Note to the Reader
Please refer to: Williams RW, Rakic P (1988) Three-dimensional counting: An
accurate and direct method to estimate numbers of cells in sectioned
material. J Comp Neurol 278:344-352 (1988). Orginal version © 1988 Alan R.
This HTML document is a substantially revised and expanded version of the
original paper. Copyright © 1998 by R. W. Williams.
NOTE: Brackets [...] and a font change are used to denote revisions and
additions by R. Williams not found in the original publication. Please also
note that the original publication contained an error in the description of
the use of Gundersen's unbiased counting frame. This error is corrected in
this revision and in our Erratum and Addendum. We thank H.J.G. Gundersen and
V. Howard for pointing out this mistake.
Updated Sept 10 1999.
Three-Dimensional Counting: An Accurate and Direct Method to Estimate
Numbers of Cells in Sectioned Material
Robert W. Williams and Pasko
Section of Neuroanatomy, Yale University School of Medicine, New Haven,
A Critique of
Implementation of Three-dimensional Counting
The Problem of
Non-uniform Z-axis Shrinkage
1946 Paper: Full text with annotation
Counting on the Internet using the iScope
Analysis on the Internet using the Mouse Brain Library
Links on Stereology
We introduce a way to count and measure
cells in an optically defined volume of tissue called a counting box.
This method—direct three-dimensional counting (3DC)—eliminates the need
for correction factors, such as that introduced by Abercrombie (1946),
to determine the number of cells per unit volume (Nv).
Problems caused by irregular cell shape and cell size, nonrandom
orientation, and splitting of cells by the knife during sectioning are
overcome. Furthermore, 3DC is insensitive to large variations in section
The innovation feature of 3DC is the definition of a counting box
with top and bottom sides located inside the section and at precise
distances away from the cut surfaces of the tissue. The positions of the
top and bottom sides of the counting box are delimited by using a digital
length gauge in conjunction with a Z-axis control unit.
Sections of tissue between 8 and 100 µm thick are examined with a
high numerical aperture objective in combination with video-enhanced
differential interference contrast optics (DIC). Cells are marked on a
television screen while the microscopist scans systematically from the top
to the bottom of the counting box. Cells that are located completely
inside the box and cells that only cross through its top, right, or back
sides are counted. All cells that cross the planes that define the bottom,
left, and front sides of the counting box are not counted.
Direct 3DC provides an accurate, simple and reliable way to count
cells, nuclei, nucleoli, or other objects in sectioned material. To build
the counting system requires a standard transmitted light microscope
equipped with DIC optics, a video camera, a position sensor, and a Z-axis
Neuron number is a fundamental determinant of brain function
(Donaldson, 1895; Bok and Van Erp Taalman Kip, 1939, Frankhauser et al.,
1955; Vernon and Butsch, 1957; Jerison, 1963, reviewed in Williams and
Herrup, 1988). Even a small deficit of these cells can have long-lasting
effects on behavior, primarily because neurons, unlike most other cell
types are unable to proliferate throughout life (Sidman, 1970; Rakic,
1985a,b). Systematic changes in neuron number also appear to fine-tune and
balance connections between different parts of the brain and body during
normal development (reviewed in Prestige, 1970; Hamburger and Oppenheim,
1982; Cowan et al., 1984; Finlay et al., 1987). While progress in
understanding the control of neuron number has been rapid during the past
decade, a serious problem has limited the scope of studies. This problem
has to do with the methods that are used to determine the numerical
density of cells in material that has been sectioned (reviewed in Agduhr,
1941; Konigsmark, 1970; Colonnier and Beaulieu, 19985; Haug, 1986). Even
the most rigorous studies can rarely claim precision greater than ± 10%.
Consequently, only marked changes or effects can be analyzed with
confidence. Problems that demand greater accuracy and reliability cannot
be resolved with current methods.
Several solutions to this problem have been proposed (Ahrens, 1967;
Clarke, 1967; Sterio, 1984; Gundersen, 1986; Rose and Rohrlich, 1987), but
as a cynic might predict, the better the solution, the harder the
implementation. As a result, a practical but biased method developed by
Abercrombie (1946) is now used most frequently. In the next section we
consider in detail several limitations of Abercrombie's method. The main
purpose of this paper, however, is not to criticize his method but to
introduce an alternative way of counting that makes it possible to get a
direct and unbiased estimate of cell number by using standard histological
sections and a conventional light microscope. The essential idea is to
define a sample volume within a section of tissue and then count all cells
within that volume as well as an average of half of all cells that
intersect its outer surface. Several technical problems stand between this
ideas and its implementation.
