GRAPPA Operator for Wider Radial Bands (GROWL) with Optimally Regularized Self-Calibration

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1 GRAPPA Operator for Wider Radial Bands (GROWL) with Optimally Regularized Self-Calibration Wei Lin,* Feng Huang, Yu Li, and Arne Reykowski Magnetic Resonance in Medicine 64: (2010) A self-calibrated parallel imaging reconstruction method is proposed for azimuthally undersampled radial dataset. A generalized auto-calibrating partially parallel acquisition (GRAPPA) operator is used to widen each radial view into a band consisting of several parallel lines, followed by a standard regridding procedure. Self-calibration is achieved by regridding the central k-space region, where Nyquist criterion is satisfied, to a rotated Cartesian grid. During the calibration process, an optimal Tikhonov regularization factor is introduced to reduce the error caused by the small k-space area of the self-calibration region. The method was applied to phantom and in vivo datasets acquired with an eight-element coil array, using radial views with 256 readout samples. When compared with previous radial parallel imaging techniques, GRAPPA operator for wider radial bands (GROWL) provides a significant speed advantage since calibration is carried out using the fully sampled k-space center. A further advantage of GROWL is its applicability to arbitrary-view angle ordering schemes. Magn Reson Med 64: , VC 2010 Wiley-Liss, Inc. Key words: projection reconstruction; partial parallel imaging; GRAPPA; self-calibration; regularization In recent years, there has been an increased interest in radial MRI due to its potential for highly accelerated dynamic imaging (1,2), advantages in motion artifacts reduction (3,4), and ability to achieve ultrashort echo times (5). In many applications, radial images are acquired using a phased-array coil, providing the opportunity for further acceleration using partial parallel imaging techniques. General-purpose parallel imaging reconstruction methods for non-cartesian datasets have been previously proposed and applied to undersampled radial MRI, including the iterative conjugate gradient sensitivity encoding (SENSE) (6,7) and parallel MRI with adaptive radius in k-space (PARS) (8). Streaking artifacts resulting from azimuthal undersampling can be significantly reduced with these parallel imaging reconstruction methods. However, the application of these techniques is limited by long reconstruction time and a dependence on sensitivity map measurements. Several parallel imaging techniques have been developed for radial MRI, exploiting the k-space locality principle that was first presented in the PARS method. In radial generalized auto-calibrating partially parallel acquisition (GRAPPA) (9), k-space is divided into many Invivo Corporation, Philips Healthcare, Gainesville, Florida, USA. *Correspondence to: Wei Lin, Ph.D., Invivo Corporation, Philips Healthcare, 3545 SW 47th Ave, Gainesville, FL wei.lin2@philips.com Received 25 June 2009; revised 18 February 2010; accepted 19 February DOI /mrm Published online 25 May 2010 in Wiley Online Library (wileyonlinelibrary.com). VC 2010 Wiley-Liss, Inc. 757 concentric rings (segments), and missing data within each segment are estimated using a relative shift operator, i.e., a GRAPPA convolution kernel. The advantage of radial GRAPPA over conjugate gradient SENSE is a shorter reconstruction time since there is no need to solve a large system of linear equations. When compared with PARS, radial GRAPPA requires solving a smaller number of linear equations. One limitation of the radial GRAPPA technique when compared with its Cartesian counterpart is the need for a separate full k-space for the calibration of convolution kernels. This limits the usefulness of radial GRAPPA in some static imaging applications, such as ultrashort echo time imaging (5). The need for an extra sensitivity reference scan also makes the technique more susceptible to motion-induced coil sensitivity discrepancy between the reference and the accelerated scan. This limitation has been partially addressed by later work generating pseudo full k-space using image-support constraints (10,11). A second drawback for radial GRAPPA that is more difficult to overcome, as will be demonstrated later in this work, is some residual blurring artifact at higher acceleration factors. A k-space implementation of PARS reconstruction method was also introduced that is self-calibrated and uses interpolated weights to reduce the computation cost (12). More recently, GRAPPA operator gridding (GROG) has been proposed to overcome some difficulties associated with radial GRAPPA (13 15). The data on the Cartesian grid, near the acquired radial trajectory, are first estimated using GROG, followed