Page 1

APMIS 96: 379-394, 1988

379

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. MØLLER1, K. NIELSEN2,

J. R. NYENGAARD, B. PAKKENBERG1, F. B. SØRENSEN, A. VESTERBY3 and M. J. WEST

Stereological Research Laboratory, University Institute of Pathology and 2nd University Clinic of Internal

Medicine, Institute of Experimental Clinical Research, University of Århus, Neurological Research Laboratory1,

Hvidovre Hospital, Institute of Pathology2, University of Copenhagen, Rigshospitalet, University Institute of

Pathology3, Århus Amtsygehus, Denmark

Gundersen, H. J. G., Bendtsen, T. F., Korbo, L., Marcussen, N., Møller, A., Nielsen, K., Nyengaard, J. R., Pak-

kenberg, B., Sørensen, F. B., Vesterby, A. & West, M. J. Some new, simple and efficient stereological

methods 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 microscopic

structures which is specifically tuned to provide reliable data from sections. Within the last few years, a

number of new methods has been developed which are of special interest to pathologists. Methods for

estimating the volume, surface area and length of any structure are described in this review. The principles

on which stereology is based and the necessary sampling procedures are described and illustrated with

examples. 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 Århus C,

Denmark.

THE BASIC IDEAS AND TOOLS

OF STEREOLOGY

At a practical level, stereological methods are

precise tools for obtaining quantitative informa-

tion about three-dimensional, microscopic struc-

tures, based mainly on observations made on

sections. Two-dimensional sections contain quan-

titative information about

structures only in a statistical sense. For this

statistical information to be "true" or unbiased a

few requirements must be fulfilled about the sec-

tions and the way they are made. In practice it is

nearly always very easy to fulfil these requirements

(often as easy as doing it the wrong way!) and

throughout the text the sampling methods are

described and references are given to full descrip-

tions of the techniques.

three-dimensional

Two concepts are often referred to in this and

other papers dealing with stereology: "Unbiased"

and "efficient". The concepts are used in a statisti-

cal sense and mean "without systematic deviation

from the true value" and "with a low variability

after spending a moderate amount of time", re-

spectively.

In the following, two-dimensional information

and the way to obtain it effeciently are dealt with

first, then the basic methods for measuring vol-

ume, surface area, and length are described. For

discrete particles like cells, nuclei, glomeruli etc., a

number of new methods are available for measur-

ing their number and mean sizes in a very direct

way. These methods are described in the second,

forthcoming part of the review. Al1 the methods

are used extensively in biological research and

pathology, as indicated by a number of references

Page 2

APMIS 96: 379-394, 1988

380

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 mean

the difference between "interesting observations"

and real knowledge.

TWO-DIMENSIONAL QUANTITATION

The strength of stereological methods is that they

report data for three-dimensional structures in

terms of three-dimensional quantities - not just in

terms of the two-dimensional sections which are

most often used for the observations. The size of

a cell may naturally be expressed in µm3, not by the

area in µm2 which its profile happens to occupy on

a 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 section

with some profiles. Hatched profiles are completely

inside the field. Below. The same field with an unbiased

counting frame (Gundersen 1977), the hatched profiles

are counted according to the counting rule described in

the above reference and in the text.

The two-dimensional methods also serve as an

introduction 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 anything

else which imposes artificial boundaries on the

visible part of the section: How many profiles do

we see? This is not as simple a question as it may

seem: There are 31 profiles completely inside the

window, but, in addition, 10 profile parts are seen

on the edges (they may or may not represent 10

different profiles, but we have no way of knowing

that). So, what is the number of profiles; 31, 41,

36 or what? Most biologists have learned how to

count 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 practical

counting rules more than ~ 10 years old are biased.

For humans, (machines are different), the only

known unbiased frame and its counting rule is also

illustrated in Fig. 1: In addition to profiles com-

pletely inside the frame one counts all profiles

with anything inside the frame provided they do

not in any way touch or intersect the full drawn

exlusion edges or their extensions (Gun-dersen

1977).

The answer to the question is 27 profiles. This

number is not directly comparable to any of the

above figures because the area of the frame is less

than that of the window. For irregular profiles

especially there must be a "guard-area" around the

frame for counting correctly. The natural way to

report such a number is as a numerical density

in the plane, i.e. the number of profiles per

area, a quantity which in stereology is given the

symbol QA

(

(

)

)

2

2

u8 . 8

27

frameofarea

profilesof#

tsecA

profQ

−

====

−

3.1u) (prof/sect

A

Q

assuming that the area of the counting frame is 8.8

in some recognizable areal units, u2. The main

advantage of reporting the number of profiles in

such a standardized way - that it makes it possible

to draw comparisons between different techniques

and observers - is so great that pathologists always

quantify e.g. "cellularity" in terms of number of

cell profiles per unit area and never as just cells or

mitoses per field of unknown size and magnifica-

Page 3

RECENT STEREOLQGICAL DEVELOPMENTS FOR PATHOLOGY

381

Fig. 2. The same field as in Fig. 1, now superposed with

a test system with a set of regularly spaced points (and

a counting frame). For estimates of the areal fraction of

profiles or mean profile area one counts all points hitting

profiles (disregarding their relationship to the frame). A

point is considered a hit if a profile (including its

boundary) covers the upper right corner in the cross

where 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 fields

were sampled. If the fields were sampled just

because the pathologist liked them the best the

number of profiles per area may well characterize

the 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 - but

all inside a well defined region.

