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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Hum Brain Mapp. Author manuscript; available in PMC 2010 July 1.
Published in final edited form as:
PMCID: PMC2844721
NIHMSID: NIHMS183557

Collaborative Computational Anatomy: An MRI Morphometry Study of the Human Brain via Diffeomorphic Metric Mapping

Abstract

This paper describes a large multi-institutional analysis of the shape and structure of the human hippocampus in the aging brain as measured via MRI. The study was conducted on a population of 101 controls (n=57) with Clinical Dementia Rating (CDR) score 0 and subjects clinically diagnosed with Alzheimer’s Disease (AD, 28 with CDR 0.5 and 10 with CDR 1) or semantic dementia (SD, 4 with CDR 0.5 and 2 with CDR 0) with imaging data collected at Washington University in St. Louis, hippocampal structure annotated at the Massachusetts General Hospital, and anatomical shapes embedded into a metric shape space using large deformation diffeomorphic mapping (LDDMM) at the Johns Hopkins University. A global classifier was constructed for discriminating cohorts of nondemented (CDR 0) and demented (CDR 0.5 or 1) subjects based on linear discriminant analysis of dimensions derived from metric distances between anatomical shapes demonstrating class conditional structure measured via LDDMM metric shape (p < .01). Localized analysis of the control and AD subjects only on the coordinates of the population template demonstrates shape changes in the subiculum and the CA1 subfield in AD (p < .05). Such large scale collaborative analysis of anatomical shapes has the potential to enhance the understanding of neurodevelopmental and neuropsychiatric disorders.

1. Introduction

Emerging collections of imaging, clinical and functional data at multiple sites provide an unprecedented opportunity to increase statistical power in detecting disease in the human brain (e.g. Fennema-Notestine, et al. 2007). However there are several formidable challenges such as portability of software and data developed and collected at individual sites. To address these challenges, the National Institutes of Health created the Biomedical Informatics Research Network (BIRN) (Grethe, et al. 2005; Jovicich, et al.; Keator, et al. 2008). In addition, a major difficulty confronting BIRN and other neuroscientists is that the human brain is a collection of geometrically complex, interconnected, folded structures. Serious study of the function and structure of the human brain requires computational analysis that considers this complex geometry. Computational Anatomy has emerged as a discipline focused on such issues including the representation of the biological variability of the local coordinate systems of human anatomy studied via morphometric tools (Grenander and Miller 1998; Miller, et al. 2002; Thompson and Toga 2002). Computational Anatomy encompasses many forms of neuromorphometric analysis (Thompson, et al. 2004). Focusing on the inference of the statistical representations of shape, studies in the computational anatomy of growth, atrophy, and disease have literally exploded over the past few years.

In particular, applications of Computational Anatomy in normal aging and Alzheimer’s Disease (AD) have been studied in both cortex and deep brain structures by several groups (Ballmaier, et al. 2004; Buckner, et al. 2004; Csernansky, et al. 2000; Gee, et al. 2003; Good, et al. 2001; Miller, et al. 2003; Thompson, et al. 2003). AD is characterized by neuronal degeneration associated with the progressive deposition of neurofibrillary tangles and beta-amyloid plaques. Structural MRI has revealed widespread changes with some of the earliest and most robust in the hippocampal formation (Ball 1977; Fischl, et al. 2002; Haller, et al. 1996; Head, et al. 2005; Kaye, et al. 1997; Killiany, et al. 2002; Laakso, et al. 2000; Lehtovirta, et al. 1995; Petersen, et al. 2000). With the advent of large scale mult-site neuroimaging studies of AD such as the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (Jack, et al. 2008), an accurate and explicit representation of the shape of the hippocampus is essential for accurately characterizing the nature and exact location of shape changes.