In this paper we describe the procedure and equipment need to
overcome these technical problems. We call this method direct
three-dimensional counting (3DC)—direct because no correction
factor is used and because serial reconstruction is not required, and
three-dimensional because the three-dimensional position of cells is
taken into account while counting. The 3DC method can be applied
uniformly, irrespective of cell size, stage of development, or
experimental condition. However, we should point out immediately that 3DC
is limited in its use to sections thicker than 6-8 µm. [Like the optical
disector to which our method is closely related, the 3DC method does make
the assumption the section shrinkage in the Z-axis is uniform. Hatton and
von Bartheld (1999) have found that "uniform shrinkage" condition can be
difficult to meet because shrinkage in the core of sections can be
substantially less than shrinkage near the edge. This issue is taken up
again in the section titled "Section Thickness."] 3DC complements unbiased
stereological methods used to estimate numbers of particles in thin and
ultrathin sections (reviewed in Crus-Orive and Hunziker, 1986; Crus-Orive,
1986; de Groot and Bierman, 1986; Gundersen and Jensen, 1986; Braendgaard
and Gundersen, 1986; Gundersen, 1986).
Critique of Abercrombie's Correction
We present 3DC as an alternative to the counting method developed in
the 1940s by Floderus (1944) and
Abercrombie (1946, full text available). To appreciate the advantages
of 3DC we begin by critiquing five features of Abercrombie's
method—specifically, method 1 of the 1946 paper. Consider a section of
tissue 10 µm thick. The microscopist focuses up and down through the
section counting cells that fall in a 50- by 50-µm square and an average
of half of those that intersect its edges (Figs. 1, 2). The group of cells
that are counted includes entire cells embedded in the section and
fragments of cells that were split by the knife during sectioning. The
cell fragments at the upper and lower surfaces of the section are counted
as it they were entire cells so that the count is always an integer. An
unwelcome side-effect is that the count is too high. This is the split
cell error. The thinner the section and the larger the cells, the greater
the overestimate. Abercrombie (1946) reasoned that the true number of
cells should equal the number of cells and cell fragments that are counted
times a correction factor equal to the ratio of the section's thickness
divided by the sum of the section's thickness plus the mean height of the
number = count x (section thickness)/(section thickness + cell height)
If cells have an average height of 10µm thick, then the correction
factor is 0.5 and the true number is roughly half the count.
Unfortunately, there are fundamental problems that undermine the
usefulness and accuracy of this correction and its variants (Floderus,
1944; Konigsmark, 1970). Some of these problems were highlighted by
Abercrombie in his cogent and influential 1946 paper.
Fig. 1. A field of cells and a superimposed counting frame
imaged using video-enhanced differential interference contrast (DIC)
optics on a television monitor. A x100 oil immersion objective with a
numerical aperture of 1.25 was used to examine this field of cells in
layer V of primary visual cortex of a fetal rhesus monkey. The perimeters
of several neurons in the counting box (50 by 50 by 25 µm deep) have been
outlined by using a drawing program [MacMeasure] and a digitizing tablet.
Some of the outlines correspond to cells not currently in the focal plane.
QuickTime movie of a through-focus series of DIC images used for 3DC
The first problem is that the method is biased (Abercrombie,
1946; Marrable, 1962; Aherne, 1967; Konigsmark, 1970; Hendry, 1976).
Systematic errors will arise if the size, shape, and orientation of cells
are not taken into account (Snedecor and Cochran, 1980; Cruz-Orive and
Hunziker, 1986). It is difficult to determine how much bias different
methods generate (Marrable, 1962; Colonnier and Beaulieu, 1985).
Abercrombie warned that his method would tend to overestimate cell number
by 5–10%. The problem of bias becomes more serious and less predictable
when the size distribution of cells is skewed or bimodal (Hendry, 1976).
Fig. 2. A schematic of a counting frame, ABCD, that defines the
length and width of the counting box—50 µm in both dimensions. This
counting frame is a cross section through the counting box shown in Figure
3, as viewed from the top. Thus, corner A in this figure matches the back
left edge of the counting box in Figure 3. The unshaded cells are counted;
the shades cells are excluded. The three-dimensional positions of cells in
this figure are shown schematically in Figure 3. [The asterisk in the
lower right corner is intended to highlight the correction made to the
original figure in which line segment DC was extended through cell 7 to
the right of the page. This was incorrect, and as shown in this revised
figure, the line is actually reflected downward at point C.]
Fig. 3. A counting box. This schematic shows a small piece of
tissue cut at a nominal thickness of 40 µm. The thickness actually varies
± 4 µm. The counting box has dimensions of 50 x 50 x 25 µm and is embedded
entirely within the section. It is insulated from the rough-cut surfaces
of the section by upper and lower guard zones. Thus, cells cut in half by
the knife (e.g., the cell labeled cut), and the cells that are torn
away from surrounding tissue (cell labeled not cut and the pit left
by a cell torn away with the adjacent section) do not compromise the
accuracy of the counting procedure. The same cell outlines shown in Figure
2 are depicted here positioned in three dimensions. Applying the counting
rules gives the following results: Cells 11-14, which are completely
inside the box, are counted. Cells 1 and 8-10, which cross through the
forbidden planes are excluded [Cell 7 should be included]. The
transections of these cells with the forbidden planes are shaded. Note
that the forbidden planes extend outward in Figure 2. For this reason cell
1 in the upper left corner of the box is excluded. Similarly, the bottom
forbidden plane extends across the entire section. Cells 2, 4, and 5 only
cross the top, right, and back sides, and, therefore, are included in the
count. [In the original publication we stated that cell 7 in the lower
right corner should not be counted. This cell should actually be counted.
erratum and addendum explains more completely why cell 7 should be
counted, and see Figure 2.]
