by thereciprocal value of the sampling probability (henceforthreferred to as ‘fractionator estimates’). Using the Vref×NV method, an estimate of the total number of neuronsis obtained by multiplying the estimated total volumeof this brain region (6ref) by the estimated mean neurondensity (nV) in a systematically and randomly sampledpart 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 choicebetween these methods may be based on both statisticaland economical efficiency (the term ‘statistical effi-

* Corresponding author. Tel.: +49-241-8089548; fax: +49-241-8888431.

E-mail address: [email protected] (C.Schmitz).

0891-0618/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.PII: S 0 8 9 1 -0618 (00 )00066 -1

C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–11494

ciency’ refers to the precision of an estimator, which ismore precise as its variation is smaller (Bronshtein andSemendyayev, 1985). In contrast, the term ‘economicalefficiency’ refers to the amount of work needed toguarantee a given precision of the estimates (for detailssee Schmitz, 1998). However, a systematic comparisonbetween the fractionator and the Vref×NV method withrespect to statistical and economical efficiency has notbeen performed.

Second, for the Vref×NV method there are threedifferent methods available for predicting the precisionof the obtained estimates (Table 1; henceforth referredto as ‘prediction methods’). Moreover, 16 differentprediction methods were reported in the literaturewhich have been or might be applied to assess theprecision of fractionator estimates (Table 1; the early,preliminary method given by Gundersen, 1986; Equa-tion (2.11) is not considered here). Most of these pre-diction methods (V1, V2, F1–F4, F6, F8–F15) arebased on complex theoretical statistics. Others, such as

F7, have been developed mainly by computer simula-tion (Schmitz, 1998; Glaser and Wilson, 1998, 1999) orwere applied without presentation of the theoreticalbackground (V3, F5). In three recent reports some ofthese prediction methods have been compared usingcomputer simulation (Schmitz, 1998; Glaser andWilson, 1998, 1999). However, a comprehensive com-parison of all these prediction methods is not availablein the literature.

Third, to design properly an experiment, importantquantities to determine are, for example, the minimaldifference one wants to detect between the means oftwo populations under comparison on the one hand,and the biological variation and the precision of theestimates on the other. A recent attempt to find asolution to this problem (Geinisman et al., 1996) isdemonstratedly unsatisfactory (Schmitz et al., 1999b).Another, frequently used approach to demonstrate thereliability of an estimated mean total number of neu-rons is to show that the mean of the squared coeffi-cients of error of the estimates as predicted with one ofthe prediction methods summarized in Table 1 is lessthan half of the squared coefficient of variation of theestimated total neuronal numbers (henceforth referredto as ‘CE2/OCV2 approach’; for recent examples seeGeinisman et al., 1996; Begega et al., 1999; Korbo andWest, 2000; among others). This approach is based ona 20-year-old scheme designed to optimize the samplingefficiency of stereological studies in biology (Gundersenand Østerby, 1981) and has often been described inguidelines how to carry out stereological studies (forrecent examples see West, 1993; Larsen, 1998; Nyen-gaard, 1999; among others). On the other hand, thisapproach has been often criticized (Schmitz, 1997,2000) based on the relevant statistical literature(Nicholson, 1978; Searle, 1987). An in-depth evaluationof the power of the CE2/OCV2 approach is notavailable.

This study is aimed to clarify this unsettled situation.It is not intended to provide a comprehensive analyticalapproach for solving the mentioned problems. If at allpossible, an analytical approach would require detailedtheoretical–statistical considerations and would there-fore be difficult to understand by neuroscientists notfamiliar with the relevant statistics but interested inquantitative neuroanatomy. Rather, this study is in-tended to provide simple solutions for the mentionedproblems by analyzing the results of repeated estimatesof the total number of neurons in the same brainregion, or by repeated estimates of mean total neuronalnumbers of populations of individuals. For methodo-logical reasons this cannot be achieved by biologicalexperiments (Cruz-Orive, 1994; Schmitz, 1998). There-fore, we have addressed this issue by a computer simu-lation approach. The description of the computersimulation is presented in parallel to descriptions of real

Table 1Methods for predicting the precision of estimated total neuronalnumbers obtained using either the Vref×NV method (V1–V3) or thefractionator (F1–F15), available in the literature

Method Source

V1 Table 3 in West and Gundersen (1990)Table 4 in Geinisman et al. (1996)V2

V3 Appendix A in Simic et al. (1997)F1 Equation (6) in Gundersen and Jensen (1987); cf.

also Table 5 in West et al. (1991)Equation 20 in Gundersen and Jensen (1987)F2

F3 Equation 20 in Cruz-Orive (1990)F4 Discussion in Thioulouse et al. (1993)

Equation A.2 in Larsen (1998)F5Chapter 4 in Scheaffer et al. (1996)F6Equation A.4 in Larsen (1998); cf. also Appendix inGlaser and Wilson (1998); Appendix in Glaser andWilson (1999)

F7 Figure 6 [‘P6’] in Schmitz (1998); Equation A.5 inGlaser and Wilson (1998); Equation A.5 in Glaserand Wilson (1999); Equation 24 in Nyengaard(1999)

F8 ‘Explicit nugget formula’ (Equation 3.12) with m=0in Cruz-Orive (1999)

F9 ‘Explicit nugget formula’ (Equation 3.12) with m=1in Cruz-Orive (1999)

F10 ‘Implicit nugget formula’ with m=0 (Subsection 3.4)in Cruz-Orive (1999)

F11 ‘Implicit nugget formula’ with m=1 (Subsection 3.4)in Cruz-Orive (1999)

F12 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.F13(1999)Use of Equation 14; i.e. m=0 in Gundersen et al.F14(1999)Use of Equation 15; i.e. m=1 in Gundersen et al.F15(1999)

C. Schmitz, P.R. Hof / Journal of Chemical Neuroanatomy 20 (2000) 93–114 95

studies involving estimates of total numbers of neurons.We show that estimates of total neuronal numbersobtained using the fractionator are from a statisticaland economical standpoint more efficient than corre-sponding estimates obtained using the Vref×NV

method. Furthermore, the use of two simple predictionmethods (one for homogeneous and the other for clus-tered neuron distributions) permits satisfactory predic-tions about the variation of presumably any estimatesof total neuronal numbers obtained using the fractiona-tor. Finally, we show that assessing the reliability ofestimates of mean total neuronal numbers using theCE2/OCV2 approach is neither useful nor informative.The presented results may constitute a new set ofrecommendations for the rigorous usage of design-based stereology.

We are aware that many scientists interested in quan-titative neuroanatomy are not familiar with details ofcomputer simulations. On the other hand, many expla-nations are necessary to facilitate repeating the work byother laboratories. Therefore, Sections 2 and 3 of thisstudy are presented in the following manner. Readersnot familiar with stereological nomenclature or inter-ested only in a fast overview should only read the textgiven in normal fonts. Readers interested in details ofthe work should read the entire text in the presentedorder.

2. Materials and methods

2.1. Experiment 1

Experiment 1 was intended to investigate the influ-ence of the shape of the reference space, of the spatialdistribution of neurons within this reference space andof stereological sampling on the variation of estimatedtotal neuronal numbers. This was achieved by modelingvarious virtual brain regions (VBR) with different vir-tual reference spaces (VRS) and different neuron distri-butions within these reference spaces, and by modelingestimates of the total neuronal numbers of these VBRwith different stereological sampling schemes.

2.1.1. Modeling of 6irtual brain regionsSixteen different VBR were modeled and consisted of

virtual neurons (VN) in VRS. Details of these VBR aresummarized in Table 2; schematic illustrations of theVRS are shown in Fig. 1. VRSi might be interpreted asa virtual rat external globus pallidus, and VRSii as avirtual rat striatum considering only the striosomes.

For VBR1–VBR4 and VBR9–VBR12, the VRS was asphere with radius r=850 mm and a volume of 6=2.57mm3. This volume was similar to the estimated volumeof the rat external globus pallidus (Oorschot, 1996).

Table 2Details on runs A–Y of the simulations carried out in Experiment 1a

Run VBR VRS c of VN VSSSPP

i1 aA 500 1500i2B d3 1

3 iC 500 d6 14 iD 500 z3 1

aE 15 ii 500d3F 16 ii 500

1d6500G ii78 500 z3 1H ii

I 2a50 000i9d350 000 2i10K

11 50 000 d6 2L i12 iM 50 000 z3 2

N 13 ii 50 000 a 22d350 000O ii14

15 iiP 50 000 d6 2Q 16 ii 50 000 z3 2

a 3R 9 i 50 00010 iS 50 000 d3 3

T 11 i 50 000 d6 33z350 000i12U3a13 50 000V ii

14 iiW 50 000 d3 3X 15 ii 50 000 d6 3

16 ii 50 000 z3 3Y

a VBR, virtual brain region; VRS, virtual reference space; VN,virtual neurons; SPP, spatial point process; VSS, virtual samplingscheme; i, pallidal VRS; ii, striatal VRS (for 3-D sketchs of VRSi andVRSii see Fig. 1); a, homogeneous SPP; d3, centripetal SPP; d6,centrifugal SPP; z3, clustered SPP (for schematic representations ofSPPa, 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 con-sisted of 1000 small spheres with r=85 mm, which werearranged 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 of50 000 VN (VBR9–VBR16). VN were modeled aspoints and were arranged in the VRS according to fourdifferent so-called ‘spatial point processes’ (SPP) asschematically shown in Fig. 2. SPPa might be inter-preted as a homogeneous distribution of VN, SPPd3 asa centripetal VN distribution, SPPd6 as a centrifugalVN distribution and SPPz3 as a clustered VNdistribution.