In Computational Anatomy, morphometric studies of shape are carried out by metric comparison of anatomical structures via vector field displacements relating the coordinatized structures (Avants and Gee 2004; Beg, et al. 2005; Miller, et al. 2006). The morphometry study reported here focuses on a large scale mult-site shape analysis of the hippocampus in a 101-subject MRI data set consisting of controls and subjects clinically diagnosed with AD and semantic dementia (SD). The processing pipeline described here involves data collection and automated construction and visualization of anatomical manifolds from MRI data collected at Washington University in St. Louis and annotated via FreeSurfer (Fischl, et al. 2002; Fischl, et al. 2004) at the Massachusetts General Hospital; the morphometric comparison was performed at the Johns Hopkins University. Two statistical characterizations of the normal and disease cohorts based on shape change are presented here. First a localized statistical analysis is performed in the common, extrinsic, template coordinates of the atlas. Local areas of change are encoded via the momentum representation of shape change in the population indexed over the atlas. The second approach is to construct classifiers for discriminating the population directly from the metric distances between anatomical structures as generated via LDDMM (Beg, et al. 2005; Miller, et al. 2006; Vaillant, et al. 2007). Clustering is performed using discriminant analysis in the dimensions obtained from multi-dimensional scaling of the matrix of interpoint metric distances between anatomies. This provides a method for building classifiers for discriminating between cohorts.

It is demonstrated that a single data analysis pipeline allows for the direct metric analysis of human hippocampus shape going directly from the MR image to quantitative measures of morphometric shape change revealing the class conditional structure associated with cohorts of nondemented and demented subjects. Cohort clustering is reflected by localized changes in the subiculum and CA1 subfield of the hippocampus.

2. Methods

2.1 Data acquisition

MRI data from 101 individuals were collected at Washington University in St. Louis as part of ongoing studies of structural brain morphometry associated with the Alzheimer Disease Research Center (Buckner, et al. 2004; Buckner, et al. 2005; Fotenos, et al. 2008; Fotenos, et al. 2005; Gold, et al. 2005; Head, et al. 2005; Salat, et al. 2004). The present study analyzed data from individuals classified as nondemented control subjects (n=57; mean age = 76 yrs, range = 60–89; 26 female), clinically-diagnosed AD (n=38; mean age = 74 yrs, range = 62–84; 15 female) and semantic dementia (n=6; mean age = 69 yrs, range =62–80; 2 female). The classification of nondemented controls and AD employed procedures based on the Clinical Dementia Rating (CDR; (Morris 1993). The determination of AD or control status was based solely on clinical methods, without reference to psychometric performance. The diagnosis of AD is based on clinical information (derived primarily from a collateral source) that the subject has experienced gradual onset and progression of memory decline and other cognitive domains; see Fotenos et al. (2005) and Marcus et al. (2007) for more details of selection criteria for these subjects. All nondemented control subjects were CDR 0 which excludes any indication of memory impairment. Individuals meeting criteria for MCI would not be included in our CDR 0 group. For AD, 28 were CDR 0.5 (very mild dementia) and 10 were CDR 1 (mild dementia). For SD, 4 were CDR 0.5 and 2 CDR 1 at their nearest assessment to the MRI. The six individuals with SD have been described previously (Gold, et al. 2005) and were classified based on a specifically designed battery of neuropsychological tests meeting original (Hodges, et al. 1992) and consensus (Neary, et al. 1998) inclusion and exclusion criteria for SD. While a larger number of participants with SD would be desirable, we were only able to recruit six to this specific scanning protocol. Participants were excluded if they had a history of neurologic, psychiatric, or medical illness that contributed to dementia diagnosis. Participants consented to participation in accordance with guidelines of the Washington University Human Studies Committee. The imaging procedures have been described previously (Marcus et al., 2007). Data from the AD cohort can be obtained freely as part of the OASIS open-access data release (www-oasis.brains.org). Briefly, for each participant, two to four high-resolution MP-RAGE scans were motion corrected and averaged per participant (four volumes were averaged for all except five participants; Siemens 1.5 T Vision System, resolution 1×1×1.25 mm, TR = 9.7 ms, TE = 4ms, FA = 10, TI = 20 ms, TD = 200 ms) to create a single image volume with high contrast-to-noise. These acquisition parameters were empirically optimized to increase gray/white and gray/cerebrospinal fluid contrast.

