Comparison of Gradient Encoding Schemes for Diffusion-Tensor MRI

Transcription

1 JOURNAL OF MAGNETIC RESONANCE IMAGING 13: (2001) Original Research Comparison of Gradient Encoding Schemes for Diffusion-Tensor MRI Khader M. Hasan, PhD, 1 Dennis L. Parker, PhD, 2 and Andrew L. Alexander, PhD 2 * The accuracy of single diffusion tensor MRI (DT-MRI) measurements depends upon the encoding scheme used. In this study, the diffusion tensor accuracy of several strategies for DT-MRI encoding are compared. The encoding strategies are based upon heuristic, numerically optimized, and regular polyhedra schemes. The criteria for numerical optimization include the minimum tensor variance (MV), minimum force (MF), minimum potential energy (ME), and minimum condition number. The regular polyhedra scheme includes variations of the icosahedron. Analytical comparisons and Monte Carlo simulations show that the icosahedron scheme is optimum for six encoding directions. The MV, MF, and ME solutions for six directions are functionally equivalent to the icosahedron scheme. Two commonly used heuristic DT-MRI encoding schemes with six directions, which are based upon the geometric landmarks of a cube (vertices, edge centers, and face centers), are found to be suboptimal. For more than six encoding directions, many methods are able to generate a set of equivalent optimum encoding directions including the regular polyhedra, and the ME, MF and MV numerical optimization solutions. For seven directions, a previously described heuristic encoding scheme (tetrahedral plus x, y, z) was also found to be optimum. This study indicates that there is no significant advantage to using more than six encoding directions as long as an optimum encoding is used for six directions. Future DT-MRI studies are necessary to validate these observations. J. Magn. Reson. Imaging 2001;13: Wiley-Liss, Inc. Index terms: diffusion tensor; icosahedron; encoding optimization; heuristic polyhedra; numerically optimized polyhedra; regular polyhedra MEASUREMENTS OF WATER self-diffusion in biological tissues is related to the underlying tissue microstructure (1). Recently, the self-diffusion tensor has become a widely used model for describing diffusion measurements (2). Although there is recent evidence for more complex diffusion behavior for the human brain 1 Department of Physics, University of Utah, Salt Lake City, Utah. 2 Department of Radiology, University of Utah, Salt Lake City, Utah. Contract grant sponsor: NIH; Contract grant number: P30 CA42014; RO1 MH *Address reprint requests to: A.L.A., PhD, W.M. Keck Laboratory for Functional Brain Imaging, The Waisman Center, 1500 Highland Ave, University of Wisconsin, Madison, WI alexander@psyphw.psych.wisc.edu Received June 27, 2000; Accepted November 7, (3,4), the single diffusion-tensor is a simple and elegant model that appears to be appropriate for many applications. Diffusion-tensor MRI (DT-MRI) measurements describe the magnitude of diffusion as a function of direction. In fibrous tissues such as white matter and muscle, the diffusion appears to be directionally anisotropic and the direction of greatest diffusion appears to be aligned with the direction of the fiber (2). A minimum of six diffusion encoded images is necessary to completely describe the 3 x 3 diffusion tensor (5 11) D D xx D xy D xz D yx D yy D yz D zx D zy D zz (1) because the tensor is symmetric about the diagonal (e.g., D ij D ji ). Noise in the raw diffusion encoded measurements may lead to errors in the estimates of the diffusion tensor. The gradient encoding scheme may also influence the degree to which the measurement noise affects diffusion tensor estimates. Several gradient encoding schemes have been described with six or more encoding directions, N e (12 25). In this paper, we will attempt to answer two questions related to diffusion tensor encoding. What is the optimal gradient encoding scheme for six encoding directions (N e 6)? Is there an advantage to using more than six encoding directions (N e 6)? Previously described encoding schemes and several novel methods for optimizing the set of gradient encoding directions will be evaluated. Three approaches to selecting the gradient orientations for DT-MRI are discussed in this paper. They include heuristic directional selection, numerical optimization, and use of predefined geometric polyhedrons. The diffusion tensor accuracy associated with these schemes will be compared. THEORY Diffusion Tensor Encoding The encoding formalism in this study applies specifically to a single tensor model of diffusion. A discussion of more complex models of diffusion is beyond the scope of this paper (3,4). A single diffusion tensor, D, may be reconstructed using 6 or more diffusion encoded measurements acquired with diffusion weighting, b, and a 2001 Wiley-Liss, Inc. 769