The SPP applied here were the same as described indetail 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 pro-cesses’, in which the point density was allowed to varyas a function of distance from the center of the refer-ence space (radius). The density function of SPPd3

decreased as a 6th degree polynomial function of ra-dius, whereas the density function of SPPd6 increased as


Fig. 1. 3-D sketch of VRS used to model VBR. VRSi is a sphere withradius r=850 mm and volume 6=2.57 mm3. VRSii consists of 1000small 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 asa virtual rat external globus pallidus and VRSii as a virtual ratstriatum considering only the striosomes.

Fig. 3. Schematic summary of modeling estimates of total neuronalnumbers using either the fractionator or the Vref×NV method. (a)Schematic representation of S=13 parallel, systematically and ran-domly sampled sections of a VBR with pallidal VRS. (b) Rectangularlattice with side lengths sl, systematically and randomly placed on theupper surface of a section of the VBR (shown enlarged for betterdemonstration of details). This lattice determines the positions ofcubic counting spaces (dark squares) for counting neurons. Such alattice is also used to estimate the surface area of this section bycounting the intersections of the lattice situated within the VBR(arrows), and is used to estimate the boundary length of this sectionby counting the intersections of the lattice with the boundaries of thesection (arrowheads). For clarity only one lattice is shown, althoughin the computer simulation three different lattices were used forcounting neurons, estimating surface areas and boundary lengths. (c)Cubic counting spaces with edge e, systematically and randomlyplaced in regular intervals sl in the central part of the sectionthickness t. VN situated within the counting spaces are counted.

a 6th degree polynomial function of radius. SPPz3 wasa ‘Poisson cluster process’, based on 250 so-called‘parent points’ (see Schmitz, 1998, for details). Detailedapproaches to SPP as well as techniques for simulatingthem may be found in Cox and Isham, 1980; Diggle,1983; Ripley, 1987; among others).

2.1.2. Modeling of estimates of total neuronal numbersEstimates of total neuronal numbers were modeled as

recently described in detail (Schmitz, 1998). All stepscarried out in real estimates of total neuronal numbersof brain regions of interest using the fractionator or the

Fig. 2. Spatial distributions of VN in VRS used to model VBR. VNare modeled as points and are arranged in the VRS shown in Fig. 1according to so-called SPP. The figure shows modeled 100 mm thicksections 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. TheSPP applied here are the same as described as SPPa, SPPd3, SPPd6 andSPPz3 in Schmitz (1998).

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 Cartesiancoordinate system (V) and dissected to a total numberof S parallel, isotropic uniform random (IUR) sectionswith section thickness t and normal vectors parallel tothe z-axis of V (Fig. 3a).

For modeling fractionator estimates, rectangular lat-tices with uniform side length slN were placed in asystematic–random manner on the upper surface of thesections (Fig. 3b). These lattices determined the posi-tions of cubic counting spaces with edge e in the centralpart of the section thickness (Fig. 3c). All VN situatedwithin the counting spaces (Q−; for the formal defini-tion of the mnemonic − in the context of estimates oftotal neuronal numbers see Gundersen, 1986) werecounted. Estimates of total numbers of VN (nF) werecalculated as shown in Eq. (1) (Gundersen, 1986):


nF=% Q−(slN)2t

e3 (1)

For modeling Vref×NV estimates, the same rectangu-lar lattices for placing cubic counting spaces in the centralpart of the section thickness were used as if applying thefractionator. Also, all VN situated within the countingspaces (Q−) were counted. From the number of countedVN (� Q−) and the number of counting spaces used(� F), estimates of the mean VN density (nV) werecalculated as shown in Eq. (2) (West and Gundersen,1990):

nV=% Q−

e3% F(2)

For modeling estimates of the average surface area ofthe sections (A( ), a second set of rectangular lattices withuniform side length slA was placed in a systematic-ran-dom manner on the upper surface of the sections (Fig.3b). All intersections of the lattices situated within theVRS were counted (P ; arrows in Fig. 3b). Estimates ofA( (i.e. a) were calculated as shown in Eq. (3) (Gundersenand Jensen, 1987):

a=% P(slA)2

S(3)

Volume estimates (6ref) were calculated according tothe 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 calcu-lated as shown in Eq. (5) (West and Gundersen, 1990):

nV×N=6ref×nV (5)

To obtain estimates of the average total boundarylength of the sections (B( ), a third set of rectangularlattices with uniform side length slB was placed in asystematic–random manner on the upper surface of thesections (Fig. 3b). All intersections of the lattices with theboundaries of the sections (IS) were counted. Estimatesof B (i.e. b) were calculated according to the Buffon’sprinciple (Buffon, 1777) as shown in Eq. (6) (Cruz-Orive,1997):

b=% IS×0.25×p×slB

S(6)

From the estimated average surface area (a) and theestimated average total boundary length of the sections(b), estimates of the shape coefficient (SC=B/ A) werecalculated as shown in Eq. (7) (cf. Roberts et al., 1994):

sc=b

a(7)

The estimates of total VN numbers using the fraction-ator or the Vref×NV method were carried out using threedifferent virtual sampling schemes (VSS), resulting indifferent numbers of counted neurons and thereforedifferent variation of the estimates. Details on these VSSare provided in Table 3.

VSS1 and VSS2 were intended to simulate samplingapproximately (: ) 150 VN with :150 counting spaceswhen 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 VNnumbers of VBR9–VBR16. The average numbers ofpoints counted for estimating the volume of the corre-sponding reference spaces were 150 (VSS1 and VSS2) or750 (VSS3). The average numbers of IS counted fordetermining the shape coefficient of the correspondingreference 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 thecomputer simulation were performed (A–Y; see Table 2).

Each run consisted of 1000 repetitions of a simulatedstereological procedure, resulting in 1000 fractionatorestimates of the total number of VN of the investigatedVBR as well as in 1000 Vref×NV estimates of the totalVN number of this VBR.

This was equal to that what has been described byCruz-Orive (1994) as a ‘rewinding of a video movie ofthe splitting process and repeating the sampling proce-dure again with fresh random numbers to select thedifferent sampling units’. The position of the investi-gated VBR in space as well as the positions of thelattices on the sections were changed for each repetitionof the simulation, whereas the distributional pattern ofthe VN in the VRS was changed after each 50 repeti-tions. This was carried out to prevent dependence ofthe obtained results on a single realization of the ap-

Table 3Details on the VSS useda

VSS3VSS1 VSS2

151t (mm) 25825811–197–11S 7–1137.237.2e (mm) 172.67

slN (mm) 258 258 151257257 150slA (mm)

55015001500slB (mm)

a t, Section thickness; S, number of sections; e, edge of the cubiccounting spaces; slN, side length of the rectangular lattices for placingthe counting spaces; slA, side length of the rectangular lattices forestimating surface areas of sections of VBR; slB, side length of therectangular lattices for estimating boundary lengths of sections ofVBR.


plied SPP. Therefore, 20 distributional patterns of VNwere derived from each type of SPP, and a total of24×2×1000=48 000 estimated total numbers of VNwas obtained. For each estimated total number of VN,the corresponding predicted coefficients of error[CEpred(n)] were calculated using all methods shown inTable 1 (a note on the use of method F8–F11 is givenin Appendix A).

2.1.3. Analysis of the estimatesThe results of Experiment 1 were analyzed for the

relationship between the real variation of estimatedtotal VN numbers and the predictions of this variationobtained using the prediction methods summarized inTable 1. This was carried out by calculating ratio R4

shown in Formula (16). R4=1 indicates that the corre-sponding prediction method resulted in an exact meanprediction of the variation of the estimates. R4\1indicates that the variation of the estimated total VNnumbers was overestimated, whereas R4B1 indicatesthat the variation of the estimated total VN numberswas underestimated.

