## Abstract

The surface area of the cortex is theoretically important but more complicated to measure than cortical volume. Its theoretical importance is primarily due to a close relationship between the surface area (not the volume) of the cortex at any region and the number of neurons in that region. The surface of the cortex in humans is highly convoluted and this creates difficulties in measuring its surface area. We report here on a straightforward extension of stereologic methods to magnetic resonance images that is simple, efficient and elegant. We studied the method’s reliability and the relationship of measurement error to biological variation and suggest a simple method, using variance components analysis, for quantifying all the relevant parameters for efficient design of experiments. We applied this method to a small sample of 15 healthy young male subjects, matched individually on age and parental socioeconomic status to 15 healthy female subjects and measured the surface areas of both cerebral hemispheres. We found no evidence for gender differences or asymmetry in this small study, unlike a previous report using post-mortem material.

## Introduction

The human cerebral cortex is a thin laminar structure with a highly convoluted shape (Carpenter, 1976). Its thickness varies, with motor regions thicker than association regions and association regions thicker than sensory regions (Von Economo, 1929). Because cortical thickness is not constant, there is no simple relation between its surface area and volume.

Measurements of the volume of the cerebral cortex are legion, but measurements of the surface area of the cortex are much less frequently found. The primary reason for this discrepancy appears to be that it is easier to measure volume than surface area. Most volume measurements use the Cavalieri principle (Cavalieri, 1635) for calculating cortical volumes: the brain is sectioned into thin slices, the area of the cortex is measured on these slices and the volume is the sum of the cortical areas times the slice thickness. The convoluted shape of the brain makes measuring the cortical area on the slices slightly more difficult, but Cavalieri’s principle is still valid (Gundersen and Jensen, 1987). The orientation of sectioning is immaterial: it does not matter whether the brain is sliced in an anterior–posterior or a superior–inferior direction — the results are the same — so long as Cavalieri’s principle is applied properly [see Gundersen and Jensen (Gundersen and Jensen, 1987) for a discussion of some commonly made errors in applying Cavalieri’s principle]. Measurements of the volume of the cerebral hemispheres have shown that the brains of women are ∼10% smaller than those of men (Filapek, 1994) and there do not seem to be marked differences between the volumes of the left and right hemispheres of men or women (Batnitsky *et al*., 1981).

Measurements of surface area are more difficult because they are more intricately involved with the convoluted geometry of the brain. We distinguish three major categories of methods for measuring surface area. The first category involves direct measurement, such as covering the surface of the brain with foil of a known thickness and measuring the weight of the foil (Kopp *et al*., 1977). The second category, grounded in methods of stereology (Weibel, 1979; Baddeley *et al*., 1986; Howard and Reed, 1998; Jensen, 1998), deduces surface area from measurements on sections through the brain. The third category, the modeling method, involves either making a geometric model of the cortex, typically a tessellated network of triangles and calculates surface area by summing the area of the triangles (Batnitsky, 1981; Lorensen and Cline, 1987; Dale *et al*., 1999), or by assuming some relationship between the number of voxels on the surface and the surface area (Sisodiya and Free, 1997).

All these methods for making surface area measurements have difficulties. The direct method is obviously unsuitable for *in vivo* work and it is difficult to apply the foil to all the sulci on the surface of a brain. The stereological methods all require some degree of randomization in choosing a sectioning direction. In practice, however, this requirement for randomization has often been ignored, leading to biased estimates (Haug, 1987; Steinmetz *et al*., 1989). The modeling method depends strongly on the techniques used to extract the modeled surface.

The surface area of the cerebral cortex is of considerable theoretical interest. The number of neurons in a particular cortical region appears to be proportional to its surface area, not its volume (Rockel *et al*., 1974, 1980; Powell, 1981), except in Brodman’s area 17 in primates, where the cell counts are approximately twice that which would be expected (but still constant between species). Henery and Mayhew (Henery and Mayhew, 1989) reported that the surface area of the cerebral hemispheres differed between men and women, thus suggesting that women and men have different numbers of cortical neurons, perhaps also related to gender differences in brain volume (Blinkov and Glezen, 1968). Left and right hemisphere surface areas were not significantly different in the Henery and Mayhew study. Their study involved a small sample of six elderly women, and six elderly men and used post-mortem brains.