The Area of Profiles

The simplest way of reporting "how much there is"

of the structure in the profiles shown in Figs. 1 to 3

is to quantify the areal fraction of the structure,

generally denoted AA

(

t struct/sec

A

∑

A quantity which can be estimated by counting

points which hit the structure, P(struct), divided by

points hitting the section, here illustrated by the

example in Fig. 2. Note that in the simple situation

where the section fills the whole window one need

not count the points hitting the section, all 80

points in the test system do that. In terms of time

and effort there is no way to do it more efficiently,

)

0.23A

===

∑

=

80

18

tionssecof areatotal

structureof areatotal

) t(secP

)struct( P

not to mention the capital investment in hardware.

It takes ~ 18 seconds to count the hits in Fig. 2. To

trace 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 automatic

image analyzers can work on ordinary biological

tissue), for examples of direct comparisons see for

example Gundersen et al. 1981, Regeur & Pakken-

berg 1988, and references therein. For serious use

one must again sample the fields systematically or

by some other random mechanism and for just 5

such fields quantified in this way the coefficient of

error,

MEAN

CM =

of the estimate is in the order of 5%. A

nomogram for reading off the precision of

point-counting estimation of areas is found in Fig.

18 in Gundersen & Jensen 1987. As it is well

known, Pp(struct/sect) may also be used as an

SEM

estimate of the volume fraction Vv(struct/speci-

men) of the structure under study, described in the

next section. One must then as well fulfil some

further requirements with respect to the sampling

of the sections inside the specimen. Note that the

arrangement of the points is irrelevant to the

estimator and that the magnification need not be

known because we are only estimating a relative

area. Later on we shall see another use of a test

system for estimating absolute areas where we

must specify the above information about the

points somewhat more precisely.

Another question which a pathologist would

often ask is the area of a typical profile, i.e. the

mean profile area a(prof). Since the relative area

and 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

27

80

18

profilesof#relative

profiles ofarearelative

) t sec/prof (

a

q

) tsec/ prof8

A

A

Note that the straightforward estimator is the ratio

of two independent estimators: the points hitting

profiles are counted irrespective of whether the

profile in question is counted in the frame or not,

see Fig. 2. It is only if we need to know the

distribution of sectional profile areas that the

counting of points is restricted to profiles sampled

in the frame (and the point density should then be

somewhat higher).

The Boundary of profiles

Another way of quantifying the structures seen in a

microscopic field is to measure the length of their

Page 4

GUNDERSEN et al

382

Fig. 3. The same field as in previous figures, the test

system now includes a set of test lines: the upper edge of

the lines joining some crosses. The length of one test line

is the distance between two points with the definition

given in Fig. 2.

boundaries. Again, it is far too slow an operation

to 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 of

the integral test system. This only means that the

lines and the points - and the frame, if there is one

- are inseparable because they are drawn together

in the same physical test system, which is a piece

of paper or a transparency, Jensen & Gundersen

1982. The arrangement of the different parts of the

system is irrelevant, much to the surprise of most

investigators. In the integral test system the ratio

between the total length of test lines and the

number of test points is l/p u, i.e. the test line

length in length units u per test point, simply.

Whenever a test line intersects a profile boundary

it thereby makes an intersection point l(prof). In

addition one must know the number of test points

P(sect) hitting the section like before. The relative

profile boundary length or profile boundary length

per area of section BA is in terms of the example in

Fig. 3

1-

1

0.86u u

) (prof/sect

L

I

A

B

=⋅=

⋅=⋅=

−

⋅

80

18

41.02

tion sec onsint po

tionssec erintprofile

21

p

2

π

π

π

Note that we may estimate the total length of test

line on the section by counting points on the

section since l/p = 0.41 u, the test line length per

point in the test system, is known.

With the above estimate of the relative

boundary 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 is

globally isotropic, i.e. without any detectable pre-

ferred orientation, or that the test system or the

fields of vision are rotated at random. As it is well

known, 2 ∙ IL(prof/sect) is an estimator of surface

density, SV(structure/specimen), but only when

the orientation destribution of the sections is also

taken into account, a problem dealt with later on.

Test Systems and Efficiency

Both two- and three-dimensional quantitation has

suffered 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 the

most important aspect is to sample enough fields

of vision to climate any variation from field to

field. If that is done, the necessary total number of

events counted need never exceed 100 to 200, i.e.

for the estimation of a(prof) one counts roughly

100 P(prof) and 100 Q(prof) over the set of fields

sampled in one specimen, cf. Gundersen et al.

1980, Gundersen & Østerby 1981, Gundersen

1986, and the numerous references in the latter

review. 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 complete

covering of the plane) of the unit enclosd in stippled lines

with area u2. The three point sets are: all encircled points

(one per unit), all points at the ends of test lines (four per

unit), and all points not on test lines ( 16 per unit). The

two line sets are all lines with an encircled point at the

end (one per unit) and all lines (two per unit).

Page 5

RECENT STEREOLQGICAL DEVELOPMENTS FOR PATHOLOGY

383

sampling scheme from data of the study itself.

Such a prediction may even be read off in a simple

nomogram, see Gundersen et al. 1980 and Gun-

dersen & Østerby 1981.

Test systems are usually so simple that the easiest

way to get one is to draw it. For many purposes two

or three basic designs are suffiicient. For a new

study one then just makes a copy of a suitable test

system at a magnification where the above sam-

pling intensity is realized, see for example Gun-

dersen 1984. In many cases it is a great advantage

to have more than one set of points and lines in the

same test system. One then uses the few, "coarse"

points for counting on the section and another set

of many, "fine" points for counting on something

which is only covering a fraction of the section.

The known ratio between the point sets is then

taken into account in the calculations. One of the

most popular test systems in our laboratories over

a number of years is shown in Fig. 4.

CAVALIERI'S DIRECT ESTIMATOR OF THE

VOLUME OF ANYTHING FROM SECTIONS

This is the first of some examples of very simple

stereological estimation procedures which have

turned out to be very useful in a vast number of

situations. They are also characterized by their

strong property of unbiasedness, i.e. in order to use

them one does not have to make any unrealistic

assumptions about the structure with respect to

shape, orientation etc., etc.