by conventional Cartesian GRAPPA with various kernel shapes or conjugate-gradient-based iterative reconstruction. In this work, an alternative self-calibrated parallel imaging technique, GRAPPA operator for wider radial band (GROWL), is proposed for azimuthally undersampled radial datasets. This technique is based on expanding each radial readout line into a wider band using GRAPPA relative shift operators. The resulting k-space sampling pattern is similar to that of periodically rotated overlapping parallel lines with enhanced reconstruction technique originally proposed for motion compensation (16). As will be demonstrated, image reconstructed from such a k-space sampling pattern is less susceptible to either streaking or blurring artifacts. The central fully sampled k-space region is used for the selfcalibration of GRAPPA operators. An optimal Tikhonov factor is introduced to further reduce the GRAPPA operator error when the calibration region is small. When compared with previous radial parallel imaging method, GROWL provides a significant speed advantage due to a small GRAPPA operator kernel size and the small

2 758 Lin et al. number of operators required. A further advantage of GROWL over radial GRAPPA is its applicability to any radial view-angle ordering such as the golden-angle scheme (17), which is beneficial in dynamic and motion compensation applications. A detailed description of the method is first provided, followed by evaluation in both phantom and in vivo experiments. MATERIALS AND METHODS In an azimuthally undersampled radial dataset, the Nyquist criterion is only satisfied within a central k-space circle, which is denoted as the Nyquist circle in this study. The initial Nyquist circle has a radius of r 0 ¼ N/(p field of view [FOV]), where N is the number of radial readout lines (Fig. 1a). In the proposed GROWL method, GRAPPA relative shift operator is used to expand each radial line into a k-space segment consisting of m parallel k-space lines (Fig. 1b). As a result, the radius of the Nyquist circle increases to r ¼ mr 0, and therefore the streaking artifacts caused by azimuthal undersampling will be reduced. As the data within the initial Nyquist circle are fully sampled, they can be used for self-calibration. The data within this region are first regridded from the acquired radial lines onto the Cartesian grid. For each radial readout line, a shearing method (18) is used to rotate the regridded Cartesian data set to align with the readout line. The GRAPPA relative shift operator weights can then be computed from this calibration region and used to expand each readout line into a wider band (Fig. 1c). There are two sources of reconstruction error in the GROWL method: those introduced by the GRAPPA relative shift operator and those introduced by the regridding of non-cartesian (periodically rotated overlapping parallel lines with enhanced reconstruction like) k-space dataset. The error caused by the GRAPPA operator can be further divided into the approximation error, which depends on the kernel shape and size, and noise amplification error, which depends on the data noise level (19). In this section, an optimal Tikhonov regularization factor is first introduced for the calibration of the GRAPPA relative shift operator. Then the error caused by the GRAPPA operator is examined on a Cartesian grid, followed by the comparison of the GROWL reconstruction results with the convolution regridding, GROG, and radial GRAPPA for undersampled radial datasets. Optimal Regularization for GRAPPA Calibration In this work, an optimal Tikhonov regularization factor is introduced to minimize the error introduced by GRAPPA relative shift operator since the available autocalibration signal (ACS) only occupies a small portion of the k-space for highly undersampled radial datasets. When compared with previous methods (20,21) to deal with a small calibration region, the proposed method is much less computationally intensive. Let t and S be the vector of target and the matrix of source data points (open and gray solid circles shown in Fig. 1c) from multiple k-space locations and coil channels. Let w be the FIG. 1. The basic principle of GROWL. a: For an azimuthal undersampled radial dataset, the Nyquist criteria are only satisfied in the small circle ( the Nyquist circle ) near the center of the k- space. b: After the GRAPPA relative shift operators are used to expand each radial readout lines into a wider band, the Nyquist circle is enlarged. c: For each radial line, calibration is first performed in the initial Nyquist circle (the large gray circle), followed by GRAPPA relative shift operations from source (gray) points to target (white) points. weight vector for the GRAPPA relative shift operator. During the calibration