Spatial Normalization of Diffusion Tensor MRI Using Multiple Channels

NeuroImage 20 (2003) 1995Ð2009 www.elsevier.com/locate/ynimg

Spatial normalization of diffusion tensor MRI using multiple channels

Hae-Jeong Park,a,b Marek Kubicki,a,b Martha E. Shenton,a,b Alexandre Guimond,c Robert W. McCarley,a Stephan E. Maier,d Ron Kikinis,b Ferenc A. Jolesz,d and Carl-Fredrik Westinb,*

a Clinical Neuroscience Division, Laboratory of Neuroscience, Boston VA Health Care System-Brockton Division, Department of Psychiatry, Harvard Medical School, Boston, MA 02115, USA. b Surgical Planning Laboratory, MRI Division, Department of Radiology, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA 02115, USA. c Center for Neurological imaging, MRI Division, Department of Radiology, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA 02115, USA. d Department of Radiology, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA 02115, USA. Received 10 March 2003; revised 30 July 2003; accepted 5 August 2003

Abstract Diffusion Tensor MRI (DT-MRI) can provide important in vivo information for the detection of brain abnormalities in diseases characterized by compromised neural connectivity. To quantify diffusion tensor abnormalities based on voxel-based statistical analysis, spatial normalization is required to minimize the anatomical variability between studied brain structures. In this article, we used a multiple input channel registration algorithm based on a demons algorithm and evaluated the spatial normalization of diffusion tensor image in terms of the input information used for registration. Registration was performed on 16 DT-MRI data sets using different combinations of the channels, including a channel of T2-weighted intensity, a channel of the fractional anisotropy, a channel of the difference of the first and second eigenvalues, two channels of the fractional anisotropy and the trace of tensor, three channels of the eigenvalues of the tensor, and the six channel tensor components. To evaluate the registration of tensor data, we defined two similarity measures, i.e., the endpoint divergence and the mean square error, which we applied to the fiber bundles of target images and registered images at the same seed points in white matter segmentation. We also evaluated the tensor registration by examining the voxel-by-voxel alignment of tensors in a sample of 15 normalized DT-MRIs. In all evaluations, nonlinear warping using six independent tensor components as input channels showed the best performance in effectively normalizing the tract morphology and tensor orientation. We also present a nonlinear method for creating a group diffusion tensor atlas using the average tensor field and the average deformation field, which we believe is a better approach than a strict linear one for representing both tensor distribution and morphological distribution of the population. © 2003 Elsevier Inc. All rights reserved.

Keywords: Diffusion tensor; Spatial normalization; Tractography

Introduction ticularly useful for evaluating white matter abnormalities in the brain, because the measurement of water diffusion in Diffusion tensor magnetic resonance imaging (DT-MRI), initially proposed by Basser and colleagues (Basser et al., brain tissue is based on the movement of water molecules, 1994), is a rapidly evolving technique that can be used to which is restricted in white matter. Speciﬁcally, in pure investigate the diffusion of water in brain tissue. It is par- liquids, such as water, the motion of individual water molecules is random, with equal probability in all directions. However, within white matter, which is comprised of my- * Corresponding author. 75 Francis Street, Brigham and Women’s elinated ﬁbers, the movement of water molecules is sub- Hospital, Department of Radiology, Thorn 323, Boston, MA 02115. Fax: ϩ1-617-264-6887. stantially restricted and that restriction, when compared E-mail address: [email protected] (C.-F. Westin). with pure water, is present along all directions, but partic-