2.2 The Morphometry of Hippocampal Manifolds

Left and right hippocampi were segmented using automated whole brain Bayesian segmentation via Freesurfer (Fischl, et al. 2002; Fischl, et al. 2004) labelling each voxel of the MRI volume based on prior probabilistic information compiled from a set of manually labeled training brain volumes and the local intensity distribution of each class. A single connected representation of each hippocampus is based on a combination of geometric constraints, with components removed or added iteratively to minimize the total costs until the segmentation is modified into a single topologically correct connected component (Segonne, et al. 2003). A tessellation was constructed based on eight triangles representing each face at the interface of hippocampus voxels and differently labeled voxel. To ensure smoothness and accuracy of the surface, the surface was refined based on Gaussian curvature measurement to get rid of high-frequency errors in regions where a string of voxels with similar intensity to the hippocampus is mistakenly labeled because of the partial volume effects of the white matter and the adjacent ventricle.

We model the space of shapes I [set membership] x2110 as objects indexed over manifolds or subsets of R3, either 2-dimensional surfaces or three-dimensional subvolumes. The shape space is a Riemmannian manifold with metric structure resulting from the assumption that the shapes are an orbit under a group of diffeomorphisms (Grenander and Miller 1998) (1-1 and onto transformations with inverses that are smooth). For any pair I, J [set membership] x2110, there exists a flow of diffeomorphisms gt, t [set membership] [0,1] transforming one shape to the other g · I ~ J. The metric distance between any pair I, J is given by the length of the shortest or geodesic curve through the space of shapes connecting them. The diffeomorphisms are constructed as a flow of ordinary differential equations ġt = vt(gt), t [set membership] [0, 1] with g0 = id the identity map, and associated vector fields vt, t [set membership] [0, 1]. The metric between two shapes I, J takes the form

ρ(I,J)2=infv:g=01vt(gt)dt,g0=id01vtV2dt,
(1)

such that g transforms I to J. The norm ||·||V is chosen to ensure that the vector fields are smooth in space (derivatives exist in the squared-energy sense). To calculate the norm we use Large Deformation Diffeomorphic Metric Mapping (LDDMM) for surfaces (Vaillant and Glaunes 2005; Vaillant, et al. 2007) and for volumes (Beg, et al. 2005) by introducing a cost function measuring correspondence between mapped anatomical objects C(g · I, J) and then computing the geodesic connection to minimize the cost. The shapes come as segmented volumes or as triangulated meshes associated with the hippocampal subregion. For mapping the triangulated meshes representing the boundary of the hippocampus from Freesurfer, we use the surface matching (Vaillant, et al. 2007); for mapping the volume segmentations, we use the image matching procedure for segmentations (Kirwan, et al. 2007; Miller, et al. 2005). In each case we solve the inexact matching problem forcing one shape to map onto the other obtain a matching cost C which is small but not identically zero. In LDDMM volume mapping correspondence is based on the intensity data at the voxel level whereas in LDDMM surface mapping, it is based on the normals to the triangulated surfaces (Vaillant and Glaunes 2005). Reliability has been demonstrated for both methods (Beg, et al. 2005; Qiu, et al. 2007a; Qiu, et al. 2007b; Vaillant, et al. 2007); also pose in both cases is removed via rigid landmark prior to the mapping. The LDDMM metric shape space embedding is computationally intensive. The mapping procedures were run on clusters at JHU, the BIRN coordinating center at San Diego and the Teragrid (http://www.teragrid.org).