The irregular shape of various organs in the

human body has imposed an interesting and in

many cases unsolved problem for estimation of

reliable volumes, the human brain cortex being an

example in point with the folded outline of its

surface. With the introduction of stereological

methods some 20 years ago and a growing im-

provement of techniques since then, volume esti-

mation is no longer a problem. In the following, a

method for estimation of brain cortex, white

matter, central grey structures and brain ventricles

is presented. The principle is evidently extremely

general and therefore applicable to almost any-

thing, the mathematical basis of the estimator is

very old. For all intents and purposes it was given

by the Italian mathematician Cavalieri who lived

from 1598 to 1647. Cavalieri showed that the

volume of any object V(obj) may be estimated

from 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 a

transparent 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 by

the upper section plane, disregarding oblique pial

surface visible below the section plane. See text for

further details. The length of one test line is 2 cm.

sections of the object Σ Σ Σ Σa(prof) and multiplying this

figure by t:

V(obj) = t ∙ Σ Σ Σ Σa(proit)

There are no other conditions except that the

position of the first section must be random in the

object, see Gundersen & Jensen 1987. Specifically,

the estimate is completely independent of the

orientation of the set of sections and of the shape

of the object.

Method

The fixed brains are embedded in 20% gelatin in

2% aqueous phenol and stored cold for seven days

after which surplus gelatin is trimmed off (alterna-

lively, the brains may be embedded in 7% Agar).

The brains are then cut into parallel, coronal slices

six mm thick. Since the brain may not be sliced

with a precise distance of six mm the average slice

thickness t can be estimated by measuring the

length of the brain omitting the first and last slice

and then divide with the total number of slices

minus 2. On all cut surfaces on all sections one

then applies at random a test system with regularly

arranged points, see Fig. 5. For each structure

under study one now simply counts all points

P(struct) which hit it. The absolute area of the

cross section of the structure is

Page 6

GUNDERSEN et al.

384

a(sect) = a(p) ∙ P(struct) u2

where a(p) is the area in known units u2 associated

with each point in the regular test system, see Fig.

4. The total volume of each structure: cortex, white

matter, central grey matter, ventricles etc., is now

estimated after adding all points hitting that struc-

ture

V(struct) = t ∙ a(p) ∙ Σ Σ Σ ΣP(struct) u3

The remarkable freedom of the estimator from

assumptions regarding shape and orientation has

already been commented upon. It is also a very

useful estimator of the volume of very irregularly

shaped cells (for a range of examples, see Gun-

dersen & Jensen 1987), where one never has to

resort to serial reconstruction, a very tedious

procedure from which one gains no reliable, quan-

titative information.

Another noticeable feature of the estimator is its

effiiciency, studied in detail in Gundersen & Jensen

1987. Even for a very irregular three-dimensional

structure 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 with

a precision better than 5%, a procedure which

takes ~ 15 min. For a number of further practical

details see Pakkenberg 1987 and Regeur & Pakken-

berg 1988 wherein large series of human brains

are studied.

Most of the stereological estimators of struc-

tural quantities described in the literature, see for

example Weibel's book from 1980 or the reviews

by Gundersen 1980 and 1986, are two step proce-

cures: One estimates in the first step the density of

the structure of interest in the volume of the

containing or "reference" space V(ref). All these

densities: Volume per volume VV, surface area per

volume SV, length per volume LV, or number per

volume NV, are ratios which generally do not al-

low one to make conclusions about changes in the

absolute amount of the structure under study

(some spectacular examples of mistakes made over

the years are discussed in the section entitled "The

Reference Trap" in Brændgaard & Gundersen

1986). The greatest value of Cavalieri's estimator

is therefore that it effortlessly allows one in the

second step of an ordinary stereological estimation

procedure to obtain estimates of the volume of the

reference space even if it is enclosed in some other

tissue. 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 an

essentially unbiased estimate of its volume. The

absolute amount of the structure is then the

density times the absolute volume of the reference

space.

As already mentioned, the general estimator of

volume fraction is

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

for the use of which the sections must have

random positions in the specimen, in practice a

systematically random set of roughly equidistant

sections is likely to be the most efficient sampling

scheme. Most of the practical details of this

estimator have already been dealt with. It is, how-

ever, worth mentioning again that for almost all

specimens five systematic sections and a total of <

200 points counted will provide a very precise

estimate, see several of the references already

provided.

ESTIMATION OF LENGTH OF FIBRES,

TUBULES, CAPILLARIES ETC.

The probability that a given structure is hit by a

randomly positioned and randomly oriented sec-

tion is proportional to its linear dimension or

length. It follows that if we observe how often a

particular type of structure is hit on a section - i.e.

count the number of profiles - we have a simple

and direct estimator of its total length (if the

volume of its containing space is known). It turns

out to be as simple as

L(struct) = 2 ∙ QA(struct/ref) ∙ V(ref) u

where the numerical density in the plane QA is

estimated as described above for number of pro-

files in general. From a few EM-sections through

randomly sampled glomeruli (Østerby & Gun-

dersen 1978) and en estimate of the total volume

of glomeruli in a rat kidney of ~ 30 mm3 we may

then learn that the length of capillaries in rat

glomeruli in one kidney is (Østerby & Gundersen

1980)

L(cap) = 2 ∙ QA(struct/ref) ∙ V(ref)

= 2 ∙ 11,000 ∙ 30mm ~ 600m

Page 7

RECENT STEREOLQGICAL DEVELOPMENTS FOR PATHOLOGY

383

Fig. 6. The Orientator by Mattieldt 1988. Above. The

specimen is put roughly at the centre of the circle with

equidistant divisions along the perimeter. The specimen

must be placed on a previously cut, arbitrary surface

(which might be through some embedding material). A

random number among the divisions is selected (using

a random number table). The specimen is cut with a

section perpendicular to the plane of the circle and in the

selected direction (the 14 to 32 direction in the example).