The second problem is the difficulty of determining an appropriate
value for object height to enter into the equation. Abercrombie's
impractical solution was to measure the diameter of cells in sections cut
precisely at right angle to the remainer of the tissue. In practice, this
is almost never done. [But see Abercrombie and Johnson, 1946, J. Anat.
Lond. 80:37-50; for an example.] Instead, cell diameter is substituted for
cell height. In some cases, for instance, when cells are oblong or when
the tissue shrinks unevenly during processing, the differences between
these two measurements can be substantial.
The third problem is that the equation in based on the assumption
that the knife cleanly cuts through cells in its path. However, if the
blade plows through the tissue, cells will often be pushed aside and
remain uncut. (Fig. 3). Imagine, for instance, the difficulty of cutting a
mixture of gelatin and hazel nuts with a butter knife. When cells are not
split, the correction will seriously underestimate cell number and
density. Unfortunately, there is no easy way to determine what percentage
of cells are cut and what percentage are pushed aside during sectioning
(cf. Jones, 1937). The 10-15% underestimate of mitotic cell number in fish
blastulae that puzzled Marrable (1962) is almost certainly due to this
The fourth problem is the sensitivity of Abercrombie's method to
variation in the thickness of individual sections. The volumes of samples
counted using Abercrombie's method extend from the upper to the lower
surfaces of each section. Consequently, when thickness varies so will the
counts. This gives rise to excessive sampling errors. Graf (1948) and
Clarke (1967) found differences of up to 50% in the thickness of adjacent
paraffin sections. We have found that neighboring celloidin sections cut
at a nominal setting of 30 µm can vary from 25 to 35 µm. Frozen sections
can have equal of greater variation due to unstable block temperature and
The fifth and final problem is the difficulty of correctly
identifying fragments of cells located at the top and bottom of the
sections. Small fragments of neurons may be difficult to distinguish from
glial cells or pericytes. As a result neurons may be erroneously included
or excluded from the count.
While studying changes in neuron number in the primate visual system
(Williams and Rakic, 1986, 1988; Williams et al., 1987), we became more
and more dissatisfied with Abercrombie's correction. We began developing
an alternative that would be less sensitive to cell size and shape and
section thickness. Our new method has been refined over a period of 2
years. As described in the next section, 3DC overcomes each of the five
problems that we raised.
Method and Implementation
The essential feature of a 3DC system is that it accurately defines a
small volume of tissue called a counting box (Fig. 3). A key
feature of 3DC is that the top and bottom sides of the counting box are
inside the section, a precisely defined distance away from the cut
surfaces of the section. When the counting box has been set up, cells in
and around the box are counted by using the rules listed below:
- Cells (or nuclei, or nucleoli) completely inside the box are
- Cells that cross through the three planes that include the bottom,
front, or left side of the box are excluded. These three planes are
called forbidden planes. They extend beyond the counting box
itself (see Fig. 3). [The forbidden plane defined by the front surface
of the box actually does not extend to the right of the box, but is
reflected forward. Cell 7 should not have a gray transect. This is
somewhat difficult to envision, but should be clear from the
illustrations included in the Erratum and Addendum.]
- Cells that cross through the top, right, and back side of the box
are counted, provided they do not also cross over any of the three
As shown by Gundersen (1977) and Howard et al., (1985), this counting
protocol gives an unbiased measure of the local volumetric density of
cells. For this reason cell size does not come into play in determining
cell number or volumetric density. This simple counting procedure
overcomes the first and second problems with Abercrombie's correction.
Defining the counting box
The four walls of a counting box are defined by using a counting frame
or an eyepiece reticle. As the tissue is moved up and down through the
focal plane, the edges of the frame trace out the walls. When a video
monitor is used for counting, a plastic frame can be placed directly on
the screen, or a frame can be generated and displayed by using a computer
(Fig. 1). The top and bottom sides of the counting box are defined using a
shaft encoder or a length gauge. Ideally, the accuracy and resolution of
these devices should equal the depth of focus of the objective. A shaft
encoder has the advantage of being inexpensive but the disadvantage that
measurements are indirect: rotation of the fine-focus control is measured,
not vertical movement. For this reason, it is necessary to calibrate the
linearity, accuracy, and thermal stability of the focusing mechanism
throughout its range.1 A digital length gauge does not have
this problem. The tip of the gauge rests on the stage. Measurements are
direct and calibration of the focus control is not necessary. Resolution
of 0.1 µm and accuracy of 0.2-0.5 µm can be achieved with either method.
