k-t GRAPPA: A k-space Implementation for Dynamic MRI with High Reduction Factor

Transcription

1 Magnetic Resonance in Medicine 54: (2005) k-t GRAPPA: A k-space Implementation for Dynamic MRI with High Reduction Factor Feng Huang,* James Akao, Sathya Vijayakumar, George R. Duensing, and Mark Limkeman A novel technique called k-t GRAPPA is introduced for the acceleration of dynamic magnetic resonance imaging. Dynamic magnetic resonance images have significant signal correlations in k-space and time dimension. Hence, it is feasible to acquire only a reduced amount of data and recover the missing portion afterward. Generalized autocalibrating partially parallel acquisitions (GRAPPA), as an important parallel imaging technique, linearly interpolates the missing data in k-space. In this work, it is shown that the idea of GRAPPA can also be applied in k-t space to take advantage of the correlations and interpolate the missing data in k-t space. For this method, no training data, filters, additional parameters, or sensitivity maps are necessary, and it is applicable for either single or multiple receiver coils. The signal correlation is locally derived from the acquired data. In this work, the k-t GRAPPA technique is compared with our implementation of GRAPPA, TGRAPPA, and sliding window reconstructions, as described in Methods. The experimental results manifest that k-t GRAPPA generates high spatial resolution reconstruction without significant loss of temporal resolution when the reduction factor is as high as 4. When the reduction factor becomes higher, there might be a noticeable loss of temporal resolution since k-t GRAPPA uses temporal interpolation. Images reconstructed using k-t GRAPPA have less residue/folding artifacts than those reconstructed by sliding window, much less noise than those reconstructed by GRAPPA, and wider temporal bandwidth than those reconstructed by GRAPPA with residual k-space. k-t GRAPPA is applicable to a wide range of dynamic imaging applications and is not limited to imaging parts with quasi-periodic motion. Since only local information is used for reconstruction, k-t GRAPPA is also preferred for applications requiring real time reconstruction, such as monitoring interventional MRI. Magn Reson Med 54: , Wiley-Liss, Inc. Key words: GRAPPA; parallel imaging; dynamic imaging; MRI; cardiac imaging Research and Predevelopment, Invivo Corporation, Gainesville, FL 32608, USA. *Correspondence to: Feng Huang, Invivo Corporation, 3545 SW 47th Avenue, Gainesville, FL 32603, USA. fhuang@invivocorp.com Received 14 January 2005; revised 26 April 2005; accepted 30 May DOI /mrm Published online 28 September 2005 in Wiley InterScience ( wiley.com) Wiley-Liss, Inc Dynamic magnetic resonance imaging (MRI) is important for many clinical applications, including interventional MRI, cardiac MRI (CMRI), and functional MRI (fmri). To increase spatiotemporal resolution and reduce motion artifacts, data acquisition speed is a crucial factor in these dynamic applications. One approach to reduce the scan time is to reduce the amount of acquired data by a given factor, which is usually called the reduction factor. Strategies of this approach are able to reduce data acquisition without compromising image quality significantly because typical dynamic image series exhibit a high degree of spatiotemporal correlations. Tsao et al. (1) divided these strategies into 3 kinds. The first kind exploits spatial correlations. Partial Fourier methods (2,3), reduced field-of-view methods (4 7), parallel imaging (8,9), and prior information driven methods (10) belong to this kind. The second kind exploits temporal correlations, including keyhole (11,12), view sharing strategies (13,14), and unaliasing by Fourier-encoding the overlaps using the temporal dimension (UNFOLD) (15). Approaches in the third kind exploit both spatial and temporal correlations. k-t SENSE (1), UNFOLD-SENSE (16), and Noquist (17) are promising techniques of this kind. k-t SENSE provides an elegant way of introducing prior knowledge from a training data set into the unfolding process. UNFOLD-SENSE uses the combination of SHRUG (self, hybrid referencing with UNFOLD, and GRAPPA) for sensitivity maps, variable density SENSE for reconstruction, and UNFOLD for artifact/amplified noise suppression. Both of these 2 methods are based on UNFOLD and SENSE (8). UNFOLD applies temporal filters (or the signal support that is the region containing most of the signal strength, as explained by Tsao (18)) to unfold a wrapped image, which consists of overlapped static (relatively) and dynamic signal. Choosing the filter/support is not a simple task. For example, if the support is big, then artifacts cannot be totally removed. On the other hand, if the support is small, then partial signal may be lost and the final image may be blurred. Tsao (1) cleverly uses training data to avoid this problem. However, the training data may not provide accurate information, in case the real acquisition does not follow the exact same procedure as the training data acquisition. Some patient motion, or non-periodic dynamic MRI (such as dynamic contrast enhanced MRI, different stimulations in fmri), may cause misregistration between training data and real acquisition data. Moreover, UNFOLD based methods reconstruct after all of the time frames are acquired; hence, it is not favored for applications that need real time monitoring. Also, accurate sensitivity maps are crucial for any SENSE based reconstruction method. To produce the sensitivity maps, some extra processing is necessary for these SENSE based methods. For example, Madore proposed to use SHRUG (16) for sensitivity maps. Tsao proposed to use the method provided in TSENSE (19) for sensitivity maps. Noquist requires a static region in the FOV. In case this static region is only relatively static (as in motion due to respiration in CMRI), motion artifacts result in the dynamic region. Even if motion in the static region is insignificant, the noise from this region contributes to the reconstructed dynamic fraction. This explains the band observed along the dynamic region in the images reconstructed by Noquist. The reconstruc-