The cut surface and the bottom surface makes a straight

edge. Below. The specimen is placed on the surface just

cut with the straight edge parallel to the 0 to 0 direction

of the second circle with non-equidistant divisions along

the perimeter. The divisions are produced as shown in

Fig. 16. A new random number is selected and the

specimen is cut in this direction (65 in the figure) with a

section perpendicular to the circle. The last surface cut

is isotropic uniform, IU, in the specimen, i.e. without

regard to the position of the specimen in the first step the

last surface has an orientation which is taken from all

possible orientations with constant (uniform) probabili-

ty.

and that this length is increased by 200 m after just

4 days of diabetes!

There is a rather strong requirement to fulfil in

the above estimate, namely that the glomerular

capillary profiles were counted on isotropic EM-

sections. This is a practical problem to which a very

simple and strong solution without any assump-

tions has recently been found and is described

below. There is also the mild assumption that the

structure is "much longer than it is wide", i.e. its

length is much larger than its diameter, see Matt-

feldt et al. 1985 and Østerby et al. 1988. Contrary

to general belief it is not in any way a requirement

that the structure should be straight or cylindrical,

one still counts just the number of isolated profiles.

It follows that for a tubular structure one may

choose to count the isolated inner profiles of the

lumen or the isolated outer profiles. On a given

section their number (counted in an unbiased

frame) will generally not be the same, but averaged

over a suitable number of fields they estimate the

same length, see Seyer-Hansen et al. 1980 for an

illustrative example in which kidney tubule length

was estimated.

THE SIMPLE WAY OF MAKING

A TRULY ISOTROPIC SECTION:

THE ORIENTATOR

Isotropic sections are necessary for a number of

stereological procedures: The above mentioned

estimator of length, the very efficient estimator of

membrane thickness and its distribution ( Jensen et

al. 1979), and some of the newer estimators to be

described later. The procedure is named the Orien-

tator and is described by Mattfeldt 1988. Fig. 6

illustrates the practical steps with an example.

More generally, one might embed the specimen in

Agar and then make the two consecutive sections

illustrated in Fig. 6. If the structure under study is

highly anisotropic the estimation may be carried

out on two more section planes perpendicular to

the last section plane in Fig. 6 and also perpendic-

ular to each other, a co-called Orthrip (Mattfeldt et

al. 1985). It takes one minute to make the two

sections for the Orientator, hardly an excuse for

ignoring structural anisotropy. For a procedure for

estimating the number of capillaries in glomeruli,

also relying on isotropy, see Nyenguardet al. 1988.

Page 8

GUNDERSEN et al

386

Fig. 7. a. Simple sampling

scheme for vertical sections.

An arbitrarily shaped object

is sliced at random into a

number of slabs with the

same thickness, all parallel

to a horizontal plane chosen

a priori. A systematic set of

three slabs are selected, all

perpendicular to the vertical

axia (arrow). b. A horizontal

slab, seen from above, is cut

into vertical sections in

three systematic directions.

The first direction around

the vertical axis is selected at

random and the other direc-

tions can be determined us-

ing a wax plate with a stel-

late pattern, as shown. A set

of vertical sections have

been obtained.

ESTIMATION OF SURFACE AREA

OF ANYTHING FROM SECTIONS

On a section, the surface of a structure shows up as

a boundary. There exists a relationship between

the length of the boundary seen and the surface

area of the structure, but this is not an efficient

estimator because all the boundaries must be

traced. Instead, one counts intersection points I

between the boundary and so-called IUR test lines:

S(struct) = 2 ∙ IL(struct/ref) ∙ V(ref)

=⋅⋅

∑

∑

2p

I

I(struct)

P(ref)

2

V(ref)u

in some analogy to the estimator of profile

boundary. Considering the multitude of biological

processes related to surfaces the estimator ranks

among the most important of the stereological

estimators. IUR in the above means isotropic,

uniform random, i.e. the test lines must have

isotropic orientation (all directions equally likely)

and random position in three-dimensional space,

not just on the section. It is possible to obtain this

on isotropic sections, but on such sections one

tends to loose track of the architecture of the

tissue. However, a recently discovered method can

produce isotropic test lines while still maintaining

a preferred direction of sectioning.

VERTICAL SECTIONS,

THE EFFICIENT SOLUTION TO MOST

ANISOTROPY PROBLEMS

Most existing stereological procedures require

some kind of isotropy. Plane sections of the

structure in question must be isotropic, uniform

random (IUR) planes, or the structure itself must

be statistically isotropic. The only two exceptions

from this are estimation of volume and number.

These basic requirements give the pathologist,

working in the practical field of stereology, serious

problems, due to the fact, that biological structures

are often anisotropic. In such situations it may be

diffiicult or impossible to make IUR tissue sections.

Indeed, the pathologist often prefers to take sec-

tions at a particular orientation to obtain informa-

tion which is absent on other sections. A practical

solution to the problem of anisotropy is presented

in this section, i.e. Vertical Sections. General

remarks on the basic principle of vertical sections,

its impact on sampling procedures and how to

make measurements on vertical sections wil1 be

reviewed. An example which shows how to esti-

mate surface area from vertical tissue sections is

briefly presented.

Definition of Vertical Sections

A vertical tissue section is a plane section perpen-

dicular to a given "horizontal" plane. The meaning

of a horizontal plane is only a plane of reference,

Page 9

GUNDERSEN et al

387

Fig. 8. Sampling scheme,

with improved efficiency,

for vertical sections. Con-

sider the three slabs in Fig.