The entire analysis of the results of Experiment 1comprised calculation of the following variables:

Mean, S.D. and empirically estimated coefficient oferror of the 1000 estimates of Vref and of the 1000estimates of NV obtained using the Vref×NV method:

CEemp(6ref)=S.D.(6ref)mean(6ref)

(8)

CEemp(nV)=S.D.(nV)mean(nV)

(9)

The ratio between CEemp(6ref) and CEemp(nV):

R1=CEemp(6ref)CEemp(nV)

(10)

Mean, S.D. and empirically estimated coefficient oferror of the 1000 estimated total numbers of VN ob-tained using the Vref×NV method:

CEemp(nV×N)=S.D.(nV×N)mean(nV×N)

(11)

The ratio between [CEemp(6ref)+CEemp(nV)] andCEemp(nV×N):

R2=[CEemp(6ref) + CEemp(nV)]

CEemp(nV×N)(12)

Mean, S.D. and empirically estimated coefficient oferror of the 1000 estimated total numbers of VN ob-tained using the fractionator:

CEemp(nF)=S.D.(nF)mean(nF)

(13)

The ratio between CEemp(nV×N) and CEemp(nF):

R3=CEemp(nV×N)

CEemp(nF)(14)

The mean of the 1000 squared predicted coefficientsof error of either nV×N or nF, calculated separately foreach of the applied prediction methods (herein, themean of these data is called ‘meanE1’):

meanE1{[CEpred(n)]2}=%

1000

Run=1

([CEpred(n)]2)Run

1000(15)

The ratio between meanE1{[CEpred(n)]2} and[CEemp(n)]2, also calculated separately for each of theapplied prediction methods:

R4=meanE1{[CEpred(n)]2}

[CEemp(n)]2(16)

2.2. Experiment 2

Experiment 2 was intended to investigate the influ-ence of both stereological sampling and biological vari-ability (as well as the relationship between thesevariables) on the observed interindividual variation ofestimated total numbers of neurons of a sample ofindividuals.

2.2.1. Modeling of populations of 6irtual brain regionsFive different virtual populations (P1–P5) of 15 000

VBR each were modeled, differing in the VN distribu-tions within the VRS and the frequency distributions ofthe total VN numbers of the 15 000 VBR each (forillustration see Table 4 and Fig. 4). The mean totalnumber of VN per VBR was approximately 50 000 foreach population.

VBR in populations P1–P4 consisted of homoge-neously distributed VN in a spherical VRS with radiusr=850 mm (according to VRSi−SPPa in Fig. 2),whereas VBR in population P5 consisted of clusteredVN in a spherical VRS with the same radius (accordingto VRSi−SPPz3 in Fig. 2).

The total VN numbers of the 15 000 VBR each inP1–P5 (i.e. the frequency distributions of the total VNnumber of P1–P5; FDP1–FDP5) were obtained by usingthe integer values of 15 000 pseudorandom numberseach generated with the pseudorandom number genera-tor implemented in MS Excel for Windows 95, version7.0. This pseudorandom number generator allows thegeneration of a preselected number of pseudorandomnumbers (here, 1000, 14 000 or 15 000 as given in detailin Table 4) according to a preselected distribution(here, ‘standard’), a preselected mean (here, each time50 000) and a preselected S.D. (here, 50, 2000, 6000,7500 or 23 500 as given in detail in Table 4). FDP1 andFDP3 were generated by one realization of this pseu-dorandom number generator, and FDP2 and FDP4 bytwo realizations each of this pseudorandom numbergenerator. FDP5 was identical to FDP1.


Table 4Details on the virtual populations [P1–P5] of VBR modeled in Experiment 2a

P2 P3 P4P1 P5

15 000c of VBR 15 000 15 000 15 000 15 000i ii iVRS i

aSPP a a a z314 000 15 000c of PRN (first realization) 14 00015 000 †50 000 50 00050 000 50 000Preselected mean (first realization) †

50 6,000Preselected S.D. (first realization) 502000 †1000 –– 1000c of PRN (second realization) –

–Preselected mean (second realization) 50 000 – 50 000 –7500 –– 23 500Preselected S.D. (second realization) –

42 487c of VN-minimum 28 314 28 179 1000 42 48779 843 75 146 149 912 57 888c of VN-maximum 57 88850 023 50 02049 973 50 012c of VN-mean 49 973

1977 5999 5981 2012c of VN-S.D. 20120.040 0.120 0.120 0.0400.040c of VN-CV

a VBR, virtual brain region; VRS, virtual reference space; SPP, spatial point process; PRN, pseudorandom numbers generated with thepseudorandom number generator implemented in MS Excel for Windows 95, version 7.0. Preselected mean, preselected mean of PRN whengenerating PRN according to a ‘standard’ distribution with this pseudorandom number generator. Preselected S.D., preselected standard deviationof PRN when generating PRN according to a ‘standard’ distribution with this pseudorandom number generator. †, No use of the pseudorandomnumber generator, since the frequency distribution of the total VN number of P5 was identical to the frequency distribution of the total VNnumber 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 schematicrepresentations of SPPa and SPPz3 see Fig. 2).

2.2.2. Modeling of estimates of mean total neuronalnumbers

All steps carried out in real estimates of the meantotal neuronal number of a sample of individuals se-lected from a population were modeled. A number ofVBR was selected from the investigated population,and the total VN numbers of the selected VBR wereestimated using the fractionator. With respect to theinvestigated populations, the numbers of selected VBRand the VSS applied, 18 different virtual stereologicalstudies (VSTST) were carried out as summarized inTable 5 [c1 to c18].

Each VSTST consisted of the following steps. First,either f=6 or f=12 VBR were uniformly and ran-domly selected from the investigated population. ‘Uni-formly and randomly selected’ means that each VBRhad the same chance to be selected. From the realnumbers of VN of the selected VBR, mean, S.D. andcoefficient of variation were calculated. According tothe literature, the square of this coefficient of variationwas named ‘real inherent biological variance of theindividuals’ (West, 1993; ICV2). Second, the total num-bers of VN of the selected VBR were estimated onceusing the fractionator as explained above, using eitherVSS2 or VSS3 (Table 3). From the estimated totalnumbers of VN of the selected VBR, mean, S.D. andcoefficient of variation were calculated. According tothe literature, the square of this coefficient of variationwas named ‘observed relative variance of group’ (West,1993; OCV2). Predicted coefficients of error of theestimated total numbers of VN were calculated usingmethod F7 (Table 1). Herein, the mean of these data iscalled ‘meanE2’:

meanE2{[CEpred(n)]2}=%f

VBR=1

([CEpred(n)]2)VBR

f(17)

Each VSTST was carried out 1000 times, startingwith selecting VBR from the investigated population.

2.2.3. Analysis of the estimatesThe results of Experiment 2 were analyzed for the

relationship between the interindividual variation of theestimated total VN numbers of the selected VBR on theone hand (OCV2), and the sum of the interindividualvariation of the true total VN numbers of the selectedVBR and the predicted mean variation of the estimateson the other hand (SICV−CE). This was carried out bycalculating ratio R5 shown in Formula (19). R5=0indicates that there was no difference between OCV2

and SICV−CE. R5\0 indicates that SICV−CE wasgreater than OCV2, whereas R5B0 indicates thatSICV−CE was smaller than OCV2.

Furthermore, the relationship between the predictedmean variation of the estimates and the interindividualvariation of the estimated total VN numbers of theselected VBR was analyzed. This was carried out bycalculating ratio R6 shown in Formula (20).

The entire analysis of the results of Experiment 2comprised calculation of the following variables foreach run of the VSTST.

The sum of ICV2 and [meanE2{[CEpred(n)]2}]:

SICV−CE=ICV2+meanE2{[CEpred(n)]2} (18)

The difference between SICV−CE and OCV2:

R5=SICV−CE−OCV2

SICV−CE+OCV2 (19)


The ratio between [meanE2{[CEpred(n)]2}] and OCV2:

R6=meanE2{[CEpred(n)]2}

OCV2 (20)

The mean of the 1000 values of OCV2:

mean(OCV2)=%

1000

Run=1

(OCV2)Run

1000(21)

The mean of the 1000 values of SICV−CE:

mean(SICV−CE)=%

1000

Run=1

(SICV−CE)Run

1000(22)

The difference between mean(SICV−CE) andmean(OCV2):

R7=mean(SICV−CE)−mean(OCV2)mean(SICV−CE)+mean(OCV2)

(23)

Further analysis of the data is presented in AppendixB.

2.3. Source of randomness

A pseudorandom number generator provided byL’Ecuyer (1988) (Figure 3 in this study; PRNG1) wasused as source of randomness (details of PRNG1 aregiven in Appendix C). Repeating the entire simulationwith another pseudorandom number generator pro-vided by L’Ecuyer (1988) (Figure 4 in this study;PRNG2) led to almost identical results (details ofPRNG2 are also given in Appendix C). Therefore, onlyresults obtained using PRNG1 are presented.