We wanted to see if we could replicate Henery and Mayhew’s findings using magnetic resonance imaging (MRI) data in middle-aged individuals rather than in post-mortem material from the elderly as part of a plan in our laboratory to develop methods for measuring the surface area of regions of the cortex. Compared with post-mortem studies, MRI studies of the brain are non-invasive, obtainable at any point during the subject’s life and relatively inexpensive.

We extended the stereologic methods used by Henery and Mayhew to MRI, aiming to answer two scientific questions.

Could we also demonstrate gender differences in the surface area of the cerebral cortex?

Is there any difference between the surface areas of the left and right hemispheres?

## Materials and Methods

### Subjects

As part of our ongoing investigations of brain structures in individuals with mental disorder, we have obtained MRI studies of the brains of many healthy unaffected individuals. All of these unaffected individuals provided written informed consent to participate in these studies. The subjects were recruited from hospital staff and the local community by word of mouth or by advertisement. They had no personal or family psychiatric history as assessed by a structured interview using DSM-IV (American Psychiatric Association, 1994) criteria. No subject had a history or MRI evidence of overt cerebral pathology as judged by a radiologist and none had a history of alcohol or substance abuse or of any medical illness known to affect the brain. Additional exclusion criteria included: history of head injury sufficient to have caused coma >1 h duration; past or current history of CNS disease (e.g. stroke, multiple sclerosis), or focal lesion incidentally discovered on our MRI; current (urine screen) or past history of substance abuse or dependence or significant major medical disease; history of hypertension, diabetes or current treatment with antihypertensives, insulin or oral hypoglycemics (to minimize stroke risk); current pregnancy (all pre-menopausal females were tested); history of any drugs of abuse or stimulant use (e.g. amphetamines, cocaine) in 12 months before current admission; any major mental illness as assessed by SCID-P (First *et al*., 1997).

From this collection of subjects, we selected two groups of 15, one male and the other female. Subjects in each group were chosen to be similar with respect to age and parental socioeconomic status (SES) (Hollingshead, 1975) factors which are known to affect the size of brain structures (Pearlson *et al*., 1981) and which could potentially confound our results.

All subjects were strongly right-handed (Annett, 1970). This requirement was included as part of the experimental design because of the known relationship between handedness and the laterality of certain cerebral structures, such as the planum temporale (Geschwind and Levitsky, 1964). Although this requirement means that the potentially important relationships between degree of handedness were not studied, we believed that we first needed to establish any lateral asymmetry in right-handed individuals, whom we expected to show the greatest asymmetry (based on the planum temporale results), rather than complicating the experimental design by including left-handed subjects.

The mean age of the male group was 39.7 years with a standard deviation of 11.1 years, while the mean age in the female group was 38.4 years with a standard deviation of 9.8 years. There was no significant difference between the two groups with regard to age (*t =* 0.32 with 28 d.f., *P* = 0.74, two-tailed *t*-test). In the male group, three subjects had familial SES level = 2 and 12 subjects had familial SES level = 3. In the female group, two subjects had familial SES level 2 and 11 had familial SES level = 3. Because of lack of sufficient demographic information, the familial SES level could not be accurately ascertained for two of the women, but the familial SES for both was either 2 or 3. A χ^{2} test on the 15 men and 13 women was not significant (Pearson’s χ^{2} = 0.101, *P* = 0.75 with 27 d.f.).

### Imaging Parameters

The subjects’ brains were examined with a 1.5 T General Electric Signa MRI scanner. We acquired a contiguous series of 124 spoiled gradient recall acquisition in the steady state (SPGR) images in the coronal plane. The repetition time (*T*_{R}) was 35 ms, echo time (*T*_{E}) was 5 ms, the field of view was 24 cm and slice thickness was 1.5 mm. All images were obtained using a matrix of 256 × 256 voxels with one excitation at each phase encoding step. Digital MRI images were archived on CDROM.

### Method of Vertical Sections

The method of vertical sections (Baddeley *et al*., 1986) is a simple, yet efficient way to measure the surface area of an object.

In brief, the method is as follows.