7a, as a set of parallel, sys-

tematic sections. Each slab

is placed on the table. a. The

firct slab is sliced into paral-

lel "chips" of the same thickness in an arbitrary direction. Likewise, the next slabs are cut into chips, however, in an-

other systematically selected direction obtained by rotating the slabs lying flat on the table. b. Parallel, equidistantly

spaced chips are systematically selected (three in this example) for further processing. Now the vertical axis is defined

for these three chips as running longitudinally through the chips. Then the chips are allowed to rotate freely around

their vertical axis. c. Embedding of the chips has now been accomplished at random with respect to the rotation around

the vertical axis (e.g. in Agar or paraffin). Sections from the block then provide a material, where the vertical axis is

easy to identify (the long axis of the chips). Taking another set of chips at the stage shown in b, another vertical direction

in the specimen will be defined, and thus, eventually a set of sections with a high number of vertical axes is obtained!

for highly efficient stereological estimation.

which defines the orientation of the section. This is

a rather handy definition, since the horizontal

plane can be given by the tissue itself or generated

artificially by the observer. Examples of tissues

naturally possessing a horizontal plane are for

instance tissues covered with flat epithelium like

skin. The possibility to produce a horizontal,

reference plane of any organ of arbitrary shape

makes the principle of vertical sections applicable

to every experimental setting.

Sampling Procedures for Vertical Sections

In the practical handling of tissue for use in vertical

section stereology, four requirements must be

fulfilled, see Baddeley et al. 1986. The first three

deals with the cutting of tissue specimens and are

discussed here, whereas the fourth requirement

concerns the stereological test system employed

for the measurements. The number of steps in-

volved in the handling of specimens may vary

considerably in different applications.

Requirement 1: Either the tissue must intrinsi-

cally possess an identifiable directional (vertical)

axis, or the pathologist must generate such a di-

rection.

The axis only needs to be identifiable, and no

symmetry with respect to the biological structure is

required. It is very easy to define a vertical axis in

muscle at the macroscopical level. In skin the ver-

tical axis can be defined by the normal to the hori-

zontal, reference plane represented by the macro-

scopical cutaneous surface. Tubular organs like

gut and large blood vessels may be opened along

their axis, flattened and thus made into flat speci-

mens with an identifiable horizontal plane

(see Fig. 4 in Baddeley et al. 1986). The effiiciency

of the method is greatest, when the vertical sec-

tions contain the preferred directional axis of the

biological structure in question. However, the

pathologist is completely free to generate an artifi-

cial vertical axis, which may be of special interest

in the experiment. If an artificial vertical axis is

desired, the arbitrary tissue specimen is sliced in

several planes, all parallel to an arbitrary starting

plane of arbitrary direction, i.e. the horizontal base

plane. Vertical sections can then be sliced normal

to the horizontal plane, see Fig. 7. The "horizontal

slabs" should be of the same thickness, to make the

Fig. 9. "Object rotator,', which can be mounted on the

object table of an ordinary miscoscope. After the histo-

logical specimen has been attached, the instrument

enables free rotation of the specimen in the section plane

and alignment with the vertical axis is easy. (Proto type

manufactured bv Olvmous. Denmark).

Page 10

GUNDERSEN et al.

388

Fig. 10. A segment of a cycloid arc with orientation

distribution proportional to the sine of the angle to the

vertical axis. The length along the test curve is twice its

height (parametric equation of a cycloid: x = θ θ θ θ-sinθ θ θ θ,

y = 1-cosθ θ θ θ, where 0 ≤ ≤ ≤ ≤< < < <θ θ θ θπ π π π .

estimation procedure most efficient. Another de-

sign for generating vertical sections is shown in Fig.

8. where optimal conditions to increase efficiency

are given.

Requirement 2: All the vertical sections must be

parallel to the vertical, i.e. normal to the horizontal,

and the vertical direction must be identified in

each section.

In tissue sections showing a natural orientation,

the vertical axis is easily identified. If an artificial

vertical axis has been generated, as described

above, this requirement can be satisfied by taking

all vertical sections normal to the common hori-

zontal plane, and including the lower face of the

slabs in the sections. The vertical direction is then

always perpendicular to the lower edge of the slab

as seen on the vertical histological section. Howev-

er, the vertical sections need to be truly normal to

the horizontal plane, and care has to be taken to

identify the horizontal plane during the processing

of the vertical tissue slabs. At the microscopical

stage it might not always be possible to identify the

vertical axis within the field of vision. In this case

low magnification micrographs of the section,

including the horizontal plane normally solve this

problem. For light microscopy a very useful device

has been constructed, enabling rotation of the

histological tissue specimen to align the vertical

axis, see Fig. 9.

Requirement 3: Relative to the common horizon-

tal plane, the vertical sections must have random

positions and random (i.e. isotropic) orientations.

This is the first step in the sampling scheme

where randomness is required. A set of systematic

randomly positioned sections can be sliced normal

to the horizontal plane by a semiautomatic tissue

slicer, see Fig. 6 in Baddeley et al. 1986. Systematic

random orientation is achieved by selecting the

first direction at random, and the succeeding

directions systematically with respect to the first,

see Fig. 7. The pathologist has at this stage no

freedom to select an especially advantageous direc-

tion. This freedom was exhausted at the first stage

when the vertical direction was defined.

Test Systems for Vertical Sections

Unbiased stereological estimates relying on point

counting can freely be obtained from vertical

sections, a fact, which enables the pathologist to

make a number of stereological estimates from the

same set of vertical sections. However, stereologi-

cal estimates obtained by the use of test line

systems must take into consideration the "vertical

setting" of the sampling scheme described above.

This lead to:

Requirement 4: On the verticalsection, a test line

is given a weight proportional to the sine of the

angle between the test line and the vertical direc-

tion.

