Accelerated MRI by SPEED with Generalized Sampling Schemes

Transcription

1 Magnetic Resonance in Medicine 70: (201) Accelerated MRI by SPEED with Generalized Sampling Schemes Zhaoyang Jin 1 * and Qing-San Xiang 2 Purpose: To enhance the fast imaging technique of skipped phase encoding (PE) and edge deghosting (SPEED) for more general sampling options, and thus more flexibility in implementations and applications. Methods: SPEED uses skipped PE steps to accelerate MRI scan. Previously, the PE skip size was chosen from prime numbers only. This restriction has been relaxed in this study to allow choice of any integers rather than merely prime numbers. Various sampling patterns were studied under all possible combinations of PE skip size and PE shifts. A criterion based on the rank values of ghost phasor matrices was introduced to evaluate SPEED reconstruction. Results: The reconstruction quality was found to correlate with the rank value of the ghost phasor matrix and the skipped PE size N. A low-rank value indicates a singular matrix that causes failure of the SPEED reconstruction. Composite numbers combined with appropriately chosen PE shifts yielded satisfactory reconstruction results. Conclusion: With properly chosen PE shifts, it was found that any integers, including both prime numbers and composite numbers, could be used as PE skip size for SPEED. This finding allows much more flexible data acquisition options that may lead to more freedom in practical implementations and applications. Magn Reson Med 70: , 201. VC 201 Wiley Periodicals, Inc. Key words: fast imaging; rank; edge deghosting; SPEED; compressed sensing; optimized sampling 1 Institute of Information and Control, Hangzhou Dianzi University, Hangzhou, Zhejiang, People s Republic of China. 2 Department of Radiology, University of British Columbia, Vancouver, British Columbia, Canada. Grant sponsor: National Natural Science Foundation of China; Grand number: Grant sponsor: Children s and Women s Health Centre of British Columbia. *Correspondence to: Zhaoyang Jin, Ph.D., Institute of Information and Control, Hangzhou Dianzi University, Hangzhou, Zhejiang, People s Republic of China. jinzhaoyang@hdu.edu.cn Received 24 August 2012; revised 28 November 2012; accepted 28 November 2012 DOI /mrm Published online 0 January 201 in Wiley Online Library (wileyonlinelibrary.com). VC 201 Wiley Periodicals, Inc MRI is often limited by its relatively long data acquisition time in clinical applications. Various approaches have been proposed for both reduced imaging time and improved patient comfort level (1 21). The recently developed fast imaging method of skipped phase encoding (PE) and edge deghosting (SPEED) is able to not only effectively reduce scan time by acquiring skipped k-space data lines (22) but also reconstruct images very fast based on analytical solutions. Typically, three interleaved datasets with different shifts in PE allow reconstruction by using a two-layer signal model for ghosted edge maps. A few variations and applications have been developed for SPEED, given its simplicity and flexibility. SPEED has been further developed with array coil enhancement (ACE) by combing itself with parallel imaging (2). SPEED-array coil enhancement partially samples k-space with a PE skip size of N by using multiple receiver coils in parallel, achieving scan reduction factors even greater than the number of receive coils. Simplified SPEED (S-SPEED) was proposed to accelerate magnetic resonance angiography by taking advantage of the sparsity of vasculature (24,25). For sparse objects with dark or modest tissue background, simplified SPEED partially samples k-space with only two interleaved datasets and models the sparse ghosted images with a single-layer structure. The efficient multiple acquisitions-speed method was proposed for more efficient scan time reduction by sharing similar