process, weight vector w is determined by solving the overdetermined linear equation t ACS ¼ S ACS w: Here the subscript ACS indicates that both target and source data points are collected in the ACS region. The standard method to solve Eq. 1 is the linear least square approach, which seeks to minimize the residual error: ½1Š w 0 ¼ arg min w ðkt ACS S ACS wk 2 Þ: ½2Š Here kk is the L 2 norm. The optimal weight vector, however, should minimize the error over the entire k-space: w opt ¼ arg min w ðkt E S E wk 2 Þ: ½3Š Here the subscript E indicates that both target and source data points are collected in the entire k-space. In other words, the ideal weight vector is the solution to following equation: t E ¼ S E w: In reality, only S E is known and t E is unknown. Therefore, Eq. 4 cannot be solved. One key observation is that Eqs. 1 and 4 will have different condition numbers, which is defined as the ratio between the maximal and minimal singular values of the source data matrix S using the singular value decomposition. The condition number measures the stability in the ½4Š

3 GRAPPA Operator for Wider Radial Bands 759 GRAPPA operation by giving the maximum ratio of the relative error in weight w divided by the relative error in the target data t. If each GRAPPA operator kernel contains N S source data points (from multiple k-space location and coil channels) and the ACS region contains N ACS k-space points, then the size of the source data matrix S is N ACS N S. Typically, N ACS N S. The maximal singular value of S is determined by coil sensitivity profiles and therefore is essentially independent of N ACS. Assuming that each channel of MR data contains independent gaussian noise with a standard deviation (SD) of s, it can be shown that the minimal singular value s min of a random N ACS N S (N ACS N S ) rectangular matrix is approximately (22): p s min ffiffiffiffiffiffiffiffiffiffiffi s: ½5Š N ACS Therefore, a smaller ACS region results in a lower s min value and a higher condition number. On the other hand, a higher noise level in the data helps to stabilize the system by lowering the condition number (23). A well-established method to deal with ill-conditioned linear systems is the Tikhonov regularization (24,25), which solves Eq. 1 by minimizing: w opt ¼ arg min w kt ACS S ACS wk 2 þl 2 kwk 2 : ½6Š Here l is known as the Tikhonov factor. The solution to Eq. 6 is: w opt ¼ Xn j¼1 s j s 2 j þ l 2uH j tv j : Here u j, v j, and s j are the left singular vectors, right singular vectors, and singular values of S, respectively, generated by singular value decomposition, with singular vectors and singular values indexed by j. The hypothesis used in this work is that optimal Tikhonov factor for the GRAPPA operator calibration can be determined from the full k-space calibration equation (Eq. 4) in order to minimize the residual error root mean square error (RMSE) computed from all the k-space data. It should be noted that the error minimized here contains both contributions from the approximation error, which depends on the kernel shape and size, and the noise amplification error. The goal here is to strike a balance between these two error sources and to minimize the overall error. If the entire k-space contains N E data points, we have determined empirically using experimental data that the optimal Tikhonov factor l opt is approximately the minimal singular value of the full k-space calibration equation (Eq. 4): p l opt ffiffiffiffiffiffi N Es ½8Š Evaluation of GRAPPA Relative Shift Operator Simulations were carried out to examine the error introduced by the GRAPPA relative shift operator and the effectiveness of the proposed regularization scheme. For this purpose, a noise-free T 1 -weighted Cartesian brain MR ½7Š dataset was downloaded from a simulated brain database ( The complex sensitivity profile of a head coil array with eight coil elements equally spaced around a cylinder was computed using an analytic Biot-Savart integration. The k-space data for each individual channel were then derived using the Fourier transform. To simulate different noise levels, gaussian distributed random noise was added to both real and imaginary components of each channel of k-space data, resulting in a noise SD in the range of 0.1%-5.0% of signal intensity of the white matter (the dominant tissue) in the final images reconstructed using the square-root-ofsum-of-square channel combination. To evaluate the error introduced by GRAPPA relative shift operators, the entire set of Cartesian k-space (source k-space) was shifted by operators to generate another set of Cartesian k-space (target k-space). The difference between two k-space datasets was evaluated by the RMSE of their corresponding images using the square-root-ofsum-of-square