This requirement is needed to make the test lines

IUR lines in three-dimensional space, when used

on a set of uniform random positioned vertical

sections. The mathematical outline of this require-

ment has been decribed by Baddeley et al. 1986.

Briefly, the probability of finding a IUR line at a

given angle away from the vertical axis is propor-

tional to the sine of this angle, (which increases

from O at 0° to 1 at 90° away from the vertical axis).

Fortunately, test systems based on these principles

are rather easy to use. The weighting of test lines

can most easily be obtained by construction of a

test system in which test lines at a given angle to the

vertical axis have a length proportional to the sine

of this angle. Such test systems are needed for the

unbiased estimation of surface area and their

composition and application are described below,

illustrated by an example.

Estimation of Surface Area from Vertical Sections

When used on vertical sections, test line systems,

which fulfil requirement 4, mentioned above, are

IUR in space. There are many ways to create

correct test lines, see Baddeley et al. 1986, but for

the practical purpose of surface estimation on ver-

tical sections cycloids are the most suitable,

Page 11

RECENT STEREOLOGICAL DEVELOPMENTS FOR PATHOLOGY

389

since only the total intersection number is then

counted, and no numerical weighting is required,

see Fig. 10. Test systems based on cycloids are

made by computer assisted drawing and are most

efficient when the test curves are "staggered" as

shown in Fig. 11, thus avoiding unnecessary

stastistical variation, e.g. caused by periodic effects

in the biological structure under study, see Fig. 12.

Surface area estimated on vertical sections, ac-

cording to the principles outlined here, is free of

any assumptions about shape. A very instructive

example is given in the paper by Baddeley et al.

1986, illustrating both the production of vertical

sections and the estimation by a cycloid test line

system. However, the method has already been

used within a range of biological settings. Chon-

drocytes of the epiphysal cartilage plate are an-

isotropically arranged, but vertical sections has

solved the problem of unbiased estimation of cell

surface area (Cruz-Orive & Hunziker 1986). Ex-

amples, where vertical sections have been generat-

ed for practical reasons, have also been carried out.

For instance, estimation of capillary surface in the

visual cortex has been done on vertical sections,

and it is still possible to carry out other stereologi-

cal estimates on the same tissue sections, since the

cortical structure and orientation have been pre-

served, see Brændgaard & Gundersen 1986. A

recent investigation includes the estimation of

lung surface area on vertical sections, a practical

solution in this case, where IUR sections are hard

to make, Michel & Cruz-Orive 1988.

STAR VOLUME, A MEASURE OF SIZE

OF VERY COMPLEX STRUCTURES

The star volume is defined as the mean volume of

all parts of an object which can be seen unobscured

in all directions from a particular point, see Fig. 13

(Serra 1982, Gundersen & Jensen 1985). The

averaging in the above mean is over all points

inside the structure. The star volume may be

defined for any type of structure including cavities

like bone marrow space and networks like the

trabecular system. It owes its name to the fact that

the structural characteristic it reports is clearly a

volume and in very complex structures like bone

it is almost the only way to obtain a measure of

size with a strict mathematical definition.

Knowledge of changes in the architecture of the

human skeleton is important because the strength

Fig. 11. a. Simple test system for estimation of surface

based on cycloids. The left edge of the frame must be

aligned with the vertical axis of the section under study.

SV is estimated by counting the number of intersections

with the cycloid arcs, Vv of the reference space by point

counting using the test points, whereas the straight lines

are only introduced to make the following of the test

system across the section easier. b. "Staggered" cycloid

test system for improvement of statistical efficiency

in layered structures. Produced by and available from

Cruz-Orive, Anat. Inst., Berne University, Schweitz.

of bone depends not only on the amount of bone

present but also on the bone structure. Bone mass

declines with age in both sexes due to a negative

bone balance, that is, more bone is resorbed than

formed. This loss of bone is not only due to a sim-

ple thinning of trabecular bone but there seems

also to be a loss of bone caused by perforation or

resorption of the trabeculae (Parfitt 1984). This

leads to a discontinuity in the trabucular network

and thereby to a decrease in bone strength.

Page 12

GUNDERSEN et al.

388

Fig. 12. Vertical section of human skin projected onto a staggered cycloid test system for estimating surface area

of the dermoepidermal interface. The vertical axis is normal to the macroscopical cutaneous surface. Intersections

between the dermoepidermal junction and cycloid arcs are counted. Due to the layering of the biological structure,

staggered cycloids make the estimation more efficient. Points falling in the reference space (either epidermis or corium

in this example) are also counted. (H + E, original magnification: 40X).

Conventional bone histomorphometry has hith-

erto not been able to demonstrate this pathogene-

sis - loss of trabecular bone components - due

to a lack of unbiased stereological parameters. Such

parameters can, however, now be obtained by

using vertical sections (Vesterby et al. 1987). One

of the most informative stereological parameters

in bone appears to be the star volume.

The star volume of the marrow space and the

trabeculae is very easily estimated: Whenever a

Fig. 13. Two-dimensional schematic drawing of the

star volume of the trabeculae and the marrow space. In

arbitrary units of volume, u3, the marrow space star

volume at A is 109 u3 with a coefficient of variation (CV)

of 1.24, at B: 35 u3 (CV: 0.32), at C: 44 u3 (CV: 0.65). The

trabecular star volume at D is 13 u3 (CV: 0.81), at E: 16

u3 (CV: 1.5O) and at F: 3 u3 (CV: 1.79). The estimate of

star volume varies very much, mainly as a function of

orientation. The mean star volume of a structure is

times the average

random directions chosen uniformly among all three-di-

mensional orientations.