3. Results

3.1. Results of Experiment 1

To describe the results of Experiment 1, it is neces-sary to compare results from different runs of thecomputer simulations. For the sake of clarity, thesecomparisons will be presented in an abbreviated for-mat. For example, a comparison between runs A–D ofExperiment 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 VBRin population P5 are modeled as clustered VN distribution in pallidal VRS (according to VRSi−SPPz3 in Fig. 2). For P1 and P5, total VN numberis 49 97392012 (mean9S.D.), for P2 it is 50 02391977, for P3 50 02095999, and for P4 50 01295981. The frequency distributions of totalVN number of P1, P3 and P5 approximate a Gaussian distribution, whereas the frequency distributions of P2 and P4 do not. Notation of the datain brackets means that the smaller value was not included in the corresponding class.


Table 5Details on the VSTST modeled in Experiment 2a

fVSTST VSSP

6 2c1 162 2c2

3c3 6 24c4 6 2

121 2c512 2c6 2123 2c7

4c8 12 21c9 6 3

62 3c106c11 3364 3c12

1c13 12 32c14 12 3

123 3c1512c16 34

65 2c17c18 65 3

a P, investigated population; f, number of selected individuals;VSS, applied sampling scheme. Details on VSS2 and VSS3 are givenin Table 3.

was approximately 500 [VBR1–VBR8] or approxi-mately 50 000 [VBR9–VBR16].

Fig. 5a shows the empirically estimated coefficientsof 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 ob-tained if estimating Vref of VBR with striatal VRS by

Fig. 5. Results of Experiment 1, shown as a function of the runs ofthe simulation [A–Y; details of these runs are given in Table 2]. Eachrun consists of 1000 modeled estimates of the total number of VN ofone 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 thesame VBR using the fractionator. The ordinates of all graphs aretruncated at the shown values. (a) Empirically estimated coefficient oferror of the volume estimates (6ref) of the investigated VBR obtainedusing the Vref×NV method [CEemp(6ref)]. The highest values ofCEemp(6ref) are obtained for VBR with striatal VRS by using thevirtual 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 forVBR with pallidal VRS by using VSS3 (runs R, S, T, U; for VSS3 seealso Table 3). (b) Empirically estimated coefficient of error of the VNdensity estimates (nV) of the investigated VBR obtained using theVref×NV method [CEemp(nV)]. The highest values of CEemp(nV) areobtained for VBR with clustered VN distribution (runs D, H, M, Q,U and Y), and the smallest values for VBR with homogeneous VNdistribution (runs A, E, I, N, R and V). (c) Results obtained for ratioR1 [Eq. (10); i.e. ratio between CEemp(6ref) and CEemp(nV)]. Thehighest values of ratio R1 are obtained for VBR with striatal VRSand homogeneous VN distribution (runs E, N and V), and thesmallest values for VBR with pallidal VRS and clustered VN distribu-tion (runs D, M and U). (d) Empirically estimated coefficient of errorof estimated total VN numbers [nV×N] obtained using the Vref×NV

method [CEemp(nV×N)]. The highest values of CEemp(nV×N) areobtained for VBR with clustered VN distribution (runs D, H, M, Q,U and Y), and the smallest values for VBR with homogeneous VNdistribution (runs A, E, I, N, R and V). Note that the values obtainedfor 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. ratiobetween [CEemp(6ref)+CEemp(nV)] and CEemp(nV×N)]. This ratio isalways greater than 1. (f) Empirically estimated coefficient of error ofestimated total VN numbers [nF] obtained using the fractionator[CEemp(nF)]. The highest values of CEemp(nF) are obtained for VBRwith clustered VN distribution (runs D, H, M, Q, U and Y), and thesmallest values for VBR with homogeneous VN distribution (runs A,E, I, N, R and V). Note that the values obtained for CEemp(nF) aresimilar 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 withcentrifugal VN distribution of 50 000 VN in pallidal VRS (runs Land T). There are three essential findings of Experiment 1 shown inthis figure. First, CEemp(nV×N) and CEemp(nF) vary as a function ofthe 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, thevariation of Vref×NV estimates is not simply the sum of the variationof the number of counted neurons and the variation of the volumeestimates (e). Rather, the covariance between these variables has to beconsidered if calculating the variation of Vref×NV estimates (Gun-dersen and Jensen, 1987). Third, except for VBR with centrifugal VNdistribution of 50 000 VN in pallidal VRS (runs L and T), thefractionator estimates have a greater precision than the correspondingVref×NV estimates.

For all 24 runs of the computer simulation, the meanof the 1000 estimated total numbers of VN obtainedusing either the fractionator or the Vref×NV method

Fig. 5.


using VSS1 or VSS2 [runs E, F, G, H and N, O, P, Q],and the smallest values if estimating Vref of VBR withpallidal VRS by using VSS3 [runs R, S, T, U].

Fig. 5b displays the empirically estimated coefficientsof error of nV obtained using the Vref×NV method.CEemp(nV) varied as a function of the applied SPP [runsA–B–C–D, E–F–G–H, etc.], as a function of theVRS [runs A–E, B–F, etc.], as a function of the totalnumber of VN [runs A–I, B–K, etc.] and as a functionof the VSS used [runs I–R, K–S, etc.]. The highestvalues of CEemp(nV) were obtained for VBR with clus-tered VN distribution [runs D, H, M, Q, U and Y], andthe smallest values for VBR with homogeneous VNdistribution [runs A, E, I, N, R and V].

Fig. 5c shows the results obtained for ratio R1. Thisratio varied as a function of the applied SPP [runsA–B–C–D, E–F–G–H, etc.], as a function of theVRS [runs A–E, B–F, etc.], as a function of the totalnumber of VN [runs A–I, B–K, etc.] and as a functionof the VSS used [runs I–R, K–S, etc.]. The highestvalues of R1 were obtained for VBR with striatal VRSand homogeneous VN distribution [runs E, N and V],and the smallest values for VBR with pallidal VRS andclustered VN distribution [runs D, M and U].

Fig. 5d displays the empirically estimated coefficientsof error of nV×N obtained using the Vref×NV method.Like CEemp(nV), CEemp(nV×N) varied as a function ofthe applied SPP [runs A–B–C–D, E–F–G–H, etc.], asa function of the VRS [runs A–E, B–F, etc.], as afunction 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 obtainedfor VBR with clustered VN distribution [runs D, H, M,Q, U and Y], and the smallest values for VBR withhomogeneous VN distribution [runs A, E, I, N, R andV].

Fig. 5e shows the results obtained for ratio R2. Thisratio was always greater than 1.

Fig. 5f shows the empirically estimated coefficients oferror of nF obtained using the fractionator. LikeCEemp(nV×N), CEemp(nF) varied as a function of theapplied SPP [runs A–B–C–D, E–F–G–H, etc.], as afunction of the VRS [runs A–E, B–F, etc.], as afunction 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 forVBR with clustered VN distribution [runs D, H, M, Q,U and Y], and the smallest values for VBR withhomogeneous VN distribution [runs A, E, I, N, R andV].

Fig. 5g shows the results obtained for ratio R3. Thisratio was greater than 1, except for VBR with centrifu-gal VN distribution of 50 000 VN in pallidal VRS [runsL 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 ornF (additional data are given in Appendix A). Theresults can be summarized as follows. If defining arange of 0.75BR4B1.25 as satisfactory mean predic-tion of the precision of nV×N or nF, no method led tosatisfactory predictions in any runs of the simulation.For VBR with a total number of 500 VN, applicationof all prediction methods resulted either in underesti-mation or overestimation of CEemp(nV×N) or CEemp(nF)[runs A–H]. Note in particular that F5 resulted inunderestimations of CEemp(nF). The methods V1 andV3 considerably underestimated CEemp(nV×N), exceptin 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 underestimatedCEemp(nF), except in cases such as V1 and V3. Satisfac-tory mean predictions of the precision of nF wereobtained using F7 for investigating VBR9–VBR11 andVBR13–VBR15 [runs R–T and V–X; in these runs VNdistributions were modeled according to homogeneousor inhomogeneous Poisson processes]. Satisfactorymean predictions were also obtained using F6 for inves-tigating VBR12 and VBR16 [runs U and Y; in these runsVN distributions were modeled according to Poissoncluster processes]. Furthermore, satisfactory mean pre-dictions of the precision of nF were also be obtainedusing F9 or F13 for investigating VBR9–VBR11 andVBR13–VBR15 [runs R–T and V–X], and using F11 orF15 for investigating VBR12 and VBR16 [runs U andY].

3.2. Results of Experiment 2

For all 18 VSTST, the mean of the 1000 estimatedmean total numbers of VN was approximately 50 000.Fig. 7a displays the frequency distributions of the 1000values of ratio R5 each, and Fig. 7b of ratio R6. It wasfound that both ratios varied in a broad range. Forratio R5 the largest range was found for c18 (−0.963to +0.763) and the smallest for c15 (−0.272 to0.353). The frequency distributions of R6 were com-posed of values between 0 and over 100% except c15and c17. The largest range was found for c12 (0.5–4814%), and the smallest for c15 (3.1–53.6%).