2 k-t GRAPPA 1173 FIG. 1. The basic idea of GRAPPA. In GRAPPA, more than 1 line acquired in each of the coils in the array are fit to an ACS line acquired in a single coil of the array. In this case, 4 acquired lines are used to fit a single ACS line in coil #4. In GRAPPA, a block is defined as a single acquired line plus the missing lines adjacent to that line. tion time is also a limitation for the Noquist technique. The purpose of k-t GRAPPA is to avoid the necessity of training data, sensitivity map calculation, static region assumption, and choice of filters/parameters, while preserving the reconstruction quality with high reduction factors by using local correlation. The proposed k-t GRAPPA implementation also belongs to the third kind of techniques. This method uses the idea of GRAPPA to exploit local correlations along both spatial and temporal dimensions. Because the signal correlation is derived from the acquired data, there is no requirement for acquisition of training data, or the calculation of sensitivity maps. Similar to GRAPPA, there is no special data assumption, choice of filter, or any parameter in k-t GRAPPA. The only inputs for k-t GRAPPA are the partial k-t space data and the positions of auto calibrate signal (ACS) lines. In the present work, the k-t GRAPPA technique is introduced after a review of GRAPPA. As an extension of GRAPPA, techniques for improvement of GRAPPA can be applied to k-t GRAPPA as well. In the Results section, CMRI results and comparison with other reconstruction methods are used to affirm the idea of k-t GRAPPA. And the temporal signal profile is used to evaluate the performance of k-t GRAPPA for temporal resolution. More experimental results and applications can be referred to Huang et al. (20) and Vijayakumar et al. (21). METHODS In this section, we first review the conventional GRAPPA method. Then the extension of GRAPPA in k-t space is presented. Several modifications are explained for the purpose of implementation of improvements. Review of GRAPPA In GRAPPA, uncombined images are generated for each coil in the array by applying multiple block wise reconstructions to generate the missing lines for each coil. This process is shown in Fig. 1 (copy from (9)). Data acquired in each coil of the array (black circles) are fit to the ACS line (checkerboard circles). However, as can be seen, data from multiple lines from all coils are used to fit an ACS line in a single coil, in this case an ACS line from coil 4. This fit gives the weights, which can then be used to generate the missing lines (gray circles) from that coil. Once all of the lines are reconstructed for a particular coil, a Fourier transform can be used to generate the uncombined image for that coil. Once this process is repeated for each coil of the array, the full set of uncombined images can be obtained, which can then be combined using a conventional square root of sum-of-squares reconstruction (SSoS). In general, the process of reconstructing data in coil j at a line mk y offset from the normally acquired data using a block wise reconstruction can be represented by: L N b1 S j k y mk y nj,b,l,ms l k y bak y [1] l1 b0 where A represents the acceleration factor. N b is the number of blocks used in the reconstruction, where a block is defined as a single acquired line and A 1 missing lines. In this case, n(j,b,l,m) generated by fitting the ACS lines represents the weights used in this now expanded linear combination. In this linear combination, the index l counts through the individual coils, while the index b counts through the individual reconstruction blocks. This process is repeated for each coil in the array, resulting in L uncombined single coil images that can then be combined using a conventional SSoS or any other optimal array combination. k-t GRAPPA Dynamic MRI acquires raw data in k-space at different time points t. The raw data can be equivalently viewed as

3 1174 Huang et al. FIG. 2. The basic idea of k-t GRAPPA. The row is the phase encoding direction. The column is for different t. Black dots are equally spaced, time interleaved acquired data. Stars show the fully acquired central band that is ACS lines. Circles are missing data. The arrows give 3 examples of the adjacent data used to interpolate a missing data. This figure shows the 1 channel case. In multi-channel cases, all channels apply the same sampling pattern. For each missing data point, adjacent data from all channels in the same place as shown in this Figure are utilized for interpolation. In cases where the reduction factor is 4 and the number of channels is 4, 16 data points (4 from each channel) are utilized for interpolation. being acquired in a higher dimensional k-t space. The arrangement of these discrete samples in k-t space is referred to as the k-t sampling pattern. Fig. 2 shows the sampling pattern for k-t GRAPPA when the acceleration factor is 4. At each time point, k-space is sampled in a Cartesian manner. The frequency-encoding direction is omitted for simplicity. It is oriented perpendicular to the page. k refers