3

π π π π

3

0l for many random points and many

Page 13

RECENT STEREOLOGICAL DEVELOPMENTS FOR PATHOLOGY

391

Fig. 14. Sketch of a lumbar vertebra, indicating the

sampling procedure for selection of two random vertical

section planes, perpendicular to each other. The whole

vertebra is positioned randomly with the inferior surface

on a piece of paper which contains horizontal and

longitudinal parallel lines spaced by 5 mm. The center of

the vertera is marked and a rotation of the specimen is

performed. The first random section plane is given by the

longitudinal lines to the right of the center of the

vertebra. (A1). The other perpendicular section plane is

given by the horizontal lines, starting with the first one

to the left of the center of the vertebra (B1). In this way

approximately eight sections are obtained per vertebra.

The section planes are parallel to the "cylindrical" axis

of the vertebra which is also the vertical axis.

random point on a random section hits an object

one measures the length l0 of the unbroken linear

intercept in the structure through the point in a

three-dimensionally isotropic direction. Each in-

tercept length is then raised to the third power and

the star volumen, v* is estimated unbiasedly by

= = = =

I

3

0

3

*

v

π π π π

The statistical variation of the estimate is often

rather large, and in order to obtain reasonably

narrow confidence limits it is likely to be necessary

to measure 100 to 200 intercepts in as many

different directions as possible - in three-dimen-

signal space. The unbiasedness of the estimator is

most critically dependent upon the three-dimen-

signal isotropy of the chosen direction in which the

Fig. 15. Test system for estimating star volume from

vertical sections consisting of a circular frame for se-

lecting directions and an inner system of lines and

points. The direction of the outer frame axis is indicated

by the arrow at the top, the frame is fixed with this

direction parallel to the vertical direction. The construc-

tion of the intervals is illustrated in Fig. 16. The frame is

divided into 97 directions; for the first field a random

number is selected (e.g. 3) and the transparent, inner

grid which contains parallel lines and test points is

rotated into that direction. The next direction in the same

or the following fields is selected by adding a predeter-

mined number - for instance 20 - to the number of the

previous selected direction (3 + 20 = 23). When the sum

exceeds 97 (e.g. l13) it is reduced by 97 (113 - 97 = 16)

and the remainder (16) is the new number and thereby

the new direction.

intercept is measured. Finally, the estimate is only

biologically meaningful if sampling is uniform

over a suitably defined reference space, e.g. a

complete bone. The sampling scheme developed

for vertebral corpora aims specifically at these

problems.

The practical sampling procedure is as follows:

In order to produce random section planes parallel

to the vertical axis of the vertebrae each specimen

is placed with the inferior surface on a sampling

grid with horizontal and longitudinal parallel lines

spaced 5 mm, see Fig. 14. The vertebra is then

randomly positioned and randomly rotated to find

the first plane of sectioning defined by the lines of

the sampling grid, the center, and the cylindrical

axis of the vertebra. In this randomly selected

orientation three to five 5-mm-slices are cut con-

Page 14

GUNDERSEN et al.

390

Fig. 16. Construction of non-equidistant, sine-weighted

orientations for selecting three-dimensionally isotropic

directions on vertical sections. On the radius shown to

the left of the quarter circle a number of equidistant

intervals are marked off. It is most efficient to use a prime

number. The end-point of each interval is transferred

horizontally to the circular arc. Directions from the

centre of the circle through these points are sine-weight-

ed, i.e. their density is proportional to the sine of the

angle from the vertical axis to the direction itself. See also

Figs. 6 and 15.

secutively on a sawing machine. In a plane perpen-

dicular to the first randomly selected plane three to

four similar slices are cut. That is, from each verte-

bra an average of eight 5-mm-thick slices are cut in

two mutually perpendicular planes and all slices

are parallel to the cylindrical axis of the vertebra,

which also is the vertical axis.

Bones slices are embedded, undecalcified, in

methylmetacrylate. From each direction two slices

are selected systematically randomly and from

each of these four slices per vertebra one 8-µm-

thick section is cut on a heavy duty microtome

(Jung, model K) and stained with Goldner-

Trichrome.

Using a recently developed special projection

microscope (Olympus, BHS) the image of the

sections is projected onto a test system placed on

the table at a final magnification of 16X for the

estimation of marrow space star volume and of

65X for the estimation of the star volume of trab-

ecular bone. Starting at a random position

outside the section the microscope stage is moved

in one direction at a time to provide a systematic

pattern of touching fields. On an average, nine

fields per section are used for estimating the star

volumen of trabeculae and two fields per section

for the marrow space. The test system consists of a

frame for systematic selection of random three-

dimensional isotropic directions on vertical sec-

tions and of a set of points and parallel lines on a

separate, transparent sheet, see Figs. 15 and 16.

The axis of the frame is kept parallel to the vertical

axis of the sections. The intercept lengths are

measured using a ruler separated equidistantly

into 20 classes. On an average, 172 measurements

for marrow space and 155 for trabeculae are per-

formed per specimen, see Fig. 17.

The material for the preliminary study reported

here consisted of the first lumbar vertebra which

was obtained at autopsy from two females and

seven men with a mean age of 57 ages (range 26 to

79 years) without malignant diseases or metbolic

bone diseases. The star volume of the marrow

space increased from 11.5 mm3 in the youngest to

Fig. 17. Two-dimensional schematic drawing of marrow

space and trabeculae of a lumbar vertebra at a magnifi-

cation of approximately 16X, at which magnification

the star volumen of the marrow space is measured. All

12 encircled points are used for sampling in the marrow

space. In the shown example, 10 sampling points fall on

the marrow space and through all these the linear

intercept length - in the direction of the lines - of the

marrow space are measured and raised to the third power

(

marrow space the intercept is measured twice, see also

Vesterby et al. 1988.

