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Morphological Appearance Manifolds in Computational Anatomy: Groupwise Registration and Morphological Analysis

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Morphological Appearance Manifolds in Computational Anatomy: Groupwise Registration and Morphological Analysis NeuroImage 45 (2009) S73–S85 Contents lists available at ScienceDirect NeuroImage journal homepage: www.elsevier.com/locate/ynimg Morphological appearance manifolds in computational anatomy: Groupwise registration and morphological analysis Sajjad Baloch ⁎, Christos Davatzikos University of Pennsylvania, Philadelphia, PA, USA article info abstract Article history: Existing approaches to computational anatomy assume that a perfectly conforming diffeomorphism applied Received 15 October 2008 to an anatomy of interest captures its morphological characteristics relative to a template. However, the Accepted 15 October 2008 amount of biological variability in a groupwise analysis renders this task practically impossible, due to the Available online 12 November 2008 nonexistence of a single template that matches all anatomies in an ensemble, even if such a template is constructed by group averaging procedures. Consequently, anatomical characteristics not captured by the Keywords: Computational anatomy transformation, and which are left out in the residual image, are lost permanently from subsequent analysis, Groupwise registration if only properties of the transformation are examined. Deformation based morphometry This paper extends our recent work [Makrogiannis, S., Verma, R., Davatzikos, C., 2007. Anatomical Voxel-based morphometry equivalence class: a computational anatomy framework using a lossless shape descriptor. IEEE Trans. Biomed. Morphological appearance manifolds Imag. 26(4), 619–631] on characterizing subtle morphological variations via a lossless morphological descriptor that takes the residual into account along with the transformation. Since there are infinitely many [transformation, residual] pairs that reconstruct a given anatomy, we treat them as members of an Anato- mical Equivalence Class (AEC), thereby forming a manifold embedded in the space spanned by [transformation, residual]. This paper develops a unique and optimal representation of each anatomy that is estimated from the corresponding AECs by solving a global optimization problem. This effectively determines the optimal template and transformation parameters for each individual anatomy, and eliminates respective confounding variation in the data. It, therefore, constitutes the second novelty, in that it represents a group-wise optimal registration strategy that individually adjusts the template and the smoothness of the transformation according to each anatomy. Experimental results support the superiority of our morphological analysis framework over conventional analysis, and demonstrate better diagnostic accuracy. © 2008 Published by Elsevier Inc. Introduction Such approaches date back to D'Arcy Thompson (Thompson, 1917), who, in 1917, studied differences among species by measuring Computational anatomy typically characterizes anatomical differ- deformations of the coordinate grids from images of one species to ences between a subject S and a template T by mapping the template those of another. (Grenander, 1983) later capitalized on this idea to space ΩT to the subject space ΩS through a diffeomorphic shape propose the deformable template based morphometrics, which were transformation h : ΩTYΩS; xihðÞx . The resulting transformation h further extended by the landmark-based approach (Bookstein, 1989). carries information about morphological differences between a Later work (Christensen et al., 1993; Miller et al., 1997) provided a subject and a template. Various descriptors may then be derived foundation for numerous approaches (Grenander and Miller, 1998; from h for quantifying these morphological characteristics (Davatzikos Miller and Younes, 2001; Miller et al., 2002; Geng et al., 2005; et al., 1996, 2001; Ashburner et al., 1998; Ashburner and Friston, 2000; Lorenzen et al., 2006) based on the diffeomorphic transformations. Thompson et al., 2000; Chung et al., 2001; Chetelat and Desgranges, Deformation based morphometry (DBM), for instance, establishes 2002; Leow et al., 2006). group differences based on the local deformation, and displacement, (Ashburner et al., 1998; Gaser et al., 1999; Joshi, 1998; Collins et al., 1998; Cao and Worsley, 1999), or the divergence of the displacement Abbreviations: DBM, Deformation based morphometry; TBM, Tensor based of various anatomical structures (Thirion and Calmon, 1999). A feature morphometry; VBM, Voxel based morphometry; AEC, Anatomical equivalence class; of particular interest in this case is the Jacobian determinant (JD), CMD, Complete morphological descriptor; OMS, Optimal morphological signature; LJD, which identifies regional volumetric changes (Davatzikos