2 770 Hasan et al. reference image, S o, acquired with b 0. The diffusion attenuated signal, S, is (7,8) S S o e bĝ Dĝ, (2) where ĝ [g x g y g z ] is the unit gradient vector representing the diffusion-encoding direction. The diffusion measurement in Eq. [2] can be rewritten S S o e bd h, (3) where d is a 6 x 1 vector of the unique diffusion tensor elements (9) d D xx D yy D zz D xy D xz D yz, (4) and h is the 6 x 1 diffusion encoding vector for the measurement h g x 2 g y 2 g z 2 2g x g y 2g x g z 2g y g z. (5) For N e diffusion encoded measurements, the set of encoding gradients is and the encoding matrix is G ĝ 1...ĝ Ne, (6) H h 1...h Ne. (7) In this study, several methods for determining the diffusion tensor are presented. For a single diffusion measurement, the projected diffusion value described by Eq. [3] for a given encoding vector is y d h. (8) The entire set of N e diffusion-encoded projections is Y y 1...y Ne Hd. (9) This equation describes a linear relationship between the measurements y and the tensor elements d. The diffusion tensor elements may be determined using a least squares approach d H H 1 H Y. (10) Alternatively, singular value decomposition can be used to determine the pseudo-inverse of H to solve Eq. [9] directly (see Appendix I). Measurement Noise The effects of measurement noise can be described by adding an N e x 1 independent noise vector,, tothe projected diffusion measurements so that Eq. [9] becomes Y Hd. (11) The mean square error (MSE) of the diffusion measurements can be written MSE d Y Hd 2. (12) The tensor elements can be estimated using Eq. [10]: dˆ H H 1 H Y. (13) If the noise for each measurement is assumed to be independent, the covariance matrix for Y can be written cov Y I Ne N e 2 y, (14) where I is the N e x N e identity matrix. It should be noted that this definition will be most accurate for isotropic diffusion tensors; however, highly anisotropic diffusion tensors may demonstrate different variances for each of the encoding directions. By combining Eqs. [13] and [14], it can be shown that the covariance matrix of the diffusion tensor elements dˆ depends on the covariance of the measurements (10): cov dˆ H H 1 H cov Y H H 1 H H H 1 2 y. (15) This relationship demonstrates that the amount of random uncertainty or noise in the estimates of the diffusion tensor elements is a function of the encoding matrix, H. Heuristic Encoding Schemes Heuristic encoding schemes use a base set of directions defined by the logical x, y and z gradients of the MRI scanner, which correspond to the faces of a cube. The unit cube with an edge length 2, originating at (0, 0, 0) defines 13 non-colinear directions at the face centers, edge bisectors, and body diagonal directions. The following sets of non-colinear gradient directions [g x, g y, g z ] are commonly selected to be the building blocks of heuristic encoding: B0 {[1, 0, 0]; [0, 1, 0]; [0, 0, 1]}, B1 ( 2) 1 {[0, 1, 1]; [1, 0, 1]; [1, 1, 0]}, B2 ( 2) 1 {[0, 1, 1]; [1, 0, 1]; [1, 1, 0]}, and B3 ( 3) 1 {[1, 1, 1]; [1, 1, 1]; [ 1, 1, 1]; [1, 1, 1]}. Heuristic schemes are probably the most commonly used for DT-MRI. The following heuristic tensor encoding schemes have been described. In each of the schemes, the applied gradient amplitude in each direction is normalized to a constant so that the diffusion-weighting, b, does not change. Orthogonal (ORTH) Gradient Encoding (N e 6) This encoding scheme combines measurements along the three orthogonal gradient axes {B0} with off-diagonal edge measurements {B1} to estimate the entire tensor. This approach is called Pyramidal encoding in chemical shielding tensor spectroscopy (11).

3 Gradient Encoding for Diffusion Tensor MRI 771 Table 1 Normalized Gradient Schemes With N e 6, 21 and 30 Scheme Encoding vectors {[g x, g y, g z ]} ME21 {[0.506, 0.708, 0.492], [0.834, 0.544, 0.087], [0.391, 0.697, 0.601], [ 0.066, 0.965, 0.253], [0.859, 0.462, 0.220], [0.996, 0.048, 0.078], [0.248, 0.229, 0.941], [0.544, 0.839, 0.019], [ 0.268, 0.661, 0.745], [0.753, 0.48, 0.45], [0.599, 0.22, 0.77], [0.223, 0.727, 0.65], [ 0.448, 0.271, 0.852], [ 0.368, 0.259, 0.893], [0.451, 0.887, 0.973], [ 0.843, 0.19, 0.503], [0.768, 0.112, 0.63], [0.009, 0.288, 0.953], [ 0.857, 0.401, 0.323], [0.489, 0.66, 0.57], [0.0236, 0.956, 0.292]} ME30 {[0.767, 0.597, 0.237], [0.868, 0.452, 0.204], [0.662, 0.667, 0.342], [ 0.283, 0.947, 0.151], [0.604, 0.786, 0.134], [0.922, 0.382, 0.066], [0.250, 0.284, 0.926], [0.178, 0.921, 0.346], [ 0.366, 0.851, 0.376], [0.377, 0.194, 0.906], [0.448, 0.422, 0.788], [ 0.265, 0.769, 0.581], [ 0.084, 0.559, 0.825], [ 0.512, 0.533, 0.673], [0.152, 0.988, ], [ 0.824, 0.222, 0.521], [0.95,.0384, 0.309], [0.545, , 0.838], [ 0.821, 0.253, ], [0.431, 0.641, 0.635], [ 0.034, 0.703, 0.711], [0.795, 0.471, 0.382], [ 0.089, 0.146, 0.985], [0.514, 0.624, 0.589], [0.987, 0.104, 0.126], [ 0.655, 0.194, 0.73], [ 0.748, 0.248, 0.615], [ 0.566, 0.813, 0.136], [ 0.166, 0.882, 0.441], [0.192, 0.313, 0.93]} Oblique Double Gradient (ODG) Encoding (N e 6) The ODG encoding technique, described by Basser and Pierpaoli (12), is composed of the entire set of edge bisectors: {B1 B2}. This technique has also been applied in chemical shielding tensor spectroscopy (11). Orthogonal/Tetrahedral Hybrid (S7) Encoding (N e 7) The combination of orthogonal {B0} and tetrahedral encoding {B3} (13) has been described for both DT-MRI (14) and thermal expansivity tensor measurements (9). Decahedral Gradient (S10) Encoding (N e 10) Skare and Nordel (15) proposed combining the merits of tetrahedral encoding with the ODG encoding scheme: {B1 B2 B3}. Complete Heuristic Gradient (S13) Encoding (N e 13) This encoding scheme combines all face, edge and corner directions: S13 {B0 B1 B2 B3}. Numerically Optimized Encoding Schemes An alternative to using heuristic directions is to optimize the set of encoding directions based upon a specified criterion. These optimization procedures can be used to obtain a set of encoding directions for any arbitrary N e. A number of optimization criteria are presented below. Several of the encoding gradient schemes obtained using the numerical optimization criteria are listed in Table 1. Minimum Total Variance (MV) The total variance of the diffusion tensor is a sum of all the variances in the