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APMIS 96: 379-394, 1988

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Some new, simple and efficient stereological methods _

and their use in pathological research and diagnosis _

Review article

H. J. G. GUNDERSEN, T. F. BENDTSEN, L. KORBO1, N. MARCUSSEN, A. MLLER1, K. NIELSEN2,J. R. NYENGAARD, B. PAKKENBERG1, F. B. SRENSEN, A. VESTERBY3 and M. J. WEST

Stereological Research Laboratory, University Institute of Pathology and 2nd University Clinic of InternalMedicine, Institute of Experimental Clinical Research, University of rhus, Neurological Research Laboratory1,

Hvidovre Hospital, Institute of Pathology2, University of Copenhagen, Rigshospitalet, University Institute ofPathology3, rhus Amtsygehus, Denmark

Gundersen, H. J. G., Bendtsen, T. F., Korbo, L., Marcussen, N., Mller, A., Nielsen, K., Nyengaard, J. R., Pak-kenberg, B., Srensen, F. B., Vesterby, A. & West, M. J. Some new, simple and efficient stereologicalmethods and their use in pathological research and diagnosis. APMIS 96: 379-394, 1988.

Stereology is a set of simple and efficient methods for quantitation of three-dimensional microscopicstructures which is specifically tuned to provide reliable data from sections. Within the last few years, anumber of new methods has been developed which are of special interest to pathologists. Methods forestimating the volume, surface area and length of any structure are described in this review. The principleson which stereology is based and the necessary sampling procedures are described and illustrated withexamples. The necessary equipment, the measurements, and the calculations are invariably simple and easy.

Key words: Cavalieri's principle; isotropic sections; star volume; surface area; stereology; vertical sections;volume.

H. J. G. Gundersen, Stereological Research Laboratory, University Bartholin Building, DK-8000 rhus C,Denmark.

THE BASIC IDEAS AND TOOLSOF STEREOLOGY

At a practical level, stereological methods areprecise tools for obtaining quantitative informa-tion about three-dimensional, microscopic struc-

tures, based mainly on observations made onsections. Two-dimensional sections contain quan-titative information about three-dimensionalstructures only in a statistical sense. For this

statistical information to be "true" or unbiased afew requirements must be fulfilled about the sec-tions and the way they are made. In practice it isnearly always very easy to fulfil these requirements(often as easy as doing it the wrong way!) andthroughout the text the sampling methods aredescribed and references are given to full descrip-tions of the techniques.

Two concepts are often referred to in this andother papers dealing with stereology: "Unbiased"and "efficient". The concepts are used in a statisti-cal sense and mean "without systematic deviationfrom the true value" and "with a low variabilityafter spending a moderate amount of time", re-spectively.

In the following, two-dimensional informationand the way to obtain it effeciently are dealt withfirst, then the basic methods for measuring vol-ume, surface area, and length are described. Fordiscrete particles like cells, nuclei, glomeruli etc., anumber of new methods are available for measur-ing their number and mean sizes in a very directway. These methods are described in the second,forthcoming part of the review. Al1 the methodsare used extensively in biological research andpathology, as indicated by a number of references


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to recent studies throughtout the text. Hopefully,the examples will make many pathologists appre-ciate that, especially in their own interesting tissue,effiicient and unbiased quantitative data may meanthe difference between "interesting observations"

and real knowledge.

TWO-DIMENSIONAL QUANTITATION

The strength of stereological methods is that theyreport data for three-dimensional structures interms of three-dimensional quantities - not just interms of the two-dimensional sections which aremost often used for the observations. The size ofa cell may naturally be expressed in m3, not by thearea in m2 which its profile happens to occupy ona section. Nevertheless, there are many situations

where simple, fast and reliable methods for quan-titation of two-dimensional structures are needed.

Fig. 1. Above. A restricted field of vision of a sectionwith some profiles. Hatched profiles are completelyinside the field. Below. The same field with an unbiasedcounting frame (Gundersen 1977), the hatched profilesare counted according to the counting rule described inthe above reference and in the text.

The two-dimensional methods also serve as anintroduction to the full set of stereological meth-ods, since the same principles are involved.