1053-8119/$ Ð see front matter © 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.neuroimage.2003.08.008 1996 H.-J. Park et al. / NeuroImage 20 (2003) 1995–2009 ularly perpendicular to the fiber axis. Consequently, in As mentioned above, an appropriate atlas for diffusion white matter fiber tracts, the major axis of the diffusion tensors is also required in order to represent a population of tensor ellipsoid is much larger than the other two axes, and diffusion tensors. Jones and colleagues (Jones et al., 2002) it coincides with the direction of the fibers. Thus, linking the created a diffusion tensor average brain by normalizing major diffusion axes in white matter makes possible the tensors with a linear transformation derived from the indi- visualization and appreciation of white matter tracts within vidual fractional anisotropy images. Their work demon- the brain. Such fiber-tracing schemes, also called DT-MRI strates the feasibility and the importance of an atlas template tractography (Basser, 1998; Conturo, et al. 1999; Jones et derived from the studied population. al., 1999; Mori et al., 1999; Westin et al., 1999, 2002; As a registration method for the diffusion tensor, Alex- Basser et al., 2000; Poupon et al., 2000; Gossl et al., 2002; ander and coworkers (Alexander and Gee, 2000) have pro- Mori et al., 2002), where fiber tracts follow the major posed a multiresolution elastic matching algorithm using eigenvector of the diffusion tensor, provide important infor- similarity measures of the tensor in order to manage tensor mation about the connectivity between brain regions. data instead of scalar data. Our group has proposed diffu- The quantification of diffusion tensor data for investigat- sion tensor registration methods using tensor similarity ing white matter abnormalities is generally approached us- (Ruiz-Alzola et al., 2000, 2002) and multiple channel infor- ing one of two methods: (1) region-of-interest-(ROI) based mation (Guimond et al., 2002). Ruiz-Alzola and colleagues methods or (2) voxel-based methods. Most researchers have (Ruiz-Alzola et al., 2000, 2002) extended the general con- used ROI methods that begin by identifying anatomical cept of intensity-based similarity in registration to the tensor brain regions and by comparing the anisotropy or the extent case and also proposed an interpolation method by means of of the region having thresholded high anisotropy (Peled et the Kriging estimator. Their work is based on template al., 1998; Kubicki et al., 2002; McGraw et al., 2002). For matching by locally optimized similarity function. Of fur- studies where the region of interest (ROI) of potential ab- ther note, Guimond and colleagues (2002) have noted the normality is difficult to define precisely, voxel-based meth- importance of channel information used for registration and ods tend to be used (Eriksson et al., 2001; Rugg-Gunn et al., they introduced multiple channel registration for tensor im- 2001). A voxel-based strategy is more exploratory and is ages by, for example, using all components of the tensor suitable for identifying unanticipated or unpredicted/unhy- simultaneously in the registration process with successively pothesized areas of abnormal white matter morphology. updating tensor orientation. Voxel-based quantification of diffusion tensor, however, The performance of spatial registration is worth evaluat- requires more sophisticated spatial normalization in order to ing in terms of the input information used for registration, remove anatomical confounds, such as the extent of white i.e., either a univariate scalar data (for instance, FA or T2) matter, misalignment of fiber tracts, and slight variations in or multivariate tensor data, because most neuroimaging fiber bundle thickness. groups are familiar with a univariate registration due to its Spatial normalization involves the registration of images easy access to the registration algorithms, such as SPM and the generation of a stereotaxic atlas that represents the (Ashburner and Friston, 1999) or AIR (Woods et al., 1998). statistical distribution of the group at each voxel (Friston et When evaluating the spatial registration of diffusion tensor al., 1995; Mazziotta et al., 1995; Thompson and Toga, data, it is important to define the concept of anatomical 1997; Grenander and Miller, 1998; Guimond et al., 2000). correspondence between data sets that are to be spatially The registration of diffusion tensor images has generally registered. Because the anatomical correspondence is deter- been performed in a similar way to the registration of mined by the intensity distribution of channels (i.e., types of T1-weighted or SPGR MR images. That is, T2-weighted information) used for registration, appropriate channels MR or fractional anisotropy (FA) images, which are scalar should be used to achieve the definition of anatomical cor- data, have been used to estimate deformation fields or trans- respondence in the registration process. In this article, we formation functions to minimize the intensity difference define anatomical correspondence between white matters in between the template and the normalized image. With the terms of both spatial location of the fibers and their local estimated deformation field or transformation function, the tensor orientation, and we evaluate the registration perfor- morphology of the diffusion tensor is deformed to fita mance to determine an optimal set of information types to stereotaxic space. This transformation function can be either meet these criteria. an affine transformation or a nonlinear elastic warping. For To evaluate the registration performance, we have pro- the accurate registration of diffusion tensor images, how- posed an evaluation method based on tractography, where ever, an additional step is required to adjust the orientation we compare the similarity/dissimilarity of pairs of fiber of the tensor according to the transformation. As was pro- bundle maps, i.e., the results of the fiber tractography, de- posed by Alexander and colleagues (Alexander and Gee, rived from both the registered diffusion images and the 2000; Alexander et al., 2001), the rotational component of target (template) diffusion images. The assumption here is linear transformation, or the local rotational component of that small local registration errors will accumulate and be- deformation field, can be applied in order to reorient tensors come visible in the tractography results. Such a method will in the whole tensor field. also be simultaneously sensitive for both spatial errors in the H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009 1997 registration as well as geometrical tensor alignment errors. ment v(x) for each voxel, x, of a target image, T, to match The better the images are registered, the more similar the the corresponding location in a source image, S. The opti- fiber bundle maps will be. We have also evaluated the mal solution can be found by using the iterative scheme as registration performance with voxel-based measures by ex- follows: amining the dispersion of tensors at each voxel after registration. The evaluation of registration performance in terms v ϩ ͑ x͒ ϭ G␴*ͩv ͑ x͒ of the type of similarity function and the type of matching n 1 n process specific to registration algorithms are not covered in the article. Additionally we have proposed a method for C ˜ Ϫ ͑ ͒ creating a group diffusion tensor atlas utilizing the average 1 Scn Tc x ,ϩ ͸ ٌS˜ ͑ x͒ͪ cn brain morphology as well as the average tensors derived C ʈٌ ˜ ʈ2 ϩ ʈ ˜ Ϫ ͑ ͒ʈ2 cϭ1 Scn Scn Tc x from the registration using the whole tensor information. (1) where S˜ ϭ S ⅙h (x), ⅙ denotes the composition with the Materials and methods cn c n local tensor reorientation, and the transformation h(x)isthe sum of current position and its deformation, i.e., h(x) ϭ x ϩ Subjects and data acquisition v(x). C is the number of channels in the images. G␴ is a Gaussian filter with a variance of ␴2; * denotes the convo- Sixteen normal healthy subjects, with a mean age of 42 lution. A more detailed formulation of this model, and its (range ϭ 30Ð51), were recruited from the general commu- association with other registration methods, such as the nity at the VA Boston Healthcare System, Brockton, MA. minimization of the sum of squared difference criterion, This research was approved by the local Institutional Re- optical flow and the demons algorithm, can be found in view Board of the VA Boston Healthcare System, and all Guimond and Roche (1999). subjects signed written informed consent prior to participa- The Gaussian filter used to smooth the displacement has tion. Subjects were scanned using Line Scan Diffusion a progressively decreasing standard deviation ␴. This pro- Imaging (LSDI) (Gudbjartsson et al., 1996; Maier et al., gressive method constrains deformation strongly in the be- 1998; Mamata et al., 2002), which is comprised of a series ginning of the registration procedure to correct for gross of parallel columns lying in the image plane. The sequential displacements, whereas later it applies weaker constraints collection of these line data in independent acquisitions near the end of the procedure for finer displacements. The makes the sequence largely insensitive to bulk motion arti- multiresolution process of the registration was performed at fact because no phase encoding is used and shot-to-shot four resolution levels to accelerate convergence. A trilinear phase variations are fully removed by calculating the mag- interpolation was used for the resampling process. When the nitude of the signal. full tensor field was used during the registration, the tensor A quadrature head coil was used on a 1.5-T GE Echo- orientations were adjusted iteratively according to deforma- speed system (General Electric Medical Systems, Milwau- tion field using the Preservation of Principal Direction al- kee, WI), which permits maximum gradient amplitudes of gorithm, as described by Alexander et al. (2001). 40 mT/m. For each slice, six images with high diffusion- The advantage of the demon’s algorithm compared with weighting (1000 s/mm2) along six noncollinear and non-co- several low dimensional registration algorithms is the high planar directions were collected. Two base line images with dimensional warping which renders local registration. Ad- low diffusion weighting (5 s/mm2) were also collected and ditional advantages of this modified demons algorithm in- averaged. Scan parameters were as follows: rectangular clude multiple channel registration and its iterative adjust- FOV (field of view) 220 ϫ 165 mm; 128 ϫ 128 scan matrix ment of tensor orientation during tensor registration. (256 ϫ 256 image matrix); slice thickness 4 mm; interslice Because the demons algorithm matches voxel values distance 1 mm; receiver bandwidth Ϯ 4kHz; TE (echo time) between the source and target images, the resulting trans- 64 ms; effective TR (repetition time) 2592 ms; scan time 60 formation is determined by the type of information con- s/slice section. A total of 31Ð35 coronal slices covering the tained in these voxels. Below we compare the registration entire brain were acquired, depending on brain size. performance by combining into these voxels information obtained from various channels. Multiple-channel spatial registration