2.3 Shape Analysis via Random Momentum Fields

To localize geometric changes in the anatomical structures of the group, statistical shape analysis is performed on the diffeomorphic maps. The geodesic flow ġt = vt(gt) between any pair of shapes I, J is encoded by the initial vector field v0 (Miller, et al. 2006); along the geodesic there is a conservation law which implies that the entire flow can be generated from the initial vector field in the tangent space at the identity of the shape. Localized statistical shape analysis is performed on these initial vector fields as first done for landmarked shapes (Vaillant, et al. 2004). It is natural to compute the statistically significant locations of the shape change between the populations characterized by these initial vector fields. Defining the norm ||vt||V2=Avt,vt, then along the geodesic the momentum is defined as Mt = Avt where A is the inverse of the Green’s kernel that makes V a reproducing kernel Hilbert space. Assuming the momentum is smooth enough, it satisfies

Mt=(Dgt1)M0gt1Dgt1.
(2)

The initial momentum M0 completely determines the LDDMM maps from the template onto the target shapes and has the added attractive property that it is normal to the level lines of the template (Miller, et al. 2006). For surface mapping M0 is normal to the template and specified by a scalar field indexed over it according to

M0(x)=μ(x)N(x),xStemp,
(3)

with N(·) the normal field to the surface. Population shape variation is represented by the size of the scalar fields μ(x), x [set membership] Stemp with positive sign pointing outward motion and negative pointing inward motion relative to the template coordinates. For statistics we model μ(x), as a Gaussian field in the form of

μ(x)=kUkφk(x),xStemp,
(4)

where the Uk are Gaussian random variables and [var phi]k(x) are chosen as the k-th eigenfunction of a complete orthonormal base generated from the Laplace-Beltrami operator on Stemp (Qiu, et al. 2006).

2.4 Template Construction

We assume the orbit I [set membership] x2110 of anatomical shapes is generated from an exemplar or template Itemp [set membership] x2110 which must be estimated. All elements I [set membership] x2110 are modelled as generated by the flow of diffeomorphisms from the template for some ġt = vt(gt) with I = g1 · Itemp. Our random model assumes the anatomies I(i) [set membership] x2110, i = 1, 2, …, n, are generated via geodesic flows of the diffeomorphism equation with the conservation equation holding and the flow satisfies the conservation Eqn. 2 so that when M0(i), i = 1, 2, ···, n are considered as hidden variables, our probability law on I(i) [set membership] x2110 is induced via the random law on the initial momenta M0(i). We model this as independent and identically distributed Gaussian random fields with zero mean and fixed covariance matrix. The goal is to estimate Itemp from the set of observations which are taken as conditional Gaussian random fields with mean fields I(i)=g1(i)·Itemp. An iterative Expectation-Maximization (EM) procedure (Allassonniere, et al. 2007; Ma 2006; Ma, et al. 2008) is used to generate the template. An initial manually constructed left and right hippocampus surface was mapped to each of the observations, and transformed via 10 iterations over the entire population.

2.5 Classification

Classification is achieved by applying multi-dimensional scaling (MDS) (Cox and Cox 1994) to the n × n matrix of inter-subject metric distances [rho with circumflex](I, J) obtained for n subjects. MDS aims to detect the finite d-dimensions that explain the observed distances between the shapes. Clearly, given a population in an n-dimensional space, having the metric distances between them allows for the categorization of the n-dimensions upon which they are laid out. If all the anatomies are connected along several axes, then these axes can be discovered via direct examination of the metric distances between every element in the population. The metric distance matrix is expanded as a completely orthonormal basis ρ^(I,J)=i=1dλiϕi(I)ϕi(J) where {[var phi]i, i=1, …, d} is generated via singular value decomposition. This finite dimensional Euclidean embedding is then used in statistical tests via linear discriminant analysis.