estimated tensor elements D ij, 3 3 TV 2 D ij. (16) i 1 j 1 This optimization criterion was originally described by Papadakis, et al (16,17). In their study, they used the MV optimization to generate sets with N e 6, 12, and 24, and showed that the TV was reduced for the numerically optimized encoding relative to both the ORTH and the ODG encoding schemes described above. They also hypothesized that the variance decreased as N e was increased. Minimum Force (MF) The gradient direction distribution problem can also be described in terms of a criterion which maximizes the separation between points constrained to the surface of a sphere. Jones, et al (18) described a minimum force criterion based upon charges on a unit sphere to generate the optimal encoding sets. A total of N e charge pairs on the surface of a sphere are placed by reflecting the coordinates through the origin. The charge locations are selected to minimize the forces between each of the charges and its neighbors. Minimum Energy (ME) The ME, related to the MF criterion above, is the minimum total interaction (Coulombic) energy of 2*N e unit charges on the unit sphere. The charges experience vertex repulsion and prevent clustering (19). The ME optimization criterion for gradient encoding sets is defined by the following expression analogous to the total Coulombic energy: 2N e 2N e E 1 ĝ i ĝ j. (17) i 1 j i An encoding for N e 46 obtained using the ME criterion is illustrated in Fig. 1. Singular Value Decomposition (SVD) The inverse of the encoding matrix, H, can be estimated using singular value decomposition (SVD) as shown in Appendix I. The variance of the tensor elements is inversely related to the square of the singular values, s i (Eq. [A5]). We have also found that the encoding matrix that yields the minimum inverse pseudo-determinant

4 772 Hasan et al. at each vertex (21). Face triangulation can provide sets with N e 5n 2 1 for n 1, 2, 3,... (22 24). This results in ICOSA6, ICOSA21, ICOSA46, ICOSA81, ICOSA126, etc. The ICOSA6 and ICOSA21 encoding schemes are illustrated in Fig. 1. The larger ICOSA sets (N e 6) can be generated by edge bisection and tesselation of the ICOSA6 directions and projection onto a unit sphere (21). Muthupalli, et al (25) concluded that ICOSA6 is the optimum encoding scheme for DT-MRI. The icosahedron has also been applied in encoding for chemical shielding tensor spectroscopy (7,11). Tuch et al (4) used the ICOSA126 for diffusion measurements with high angular sampling. Figure 1. Examples of four regular polyhedra encoding sets. a: ICOSA6, b: ICOSA21, c: DODECA10 and d: ICOSA Q (18) s i i 1 of the encoding matrix will minimize the variance of the tensor elements. Minimum Condition Number (MCOND) The condition number of the encoding matrix, H, is often used as a measure of independence of columns (or rows) in a matrix. Skare, et al recently used this criterion to generate encoding sets (20). The condition number of the encoding matrix is defined in terms of the corresponding singular value matrix S (defined in Appendix I) cond H max diag S min diag S. (19) Note that the condition number only depends upon two of the singular values the minimum and the maximum. Geometric Polyhedra Schemes Another potential solution to the encoding problem is to use geometric polyhedra with the highest symmetry. For example, the tetrahedron (13) is the first Platonic solid with N e 4. However, tetrahedral encoding (equivalent to heuristic building block {B3}) is not enough to obtain the six independent diffusivities for estimating the diffusion tensor. The encoding gradients for several polyhedra are listed in Appendices II and III. Icosahedral (ICOSA) Polyhedra The regular icosahedron polyhedra family is defined by three polygon edges per face with five faces intersecting Other Polyhedra Numerous other polyhedra may also be used for encoding. These polyhedra can be generated by either dual relationships (face / edge centers) or truncation of the regular icosahedra. The dodecahedron with pentagonal faces is the dual to the icosahedron with triangular faces. Both have the same surface-to-volume ratio in the unit sphere (26,27). The dodecahedron (DODE10 with N e 10; depicted in Fig. 1) is defined by five polygon edges per face with three faces intersecting at each vertex (11,21). Bisection of the edges of the ICOSA6 leads to another polyhedron set with N e 15 (POLY15) (11). A BuckyBall encoding set can be obtained by ICOSA6 vertex truncation (27 29). For example, the vertices pointing in the north hemisphere of the BuckyBall cage can provide the BUCKY30 encoding set (30). Analytic (AN) Approach Wong and Roos (31) applied an analytical differential approach to select uniformly distributed vertices on the unit sphere for RF pulse design. The analytical solution approximately sweeps the surface of a unit sphere at a constant rate. The parameterized normalized analytical set encoding directions [g x,g y,g z ] can be written as: g z n 2n N e 1 N e (20) g x cos g z N e 1 g z 2 g y 1 g z 2 g x 2 (21) (22) where n 1, 2,..., N e. If the number of encoding directions is large enough, the placement of points should be spaced nearly uniformly on the sphere. METHODS Numerical Optimization of DT-MRI Encoding DT-MRI encoding sets were determined numerically for N e between 6 and 128. In each case the directions were selected as N e noncolinear vectors on a unit sphere. No information regarding either the shape or the orientation of the diffusion tensor was assumed. The encoding