The Number of Profiles per Area

In Fig. 1 is shown a section seen through a "win-dow": a photograph, a field of vision or anythingelse which imposes artificial boundaries on thevisible part of the section: How many profiles dowe see? This is not as simple a question as it mayseem: There are 31 profiles completely inside thewindow, but, in addition, 10 profile parts are seenon the edges (they may or may not represent 10different profiles, but we have no way of knowingthat). So, what is the number of profiles; 31, 41,36 or what? Most biologists have learned how tocount erythrocytes inside countingchambers; i.e.counting on certain edges and corners of the

counting frame, but since it systematically overes-timates the number, it is a biased rule, see the re-view by Gundersen (1978). All known practicalcounting rules more than ~ 10 years old are biased.For humans, (machines are different), the onlyknown unbiased frame and its counting rule is alsoillustrated in Fig. 1: In addition to profiles com-pletely inside the frame one counts all profileswith anything inside the frame provided they donot in any way touch or intersect the full drawnexlusion edges or their extensions (Gun-dersen1977).

The answer to the question is 27profiles. Thisnumber is not directly comparable to any of theabove figures because the area of the frame is lessthan that of the window. For irregular profilesespecially there must be a "guard-area" around theframe for counting correctly. The natural way toreport such a number is as a numerical densityin the plane, i.e. the number of profiles perarea, a quantity which in stereology is given thesymbol Q

A

( )( )

22

u8.8

27frameofarea

profilesof#

tsecA

profQ

==== 3.1u)(prof/sectA

Q

assuming that the area of the counting frame is 8.8

in some recognizable areal units, u2. The mainadvantage of reporting the number of profiles insuch a standardized way - that it makes it possibleto draw comparisons between different techniquesand observers - is so great that pathologists alwaysquantify e.g. "cellularity" in terms of number ofcell profiles per unit area and never as just cells ormitoses per field of unknown size and magnifica-


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Fig. 2. The same field as in Fig. 1, now superposed witha test system with a set of regularly spaced points (anda counting frame). For estimates of the areal fraction ofprofiles or mean profile area one counts all points hittingprofiles (disregarding their relationship to the frame). Apoint is considered a hit if a profile (including its

boundary) covers the upper right corner in the crosswhere the two lines cross each other.

tion.For the number of profiles per area to be of any

use one must, however, also specify how the fieldswere sampled. If the fields were sampled justbecause the pathologist liked them the best thenumber of profiles per area may well characterizethe pathologist more than the section. The sim-plest sampling system which also fulfils the statisti-cal requirements for a scientific study is one where

the fields are sampled systematically and indepen-dent of their content - and of the observer - butall inside a well defined region.

The Area of ProfilesThe simplest way of reporting "how much there is"of the structure in the profiles shown in Figs. 1 to 3is to quantify the areal fraction of the structure,generally denoted AA

( ) 0.23tstruct/secA

A ==== 80

18tionssecofareatotal

structureofareatotal

)t(secP

)struct(P

A quantity which can be estimated by counting

points which hit the structure, P(struct), divided bypoints hitting the section, here illustrated by theexample in Fig. 2. Note that in the simple situationwhere the section fills the whole window one neednot count the points hitting the section, all 80points in the test system do that. In terms of timeand effort there is no way to do it more efficiently,

not to mention the capital investment in hardware.It takes ~ 18 seconds to count the hits in Fig. 2. Totrace the boundaries of all profiles manually with-out a large bias takes 5 to ten times more time,without any gain in real precision (no automaticimage analyzers can work on ordinary biological

tissue), for examples of direct comparisons see forexample Gundersen et al. 1981, Regeur & Pakken-berg 1988, and references therein. For serious useone must again sample the fields systematically orby some other random mechanism and for just 5such fields quantified in this way the coefficient oferror, MEANSEMCM= of the estimate is in the order of 5%. Anomogram for reading off the precision ofpoint-counting estimation of areas is found in Fig.18 in Gundersen & Jensen 1987. As it is wellknown, Pp(struct/sect) may also be used as an

estimate of the volume fraction Vv(struct/speci-men) of the structure under study, described in thenext section. One must then as well fulfil somefurther requirements with respect to the samplingof the sections inside the specimen. Note that thearrangement of the points is irrelevant to theestimator and that the magnification need not beknown because we are only estimating a relativearea. Later on we shall see another use of a testsystem for estimating absolute areas where wemust specify the above information about thepoints somewhat more precisely.

Another question which a pathologist wouldoften ask is the area of a typical profile, i.e. themean profile area a(prof). Since the relative areaand the relative number of profiles may be esti-mated by the above techniques, this is a very sim-ple problem to solve, see also Fig. 2:

2

0.073u(prof)a ====2

u8.8

2780

18

profilesof#relative

profilesofarearelative

)tsec/prof(a

q

)tsec/prof8A

A

Note that the straightforward estimator is the ratioof two independent estimators: the points hittingprofiles are counted irrespective of whether theprofile in question is counted in the frame or not,

see Fig. 2. It is only if we need to know thedistribution of sectional profile areas that thecounting of points is restricted to profiles sampledin the frame (and the point density should then besomewhat higher).