Within the three-dimensional anatomic specimen, choose any particular fixed direction, called the ‘vertical axis,’ marked on Figure 1 by the arrow.

For each slice, use the location of the random point P to position a cycloidal grid, such as that shown in Figure 1, over the slice and count the number of intersections that the surface of the object makes with the grid. Each cycloid on the grid shown in Figure 1 is equivalent to the path traced by a point on the circumference of a wheel as the wheel rolls a flat plane for a distance equal to half its circumference. The orientation of the cycloids relative to the vertical axis must be as shown in Figure 1 or a right-to-left mirror image of that orientation — the short axis of the cycloids used in the grid must be parallel to the vertical axis for the method to work. The importance of the cycloidal grid’s orientation is that the cycloid effectively weights the orientation of line segments constituting the cycloid by the sine of the angle with the vertical axis. This weighting is what makes it possible to obtain an unbiased estimate of surface area from vertical sections that does not involve randomizing the orientation about axes other than the vertical axis.

From elementary stereology (Weibel, 1979), the following equation holds:

where *S* is the surface area of the object, *V* is the volume spanned by all the slices (not the volume of the object itself), *I* is the number of intersections counted and *L* is the total length of all cycloids in all the slices. By rearranging,

and thus *S* is proportional to the number of intersections counted, because *V* and *L* are properties of the sectioning procedure, not the object itself.

It is relatively straightforward to apply the method of vertical sections to MRI images. The method of vertical sections is known to be unbiased (Baddeley *et al*., 1986), but the bias of the direct and surface modeling methods is unknown. The complexity of the surface being measured (so long as it is visible) does not bias the results. Also, we note that sectioning with a knife (as is usually the case in the anatomic studies for which stereologic methods were developed) can only be done once because something that has been sliced cannot be reassembled and re-sliced from another direction. In MRI images, by using interpolation methods, it is easy to reformat the image data so that we get the appearance of having obtained slices in any orientation that we might desire. Furthermore, this reformatting can be done as often as we wish, so long as the original image data are available. A significant advantage of the vertical sections approach over other stereological methods for measuring surface area is that it enables the researcher freely to choose the vertical axis and generate a uniform rotation about it, rather than generate a random rotation in three dimensions. The preservation of the vertical axis (here we will use the inferior–superior axis of the MRI data) makes visual interpretation of the randomly rotated images easier.

This difference between physical objects and MRI images suggests that, for MRI images, it would potentially increase the accuracy of the method if we chose more than one rotation in step 2 above, repeat steps 3–5 for each rotation and average over rotations. Two possibilities for choosing additional rotations present themselves at once: one option is to generate additional random samples in step 2 and use those (‘random rotations’); the other is to choose the first orientation as in step 2 and to choose additional orientations at angles relative to the first that are in some orderly progression such as 𝛉_{r}, 𝛉_{r} + π/6, 𝛉_{r} +2π/6, 𝛉_{r} + 3π/6, 𝛉_{r} + 4π/6 and 𝛉_{r} + 5π/6, where 𝛉_{r} is the original random rotation chosen in step 2 (‘systematic rotations’). We note that, in this example, additional rotations in this series (such as 𝛉_{r} +6 π/6) are just left–right mirror images of those with a rotation of π less and we do not routinely measure these mirror images because we feel they yield little additional information. (No bias is introduced because 0 ≤ 𝛉_{r} < 2π, so that either ‘mirror image’ is just as likely to be encountered.) We chose the systematic rotations option because of known results in another, related stereologic method for measuring surface area — the orotrip (Mattfeldt *et al*., 1985)—in which it is known that a systematic choice of multiple sectioning direction is more efficient than choosing the same number of sectioning directions at random.

### ‘Grid Settings’ and ‘Direction’

Seven numbers are required to describe the parameters of any particular measurement: the random rotation about the vertical axis; the number of ‘systematic rotations’ about this axis; the *x*-, *y*- and *z*-coordinates of the random location within the anatomic specimen; the spacing between successive slices; and the height of one particular cycloid (the width is always π/2 times the height from elementary geometry). For efficiency, we arranged the individual cycloids as closely as we were able, subject to the constraint that the rectangles bounding a particular cycloid did not overlap and that the bottom left hand corner of one cycloid lay in the center of the voxel at the random location P and that the horizontal spacing between adjacent cycloids was an integral number of voxels. The effect of this constraint was that the right-hand end of a particular cycloid did not always join with the bottom left hand corner of its neighbor located diagonally to the upper right. We refer to the *x*-, *y*- and *z*- coordinates of the random location, distance between slices and the height of the particular cycloid as a ‘grid-setting’.

