Computational Neuroanatomy Using Shape Transformations

Christos Davatzikos 1 Quantifying Anatomy via Shape Transformations 250

fohns Hopkins University 2 The Shape Transformation 251

3 Measurements Based on the Shape Transformation 253

3.1 Measurements from Volumetric Images 3.2 Measurements on Surfaces

4 Spatial Normalization of Image Data 255

4.1 Structural Images 4.2 Functional Activation Images 4.3 Other Applications

5 Conclusion 258

References 259

The explosive growth of modern tomographic imaging methods has provided clinicians and scientists with the unique opportunity to study the structural and functional organization of the human brain, and to better understand how this organization is disturbed in many neurological diseases. Although the quest for understanding the anatomy and function of the brain has been very old, it has previously relied primarily on qualitative descriptions. The development of modern methods for image processing and analysis during the past 15 years has brought great promise for describing brain anatomy and function in quantitative ways, and for being able to characterize subtle yet important deviations from the norm, which might be associated with or lead to various kinds of diseases or disorders. Various methods for quantitative medical image analysis seem to be converging to the foundation of the emerging field of computational neuroanatomy, or more generally, computational anatomy.

Despite the promises of modern imaging technology, there are many difficulties involved in quantitative studies of brain morphology. First, the structural and functional organization of the human brain is very complex and variable across individuals, which necessitates the development of highly sophisticated methods. Second, brain function often has very focal character. For example, an abnormally shaped cortical gyrus might be completely unrelated to a neighboring normally shaped gyrus that might perform a totally different function. Moreover, subtle localized abnormalities can have large effects on brain function. Therefore, gross anatomical descriptions are of very limited use. Finally, the exploding volume of image data acquired throughout the world makes it imperative to develop highly automated computerized image analysis methodologies.

Although plenty of quantitative methods have been used in the past for analyzing tomographic images, they have been often limited by the lack of sophistication. As an example we will consider a brain structure called the corpus callosum, which includes the majority of the nerve fibers connecting the two hemispheres of the brain, and which is shown schematically in Fig. 1. The corpus callosum has been believed to be implicated in several neurological diseases and in normal aging. It also is believed to display sex differences. A widespread method for obtaining local measurements of callosal size from tomo-graphic images has been to divide the anteroposterior extent of the structure in five partitions of equal length, and measure the corresponding areas of the callosal subdivisions, as depicted in Fig. 1. Area measurements of these compartments have been used as indicators of interhemispheric connectivity of the corresponding cortical regions. Figure 1 demonstrates some limitations of this method. In particular, the partitioning of the structure depends on its curvature and shape. Therefore, the subdivisions of the callosum in two different brains might differ, depending on each individual's morphology (Fig. 1, top). Moreover, a region of reduced interhemispheric connectivity, which presumably is a region of relatively smaller area, might fall in between two partitions (Fig. 1, bottom), or it might only be a part of a partition. Therefore, when examining area measurements of the whole partition, the results might be washed out. Finally, the extension of such measurements methods to 3D is extremely difficult, since they would require the manual outlining of regions of interest, which often have complex shapes, so that measurements from these regions can be obtained. Mathematical methods, such as the ones described in the following sections, coupled with computer algorithms

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Region of Interest

FIGURE 1 A demonstration of some of the limitations of traditionally used methods for measuring the morphology of a brain structure, the corpus callosum. The anterior-posterior (left-right) extent of the structure is divided into 5 equal intervals, and each of the 5 corresponding areas are measured. As the top row demonstrates, the partitioning resulting from this procedure might be affected by the curvature or, in general, the shape of the structure under analysis. Clearly, the first and the last compartments of the structure are not divided similarly in the two structures. Moreover, a region of interest, e.g., a region that might be affected by some disease process, might fall in between two partitions, as in the bottom image. Since the shaded region occupies only part of two of the partitions of the structure, an effect in that region might be washed out.

FIGURE 1 A demonstration of some of the limitations of traditionally used methods for measuring the morphology of a brain structure, the corpus callosum. The anterior-posterior (left-right) extent of the structure is divided into 5 equal intervals, and each of the 5 corresponding areas are measured. As the top row demonstrates, the partitioning resulting from this procedure might be affected by the curvature or, in general, the shape of the structure under analysis. Clearly, the first and the last compartments of the structure are not divided similarly in the two structures. Moreover, a region of interest, e.g., a region that might be affected by some disease process, might fall in between two partitions, as in the bottom image. Since the shaded region occupies only part of two of the partitions of the structure, an effect in that region might be washed out.

that implement these methods efficiently, will eventually help us overcome most of the difficulties just described.

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