Morphological Analysis of Brain Structures Using Spatial Normalization

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1 Morphological Analysis of Brain Structures Using Spatial Normalization C. Davatzikos 1, M. Vaillant 1, S. Resnick 2, J.L. Prince 3;1, S. Letovsky 1, and R.N. Bryan 1 1 Department of Radiology, Johns Hopkins University 2 Laboratory for Personality and Cognition, National Institute on Aging 3 Department of Electrical and Computer Engineering, Johns Hopkins University Abstract. We present an approach for analyzing the morphology of anatomical structures of the brain, which uses an elastic transformation to normalize brain images into a reference space. The properties of this transformation are used as a quantitative description of the size and shape of brain structures; inter-subject comparisons are made by comparing the transformations themselves. The utility of this technique is demonstrated on a small group of eigth men and eight women, by comparing the shape and size of their corpus callosum, the main structure connecting the two hemispheres of the brain. Our analysis found the posterior region of the female corpus callosum to be larger than its corresponding region in the males. The average callosal shape of each group was also found, demonstrating visually the callosal shape differences between the two groups. 1 Introduction The morphological analysis of the brain from tomographic images has been a topic of active research during the past decade [1, 2, 3]. It is important for understanding the normal brain as well as for identifying specific anatomical structures affected by diseases. In this paper we propose a methodology for studying the size and shape of brain structures, based on a spatial normalization procedure [4, 5] which transforms images into a common reference system, bringing them into correspondence with a template image. Associated with this normalization transformation is a deformation function, which is defined at each point of a structure of interest in the template, and which reflects the size difference between an infinitesimal region around a point in the template and its corresponding infinitesimal region in the subject space. Inter-subject comparisons are made by comparing the corresponding deformation functions. This normalization procedure reduces the effect of the shape variability on size measurements, and provides a common reference system for inter-subject comparisons. More importantly, it does not require the a-priori knowledge of the location of a region of interest within a structure, e.g. a region of abnormality; such regions are readily identified through the deformation function. Finally, as shown in Section 2, our methodology provides a means for obtaining average shapes. The primary purpose of this paper is to present a morphometric tool. However, we demonstrate its utility in investigating sex differences in the corpus callosum, by applying it to two small groups of right-handed subjects.

2 2 Materials and Methods Template/Subject Registration. Consider a structure of interest, R, bounded by the boundary B. Let, also, R a be the corresponding structure in a template, which herein is taken from a brain atlas [6], and B a be its boundary. We obtain a map between each point in R a and a point in R in two steps. In the first step we find a parameterization, x(s); s 2 [0; 1]; of B, and a parameterization x a (s) of B a. In conjunction, these two parameterizations define a map from B a to B, which corresponds B a 3 x a (s); s 2 [0; 1] to x(s) 2 B. Depending on the structure of interest and on the data, we obtain x(s) and x a (s) in the following two ways. 1) If specific landmarks can be identified along B and B a, then x(s) and x a (s) are curves parameterized piece-wise by arclength, preserving the landmark correspondences. 2) If specific landmarks cannot be identified along the two boundaries reliably, then x(s) and x a (s) are curves parameterized by arc-length without any intermediate break-points. Since in this case there are no landmarks, the best point-to-point correspondence between x(s) and x a (s) is determined by finding the circular shift of x(s) which brings its curvature in best agreement with x a (s). In the second step of our registration procedure an elastic warping is applied to the template bringing it into registration with the subject image. This warping is obtained by first deforming B a into registration with B; the interior region R a in the template is then deformed elastically like a rubber sheet, following the deformation of its boundary (see [4, 5] for details). Deformation Function. The warping applied to a structure in this registration procedure is reflected in the deformation function, denoted by d(u; v), which is defined at each point (u; v) 2 R a. If the point (u; v) 2 R a is mapped to the point U(u; v) 2 R in the subject space, then d(u; v) is defined by d(u; v) = det (ru(u; v)) ; (1) where r denotes the gradient of a vector function and det() denotes the determinant of a matrix. The deformation function d(u; v) measures the stretching or shrinking which brings the template into registration with a subject. In particular, from the principles of continuum mechanics [7], if d(u; v) > 1 at a point (u; v), then an infinitesimal area around (u; v) expands as a result of the template warping. Similarly, if d(u; v) < 1, then a local shrinking around (u; v) occurs. Consider now a region P R a in the template, and its corresponding region U(P) in the subject image. The area, A, of U(P) is given by [7] A = ZZ P d(u; v)dudv : Note that, since dudv is the area of an infinitesimal region in the reference space, d(u; v) is a local scaling factor representing area enlargement/shrinkage in the neighborhood of (u; v) with respect to the reference template. Therefore, d(u; v) is a means for quantifying the size of a structure and its subregions by using the template as metric, and is used herein for inter-subject comparisons of shapes. Population Analyses. Based on the spatial normalization procedure described above, qualitative and quantitative population comparisons can be readily made. Specifically, let U 1 (u; v); ; U N (u; v) be the maps from the template to each of the N subjects of