The Boundary of profilesAnother way of quantifying the structures seen in amicroscopic field is to measure the length of their


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Fig. 3. The same field as in previous figures, the testsystem now includes a set of test lines: the upper edge ofthe lines joining some crosses. The length of one test lineis the distance between two points with the definitiongiven in Fig. 2.

boundaries. Again, it is far too slow an operationto trace the outline of all profiles for that purpose,the setup shown in Fig. 3 is much more efficient.A test system with lines of known length is part ofthe integral test system. This only means that thelines and the points - and the frame, if there is one- are inseparable because they are drawn togetherin the same physical test system, which is a pieceof paper or a transparency, Jensen & Gundersen1982. The arrangement of the different parts of thesystem is irrelevant, much to the surprise of mostinvestigators. In the integral test system the ratiobetween the total length of test lines and the

number of test points is l/p u, i.e. the test linelength in length units u per test point, simply.Whenever a test line intersects a profile boundaryit thereby makes an intersection point l(prof). Inaddition one must know the number of test pointsP(sect) hitting the section like before. The relativeprofile boundary length or profile boundary lengthper area of section BA is in terms of the example inFig. 3

1-1

0.86uu

)(prof/sectL

IA

B

==

==

8018

41.02

tionseconsintpo

tionssecerintprofile

21

p

2

Note that we may estimate the total length of testline on the section by counting points on thesection since l/p = 0.41 u, the test line length perpoint in the test system, is known.

With the above estimate of the relativeboundary at hand one may then estimate the mean

profile boundary b (prof)just as before.When estimating the boundary length on sec-

tions one must ensure that either the boundary isglobally isotropic, i.e. without any detectable pre-ferred orientation, or that the test system or thefields of vision are rotated at random. As it is well

known, 2 IL(prof/sect) is an estimator of surfacedensity, SV(structure/specimen), but only whenthe orientation destribution of the sections is alsotaken into account, a problem dealt with later on.

Test Systems and EfficiencyBoth two- and three-dimensional quantitation hassuffered a lot from the very widespread misunder-standing that one has to count thousands of points- but this is never the case.

With respect to the precision of the estimate themost important aspect is to sample enough fields

of vision to climate any variation from field tofield. If that is done, the necessary total number ofevents counted need never exceed 100 to 200, i.e.for the estimation of a(prof) one counts roughly100 P(prof) and 100 Q(prof) over the set of fieldssampled in one specimen, cf. Gundersen et al.1980, Gundersen & sterby 1981, Gundersen1986, and the numerous references in the latterreview. It is also possible to predict the optimal

Fig. 4. A test system with three sets of points and two sets

of test lines, arranged in a regular tessellation (a completecovering of the plane) of the unit enclosd in stippled lineswith area u2. The three point sets are: all encircled points(one per unit), all points at the ends of test lines (four perunit), and all points not on test lines ( 16 per unit). Thetwo line sets are all lines with an encircled point at theend (one per unit) and all lines (two per unit).


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sampling scheme from data of the study itself.Such a prediction may even be read off in a simplenomogram, see Gundersen et al. 1980 and Gun-dersen & sterby 1981.

Test systems are usually so simple that the easiestway to get one is to draw it. For many purposes two

or three basic designs are suffiicient. For a newstudy one then just makes a copy of a suitable testsystem at a magnification where the above sam-pling intensity is realized, see for example Gun-dersen 1984. In many cases it is a great advantageto have more than one set of points and lines in thesame test system. One then uses the few, "coarse"points for counting on the section and another setof many, "fine" points for counting on somethingwhich is only covering a fraction of the section.The known ratio between the point sets is thentaken into account in the calculations. One of themost popular test systems in our laboratories over

a number of years is shown in Fig. 4.

CAVALIERI'S DIRECT ESTIMATOR OF THEVOLUME OF ANYTHING FROM SECTIONS

This is the first of some examples of very simplestereological estimation procedures which haveturned out to be very useful in a vast number ofsituations. They are also characterized by theirstrong property of unbiasedness, i.e. in order to usethem one does not have to make any unrealisticassumptions about the structure with respect toshape, orientation etc., etc.