Because the cycloidal grid may be applied from any particular orientation, we used the following rule of thumb to set the grid settings. First, we chose a fixed value for the cycloidal height. Because of the properties of the cycloid, the width of the cycloid is given (its width is π/2 times its height). We then set the spacing between slices to the height of the cycloid. So, by choosing a given value for the cycloidal height, the other mutable sampling parameters were given. Choosing a small distance cycloidal height gives more intersections with the cycloid than choosing a large distance for the cycloidal height.

### Voxel Anisotropy and Surface Area Calculations

The voxels in our study were not cubes, but cuboids (the lengths of all edges were not the same.) Our software corrects for this anisotropy by using tri-linear interpolation to interpolate the voxel data to isotropic pixels on the computer display. All surface area calculations (which amounted to making sure that the cycloid grips were not ‘warped’ by the voxel anisotropy and that the total length *L* in previous equation for *S* was correctly calculated) were done in the isotropic tri-linear interpolated domain.

### Digital Phantom

To test our software for measuring surface area by the method of vertical sections, we created a digital phantom which corresponded to a sphere with radius 7.5 cm and thus a surface area of 706.9 cm^{2} (= 4π*r*^{2}, approximately the same surface area as a cerebral hemisphere) and which was contained in a volume that was 30 × 25 × 20 cm and 128 × 128 × 128 voxels. We chose these voxel sizes to be approximately those of our data and chose to use voxel dimensions that were not uniform in order to exercise the software routines involved in making the measurements. The sphere was centered in the MRI volume. Voxels with locations inside the sphere were set to 1, voxels with locations outside the sphere were set to 0. This noiseless phantom was not intended to mimic MR data, but to provide a simple way to verify that the software calculations were correct.

### Volume Measurements

Our methods for measuring cerebral volumes have been described elsewhere (Buchanon *et al*., 1998). In brief, our software allowed us to interactively edit a mask indicating which voxels are in the desired structure (in this case, the brain). Volumes were calculated by multiplying the number of voxels in the cerebrum and cerebellum by the volume of an individual voxel. In passing, we note that these volume measurements could also have been made by stereology, but we chose to use this method because of its value (in unrelated frontal lobe studies) in helping us to visualize the surface anatomy of the cortex.

### Studies

We performed three separate studies: one testing validity, one testing reliability and the third testing our hypotheses regarding gender and laterality.

### Statistics

All statistical calculations were performed with JMP statistical software from the SAS institute (Saul *et al*., 2001).

## Results

### Study 1. Validity

To test our methods and software for measuring surface areas, we performed an experiment in which the surface area of the digital phantom was measured using two different grid settings with different values for cycloidal height (and thus, by our rule of thumb, also setting cycloidal width and spacing between slices). For each cycloid size, we chose two pairs of values for starting orientation and cycloidal starting location. Six different slicing orientations (separated by π/3) were chosen for measurement of surface area, four or five sections were measured for each orientation and the total number of intersections counted per orientation was ∼75 for the first cycloid size and 100 for the second cycloid size. Each measurement was repeated on a second day. Thus, this study nests repetition within starting location and starting orientation within cycloidal size. In the analysis of variance for this study, the effect of cycloid size was taken to be a fixed effect, while starting location and starting orientation and repetition were taken to be random effects. We sought to estimate the difference between surface areas using two different cycloid sizes, μ_{1} − μ_{2}, and the variance components due to starting location and starting orientation, σ_{o}^{2} and repetition, σ_{r}^{2}.