spatial information among multiple acquisitions, leading to acceleration factors greater than that achievable with single acquisition (26). The PE skip size N in previous versions of SPEED except SPEED-array coil enhancement, was restricted to be prime numbers only, such as N ¼ 5, 7, or 11, avoid reconstruction difficulties due to possible ghost phase degeneracy (22 26). In this study, this restriction was very much relaxed, and more generalized sampling schemes of SPEED (G-SPEED) were proposed. This general approach will be called G-SPEED. It was demonstrated that the PE skip size N does not have to be limited to prime numbers, and composite numbers with appropriately selected PE shifts can also result in satisfactory reconstruction. In fact, combinations between the PE skip size N and PE shifts determine the quality of reconstruction, which is reflected in the rank value of a ghost phasor matrix. High quality reconstruction was achieved with composite skip sizes at N ¼ 4, 6, 8, 9 and 10. These new possibilities in data acquisition provide SPEED with more flexibility in terms of practical implementations and applications. METHODS SPEED with Generalized Sampling Schemes Similar to the original SPEED (22), the full k-space data were selectively sampled into three interleaved datasets S 1 (k), S 2 (k), and S (k), denoted as N (d 1, d 2, d ), where N is the PE skip size and d 1, d 2, and d are different PE shifts. We have used k ¼ (k x, k y ) to indicate k-space data location. As the data-sampling pattern is periodic, only the relative PE shifts are important. With d 1 ¼ 0, d 1 <

2 Generalized SPEED 1675 d 2, and d 2 < d, there were a total number of (N 2)(N 1)/2 possible combinations of N (d 1, d 2, d ) for each N. The sampled datasets were first Fourier transformed into three ghosted images, I 1 (r), I 2 (r), and I (r), each of which is associated with N-fold aliasing ghosts and thus has ghost overlapping up to N layers. Now, r ¼ (x, y) represents position in image space. Three sparse ghosted-edge maps, E 1 (r), E 2 (r), and E (r), were generated after a differential operation applied along the PE direction for each dataset. By modeling them with a double-layer structure, each pixel in ghosted edge maps can be described by three ghost equations, written as E 1 ¼ P n1 d1 G n1 þ P n2 E 2 ¼ P n1 d2 G n1 þ P n2 E ¼ P n1 d G n1 þ P n2 d1 G n2 [1] d2 G n2 [2] d G n2 [] where G n1 and G n2 are two complex-valued ghosts of different orders, P d n are ghost phasors, representing possible vector rotations, or relative phase shifts of the complex-valued ghosts in the three edge maps E 1 (r), E 2 (r), and E (r). The ghost phasor P d n is known to have a form of Pd n ¼ eið2pdn=nþ ; d ¼ 0; 1; 2; ; N 1; n ¼ 0; 1; 2; ; N 1; [4] where d, n, and N are all integers representing the relative sampling shift in PE, the order of ghost, and the PE skip size, respectively. Equations [1] [] can also be written in matrix form as E ¼ PG [5] where E ¼ (E 1, E 2, E ) T is a known vector, G ¼ (G n1, G n2 ) T is an unknown vector, and P is a 2 ghost phasor matrix defined as 2 6 P ¼ 4 P n1 d1 P n1 d2 P n1 d Pd1 n2 Pd2 n2 Pd n2 7 5 [6] To simplify the equations, let 2 Pd1 n1 P n2 2 d1 R 1 R 2 6 P ¼ 4 Pd2 n1 P n2 7 5 ¼ 4 R R 4 5 R 5 R 6 P n1 d d2 Pd n2 Equations [1] [] or 5 describe an overdetermined linear system with three equations and two unknowns. An analytical approach to solving such an overdetermined system can begin by multiplying both sides by P þ, P þ ¼ R 1 R R 5 R 2 R 4 R 6 where the superscript þ represents conjugate-transpose of a matrix, and the superscript * means conjugate. This reduces the system to a square, 2 2 system known as the normal equations: [7] P þ E ¼ P þ PG [8] If the column vectors of P are linearly independent, then P þ P is nonsingular and G ¼ LE ¼ ½P þ PŠ 1 P þ E [9] where the superscript 1 represents inverse of a matrix, and L ¼ ½P þ PŠ 1 