Spatial registration was used to determine anatomical Evaluation of fiber bundle registration correspondences between source and target (template) images or, in statistical terms, to remove anatomical con- Evaluation data founds. We used a multiple-channel demons algorithm for The registration schemes were evaluated using both ar- estimating deformation fields in the spatial normalization tificially deformed DT-MRI data and real DT-MRI data (Guimond et al., 2002). This algorithm finds the displace- from different subjects. 1998 H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009

Registration of artificially deformed DT-MRI data to its respective deformation fields, fx, to the registered tensor original DT-MRI data. The artificially deformed diffusion image, Vr. The procedure can be summarized as follows: tensor data were created by applying nonlinear transforma- 3 3 tions to 10 DT-MRI data sets. To avoid bias to any of our g:T2Vs T2Vt fx:Vr Vs Vs O¡Vr O¡Vs . (3) registration schemes, and for the easy control of the defor- X mation for generating the test data, we chose to apply a low-order (in this study, using 7 ϫ 8 ϫ 7 basis functions) We compared the tensor field, Vs, with each of the

nonlinear registration method from SPM99 (Ashburner and inversely transformed tensor images, VsT2, VsFA, VsDE, Friston, 1999). For each subject, the nonlinear transforma- VsAT, VsEV, and VsTC, using dissimilarity measures of ﬁber tion g was generated using SPM99 by registering the sub- tracts and a voxel-based overlap measure, which are de- ject’s T2-weighted image of diffusion tensor image (Vs) to scribed below under “Performance index of ﬁber bundle the T2-weighted image of a target diffusion tensor image registration” and “Voxel-based evaluation of registration (Vt) arbitrary chosen from the extra DT-MRI data. A reg- performance.” istered diffusion tensor image, Vr, was created by applying the transformation g to the original tensor image, Vs, fol- Registration of DT-MRI data from different subjects. lowed by the regional adjustment of tensor orientation, For the evaluation of the registration of DT-MRI data from using the Preservation of Principal Direction method (Al- different subjects, we normalized 15 diffusion tensor im- exander et al., 2001). In this case, the registered image, Vr, ages, Vs, to one target diffusion image, Vt, using multiple keeps the same topology but changed morphology of the channel information, using the above procedure. The regis- original diffusion tensor, Vs. tration performance was evaluated by comparing the target

The performance of fiber tract registration according to image, Vt, and the registered images, VrX (i.e., VrT2, VrFA, the channel information was evaluated by inversely trans- VrDE, VrAT, VrEV, and VrTC), derived from the various types forming the registered image, Vr, to the source image, Vs. of channel information. Here, the tensor fields, VrX,may For the inverse registration, we applied the demons algo- have differences in topology compared with target image, rithm with the following six different combinations of in- Vt. This procedure can be summarized as follows: formation: (1) T2-weighted image (T2); (2) fractional an- 3 gX:Vs Vt isotropy (FA), i.e., O¡ Vs VrX. (4) ͱ͑␭ Ϫ ␭ ͒2 ϩ ͑␭ Ϫ ␭ ͒2 ϩ ͑␭ Ϫ ␭ ͒2 1 2 2 3 1 3 FA ϭ , (2) ͱ ͱ␭2 ϩ ␭2 ϩ ␭2 2 1 2 3 The above evaluation processes are shown in Fig. 2. We compared the original images and the reverted images, ␭ ␭ ␭ where 1, 2, 3 are eigenvalues of the tensor from bigger which were registered to a target image and then inversely to lower values, respectively: (3) difference of the first and registered to the original images (bottom left side in the ␭ Ϫ ␭ second eigenvalues (DE), i.e., 1 2; (4) fractional an- figure). We also compared the target image and the regis- ␭ ϩ ␭ ϩ ␭ isotropy and trace of tensor (AT), i.e., FA and 1 2 3; tered images of original images that were registered to the ␭ ␭ ␭ (5) three channel eigenvalues (EV), i.e., 1, 2, 3; and (6) target image (bottom right side in the figure) using different six channel tensor components (TC), i.e., Dxx, Dxy, Dxz, intensity information: T2, FA, DE, AT, EV, and TC. Dyy, Dyz, and Dzz components of tensors. FA, DE, AT, and EV are scalar indices without directional information of Generation of fiber bundle maps the tensor, whereas the TC comprises both direction and In order to evaluate the registration performance in terms magnitude of the diffusivity. Figure 1 shows the types of of minimization of tensor field distortion, we used a tradi- intensity information used for registration. T2, FA, and tional eigenvector tracking method. In the tract calculation, mean ADC (i.e., trace) images are displayed in the first row we used a fourth-order RungeÐKutta method for the inte- ␭ ␭ ␭ and 1, 2, 3 images, i.e., the three channels of EV, are gration solver (Press et al., 1992; Basser et al., 2000; Tench ␭ Ϫ ␭ displayed in the second row. DE image, i.e., 1 2,is et al., 2002). This gains both speed due to a larger step displayed in the right side of the third row and all tensor images length and stability compared with a direct Euler method. A of TC, i.e., Dxx, Dxy, Dxz, Dyy, Dyz, and Dzz, are shown trilinear interpolation method was used for obtaining sub- with absolute values of diffusion in the left bottom three rows. voxel estimation with a 1-mm step size. In order to bias the registration to the brain region, i.e., A straightforward implementation of the eigenvector track- to gray and white matter, we multiplied all channels by the ing method is highly dependent on the major eigenvector and brain mask of the T2-weighted image derived from SPM99. will have problems following fibers at locations where other

Thus, we derived deformation ﬁelds, fX (i.e., fT2, fFA, fDE, ﬁbers are crossings and the anisotropy is low. To reduce the fAT, fEV, and fTC), according to the type of information used impact of this problem, we adopted the regularization scheme for registration (T2, FA, DE, AT, EV, and TC), respec- proposed by Bjornemo and colleagues (Bjornemo et al., 2002)

tively. We created inversely transformed tensor images, VsX and added a small bias toward the previous tracking direction (i.e., VsT2, VsFA, VsDE, VsAT, VsEV, and VsTC), by applying to the current tensor as follows: H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009 1999