### Study 2. Reliability

To test the reliability of our methods and software for measuring surface areas of the cortical gray–CSF boundary, we performed a study in which one rater measured six brains, each with two different settings for cycloidal starting points, initial starting orientation and two orientations about the vertical axis, and repeated all measurements on two separate days using the same grid settings. Depending on the random orientation, four or five sections through each hemisphere were studied and point counts on all sections in one hemisphere were on the order of 100 intersections noted per subject per hemisphere, or ∼200 intersections counted per subject. The volume of this randomly oriented block of images (*V* in the vertical sections equation) was determined by taking the area of the field of view of the interpolated images and then multiplying by the slice thickness (1 mm) and the number of slices in the interpolated data. Care was taken to make sure that the entire brain was shown in the interpolated data. The vertical axis was taken to be the superior–inferior axis, and random rotations were taken uniformly around this axis. This study nested repetition within grid setting within brain. In the analysis of variance for this study, the effects of grid setting and repetition were taken to be random. This study allowed us to estimate the variance components due to brain, σ_{b}^{2}, grid, σ_{g}^{2} and repetition, σ_{r}^{2}. Here, σ_{b}^{2} measures the biological variability in the measurement, while σ_{g}^{2} measures the variability due to the stereological method and σ_{r}^{2} measures the errors made by rater inconsistencies. We note that any variance due to partial volume effects, grid parameters such as cycloidal grid spacing or number of orientations were in either σ_{r}^{2} (the rater didn’t mark exactly the same points because of troubles in finding the gray-matter/ CSF edges) or in σ_{g}^{2} (the particular orientation of the brain may have made some gray-matter/CSF edges more or less distinct), but not in σ_{b}^{2}, thus our identification of σ_{b}^{2} with the intrinsic biological variability, at least to the extent to which the MRI images (the raw data here) reflected the true proportions of the cortex. Given that only one rater made all measurements, we cannot estimate how much additional variance might be added in a hypothetical study of cortical surface area in which some brains were measured by one rater and other brains were measured by another.

Table 2 shows the results of the reliability study. From the relative magnitudes of the variance components, we concluded that σ_{b}^{2}, which is >10 times σ_{r}^{2}, dominated the precision of the experiment using these grid parameters. Thus, most of the variation we observe was not due to factors related to our method of measuring surface area.

### Gender and Laterality Study

To test our primary hypotheses, we first measured the gray–CSF surface area values of both hemispheres combined, using the same grid parameters as outlined in study No. 2. Our software allowed editing the intersection points to be counted, so we then erased all intersections corresponding to the left hemisphere, but allowed those intersections with the right hemisphere to remain, thus obtaining a measurement of the right hemisphere surface area alone. The surface area of the left hemisphere was calculated (not measured independently) by subtracting right hemisphere surface area from total cortical surface area.

We believe that our method of calculating left hemisphere surface area by subtraction of the right hemisphere surface area from the total cortical surface area is superior to the perhaps more straightforward method of measuring each hemisphere separately, then adding the hemispheric surface areas to get the total cortical surface area. Cycloidal lines crossing the inter-hemispheric fissure necessarily give rise to two intersections, one for each hemisphere. Our approach guarantees that total cortical surface area equals left plus right hemispheric surface area, while measuring the left and right hemisphere individually can lead to additional variance in the total because some intersections (those on the interhemispheric fissure) could potentially be counted twice (once on the left, again on the right) or not at all. To avoid this, both measurements could be made simultaneously, but we believe the method we used here is easier because it requires the observer to look at only one MR image at a time.

### Age Effects

The correlation coefficient between age and surface area in all hemispheres was−0.27 [*F*(1,58) = 5.12, *P* = 0.03] (see Figure 3). The correlation coefficient between total surface area (left + right hemisphere together) and age was −0.29 [*F*(1, 28) = 2.63, *P* = 0.116]. These results show that age appears to affect brain surface area, conclusively in the case of individual hemispheric area, with at least a trend toward an effect in the case of total cortical surface area.

The correlation coefficient between age and brain volume was −0.24 [*F*(1,28)=1.65, *P* = 0.21].

### ANCOVA Analysis

Of course, the left and right hemisphere surface areas are likely to be highly correlated and thus we chose a standard repeated-measures design, with side as a repeated measure. Because of its intrinsic biologic meaning, we used total surface area (= left + right hemisphere surface area) to test for the overall effect of side.