P þ. 2 As P þ P ¼ R 1 R R R 1 R R R 2 R 4 R R R 5 R 6 R ¼ 1 R 2 þ R R 4 þ R 5 R 6 R 1 R 2 þ R R ; 4 þ R 5 R 6 if we define M ¼ R 1 R 2 þ R R 4 þ R 5 R 6, then, P þ P ¼ M M ; [10] ½P þ PŠ 1 1 M ¼ 9 M M M ; [11] 1 R L ¼ 1 MR 2 R MR 4 R 5 MR 6 9 M M M R 1 þ R 2 M R þ R 4 M R : [12] 5 þ R 6 The solution candidate of G with constant pair (n 1, n 2 ) can be calculated directly based on Eqs. [9] and [12]. There can be a least-square error (LSE) solution for such overdetermined system, for example, LSE solution finds G which minimizes r PG E [1] where r is the residual of the solution, which is defined as the two-norm or square sum of all the vector elements. The correct solution of (G n1, G n2 ) along with parameters (n 1, n 2 ), can be simultaneously determined by testing all possible (n 1, n 2 ) pairs while comparing the resulting LSE until a minimum r is found. By minimizing LSE, the ghost-order index pair (n 1, n 2 ) is determined through a multiple, but limited number of trials. Considering the symmetric positions of n 1 and n 2 in Eqs. [1] [], we have a total of N(N 1)/2 trials to run through with LSE solution, for example, there are 15 trials for N ¼ 6. With the determined (n 1, n 2 ) pair, ghost solutions (G n1, G n2 ) are then simultaneously obtained. Subsequently, the edge ghosts (G n1, G n2 ) can be sorted out according to their associated (n 1, n 2 ) values and can thereby yield N separate edge ghosts. For example, there are six G n maps when N ¼ 6, with n ¼ 0, 1, 2,, 4, and 5. They can be individually shifted along PE direction by n/n field of view to its source location. The six separate edge ghosts

3 1676 Jin and Xiang can therefore be registered and summed into a single deghosted edge map E 0 (r) with reduced noise and artifacts. The deghosted edge map E 0 (r) is further inverse-filtered and reconstructed into a final deghosted image I 0, as shown in Eq. [14], DFT½E 0 ðrþš I 0 ¼ IDFT [14] e j2pðky =Ny Þ 1 where DFT and IDFT means discrete Fourier transform and its inverse transform. To avoid the problem of dividing by zero at the very center of k-space when k y ¼ 0, a band of 8 2 lines of central k-space was fully collected and used to simply replace the inverse-filtered data in Eq. [14]. The above process is summarized in Figure 1. A Rank-Based Criterion The ghost phasors depend on such factors as the displacement or the order of the ghost denoted by n, the PE skip size N, and the relative shift d in PE. Previously, PE skip size N was limited to be a prime number to avoid reconstruction difficulty in matrix inversion due to potential ghost phase degeneracy (22). The key idea of this work is that while the prime number requirement is helpful, this restriction can be very much relaxed. Composite numbers combined with appropriately chosen PE shifts can also yield good reconstruction results. The solution of ghosts G given by Eq. [9] does not always exist. If the column vectors of ghost phasor matrix P are not linearly independent or P is rank deficient, then P þ P is singular, and (P þ P) 1 does not exist, and the formula [9] leading to the LSE solution of G breaks down. In this study, the ghost phasor matrix P was studied using all possible combinations of skipped PE size and PE shifts. For a given N (d 1, d 2, d ) combination, there are a total number of N(N 1)/2 possibilities of (n 1, n 2 ) pair, and thus the same number of ghost phasor matrices. The rank of P was used to gauge the reconstruction quality. There are a total of N(N 1)/2 possible rank values for each N (d 1, d 2, d ) combination. For any matrix, its column rank must be equal to its row rank. The phasor matrix P has a dimension of 2. Its rank (P) min (, 2), and its full rank equals 2. If the rank of P equals 1, the column vectors in P are not linearly independent and P is a singular matrix, which will cause failure in the calculation of G in Eq. [9]. For example, for