Fig. 8a shows the results obtained for mean(OCV2),and Fig. 8b the results obtained for mean(SICV−CE).Except c17 and c18, nearly identical results werefound for mean(OCV2) and mean(SICV−CE). Both vari-ables depend on the interindividual variation of thenumber of VN among the VBR of the investigatedpopulation [compare c1 with c3; c2 vs. c4; etc.],and on the applied VSS [c1 vs. c9; c2 vs. c10;etc.]. Fig. 8c displays the results obtained for the differ-ence between mean(SICV−CE) and mean(OCV2). Exceptfor c17 and c18, there was virtually no differencebetween these variables.


Fig. 6.

4. Discussion

4.1. Validity of the results

Modeling of stereological estimates involves variousrandom elements (actual spatial VN distributions of

VBR, planes of section, thickness of the first sections,positions of the lattices onto the sections). Therefore, itrepresents a so-called ‘stochastic simulation’ (see Rip-ley, 1987). For each stochastic simulation a source ofrandomness is required. Here, a pseudorandom numbergenerator (L’Ecuyer, 1988; PRNG1) was applied as


source of randomness (details on PRNG1 are given inAppendix C). Formally, pseudorandom numbers aregenerated by a computer using a simple numericalalgorithm. Consequently, pseudorandom numbers arenot truly random. Rather, any given sequence of pseu-dorandom numbers is supposed to appear random tosomeone who does not know the algorithm. Further-more, pseudorandom numbers are considered ‘random’if a sequence of pseudorandom numbers has the sameprobability of passing certain statistical tests as trulyrandom numbers would have (Knuth, 1981). L’Ecuyer(1988) demonstrated the ‘randomness’ of PRNG1 byusing 21 different tests of randomness, details of whichmay be found in Knuth (1981) or Marsaglia (1985).Based on L’Ecuyer’s (1988) evaluation, James (1990)has recommended the use of PRNG1 for stochasticsimulations. However, despite the demonstration of the‘randomness’ of the applied pseudorandom numbergenerator, one cannot rule out the possibility that otherresults would have been obtained if another pseudoran-dom number generator would have been used (seeRipley, 1987). Aside from using only such generatorswhich have been exhaustively tested (Knuth, 1981;Marsaglia, 1985), it is therefore recommended to carryout any stochastic simulation with different pseudoran-dom number generators (Ripley, 1987). This wasachieved here by repeating the entire simulation usinganother pseudorandom number generator developedand tested by L’Ecuyer (1988); PRNG2; see AppendixC for details). PRNG2 yielded nearly identical results asPRNG1.

4.2. Rele6ance of the results to quantitati6eneuroanatomy

Brain regions as modeled in these simulations do notoccur naturally. Nonetheless, these simulations havetheir biological relevance if the following points areconsidered. First, for methodological reasons, it is vir-

tually impossible to determine the exact 3-D distribu-tion pattern of neurons in a brain region of interest(Reed and Howard, 1997). Second, detailed informa-tion on the frequency distributions of total neuronalnumbers is currently not available from the literature.Therefore, in investigations comparable to those pre-sented here, the common usage is to apply exactlydefined point distribution patterns for modeling anyindividuals (here, VBR) containing any type of particles(here, VN; Konig et al., 1991; McShane and Palmatier,1994; Schmitz, 1998; Glaser and Wilson, 1998, 1999).Third, the volume of the reference spaces and the meantotal numbers of VN were selected to be similar toestimates of the mean volume and the mean totalneuronal number of the rat external globus pallidus asreported by Oorschot (1996). Fourth, the number ofVBR investigated in Experiment 2 (i.e. 6 or 12), theaverage number of counted VN (i.e. 150 or 750), andthe values of OCV obtained in Experiment 2 cover theranges of these variables reported in most stereologicalstudies published to date. In summary, the computersimulations presented here may serve as a useful substi-tute for quantitative neuroanatomical studies.

4.3. Statistical and economical efficiency of theVref×NV method and the fractionator in estimatingtotal neuronal numbers

Estimates of total neuronal numbers using the frac-tionator require only counting of neurons in a part ofthe brain region of interest. In contrast, estimates oftotal neuronal numbers using the Vref×NV methodrequire counting of neurons in a part of the brainregion of interest and estimating the total volume ofthis brain region. Therefore, already from a theoreticalpoint of view the Vref×NV method has the smallereconomical 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 modeledestimates 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 estimatesof 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 appliedprediction method [Eq. (16); i.e. ratio between the mean of the 1000 squared predicted coefficients of error [meanE1{[CEpred(n)]2}]] and theempirically estimated squared coefficient of error [[CEemp(n)]2] after estimating the total VN number of the investigated VBR 1000 times andpredicting the variation of the estimates with one of the prediction methods shown in Table 1. The ordinates of all graphs are limited to0.45R451.6. (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 byusing VSS2 (Table 3) resulting in counting of approximately 150 VN per estimate. (c) Results obtained for runs R–Y, that is, investigating VBRwith 50 000 VN by using VSS3 (Table 3) resulting in counting of approximately 750 VN per estimate. There are three essential findings ofExperiment 1 shown in this figure. First, if defining a range of 0.75BR4B1.25 as satisfactory mean prediction of the precision of the estimatedtotal VN numbers, no prediction method leads to satisfactory predictions in any runs of the simulation. Second, for runs A–H, application ofall prediction methods results either in underestimation or overestimation of CEemp(nV×N) or CEemp(nF). Third, for runs I–Y, the use of threepairs of prediction methods (F7–F6, F9–F11 and F13–F15; each time the first prediction method used when investigating VBR with VNdistributions according to homogeneous or inhomogeneous Poisson processes and the second prediction method used when investigating VBRwith VN distributions according to Poisson cluster processes) results in satisfactory predictions of the variation of most of the modeledfractionator estimates.


Fig. 7. Results of all VSTST carried out in Experiment 2 (c1 to c18; details of these VSTST are given in Table 5). Each VSTST consists of1000 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), estimatingthe 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 squaredcoefficient of variation of the real total VN numbers of the selected VBR is calculated [ICV2], the mean squared predicted coefficient of error ofthe estimated total VN numbers [meanE2{[CEpred(n)]2}], and the squared coefficient of variation of the estimated total VN numbers [OCV2]. Theordinates of all graphs are truncated at the shown values. (a) Frequency distributions of the results obtained for ratio R5 (Eq. (19)), describingthe difference between [ICV2+meanE2{[CEpred(n)]2}] and OCV2. (b) Frequency distributions of the results obtained for ratio R6 [Eq. (20); thatis, 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 c18 (−0.963 to +0.763) and the smallest for c15 (−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 c15 and c17. The largest rangeis found for c12 (0.5–4814%), and the smallest for c15 (3.1–53.6%).

The results of Experiment 1 show that the statisticalefficiency of both fractionator estimates and Vref×NV

estimates depends on the shape of a brain region ofinterest, on the spatial distribution of neurons withinthis brain region, and on the sampling scheme used forestimating total neuronal numbers. This confirms re-sults of previous studies (Schmitz, 1998; Glaser and

Wilson, 1998, 1999). Furthermore, the results of Exper-iment 1 demonstrate that the statistical efficiency ofboth fractionator estimates and Vref×NV estimates de-pends on the ratio between the mean number ofcounted neurons and the total number of neurons inthe brain region of interest (ratio R8). Interestingly, forneuron distributions corresponding to homogeneous or


inhomogeneous Poisson processes, there was only asmall difference in the statistical efficiency of bothfractionator estimates and Vref×NV estimates betweenR8=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 andL). In contrast, for neuron distributions correspondingto Poisson cluster processes there was a high differencein the statistical efficiency of both fractionator esti-mates and Vref×NV estimates between R8=0.3 andR8=0.003 (Fig. 5d and f; compare run D with run M).

Moreover, the results of Experiment 1 show thatfractionator estimates have a greater statistical effi-ciency than Vref×NV estimates (Fig. 5g). This is due tothe fact that the variation of fractionator estimatesdepends solely on the variation of the number ofcounted neurons, whereas the variation of Vref×NV

estimates depends on the variation of the number ofcounted neurons and on the variation of the volumeestimates. However, the variation of Vref×NV estimatesis not simply the sum of the variation of the number ofcounted neurons and the variation of the volume esti-mates, as demonstrated in Fig. 5e. Rather the covari-ance between these variables has to be considered ifcalculating the variation of Vref×NV estimates (Gun-dersen and Jensen, 1987). If volume estimates are ob-tained by using the point counting method and theCavalieri’s principle, the variation of the estimates de-pends on the number of counted points and on theso-called ‘average shape coefficient’ (SC) of the sectionsof the brain region of interest (Gundersen and Jensen,1987; Roberts et al., 1994). This average shape coeffi-cient is defined as the ratio between the average total

Fig. 7.