Fig. 1. Types of intensity information used for registration. T2, FA (fractional anisotropy), and mean ADC (apparent diffusion coefficient, the trace of tensor) ␭ Ϫ ␭ images in the first row, DE (difference of the first and second eigenvalues, i.e., 1 2) in the right side of the third row are used as a single channel registration respectively. AT contains two channels information, i.e., fractional anisotropy and trace of tensor (mean ADC). Three channels of information ␭ ␭ ␭ in EV (eigenvalues, i.e., 1, 2, and 3 images) are displayed in the second row. Absolute intensity of each channel of TC (tensor components, i.e., Dxx, Dxy, Dxz, Dyy, Dyz, and Dzz gradient images) is seen in the last three rows. Areas with very low signal (surrounding air) or with mean apparent diffusion coefficient (ADC) values higher than 2.0 ␮s/mm2 (cerebrospinal fluid) were excluded.

ϭ ϩ ␣ T Dreg D xú tϪ1xú tϪ1 vector from the current point xt, i.e., major eigenvector of D, denoted as eˆ (D). ␣ controls the regularization strength and ϭ ͑ ͒ 1 xú t eˆ1 Dreg , (5) a larger value of ␣ will smooth the proposed ﬁber paths to ␣ where D is the current tensor at xt and xút is the direction a greater extent. was adaptively applied according to the 2000 H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009

Fig. 2. Evaluation of registration with artificially deformed DT-MRI. For evaluation on the registration of artificially deformed data, images registered to a target image were inversely transformed to source images using different intensity information: T2, FA, DE, AT, EV, and TC (reverted images, bottom left). Evaluation of the registration to different subject was conducted on the registered images on a arbitrary chosen subject image (registered images, bottom right). The dissimilarity between two comparing tensor images was evaluated using fiber bundle map that was derived using regularized RungeÐKutta fourth-order integration method. Regularization was incorporated to render fiber tracking at the region of crossing fibers. Streamtubes were used for the tract visualization. fractional anisotropy, as in Eq. (6), so that at the high weighted image was created as a seed mask for tractography anisotropy tensor fields, no regularization was applied: using the SPM99 segmentation algorithm (Ashburner and ␣͑ ͒ ϭ ␣ ϫ ͓ Ϫ ͑ ͔͒ Friston, 2000). The voxels that had a higher value than 0.9 v g 1 FA v , in the white matter probability map were used as initial Յ ͑ ͒ Յ ϭ seeds for the fiber tracking. As stopping criteria for the fiber THR_FAlow FA v THR_FAhigh 0, elsewhere, tracking, we used the criteria of low FA (0.2) and a rapid (6) change of direction (30¡ per 1 mm), as well as white matter where ␣(v) and FA(v) are alpha value and fractional anisot- mask. ␣ ropy at a voxel v with a global constraint g, a lower bound threshold of FA, THR_FAlow and a higher bound threshold Performance index of fiber bundle registration ␣ THR_FAhigh g, THR_FAlow, and THR_FAhigh were deter- The accuracy of the deformation was evaluated by cal- mined manually according to the tensor images case by culating the dissimilarity between fiber tracts derived from case. Fiber tracts were visualized by the stream tube method the tensor images at all the seed points within white matter (Schroeder et al., 1991), as used for example by Zhang et al. segmentation. We calculated mean dissimilarity between (2000). the tracts in the original tensor field, Vs, and tracts in the

A probability map of white matter derived from T2- inversely transformed tensor ﬁelds, VsX (i.e., VsT2, VsFA, H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009 2001

VsDE, VsAT, VsEV, and VsTC). We also measured the dis- Voxel-based evaluation of registration performance similarity between tracts in the target field, Vt, and tracts in We also applied a voxel-based measure of tensor overlap the normalized tensor fields, VrX (i.e., VrT2, VrFA, VrDE, between the source images and the registered images. VrAT, VrEV, and VrTC). Mean dissimilarity was defined by the following equa- Overlap (OVL). Overlap of eigenvalue-eigenvector pairs tion: between tensors was defined by Basser and Pajevic (2000) by the following equation: 1 DISSIM ϭ ͸͓d ͕tr ͑ x, Vs͒, tr ͑ x, Vs ͖͔͒ , X n͑S͒ X S X X 1 xʦS OVL ϭ ͸C ͑ x, Vs, Vs ͒ (10) X n͑S͒ X X xʦS (7) 3 where tr (x, Vs) indicates a tract in the original tensor field, S ͸␭ ␭ Ј͑␧ á ␧ Ј͒2 Vs, with a seed point x and tr (x, Vs ) indicates a tract in the i i i i X X ϭ ͑ Ј͒ ϭ i 1 inversely transformed tensor field, VsX, with the same seed CX x,V,V , (11) point x. We calculated the dissimilarity function, d ,offiber 3 X ͸␭ ␭ Ј bundles at all the seed points of the white matter segmen- i i tation, S, with the total number of seed points, n(S), in the iϭ1 original tensor field, Vs. In the calculation, we ruled out ␭ ␧ ␭ Ј ␧ Ј where i, i and i , i are eigenvalueÐeigenvector pairs of those seed points that had both fibers in comparison shorter the x-th tensors in the tensor fields V and VЈ respectively. S than 5 cm in length. The dissimilarity function between fiber is the white matter segmentation of the original tensor field, bundles was defined by the two following methods: Vs, with a total number of tensors, n(S). The minimum value 0 indicates no overlap and maximum value 1 indicates Mean square error (MSE). We defined the mean square complete overlap of the principal axes of the diffusion error between tracts as the mean distance normalized by the tensor between voxels. tract length in the following equation: The alignment of tensors in the samples of normalized images can be a measure of registration performance. As a T measure of alignment, we calculated the mean dispersion ͸ʈ ͑ ͉ ͒ Ϫ ͑␭ ͉ ͒ʈ 1/T trS t x, Vs trX t x, Vsx index of normalized tensor images inside the white matters tϭ1 and the mean fractional anisotropy of the white matters in dMSE ϭ , (8) X ͱ͉ ͉͉ ͉ the average image: trS trX ͉ where trS(t x, Vs) indicates spatial position on trace line trS Mean dispersion index (MDI). Dispersion index indicat- at the offset t from the starting point. ʈ á ʈ denotes the square ing the alignment of principal eigenvectors was also pro- ͉ ͉ ͉ ͉ error between trace lines trS and trX. trS and trX indicate posed by Basser and Pajevic (2000) and was defined as ͉ ͉ the lengths of the trace lines and T is equal to trS . The tract follows: trX was resampled to have the same number of tracking ␭ 1 points with trS with the resampling ratio of . MDI ϭ ͸di ͑ x͒ (12) X n͑S͒ X xʦS Endpoint divergence (EDIV). We defined the end point ␤ ϩ ␤ divergence as the distance between end points of both tracts ͑ ͒ ϭ ͱ 2 3 diX x ␤ , (13) normalized by mean tract length, T, seeded from a same 2 1 point x. ␤ ␤ ␤ where 1 (largest), 2, and 3 are eigenvalues of the mean ͉ ͑ ͉ ͒ Ϫ ͑␭ ͉ ͉͒ ͗␧ ␧ T͘ trS tT x, Vs trX tT x, VsX dyadic tensor 1 1 derived from the eigenvectors associ- dEDIV ϭ , (9) ␧ X T ated with the largest eigenvalue ( 1) of the tensors, for the N subjects, in the voxel, x, that belongs to the white matter ͉ ͉ where T is equal to trS , length of fiber tract trS, and tT is the segmentation, S, of the average tensor field with a total ␭ ͗␧ ␧ T͘ location of the end point of trS, whereas tT is the location number of tensors, n(S). The mean dyadic tensor 1 1 can of the end point of trX. Because fiber tracking was per- be written as the following equation: formed bidirectionally from the seed point, we calculated ␧2 ␧ ␧ ␧ ␧ the end point divergence in both directions and chose max- iX iX iY iX iZ 1 N ͳ␧ ␧ Tʹ ϭ ͳ ␧ ␧ ␧2 ␧ ␧ ʹ ϭ ͸␧j␧jT imum divergence. i i ͩ iX iY iY iY iZ ͪ i i , 2 N ␧ ␧ ␧ ␧ ␧ jϭ1 Note that we did not transform the points of the original iX iZ iY iZ iZ fiber bundle maps to a stereotaxic space as was done by Xu (12) et al. (2002), instead we performed tractography in the ␧j transformed diffusion fields. where i is the ith component of the principal eigenvector in 2002 H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009