5 Gradient Encoding for Diffusion Tensor MRI 773 optimization was performed using different optimization criteria - MV, MF, ME, SVD, and MCOND. For each criterion, the simplex algorithm (32) was used to determine the best set of encoding directions. To facilitate comparison between encoding sets, H, we adopt a relative encoding advantage factor REAF H 6/N e TV ICOSA6 /TV H, (23) which is defined here in terms of the relative total variance of the tensor components (11). The 6/N e factor normalizes the figure of merit to constant imaging time. A REAF value larger than unity indicates an advantage over the ICOSA6. The encoding with the largest REAF should be optimum. Monte Carlo Noise Simulation Monte Carlo computer simulations of DT-MRI imaging experiments were performed using Matlab (The Mathworks, Inc., Natick, MA). These simulations were done to evaluate the effects of image SNR, tensor shape, and orientation (6). These simulations were performed in a manner similar to the Monte Carlo methods described by Pierpaoli and Basser (33) and Bastin, et al (34). For a given diffusion tensor shape with a specified orientation and image signal-to-noise ratio (SNR), a simulated diffusion tensor imaging experiment was performed in which the noise-free signals for the specified gradient encoding were first calculated. Gaussian random noise with zero mean and a specified variance (e.g., amount of noise) was independently added to the reference signals and each of the encoded signals. The absolute value of the simulated measured signal was used in subsequent calculations. The simulation was repeated 5000 times for 46 diffusion tensor orientations. The orientations of the major eigenvectors were generated using the vertices of the ICOSA46 encoding scheme. The total tensor variance, TV, (Eq. [16]) was estimated for each encoding scheme and tensor orientation. A measure of the normalized tensor error ε 3 TV, trace D (24) was used, where is the expectation value of TV over all orientations of the diffusion tensor, and. The effects of noise on measurements derived from the diffusion tensor the effective apparent diffusion coefficient (ADC eff trace{d}/3), fractional anisotropy (FA) (1), and the major eigenvector direction e 1 were also determined from the simulations. The error in direction is defined by the angular dispersion acos e 1true e 1est. (25) Eight different encoding schemes were examined with Monte Carlo simulations ORTH, ODG, ICOSA6, ME6, MV6, MV12, MV24, and MV48. The SNR of the reference (b 0) image was set at 25. This is similar to the SNR that we obtain in our imaging experiments. Each simulated DT-MRI experiment was configured to have one reference image and 48 diffusion-weighted images. For encoding schemes with N e 48, the encoded images were repeated 48/N e times and averaged to maintain a constant imaging time. Simulation studies were performed to determine the effects of noise for three shapes of diffusion tensors [ 1, 2, 3 ] prolate ( cigar - shaped) [1600, 400, 400], oblate ( cushion -shaped) [1000, 1000, 400] and spherical [800, 800, 800] defined by the three eigenvalues 1, 2, and 3. Diffusion Tensor Imaging Experiments DT-MRI measurements of the brain of a normal human subject were obtained with a 1.5 T SIGNA MRI system (General Electric, Milwaukee, WI; 5.6 Horizon with Echospeed gradients 22 mt/m max amplitude). A single shot spin-echo EPI pulse sequence was modified to obtain DT-MRI measurements using the ORTH, ODG, and MV6 encoding schemes. Single axial slices with image matrix 128 x 128 were acquired with FOV 20 cm. The total number of images for each encoding measurement was N t 55 [ 7 (8 x 6)], which was partitioned into 7 reference images, and 8 diffusion-weighted averages in six encoding directions. The imaging parameters were TE 100 msec, TR 3000 msec, 30 msec, 36 msec, slice thickness 5 mm, and b 953 sec mm 2. Maps of the ADC eff and the FA were generated for each of the DT-MRI encoding schemes. RESULTS Numerical Determination of Encoding Directions for Any N e Figure 2 shows the relative variance and energy values for the encoding sets with N e between 6 and 50 as obtained by three different optimization procedures MV, ME and MSVD. The results demonstrate that the numerical encoding sets are functionally equivalent. The TV in Fig. 2a for all optimized sets decreases with N 1 e (R ). The REAF measures in Fig. 2a are normalized for imaging time and demonstrate that there is basically no difference between the MSVD, MV and ME optimizations (R ). The REAF is also constant for all N e, indicating that there is no significant advantage to using more than six encoding directions. In Fig. 2b, the energy index, E, increases quadratically with N e (R ). It should be noted that the ME criterion resulted in slightly lower energy than the MV and MSVD criteria. In contrast, both the ME and MV criteria resulted in equivalent total variances. This may indicate that the ME criterion is slightly more sensitive to the encoding orientations. Analytical Comparison of All Encoding Schemes A comparison of measures related to encoding are listed in Tables 2, 3, and 4 for many of the heuristic, numerical, and polyhedra encoding schemes, respectively. The tables show that the energy increases as N e increases. For fixed N e, the encoding with the lowest E