to the index of the phase-encode line. Similar to the sampling pattern described in UNFOLD (15) and TSENSE (19), the acquired k-space data are time interleaved. However, at each time point t, several ACS lines are also acquired along with the regularly spaced phaseencode lines. Black dots in Fig. 2 show the acquired data that are undersampled. Since the center of k-space has high energy, the ACS lines are generally located at the center of k-space. Hence, for each time frame in k-space, there is a fully acquired central band. Different sets of phase-encode lines are acquired at successive time points. Stars in Fig. 2 show the fully acquired central band. The circles in Fig. 2 show the missing data. If there are multiple coils, then all of those coils have the same sampling pattern. Reconstruction of a dynamic image series involves the determination of object signals in k-t space from the discretely sampled data. In k-t GRAPPA, uncombined images are generated for each coil in the array by applying multiple block wise reconstruction to generate the missing lines for each coil. This process is shown in Fig. 2. Unlike conventional GRAPPA, k-t GRAPPA utilizes data from adjacent time frames as well, to interpolate the missing data. Fig. 2 illustrates 3 examples for interpolation in case there is only 1 coil. When the reduction factor is A, to interpolate the data at a line k y mk y in time t, where m is the offset from the normally acquired data, the data at line k y,k y Ak y in time t, and the data at line k y mk y in time t m, t A m are utilized. Those data points are actually the closest acquired neighbors on the same row or column in k-t space. When there are multiple coils, data from multiple lines (same position as described above in k-t space) from all coils as well as adjacent time frames are used to interpolate a missing line in a single coil. The described data are first used to linearly fit the ACS lines. This fit gives the weights, which can then be used to generate the missing lines from that coil. Once all of the lines are reconstructed for a particular coil, a Fourier transform can be used to generate the uncombined image for that coil. Once this process is repeated for each coil of the array, the full set of uncombined images can be obtained. In general, the process of reconstructing data in coil j at a line k y mk y in time t using a block wise reconstruction can be represented by: L S t j k y mk y l1 N b1 b0 vtm,tam n b j,l,ms l t k y bak y n v j,l,ms l v (k y mk y, [2] where A represents the acceleration factor. N b is the number of blocks used in the reconstruction, where a block is defined as a single acquired line and A 1 missing lines, and L is the number of coils. In this case, n b (j,l,m) and n v (j,l,m) generated by fitting the ACS lines represent the weights used in this now expanded linear combination. In this linear combination, the index l counts through the individual coils, while the index b counts through the individual reconstruction blocks; the index v counts through the adjacent frames (time), which acquired data at line k y mk y. This process is repeated for each coil in the array, resulting in L uncombined single coil images at each time instant t, which can then be combined using a conventional SSoS or any other optimal array combination. k-t GRAPPA can be treated as a combination of GRAPPA (9) and sliding window (22) techniques. If the correlation along the time dimension were not used for interpolation, then k-t GRAPPA would work out to be exactly the same as GRAPPA. If the correlation along the phase encoding dimension were not used, then k-t GRAPPA would be equivalent to a weighted sliding window technique. Hence, this proposed method has some similarity to both GRAPPA and sliding window techniques. On the one hand, images reconstructed by k-t GRAPPA can present less noise than those by GRAPPA because of its similarity to sliding win-

4 k-t GRAPPA 1175 L S tm j k y mk y or l1 N b1 b0 n b j,l,ms l t k y bak y 2 n tam j,l,ms tam l k y mk In case m A, y L S tam j k y mk y l1 N b1 b0 n b j,l,ms l t k y bak y [3] 2 n tm j,l,ms tm l k y mk In case m A. [4] y Correspondingly, the formulae for interpolation are L S t j k y mk y l1 N b1 b0 n b j,l,ms l t k y bak y FIG. 3. The basic idea of k-t GRAPPA without ACS lines. The row is the phase encoding direction. The column is for different t. Black dots are equally spaced, time interleaved acquired data. Circles are missing data. The points connected with solid lines are points contained in ACS lines. The points connected with dotted lines are points used in the linear interpolation. dow and these images have less residue artifacts, that is, better temporal fidelity, than those by sliding window because of its similarity to GRAPPA. On the other hand, the images reconstructed by k-t GRAPPA may have narrower temporal bandwidth than those by GRAPPA because of the interpolation along the time direction. The experimental results in the Results section support this prediction. Modifications of k-t GRAPPA To eliminate the acquisition of ACS lines, and hence further increase the reduction factor, Breuer et al. (23) proposed to use the lines from adjacent time frames to form the necessary ACS lines. If consecutive dynamic images vary smoothly, it is safe to use the central lines from temporal neighbors as ACS lines. This is similar to using the information from adjacent time frames to approximate sensitivity maps. This technique can be adopted for k-t GRAPPA. The basic idea is to utilize information from the nearest adjacent time frame as ACS lines, then use the other 3 neighbors, instead