Footnote 1. There are no commercially available vertical
calibration standards for transmitted light microscopy. We have
constructed our own standard by using 0.25- and 0.48-thousandth-of-an-inch
Mylar plastic sheets calibrated with a length gauge to an accuracy of
better than 0.2 µm. [Including calibrated microspheres in embedding medium
or tissue, may be an alternate method.]
Signals from the shaft encoder or the digital length gauge are decoded
and displayed using a Z-axis control unit. This custom-made unit has two
programmable relays that trigger high and low frequency tone generators.
The vertical positions at which tones are produced correspond to the top
and bottom sides of the counting box. As a result, deciding whether a cell
is inside the counting box, above or below the box, or crosses through the
top or bottom side of the box is simple and can be done without having to
look away from the monitor. An alternative method of defining the top and
bottom sides of the counting box is to use the relays to change the color
background of the monitor. This can be done with an RGB monitor by using
the two relays to interrupt the red and green video signals.
Having a counting box with top and bottom surfaces submerged in the
section and defined by an optical plane eliminates the third problem we
raised with Abercrombie's method—the fact that cells may not be split in
two as frequently as expected during sectioning.
Optical height vs. true height
The accuracy of the 3DC method depends on the precision with which the
height of the counting box is defined. The measured height of objects
imaged with a microscope depends on differences in the refractive indices
of the tissue, the objective, and the material in which the tissue and
slide are immersed (Snell's law of refraction). If a dry objective is used
to focus first on the top and then on the bottom of an object, the
distance the stage travels will only be 66% of the true distance
separating the top and bottom (Galbraith, 1955; Glaser, 1982; Gahm and
Witte, 1986). The difference is accounted for by the ratio of the
refractive indices of air and crown glass (1.000:1.524). An object that is
45 µm high will be imaged over a vertical distance of only 30 µm.
Obviously, when one uses dry objectives, refraction must be taken into
account in order to define the height of a counting box. The Z-axis
control unit we have built can compensate for such optical foreshortening.
A much better solution is to avoid this problem by using oil immersion.
When an oil immersion objective is used, the tissue, the mounting medium,
and coverglass, the immersion oil , and the glass of the objective have
refractive indices very close to 1.52 ± 0.02 (Emil, 1943; Loveland, 1981,
pp. 64, 65). Thus, the optical height and the true height of objects
correspond almost exactly. 2
Footnote 2. The refractive index of living tissue is
substantially lower than that of dead or fixed tissue (Gahm and Witte,
1986) and consequently measurements taken from living tissue will be
foreshortened by about 9%.
Depth of field
Ultimately, the depth of field limits the minimum height of a counting
box. If the box is 2 µm high and the depth of field is 0.5 µm, it will not
be possible to determine with precision the vertical position of cells
with respect to top and bottom sides. The depth of field of an objective
depends on its [effective] numerical aperture (NA), the wavelength (L) of
light (about 0.55 µm), and the refractive indes (ri) of
immersion oil (Beadle, 1940; Loveland, 1981; his appendix 1):
|depth of field =
The higher the aperture of the lens, the shallower the depth of field.
The depth of field of an oil objective with a [realized] NA of 1.4 is 0.16
µm. A section that is 6 µm thick can be sectioned optically into 20–40
levels. It is, therefore, possible to determine accurately which cells
cross through the top and bottom sides of the counting box. Infrequently,
however, errors will be made at these sides. For instances, cells above
the box may have very small polar segments that extend through the top
side of the box. If these small polar segments are not detected, the cells
will not be counted (Floderus, 1944; Konigsmark, 1970). Such errors of
omission will be counterbalanced somewhat by errors of commission made at
the bottom side of the counting box. These counterbalancing errors are
generated by cells that are included in the count even though they have
polar segments that extend through the forbidden bottom surface of the
box. To reduce any possible asymmetry in the incidence of the errors at
top and bottom sides of the box due to the finite depth of focus or to the
polar caps, one can switch the relative roles of the top and bottom sides
of the counting box and average the two resulting counts. In practice, we
have never found this necessary.
[It is sometimes important to coverslip the relatively thick
sections used for 3DC using the thinner Number 1 (1 ounce)
coverslips. These have a thickness of less than 130 µm. This makes it
possible to focus on the bottom of thick sections even when using high NA
objectives that have modest working distance (90 µm). If you use 1.5 ounce
coverslips you may not have sufficient working distance to define a
couting box. However, note that most dry objectives are designed for
minimal spherical aberration when used in conjunction with the 1.5 ounce
The fourth problem we pointed out with Abercrombie's correction is its
sensitivity to variation in the thickness of sections. 3DC largely
overcomes this problem (Fig. 3). The height of the counting box is fixed
by the investigator—not by idiosyncratic interactions between the block of
tissue and the knife. The counting box is insulated from variation in
section thickness that occur during sectioning by the guard zones that lie
above and below the box (Fig. 3). Ideally, the guard zone above the top
side of the counting box should be higher than the largest cells. Cells
that cross through the top side of the box are potentially included in the
count, so it is important that they can be identified unambiguously, if
necessary, by focusing above the top side. If the guard zone is thick
enough, 3DC will be immune to the fifth problem—cell identification. In
practice, the guard zone may be shorter than the height of the largest
cells. For instance, nuclei of neurons often have a simple, nearly
spherical shape and can be distinguished easily from glia and pericytes
(Williams and Rakic, 1988). As a result, in many cases the upper guard can
be safely set at one-half the height of the larger cells or nuclei: 4 µm
is adequate. The lower guard zone can always be shallow (often less than 2
µm), because all cells or other objects that intrude into this zone are
excluded from the count (Fig. 3). Identification is not an issue.
