Building an Average Population HARDI Atlas

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1 Building an Average Population HARDI Atlas Sylvain Bouix 1, Yogesh Rathi 1, and Mert Sabuncu 2 1 Psychiatry Neuroimaging Laboratory, Brigham and Women s Hospital, Harvard Medical School, Boston, MA, USA. 2 A. A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, USA. Abstract. We present a framework to build a High Angular Resolution Diffusion Image (HARDI) Atlas based on a population of HARDIs from different subjects. The method relies on a non-linear group-wise registration algorithm as well as a set of tools to re-orient the gradient directions of individual subjects and re-interpolate their diffusion weighted signal in a canonical gradient direction set in atlas space. We test the technique on a population of forty six HARDIs acquired on a 3 Tesla scanner. Our experiments show that our atlas can capture information beyond that of a single tensor, even with the inherent smoothing effect of averaging many different brains together. 1 Introduction Diffusion Magnetic Resonance Imaging (dmri) needs no introduction. It is one of the most active research areas in medical image analysis in particular in neuroimaging, where its ability to provide information about the location, orientation and integrity of white matter has become indispensable [1]. One popular technique for neuroimaging studies is the use of an atlas to describe a population, and several methods have been proposed to build a dmri atlas from a population of subjects [2 4]. To our knowledge, all of these techniques use the Diffusion Tensor (DT) model to represent the diffusion information captured in the diffusion signal [5]. While DT imaging is still the de facto standard for neuroimaging analysis, more accurate diffusion models, capable of capturing more than a single primary orientation of diffusion, have been proposed and are becoming more prevalent [6 8]. In this article, we introduce a framework to build an average population atlas of the original diffusion signal without the use of a parametric model such as the DT (see for example [2, 9]). To our knowledge, only a couple attempts have been made to work with higher order signal [10, 11]. In [10] the authors rely on the 4th order tensor model to represent diffusion, whereas [11] uses a single tensor registration pipeline to transform the original dmris that are later used to fit a two tensor model. In contrast, we build a template of the diffusion signal directly, without the need for a model. Our work is inspired by the DT atlas building framework of [2] where an unbiased atlas building method designed for scalar data is used to compute diffeomorphic maps from each subject s space to

2 2 Sylvain Bouix, Yogesh Rathi, and Mert Sabuncu template space and diffusion tensor images are resampled and reoriented using the diffeomorphic maps. The main challenge when working with HARDI data is that one does not have a model such as DT that can be reoriented easily through the diffeomorphic map. Our approach, described in detail in section 2, is to, for each voxel in each subject: (i) transform the original signal through the diffeomorphic maps; (ii) reorient the gradient directions associated with this voxel; (iii) estimate a spherical harmonic representation of the signal; and (iv) resample the diffusion signal in a canonical set of gradient directions. Our experiments show (section 3) that building a HARDI atlas from a large population of subjects can capture enough information to represent complex neuronal fiber architecture, which cannot be handled by a single diffusion tensor model. 2 Methods 2.1 Group-wise Non-linear Registration In order to establish correspondence between all subjects, we use the Asymmetric Image-Template Registration method presented in [12]. This algorithm performs a non-linear alignment between a scalar individual image and a scalar template image that represents a population average. The heart of the method lies in using a bi-directional objective function that takes the asymmetric relationship between the individual image I and the template T into account by introducing a correction factor: the Jacobian that quantifies the deformation of the spatial grid. In mathematical terms, given an image I and a template T, the following objective function is minimized: 1 ( ˆΦ = arg min [I(x) T (Φ(x))] 2 + [I(Φ 1 (x)) T (x)] 2 det[ Φ 1 (x)] ) dx Φ 2 R 3 + Reg(Φ), (1) where Φ : R 3 R 3 is a smooth and invertible (i.e., diffeomorphic) transformation from the atlas coordinates to the coordinates of the individual image, denotes the Jacobian operator with respect to spatial coordinates, det denotes the determinant, and Reg is a regularization term that penalizes non-smooth transformations. The transformation Φ is parametrized using a stationary velocity field via an ordinary differential equation [12]. An efficient solution to Eq. 1 is then computed by adapting the so-called log-domain diffeomorphic demons registration framework of [13]. For a given population of images {I i } and transformations {Φ i } from the atlas coordinates to the corresponding image coordinates, an estimate of a template image that reflects the population average can be computed as [14]: i T (x) = I i(φ 1 i (x)) det( Φ 1 i (x)). (2) i det( Φ 1 i (x))