reconstruction. This process was repeated for operators with different kernel shapes by changing both the number of source data points N x along the readout direction and the distance shifted perpendicular to the readout direction Dk y. GROWL bears some similarity to a k-space implementation of PARS method (12). In GROWL, GRAPPA extrapolation operator use source points from a single radial line. In the modified PARS method, sources points from multiple radial lines were used in the GRAPPA interpolation operator. The errors of these two types of GRAPPA operators were compared at different noise levels and kernel sizes. To evaluate the effectiveness of the proposed regularization method, ACS regions with sizes in the range of to were used for the calibration of the GRAPPA weights, which corresponds to the available fully sampled central k-space region when radial lines are acquired. GROWL Reconstruction The performance of GROWL reconstruction and its dependence on parameter selections at different noise levels were first evaluated using simulated radial datasets. The noise-free brain dataset previously used for the evaluation of the GRAPPA operator was used for this purpose. The Cartesian dataset was first inverse regridded to generate a 384-view radial dataset, which approximately satisfies the Nyquist criteria in the entire k-space for a 256-point readout matrix size. Gaussian-distributed random noise with different SD in the range of 0.1%-5.0% was then added to both real and imaginary components of each channel and every radial data point. Undersampled 64- and 32-view data sets were subsequently generated by extracting every 6th and 12th radial k-space line from the full dataset. To examine the effect of width of radial band (N y ), GROWL reconstructions were carried out for N y ¼ 3, 5, 7, and 9. To examine the effect of regularization, Tikhonov factors in the range of times of the proposed optimal value (Eq. 8) were used for GRAPPA operator calibration. The performance of the optimal GROWL algorithm was compared with convolution gridding, GROGþSENSE, and radial GRAPPA using both 64- and 32-view undersampled

4 760 Lin et al. radial datasets, with noise standard deviations ranging from %. In our implementation of GROGþSENSE (26), after radial data are used to estimate data on the nearest Cartesian grid point, total variation is used as the sparsity constraint term for the subsequent iterative SENSE reconstruction. For radial GRAPPA, k-space was divided into 16 concentric rings, with separate calibration and GRAPPA interpolations. The size of the convolution kernel is 9 (readout points) 4 (radial views). The fully sampled, 384-view k-space was used for the calibration of radial GRAPPA kernels. The iterative method first proposed by Pipe and Menon (27) and Pipe (28) was used to weight data prior to regridding using a Kaiser-Bessel kernel with a width of 4. The RMSE of the square-root-ofsum-of-square reconstruction for all algorithms was computed for comparison. The performance of GROWL with actual radial datasets was examined and compared with radial GRAPPA. In vivo brain datasets were acquired from a healthy volunteer on a 3.0-T Achieva scanner (Philips, Best, The Netherlands) with an eight-channel head coil array (In vivo, Gainesville, FL), using a multislice two-dimensional radial gradient echo sequence. Scan parameters were FOV mm 2, slice thickness 5mm, matrix size 256 (readout) 256 (view no.), pulse repetition time/echo time ¼ 250/4.2 ms, flip angle ¼ 80. Two view-angle ordering schemes were used: a linear scheme with uniform view-angle distribution and a golden-angle scheme where adjacent views differs by The golden-angle view-ordering scheme results in a nearly uniform view-angle distribution for arbitrary view numbers. Every 4th and 8th radial readout line was extracted from the first dataset to generate undersampled 64- and 32-view data, while the first 55 views from the second dataset were used for the reconstruction of undersampled golden-angle images. To further demonstrate the application of GROWL, a three-dimensional (3D) radial dataset was generated from the T 1 -weighted brain images used in the previous simulation. The sensitivity profile of an eight-channel coil was simulated using one-sided gaussian functions. For each of the eight channels, a total of 2800 radial projections were generated from the image volume, using the inverse regridding. The tips of the radial projections form a two-dimensional spiral on the spherical surface, from the pole to the equator. The amount of available data represents a reduction factor of R ¼ 23.4 from a fully sampled 3D Cartesian trajectory. GROWL was then applied to expand each radial line into a 3D rod. Each 3D rod consists of the original acquired radial line, two