We should point out that 3DC is sensitive to differential shrinkage of
sections that occus after cutting. This will increase count variation. The
degree of differential shrinkage can be assessed with a 3DC system by
measuring the thickness of sets of sections immediately after they have
been cut and again after they have been processed.
[Special caution applies to the analysis of frozen-sectioned
material. Frozen tissue cut at 50 µm that is mounted on a gelatinized
slide will shrink in thickness to 20 µm or less when dehydrated through
alcohol and xylene (a residual thickness of merely 10 µm is not uncommon).
There is no guarantee that this shinkage will be uniform, particularly if
tissue is processed rapidly. Tissue close to the exposed surface may
shrink more than tissue close the glass. If the counting box is
consistently defined in a region with greater or lesser shrinkage, then
systematic bias may be introduced. There may also be substantial
section-to-section and batch-to-batch variation in frozen section. As a
result, the effective volume of the counting box may be quite variable. (RW
Variation across sections can also bias counts if the non-uniformity
is due to inherent differences in tissue type. For example, the degree of
shrinkage of myelin-rich areas may differ significantly from that of
densely packed granule cell layers, and this regional difference could
bias a ratio of glial cells to granule cells. (RW Sept 1999)
As recently demonstrated by Hatton and von Bartheld (1999), and as
mentioned above, non-uniform Z axis shrinkage can lead to a systematic
bias in density estimates. The source of this bias is simple to visualize.
Assume that a 100 -µm-thick section is stuck to a slide after cutting but
before dehydration. The section is dried on a slide warmer and eventually
dehydrated and cleared in alcohol and xylene. The outer 20 µm of this
section may shrink to approximately5 µm, while the inner 20 µm may shrink
to only 10 µm. The profile of percentage shrinkage as a function of Z-axis
location may be highly non-linear and simply selecting the center of the
section is not a good solution. Clearly, in a situation like this the
particular position of the counting box will have an enormous impact on
the eventual count.
It is practical to assess cumulative differences in Z-axis shrinkage
by measuring total section residual thickness across a set of sections or
across different parts of single sections. If substantial regional
variation in the residual thickness is discovered then the simplest
expedient to avoid introducing a counting bias in to use a counting box
that extends through the full thickness of all sections. Do not use the
guard zones at the top and bottom of the count box. Now you will have to
assume in your calculations that the counting box height is equal to the
original thickness at which the sections were cut. Using this quick-fix
method you do not even have to have a Z-axis encoder, but you do need to
have excellent optics and you do have to use the same counting rules for
the counting box. This quick-fix 3D counting method is actually the first
way in which we implemented the counting box, prior to equipping our
microscopes with Z-axis encoders. This method is particularly well adapted
to counts of frozen sections. (RW Sept 1999)
There are several solutions to this problem: one is to revert to the
quick-fix method of using a counting box that is as high as the entire
section. A better alternative is to analyze estimates obtained using
counting boxes at different mean Z-axis heights. This analysis can
simultaneously be carried out in cell-dense and cell-sparse zone to
determine whether differential Z-axis shrinkage is affected by regional
tissue factors. If differential shrinkage is a problem then boxes located
at different heights in the section will give different mean density
estimates. This method will also pick up counting bias introduced by
differences in staining intensity and image quality. Often the centers and
the mounted-side of sections are much more lightly stained that the
external surfaces of sections. A final suggestion for circumventing
differential Z-axis shrinkage is to use special cutting and processing
methods. Von Bartheld (1999) recommends cyrostat sectioning, but our own
favorite method to minimize this problem is to embed tissue in
Reasons for differential interference contrast (DIC) optics
It can be difficult to get high-definition images from thick sections.