3 a population. Let, also, U p (u; v) be the average of the N functions U 1 ; ; U N. Then the average shape of a structure, denoted by C p, in that population is C p = [ U p (u; v) ; (2) (u;v)2ca where C a is the collection of points in the template belonging to the structure of interest. Let p (u; v) be the point-wise mean of the deformation function of the population. Then the difference between two populations, denoted with subscripts 1 and 2, can be measured as an effect size defined as [8] e(u; (u; p1 v) = v)? p2 (u; v) ; (3) (u; v) where (u; v) is the point-wise standard deviation of the two populations combined. We will use the effect size as a measure of the statistical significance of our experimental results in Section 3. Impact of the Template. In this section we show that, in principle, any template can be used, without this affecting shape comparisons. Specifically, let U(u; v) be the map from Template 1 to a subject, and let the corresponding deformation function be d(u; v). Let, also, T(u; v) be the map from Template 1 to Template 2, and let s(u; v) be the corresponding deformation function. Finally, let U 0 (T(u; v)) be the map from Template 2 to the same subject, and d 0 (T(u; v)) be the corresponding deformation function. Assuming that U(; ), T(; ), and U 0 (; ) map homologous points to each other, then U(u; v) = U 0 (T(u; v)). From the principles of mathematical analysis it then follows that ru = ru 0 rt ; which together with (1) yields d(u; v) = d 0 (T(u; v))s(u; v) : (4) Now consider two populations having average deformation functions p1 (u; v) and p2 (u; v), respectively, with respect to Template 1. Let, also, (u; v) be the point-wise standard deviation of the two populations combined. Finally, let the average deformation functions and the combined standard deviation of these populations with respect to Template 2 be 0 p1 (T(u; v)), 0 p2 (T(u; v)), and 0 (T(u; v)). Using Equation (4) it can be readily shown that 0 1 (T(u; v)) = p1 s(u; p1 (u; v) ; (5) v) with analogous expressions holding for 0 p2 (u; v) and 0 (u; v). Multiplying the numerator and denominator of (3) by 1=s(u; v), and using (5), we conclude that e 0 (T(u; v)) = e(u; v) ; which implies the invariance of the effect size from the selection of the template. A key assumption in the development above is that the warping transformation maps homologous points to each other. In practice there are deviations from this assumption, due to registration errors. These errors, however, are fairly small, assuming that matching the boundaries of two homologous structures yields a good match of the interior of the structures, as well. Moreover, the inherent smoothness of the elastic transformation results in smoothly varying deformation functions, and therefore it reduces the effect of registration errors since neighboring points have very similar deformation functions. This is bolstered by experimental evidence provided in the following section.

4 3 Results 3.1 Synthetic Images In order to demonstrate the relationship between the deformation function and shape differences of the underlying structures, in this section we apply our technique to the two synthetic images shown in Fig. 1. In this example we elastically warped the shape in (a) (b) Fig. 1. Two synthetic images. The shape in (b) was created from the shape in (a) by expanding the lower left part of the boundary downwards, by squashing the lower right part of the boundary upwards, and by pulling the upper middle part of the boundary downwards. Fig. 1a to that in Fig. 1b. The resulting deformation function is displayed in Fig. 2a. In Fig. 2a the brighter the deformation function is the more expansion the shape in Fig. 1a underwent to match that in Fig. 1b. Darker regions imply contraction. In Fig. 2b and (a) (b) (c) Fig. 2. (a) The deformation function displayed as a gray scale function superimposed on the boundary of Fig. 1a. The regions in which the deformation function is larger than 1.15 and 0.85 are shown in (b) and (c), respectively, and are consistent with the differences in the two shapes in Fig. 1. Fig. 2c we show, in white, the regions in which the deformation function was larger than 1.15 and 0.85, respectively; these values correspond to a 15% expansion and shrinkage, respectively. Figs. 2b and Fig. 2c reflect the shape differences between the shapes in Fig. 1a and Fig. 1b. Specifically, the high value of the deformation function at the lower left part of the structure is in agreement with the fact that that part of the boundary was stretched downwards causing the expansion of the structure in that region. Similarly the contraction at the lower right and upper middle is reflected in Fig. 1c in the low values of the deformation function.