Denote the embedded feature vector of a shape as Xi [set membership] RdL + dR and its class label as Ci [set membership] C. Given labelled training populations with labels (X1, C1), ···, (Xn, Cn) independent and identically distributed, Fisher’s linear discriminant involves the selection of a “best” hyperplane for partitioning the feature space into discriminant regions. This involves estimation of class-conditional prior probabilities πj = P[C = j] and class-conditional mean vectors μj and covariance matrices Σj. Test observation X is classified as belonging to that class which maximizes the class-conditional posterior probability. These classifiers can be understood as projection into (C − 1) -dimensional Euclidean space and subsequent piecewise linear partitioning.

Conditioned on training data (Xi, Ci), i = 1, … n, we choose the number of discriminating dimensions dL, dR for the MDS by minimizing an empirical estimate of the conditional probability of misclassification (Devroye, et al. (1996), page 3) given by

P[C^jCj(Xi,Ci),i=1,,n].

Given m new feature vectors Xj [set membership] RdL + dR, j = n + 1, …, n + m, the optimum selection of dimensions minimize the empirical estimator

P^(dL,dR)=1mj=n+1n+mI{C^jCj(X1,C1),,(Xn,Cn)},
(5)

where the classifiers are indexed by the MDS embedding dimensions dL and dR.

Concerning the model selection criteria, we have considered several involving empirical risk minimization. In particular, Fisher’s linear discriminant can be understood as involving estimation of class-conditional prior probabilities πj = P[C = j] and class-conditional mean vectors and covariance matrices. “Linear” implies constant covariance for all classes j; model bias can be reduced by relaxing this constraint, at the expense of increased variance. Test observation X is then classified as belonging to that class which maximizes the class-conditional posterior probability. These classifiers can also be understood as projection into (C − 1) -dimensional Euclidean space and subsequent piecewise linear partitioning. Fisher’s procedure, and associated optimality results, can be derived from Bayes theory and from likelihood ratio theory. The approach is perhaps most suitable for applications in which the class-conditional distributions are unimodal, and can be seen to be optimal for the case of spherically symmetric class-conditional distributions. General linear discriminant analysis can be more difficult to employ but can be shown to out-perform Fisher’s version (Duda and Hart 1973).

3. Results

3.1 Localized shape analysis in template coordinates

Figure 1 provides an intuitive notion of metric distances computed from a flow of sequence of diffeomorphisms satisfying dgt/dt =vt(gt), g0 = id matching I (control) to J (AD). The numbers at each stage of the sequence are the distances obtained from ρ^(I,J)=01vtVdt.

Figure 1
The flow of a sequence of diffeomorphisms generated by LDDMM applied to the shapes I satisfying dgt/dt = vt(gt), g0 = id matching I (control) to J (AD). Shown below each is the estimate of the metric [rho with circumflex] given by 0tv ...

Figure 2 shows the template estimation of 3D hippocampus data. To illustrate the template property, the initial and final metric distances (ρ(I0, Ik), ρ(I0(10),Ik)) are shown for 8 of the 101 targets. Clearly the initial template I0(0) is further in metric distance than the estimated template I0(10).

Figure 2
Initial template and the generated BIRN template after iterations of the template estimation algorithm through the collection of shapes (left column); 8 of the 101 shapes are shown. The first row of the table shows the metric distances [rho with circumflex] ...

Once the template coordinates are constructed, the metadata can be transferred to it using the diffeomorphic mapping procedure. Figure 3 shows a partitioning on the left hippocampus surface in template coordinates Stemp obtained by diffeomorphic transfer of the Washington University hippocampus atlas (Wang, et al. 2006) to the template. The partitioning was based on the intersection of segmentations of the subiculum, subfields (CA1, CA2, CA3, CA4) and gyrus dentate with the hippocampal surface (Duvernoy 1998; Wang, et al. 2006). The three partitioned zones are inferior medial zone (IMZ) proximal to the subiculum, lateral zone (LZ) proximal to the CA1 subfield and superior zone (SZ) proximal to the gyrus dentate and the CA2, CA3 and CA4 subfields.