j the voxel for the jth subject. The minimum dispersion index diffusion images. JR{hi(x)} is the rotational component of j of 0 indicates that there is no scatter about the mean eigen- deformation derived from the deformation field hi (x) and vector and a maximum of 1 when the eigenvectors are the operator V indicates adjustment of the tensor orientation j uniformly distributed about the sphere. with JR{hi (x)}. Note that the selection of a first target image does not affect the creation of the average template with this Mean fractional anisotropy (MFA). Fractional anisotropy method (Guimond et al., 2000, 2002). of the average tensor image can be a measure of the align- For averaging tensors at each voxel from the normalized ment of the tensor, an alignment of eigenvectors associated diffusion images, we used both the mean and median of the with not only the largest eigenvalue but also of all eigen- tensors, as suggested by Jones et al. (2002). The mean and values and eigenvectors. This index is based on the fact that the median of a sample of tensors are derived from the the average of anisotropic tensors stays anisotropic if they definition of Frechet (1948), where the central location is are aligned and that the average of a set of nonaligned defined to minimize the sum of distances in the domain of anisotropic tensors will become more isotropic and thus the the samples. As a measure of the distance between two FA will decrease. We calculated the mean fractional anisot- tensors A and B, the following metric d (A,B) is used: ropy inside the white matter of the average image. ͑ Ϫ ͒2 ϩ ͑ Ϫ ͒2 A11 B11 A22 B22 ϩ ͑ Ϫ ͒2 ϩ ͑ Ϫ ͒2 d͑ A, B͒ ϭ ͱ A33 B33 2 A12 B12 , Diffusion tensor atlas as an average of tensors and ϩ 2͑ A Ϫ B ͒2 ϩ 2͑ A Ϫ B ͒2 deformation fields 13 13 23 23 (14) We created average maps of the diffusion tensor according to normalization schemes by combining mean intensi- where Aij and Bij indicate the components of tensors. ϭ ties of all the normalized images and the mean of the To calculate the median, X,ofN tensors, Di (i 1... deformation fields (Guimond et al., 2000, 2002). A diffu- N), the gradient descent of the absolute distance function ͑ ͒ ϭ ͸N ͑ ͒ sion tensor image among the group was chosen as a tem- d1 X iϭ1d X, Di is solved with the mean of a sample ␮ ϭ ͸N porary atlas and all other images were registered to the of tensors, i.e., 1/N iϭ1Di used as initial value. Due to temporary atlas with an adjustment of tensor orientation. the time complexity, the median averaging method was The average of the registered diffusion tensor images was used only at the final iteration. In our atlas generation, both resampled with the inverse of the average deformation field mean and median tensor images are resampled with the in order to achieve morphological (shape) mean as well as inverse mean deformation field. the intensity (tensor) mean. The average map was again From the normalized diffusion images registered using used for the target atlas of the next iteration. Four iterations T2, FA, DE, AT, EV, and TC, we created average maps by were used to create an average diffusion tensor map. The calculating the mean of the tensors and by calculating the following equations explain how the average of the intensity median of the tensors. In order to compare linear registra- and the average of the deformation field were incorporated tion and nonlinear registration, we created average images, into the generation of the target atlas: using the mean and median from the normalized images linearly registered using all TC channels. We compared j j 3 j͕ j͑ ͖͒ Ϸ ͑ ͒ hi:S TiϪ1, S hi x TiϪ1 x these average images in terms of the mean dispersion index (MDI) and the mean fractional anisotropy (MFA). N ៮ 1 S ͑ x͒ ϭ ͸J ͕hj͑ x͖͒ ᮏ Sj͕hj͑ x͖͒, i N R i i jϭ1 Results for averaging with mean The performance of registration according to channel ៮ ͑ ͒ ϭ ͓ ͕ j͑ ͖͒ ᮏ j͕ j͑ ͖͒ Si x median JR hi x S hi x , information was evaluated using both the dissimilarity of fiber bundles [i.e., mean square error (MSE) and divergence j ϭ 1,...,N͔, for averaging with median (DIV) index] and the voxel-based overlap measure (OVL). We applied a nonparametric sign test to evaluate the N ៮ 1 difference between registration using TC and registration h ͑ x͒ ϭ ͸hj͑ x͒ i N i using other methods. The results of this evaluation are jϭ1 displayed in Fig. 3 and summarized in Table 1. When ៮ ៮ Ϫ performing registration of artificially deformed data sets, T ͑ x͒ ϭ S ͕h 1͑ x͖͒, (13) i i i TC showed significantly increased performance in EDIV, j Ͻ where hi is the deformation field of subject j from the source MSE, and OVL (P 0.005) over T2, FA, and DE, but did j image of the subject S -to the target image TiϪ1, i.e., the not show significantly increased performance over AT and average atlas created from the previous iteration iϪ1. Ini- EV. Only improvement of TC in OVL was found over that Ͻ tially, T0 is an arbitrary image chosen from the group of of AT (P 0.05) for this simulated deformation. In the H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009 2003 registration of different subjects, TC showed significantly better performance of nonlinear registration in the align- increased performance (P Ͻ 0.005) over any other type of ment of the principal direction of tensors. From Fig. 6b, we channel information in terms of MSE, EDIV, and OVL. can see that the dispersion index histogram of nonlinear The average diffusion tensor brain was created using TC registration using TC demonstrates a shift toward zero, by four iterations of averaging the intensity and deformation compared with both nonlinear registration using T2, FA, fields. T2, FA, and 2D visualization of TC of a single brain DE, AT, and EV and linear registration using TC (linTC). used as the initial atlas are displayed in the upper row of Fig. This closeness to zero implies the better performance of 4, and corresponding images of the average brain are dis- nonlinear registration using TC. played in the lower row of the same figure. The average brain shows higher signal to noise ratio due to the averaging scheme employed to build this image. The fiber bundle maps of the average brains and a com- Discussion parative single brain are displayed in Fig. 5. Fiber bundle maps of the average brains show much smoother tracts than Evaluation of spatial normalization of white matter a fiber bundle map of a single brain. As was noted in Jones et al. (2002), most of the short fibers in the peripheral region In contrast to traditional MRI where both gray and white of brain, such as the arcuate fibers, do not appear in the matter are relatively homogeneous, the information con- average brains due to smoothing effects caused by registra- tained in the diffusion tensor components is much more tion errors, intersubject variability, or both. complex. Therefore, we need precise spatial normalization The registration performance in terms of the mean of the and interpretation focusing on the properties of tensors. The dispersion index (MDI) of normalized images using differ- registration method for tensors should be able to match the ent methods and the mean of fractional anisotropy (MFA) of spatial location of white matter structures, i.e., fiber bundles, different type of average atlas is summarized in Table 2. We and to match the orientation of tensors. Accordingly, the calculated the mean dispersion index (MDI) on two types of evaluation of the registration should include the detection of normalized tensor images, registered to a single subject and anatomical correspondence defined in terms of the spatial to a group atlas as a target image. structures of the diffusion tensor fields. In this article, we Results demonstrate that the mean of the dispersion in- proposed a new evaluation method using fiber tractography dex of nonlinear normalization using TC (0.34) is lower in combination with voxel-based correspondence metrics. than the linear normalization using TC (0.41). This result The advantage of using fiber tractography is that it imposes indicates a higher alignment of the eigenvectors associated a stronger model of what we would like to register, the with the largest eigenvalue at each voxel in the nonlinear white matter fibers. normalization. MDI of the nonlinear normalization using We evaluated both the registration of artificially de- TC is lowest in both registration to a single brain and to an formed data to its original data and registration between atlas. MFA was highest in the atlases generated by both the DT-MRI from different subjects. In the first evaluation, the mean