5 3.2 Analysis of the Corpus Callosum The procedures described in Section 2 were used to compare the shape of the corpus callosum in 8 men and 8 women. The deformation functions were calculated for each of the 16 subjects and averaged. The mean deformation functions were then normalized by the total area, so that the integral of d(u; v) was the same for the two groups. This is equivalent to scaling the original MR images so that the average total callosal area is the same for both groups. In Figs. 3a, 3b, and 3c we show in white the regions having effect size greater than 1, 0.75, and 0.5, respectively. (a) (b) (c) Fig. 3. The regions where the effect size of the difference between males and females was greater than (a) 1, (b) 0.75, and (c) 0.5. Finally we determined the average corpus callosum for each of the two groups, using Equation (2). The result is shown in Fig. 4, and it qualitatively verifies our quantitative analysis. Specifically, the posterior part of the corpus callosum appears to be more bulbous in females, as opposed to the rest of the corpus callosum which appears to be larger in males. (a) (b) Fig. 4. The average corpus callosum of (a) the male group and (b) the female group. 4 Summary In this work we developed an approach for quantifying the shape of two-dimensional brain structures, based on a spatial normalization procedure and on the measurement of a deformation function resulting from the registration of an atlas with subject images. This technique was tested by comparing a group of male with a group of female subjects.

6 Our methodology results in a continuous deformation function which reveals local differences between homologous structures in different subjects. Inter-subject comparisons of any sub-region of a structure can be readily obtained by integrating the deformation function in that sub-region. Using an atlas as template in our analysis provides a common reference system for population studies. Within this uniform framework intersubject comparisons can be readily performed by comparing the deformation functions. We have also demonstrated the utility of spatial normalization in defining average shapes for populations. The average structure of a group is found by averaging U (u; v) and V (u; v), as opposed to averaging subject images which results in fuzzy average shapes [9, 10]. The shape variability of a structure in a population is reflected in the standard deviation of the normalizing functions U (u; v) and V (u; v), rather than in the fuzziness of the average image. Several methodological issues of our approach also need to be further investigated and refined in future work. In particular, in addition to the deformation function, other properties of the elastic warping of the atlas can be measured, such as strain. These properties reflect not only size characteristics, but shape characteristics as well, since they reflect angular warping in addition to growth or shrinkage. Moreover, the extension of this methodology to 3D will allow its use in describing 3D shapes. The 3D formulation of our spatial normalization method is reported in [11, 5]. References 1. F.L. Bookstein. Principal warps: Thin-plate splines and the decomposition of deformations. IEEE Trans. on Pattern Analysis and Machine Intelligence, 11(6): , D.L. Collins, C.J. Holmes, T.M. Peters, and A.C. Evans. Automatic 3-D model-based neuroanatomical segmentation. Human Brain Mapping, pages , M.I. Miller, G.E. Christensen, Y. Amit, and U. Grenander. Mathematical textbook of deformable neuroanatomies. Proc. of the National Academy of Sciences, 90: , C. Davatzikos, J.L. Prince, and R.N. Bryan. Image registration based on boundary mapping. IEEE Trans. on Med. Imaging, 15(1): , Feb C. Davatzikos. Spatial normalization of 3D images using deformable models. J. Comp. Assist. Tomogr., To appear. 6. J. Talairach and P. Tournoux. Co-planar Stereotaxic Atlas of the Human Brain. Thieme, Stuttgart, M.E. Gurtin. An Introduction to Continuum Mechanics. Orlando: Academic Press, J. Cohen. Statistical Power Analysis for the Behavioral Sciences. Lawrence Erlbaum Associates, A.C. Evans, W. Dai, L. Collins, P. Neeling, and S. Marett. Warping of a computerized 3-D atlas to match brain image volumes for quantitative neuroanatomical and functional analysis. SPIE Proc., Image Processing, 1445: , A.C. Evans, D.L. Collins, S.R. Mills, E.D. Brown, R.L. Kelly, and T.M. Peters. 3D statistical neuroanatomical models from 305 MRI volumes. Proc. of the IEEE Nucl. Sc. Symposium and Med. Imaging Conf., 3: , C. Davatzikos. Nonlinear registration of brain images using deformable models. Proc. of the Workshop on Math. Meth. in Biom. Image Anal., June 1996.

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