3 Building an Average Population HARDI Atlas 3 Our group-wise registration procedure co-registers a population of images by iterating between computing the individual registrations by optimizing Eq. (1) and updating the template image using Eq. (2) and the latest estimates of the transformations. Crucially, at the end of each iteration we perform a normalization on the current transformation estimates so that the average transformation across the subjects is identity, i.e., i Φ i(x) = x, x. These steps are repeated until convergence. The final result consists of a template image T and a set of diffeomorphic transformations {Φ i } from the atlas coordinates to the individual image coordinates. In our experiments, similarly to [15], we used the scalar Fractional Anisotropy (FA) images derived from the individual dmris to perform the group-wise registration. 2.2 Transforming Individual HARDIs into Template space If we were working with scalar images, our atlas would simply be the template computed in the previous section. However, when dealing with HARDI signal, one must perform additional steps. Firstly, the rotational component of the non-linear transform must be extracted at each location in the input image, so that the gradient directions associated with this location can be rotated accordingly. We extract this rotation R by computing the finite strain of the Jacobian of the transformation as described in [16]: R(x) = [ Φ(x)( Φ(x)) T ] 1 2 Φ(x). (3) Note that there is a different rotation for each voxel in the image. Secondly, we transform and resample each gradient weighted component of the subject s HARDI into atlas space as we would resample a scalar image, including scaling the MR signal by det( Φ 1 (x)). The final result for each subject consists of: (i) the original set of gradient directions, (ii) R a 3D volume containing a rotation at each voxel, and (iii) the 4D HARDI signal transformed and resampled into atlas space. 2.3 Averaging the Resampled HARDIs Since the HARDI signal at a voxel is a function defined on the sphere, an appropriate representation has to be used to compute the average signal. Spherical harmonics [17] provide one such basis in which averaging becomes a linear operation. The method works by first computing the coefficients of the spherical harmonic (SH) basis of order L that best fits the measured signal. Given any bandlimited signal S defined on the sphere, one can write it as an expansion in terms of the SH basis as: S(θ, φ) = L l l=0 m= l c l,my l,m, for any direction (θ, φ), where Y l,m are the basis functions given by: Y l,m (θ, φ) = (2l + 1)(l m)! P l,m (cos θ)e imφ, 4π(l + m)!

4 4 Sylvain Bouix, Yogesh Rathi, and Mert Sabuncu where P l,m is the associated Legendre polynomial. The above equations can be written as a linear system of equations and c l,m can be computed using the Moore-Penrose pseudo inverse. If needed, a regularization term can be added as shown in [17]. Let C i be the vector of coefficients c l,m that represents signal at a voxel for the i th subject. Then, the average signal over M subjects can be computed by linear averaging: C = 1 M M i=1 C i. Thus, the average signal at each voxel of the atlas is computed using the SH coefficients. Next, the average HARDI s of the atlas are computed by evaluating the signal (in the SH basis) in a specified set of directions. Our gradient direction scheme results from sampling the sphere on the vertices of a second order tessellation of the icosahedron, leading to 81 unique directions on the unit hemisphere. 3 Experiments and Results 3.1 Subjects Forty-six HARDI scans from our Normal Control (NC) database were selected as input to our atlas building framework. The dataset contains 8 females and 38 males, the average age is ± years. Diffusion-weighted images were acquired on a 3T scanner (General Electric Company, Milwaukee, WI, USA) using an echo planar imaging (EPI) sequence, with a double echo option to reduce eddy-current related distortions. To reduce the impact of EPI spatial distortion, an 8 Channel coil and ASSET with a SENSE-factor of 2 were used. The acquisition consisted of 51 directions with b = 900s/mm 2, and 8 baseline images with b = 0s/mm 2. The scan parameters were: T R = 17000ms, T E = 78ms, F OV = 24cm, encoding steps, 1.7mm slice thickness. A total of 85 axial slices covering the whole brain were acquired. 3.2 Building the atlas We estimated Fractional Anisotropy (FA) maps for each HARDI scan and rigidly registered all FA maps to a randomly selected subject using FSL 3 linear registration with six degrees of freedom. For each subject, the resulting transformation was applied to the HARDI scan, and the gradient directions were reoriented to account for the rotation introduced by the rigid alignment. We acknowledge here that we could have used an un-biased groupwise linear registration, the bias introduce in this step using a simpler rigid registration tool is clarly minimal. We then applied our atlas building pipeline as described in Section 2, using FA maps to compute the diffeomorphic maps and estimate the group template space. Figure 1 shows the resulting atlas. As expected, the main white matter tracts are apparent, whereas the regions closer to neocortical areas are blurrier due the variability of the anatomy between individuals. 3