Tables 3–6 show the results of our study. There was no evidence for an effect of gender or age on total surface area, nor was there any effect of the repeated measure, side, implying no significant left–right differences in hemispheric surface area. A one-tailed *t*-test for the difference in means of the male and female brain volumes just failed to reach significance (*t =* 1.61, *P* < 0.055 with 28 d.f.)

## Discussion

In brief, we found, as did previous workers (Henery and Mayhew, 1989; Roberts *et al*., 2000), that it was relatively easy to adapt the method of vertical sections to the measurement of hemispheric surface area using, in this case, MRI images. Because the method of vertical sections is known to be unbiased, we believe that our validation test proves that our results are both accurate and unbiased, while the bias of the direct and surface-modeling methods is unknown. We were able to estimate the magnitude of the errors due to the method itself and also the errors due to intra-rater variability and found both to be small in comparison to the intrinsic biological variability of the hemispheric surface areas.

After our study was complete, we became aware of a slight variant of our method that may have been able to reduce our already small measurement error due to the stereologic method, σ_{g}^{2}, still further. In the method described here, we only chose orientations that started at a random orientation and continued on over a progression that extended from 0 to π, because we falsely thought that orientations between π and 2π would be entirely redundant because they would be mirror images of those slices with orientations from 0 to π. However, we realized that, because we could just as easily have replaced the cycloids in Figure 1 by left-to right-mirror images of those shown, we could also have investigated a systematic progression that went from a random starting orientation plus a progression from 0 to 2π, rather than from a random starting orientation plus a progression from 0 to π. We plan a further study to investigate this matter in more detail. However, we would like to reiterate that our results are unbiased and that the magnitude of σ_{g}^{2} is small relative to the population variance, σ_{b}^{2}, in this particular study and so this theoretical concern is of little importance to the interpretation of our results here, but might be important in other studies that adopt our methods.

We did not replicate the results of Henery and Mayhew (Henery and Mayhew, 1989), who found total brain surface area to be less in women than men, but our results are certainly in the same numerical range as both their study and that of Roberts *et al*. (Roberts *et al*., 2000). We believe that at least one of three important factors may be playing a role in our failure to replicate the Henery and Mayhew findings: statistical power, subject age and fixation artifact. Both our studies were small and had the power to detect only a relatively large effect size (an ∼0.97 standardized effect size in the Henerey study and a 0.53 standardized effect size in our study, given α = 0.05 and β = 0.8). Given that the mean cortical surface area was ∼1600 cm^{2} and that its standard deviation in men and women is ∼250 cm^{2}, our study should have been able to detect a gender difference of ∼8% — approximately the magnitude of gender differences in cortical volumes (not surface areas). Henerey *et al*. show a gender difference in surface area of ∼18%, so we can clearly say that we did not replicate their findings, even though our reported surface areas and theirs are certainly of the same order of magnitude. Henerey *et al*.’s subjects were elderly (average age at death = 80 years), whereas our subjects were much younger (average age = 32 years). We could find no studies in which any comparison of post-mortem and imaging studies is made. For post-mortem material, there are well-known artifacts associated with age-related shrinkage due to fixation (Braendgaard and Gundersen, 1986). For MRI, issues such as partial volume artifact — which caused us considerable difficulty until we developed the ‘edge view’ discussed in the Materials and Methods section — certainly represent a difficulty with almost all such studies of the brain.