Figure 3
Bottom and top views of the partitioned left hippocampal surface of the template transferred from the Washington University template (Wang, et al. 2006). Partitioning is based on the intersection of the subvolume segmentations with the surface: IMZ (inferior ...

Figure 4 compares the averaged shapes of the populations of the demented (purple, n=38) and nondemented subjects (n=57) where the SD subjects were excluded. These “average” anatomical structures were generated by finding the average vector field that transfers the template onto the nondemented (vector endpoints) and demented (purple) population groups generated by using the geodesic flow of the average vector field v0 representing each group. Note that the largest velocity vectors occur in the IMZ and LZ partitions that are proximal to the subiculum and the CA1 subfield.

Figure 4
Average shapes of the populations of demented (purple, n=38) and nondemented adults (vector endpoints, n=57) generated by shooting the template onto the two populations. The largest velocity vectors occur in the IMZ and LZ partitions (based on Figure ...

3.2 Hippocampus shape analysis via Random Momentum Fields

To compare the shape difference between the above two populations, two-sample student t-test was performed on each of the first twenty expansion coefficients Uk. It was found that group difference occurred in the 1st, 5th, 20th components at a significance level of 0.05. Figure 5 shows significant differences in the magnitude of the Jacobian measured on a logarithmic scale and averaged over all the maps between the two populations indexed over the template Stemp. The warm and cool colors respectively correspond to expansion and compression. Note that hippocampal atrophy, while not directly measured, is indicated by the location of shape change as inferred from the Jacobian. Note that the greatest expansion occurs occurs in the IMZ and LZ partitions that are proximal to the subiculum and the CA1 subfield.

Figure 5
Determinant of Jacobian (measured on a logarithmic scale) shows the difference between demented (n=38) and nondemented (n=57) groups. Largest expansion occurs in the LZ and IMZ partitions (based on Figure 3) that are proximal to the CA1 subfield and subiculum ...

3.3 Classification based on the metric distances

To demonstrate the performance of classification of two populations, we consider a training set (n = 45) of nondemented subjects (nc = 21) as one class corresponding to CDR 0 and group all AD and SD i.e. demented subjects (nd = 24) as the other class corresponding to CDR 0.5 or CDR 1 and consider the remaining m =56 subjects as test subjects corresponding to 36 nondemented subjects with CDR 0 and 20 demented subjects with CDR 0.5 or CDR 1. Figure 6 shows the empirical probability of error P from Eqn. 5 as a function of MDS embedding dimensions dL and dR for the two class version of the problem; the darker the color the smaller P is. A region of dimensionality-space yielding classification performance estimates significantly superior to chance is apparent; at (dL,dR)=(4,5) we obtain P^(dL,dR)=13/560.23.

Figure 6
Two class (nondemented CDR 0 vs demented CDR 0.5 or 1) discriminant classification error P(dL, dR) as a function of MDS embedding dimensions for left (dL) and right hippocampus (dR); darker color represents smaller P ...

Figure 6 also shows the smoothness of P(dL, dR) and the regularization inherent in the relatively low-dimensional model selected suggests that the level of performance obtained here represents a statistically significant improvement (over chance) in classification capability. A permutation test (Good 2000) puts the estimated p-value for this result at [p with hat] = 0.0095 ± 0.0010. Thus metric classifier based on LDDMM captures shape information in the MR data that is correlated with clinical diagnoses. While volume alone does provide some classification signal (P =0.30), performance using metric distances is superior.

4. Discussion

This study demonstrates the ability of both CA tools and data from multiple sites to generate results consistent with clinical findings in normal and abnormal aging in AD. In particular, CA tools based on large deformation diffeomorphic mappings (Csernansky, et al. 2000; Csernansky, et al. 2004; Miller 2004; Wang, et al. 2006; Wang, et al. 2003) have been useful in discriminating nondemented subjects and those with very mild AD in cross-sectional and longitudinal studies. Deformations of the hippocampal surface proximal to the CA1 subfield and the subiculum were also observed (Wang, et al. 2006; Wang, et al. 2003). More recently, a longitudinal study found that reduced volume and abnormal shape of the hippocampus could predict future cognitive decline in healthy elderly individuals (Csernansky, et al. 2005). The pattern of hippocampal shape variation in these subjects resembled those observed in subjects with very mild AD (Csernansky, et al. 2000; Wang, et al. 2006).