additional parallel lines along azimuthal direction, and two additional parallel lines along zenithal direction. The self-calibration of the GRAPPA operator is performed using the central fully sampled k-space region, following the inverse regridding onto a rotated Cartesian grid. RESULTS GRAPPA Relative Shift Operator Figure 2 shows the performance of GRAPPA relative shift operators with various kernel shapes at different signal-to-noise ratio (SNR) conditions. Figure 2a shows the T 1 -weighted brain image used in this and all subsequent simulation experiments. Figure 2b shows two factors that determine the geometry of a GRAPPA relative shift operator: the number of source points (N x ) along the readout direction and the distance of the shifted points from the source readout lines (Dk y ). Figure 2c,d shows the dependence of RMSE on the kernel geometry for both noise-free and noisy data. The approximation error decreases when more source points are included in the relative shift operator. This decrease is most significant when N x increases from 1 (which is the case for GROG) to 3 but levels off at around N x ¼ 9. Therefore N x ¼ 9 is used in the current implementation of GROWL algorithm to achieve a reasonable performance/speed balance. The GRAPPA operator error increases with Dk y, when the target point moves farther away from the acquired readout line. The comparison between noisefree (Fig. 2c) and noisy (Fig. 2d) data shows that the contribution of noise amplification becomes more significant for large Dk y values. Figure 2e shows the SNR performance of the GRAPPA relative shift operator. For the line closest to the acquired readout line (Dk y ¼ 0.5/FOV), operator errors remain low even for the noisy dataset. For the line far from the acquired readout line (e.g., Dk y ¼ 2.0/FOV), however, operator error could become quite high when data contain a high level of noise, indicating a high level of noise amplification. It should be pointed out that the curves shown in Fig. 2c-e are RMSEs introduced by individual GRAPPA relative shift operators instead of the overall GROWL reconstruction algorithm. In the GROWL reconstruction, many GRAPPA operators are used to estimate several parallel lines from a single acquired radial line. Therefore, GROWL reconstruction error is a weighted sum of different GRAPPA relative shift operators with different Dk y values. Figure 2f further compares the error for two different types of GRAPPA operators, those using source points from a single radial readout (solid, GROWL) and those using source points from multiple readouts (dashed, modified PARS). For the modified PARS method, the distance between two parallel source lines is 6/FOV, which is approximately the gap between adjacent radial lines at the outermost k-space for a 64-view radial dataset with 256 readout points. Results show that the GRAPPA extrapolation operator yields error levels similar to those of the interpolation operator for target data points at distances Dk y ¼ 1-2/FOV. At Dk y ¼ 3/FOV, extrapolation operators gives higher error than interpolation operators. For both types of GRAPPA operator, noise amplification is significant for large Dk y values. Figure 3 shows the results regarding the proposed regularization scheme. The condition number of the GRAPPA fitting equation increases dramatically when a smaller ACS region is used for calibration, as shown in Fig. 3a. As a result, without regularization the GRAPPA operator error become very high when a small ACS region is used (Fig. 3b). With the proposed regularization scheme, however, the GRAPPA operator errors are reduced and become independent of the area of the ACS region (Fig. 3c). Figure 3d further shows that the minimal GRAPPA operator error is always achieved when the

5 GRAPPA Operator for Wider Radial Bands 761 FIG. 2. Results from the simulation experiment evaluating the performance of GRAPPA relative shift operator. a: Noise-free digital brain phantom image used for the simulation. b: The kernel shape of the GRAPPA relative shift operator, showing both the number of source readout points (N x ) and the distance Dk y between the target point and the source readout line. c,d: GRAPPA operator error versus N x for noise-free (c) and noisy data (d). The four curves, from top to bottom, correspond to Dk y ¼ 2/FOV, 1.5/FOV, 1.0/FOV and 0.5/FOV. In (d), the SD of gaussian noise is 1.0% of the signal intensity of white matter tissue. e: GRAPPA operator error versus noise SD. f: Error comparison of GRAPPA extrapolation (solid) and interpolation (dashed) operators. For GRAPPA interpolation operators, the distance between two parallel source lines is 6/FOV. proposed optimal Tikhonov factor l opt (Eq. 8) is used for this dataset. These results demonstrate the effectiveness