the sampling scheme of 6 (0, 2, 4), there are totally 15 possible pairs of ghost indexes, among which three pairs (n 1 ¼ 5, n 2 ¼ 2), (n 1 ¼ 4, n 2 ¼ 1) and (n 1 ¼, n 2 ¼ 0) will lead to a singular ghost phasor matrix P. The singular ghost phasor matrix from (n 1 ¼ 5, n 2 ¼ 2) is specifically given below as an example, 2 P ¼ 4 e ið2p05=6þ e ið2p25=6þ e ið2p45=6þ e ið2p02=6þ e ið2p22=6þ e ið2p42=6þ ¼ 4 0:5 0:866i 0:5 0:866i 5 0:5 þ 0:866i 0:5 þ 0:866i It can be easily seen that the two-column vectors of P are not independent and the rank of P is equal to 1. If FIG. 1. k-space sampling scheme and data flow of SPEED. Full k-space S 0 (k) is sparsely sampled into three interleaved k-space datasets, S 1 (k), S 2 (k), and S (k), respectively, each with a skip size of N and a different relative shift d in the PE direction. Three ghosted images, I 1 (r), I 2 (r), and I (r), are obtained by using traditional DFT reconstruction, followed by a differential filtering step to produce three ghosted edge maps: E 1 (r), E 2 (r), and E (r). A ghostfree edge map E 0 (r) is obtained after applying ghost separation, registration, and summation steps. E 0 (r) is further inverse filtered to produce the final deghosted image I 0 (r). any one of the N(N 1)/2 trials of (n 1, n 2 ) pairs results in a singular matrix P, Eq. [9] will fail to provide a useful solution of G. The minimum rank value among values for all (n 1, n 2 ) pairs was used as a criterion for selecting N (d 1, d 2, d ) combinations as valid sampling schemes for G-SPEED. The N (d 1, d 2, d ) combinations with minimum rank value of 1 will cause the break down of Eq. [9] and thus should be avoided. It should be noted that the rank of ghost phasor matrix P only depends on the sampling

4 Generalized SPEED 1677 FIG. 2. Deghosted edge maps from a spin-echo sagittal knee scan using all possible SPEED sampling combinations for N ¼ 6, 6 (0, 1, 2) (a), 6 (0, 1, ) (b), 6 (0, 1, 4) (c), 6 (0, 1, 5) (d), 6 (0, 2, ) (e), 6 (0, 2, 4) (f), 6 (0, 2, 5) (g), 6 (0,, 4) (h), 6 (0,, 5) (i), and 6 (0, 4, 5) (j), respectively. Serious ghosted edges can be found in (f) with 6 (0, 2, 4) sampling combination. PE was applied in the horizontal direction. The bottom two rows show images reconstructed using (k) standard inverse DFT with full data, and SPEED with 6 (0, 1, 2) (l), 6 (0, 1, ) (m), 6 (0, 1, 4) (n), 6 (0, 1, 5) (o), 6 (0, 2, ) (p), 6 (0, 2, 5) (q), 6 (0,, 4) (r), 6 (0,, 5) (s), and 6 (0, 4, 5) (t) sampling scheme, respectively. The sampling combination 6 (0, 2, 4) caused failure of the reconstruction because of strong remaining major ghosts in deghosted edge maps and thus is not shown here. The short white arrows show some structure-mimicking residual minor ghosts. schemes, are independent of the MRI data, and therefore can be precalculated. A total of 11 combinations of N (d 1, d 2, d ) have been found to yield minimum rank value of 1 with N ¼ 6, 8, 9, and 10. They are 6 (0, 2, 4), 8 (0, 2, 4), 8 (0, 2, 6), 8 (0, 4, 6), 9 (0,, 6), 10 (0, 2, 4), 10 (0, 2, 6), 10 (0, 2, 8), 10 (0, 4, 6), 10 (0, 4, 8), and 10 (0, 6, 8), respectively. Although among these improper combinations of N (d 1, d 2, d ), only a small number of (n 1, n 2 ) pairs will cause singular ghost phasor matrices, the reconstructed images will contain serious remaining major ghost as a result of failed LSE solutions. These improper combinations of N (d 1, d 2, d ) should be avoided in a practical implementation. Experiments Spin-echo MRI slices in various anatomic regions and orientations were studied, including axial calves, sagittal knee, and axial head, all acquired on 1.5 T whole body clinical scanners. All data acquisitions were conducted in compliance with the institutional ethical guidelines. While other imaging parameters may vary, the k-space data matrices were all in size for simplicity. The full k-space data were retrospectively