of 4, in k-t GRAPPA with ACS lines, to approximate the missing data. Fig. 3 shows this scheme. If the same notations as in the section above are used, the formula for weight calculation can be expressed as or 2 n tam j,l,ms tam l k y mk In case m A. y L S t j k y mk y l1 N b1 b0 n b j,l,ms 1 t k y bak y [5] 2 n tm j,l,ms tm l k y mk In case m A. [6] y According to Tsao et al. (1) and our experiences (24), better results can be expected if the temporally invariant term (i.e., direct-current or DC) is calculated separately. For k-t space reconstruction, the time interleaved k-space is averaged by simply adding acquired points along the time direction and then dividing by the number of these points, and this averaged k-space is used as DC. The residual k-t space is calculated by the subtraction of this mean k-space from each k-space frame. The reconstruction algorithm is then applied to the residual k-t space. Finally, the interpolated full k-t space and the mean k-space are added back together to obtain the fully reconstructed k-t space. We call this method the residual k-space method. Since the averaged k-space is only the approximation of the DC term, using residual k-space is a kind of temporal filtering. Using residual k-space can reduce the noise but may impair temporal fidelity of the reconstructed images. k-t GRAPPA can be combined with full temporal resolution methods to increase temporal resolution and, hence, decrease the residue artifacts of the reconstructed images, especially when the reduction factor is high or no

5 1176 Huang et al. Table 1 The Average Relative Errors at Region of Interest Reconstruction method Reduction factor Acceleration factor ACS lines Average relative error at region of interest k-t GRAPPA % Sliding window % GRAPPA % k-t GRAPPA with residual k-space % GRAPPA with residual k-space % k-t GRAPPA % Sliding window % GRAPPA % k-t GRAPPA with residual k-space % GRAPPA with residual k-space % k-t GRAPPA % Sliding window % k-t GRAPPA with residual k-space % k-t GRAPPA % Sliding window % k-t GRAPPA with residual k-space % k-t GRAPPA % Sliding window % k-t GRAPPA with residual k-space % k-t GRAPPA % Sliding window % k-t GRAPPA with residual k-space % k-t GRAPPA % Sliding window % k-t GRAPPA with residual k-space % ACS lines are acquired. The basic idea is to use k-t GRAPPA to interpolate part of the missing k-space to reduce the acceleration factor, and then use a full temporal resolution method to interpolate the remaining missing lines. As a specific example, when the acceleration factor is 4, k-t GRAPPA is applied to interpolate a third of the missing lines such that the acceleration factor is decreased to 2. Particularly, the central missing line between each pair of acquired lines is interpolated. Hence, the outcome is still an equally spaced k-space, but only every other line is missing; rfov (6) is then applied to reconstruct the final image based on the data with acceleration factor 2. Because most of the missing data are generated by rfov, higher temporal resolution can be expected. Some other techniques, such as sliding block (9), segmented GRAPPA (25), non Cartesian GRAPPA (26,27), GRAPPA operator (28), LIKE (29), and so forth can also be easily applied to k-t GRAPPA in k-t space. Some modifications especially for k-t GRAPPA are also possible. For example, using prior information to find better adjacent neighbors along the time direction for interpolation, and so forth. For simplicity, the details of the methods mentioned in this paragraph are skipped and the performances are not reported in this work. RESULTS In this section, the results of k-t GRAPPA as applied to CMRI are explained. Because k-t GRAPPA is a combination of GRAPPA and sliding window techniques, the 3 reconstruction methods were compared in the first experiment. The second experiment demonstrates the performance of k-t GRAPPA with data from a single coil. The third experiment shows the performance of k-t GRAPPA without ACS lines, and the comparison with TGRAPPA. To show the accuracy, the reconstructed image was compared with the reference image, which was generated by using full k-space. Let the phrase intensity difference refer to the difference in magnitudes between the reconstructed and reference images at each pixel. The relative error or relative energy difference was defined as the square root of the sum of squares of the intensity difference divided by the square root of the sum of squares of the reference image. To evaluate temporal fidelity, temporal plots of signal intensity along a line across the ventricles were used. The result was analyzed in the fourth experiment. The last experiment in this section shows the results of the combined methods (k-t GRAPPA with rfov). The proposed method, k-t GRAPPA, was implemented in the MATLAB programming environment (MathWorks Inc., Natick, MA). At most, 4 neighbors were used for interpolation in k-t GRAPPA, 2 along the k direction and 2 along the t direction. For missing data near the boundary in k-t space, not all 4 adjacent data points are available. In this case, only the available ones were used. In all of our experiments, MATLAB codes were run on an HP workstation (xw4100) with two 3.2 GHz CPUs and 2 Gb RAM. Comparison of Reconstruction Methods with Pseudo- Partially Parallel Acquisition k-t Space In this experiment, oblique cardiac images were collected on a SIEMENS Avanto system (FOV mm, matrix , heartbeats per acquisition 10, phase