of the proposed regularization method. GROWL Reconstruction Figure 4 shows the impact of two input parameters on the GROWL performance. Figure 4a shows that there is an optimal number of parallel lines (N y ) within each radial band that gives a minimal RMSE. While a higher N y improves the k-space coverage, the additional parallel lines will also introduce errors that increase with the distance from the acquired readout lines (see Fig. 2c,d). The optimal value for N y is a function of the noise level and is higher for high SNR data (N opt y ¼ 9 when s ¼ 0.1%) than for noisy data (N opt y ¼ 5 when s ¼ 5.0%). Figure 4b demonstrates the effectiveness of the proposed regularization method. At all three noise levels, the reconstruction errors were minimized when the optimal Tikhonov factor as proposed in Eq. 8 was used. Figure 5 shows simulation results comparing GROWL reconstruction with convolution regridding, GROG, and radial GRAPPA reconstructions, using 32 radial lines to reconstruct a image. For the convolution regridding (data weighted with a RamLak filter), significant streaking artifact is present in the reconstructed image. However, the image resolution is preserved reasonably well. Combining GROG and sparsity-constrained SENSE reconstruction significantly improves the image quality, but some residual streaks still exist. Radial GRAPPA generates a k-space pattern that looks quite similar to the reference fully sampled k-space. However, some image blurring can be observed, particularly on the zoomed image. GROWL reconstruction widens each radial line into a wider band, therefore enlarging fully sampled k-space region. Data in some peripheral k-space region are still not estimated due to the limited width of the expanded radial bands. From the zoomed image, it can be clearly seen that a better image resolution is provided by GROWL when compared with radial GRAPPA, while the noise level is also higher in the GROWL image.

6 762 Lin et al. FIG. 3. Results from the simulation experiment evaluating the performance of the proposed regularization scheme for GRAPPA operator. In this experiment, the ACS is a square centering on the k-space origin with different sidelengths. The SD of gaussian noise is 1.0% of the signal intensity of white matter tissue. a: Condition numbers for the GRAPPA calibration equation (Eq. 1) versus ACS region side-length. b: GRAPPA operator error versus ACS region side-length when no regularization is applied. c: GRAPPA operator error versus ACS region side length when the proposed regularization method is applied. d: GRAPPA operator errors when different Tikhonov factors are used for the regularization during the calibration. l opt is determined using Eq. 8. The ACS region size is Figure 6 compares the RMSE of the four algorithms for both 32- and 64-view datasets at various noise levels (SD 0.1%-5.0%). RMSEs for GROWL reconstructions are much lower than those of regridding and GROGþSENSE. When compared with radial GRAPPA, RMSE for GROWL is essential identical at low noise levels (s 1.0%) but slightly higher when the data noise level becomes higher (s 2.5%). However, as shown in Fig. 5, GROWL image provides a higher image resolution. Note that GROWL did not use any additional calibration data, while radial GRAPPA used a fully sampled k-space for the calibration purpose, which is typically not available from a single accelerated scan. Figure 7 shows the results comparing GROWL with regridding and radial GRAPPA using actual in vivo radial data set. The top row shows images reconstructed with conventional regridding using 256, 64, and 32 radial views. GROWL and radial GRAPPA reconstruction, with 64 and 32 radial readout lines, is shown in middle and bottom rows. The SNR in the white matter region is around 20 for the 256-view radial dataset, as determined by the intensity ratio to a signal-free background region. Convolution regridding introduces significant streaking artifacts that are particularly severe when only 32 radial lines were used for image reconstruction. Both radial GRAPPA and GROWL algorithms were able to remove most streaking artifacts. The zoomed images of the 32- view reconstruction reveal the different characteristics of the two algorithms. The radial GRAPPA image has less noise, but some image details are lost due to image FIG. 4. Results from the simulation experiment evaluating GROWL performance using 32-view radial datasets. a: RMSE of GROWL images versus the number of parallel lines within each radial band (N y ), when different levels of noise were added to the data. b: RMSE of GROWL images when different Tikhonov factors are used for regularized calibration. l opt is determined using Eq. 8. Optimal N y values as determined from the curves shown in (a) were used in all cases.