The irregular shape of various organs in thehuman body has imposed an interesting and inmany cases unsolved problem for estimation ofreliable volumes, the human brain cortex being anexample in point with the folded outline of itssurface. With the introduction of stereologicalmethods some 20 years ago and a growing im-provement of techniques since then, volume esti-mation is no longer a problem. In the following, amethod for estimation of brain cortex, whitematter, central grey structures and brain ventriclesis presented. The principle is evidently extremely

general and therefore applicable to almost any-thing, the mathematical basis of the estimator isvery old. For all intents and purposes it was givenby the Italian mathematician Cavalieri who livedfrom 1598 to 1647. Cavalieri showed that thevolume of any object V(obj) may be estimatedfrom parallel sections separated by a known dis-tance t, by summing up the areas of all cross

Fig. 5. A coronal section of a human brain with atransparent test system superposed. For the Cavalieri-es-timate of total cortex volume one counts all encir-cled test points which hit the brain cortex sectioned bythe upper section plane, disregarding oblique pialsurface visible below the section plane. See text forfurther details. The length of one test line is 2 cm.

sections of the object a(prof) and multiplying thisfigure by t:

V(obj) = t a(proit)

There are no other conditions except that theposition of the first section must be random in theobject, see Gundersen & Jensen 1987. Specifically,the estimate is completely independent of theorientation of the set of sections and of the shapeof the object.

MethodThe fixed brains are embedded in 20% gelatin in2% aqueous phenol and stored cold for seven daysafter which surplus gelatin is trimmed off (alterna-lively, the brains may be embedded in 7% Agar).The brains are then cut into parallel, coronal slicessix mm thick. Since the brain may not be slicedwith aprecise distance of six mm the average slicethickness t can be estimated by measuring the

length of the brain omitting the first and last sliceand then divide with the total number of slicesminus 2. On all cut surfaces on all sections onethen applies at random a test system with regularlyarranged points, see Fig. 5. For each structureunder study one now simply counts all pointsP(struct) which hit it. The absolute area of thecross section of the structure is


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a(sect) = a(p) P(struct) u2

where a(p) is the area in known units u2 associatedwith each point in the regular test system, see Fig.4. The total volume of each structure: cortex, whitematter, central grey matter, ventricles etc., is now

estimated after adding all points hitting that struc-ture

V(struct) = t a(p) P(struct) u3

The remarkable freedom of the estimator fromassumptions regarding shape and orientation hasalready been commented upon. It is also a veryuseful estimator of the volume of very irregularlyshaped cells (for a range of examples, see Gun-dersen & Jensen 1987), where one never has toresort to serial reconstruction, a very tediousprocedure from which one gains no reliable, quan-

titative information.Another noticeable feature of the estimator is its

effiiciency, studied in detail in Gundersen & Jensen1987. Even for a very irregular three-dimensionalstructure like the human brain cortex a total of ~200 points counted on 10 to 15 sections will pro-vide an unbiased estimate of the true volume witha precision better than 5%, a procedure whichtakes ~ 15 min. For a number of further practicaldetails see Pakkenberg 1987 andRegeur & Pakken-berg 1988 wherein large series of human brainsare studied.

Most of the stereological estimators of struc-tural quantities described in the literature, see forexample Weibel's book from 1980 or the reviewsby Gundersen 1980and 1986, are two step proce-cures: One estimates in the first step the density ofthe structure of interest in the volume of thecontaining or "reference" space V(ref). All thesedensities: Volume per volume VV, surface area pervolume SV, length per volume LV, or number pervolume NV, are ratios which generally do not al-low one to make conclusions about changes in theabsolute amount of the structure under study(some spectacular examples of mistakes made over

the years are discussed in the section entitled "TheReference Trap" in Brndgaard & Gundersen1986). The greatest value of Cavalieri's estimatoris therefore that it effortlessly allows one in thesecond step of an ordinary stereological estimationprocedure to obtain estimates of the volume of thereference space even if it is enclosed in some othertissue. For an isolated organ like the

thyroid gland or the liver with a specific gravity ~1.0 one can in most cases simply weigh it to get anessentially unbiased estimate of its volume. Theabsolute amount of the structure is then thedensity times the absolute volume of the referencespace.

As already mentioned, the general estimator ofvolume fraction is

PP(struct/ref) = VV(struct/ref)

for the use of which the sections must haverandom positions in the specimen, in practice asystematically random set of roughly equidistantsections is likely to be the most efficient samplingscheme. Most of the practical details of thisestimator have already been dealt with. It is, how-ever, worth mentioning again that for almost allspecimens five systematic sections and a total of

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