6 774 Hasan et al. Table 2 Comparison of Encoding Measures For the Heuristic Encoding Schemes* Encoding scheme N e Total energy index (a.u) Total tensor variance REAF Condition number ODG ORTH S S S *The optimum encoding schemes have REAF values between 0.99 and 1.00 (highlighted in boldface). Figure 2. The (a) TV and REAF measures, and (b) Coulombic energy, E, are plotted versus the numerically optimized encoding results using the MV, ME, and MSVD criteria. should be most optimum according to the ME criterion. Likewise, the encoding schemes with the largest REAF should be optimum using the MV criterion. The condition number is often used to evaluate the stability of the matrix inversion. A lower condition number indicates that the singular value matrix that corresponds to the encoding matrix may be more stable. In the comparisons of the heuristic encoding schemes (Table 2), it is clear that the commonly used ORTH and ODG encoding schemes are less optimum relative to the S7 and S13 schemes. All of the numerical optimization criteria ME, MV, MSVD, and MF (Table 3) except the MCOND resulted in optimized encoding. Note that the ME criterion showed the lowest energy relative to the other schemes as expected. All of the regular polyhedra encoding schemes ICOSA, DODE, and BUCKY as well as the analytic encoding scheme AN (Table 4) demonstrated optimum encoding behavior. Monte Carlo Simulations The results of the Monte Carlo noise simulations are shown in Figs. 3 and 4. The normalized total variance for constant imaging time,, is plotted for several of the encoding schemes with spherical, oblate and prolate diffusion tensors in Fig. 3. The variance values that are plotted are the average variance for all orientations (rotations) of the simulated diffusion tensor. The error bars indicate the standard deviation associated with the rotation of the diffusion tensor. The average measurement variance in Fig. 3 illustrates that the heuristic schemes ORTH and ODG are more sensitive to measurement noise relative to the other schemes. All of the other schemes have roughly the same average variance. However, the variations as a function of the tensor orientation appear to decrease as N e is increased for the more prolate diffusion tensor (Fig. 3c). Estimates of the encoding REAF were generated from the ratios of the estimated TV in Fig. 3 for each of the encoding schemes relative to the ICOSA6. These REAF values are listed in Table 5 and demonstrate good agreement with the predicted values. There does appear to be a minor dependence of the REAF on the tensor shape. The simulations were also used to estimate the errors in tensor measures ADC eff, FA and e 1. These errors for the prolate diffusion tensor case are plotted versus the encoding schemes in Fig. 4. The error bars associated with the variances of the ORTH encoding measure- Table 3 Comparison of Encoding Measures For the Numerically Optimized Encoding Schemes MV, ME, MCOND, MF and MSVD* Encoding scheme N e Total energy index (a.u) Total tensor variance REAF Condition number MV MF ME MCOND MF MV MSVD MSVD MV ME ME MV MCOND ME MSVD *The encoding sets for MF6 and MF 10 were derived by Jones et al. (18). Boldface indicates optimum encoding scheme.

7 Gradient Encoding for Diffusion Tensor MRI 775 Table 4 Comparison of Encoding Measurements For the Geometric and Analytic (AN) Encoding Schemes Encoding scheme N e Total energy index (a.u) Total tensor variance REAF Condition number ICOSA DODE ICOSA AN BUCKY ICOSA AN ICOSA AN Boldface indicates optimum encoding scheme. ments (Fig. 4a, 4b, 4c) demonstrate a greater dependence on the tensor orientation relative to the other encoding schemes. The error in the measurements of ADC eff (Fig. 4a) demonstrate very minor dependence on the encoding method except for the ORTH encoding which demonstrates a significantly higher degree of error. The ODG, MV6, ME6 and ICOSA6 encoding schemes appear to have the smallest errors in FA (Fig. 4b). Relative to these schemes, the error in FA is significantly larger for the ORTH scheme. Additionally, there is a slight increase in the average FA error with the number of encodings; however, the dependence on the orientation appears to decrease. The major eigenvector dispersion,, (Fig. 4c) again is greatest for the ORTH encoding, is minimum for all other N e 6 encodings, and increases slightly with N e. The variance associated with the tensor orientation decreases with N e. Although not shown, the MF encoding schemes should perform equivalently to the results for either the MV or the ME schemes plotted in Figs. 3 and 4. In Vivo Human Brain Measurements Figure 5 shows a set of axial FA maps obtained using the ORTH, ODG and MV6 encoding schemes. Visual inspection of these images suggests that the FA map obtained with the ORTH encoding scheme has a greater variance relative to the maps obtained using either the MV6 or the ODG schemes. The FA map obtained using the MV6 appears to be slightly less noisy than the map for ODG, although more detailed studies must be performed to verify this observation. Visual inspection of the ADC eff maps did not reveal much difference (images not shown). DISCUSSION The results of this study demonstrated that many encoding schemes are functionally equivalent for obtaining optimum sets of DT-MRI encoding directions. In this study, the TV of the ICOSA6 encoding scheme was used as the standard against which all other encoding schemes were compared. The REAF measurement, which is the ratio of TV values, demonstrated that many of the encoding schemes were equivalently optimum to the ICOSA6 although none of the schemes were better. Figure 3. The normalized tensor error,, plotted versus encoding schemes for a: prolate, b: oblate, and c: spherical tensor shapes. The error bars indicate the variance associated with the tensor orientation.