and median methods when the nonlinear normaliza- tensor fields are topologically the same, whereas in the tion using TC was used. These results imply that the best second, the tensor fields may include topological variation. registration was done with the nonlinear registration using Our evaluations using tractography and voxel-based tensor all the tensor components. These two indices support pre- metrics showed similar results, i.e., a nonlinear registration vious findings of similarity of fiber bundle maps and overlap using TC has better performance in most evaluations and is of tensors between different subjects, suggesting that non- more robust to the difference between source and target linear registration using all the tensor information is optimal images, thus strengthening our belief that our tractography- for normalization. The mean fractional anisotropy of the based evaluation is useful to measure the performance of the median average inside white matter was higher than that of registration of DT-MRI data. The assumption here is that the mean image in our evaluation. the better the images are aligned, the more similar the fiber Maps of the dispersion index from linearly normalized bundle maps and the more overlap of the tensor orientation. diffusion images and those from nonlinearly normalized With respect to limitations of our study, we acknowledge diffusion images using TC are displayed in Fig. 6a and that errors in fiber tracking affect the evaluation of the histograms of MDI according to channel information are registration. The inherent errors arise from the limitation of displayed in Fig. 6b. In Fig. 6a, the gray level colors indi- the current technique of diffusion tensor tractography due to cate the dispersion index. The black color indicates the noise effects and the intrinsic limitations of DT-MRI, lowest dispersion, indicating complete coherence of the known as partial volume effects. Anisotropic voxel units of principal direction between tensors of the group, whereas 1.72 ϫ 1.72 ϫ 4 mm, used in the study, may add to the the white color indicates the highest dispersion, indicating errors in fiber tracking even though we used an interpolation random distribution of the principal tensor direction. Dis- scheme to continuously estimate tensors. However, the persion index maps of nonlinear registration show clear and evaluation was done systematically to all tensor images and wider black areas than linear registration, which indicates we expect that such errors were evenly distributed for all 2004 H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009 comparison procedures. Additionally, to ensure that there anisotropy (FA), or a channel of major eigenvalues differ- was no bias introduced by the choice of target image, a ence (DE). We conjecture that information contained in all smaller number of source images (five) with a different the tensor components contain is redundant and is not nec- target were evaluated. No significant changes in our results essarily required in this registration, because the deforma- could be found. tion we artificially applied contains relatively low spatial- frequency-ranged deformation field. This conjecture is supported by the fact that the registration of the artificially Using multiple channels for registration deformed data showed better performance than registration Because the anatomical correspondence is determined between different subjects. A low-dimensional warping of relative to the anatomical landmarks or local intensity dis- SPM99 has a highly smoothed deformation field and there- tribution that the matching algorithm utilizes, the type of fore the high dimensional warping could approximate the input information can determine the registration process of deformation field successfully with a larger size of Gaussian tensor images. Previous registration methods have utilized smoothing kernel. In this situation, the effect of the input mostly scalar images such as FA or T2 image or T1- channel may not be too noticeable in the estimation as long weighted image coregistered to TC. However, the anatom- as it contains the basic information to register. But in the ical correspondence derived from FA or T2 may have dif- registration of the real brain between different subjects, ferent meanings, that is, the white matter of T2 does not more precise matching using anisotropy, magnitude of dif- correspond exactly to that of FA. fusivity and orientation information is required with a lower We therefore questioned which information is optimal in Gaussian smoothing kernel size. Mean dispersion index of terms of registering fiber tracts. T2-weighted data can de- normalized images, in Table 2, registered both to a single lineate the gray and white matter as well as the CSF rela- subject and to a group atlas also suggests that registration tively well. However, white matter in T2 is relatively ho- using the whole tensor information is more robust than other mogeneous throughout the brain and contains little methods. information concerning the location and orientation of fiber The computational time grows proportional to the num- bundles. FA is a widely used index for the anisotropy of a ber of channels used for registration. In an explorative tensor, which is related to the fiber density, coherence, and study, the accuracy is more important than the complexity myelination, and thus it shows the location of fiber bundles and the time requirement is not essential as long as it stays very well. DE was chosen as a non-normalized index for within a reasonable range. In this study, the calculation time anisotropy. Because FA is a normalized anisotropic index of one 2.0 GHz Pentium 4 processor is about 5 min for one which is insensitive to the magnitude of diffusivity, adding channel and for TC calculation it takes about 30 min, which the trace of the tensor (AT) is thought to provide comple- is quite acceptable. mentary information on the registration. EV contains all the Considering the time complexity, the registration using magnitude information of the tensor due to multiple chan- three eigenvalues showed a comparable performance to that nels of eigenvalues. However, it does not guarantee that using all the tensor components. Therefore, we may choose corresponding points are in the same fiber tracts or not, to use three channels of eigenvalues in the registration with because the tensors of different fiber tracts showing differ- acceptable confidence and reduced time complexity. ent orientation may have similar eigenvalues. The peculiar- In general, registration algorithms are controlled by a ity of TC is that TC contains the directional information of similarity function as well as the input information type, tensors, including the tensor anisotropy. Using TC, the i.e., type of channels. For the registration of the tensor, optimization algorithm can search corresponding tensors of various similarity measures between tensors of source target images for the same anisotropy, magnitude of diffu- and target images can be defined. The demons algorithm sivity and direction with source images. does not use a specific tensor similarity measure during In terms of the overall performance of registration, using registration, but, instead, it simply utilizes the difference TC showed the best performance, especially in the registra- of the intensity of each channel from source and target tion of images from different subjects, and in the registra- images. Incorporating a precise tensor similarity measure tion to a group atlas. The robustness of using TC may be due and optimizing the weight of each channel might increase to utilization of tensor orientation information in the regis- the registration performance. Data reduction using a prin- tration process. In the artificially deformed registration, that cipal component method can potentially be used to effi- two channels of the fractional anisotropy and the trace of the ciently reduce the number of channels and thus decrease tensor combination (AT), three channels of eigenvalues the computational burden without compromising the reg- (EV) and six channels of tensor components (TC) showed istration error. Though these topics are of interest to more or less similar performance and these three methods further research, in this study, we restricted our evalua- showed significantly increased performance over a channel tion to only optimal choices of input channels in the of T2-weighted image (T2), a channel fractional tensor matching process. H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009 2005