et al., 1996; Log Jacobian determinant; TDM, Tissue density map. Ashburner and Friston, 1999; Chung et al., 2001). Tensor based ⁎ Corresponding author. E-mail addresses: [email protected] (S. Baloch), [email protected] morphometry (TBM) (Thompson et al., 2000; Leow et al., 2006; (C. Davatzikos). Studholme and Cardenas, 2007) utilizes the tensor information for 1053-8119/$ – see front matter © 2008 Published by Elsevier Inc. doi:10.1016/j.neuroimage.2008.10.048 S74 S. Baloch, C. Davatzikos / NeuroImage 45 (2009) S73–S85 Such transformations are typically driven by image similarity measures, in the so-called intensity-driven methods (Christensen et al., 1993; Woods et al., 1993, 1999; Ashburner and Friston,1999), either by directly employing intensity differences or via mutual information. Topological relationships among anatomical structures are main- tained by imposing smoothness constraints either via physical models (Christensen et al., 1993; Beg et al., 2005) or directly on the deformation field. Although these methods have provided remarkable results, their ability to establish anatomically meaningful correspon- Fig. 1. Effect of aggressive registration: (a) Subject; (b) Template; (c) Subject warped through a viscous fluid algorithm. Aggressively registering a bifolded sulcus to a single- dences is not always certain, since image similarity does not folded sulcus creates very thin needle like structures, which are atypical of a human necessarily imply anatomical correspondence. Alternative methods brain. (Thompson and Toga, 1996; Wang et al., 2003) are based on feature identification and matching. Hybrid methods, such as (Shen and capturing local displacement. Variants of TBM include voxel compres- Davatzikos, 2002), incorporate geometrically significant features to sion mapping (Fox et al., 2001), which describes tissue loss rates over identify anatomically more consistent transformations. time. Despite remarkable success of the above methods, their funda- Another class of methods known as voxel-based morphometry mental shortcomings result from inherent complexity of the problem. (VBM) (Ashburner and Friston, 2000; Davatzikos et al., 2001; Baron First, anatomical correspondence may not be uniquely determined from and Chetelat, 2001; Bookstein, 2001; Chetelat and Desgranges, 2002) intensity-based image attributes, which drive these algorithms. Second, factors out global differences via a relatively low dimensional exact anatomical correspondence may not exist at all due to anatomical transformation, before analyzing them for anatomical differences. variability across subjects. For instance, it may not be possible to VBM is, therefore, considered as complementary to DBM or TBM, since perfectly warp a single-folded cingulate sulcus to a double-folded the former utilizes the information not represented by the transfor- cingulate sulcus through a biologically meaningful transformation. As a mation. RAVENS maps (Davatzikos et al., 2001) addressed some of the result, the choice of a template plays an important role in the accuracy limitations of both approaches, in a mass preserving framework, by of the analysis. Anatomies closer to the template are well represented combining the corresponding complimentary features. An important by a diffeomorphism. However, large differences between an individual characteristic of the resulting maps was their ability to retain the total and the template lead to the residual information that the transforma- amount of a tissue under a shape transformation in any arbitrary tion does not capture. In other words, apriorifixing a single template region by accordingly increasing or decreasing the tissue density. for all subjects biases the analysis (Bookstein, 2001; Blezek and Miller, Irrespective of the features employed in various types of analyses, 2006); this has been a fundamental assumption in computational their accuracy largely depends on the ability of finding a spatial anatomy. Although one may argue that this residual may be eliminated transformation that allows perfect warping of anatomical structures. by a very aggressive registration, this may create biologically Fig. 2. (a) Template; (b) Mean of 31 spatially normalized brains; (c) A representative subject; (d) Spatial normalization of (b) using HAMMER; (e) Corresponding residual. While crispness of the mean brain indicates reasonably good anatomical correspondence, there are still significant anatomical differences, which may be large enough to offset disease specific atrophy in a brain. S. Baloch, C. Davatzikos / NeuroImage 45 (2009) S73–S85 S75 implausible correspondences, as illustrated in Fig.1. Aggressive warping transformation alone. Moreover, it is largely invariant to template may also lead to noisy deformation fields that are unsuitable for selection, and the amount of regularization in the transformation, thus subsequent statistical analysis.
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