6 k-t GRAPPA 1177 FIG. 4. Images reconstructed by different reconstruction methods the first frame in the sequence. The intensity scale of all images is [0 40]. (a) The reference image reconstructed with full k- space. (b) The clouded region shows the region of interest (ROI). (c) to (f) show the reconstructed images with pseudo-partially parallel acquisitions with reduction factor 3. (c) by k-t GRAPPA. (d) by sliding window; (e) by GRAPPA; (f) by GRAPPA with residual k-space; (g) and (h): by k-t GRAPPA with residual k-space with reduction factors 4 and 5.17 (acceleration factor 7). encodes per segment 15, number of phases 14, TR 2.86 ms, TE 1.43 ms, flip angle 46, slice thickness 6 mm, number of averages 1) using a cine true FISP sequence with the 12- channel TIM cardiac coil (SIEMENS Medical System Erlangen, Germany). Full k-space data were acquired. But only partial k-space data (simulated partially parallel acquisitions (PPA) with sampling pattern as in Fig. 2) were used for reconstruction. If 1 line was used out of every A lines, excluding the central ACS lines, then the acceleration factor was defined to be A. The phase encoding direction was anterior-posterior. k-t GRAPPA was implemented based on Eq. [2]. The Matlab reconstruction times, which were not optimized for real time applications, were approximately 34s per frame. The reconstruction time depends on the number of channels, matrix size, acceleration factor, number of ACS lines, and other related factors. For comparison, GRAPPA was implemented as proposed in (9). Four blocks as shown in Fig. 1 were used to reconstruct each missing line. The same ACS lines for k-t GRAPPA were used for GRAPPA. Both original and residual k-space were applied to GRAPPA and k-t GRAPPA. Sliding window was also implemented (22) and applied to the same k-space at each time frame t. Table 1 and Figs. 4, 5, 6, and 7 show the results. Table 1 presents the average relative error of the 14 time frames in the region of interest (ROI, Fig. 4b). Because k-t GRAPPA and GRAPPA make use of ACS lines, the real reduction factor is smaller than the acceleration factor. The upper segment of Table 1 shows the comparison of these reconstruction methods with acceleration factors 3 and 4. With ACS lines indexed in the ranges [57 93] and [68 83] out of the total 150 PE lines, the real reduction factors were 2 and 3. It can be seen that k-t GRAPPA worked better with residual k-space and generated less error than sliding window and GRAPPA did; GRAPPA with residual k-space generated less error than GRAPPA did. The lower segment of Table 1 used less ACS lines but higher acceleration factors. GRAPPA does not work with that few ACS lines when the acceleration factor is greater than 4. Hence, it is not illustrated in this portion of Table 1. It can be seen that k-t GRAPPA does not require as many ACS lines as GRAPPA does; k-t GRAPPA with residual

7 1178 Huang et al. FIG. 5. Absolute difference map between Fig. 4a and reconstructed images. The intensity scale of all images is [0 4]. (a) to(d): by k-t GRAPPA, sliding window, GRAPPA, and GRAPPA with residual k- space with reduction factor 3. (e) to (h): by k-t GRAPPA with residual k-space and reduction factor 2, 3, 4, and 5.17 (acceleration factor 7). k-space generated the least error in all cases. Fig. 6 plots the table. From Table 1 and Fig. 6, it can be seen that the error in images reconstructed by GRAPPA increases abruptly with the increase in the reduction factor; the error in images reconstructed by other methods (using time dimension correlation) increases quasi-linearly with slope (k-t GRAPPA), 0.03 (sliding window), (GRAPPA with residual k-space), and (k-t GRAPPA with residual k-space), where the slope was defined as the gradient of the line that best approximated the relative error plots in the sense of least squares. Hence, k-t GRAPPA with residual k-space is least sensitive to the increase in the reduction factor in this experiment. Figs. 4 and 5 show the image of the first time frame. The comparison of k-t GRAPPA performance for different time frames is shown in Fig. 7 and is discussed in the next paragraph. Fig. 4a is the reference image reconstructed with full k-space. The clouded region in Fig. 4b shows the region of interest, which was used to calculate the relative error. Figs. 4c to 4h show the reconstructed images. Fig. 5 shows the absolute difference between 4a and the reconstructed images. All of the reconstructed images used the same intensity scale [0 40]. All of the difference maps used the same intensity scale [0 4]. As predicted, it can be seen that the images reconstructed by k-t GRAPPA (Fig. 5a and 5f) have less residue artifacts than the images reconstructed by sliding window (Fig. 5b), and less noise than the images reconstructed by GRAPPA (Fig. 5c and 5d); k-t GRAPPA with residual k-space produced very little error when the reduction factor was 2 (Fig. 5e); k-t GRAPPA produced less error with residual k-space (Fig. 5f) than with original k-space (Fig. 5a); low error images can be reconstructed by k-t GRAPPA with acceleration factors as high as 7 (Figs. 4h and 5h); and images reconstructed by GRAPPA with residual k-space have less noise than images reconstructed by GRAPPA with original k-space, but have more residue artifacts (Fig. 5c and 5d). Since k-t GRAPPA with residual k-space generates better results, from now on, k-t GRAPPA refers to k-t GRAPPA with residual k-space. Fig. 7 shows the relative error of all time frames by k-t GRAPPA with reduction factors 2, 2.78, 3, 3.57, 4.05, 4.28, 4.41, 4.84, and 5.17 from bottom to top