Fig. 8. Results of all VSTST carried out in Experiment 2 (c1 toc18; details of these VSTST are given in Table 5). Each VSTSTconsists of 1000 repetitions of selecting either 6 or 12 VBR of one ofthe modeled populations P1–P5 of 15 000 VBR each (for P1–P5 seeTable 4), estimating the total number of VN of each selected VBRonce using the fractionator and either virtual sampling scheme VSS2

or VSS3 (for VSS2 and VSS3 see Table 3), and predicting the variationof the estimates using prediction method F7 (for F7 see Table 1).Fractionator estimates are modeled using either the virtual samplingscheme VSS2 (VSTST c1 to c8 and c17) or using VSS3 (VSTSTc9 to c16 and c18; for VSS2 and VSS3 see Table 3). For eachrepetition, the squared coefficient of variation of the real total VNnumbers of the selected VBR is calculated [ICV2], the mean squaredpredicted coefficient of error of the estimated total VN numbers[meanE2{[CEpred(n)]2}], and the squared coefficient of variation of theestimated total VN numbers [OCV2]. The ordinates of all graphs aretruncated at the shown values. (a) Mean of the 1000 values obtainedfor OCV2 [mean(OCV2)]. This variable depends on the interindividualvariation of the number of VN among the VBR of the investigatedpopulation (compare c1 with c3; c2 vs. c4; etc.), and on theVSS used (c1 vs. c9; c2 vs. c10; etc.). (b) Mean of the 1000values obtained for the sum of ICV2 and meanE2{[CEpred(n)]2}[mean(SICV−CE)]. Like mean(OCV2), mean(SICV−CE) depends on theinterindividual variation of the number of VN among the VBR of theinvestigated population (compare c1 with c3; c2 vs. c4; etc.),and on the VSS used (c1 vs. c9; c2 vs. c10; etc.). (c) Mean ofthe 1000 values obtained for ratio R7 (Eq. (23)), describing thedifference between mean(SICV−CE) and mean(OCV2). There is oneessential finding of Experiment 2 shown in this figure. Except forc17 and c18, there is virtually no difference between mean(OCV2)and mean(SICV−CE).

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 thegreater economical efficiency as well as the greaterstatistical efficiency. We therefore recommend to esti-mate total neuronal numbers preferably using thefractionator.

4.4. Prediction of the 6ariation of estimated totalnumbers of neurons obtained using the Vref×NV

method or the fractionator

The results of Experiment 1 show complex interrela-tions between the shape of a brain region of interest,the number and the spatial distribution of neuronswithin this brain region, the variation of estimates oftotal neuronal numbers, and the precision of predic-tions of this variation (henceforth abbreviated as ‘pre-dictions’) using the various prediction methods listed inTable 1. It is beyond the scope of this study to providea complete analysis of these interrelations. Rather,some aspects relevant for the use of the fractionator orthe Vref×NV method in quantitative neuroanatomy willbe briefly discussed in the following.

We defined a range of 0.75BR4B1.25 as satisfac-tory mean prediction of the precision of estimated totalneuronal numbers (for R4 see Eq. (16)). Using thisdefinition, we found that no prediction method led tosatisfactory predictions in any runs of the simulation(Fig. 6). Rather, for each prediction method, situationscould be modeled in which the variation of the corre-sponding fractionator estimates or Vref×NV estimateswas considerably overestimated. As well, except formethod F6, F11 and F15, situations could be modeledin which the variation of the corresponding neuronnumber estimates was considerably underestimated.Therefore, no prediction method can be regarded per-fect. This is in line with recent theoretical work con-cerned with the variation of stereological estimatesobtained using the fractionator or the Vref×NV method(Cruz-Orive, 1999; Gundersen et al., 1999). On theother hand, there was no modeled situation for which itwas impossible to obtain any satisfactory prediction.Therefore, we looked for pairs of prediction methods,the use of which resulted in satisfactory predictions ofthe variation of neuron number estimates of as manymodeled situations as possible. For the Vref×NV

method, it was not possible to find such a pair ofprediction methods. In contrast, for fractionator esti-mates with R8=150/50 000=0.003 (Fig. 6b) or R8=750/50 000=0.015 (Fig. 6c), three pairs of predictionmethods were found leading to this goal. These pairswere F7–F6, F9–F11 and F13–F15. Each time, use ofthe first method (F7, F9 and F13) resulted in satisfac-tory mean predictions if the spatial neuron distribution

boundary length of the sections and the square root ofthe average surface area of the sections of the brainregion of interest (see above; Eq. (7)). The greater thisaverage shape coefficient is, the greater is the variationof volume estimates by using the point countingmethod and the Cavalieri’s principle (Gundersen andJensen, 1987). SC was approximately 3.4 for the palli-dal VRS [VRSi; VBR 1–4 and 9–12], and was approx-imately 28.5 for the striatal VRS [VRSii; VBR 5–8 and13–16]. Therefore, the variation of the Vref×NV esti-mates was greater if investigating VBR with striatalVRS than investigating VBR with pallidal VRS [Fig.5d, runs A–E, B–F, etc.]. The difference between thestatistical efficiency of fractionator estimates and Vref×NV estimates is a function of the contribution of thevariation of volume estimates to the variation of esti-


was modeled according to homogeneous or inhomoge-neous Poisson processes, whereas use of the secondmethod (F6, F11 and F15) resulted in satisfactory meanpredictions if the spatial neuron distribution was mod-eled according to Poisson cluster processes. The com-puter simulation did not show any advantage of one ofthe mentioned pairs of prediction methods over theother pairs. However, a clear advantage of F7–F6 overboth F9–F11 and F13–F15 is the fact that predictionsof the variation of fractionator estimates are manytimes easier to calculate using F7–F6 than using F9–F11 or F13–F15 (details are shown in Appendix D). Itshould be mentioned that the use of F9 and F13resulted in satisfactory predictions even if modelingfractionator estimates of VBR with centripetal VNdistribution (Fig. 6b and c, runs K and S). F9 and F13(as well as V1–V3, F1–F4, F8, F10–F12, F14 andF15) are based on complex theoretical statistics, namelyon Matheron’s ‘theory of regionalized variables’(Matheron, 1965, 1971). In a former study, which waspublished before F9 and F13 had been reported in theliterature, it was found that the use of all predictionmethods based on Matheron’s (1965, 1971) theory re-sulted in considerable overestimation of the precision offractionator estimates if modeling VBR with centripetalVN distribution (Schmitz, 1998; see also Fig. 6b and chere, runs K and S). This finding was interpreted asindicating ‘that Matheron’s (1965, 1971) theory can inprinciple not serve as the optimum basis for predictingthe precision of fractionator estimates, independent ofthe manner of how it is adapted to the fractionator’(Schmitz, 1998). The development of F9 and F13 hasshown that this interpretation can no longer bemaintained.

For the fractionator estimates with R8=150/500=0.3, we found that there was no prediction method theuse of which led to satisfactory predictions of thevariation of the estimates (Fig. 6a). Particularly, F5resulted in underestimations of this variation. F5 is aslight modification of F7, considering the samplingfraction of fractionator estimates (i.e. if the space occu-pied by the counting spaces is, say, the 1000th part ofthe reference space of the brain region of interest, thesampling fraction sf is 1/1000). At first glance it seemsuseful to consider sf if R8 is small. This may be seenfrom the borderline case if sf=1 and thus, R8=1. Inthis case there is no variation of fractionator estimates,but the use of all prediction methods except F5 resultsin mean predictions of this variation greater than 0.However, already with R8=150/500=0.3, F5 resultedin underestimations of the variation of fractionatorestimates, whereas F7 resulted in overestimations ofthis variation (Fig. 6a). With R8=150/50 000=0.003(Fig. 6b) or R8=750/50 000=0.015 (Fig. 6c), F5 led toalmost identical predictions as F7 did. Therefore, thereis no advantage of using F5 rather than F7.

In summary, using F7 when investigating VBR withVN distributions according to homogeneous or inho-mogeneous Poisson processes and using F6 when inves-tigating VBR with VN distributions according toPoisson cluster processes facilitated satisfactory predic-tions of the variation of the modeled fractionator esti-mates. We recommend to predict the variation offractionator estimates in quantitative neuroanatomy al-ways using these simple prediction methods. Mostlikely, inspection of the investigated sections on themicroscope will be sufficient to decide whether theneuron distribution is homogeneous (warranting theuse of F7) or clustered (warranting the use of F6).Otherwise, there are methods available in the literaturefor investigating the distribution pattern of neurons in abrain region of interest (for example, see Duyckaertsand Godefroy, 2000). In any case, however, it appearsnecessary to interpret predictions of the variation ofestimated total neuronal numbers carefully.