undersampled with all possible combinations of N (d 1, d 2, d ) and processed by the SPEED algorithm and evaluated for reconstruction quality. In practice, it is suggested to sample the three datasets, S 1 (k), S 2 (k), and S (k) in an interleaved fashion, to minimize the impact of potential motion during the scan. Data processing was implemented using MATLAB programming language on a personal computer with 2 GB random access memory and 2.26 GHz duo central processing unit. The image quality from G-SPEED was quantitatively evaluated with a previously proposed measure termed total relative error (TRE) (2 26). The full k-space data were

5 1678 Jin and Xiang Table 1 Minimum Rank Value of Ghost Phasor Matrices N (d1, d2, d) Minimum rank value Improper combination N (d1, d2, d) Minimum rank value 4 (0, 1, 2) 2 8 (0, 2, 5) 2 4 (0, 1, ) 2 8 (0, 2, 6) 1 4 (0, 2, ) 2 8 (0, 2, 7) 2 6 (0, 1, 2) 2 8 (0,, 4) 2 6 (0, 1, ) 2 8 (0,, 5) 2 6 (0, 1, 4) 2 8 (0,, 6) 2 6 (0, 1, 5) 2 8 (0,, 7) 2 6 (0, 2, ) 2 8 (0, 4, 5) 2 6 (0, 2, 4) 1 8 (0, 4, 6) 1 6 (0, 2, 5) 2 8 (0, 4, 7) 2 6 (0,, 4) 2 8 (0, 5, 6) 2 6 (0,, 5) 2 8 (0, 5, 7) 2 6 (0, 4, 5) 2 8 (0, 6, 7) 2 8 (0, 1, 2) (0, 1, ) 2 9 (0,, 6) 1 8 (0, 1, 4) 2 10 (0, 2, 4) 1 8 (0, 1, 5) 2 10 (0, 2, 6) 1 8 (0, 1, 6) 2 10 (0, 2, 8) 1 8 (0, 1, 7) 2 10 (0, 4, 6) 1 8 (0, 2, ) 2 10 (0, 4, 8) 1 8 (0, 2, 4) 1 10 (0, 6, 8) 1 Improper combination reconstructed by standard inverse DFT into gold-standard images I g (x, y) for comparison. Reconstruction errors were quantified by using TRE as defined by Eq. (15). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2 TRE ¼ I 0 ðx; yþ I g ðx; yþ = X I g ðx; yþ [15] x;y x;y where I 0 (x, y) is the image reconstructed from undersampled data; both I 0 (x, y) and I g (x, y) are in magnitude form. RESULTS Figure 2 shows deghosted edge maps from a spin-echo sagittal knee scan with all 10 possible sampling combinations for N ¼ 6,namely,6(0,1,2)(a),6(0,1,)(b), 6 (0, 1, 4) (c), 6 (0, 1, 5) (d), 6 (0, 2, ) (e), 6 (0, 2, 4) (f), 6 (0, 2, 5) (g), 6 (0,, 4) (h), 6 (0,, 5) (i), and 6 (0, 4, 5) (j), respectively. Strong remaining major ghosts can be seen only in Figure 2f with sampling combination of 6 (0, 2, 4). Most other sampling combinations of N ¼ 6 can produce successful deghosted edge maps, and thus result in final reconstructions with comparable quality, as shown in 6 (0, 1, 2) (Fig. 2l), 6 (0, 1, ) (Fig. 2m), 6 (0, 1, 4) (Fig. 2n), 6 (0, 1, 5) (Fig. 2o), 6 (0, 2, ) (Fig. 2p), 6 (0, 2, 5) (Fig. 2q), 6 (0,, 4) (Fig. 2r), 6 (0,, 5) (Fig. 2s), and 6 (0, 4, 5) (Fig. 2t), respectively. Some structure-mimicking residual minor ghosts can be seen in the center of the field of view in Figure 2m o,r,t as indicated by short white arrows. Figure 2k displays the gold standard image from the full k-space data using traditional Fourier reconstruction. The difference between Figure 2k and other reconstructions is very little, suggesting a satisfactory result from SPEED with N ¼ 6except6(0,2,4)sampling combination. Table 1 displays the minimum rank values of P for all sampling combinations with composite values at N ¼ 4, 6, and 8. As the combinations of 6 (0, 2, 4), 8 (0, 2, 4), 8 (0, 2, 6), 8 (0, 4, 6) have low-rank value of 1, their corresponding LSE ghost solution will break down, resulting in strong remaining major ghost in edge maps, causing failure in the SPEED reconstruction. Table 1 also contains minimum rank values of combinations for N ¼ 9 and 10 with low-rank values, these combinations are 9 (0,, 6), 10 (0, 2, 4), 10 (0, 2, 6), 10 (0, 2, 8), 10 (0, 4, 6), 10 (0, 4, 8), and 10 (0, 6, 8), respectively, among all 21 combinations for N ¼ 9, and 81 combinations for N ¼ 10. There are totally 11 improper combinations associated for N ¼ 6, 8, 9, and 10, all with a minimum rank value of 1, indicated by sign as shown in Table 1. They should not be used in SPEED sampling. Table 1 does not include cases when