To increase contrast and definition and improve the vertical resolution of
images we recommend using DIC optics. With this method it is easy to
decide whether the edges of cells or nuclei extend beyond the boundaries
of the counting box (Fig. 1). Furthermore, at high magnification,
unstained or weakly stained tissue can be readily counted. DIC optics
offer another significant advantage: the iris diaphram of the condenser
can be opened to match teh NA of the objective. This gives the best
possible resolution and a narrow depth of filed without loss of contrast (Inoué,
1986, pp. 122-133). If fluorscent cells are to be counted, it may be
necessary to reduce out-of-focus blur by using an image procesor or
[DIC optics greatly improve the effective Z-axis resolution of a
bright-field microscope. By stripping away low spatial frequencies and by
enhancing high spatial frequencies, the definition of all boundaries is
greatly improved. This is true even when imaging conventional Nissl-stained
tissue. DIC optics may cost $6,000 to $12,000, but it is well worth the
expense. An excellent but more costly alternative, particularly when
counting fluorescently labeled cells, is to use an optical or digital
confocal microscope system]
Count accuracy and video mixing
Curcio and Sloan (1986) point out that a major deterrent to the use of
quantitative methods is the"difficulty, tedium, and human error often
associated with analysis of the large tissue samples necessary for
statistically useful conclusions." In our experience, counts made directly
at the microsope through the eyepiece are often in error by one or two
cells in a frame of merely 20 cells (5-10% error). This error is also
systematic; counts are almost invariably low. This problem is even more
serious when thick sections are counted. Not only can the number of cells
in a counting box be high (>50), but cells are often stacked on top of
each other. Accuracy can be improved signficiantly by marking cells as
they are counted. We used a video/computer mixing system to do this. The
computer/video mixer is not an essential part of the 3DC system, but after
having used such a system we would never consider reverting to counting at
the eyepiece. The computer/video system improves speed and accuracy and
Two video signals—one from a video camera mounted on a microscope and
one from a computer—are synchronized, mixed, and displayed on a video
monitor. A digitizing tablet and computer are used to mark, trace, and
count the cells. Even with this semi-automated system, accuracy suffers
when a large number of cells are counted. To make sure that counting is
accurate to within 1 or 2% we recommend that the size of the box be
adjusted so that the number of elements counted per box is on average less
Quality of the video image
We have used a Newvicon video camera that has a central resolution of
700 scan lines. The image is displayed on a monitor with with comparable
resolution (see Inoué, 1986, for a lucid discussion of video systems and
the measurement of video resolution). Given the magnificiation of the
image projected onto the facplate of the camera, a scan line corresponds
to a 0.15-µm-wide strip of tissue, and consequently the system can resolve
objects separated by a distance of about 0.4 µm (twice the equivalent scan
width divided by the Kell factor of 0.7; Inoué, 1986). The great advantage
of the video system is the ability to adjust the contrast and brightness
of the image by using a video processor. With DIC optics, sharp images can
be obtained from poorly and unevenly stained specimens.
Step-by-step counting procedure
In this section we give an example of the steps we use to determine the
volumetric cell density and absolute cell number.
Step 1. We start by measuring the thickness of several
sections. Based on these measurements and their variation we decide what
height the counting box should be. If the sections are between 36 and 44
µm thick, then to define a box with a height of 25 µm, we set one relay
to trigger 3 µm above the bottom surface of the section and the other
relay to trigger 28 µm above the bottom surface of the section. In this
particular example, the lower guard zone will be 3 µm thick at all sites
in all sections, while the height of the upper guard zone will vary in
thickness from 8 to 16 µm.
Step 2. We next adjust the length and width of the counting
box. This will depend on the sampling strategy and one the density of
cells (Snedecor and Cochran, 1967, Chapter 21; Gundersen, 1981)3.
Footnote 3: Parts of the brain with steep gradients in cell
density—retina, for instance, with its 50-100 fold regional differences
(Lia et al., 1987)—are particularly difficult to deal with an require a
stratified sampling protocol. Parts of the brain with lesser gradients
may be sampled adequately by using a simple procedure. In cases in which
a laminate structure is to be counted (e.g. the dorsal lateral
geniculate nucleus, the optic tectum, or a part of the cortex), a series
of adjacent counting boxes that extend fully across the laminae should
be analyzed, and this series of boxes should then be treated as a
single, but large, counting box.
Step 3. Before making and counting cells, we focus on the
lower surface of the section at the particular site and reset the Z-axis
control unit to zero. We confirm that the relays trigger at the correct
vertical levels and that the slide is not drifting. It is usually
necessary to reset the zero position when the slide is moved laterally.
Step 4. We begin counting by marking all cells that are
completely inside the box. As each cell is marked, the computer counts
it. We typically count cell nuclei rather than cell bodies or nucleoli.
The edges of nuclei are more distinct than the edges of cell bodies. And
in contrast to nucleoli, there is one and only one nucleus per cell in
the nervous system. We mark nuclei by tracing their perimeters.
Step 5. We next mark and count all cells that extend beyond
the counting box but that do not cross through the lines that define the
left and bottom edges of the counting frame (see Figs. 1, 2). We do this
while moving the focus back and forth from top to bottom. If a cell
crosses over one of these two forbidden lines but at a level above the
top of the counting box then it still must be excluded (for instance,
cell 1 in Figure 3).
Step 6. Finally, we mark and count cells that cross the top
surface of the counting box. These are the cells that are visible when
the high tone is first generated by the Z-axis control unit.