Although SD is primarily a semantic memory disease and has a different regional pattern of neuronal loss than in AD, this study demonstrated how embedding anatomical configurations in a metric shape space via metric distances [rho with circumflex] between shapes permits classification via clustering. The approach constructs the metric classifier via multi-dimensional scaling and linear discrimination analysis. Further, class conditional discrimination between demented (CDR 0.5 or CDR 1) and nondemented (CDR 0) can be performed based on the metric structure of LDDMM.

As with landmark and dense image mappings (Vaillant, et al. 2004; Wang, et al. 2007), the GRF representation of momentum was shown to provide a compact and efficient representation of anatomical variation. Prominent shape changes were observed in both the IMZ and LZ partitions proximal to the subiculum and CA1 subfield respectively (Figures 4 and and5)5) which is consistent with several histopathological findings (e.g. Rossler, et al. 2002; West, et al. 1994; West, et al. 2004). However these shape changes do not reflect actual atrophy. It is possible that atrophy in other subregions of the hippocampus could have induced the observed shape changes. Some expansion in the lateral aspect of the subiculum (Figure 5) has not been observed in histopathological studies. These observations need to be resolved by either a larger population or longitudinal study which is not the purpose of this study. In addition, it should be emphasized that the observed shape changes take place on the surface of the hippocampus via the momentum along the normal to the boundary defined by the image contrast and do not reflect the changes within the hippocampus or neighboring structures such as the gyrus dentate. However, as has been demonstrated in our previous work as well as others (e.g. Shi, et al. 2007) the surface based approach encodes the localized shape changes in the hippocampus in Alzheimer’s Disease or neuropsychiatric diseases.

The methodology demonstrated here goes directly from dense segmented images to metric distances. Originally LDDMM (Beg, et al. 2005) worked directly on the dense MR imagery, with no segmentations involved requiring the contrast between the images to be modeled so that the image matching is well defined. This contrast modelling is of course similar to the segmentation approach. Thus the efficacy of the segmentation would imply efficacy in the direct matching of MR intensities. Other high-dimensional diffeomorphic metric shape space embeddings now exist for anatomical shapes measured in other ways, including labelled landmarks (Joshi and Miller 2000); and unlabelled landmarks (Glaunes, et al. 2004), and dense image volumes measured as diffusion tensor images (Cao, et al. 2006; Cao, et al. 2005). Collaborative analysis of shapes via diffeomorphic metric mappings has the potential to enhance the understanding of disease in large scale studies such as ADNI.

Acknowledgments

Support for the JHU group was provided in part by P41-RR015241 (JHU) and U24 RR021382 (MGH) to BIRN, National Center for Research Resources (NIH), P01-AG0568 and P01-AG03991 (WU), and the NSF via DMS-0456253. As well support for the MGH group was provided in part by the National Center for Research Resources (P41-RR14075, R01 RR16594-01A1 and the NCRR BIRN Morphometric Project BIRN002, U24 RR021382), the National Institute for Biomedical Imaging and Bioengineering (R01 EB001550), the National Institute for Neurological Disorders and Stroke (R01 NS052585-01) as well as the Mental Illness and Neuroscience Discovery (MIND) Institute, and is part of the National Alliance for Medical Image Computing (NAMIC), funded by the National Institutes of Health through the NIH Roadmap for Medical Research, Grant U54 EB005149. MIM is grateful to Faisal Beg and Marc Vaillant for the development of the LDDMM algorithms.

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