Fig. 3. Scattergram of registration performance. The evaluation results of both registration of artiﬁcially deformed data and registration of different subjects are displayed in scattergram. The midline indicates the mean. EDIV (end point divergence), MSE (mean square error), and OVL (overlap of tensors) were calculated according to the input channel type used for registration, i.e., T2 (T2-weighted), FA (fractional anisotropy), DE (difference of eigenvalues), AT (fractional anisotropy and trace), EV (three eigenvalues), and TC (total six tensor components). Ten images were used for evaluation on artiﬁcially deformed data and 15 images were used for evaluation on different subjects.

Creation of a group atlas of diffusion tensor brain tensor components. In terms of the alignment of the major eigenvector of the tensor, nonlinear normalization showed For creating the group average diffusion atlas, we used better performance than linear normalization. As mentioned an iterative scheme similar to the one presented by by several investigators (Alexander and Gee, 2000; Jones et Guimond et al. (2000) for averaging MR images. Instead of al., 2002), due to noise effects, there may exist a possibility using structural MRI, diffusion tensor images were regis- of misorientation of the diffusion tensors in nonlinear high- tered to create a tensor diffusion atlas. Tensor orientations dimensional warping. However, this potential misorienta- were successively adjusted during each iteration. The com- tion does not necessarily pose a larger problem than the bination of the intensity average and the shape average misalignment obtained by an affine transformation. For derived from mean of deformation fields renders the topol- voxel-based statistics, where finding corresponding posi- ogy and morphology representative of the group. tions is important, nonlinear warping technique is thought to In making an atlas, Jones et al. (2002) evaluated several be appropriate. More specifically, the nonlinear warping averaging methods such as mean, median, and mode. The using all diffusion tensor information showed a lower dis- median tensor was obtained in order to be robust to outliers persion index implying better registration of tensors com- by finding a tensor that minimizes the distance to the sample pared with nonlinear registration using other types of input of tensors in an absolute distance sense. We applied these information. This provides further arguments supporting the mean and median methods in the averaging of normalized thesis that the orientation information of diffusion tensor as DT-MRIs, but instead of linear registration using FA infor- well as anisotropy is necessary for the registration of fiber mation, we used the nonlinear registration using all the tracts. 2006 H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009

Table 1 Performance of diffusion tensor registration according to intensity information