7 GRAPPA Operator for Wider Radial Bands 763 FIG. 5. Simulation results comparing four different reconstruction methods: convolution regridding, GROG, and SENSE reconstruction, radial GRAPPA, and GROWL. The left column is the reference Cartesian data, while other columns use 32-view radial data. The noise SD is 1.0% of the signal intensity of white matter tissue. Top row: k-space of the 1st channel signal shown in the logarithmic scale. Middle row: Square-root-of-sum-of-square reconstruction (except GROGþSENSE). Bottom row: Zoomed-in images. blurring. In contrast, image details are preserved better in the GROWL image, but the noise level is higher. For these datasets, the SNR limit as defined by the SNR on the fully sampled image divided by the square root of the acceleration factor becomes fairly low (about 5-7). Figure 8 shows in vivo brain images when radial views were collected using a golden-angle ordering scheme. When only 55 views are used for image reconstruction, conventional regridding generates an image with severe streaking artifacts, while the GROWL image is comparable with the 256-view reference image. Due to the variable angular spacing between adjacent views and pseudorandom ordering of successive views, the radial GRAPPA method cannot be used to reconstruct such a dataset. Figure 9 shows images from the 3D radial dataset reconstructed using two regridding methods and GROWL. When the Voronoi diagram is used for density compensation prior to the regridding, significant streaks are present in the reconstructed images (first row) due to data undersampling. When a small kernel width is used to compute the Pipe s density compensation function (28), the streaks can be significantly reduced at a cost of reduced image resolution (second row). In comparison, GROWL images (third row) reduce streaks while preserving the image resolution. The entire GROWL reconstruction time is 10 min for the entire image volume. DISCUSSION In this work, a self-calibrated parallel imaging reconstruction method is proposed for azimuthally undersampled radial datasets. Each radial view is widened FIG. 6. Simulation results comparing RMSE at different noise levels for four different reconstruction methods: convolution regridding, GROGþSENSE, radial GRAPPA, and GROWL, using undersampled radial datasets; (a) 32-view datasets; (b) 64-view datasets.

8 764 Lin et al. FIG. 7. Reconstruction results using an in vivo radial brain dataset with the uniform view angle spacing. Top row: regridding reconstruction. Middle row: GROWL images. Bottom row: Radial GRAPPA images. The number of radial views used for reconstruction is shown in the right upper corner of each image. into a band consisting of several parallel lines, using GRAPPA relative shift operator and therefore enlarging the fully sampled k-space region. There are two adjustable parameters in GROWL: the number of source points N x for GRAPPA relative shift operator and the number of parallel lines within each widened radial band N y. As shown in Fig. 2c, the GRAPPA operator error is reduced with a larger N x (no. of source points), but the rate of error reduction gradually levels off. However, the computation cost increases with the kernel size. Therefore, five to nine sources points can be used, depending on the tradeoff between accuracy and speed. A larger N y value (no. of parallel lines within each radial band) will result in a larger Nyquist circle and therefore a further reduction in streaking artifacts. However, with the increasing N y values, the outer parallel lines within each radial band will have larger Dk y (distance between the target and source points) values. As shown in Fig. 2d, the error from the GRAPPA relative shift operator could become very high for large Dk y values, when the noise level is high. As a result, there is an optimal N y value that gives the minimum reconstruction error, depending on both the number of acquired radial views and the image SNR (as demonstrated in Fig. 4a). For acceleration factors of 4-8, five to nine lines can be included in each radial band, corresponding to a Dk y ¼ 1/FOV to 2/FOV from the acquired readout line and the outermost shifted line. A second contribution of this work is a method to determine the optimal Tikhonov factor (Eq. 8) to FIG. 8. Reconstruction results using an in vivo radial brain dataset with the golden-angle view angle ordering. Left: Regridding reconstruction with 256 views. Center: Regridding reconstruction with 55 views. Right: GROWL reconstruction with 55 views.