8 776 Hasan et al. Figure 4. The measured error as a function of the encoding scheme for a: ADC eff, b: FA, and c: e 1. The error in ADC and FA is defined in terms of the variance. The angular dispersion,, in degrees describes the error in e 1. The REAF comparisons indicated that there was no advantage to using more than 6 encoding directions. In general, regular polyhedra encoding schemes (e.g., the icosahedron), which by definition have uniform angular sampling, are optimum in terms of minimizing the total tensor variance. The regular polyhedra are well defined for specific N e 6, 10, 15, 21, 30, 46, 126, etc. In our studies of N e 21, the analytic (AN) gradient encoding method described by Wong and Roos (31) yielded optimum encoding sets. For any arbitrary N e 6, numerical optimization using either the MV, the MF, the MSVD or the ME minimization criteria was able to determine the optimum encoding set. In general, the ME criterion appeared to converge to the optimum solution more quickly and accurately relative to the MV and MF criteria. Only the minimum condition number criterion did not yield an optimum encoding set as defined by the tensor variance. A large condition number indicates that a given matrix is ill-conditioned; however, it is not clear whether very small condition numbers reflect the degree of mutual orthogonality in the encoding. These studies appear to indicate that the condition number is not a sensitive indicator of encoding mutual orthogonality. One of the heuristic encoding schemes investigated here, S7, was optimum, although S13 was only slightly suboptimal. The other heuristic schemes ORTH, ODG, and S10 were less than optimum. Both the ORTH and ODG schemes are commonly used for DT-MRI studies by numerous investigators. It is apparent that the MV, MSVD, MF and ME criteria should lead to the minimum TV solutions similar, but not necessarily identical, to the ICOSA polyhedra. In Appendix II, it is shown that the MV6, MSVD6, and ME6 solutions in a constrained space are identical to the ICOSA6 solution. Although not tested, the ICOSA6 would satisfy the MF6 solution as well. One interesting observation is that the ODG encoding occurs at a local maxima in this constrained space. The ODG encoding, which is probably the most commonly used encoding scheme, has a roughly 11% higher TV relative to the ICOSA6 encoding. This advantage factor has also been reported by Papadakis et al (16) and Alderman et al (11) (for chemical shielding tensor encoding). It should be noted that all of the encoding comparisons in this study were made with all other imaging parameters (i.e.,,, G d, TE, TR) constant. This is equivalent to using encoding gradients confined to a sphere of constant gradient amplitude. However, in most imaging systems, the maximum gradient amplitude is defined by a cube with the faces oriented in the logical gradient directions. This means that larger diffusion gradient amplitudes, (G d ), can be achieved by simultaneous application of more than one gradient axis. Consequently, the maximum gradient amplitude increases by 2 and 3 at the centers of the cube edges and the cube corners, respectively. The increased gradient amplitudes in these directions could be used to either increase b, or decrease,, and consequently, TE. For example, if the maximum gradient amplitude is used for ODG encoding, the echo time can be significantly reduced for the same diffusion-weighting. On the gradient system used in the imaging experiments, the TE can be reduced from 100 msec to 90 msec by using the maximum gradient strength with ODG encoding. If the T2 is 70 msec, this decrease in TE will correspond to Table 5 Estimated REAF Values Obtained From the Monte Carlo Simulations in Fig. 4* Encoding scheme Spherical Oblate Prolate ORTH ODG ICOSA ME MV MV MV MV *REAF values greater than 0.95 are highlighted in boldface.

9 Gradient Encoding for Diffusion Tensor MRI 777 Figure 5. Axial FA images for a: ORTH, b: ODG, and c: MV6 encoding schemes. a 17% increase in image SNR, which will decrease the TV. It should be noted that the maximum gradient amplitude of the ICOSA6 can likewise be increased using the formalism described in Appendix II (by removing the gradient normalization factor). Further studies are needed to evaluate the use of maximum gradient amplitudes for these encoding schemes. Another interesting observation is that the REAF did not increase for larger N e. Previous studies that have investigated N e 6 encoding schemes either did not compare optimized schemes with N e 6 and N e 6 (16,17) or did not fix the other imaging parameters such as the diffusion weighting b (18). However, the results of this study do suggest that the TV dependence on the orientation of prolate tensors (Fig. 3a and Fig. 4) decreases as N e is increased. This is likely to be the result of more uniform angular sampling of the encoding space for higher N e. When the encoding angular sampling is sparse (e.g., ICOSA6), it is likely that the TV of prolate tensors is more dependent on the orientation. However, in general, the orientation dependence on N e appears to be relatively small. Carefully controlled DT- MRI studies, as a function of N e, will be necessary to study this effect in more detail. It should be noted that a larger number of encodings will be necessary to characterize the the diffusion when a single tensor model is not valid, such as in the case when there are multiple diffusion tensor compartments in a single voxel (e.g., for crossing white matter fiber trajectories). Tuch, et al showed that it is possible to observe non-gaussian (non-tensorial) diffusion behavior when a large number of encoding directions are obtained (4). While the TV and REAF are good measures of the diffusion tensor accuracy, they are indirect indicators of how noise influences common DT-MRI measurements, such as ADC eff, FA and the eigenvector directions. Our Monte Carlo studies verified that the encoding scheme does influence the accuracy of common DT-MRI measurements. In general, the ORTH encoding scheme demonstrated a larger error (see Fig. 4) in these measurements relative to