We did not compare our surface area measures with either the direct or modeling methods discussed in the Introduction. However, we note that the vertical sections method requires ∼30 min to measure both hemispheres and that it has the advantage of being simple to implement. The assumptions involved with the vertical sections method are minimal and do not involve the extraction of a surface (as is required for the surface modeling methods). Surface modeling methods require segmentation of the image and (usually) a method for triangulating the surface. Both of the steps involve some complications — in the case of segmentation, the validity of the segmentation method is seldom known and in the case of triangulation, there are a number of problems in deciding precisely how to build up the surface from triangles (Schroeder *et al*., 2000) and there are known deficiencies in current algorithms such as the ‘marching cubes’ algorithm (Lorensen and Cline, 1987). Of course, to use the method of vertical sections, the observer needs to be able to tell where the gray matter–CSF boundary is (something done by the rater in our method and by the computer in ‘marching cubes’), but our data prove that we are reliable and we would like to point out that in the majority of cases, a little neuroanatomical knowledge goes a long way in accurately counting the number of intersections of the cycloidal grid with the surface of the brain. For example, it is relatively easy for an observer to note that the cycloid passes through two closely applied gyri, even though the MR image blurs the apposed edges — the edge view makes it relatively simple to determine that the cycloid must have gone through the cortical surface twice. Closely apposed gyri present a particular disadvantage for the tessellation methods and this is probably why most researchers using these methods tend to display the gray–white matter surface rather than the gray–CSF surface (the true cortical surface) Even if the gray matter–CSF boundaries are fuzzy — and in MRI data of the brain this is very common — the stereological method is easier because all that is necessary is for the intersection count to be correct, while surface areas derived by the tessellation methods are highly sensitive to the precise spatial location of the tessellating triangles.

Our data show a clear trend (missing significance by only a small amount) for women’s brains to have slightly lower volumes than men (as has been found by others). Our brain volume data differ from most others — we chose to include the cerebellum in our brain volume measures in this study, while most others report only cerebral volumes. This may account for our report of smaller gender differences, or may be due to the small sample size. We did not find a compelling decrease in brain volume with age in this study, but believe that this may be accounted for — our mean subject age was much younger than most studies of brain volumes in aging.

We plan to extend our studies to a larger sample and to other subject groups, particularly to those with known brain developmental abnormalities, such as Down’s syndrome (where we expect to find significant differences) and to a group of psychiatric patients.

**Table 1**

Source of variation | Sum of squares | Degrees of freedom | Mean square | Parameters estimated | F-ratio | Prob. > F |
---|---|---|---|---|---|---|

Variance estimates: σ_{r}^{2} = 18; σ_{o}^{2} = (333 − 18)/2 = 158. | ||||||

Intraclass correlation coefficient = σ_{o}^{2} /(σ_{o}^{2} + σ_{r}^{2}) = 0.90. | ||||||

Cycloid size | 15 | 1 | 15 | 12(μ_{1} − μ_{2}) + 2σ_{o}^{2} + σ_{r}^{2} | 0.045 | 0.836 |

Orientation and starting location within cycloid size | 3331 | 10 | 333 | 2σ_{o}^{2} + σ_{r}^{2} | 18.9 | <0.0001 |

Repetition within orientation and starting location within cycloid size | 212 | 12 | 18 | σ_{r}^{2} | ||

Total | 3558 | 23 |

Source of variation | Sum of squares | Degrees of freedom | Mean square | Parameters estimated | F-ratio | Prob. > F |
---|---|---|---|---|---|---|

Variance estimates: σ_{r}^{2} = 18; σ_{o}^{2} = (333 − 18)/2 = 158. | ||||||

Intraclass correlation coefficient = σ_{o}^{2} /(σ_{o}^{2} + σ_{r}^{2}) = 0.90. | ||||||

Cycloid size | 15 | 1 | 15 | 12(μ_{1} − μ_{2}) + 2σ_{o}^{2} + σ_{r}^{2} | 0.045 | 0.836 |

Orientation and starting location within cycloid size | 3331 | 10 | 333 | 2σ_{o}^{2} + σ_{r}^{2} | 18.9 | <0.0001 |

Repetition within orientation and starting location within cycloid size | 212 | 12 | 18 | σ_{r}^{2} | ||

Total | 3558 | 23 |

**Table 2**

Source of variation | Sum of squares | Degrees of freedom | Mean square | Parameters estimated |
---|---|---|---|---|

Variance estimates: σ_{r}^{2} = 398; σ_{g}^{2} = (1596 − 398)/2 = 600; σ_{b}^{2} = (32953 − 1596)/4 = 7839. | ||||

Intraclass correlation coefficient = σ_{b}^{2}/(σ_{b}^{2} + σ_{g}^{2} + σ_{r}^{2})= 0.89. | ||||

Brain | 164 765 | 5 | 32 953 | 4σ_{b}^{2} + 2σ_{g}^{2} + σ_{r}^{2} |

Grid within brain | 9578 | 6 | 1596 | 2σ_{g}^{2} + σ_{r}^{2} |

Repetition within brain and direction | 4776 | 12 | 398 | σ_{r}^{2} |

Total | 179 119 | 23 |

Source of variation | Sum of squares | Degrees of freedom | Mean square | Parameters estimated |
---|---|---|---|---|

Variance estimates: σ_{r}^{2} = 398; σ_{g}^{2} = (1596 − 398)/2 = 600; σ_{b}^{2} = (32953 − 1596)/4 = 7839. | ||||

Intraclass correlation coefficient = σ_{b}^{2}/(σ_{b}^{2} + σ_{g}^{2} + σ_{r}^{2})= 0.89. | ||||

Brain | 164 765 | 5 | 32 953 | 4σ_{b}^{2} + 2σ_{g}^{2} + σ_{r}^{2} |

Grid within brain | 9578 | 6 | 1596 | 2σ_{g}^{2} + σ_{r}^{2} |

Repetition within brain and direction | 4776 | 12 | 398 | σ_{r}^{2} |

Total | 179 119 | 23 |

**Table 3**

Gender | Structure | Mean surface area ± SD (cm^{2}) |
---|---|---|

Female | Left hemisphere | 795 ± 119 |

Right hemisphere | 813 ± 142 | |

Total cortex | 1608 ± 256 | |

Male | Left hemisphere | 831 ± 128 |

Right hemisphere | 831 ± 121 | |

Total cortex | 1662 ± 240 |

Gender | Structure | Mean surface area ± SD (cm^{2}) |
---|---|---|

Female | Left hemisphere | 795 ± 119 |

Right hemisphere | 813 ± 142 | |

Total cortex | 1608 ± 256 | |

Male | Left hemisphere | 831 ± 128 |

Right hemisphere | 831 ± 121 | |

Total cortex | 1662 ± 240 |

**Table 4**

Gender | Mean brain volume ± SD (cm^{3}) |
---|---|

Female | 1216 ± 81 |

Male | 1288 ± 153 |

Gender | Mean brain volume ± SD (cm^{3}) |
---|---|

Female | 1216 ± 81 |

Male | 1288 ± 153 |

**Table 5**

Source of variation | Sum of squares | Degrees of freedom | Mean square | F-ratio | Prob. > F |
---|---|---|---|---|---|

Gender | 2067 | 1 | 2067 | 0.58 | 0.45 |

Age | 1799 | 1 | 1799 | 0.51 | 0.48 |

Residual | 94752 | 27 | 3510 | ||

Total | 98875 | 29 |

Source of variation | Sum of squares | Degrees of freedom | Mean square | F-ratio | Prob. > F |
---|---|---|---|---|---|

Gender | 2067 | 1 | 2067 | 0.58 | 0.45 |

Age | 1799 | 1 | 1799 | 0.51 | 0.48 |

Residual | 94752 | 27 | 3510 | ||

Total | 98875 | 29 |

**Table 6**

Source of variation | Sum of squares | Degrees of freedom | Mean square | F-ratio | Prob. > F |
---|---|---|---|---|---|

Gender | 29 383 | 1 | 29 383 | 0.5 | 0.48 |

Age | 158 018 | 1 | 158 018 | 2.7 | 0.11 |

Residual | 1 569 669 | 27 | 58 136 | ||

Total | 1 749 239 | 29 |

Source of variation | Sum of squares | Degrees of freedom | Mean square | F-ratio | Prob. > F |
---|---|---|---|---|---|

Gender | 29 383 | 1 | 29 383 | 0.5 | 0.48 |

Age | 158 018 | 1 | 158 018 | 2.7 | 0.11 |

Residual | 1 569 669 | 27 | 58 136 | ||

Total | 1 749 239 | 29 |

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

**Figure 3.**

**Figure 3.**

This research was partially supported by the National Institute of Mental Health grant RO1 MH60590. The authors would like to thank Drs Godfrey Pearlson and Vince Calhoun for their assistance. We would also like to acknowledge the contribution of an anonymous reviewer who pointed out that, although our surface area measures were unbiased, a slight modification in our method might have decreased the measurement error even more.