4.5. Optimization of stereological sampling schemes

A frequently used approach to demonstrate the reli-ability of an estimated mean total number of neurons isto show that the mean of the squared predicted coeffi-cients 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 neuronalnumbers (i.e. OCV2; for recent examples see Geinismanet al., 1996; Begega et al., 1999; Korbo and West, 2000;among others). For example, in West (1993) this ap-proach was explained as follows. If for a number ofindividuals the mean total number of neurons in agiven brain region is estimated using the fractionator orthe Vref×NV method, the squared coefficient of varia-tion of the estimated total numbers of neurons (OCV2)is affected not only by the real inherent biologicalvariance of the individuals (ICV2), but also by thevariance of the estimates (CE2), which is related to theamount of sampling performed in each individual. Ac-cording to West (1993) the relation between OCV2,ICV2 and CE2 can be calculated as shown in Eq. (24):

OCV2=ICV2+CE2 (24)

If the major contributor to OCV2 is CE2 (in this caseis CE2 greater than ICV2, and ratio R6 is greater than50%), the most efficient way to reduce OCV2 would beto reduce CE2 by increasing the precision of the esti-mates (West, 1993). If the major contribution to OCV2

is ICV2 (in this case is CE2 smaller than ICV2, and ratioR6 is smaller than 50%), the most efficient way toreduce OCV2 would be to reduce ICV2 by increasingthe number of investigated individuals (West, 1993).

The results of Experiment 2 reveal however that thisrecommendation be considered critically. Both OCV2

and (ICV2+CE2) are random variables, and the differ-


ence between OCV2 and (ICV2+CE2) may vary con-siderably if the same stereological study is carried outrepeatedly (Fig. 7a). In consequence, the ratio betweenCE2 and OCV2 is a random variable as well, which mayalso vary considerably (Fig. 7b). The actual relationbetween 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 studywould be repeated unlimited times. Provided that thepredictions of the variation of the estimated total neu-ronal numbers equal the real variation of the estimates,the mean of the observed values of OCV2 would equalthe mean of the observed values of (ICV2+CE2) (Fig.8c). In this case, the ratio between the mean of CE2 andthe mean of OCV2 would indeed provide a basis forevaluating the reliability of estimated mean total neu-ronal numbers (details are given in Appendix B). How-ever, this is obviously not the case in quantitativeneuroanatomy. Without often repeating the same stere-ological study, the CE2/OCV2 approach is neither use-ful nor informative.

However, this CE2/OCV2 approach served as thebasis for statements such as ‘there are no prior reasonsat all for expecting that the counting of more than 50or 100 items (i.e. neurons) per individual or organ isnecessary’ (Gundersen, 1986), or ‘an advantage of sys-tematic random sampling is that one need count onlyabout 100 cells or synapses to get sampling variances tobe negligible in comparison with interanimal variances’(Coggeshall and Lekan, 1996). As a consequence,counting of no more than 100–200 cells per individualhas become a general recommendation in design-basedstereology (Gundersen et al., 1988; West, 1993; May-hew and Gundersen, 1996; among others) and has beenapplied in many stereological studies published in theliterature. The results of Experiment 2 show the need tohandle this recommendation very carefully. At present,there is no method available to perform valid compari-sons between sampling variances and interanimal vari-ances in quantitative neuroanatomy.

In summary, assessing the reliability of estimates ofmean total neuronal numbers using the CE2/OCV2

approach is neither useful nor informative. We feel thatthis approach is not optimal. Finally, there is urgentneed to find a new, analytical solution of this majorproblem in design-based stereology. Based on our ownexperience, we have decided to increase the number ofcounted neurons to at least 700–1000 per individualwhenever possible (Schmitz et al., 1999a; Heinsen et al.,1999, 2000). For example, for brain regions with ahomogeneous neuron distribution, counting of approxi-mately 900 neurons results in a predicted coefficient of

error of 0.033. Accordingly, one may expect the truetotal number of neurons in the investigated brain re-gion with a probability of approximately 95% in arange of approximately 97% about the estimated totalneuronal number. The amount of time necessary tocarry out these estimates is about 1 day per brainregion of interest, which appears justified as a reason-able compromise between the amount of time dedicatedto the analysis and the precision of the obtained esti-mates. We are aware that this can be accepted only asone possible empirical solution to the mentioned prob-lem, and that an analytical solution is still lacking. Weexpect however that our study will initiate new discus-sions and, hopefully, will stimulate new approaches tosolve this important issue of quantitativeneuroanatomy.

Acknowledgements

We thank our many colleagues who prompted us tocarry out this study. We gratefully acknowledge HubertKorr and Helmut Heinsen for their constructive andhelpful comments. This study was supported by theSTART-program of the Faculty of Medicine at theRWTH University of Aachen, Germany (C.S.), and byNIH grants AG02219, AG05138, and MH58911(P.R.H.).

Appendix A. Note on the use of the prediction methodsF8–F11

For F8–F11, a term t must be calculated (Cruz-Orive, 1999). In this study, the use of F8–F11 wasdemonstrated by means of an example, i.e. by applica-tion of F8–F11 on data presented by West et al. (1996).In the latter study, which was concerned with thenumber of somatostatin neurons in the striatum of rats,neurons were counted on every tenth section through-out the striatum. Section thickness was 20 mm, and theheight of the counting spaces was 15 mm. Therefore, theso-called ‘section sampling fraction’ (ssf) was 0.1, andthe so-called ‘thickness sampling fraction’ (tsf) was0.75. Accordingly, in the example given by Cruz-Orive(1999), t was 0.1. In the computer simulations pre-sented here, every section was analyzed, and ssf wastherefore 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 F9equal F7. Therefore, we decided in the present study tocalculate t= tsf, yielding t=0.669 for VSS1, t=0.144for VSS2, and t=0.246 for VSS3. The results obtainedthis way are presented in Fig. 6. However, resultsobtained using t=1 was very similar, as shown in Fig.9.


Appendix B. Bias in predicting the relative contributionof stereological sampling to the variation of anestimated mean total neuronal number

Suppose a population of F VBR containing VN. Toevaluate the mean total number of VN in this popula-tion, select a small, random sample of f VBR andestimate the total numbers of VN of the selected VBRusing the fractionator. Since the estimates are unbiased,their mean (g) is an unbiased estimator of the truemean total number of VN of the selected VBR (G. ) aswell as of the true mean total number of VN in thepopulation (G ; notations of g, G. and G as introducedby Nicholson, 1978). This is shown in Eqs. (B.1) and(B.2))

E [g ]=E [G. ]=G (B.1)

with

g=� %

f

j=1

nj

f

n; G. =� %

f

j=1

Nj

f

n; G=

� %F

j=1

Nj

F

n(B.2)

where E […] is the expected value, n is the estimatedtotal number of VN, and N is the ‘true total number ofVN.

The variation of g depends on interindividual vari-ability and on stereological sampling, as shown in Eq.(B.3) (Nicholson, 1978):

s g2=sG.

2 +E [s g�G.2 ]=

�sN2

f

�+E [s g�G.

2 ] (B.3)

where s g2 is the variance of the distribution of g about

G ; sG.2 the variance of the distribution of G. about G ; sN

2

the variance of N among the VBR in the population;

and E [s g�G.2 ] is the expected value of the variance of g

about G. for all possible independent random samplesof f VBR from the population and all possible esti-mates of G. of the selected VBR. If sG.

2 is greater thanE [s g�G.

2 ], the ratio R9 shown in Eq. (B.4)

R9=(E [s g�G.

2 ] )s g

2 (B.4)

is smaller than 50%. In this case the variation of g ismainly due to interindividual variability. If the ratio R9

is greater than 50%, the variation of g is mainly due tostereological sampling. The relative contribution ofstereological sampling to the variation of g can becalculated as shown in Eq. (B.5):

Contrel=1−sG.

2

s g2 (B.5)

In real experiments using the fractionator, usuallyonly one random sample of f individuals is selectedfrom a population. Therefore, s g

2, sG.2 , sN

2 and E [s g�G.2 ]

are unknown, and R9 or Contrel cannot be calculated.Rather ratio R6 (see above, Section 2.2.3) is frequentlyused to predict the relative contribution of stereologicalsampling to the variation of an estimated mean totalnumber of neurons (for recent examples see Geinismanet al., 1996; Begega et al., 1999; Korbo and West,2000).

As explained in the main text, ratio R6 shows consid-erable variation if the same stereological study is re-peated (Fig. 7b). Moreover, ratio R6 is a biasedestimator of Contrel, as may be deduced from theliterature (Searle, 1987). This bias depends on the fre-quency distribution of the number of VN of the corre-sponding 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 ofthe 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 0.45R451.6. Note that the resultsobtained using t= tsf are very similar to the corresponding results using t=1.


Fig. 10. Results of all VSTST carried out in Experiment 2 (c1 toc18). The graphs display the results obtained for sG.