8 k-t GRAPPA 1179 FIG. 6. Relative errors in the region of interest by different reconstruction methods with different reduction factors. for each style of line. Unlike sliding window, k-t GRAPPA does not have the worst performance for the beginning and the ending time frames (Figs. 4, 5, and 7). The performance depends on the similarity of adjacent frames. In Fig. 7, the dashed lines show the relative error on the whole image, and the solid lines show the relative error in the region of interest. The dashed lines are much lower than the solid lines for the same reduction factor. This means most of the error occurs in the dynamic region. This can also be seen from the difference maps in Fig. 5. k-t GRAPPA with Single Channel Data Unlike GRAPPA, the number of channels does not limit the application of k-t GRAPPA. When there is only one channel available, k-t GRAPPA becomes a weighted sliding window and the ACS lines decide the weights. Fig. 8 shows the result by using only the 10th channel data in the previous section. k-t GRAPPA and sliding window were applied to the same k-space at each time frame. Fig. 8c shows k-t GRAPPA can reconstruct a high quality image with an acceleration factor as high as 7 with only 1 channel data. By comparing Figs. 8d and 8e with 8f and 8g, it can be seen that sliding window generated much more residue artifacts and folding artifacts with the same data set. With the increase in the reduction factor, the relative errors of images reconstructed by k-t GRAPPA still rise quasi-linearly with a lower slope (0.023) than those by sliding window (0.0267). k-t GRAPPA Without ACS Lines In this experiment, data from the previous experiment was first used to get the information of relative errors for comparison. Then undersampled k-space data were acquired. Oblique cardiac images were collected on a SIE- MENS Avanto system (FOV mm, matrix , heartbeats per acquisition 10, phase encodes per segment 7, number of phases 29, TR 2.86 ms, TE 1.43 ms, flip angle 46, slice thickness 6 mm, number of averages 1) using a cine true FISP sequence with a SIEMENS TIM 12 channels cardiac coil. The phase encoding direction was anterior-posterior. Acceleration factors were 2 and 4, and the sampling pattern followed Fig. 3. Since no ACS lines were acquired, the reduction factors were 2 and 4. k-t GRAPPA was implemented based on Eqns. [3] to [6]. Seventeen lines in the center were filled with lines from adjacent time frames and used as ACS lines. For comparison, TGRAPPA was implemented as proposed in (23). In our implementation, TGRAPPA used the same set of coefficients for the whole k-space of each time frame. Different from k-t GRAPPA, lines from adjacent time frames were combined to form a complete set of ACS lines for TGRAPPA, as suggested by Breuer et al. (23). Because more ACS lines were used, it took longer to reconstruct using TGRAPPA. TGRAPPA with residual k-space was also implemented. Fig. 9 shows the results by using pseudo-ppa k-space from the last experiment (where fully acquired k-space was decimated by us to mimic partially parallel acquisitions). From Fig. 9a, it can be seen that TGRAPPA had the best performance when the reduction factor was 2; the noise in the images reconstructed by TGRAPPA increases dramatically with the increase in the reduction factor; and TGRAPPA with residual k-space contained the increase of noise but still generated more errors than k-t GRAPPA did. Fig. 9c is the zoomed image reconstructed by TGRAPPA with original k- space at reduction factor 4; the noise damages visualization. Figs. 9d and 9f are the zoomed images reconstructed by TGRAPPA with residual k-space at reduction factor 4 and 6. SNRs are improved in these images, but the residual artifacts are significant (Fig. 9h); hence, Figs. 9d and 9f look blurred. Relatively, the images reconstructed by k-t GRAPPA have better SNR and less residue artifacts (Figs. 9e, 9g, and 9i). When the reduction factor is 2 by using the pseudo-ppa k-space, the mean relative errors in the ROI of images reconstructed by k-t GRAPPA (acceleration factor 3, ACS lines indexed [57 93]), TGRAPPA, k-t GRAPPA without ACS lines, and TGRAPPA with residual k-space are 4.8%, 5.1%, 6.1%, and 6.2%. Hence, k-t GRAPPA is the best, while TGRAPPA with residual k-space is the worst, when the reduction factor is low. When the reduction factor is 4, the mean relative FIG. 7. Relative errors of all time frames by k-t GRAPPA with different reduction factors. The dashed lines show the relative error on the whole image. The solid lines show the relative error in the region of interest. For each line style, the reduction factors are 2, 2.78, 3, 3.57, 4.05, 4.28, 4.41, 4.84, and 5.17 from bottom to top.

9 1180 Huang et al. FIG. 8. Results with single channel. The 10th channel data were used. The intensity scale of (b) and (c) is [0 5]. The intensity scale of (d)to(g) is [0 0.5]. (a) Comparison of relative errors at ROI. (b) The reference image reconstructed with full k-space. (c) the image reconstructed by k-t GRAPPA with reduction factor 5.17 (acceleration factor 7). (d) to (g) are the absolute difference between reference image b and the reconstructed images. (d) and (e) are by k-t GRAPPA with reduction factor 3 and (f) and (g) are by sliding window with reduction factor 3 and FIG. 9. Results of TGRAPPA and k-t GRAPPA without ACS lines. The intensity scale of (b)to(g) is [0 40]. The intensity scale of (h) and (i) is[04].(a) The relative errors of reconstructed images in the region of interest. (b) The reference image at time frame 6. (c) to (g) are reconstructed images: (c) by TGRAPPA at reduction factor 4, (d) and (f) by TGRAPPA with residual k-space at reduction factor 4 and 6, (e) and (g) by k-t GRAPPA at reduction factor 4 and 6. (h) and (i) are the absolute differences between b and f, and g.

