## Introduction

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 ¡¾ 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 cells:

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. [A QuickTime movie of a through-focus series of DIC images used for 3DC is available.]

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 ¡¾ 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. Our 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 effect.

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 sectioning chatter.

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 counted.
- 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 forbidden planes.

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 ¡¾ 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 (r_{i}) of immersion oil (Beadle, 1940; Loveland, 1981; his appendix 1):

depth of field = |
L( r_{i}^{2}-NA^{2})^{0.5} ————————— NA^{2} |

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 coverslips].

### Section thickness

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 June 1998)

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 celloidin

### 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 confocal optics.

[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 reduces fatigue.

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 than 25.

### 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, N_{v}.

**Step 8.**We repeat steps 4 to 6 at different sites and calculate the 95% confidence interval of the mean of the measurement of N_{v}. 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 television monitor.

### 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 image plane.

## Discussion

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 levels.

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 objective.

### Accuracy of 3DC

Three main factors determine the accuracy of an estimate of cell number:

- The accuracy of the estimate of local cell density, N
_{v}
- The accuracy of the estimate of the total volume from which samples are taken
- 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 N_{v}. 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 N_{v} 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

Absolute 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 N_{v} 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}

Footnote 4. N_{v} measured in fixed tissue may differ greatly from N_{v} in vivo. To estimate N_{v} in vivo, the volumetric shrinkage must be determined.

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.

### Acknowledgements

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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