the PE skip size N is a prime number. In such cases, the minimum rank value of ghost phasor matrices for all possible sampling combinations is always the full rank value of 2, as can be expected. Figure a is the gold-standard image from full k-space data of an axial head scan. Figure b is reconstructed from one of the three interleaved datasets with a direct inverse DFT, followed by a differential operation (Fig. c), with 6-fold ghosting because of the PE skip size N ¼ 6. Figure d is the edge map E 0 (r) after deghosted by LSE solution, followed by registration and summation. Figure e is the final deghosted image I 0 (r), Figure f is a residual map of Figure e. The TRE of Figure e is (6.72e 4). Figure g l is final deghosted images with sampling combinations of 4 (0, 1, ), 5 (0, 1, 4), 7 (0, 2, ), 8 (0,, 7), 9 (0,, 8), and 10 (0, 7, 8). Their TRE values are 4.1e 4, 4.44e 4, 6.66e 4, 8.77e 4, 8.45e 4, and 8.95e 4, respectively. These TRE values grow progressively with N as they should be, but all represent a reasonable reconstruction quality.

6 Generalized SPEED 1679 FIG.. Results from a spin echo axial head scan. a: This is the gold-standard image from full k-space data. b: This image is reconstructed from one of the three interleaved datasets with a direct inverse DFT, followed by a differential operation (c), with 6-fold ghosting due to PE skip size N ¼ 6. d: This is the edge map E 0 (r) after deghosted by LSE solution, followed by registration and summation. e: This is the final deghosted image I 0 (r), (f) is residual maps of (d). g l: These are final deghosted images using sampling combinations 4 (0, 1, ), 5 (0, 1, 4), 7 (0, 2, ), 8 (0,, 7), 9 (0,, 8), and 10 (0, 7, 8), respectively, all with reasonable reconstruction quality. The short white arrows show some residue minor ghosts. Figure 4a shows the mean TRE values of all proper combinations for each N using axial calves, axial head, and sagittal knee datasets, respectively. These values all grow progressively with N as expected. Figure 4b shows the TRE values of all proper combinations for N ¼ 6, corresponding to combinations of 6 (0, 1, 2), 6 (0, 1, ), 6 (0, 1, 4), 6 (0, 1, 5), 6 (0, 2, ), 6 (0, 2, 5), 6 (0,, 4), 6 (0,, 5), and 6 (0, 4, 5), respectively. The combination with minimum TRE value varies with datasets for N ¼ 6. DISCUSSION It was previously known that when the PE skip size N is a prime number (N ¼ 5, 7, 11), PE shifts can be selected arbitrarily (21 25). However, N does not have to be limited to prime numbers. When N is a composite number (N ¼ 4, 6, 8, 9, 10), appropriate choice of PE shifts can still ensure reconstruction quality. Only a small number of N (d 1, d 2, d ) improper combinations were found to result in failed reconstructions, corresponding to strong remaining major ghosts and low minimum rank value of the ghost phasor matrix P. There are a total of 11 improper combinations for N ¼ 6, 8, 9, and 10, all indicated in Table 1 with a minimum rank value of 1. The ghost phasor matrices depend only on the sampling schemes, not the specific data, and therefore can be analyzed in advance. With the extension of N from prime numbers to all integers, more general and flexible sampling schemes can be used in SPEED, offering considerably more freedom in practical implementations and applications. FIG. 4. Reconstruction errors are plotted as the mean TRE values of all candidate combinations for each N (a). For any N, reconstruction error of sagittal knee dataset is much smaller than that of axial head and axial calve datasets, but these TRE values all grow progressively with N. b: plots the TRE values of all proper combinations for N ¼ 6, corresponding to combinations of 6 (0, 1, 2), 6 (0, 1, ), 6 (0, 1, 4), 6 (0, 1, 5), 6 (0, 2, ), 6 (0, 2, 5), 6 (0,, 4), 6 (0,, 5), and 6 (0, 4, 5), respectively. The combination with minimum TRE value varies with datasets for N ¼ 6.