10 k-t GRAPPA 1181 FIG. 10. Results of TGRAPPA and k-t GRAPPA with partially parallel acquisitions. The intensity scale of all images is [0 30]. The first 3 rows are reconstructions with reduction factor 2. The next 3 rows are reconstructions with reduction factor 4. The first and fourth rows are the results by TGRAPPA with original k-space; the second and fifth rows are the results by TGRAPPA with residual k-space; the third and sixth rows are results by k-t GRAPPA. The 4 columns illustrate time frames 5, 10, 15, and 20, respectively. errors in the ROI are 9.6% (k-t GRAPPA, acceleration factor 5, ACS lines indexed [71 79]), 22% (TGRAPPA), 11% (k-t GRAPPA without ACS lines), and 13% (TGRAPPA with residual k-space). TGRAPPA becomes the worst and k-t GRAPPA remains the best when the reduction factor is 4. When the reduction factor is higher, the mean relative errors are 12% (k-t GRAPPA with ACS lines indexed [71 79], acceleration factor 7, and reduction factor 5.17), 13% (k-t GRAPPA without ACS lines, reduction factor 5), and 15% (TGRAPPA with residual k-space, reduction factor 5). k-t GRAPPA with several ACS lines is still the best when the reduction factor is high. When the reduction factor is high, the first and last several time frames reconstructed by k-t GRAPPA without ACS lines tend to be similar to the images reconstructed by TGRAPPA because fewer adjacent neighbors along the time direction are available for interpolation. Fig. 10 shows the reconstruction with undersampled data at acceleration factors 2 and 4. Reconstructed images (zoomed) for frames 5, 10, 15, and 20 are illustrated. These images also show that k-t GRAPPA can produce high SNR reconstruction with high reduction factors; the noise in the images reconstructed by TGRAPPA grows fast with the increase in the reduction factor; TGRAPPA with residual k-space restricts the increase of noise; and the images reconstructed by k-t GRAPPA are sharper than images reconstructed by TGRAPPA with residual k-space with reduction factor 4. Evaluation of Temporal Fidelity k-t GRAPPA uses interpolation along the time direction; hence, it potentially reduces the temporal resolution. The residue artifacts shown in the difference maps evaluate temporal fidelity in one way. To further evaluate the performance of k-t GRAPPA for temporal fidelity, temporal plots of the signal intensity along a line across the ventricles is used. The white line on Fig. 11a shows the position of this line. The temporal intensity profile of an image sequence along this line is composed of the signal intensity along this line at all time frames. Figs. 11b to p are temporal intensity profiles of reconstructed images. Each column of the temporal intensity profile is the signal intensity along this line at one particular time. The shape and sharpness of the temporal intensity profile show the temporal fidelity of the reconstructed image sequence. Figs. 11b to d show the temporal intensity profiles of reconstructed images with single channel data. Fig. 11b is the temporal intensity profile of the reference image. Figs. 11c and d are the temporal signal profiles of images reconstructed by k-t GRAPPA and sliding window with reduc-

11 1182 Huang et al. FIG. 11. Temporal signal profiles. (a) The white line shows the position of the chosen segment. (b) to(d): the temporal intensity profiles of reconstructed images with single channel data. (b) with reference images. (c) and (d): with images reconstructed by k-t GRAPPA and sliding window with reduction factor 3. (e) to(p): the temporal signal profiles of reconstructed images with multiple channel data. (e) with the reference image sequence. (f) to (i): with images reconstructed by k-t GRAPPA with reduction factor 3, 4.05, 5, and (j) with images reconstructed by the combined method described above with reduction factor 5. (k) to (p): temporal signal profiles of images reconstructed by GRAPPA, GRAPPA with residual k-space, TGRAPPA, TGRAPPA with residual k-space, k-t GRAPPA without ACS lines, and sliding window, all with the same reduction factor 3. tion factor 3. Both Figs. 11c and 11d have similar shape to Fig. 11b, but Fig. 11d is more blurred than Fig. 11c. This means k-t GRAPPA generates images with better temporal fidelity than sliding window in this experiment. Difference maps in Fig. 8 also confirm this conclusion. Figs. 11e to p are the temporal signal profiles of reconstructed images with multiple channel data. Fig. 11e is the reference image sequence. Figs. 11f to 11i are images reconstructed by k-t GRAPPA with reduction factor 3, 4.05, 5, and Figs. 11 h and 11i become blurred. This means k-t GRAPPA does not generate images with high temporal fidelity when the reduction factor is higher than 4 for this data set. Fig. 11j used a combined method described in the following sub-section. Figs. 11k to p are the temporal signal profiles of images reconstructed by GRAPPA, GRAPPA with residual k-space, TGRAPPA, TGRAPPA with residual k-space, k-t GRAPPA without ACS lines, and sliding window, all with the same reduction factor 3. Both Figs. 11f and 11o have very similar shape to Fig. 11e; images reconstructed by GRAPPA (Fig. 11k) and TGRAPPA (Fig. 11m) are noisier than those by k-t GRAPPA (Fig. 11f and 11o); with residual k-space, GRAPPA and TGRAPPA reconstructed images with less noise but the temporal signal profiles (Fig. 11l and 11n) are more blurred than those of k-t GRAPPA (Fig. 11f and 11o); Fig. 11p is smoother than Figs. 11f and 11o, and this means images reconstructed by sliding window have worse temporal fidelity than images reconstructed by k-t GRAPPA. k-t GRAPPA with Full Temporal Resolution Method In this experiment, the pseudo-ppa data from the first experiment was used. k-t GRAPPA was