the other schemes that were evaluated. Our results also indicate that there is not a significant advantage to using more than six encoding directions, although there does appear to be a slight decrease in sensitivity to the tensor orientation with larger N e. In this study, only the effects of the encoding direction sets were considered. Obviously, other parameters such as the diffusion-weighting (amplitude and number of steps), the TR and the TE may also have a significant impact on the tensor measurement accuracy (18,38). However, the encoding analysis presented here is independent of these other factors. Preliminary DT-MRI imaging studies on a human brain were performed to compare the accuracy between several of these encoding schemes (35). In this study, the ORTH, ODG, MV6, MV12, MV24, and MV48 schemes were compared using a bootstrap Monte Carlo technique (35,36). These studies, which were performed with constant imaging times, demonstrated that the TV was lower for the MV encodings relative to either of the ORTH or the ODG encodings. The experiments did not reveal any significant change in the TV as a function of N e. In summary, this study demonstrated that the regular polyhedra encoding schemes were the optimum encoding sets for specific cases of discrete N e (e.g., ICOSA(5n 2 1), where n 1,2,3,... ). For these cases, the iterative optimization techniques, such as the MV, MF, ME and MSVD, yielded functionally equivalent encoding sets. For the N e cases where the regular polyhedra do not exist, the numerically optimized schemes resulted in optimum encoding schemes. The S7 heuristic encoding scheme and the analytic AN encoding schemes (for large N e ) also performed optimally. In this study, the ME optimization criterion appeared to converge most quickly and accurately to an optimum solution relative to the other numerical optimization techniques. The formalism of this paper is based upon the assumption that the diffusion behavior in each voxel of an image can be represented by a single self-diffusion tensor. In general, the single tensor model appears to be a good assumption for describing diffusion measurements in the human brain. However, several recent studies have demonstrated complex diffusion behavior in brain tissues, which cannot be described using a simple single-tensor model (3,4). These complexities in

10 778 Hasan et al. the diffusion behavior can arise from multiple diffusing compartments (3,4,39). If the diffusion measurements are not well described by a single tensor model, then the tensor analysis (e.g., TV, REAF, etc.) presented here is no longer valid. In these cases, it is likely that measurements at multiple diffusion-weighting values (3,40) and many directions (4,40) will be necessary to characterize the diffusion behavior. Regardless, it is likely that optimum encoding schemes will correspond to the set of encodings that most uniformly sample the encoding space, such as the optimum DT-MRI encoding schemes described in this paper. CONCLUSIONS In this study, a number of DT-MRI encoding schemes have been compared in terms of the measured tensor variance. Analytical comparisons and Monte Carlo simulations demonstrated that the icosahedron is the optimum encoding scheme for six directions. These studies also showed that equivalently optimum encoding schemes can be obtained with more than six directions, although there does not appear to be a significant advantage to using more than six directions. Detailed DT-MRI experiments need to be performed to verify these observations. ACKNOWLEDGMENTS The authors wish to thank Sean Webb for editorial assistance and John Roberts for technical discussion and references. APPENDIX I: SINGULAR VALUE DECOMPOSITION OF THE ENCODING MATRIX As described in Eq. [9], the diffusion tensor elements, d, and the set of encoded diffusion measurements, Y, are related by the encoding matrix, H. For N e 6, H isa6x 6 matrix and the diffusion tensor elements may be determined directly by d H 1 Y. (A1) However, when N e 6, the encoding matrix (N e 6) is overdetermined. This equation can be solved by using the singular value decomposition (SVD) of the encoding matrix H USV, (A2) where S is the matrix of singular values {s} and the V and U square unitarian matrices: VV I 6 6, and UU I Ne N e. (A3) In this case, H 1 can be replaced by the pseudo-inverse of H: A VS 1 U, such that Eq. [A1] can be rewritten as d AY. (A4) The singular value decomposition (SVD) of the encoding matrix H can also be used to estimate the variance in the diffusion tensor measurements assuming noise independence. By inserting Eq. [A2] into Eq. [15], it can be shown that the variance in the diffusion tensor measurements is inversely related to the square of the singular values 6 i 1 2 dˆ i 6 2 y 2. (A5) s i i 1 APPENDIX II: ANALYTICAL COMPARISON OF ENCODING SCHEMES FOR N E 6 For N e 6,the6x3encoding gradient matrix G with gradient elements ĝ g x g y g z is defined by 1 G 1 x 2 x x x x x x, (A6) where x is the independent variable. Each row of this equation corresponds to the encoding gradient, ĝ, for one measurement. The MV, ME and MSVD criteria potential functions were applied symbolically as a function of the variable x. For each of the optimization criteria, the solution minima occurred for the Golden ratio (37) and its inverse x cos 5 x (A7) (A8) as shown in Fig. A1. These minima also correspond to the N e 6 icosahedron, ICOSA6. This indicates that the optimum encoding sets obtained with the MV, ME and MSVD criteria are equivalent to each other and the icosahedron. Another interesting observation is that the ODG encoding occurs at x 1, a local maxima indicating that this encoding is suboptimal. APPENDIX III: ICOSAHEDRAL ENCODING FOR MORE THAN SIX DIRECTIONS The encoding matrices for N e 10, 15 and 21 can be derived in a similar fashion to the N e 6 case in Appendix II. The encoding schemes can be written as: G ICOSA10 C C C C 0 0 B B A A C C C C A A 0 0 B B, C C C C B B A A 0 0 (A9)