Step 7. We divide the total count (the sum of steps 4 to 6) by
the volume of the counting box. This gives us an unbiased estimate of
the local cell density, Nv.
Step 8.We repeat steps 4 to 6 at different sites and calculate
the 95% confidence interval of the mean of the measurement of Nv.
This value is a conservative estimate of the precision of the average
cell density and does not take into account systematic regional
differences. To get a more precise estimate of the confidence interval,
it may be necessary to perform a regression analysis (Snedcor and
Cochran, 1967, pp. 186, 444-446.)
Step 9. We estimate the total cell number by multiplying the
mean density by the total volume of the region in which the counting
boxes were distributed.
Fig. 4. A typical setup for direct 3DC. On the left side of the
figure, a microscope is shown equipped with a shaft encoder
(Hewlett-Packard HEDS-6000B-06) on the fine-focus control, and a digital
length gauge (Heidenhain Metro-25 digital length gauge; Heidenhain Corp.,
Elk Grove Village, IL) is shown positioned over the stage. Only one of
these sensing devices is required. The Z-axis control unit decodes and
displays the vertical position of the stage with a resolution of 0.1 µm
and generates high and low tones when the stage moves above or below the
top and bottom sides of the counting box. The video mixing equipment
includes a video camera, a video processor, a video mixer, and a
Using 3DC to measure cell size
It is often useful to determine the size distribution of cell
populations. This can be hard, particularly when an unbiased sample of
cells cannot be selected easily. To overcome this problem, the 3DC
counting rules can be applied. The unbiased sample of cells is then
measured. In many cases, it is useful to determine mean cell diameter.
Typically, the focus is adjusted until the cross-sectional area of each
cell reaches a peak. The perimeter of this near-equatorial transection is
traced and the area is computed. This measurement is converted to the
'equivalent' diameter—the diameter of a circle that has the same area as
the cell's equatorial transection. The distribution of a set of these
measurements gives an unbiased and direct estimate of cell size in the
Given the level of interest in neuron number and its control, it is
surprising how little solid information we have (reviewed in Blinkov and
Glezer, 1968). The most obvious reason for this is that research is
hampered by serious methodological problems. Estimating absolute numbers
of neurons and rates of change in neuron populations is difficult, usually
imprecise, and is the worst cases, even subjective. While the number of
neurons in the nervous systems of several species of coelenterates,
nematodes, and annelids is known with precision (Bullock and Horridge,
1965), estimates in vertebrates are often woefully inaccurate (reviewed in
Jones, 1937; Agduhr, 1941; Blinkov and Glezer, 1968; Konigsmark, 1970;
Haug, 1986; Peters, 1987; Williams and Herrup, 1988). And as a rule, the
larger the brain, the greater the margin of error. For instance, estimates
of the number of neurons in the human brain have an almost absurd
range—from 10 billion to 1 trillion. This problem is not a minor
impediment that prevents us from solving trivial problems in brain
statistics and cell demographics. It is a major barrier that blocks our
way to a better understanding of the development, function, pathology, and
evolution of the nervous system—both at descriptive and mechanistic
Fig. 5. Citation frequency of Abercrombie's 1946 paper in
neuroscience journals (gray bars) and in all other journals (black bars).
These data were extracted from the Science Citation Index,
Institute of Scientific Information, Inc., 1955-1987. Roughly 500 of the
1,205 citations listed in the index were published in The Journal of
Comparative Neurology, the largest number for any journal. The
incidence of miscitation of Abercrombie's paper is high. For example, the
first page number is given incorrectly by more than 133 (11%) of the
papers.[Ironically, even this paper (Williams and Rakic, 1988) miscited
the title of Abercrombie's paper: the correct title is Estimation of
Nuclear Population from Microtome Sections.
Despite significant drawbacks,
Abercrombie's Method 1 (1946) continues to gain adherents and
influence. The 1946 paper is one of the most widely cited publications in
science, with a total of more than 1,200 citations (Fig. 5). Even more
impressive, the rate of use of this particular correction has increased
greatly in the past few years: the paper is more cited 100 times a year.
Evidently the method's practicality has outweighed its problems. In this
paper we have described an equally practical, more accurate, and more
reliable way to determine cell number. Each of the five problems that we
raised with Abercrombie's method are overcome. The reduced effort and
improved accuracy more than offset the modest investment in equipment.
Precedents of 3DC
The first suggestion that an ideal way to count cells would be to keep
track of their three-dimensional position in sectioned material was made
by T.W. Blackstad in a discussion appended to Konigmark's well-cited
review (1970). Blackstad suggested that a microscope developed by Glaser
and Van der Loos (1965) capable of encoding the position of cells in three
dimensions could be used to obtain direct estimates of cell number: "I can
easily imagine myself doing the identification (of neurons and glia) and
feeding the X-Y-Z coordinates of each cell into a computer, going through
the entire, length, wide, and depth of the tissue section." The 3DC system
we have developed accomplishes Blackstad's goal, but instead of
determining the precise three-dimensional positions of cells, we define
the position of six sides of a counting box and then simply mark and count
cells in and around the box.