combined with the modified rfov technique by Parrish et al. (hybrid technique for dynamic imaging (6)) for reconstruction. Since k-t GRAPPA generates high temporal resolution images when the reduction factor is lower than 5, the combined method was only applied to data with reduction factor 5 (acceleration factor 6, ACS lines 6). Fig. 12 shows the results. Fig. 12a demonstrates that the combined method generated less error than k-t GRAPPA did. Figs. 12b and c are the reconstructed images by the combined method at time frame 8 (lowest relative error) and 10 (highest relative error). Figs. 12d and e are the absolute difference maps between reference images and Figs. 12b and c. These images show that the combined method generated low residual artifact when the reduction factor is 5. Fig. 11j shows the temporal signal profile of the images reconstructed by the combined method. As predicted, image sequence reconstructed by the combined method has higher temporal

12 k-t GRAPPA 1183 FIG. 12. Results of the combined method (k-t GRAPPA and rfov) with reduction factor 5 (acceleration factor 6, ACS lines 6). The intensity scale of (b) and (c) is [0 40]. The intensity scale of (d) and (e)is [0 4]. (a) The relative errors of reconstructed images in the region of interest. (b) and (c): the reconstructed images by the combined method at time frame 8 (lowest relative error) and 10 (highest relative error). (d) and (e): the absolute difference map between reference images, and b and c. resolution than k-t GRAPPA with the same reduction factor (Fig. 11h). This experiment shows that the combined method can generate images with both high SNR and temporal resolution when the reduction factor is as high as 5. DISCUSSION Similar to k-t BLAST/SENSE, k-t GRAPPA exploits the correlations in both spatial and temporal direction. But k-t GRAPPA works on k-t space instead of the x-f space that k-t BLAST/SENSE works on. k-t GRAPPA is an extension of GRAPPA from k-space to k-t space. Hence, it has many advantages of GRAPPA. It has no requirement for sensitivity maps or prescan; several ACS lines are acquired for the signal correlation, and those ACS lines can be directly used for reconstruction. Another interesting property of k-t GRAPPA is that, unlike k-t BLAST/SENSE, the first and last several time frames can also have accurate reconstruction even though they may not have information from one time side for interpolation. Compared with our implementation of GRAPPA, images reconstructed by k-t GRAPPA have much less noise, though more residue artifacts are present. When GRAPPA does not work because of a high reduction factor or too few ACS lines, k-t GRAPPA can still generate accurate results. Residual k-space can be used by GRAPPA to reduce noise, but the reduced noise is still higher than the noise level in images reconstructed by k-t GRAPPA. Furthermore, the application of residual k-space on GRAPPA dramatically impairs the temporal resolution, such that the temporal fidelity of the reconstructed image sequence becomes worse than the image sequence reconstructed by k-t GRAPPA. Compared with the sliding window technique, images reconstructed by k-t GRAPPA have less folding artifacts and higher temporal resolution. Therefore, we may conclude that k-t GRAPPA reconstructs more accurate results with higher reduction factors than GRAPPA and sliding widow when we consider SNR and temporal fidelity simultaneously. To further increase the temporal bandwidth of the reconstructed images, k-t GRAPPA can be combined with full temporal resolution methods. k-t GRAPPA can also work without ACS lines. Notice that fewer lines were used as ACS lines for k-t GRAPPA without ACS lines than for TGRAPPA. Hence, reconstruction by k-t GRAPPA without ACS lines consumes a shorter time for calculating coefficients. TGRAPPA reduces temporal resolution less than k-t GRAPPA does, but it also suffers from high noise when the reduction factor is high.

13 1184 Huang et al. If residual k-space is applied to TGRAPPA, the temporal fidelity of the reconstructed image sequence becomes worse than the image sequence reconstructed by k-t GRAPPA. With the observation that images reconstructed by k-t GRAPPA with some ACS lines have less error than those by k-t GRAPPA without ACS lines at the same reduction factor, and that the first and last several time frames reconstructed by k-t GRAPPA without ACS lines have high relative errors when the reduction factor is 5 due to the availability of fewer adjacent neighbors for interpolation, reconstruction with ACS lines is preferred. Because k-t GRAPPA is not as sensitive as GRAPPA to the number of ACS lines, fewer ACS lines can be acquired for k-t GRAPPA to increase the reduction factor with a given acceleration factor. In general, at least A 2 ACS lines are required for k-t GRAPPA when the acceleration factor is A. As we mentioned above, at most 4 neighbors (2 along the k direction and 2 along the time direction) were used for interpolation in all of our experiments. To test if more neighbors result in more accurate results, experiments with different number of neighbors were processed. According to our experience, more neighbors along the time direction may generate worse results; more neighbors along the k direction cannot improve image quality much but take longer reconstruction time. k-t GRAPPA can be applied to wide dynamic imaging areas. It is applicable not only to areas exhibiting quasiperiodic behaviors (such as imaging of the heart, the lungs, and the abdomen), but also to kinematics motion studies, dynamic contrast uptake studies for tumor differentiation, and brain under non-periodic stimulation, where the change is not periodic. 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