5 Building an Average Population HARDI Atlas 5 Fig. 1. An axial (top) and coronal (bottom) slice of the population HARDI atlas. Left: b=0 image, center: one of the resampled gradient directions, right: FA map of the tensor map estimated from the HARDI atlas. 3.3 Evaluation with tractography In order to evaluate the benefits of building a HARDI instead of a DT template, we ran single tensor as well as two-tensor tractography on the atlas. The algorithm we used simultaneously estimates the diffusion model and performs tractography. Starting from a seed point, each fiber is traced to its termination using an un-scented Kalman filter to simultaneously fit the local model of diffusion and propagate in the most consistent direction [18]. We seeded the tractography in the midsagittal slice of the corpus callosum. The experiment was run twice, the first time using a single tensor as the diffusion model and the second time using two-tensor as the diffusion model. Results are shown in figure 2. We are pleased to report that although averaging over many subjects significantly smoothes the diffusion signal, the atlas captures enough information for the two-tensor tractography to accurately handle fiber crossings. We further illustrate the accuracy of the estimation in figure 3. Observe that the algorithm correctly aligns both the first and second tensor to the same direction in the central section of the corpus callosum (fig. 3 center). Additionally, and perhaps most importantly, in the area where the cortico-spinal tract and the corpus callosum fibers intersect, one can see that the first and second tensor are

6 6 Sylvain Bouix, Yogesh Rathi, and Mert Sabuncu Fig. 2. Tractography of the Corpus Callosum of the atlas using single tensor tractography (left) and two tensor tractography (right). Note the atlas captures enough information for the two tensor tractography to resolve fiber crossings. in perpendicular directions, thus correctly reflecting the anatomy in this region (fig. 3 right). 4 Discussion In this paper, we present a framework to build a HARDI atlas from a population of individual HARDIs. Our method relies on an unbiased, bi-directional, asymmetric group-wise registration algorithm which computes a template space and a set of diffeomorphic maps to bring individual subjects into this template space. The resulting maps are used to transform and resample the original HARDIs into template space and extract rotation maps which capture the rigid transformation that needs to be applied the the gradient direction set at each voxel in each subject. In order to average the diffusion signal in template space, we first estimate a spherical harmonic representation of the HARDI signal for each transformed subject and resample the signal in a canonical set of gradient directions. The resulting image is a 4D HARDI volume, which can then be used for further processing using any model to describe the Orientation Distribution Function. There are some limitations to such an atlas. Firstly, using a scalar image to derive the diffeomorphic maps may be suboptimal, although recent work suggest that using a high Contrast to Noise Ratio image (such as a high resolution anatomical T1 weighted) can lead to very accurate dmri registrations [19]. The use of FA in particular is controversial as it is often used as a primary measure in further statistical analyses. In such cases, we would likely use a different measure such as Goodlett s C measure [2]. Secondly, no matter how accurate

7 Building an Average Population HARDI Atlas 7 the registration is, averaging over a large number of subjects inevitably leads to smoothing the diffusion signal and losing some of the sharpness of the ODF in particular in regions close to the neocortex. Nevertheless, we have shown that sufficient information is captured in the atlas to resolve complex fiber architecture. Thirdly, our registration procedure utilized a method originally derived for scalar images. Obviously, here we are dealing with HARDIs and a rigorous derivation of an appropriate cost function for this type of signal for group-wise registration and template estimation is left to future work. Fig. 3. Close up views of two tensor tractography on a coronal slice of the atlas. The green tubes represent the estimation of the second tensor superimposed on the (red) streamline obtained by tracking along the first tensor. In the central section of the CC, the first and second tensors are aligned (center), whereas at the location where cortico-spinal tract and CC intersect, the second tensor is perpendicular to the first (right). To summarize, we have presented in this article a new method to build an unbiased group-wise HARDI population atlas. We believe such a template to be of great relevance in the dmri world, especially in neuroimaging studies where statistics of diffusion properties over clinical populations are of the utmost interest. Deriving meaningful statistics from our HARDI atlas will be the focus of our future work. Acknowledgements. Support for this research was provided in part by the National Institutes of Health Roadmap for Medical Research (U54EB005149), the National Center for Research Resources (P41RR14075), the National Institute for Biomedical Imaging and Bioengineering (R01EB006758), the National Institute on Aging (AG02238), and the National Institute for Neurological Disorders and Stroke (R01NS052585), the National Institute of Mental Health (R01MH50740, K05MH070047, P50MH080272, R01MH082918, R03TW008134), Department of Veteran Affairs Merit Awards, VA Schizophrenia Center grant, and Center for Integration of Medicine and Innovative Technology Soldier in Medicine Award. References 1. Bihan, D.L.: Looking into the functional architecture of the brain with diffusion mri. Nature Reviews Neuroscience 4 (2003)

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