2 (a), s g2 (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) andratio R12 (Eq. (B.9); g). The mentioned variables are explained inAppendix B. The ordinates of all graphs are truncated at the shownvalues. Note that (except for c17 and c18) the results obtained forEst[Contrel] are very similar to the corresponding results obtained forratio R11. Accordingly, except for c17 and c18, there is virtuallyno difference between ratio R11 and Est[Contrel], as shown by calcu-lating ratio R12.

Est[Contrel]=1−sG.

2

s g2 (B.6)

This estimate of Contrel was compared with the meanof 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 1000values of [meanE2{[CEpred(n)]2}] (see above, Section2.2.3) and the mean of the 1000 values of OCV2 wascalculated, as shown in Eq. (B.8):

R11=%

1000

Run=1

(meanE2{[CEpred(n)]2})Run

%1000

Run=1

(OCV2)Run

(B.8)

Like R6, R11 was compared with the estimate ofContrel, as shown in Eq. (B.9):

R12=R11−Est[Contrel]R11+Est[Contrel]

(B.9)

Fig. 10 shows the obtained results. As expected, foreach VSTST, sG.

2 (Fig. 10a) was smaller than s g2 (Fig.

10b). Both sG.2 and s g

2 depend on the interindividualvariation of the number of VN among the VBR of theinvestigated population (compare c1 with c3; c2vs. c4; etc.), and on the number of selected VBR(compare c1 with c5; c2 vs. c6; etc.). For theVSTST c3, c4 and c7 to c16, Est[Contrel] wassmaller than 50% and therefore the variation of g wasmainly due to interindividual variability (Fig. 10c). Forc1, c2, c5, c6, c17 and c18, Est[Contrel] wasgreater than 50% and therefore the variation of g wasmainly due to stereological sampling (Fig. 10c). Asexpected, ratio R6 was a biased estimator of Est[Contrel](the mean of the 1000 values of R6 each is shown inFig. 10d; Fig. 10e shows ratio R10). This bias wasrelated to the number of selected VN (compare c1with c5; c2 vs. c6; etc.), on the frequency distribu-tion of N of the investigated population (c3 vs. c4;c7 vs. c8; etc.) and on the accuracy of CEpred(n) inpredicting the coefficient of error of estimated totalnumbers of VN (c1 vs. c17; c9 vs. c18). Incontrast to this, ratio R11 was an unbiased estimator ofEst[Contrel], except for c17 and c18 in which thevariation of the fractionator estimates was considerablyunderestimated (R11 is shown in Fig. 10f; Fig. 10gshows ratio R12). Therefore, in principle it is quitepossible to use predictions of the variation of fractiona-tor estimates and variations among estimated totalnumbers of neurons for drawing conclusions on thecontribution of stereological sampling to the total vari-ation of estimated mean total numbers of neurons.However, this is only possible if the same stereologicalstudy would be carried out repeatedly.

selected VBR, the stereological sampling scheme used,and the accuracy of meanE2{[CEpred(n)]2} in predictingthe mean squared coefficient of error of the estimatedtotal numbers of VN.

To the best of our knowledge, the bias of ratio R6 asan estimator of Contrel has not been investigated. Thiswas achieved here by using the results of Experiment 2.After selecting either f=6 or f=12 VBR from theinvestigated population, the mean of the (known) totalnumbers of VN of the selected VBR was calculated (G. ),as well as the mean of the estimated total numbers ofVN (g). After carrying out the simulation 1000 times,the variance of the 1000 values of G. (sG.

2 ) was calculatedas the empirical estimator of sG.

2 , as well as the varianceof the 1000 values of g (sG.

2 ) as the empirical estimatorof s g

2. From these data an estimate of Contrel could becalculated as shown in Eq. (B.6):


The results of c17 and c18 illustrate the fatalconsequences of underestimating the variation of esti-mated total numbers of VN if evaluating the appropri-ateness of sampling schemes in modern, design-basedstereology with ratio R6. Although the relative contri-bution of stereological sampling to the variation of gwas approximately 99% in c17 (94% in c18), themean of the 1000 values of ratio R6 was merely 8.6% inc17 (8.7% in c18).

Finally, about 20 years ago a so-called ‘conciselycoined rule of thumb’ was established for a broad classof biological experiments employing stereologicalcounting techniques, ‘one should look at more individu-als […], rather than measure them more precisely (andin a more time-consuming way)’ (Gundersen andØsterby, 1981). However, for the VSTST involving P1

and P2, s g2 was smaller when investigating f=6 VBR

with VSS3 compared with the investigation of f=12VBR with VSS2 (c5 and c6 vs. c9 and c10).Consequently, there should be some caution with theapplication of this ‘rule of thumb’. Its use should belimited to investigations of mean total numbers ofneurons for which the variation of g is mainly due tointerindividual variability (as shown with c7 vs. c11or c8 vs. c12). Unfortunately at present, there is noavailable method to evaluate this in real stereologicalstudies.

Appendix C. Pseudorandom number generators used inthe present study

The pseudorandom number generator PRNG1

(L’Ecuyer, 1988) consists of the combination of twoso-called ‘multiplicative congruential linear pseudoran-dom number generators’ (MLCG), the basic form ofwhich is shown in Eq. (C.1):

f(l)= (a× l)MODm ; g(l)=l

m; l=1, 2, …

(C.1)

For the first MLCG, m=2 147 483 563 and a=40 014. For the second MLCG, m=2 147 483 399 anda=40 692. Before the first call, the integer l must beinitialized. In the present study, the initial seeds werel=12 345 for the first MLCG, and l=67 890 for thesecond MLCG. The combined generator takes valuesevenly spread in [0, 1] and has a so-called ‘period’ ofapproximately 2.30584×1018, which is far more thanthe period of 108 as stated by Ripley (1987) as mini-mum period of a good pseudorandom number genera-tor (the period of a pseudorandom number generator isthe sequence of pseudorandom numbers after which thegenerator begins to generate the same sequence ofnumbers over again). A portable implementation ofPRNG1, which works as long as the computer can

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 thecombination of three MLCG as shown in Eq. (C.1).For the first MLCG, m=32 363 and a=157. For thesecond MLCG, m=31 727 and a=146, and for thethird MLCG, m=31 657 and a=142. The initial seedswere l=123 for the first MLCG, l=456 for the secondMLCG, and l=789 for the third MLCG. The com-bined generator takes values evenly spread in [0, 1] andhas a period of approximately 8.12544×1012. Aportable implementation of PRNG2, which works aslong as the computer can represent all integers in therange (−32 363, +32 363), can be found in Figure 4in L’Ecuyer (1988). Following a recommendation byRipley (1987), the complete simulation was based onone single sequence of PRNG1, and the repetition wasbased on one single sequence of PRNG2.

Appendix D. Calculations necessary to predict thevariation of fractionator estimates using the predictionmethods F6, F7, F9, F11, F13 and F15

Suppose that a fractionator estimate is carried out bycounting neurons in R counting spaces, which are dis-tributed over S sections of a brain region of interest.The S sections are a systematic and random sample ofevery uth section of an entire series of sections throughthis brain region with unique section thickness t. Qr

− isthe number of neurons counted in the rth countingspace, and Qs

− the number of neurons counted in thesth section. Furthermore, Qr

− is the mean number ofcounted neurons per counting space, and Var(Qr

−) isthe variance of the number of counted neurons amongthe counting spaces. Predictions of the variation of thisfractionator estimate are then obtained as follows:

F6: CEpred(nF)='Var(Qr

−)

R(Q r−)2

(D.1)

F7: CEpred(nF)=1' %R

r=1

Qr−

=1' %S

s=1

Qs−

(D.2)

For F9, F11, F13 and F15, first of all the followingequations must be calculated:

t=t

u× t=

1u

(D.3)

a=16

[1+2t−2(t2)][1−t ]2

40−10(t2)+3(t3)(D.4)

N= %S

s=1

Qs− (D.5)

A= %S

s=1

(Qs−×Qs

−) (D.6)


B= %S−1

s=1

(Qs−×Qs+1

− ) (D.7)

C= %S−2

s=1

(Qs−×Qs+2

− ) (D.8)

D= %S−3

s=1

(Qs−×Qs+3

− ) (D.9)

Using these equations, predictions of the variation offractionator estimates are obtained as follows:

F9 CEpred(nF)='a [3(A−N)−4B+C ]

N2 +1N(D.10)

F11 P=(431−32t2+2t3+t4)B

12(D.11)

U=(268−37t2+4t3+2t4)C

15(D.12)

V=(79−16t2+2t3+t4)D

20(D.13)

CEpred(nF)=A− [1/22−t2(P−U+V)]

N(D.14)

F13 CEpred(nF)=(3[A−N ]−4B+C)/240+N

N(D.15)

F15

CEpred(nF)=(1320A−2155B+1072C−237D)/1320

N(D.16)

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