A more recent precedent of the 3DC method is described by Howard et
al., (1985). These authors have extended Gundersen's (1977) unbiased
two-dimensional counting rule, as we have, to three dimensions by using a
sophisticated tandem scanning reflected light microscope. The theoretical
underpinnings of their method are identical to those of the 3DC method:
each of their "bricks" is equivalent to a counting box. However, the
implementation of their method differs from ours and as they point out is
not suitable for use with conventional histological material.
Practicality of 3DC
Counting is tiresome. It is no surprise that an accurate but difficult
method will not be used. In comparison to Abercrombie's method and its
variants (Konigsmark, 1970), 3DC is simple. It is less work because cell
size is not a factor and therefore does not need to be measured, and the
data do not have to be corrected (Hendry, 1976) 3DC, however, does involve
a more complex counting procedure. Cells must be categorized in relation
to the counting box and its sides. However, the added effort is not
significant and is offset by improved reliability and accuracy. The
hardware requirements of the 3DC system, summarized in Figure 4, are
modest. The only equipment that is dedicated to the system is the Z-axis
control unit and the position encoder.
Applicability of 3DC
One limitation of our method is that sections must be thick enough to
accommodate a counting box that is substantially thicker than the depth of
field of the objective. We recommend that sections be at least 6-8 µm
thick. At the other extreme, there is not theoretical limit. The upper
parts of slabs of tissue, 100 to 250 µm thick, can still be counted if
they can be imaged. The depth to which the counting box can extend into
the section will, however, be limited by the working distance of the
Accuracy of 3DC
Three main factors determine the accuracy of an estimate of cell
- The accuracy of the estimate of local cell density, Nv
- The accuracy of the estimate of the total volume from which samples
- The degree to which those samples represent the entire region
Up until now the weak link in determining total cell number in a
particular part of the nervous system has been the method used to
determine Nv. The direct three-dimensional
method strengthens this link. The counting protocol makes no assumptions
about cell size, shape, or orientation (Gundersen, 1977; Howard et al.,
1985). Consequently, a correction is not required. This is why we call 3DC
a direct and inherently accurate method.[While a local estimate of Nv
may be "technically" unbiased, this does not mean that the final estimate
for an entire tissue will be unbiased. Tissue preparation, poor staining
or counting, differential shrinkage and compression, and numerous other
factors can introduce bias into regional density estimates and final cell
counts.] (RW Sept 99)
While we have pointed out several problems with Abercrombie's
correction, we must also point out that in many situations this method and
its variants can give accurate results. We do not mean to impugn the
validity of the many studies that have used this method. However, to get
an accurate result using Abercrombie's method simply is much more work and
requires verification of the validity of the assumptions that underlie the
correction. The mathematical treatment may become complex (Hendry, 1976).
In sum, the accuracy is derived secondarily.
Absolute vs. comparative accuracy of 3DC
cell number is often not important. Relative differences in cell density
between experimental and control cases, between males and females, between
right and left sides of the brain may be what matter. But even in these
situations, reliable methods are needed. Whenever a correction is applied
that make an untenable assumption about the size, shape, or orientation of
cells, the results [may] will be unreliable. For instance, an experiment
may have unsuspected effects on the size, shape, or even orientation of
cells, and consequently, a rise or fall in Nv may be an
artifact. 3DC overcomes these potential errors because no assumptions are
made about size, shape, or orientation. Numbers of cells can now be
compared with greater confidence.4
4. Nv measured in fixed tissue may differ greatly from Nv
in vivo. To estimate Nv in vivo, the volumetric shrinkage must
In many other situations, absolute cell number does matter. Absolute
cell number, unlike numerical density, is in theory independent of the
method used to process the material. However, the inaccuracy of current
counting methods has meant that even absolute numbers cannot be given much
weight. Consequently, it has been difficult to confirm or refute results.
Differences in absolute numbers may hinge on the relative merits of
corrections. Certainly if we want to compare sets of data, the accuracy
with which we can estimate absolute cell numbers is crucial.
Because 3DC is an unbiased method we may eventually be able to make
comparison between studies with greater confidence. Problems that have
been considered intractable because their solution demanded an
unattainable level of precision may now warrant study. For instance, it
should be possible to use 3DC to study natural variation in the size of
neuron populations with unprecedented accuracy. It should also be possible
to detect and track subtle developmental, experimental, and pathological
changes in cell populations.
This research was supported by grant EY-0293 (to PR) and EY-06627 (to
RW). We thank Dr. Kenneth C. Wikler for comments on this paper. We thank
Richard H. Moore of du Pont Co. for samples of Mylar and Jan Hinsch of E.
Leitz, Inc. and Martin Scott of Eastman Kodak Co. for valuable suggestions
that helped us refine the 3DC system. [Support for this revised version
provided in part by NS 35485 (to RW).]
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Since 11 August 98