Measure T2 FA DE AT EV TC Simulated Deformation EDIV 0.528 (0.056)* 0.516 (0.060)* 0.518 (0.062)* 0.503 (0.070) 0.498 (0.070) 0.495 (0.066) MSE 0.599 (0.069)* 0.585 (0.075)* 0.586 (0.075)* 0.566 (0.087) 0.559 (0.087) 0.557 (0.082) OVL 0.767 (0.043)* 0.818 (0.054)* 0.812 (0.053)* 0.873 (0.058)† 0.881 (0.057) 0.887 (0.053) Different Subjects EDIV 0.569 (0.028)* 0.563 (0.037)* 0.561 (0.034)* 0.563 (0.038)* 0.546 (0.028)* 0.535 (0.030) MSE 0.641 (0.031)* 0.631 (0.041)* 0.627 (0.037)* 0.629 (0.042)* 0.610 (0.030)* 0.595 (0.033) OVL 0.574 (0.013)† 0.565 (0.034)† 0.571 (0.024)* 0.574 (0.039)* 0.597 (0.014)* 0.611 (0.015)

Note. The values are mean (standard deviation). Abbreviations: EDIV, Endpoint Divergence; MSE, Mean Square Error; OVL, Overlap; T2, T2-weighted; ␭ Ϫ ␭ ␭ ␭ ␭ FA, fractional anisotropy; DE, 1 2; AT, two channels of FA and trace of the tensor; EV, three eigenvalues ( 1, 2, 3); and TC, six tensor components (Dxx, Dxy, Dxz, Dyy, Dyz, Dzz). Lower values of EDIV and MSE imply better performance, whereas higher values of OVL imply better performance. * and † indicates signiﬁcant difference in registration performance (p Ͻ 0.005 for * and p Ͻ 0.05 for †) compared with registration using TC when nonparametric sign test was applied.

Note that we used the average brain morphology, derived matter abnormalities in schizophrenia in comparison to from the mean deformation field, as well as the average healthy control subjects, using statistics for scalar quantities tensor intensity when creating the average brain. We believe such as FA and angles between vectors at the corresponding that the average brain created by our nonlinear method position across samples. Exploration of left and right hemi- using these two types of average information is a better spheric asymmetry for both populations has also been con- approach than a strict linear one, in representing both tensor ducted using the proposed methods (e.g., Park et al., 2003). distribution and morphological distribution of the population. Though further work will be required, including compar- Concluding remarks isons between the mean and median method in terms of better group representation, we think the selection of the In this article we have discussed spatial normalization of averaging method can be dependent on the distributional DT-MRI data in terms of the input information the regis- model of sample tensors at each voxel and the use of an tration algorithm utilizes. We have evaluated the methods atlas. As was noted by Jones et al. 2002, especially for the using DT-MRI tractography. In addition, we have created a case of outliers, the median method will be advantageous DT-MRI atlas based on the average diffusion tensor images. but for normally distributed or coherently distributed sam- Nonlinear registration using all the components of the dif- ples of tensors, the mean method is thought to be sufficient fusion tensors resulted in the most reliable results and can with a much lower computational cost. be used for generation of a group atlas, where the mean of Using the multiple channel registration method and a tensors and the mean of deformation fields are combined group diffusion tensor atlas, we are further exploring white together.

Table 2 Performance of registration in creating average brain of the group

Meas T2 FA DE AT EV TC LIN ϩ TC To a subject MDI 0.39 (0.15) 0.41 (0.16) 0.40 (0.16) 0.39 (0.16) 0.36 (0.16) 0.35 (0.16) To an Atlas MDI 0.44 (0.15) 0.40 (0.18) 0.39 (0.17) 0.39 (0.17) 0.38 (0.17) 0.34 (0.17) 0.41 (0.16) MEAN MFA 278.9 (118.4) 266.2 (144.0) 287.5 (141.2) 290.5 (137.9) 296.1 (138.7) 323.6 (136) 305.8 (124.2) MED MFA 284.4 (121.7) 271.7 (146.2) 293.3 (144.1) 293.4 (140.0) 298.9 (140.9) 325.9 (137.6) 308.3 (127.6)

Note. The values are mean (standard deviation) values of voxels inside the white matter, not a mean (standard deviation) of a group sample. Abbreviations: MDI, mean dispersion index, MFA of MEAN, mean fractional anisotropy of the mean averaging method, MFA of MED, mean fractional anisotropy of the median averaging method. All means are evaluated within white matters. LIN ϩ TC, linear normalization using six tensor components. Except for LIN ϩ TC, nonlinear normalization was used for all evaluations.

Fig. 4. DT-MRI of single brain and a group averaged brain (n ϭ 15). T2-weighted image (T2, ﬁrst column) and fractional anisotropy image (FA, second column) are displayed for a single brain DT-MRI (top row) and a group average DT-MRI (bottom row). Major eigenvectors of all tensor components (TC, third column) around the corpus callosum of single brain and average brain are also visualized in 2D space by line and colors. The color level indicates the strength of the component of major eigenvector perpendicular to the slice from blue (lowest) to red (highest). Fig. 5. Fiber bundle maps of averaged DT-MRIs. Averages of 15 DT-MRIs (n ϭ 15) are created by calculating the mean and the median of nonlinearly transformed images (WARPMEAN and WARPMED respectively). A comparative ﬁber bundle map of a single brain is also displayed. H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009 2007 2008 H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009

Fig. 6. Maps and histograms of dispersion index for linear registration and nonlinear registration. Dispersion maps at three different coronal slices derived from linear registration (Linear) using TC are displayed in the upper row of (a) and those of nonlinear registration (Warp) using TC are displayed in the lower row of (a). Gray level color shows the dispersion index. Black indicates the lowest dispersion meaning complete coherence of the direction of tensors of the group, whereas white indicates the high dispersion indicating random distribution of tensor directions. Dispersion index maps of nonlinear registration shows more dark black areas than linear registration, indicating better performance of nonlinear registration in the alignment of principal direction of tensors. The dispersion index histogram (b) of nonlinear registration using TC (black line) shows shifted toward zero compared with nonlinear registration using T2 (blue dash-dot line), FA (green dash-dot line), DE (red dash-dot line), AT (cyan line), EV (blue line), and linear registration using TC (linTC, ochre dotted line). This closeness to zero implies the better performance of registration. This histogram was calculated inside the white matter. H.-J. Park et al. / NeuroImage 20 (2003) 1995Ð2009 2009

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