3

0 l ). If two sampling points on a line fall on the "same"

Page 15

RECENT STEREOLOGICAL DEVELOPMENTS FOR PATHOLOGY

391

61.5 mm3 in the oldest in the group; marrow space

star volume was related to age with a coeffiicient

of correlation of r = 0.81, 2p = 0.014 (Vesterby et al.

1988). The slope of the regression line indicated

that the marrow space star volume increases 1 to

2 mm3 per year in normal man. It is quite remark-

able that the trabecula star volume does not change

with age: it was 0.17 mm3 in the youngest and 0.23

mm3 in the oldest with no indication of any

relationship to age.

All told, the only way to get such a pronounced

increase in the size of the marrow space star

volume is by removing or perforating bone trabec-

ulae. The whole structure thereby becomes more

and more spongy with age, but the trabeculae

remaining in old people have sizes similar to those

in young people. Therefore, the star volume looks

a very promising parameter for elucidating the

pathogenesis of architectural bone changes; it is

the only parameter which gives a direct and unbi-

ased estimate of a well defined size of the cavities

in the marrow space. However, to confirm its

superiority as a tool in this type of bone research it

has to be applied to larger groups of normal per-

sons and to osteopenic diseased patients.

In addition to the above example, star volume is

likely also to be very useful for obtaining a well

defined size parameter for very irregular cells, a

problem with many aspects which are dealt with in

the next part of this review.

REFERENCES

Baddeley, A. J., Gundersen, H. J. G. & Cruz-Orive, L.

M.: Estimation of surface area from vertical sec-

tions. J. Microsc. 142: 259-276, 1986.

Brændgaard, H. & Gundersen, H. J. G.: The impact of

recent stereological advances on quantitative studies

of the nervous system. J. Neurosci. Meth. 18: 39-78,

1986.

Cruz-Orive, L. M. & Hunziker, E. B.: Stereology for

anisotropic cells: Application to growth cartilage. J.

Microsc. 143: 47-80, 1986.

Gundersen, H. J. G.: Notes on the estimation of the

numerical density of arbitrary profiles: The edge ef-

fect. J. Microsc. 111: 2 19-223, 1977.

Gundersen, H. J. G.: Estimators of the number of objects

per area unbiased by edge effects. Microsc. Acta. 81:

107-117, 1978.

Gundersen, H. J. G.. Stereologi, eller hvordan tal for

rumlig form og indhold opnas ved iagttagelse af

strukturer pa snitplaner. Bibl. Læger. 172: 43-61,

1980.

Gundersen, H. J. G. & Jensen, E. B.: Stereological

estimation of the volume-weighted mean volume of

arbitrary particles observed on random sections. J.

Microsc. 138: 127-142, 1985.

Gundersen, H. J. G.: Stereology and sampling of biolog-

ical surfaces. In: Echlin, P. (Ed.): The Analysis of

Organic and Biological Surfaces. J. Wiley & Sons,

New York 1985, chapter 19, pp. 477-506.

Gundersen, H. J. G.: Stereology of arbitrary particles. A

review of unbiased number and size estimators and

the presentation of some new ones, in memory of

Wiliam R. Thompson. J. Microsc. 143. 3-45, 1986.

Gundersen, H. J. G. & Jensen, E. B.: The efficiency of

systematic sampling in stereology and its predic-

tion. J. Microsc. 147: 229-263, 1987.

Gundersen, H. J. G., Boysen, M. & Reith, A.: Compari-

son of semiautomatic digitizer-tablet and simple

point counting performance in morphometry.

Virschows Archiv. 37: 3 17-325, 1981.

Gundersen, H. J. G., Gøtzsche, O. & Østerby, R. Sam-

pling efficiency in morphometry simplified. Metab.

Bone. Dis. and Rel. Res. 2: 443-448, 1980.

Gundersen, H. J. G. & Østerby, R.: Optimizing sampling

efficiency of stereological studies in biology or "Do

more less well"! J. Microsc. 121: 65-73, 1981.

Jensen, E. B. & Gundersen, H. J. G.: Stereological ratio

estimation based on counts from integral test sys-

tems. J. Microsc. 125: 161-166, 1982.

Jensen, E. B., Gundersen, H. J. G. & Østerby, R.:

Reconstruction of membrane thickness distribution

from orthogonal intercept lengths in a plane section.

Scand. J. Statist. 6: 182-183, 1979.

Mattieldt, T., Mall, G., Von Herbay, A. & Moller, P.:

Stereological investigation of anisotropic structures

with the orientator. Acta Stereol. 8: 671-676, 1989.

Mattieldt, T., Mobius, H.-J. & Mall, G.: Orthogonal

triplet probes: An efficient method for unbiased es-

timation of length and surface of objects with un-

known orientation in space. J. of Microsc. 139:

279-289, 1985.

Michel, R. P. & Cruz-Orive, L. M.: Application of the

Cavalieri principle and vertical sections method to

lung: Estimation of volume and pleural surface

area. J. Microsc. 150. 117-136, 1988.

Nyenguard, J. R., Bendtsen, T. B. & Gundersen, H. J. G.:

Stereological estimation of the number of capil-

laries, exemplified by the renal glomerulus. APMIS

Supl. 96: 92-99, 1988.

Parfitt, A. M.: Age related structural changes in trabecu-

lar and cortical bone: Cellular mechanisms and

biomechanical consequences. Calcif. Tissue Int. 36:

123-128, 1984.

Pakkenberg, B.: Postmortem study of chronic

schizophrenic brains. Brit. J. Psychiat. 151. 744-752,

1987.

Regeur, L. & Pakkenberg, B.: Optimazing sampling

designs for volume measurement of components of

human brain using a stereological method. J. Mi-

crosc. 155: 113-121, 1988.

Seyer-Hansen, K., Hansen, J. & Gundersen, H. J. G.: