Measurements from Volumetric Images

As we mentioned in Section 1, the transformation determined by elastically morphing a template to an individual brain carries all the morphological characteristics of the individual brain. From this transformation, various quantities, each reflecting different aspects of anatomy, can be calculated. In most of this section we focus on regional volumetric measurements, which have long been of interest in the brain imaging community. By regional volumetric measurements we mean local size measurements. For example, a locally relatively reduced size of a brain structure in an individual or in a population might be due to regional brain atrophy, i.e., regional loss of brain tissue. Several diseases have been associated with localized brain atrophy. For example, Alzheimer's patients are believed to have hippocampal atrophy.

Using techniques similar to ours, frontal lobe abnormalities also have been reported [6] for Alzheimer's patients. Similar techniques have also been used to characterize anatomical differences in the hippocampus between normal and schizophrenic brains [28].

Regional size measurements, however, are useful not only in detecting and quantifying brain atrophy, but in studying the structural organization of the normal brain, as well. An example, which will be revisited later in this section, is the corpus callosum, a structure composed of nerve fibers that connect the two hemispheres. There is evidence that the size of the corpus callosum is proportional to the number of nerve fibers crossing from one hemisphere to the other [29]. These fibers tend to be fairly clustered; the anterior region of the corpus callosum includes fibers connecting the frontal lobes of the brain, while the posterior region includes fibers connecting the posteriorly located visual cortical regions, and so forth. Accordingly, if a region of the corpus callosum tends to be relatively larger in an individual or in a group of individuals, this might imply a relatively increased interhemispheric connectivity in that individual or group, in the corresponding cortical region. Consequently, size differences between two groups might be very localized, and difficult to detect in the presence of a very high interindividual variability.

A detailed and very localized representation of the size of a structure can be obtained via the shape transformation approach described earlier. For clarity, we will use the example ofthe corpus callosum in the remainder ofthe section; however, our model is generally applicable. Specifically, consider a template of the corpus callosum, such as the one shown in Fig. 5 on the top left. Consider, also, the transformation ¡T x that maps each point of the template to the corpus callosum of an individual brain, such as the ones on the top row in Fig. 5. Finally, consider the scalar function d x , which we will refer to as the deformation function and is defined as

FIGURE 5 The elastic transformation of a template of the corpus callosum (top left) to the shape of the corpus callosum in three different brains, whose magnetic resonance images are shown on the bottom.

where det is the determinant of a matrix and V is the gradient of a multivariate vector field. The deformation function quantifies how much we locally need to stretch the template in order to adapt it to the shape of the corpus callosum in the individual brain under measurement. Therefore, the deformation function quantifies how large the corpus callosum of the individual brain is in the vicinity of each point x, relative to the template. Consequently, the function d x is a very localized measure of size, was shown in Fig. 2. More generally, the transformation ¡T x reflects shape characteristics of the brain under measurement, relative to the template, around x.

If we want to compare two different corpora callosa, we can do so by comparing the corresponding values of d x at each point x. By grouping together regions in which the deformation functions differ, and by measuring how much they differ in those regions, we can precisely define regional size differences. Typically, we are not simply interested in measuring size differences between two individuals, but in measuring possible size differences between two groups. In that case, the deformation functions of the two groups might differ on the average, but the within-group variability might be very high, possibly making any average difference uncertain. The simplest way to measure size differences between two populations is by applying pointwise t-tests [30] on the deformation functions. By grouping together the points in which a significant difference is found, we can define the region in which two brains or two populations differ, without being restricted by a priori assumptions such as the one in Fig. 1.

In order to better demonstrate the principles of this computational model, we briefly describe its application to a previously published study on sex differences of the corpus callosum [3,31]. It has been previously hypothesized that the posterior part of the female corpus callosum is more bulbous, possibly reflecting an anatomical difference in interhemi-spheric connectivity between the two sexes. In order to test this hypothesis via the computational models just described, we examined images from a population of 114 subjects, 68 men and 46 women. The deformation analysis described above was applied, and the resulting deformation functions were compared statistically at each point in the corpus callosum of the template, i.e., at each point within the measurement reference frame. Points for which the deformation functions of women differed significantly from those of men are shown as white in Fig. 6. These points form the region within which it can be hypothesized that there is a sex difference in inter-hemispheric connectivity. Note that this region falls exactly on the border between two of the partitions shown in Fig. 1, which means that conventionally performed area measurements would severely blur the results. The contradictory findings reported in the literature might be partly due to this fact (see [31] for related references).

FIGURE 6 The region in which morphological differences between men and women were found in a group of 114 normal individuals. In particular, the highlighted region of the corpus callosum was relatively more bulbous in women than in men, possibly reflecting sex differences in interhemispheric connectivity.

FIGURE 6 The region in which morphological differences between men and women were found in a group of 114 normal individuals. In particular, the highlighted region of the corpus callosum was relatively more bulbous in women than in men, possibly reflecting sex differences in interhemispheric connectivity.

The quantitative descriptions of size, or in general of shape, can be directly associated with other, nonmorphologic variables. For example, for the corpus callosum study just described, the correlation of callosal size, quantified by the deformation function, with age is shown as an image in Fig. 7 for men (left) and women (right). Here, the correlation coefficient between the value of the deformation function and age was calculated from the same 114 subjects at each point in the corpus callosum. White regions in Fig. 7 displayed statistically significant correlation between size and age. Therefore, regions in which a significant rate of loss of interhemispheric connections is present can be identified by this analysis, under the assumption that interhemispheric connectivity is reflected by callosal size. Associations between the deformation function and various other variables, including activation images obtained through functional imaging or measures of neurocognitive performance, can be examined in a similar fashion [31]. Therefore, relationships between structure and function or cognition can be examined in greater detail.

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