11 Gradient Encoding for Diffusion Tensor MRI 779 Figure A1. The TV, E, and SVD potential functions plotted versus x for N e 6 as defined by Eq. [A6]. The minima of each potential function are located at the same x location corresponding to the icosahedron (ICOSA6). and, a a a a c c c c b b b b G POLY b b b b a a a a c c c c, (A10) c c c c b b b b a a a a d d e e 0 0 a a a a c c c c b b b b G ICOSA d d e e b b b b a a a a c c c c e e 0 0 d d c c c c b b b b a a a a (A11) where a , b , c 0.50, d , e , A , B , and C REFERENCES 1. Le Bihan D. Molecular diffusion, tissue microdynamics and microstructure of tissues from diffusion-weighted images. NMR Biomed 1995;8: Basser PJ. Inferring microstructural features and the physiological state of tissues from diffusion-weighted images. NMR Biomed 1995; 8: Mulkern RV, Gudbjartsson H, Westin CF, Zengingonul HP, Gartner W, Guttmann CR, Robertson RL, Kyriakos W, Schwartz R, Holtzman D, Jolesz FA, Maier SE. Multi-component apparent diffusion coefficients in human brain. NMR Biomed. 1999;12: Tuch DS, Weiskoff RM, Belliveau JW, Wedeen VJ. High angular resolution diffusion imaging of the human brain. In: Proceedings of the 7th Annual Meeting of ISMRM, Philadelphia, p Shrager RI, Basser PJ. Anisotropically weighted MRI. Magn Reson Med 1998;40: Armitage PA. Quantifying likely errors arising when using orthogonal or tetrahedral encoding schemes to sample diffusion anisotropy. In: Proceedings of the 7th Annual Meeting of ISMRM, Philadelphia, p Conturo TE, McKinstry RC, Aronovitz JA, Neil JJ. Diffusion MRI: precision, accuracy and flow effects. NMR Biomed 1995;8: Ahn CB, Lee SY, Nalcioglu, Cho ZH. An improved nuclear magnetic resonance diffusion coefficient imaging method using an optimized pulse sequence. Med Phys 1986;13: Nye JF. Physical properties of crystals. Oxford: Clarendon Press; p Peebles PZ. Probability, random variables and signal principles. New York: McGraw Hill; p Alderman DW, Sherwood M, Grant D. Two-dimensional chemicalshift tensor correlation spectroscopy: analysis of sensitivity and optimal measurement directions. J Magn Reson 1990;86: Basser PJ, Pierpaoli C. A simplified method to measure the diffusion tensor from seven MR images. Magn Reson Med 1998;39: Conturo TE, McKinstry RC, Akbudak E, Robinson BH. Encoding of anisotropic diffusion with tetrahedral gradients: a general mathematical diffusion formalism and experimental results. Magn Reson Med 1996;35: Shimony JS, McKinstry RC, Akbudak E, et al. Quantitative diffusion-tensor anisotropy brain MR imaging: normative human data and anatomic analysis. Radiology 1999;212: Skare S, Nordell B. Decahedral gradient encoding for increased accuracy in the estimation of diffusion anisotropy. In: Proceedings of the 7th Annual Meeting of ISMRM, Philadelphia, p Papadakis NG, Xing D, Huang CL, Hall LD, Carpenter TA. A comparative study of acquisition schemes for diffusion tensor imaging using MRI. J Magn Reson 1999;137: Papadakis NG, Xing D, Houston GC, Smith JM, Smith MI, James MF, Parsons AA, Huang CL, Hall LD, Carpenter TA. A study of rotationally invariant and symmetric indices of diffusion anisotropy. Magn Reson Imaging 1999;17: Jones DK, Horsfield MA, Simmons A. Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging. Magn Reson Med 1999;42: King RB. Applications of graph theory and topology in inorganic cluster and coordination chemistry. Boca Raton, Florida: CRC Press; p Skare S, Hedehus M, Li TQ. Characteristics and stability of different gradient encoding schemes. In: Proceedings of 8th Annual Meeting of ISMRM, Denver, p Meschkowski H. Unsolved and unsolvable problems in geometry. New York: Frederick Ungar; p 1 31.

12 780 Hasan et al. 22. Tegmark M. An icosahedron-based method for pixelizing the celestial sphere. ApJ Letters 1996;470:L81 (also see ias.edu/ max/icosahedron.html). 23. Sibley TQ. The geometric viewpoint. New York: Addison-Wesley; p Fuller RB, Applewhite EJ. Synergetics: explorations in the geometry of thinking. New York: Macmillan; p Muthupallai R, Holder CA, Song AW, Dixon WT. Navigator-aided, multishot EPI diffusion images of brain with complete orientation and anisotropic information. In: Proceedings of the 7th Annual Meeting of ISMRM, Philadelphia, p Shutler P. Magnetostatics and the volumes of platonic solids. Int J Math Edu Sci Technol 1996;27: Williams R. The geometrical foundation of natural structure. New York: Dover Publications Inc.; p Coxeter HSM. Virus macromolecules and geodesic domes in a spectrum of mathematics. In: Butcher JC, editor. New Zealand: Auckland University Press; p Kenner H. Geodesic math and how to use it. Berkley: University of California Press; p Chung F, Sternberg S. Mathematics and the buckyball. American Scientist 1993;81: Wong STS, Roos MS. A strategy for sampling on a sphere applied to 3D selective RF pulse design. Magn Reson Med 1994;32: Press WH, Flannery BP, Teukolsky SA, Vettering WT. Numerical recipes in C. 2nd Edition. Cambridge: Cambridge University Press; p Pierpaoli C, Basser PJ. Toward a quantitative assessment of diffusion anisotropy. Magn Reson Med 1996;36: Bastin ME, Armitage PA, Marshall I. A theoretical study of the effect of experimental noise on the measurement of anisotropy in diffusion imaging. Magn Reson Imaging 1998;16: Hasan KM, Parker DL, Alexander AL. Bootstrap analysis of DT-MRI encoding techniques. In: Proceedings of the 8th Annual Meeting of ISMRM, Denver, p Pajevic S, Basser PJ. Non-parametric statistical analysis of diffusion tensor MRI data using the bootstrap method. In: Proceedings of the 7th Annual Meeting of ISMRM, Philadelphia, p Dunlap RA. The golden ratio and Fibonacci numbers. New Jersey: World Scientific; p Xing D, Papadakis NG, Huang CLH, Lee VM, Carpenter TA, Hall LD. Optimized diffusion weighting for measurement of apparent diffusion coefficient (ADC) in human brain. Magn Reson Imaging 1997; 15: Alexander AL, Hasan K, Lazar M, Tsuruda JS, Parker DL. Analysis of partial volume effects in diffusion-tensor MRI. In: Proceedings of the 8th Annual Meeting of ISMRM, Denver, p Wedeen VJ, Reese TG, Tuch DS, Weigel MR, Dou JG, Weiskoff RM, Chessler D. Mapping fiber orientation spectra in cerebral white matter with Fourier-transform diffusion MRI. In: Proceedings of the 8th Annual Meeting of ISMRM, Denver, p 82.