9 GRAPPA Operator for Wider Radial Bands 765 FIG. 9. 3D reconstruction results with 2800 radial views (image matrix ). Top row: Regridding using the Voronoi diagram for density compensation. Middle row: Regridding using the Pipe s (28) density compensation function with a small kernel width. Bottom row: GROWL. From left to right: axial, coronal, and sagittal images. regularize the GRAPPA calibration process. For the datasets examined in this work, it is shown that minimal errors are achieved with the proposed optimal regularization factor for all ACS region sizes and noise levels (Figs. 3d, 4b). Future work will examine whether this method can be applied to other applications that require the calibration of GRAPPA weights. GROG (13 15) provides another approach to reconstruct undersampled radial datasets. The key difference between GROG and GROWL is the kernel size of the GRAPPA relative shift operator. In the GROG-based method, only one source point is used from each coil element to estimate the target data point. In contrast, GROWL uses multiple (up to nine in this work) source points for the relative shift operator. Figure 2c shows that GRAPPA operator errors are large when the target data point moves farther away from the acquired readout line, and errors can be reduced dramatically when more than one source point is used. Therefore, in GROG, the GRAPPA operator is only used to estimate data points that are no more than Dk y ¼ 0.5/FOV from the acquired radial lines. Because of the use of multiple source points, however, GROWL can estimate data points farther away from the source points more accurately and therefore can generate a larger coverage in k-space, as demonstrated in Fig. 5. When compared with radial GRAPPA where the newly estimated data remain on radial trajectories, the k-space sampling pattern achieved using GROWL is similar to the periodically rotated overlapping parallel lines with enhanced reconstruction technique previously proposed for motion compensation. Although the k-space coverage of GROWL may not be as large as radial GRAPPA (as shown in the top row of Fig. 5), images presented show that the GROWL image has a higher spatial resolution (as shown in the bottom row of Fig. 5). However, the GROWL image does also show a higher level of noise. The overall RMSE is similar at various noise levels for both 32-view and 64-view dataset (Fig. 6). Compared with radial GRAPPA technique, GROWL has two other advantages: First, GROWL is self-calibrated. By rotating the central fully sampled k-space region, calibration data are provided for every readout line. The second advantage of GROWL is its ability to reconstruct data with arbitrary view-angle spacing and ordering schemes, which is useful for dynamic imaging and motion-compensation applications. Conventional radial GRAPPA requires equal data spacing along the azimuthal direction. Hence, it is difficult to use for radial imaging with arbitrary view-angle spacing. GROWL is also related to the modified PARS method (12). In GROWL, GRAPPA extrapolation operator uses source points from a single radial line. In the modified

10 766 Lin et al. PARS method, source points from multiple radial lines were used in the GRAPPA interpolation operator. As shown in Fig. 2f, GRAPPA extrapolation operator yields error levels similar to those of the interpolation operator, except for target data points farther away (Dk y 3/FOV for the eight-channel coil) from the acquired radial lines. Therefore GROWL may generate higher errors then the modified PARS method at higher acceleration factors. However, GROWL provides a significant speed advantage due to two factors. First, the number of weight vectors is limited to the number of acquired radial lines. Second, the smaller number of source points further reduces the time required for matrix inversion. The acceleration factor achievable with the GROWL reconstruction depends on the number of coil elements and noise levels in the data. Our results show that images with decent qualities can be reconstructed with 32 radial views, using a commercial eight-channel head coil at a low SNR condition (about 5-7). Pipe (28) previously showed that for an undersampled dataset, the tradeoff between the image resolution and reconstruction error can be adjusted by the choice of the convolution kernel used to compute the data-weighting factor. However, our tests with GROWL datasets showed that the size of the convolution kernel for regridding has very little effect on the final image quality and the total reconstruction error. This is most probably due to the fact that a large central portion of the k-space is already fully sampled after the application of the GRAPPA relative shift operator. The computational cost of GROWL is very low when compared with most existing parallel imaging methods due to the small calibration region and the small number of GRAPPA weights required. The reconstruction time of current version of GROWL is about 10 sec for an eight-channel 256 (readout) 32 (view) twodimensional dataset on a 2.2-GHz personal computer. The method to interpolate GRAPPA operator weights, as presented in the modified PARS algorithm (12), can be used to further improve the speed for GROWL. One possible way to further improve the performance of GROWL at higher acceleration factors is to combine GROWL with radial GRAPPA or the modified PARS method. As shown in Fig. 2f, estimation errors become high when Dk y increases to 3/FOV or above and the noise level is high, limiting the ability to estimate data points farther away from the acquired readout line, using GRAPPA relative shift operators. Radial GRAPPA and the modified PARS method, however, are able to estimate the radial line that lies midway between two acquired radial lines. Therefore, it is expected that the combination of these techniques will further improve the quality of image reconstruction. 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