7 1680 Jin and Xiang As SPEED acquires three datasets with PE skip size N, its scan time is reduced by a factor of approximately N/ compared with an ordinary DFT MRI. The larger the N, the greater potential savings in scan time, but also the greater probability that more than two layers of ghosts will overlap in a pixel. When N was limited to prime numbers, there are only three choices 5, 7, and 11 for N < 12, and only three scan-time reduction factors were possible. They were 5/, 7/, and 11/. With the proposed G-SPEED, more data reduction factors, for example, 4/, 6/, 8/, 9/, and 10/, become available and can be selected freely after extending the PE skip size N to composite numbers. Therefore, reasonable candidates of N would be extended to 4, 5, 6, 7, 8, 9, 10, and 11. These possibilities provide more general and flexible sampling schemes for realistic implementations. Like other scan time compression methods, higher acceleration factors are achieved at a price of lower reconstruction quality. Typically, the scan time reduction of SPEED is limited by a factor of approximately. (e.g., N ¼ 11 with 8 2 extra lines of central k-space data) (22,27), beyond which reconstruction artifacts will become rather serious. Although the strong remaining major ghosts can be avoided by proper choice of sampling combinations N (d 1, d 2, d ) guided by rank values, some minor ghosts indicated by short white arrows in Figure d,j, can still be seen in some deghosted maps, and also obvious structure-mimicking residual minor ghosts in the center of the field of view in Figure 2 indicated by short white arrows, which are often data specific. The mechanism of these minor ghosts is not completely understood and remains to be further investigated. Several factors, alone or in combinations, may be responsible for these minor ghosts. They include, for example, the data sparsity, more than two ghost layers overlapping in a pixel, and the noise. The relative ghost phase for each pixel using different N (d 1, d 2, d ) combinations may be another contributing factor. The fact that SPEED achieves ghost separation pixel by pixel can be a fundamental reason for noise sensitivity. For example, the overdetermined system is solved by LSE solution, which is implemented pixel by pixel. The noise-like artifacts appears if the minimum of LSE is not unique in testing all possible (n 1, n 2 ) pairs while comparing the resulting LSE. The LSE solution for (n 1, n 2 ) at each pixel and the residual artifacts are data dependent, and could depend on the sampling pattern. The optimal (n 1, n 2 ) pair is chosen for each pixel if the residual of the solution r is minimized, without considering global effect. If there are multiple (n 1, n 2 ) pairs that achieve similar (or identical) r, we simply select the first (n 1, n 2 ) pair encountered. Future investigations are needed to overcome the limitation of G-SPEED in terms of minor ghost. When N is a prime number, the choice of PE shift d is found to be unimportant in traditional SPEED, as any d in the range of (0, 1, 2,..., N 1) will result in the same set of N ghost phasors uniformly distributed along the unit circle, leading to similar results (22). However, in this study without the prime number restriction for N, d becomes important to prevent the strong remaining major ghosts. These choices even affect the appearance of minor ghosts. Both the PE skip size N and PE shift d, that is, the combination of N (d 1, d 2, d ), were found to be important to get a high quality reconstruction. Generally, the sampling pattern with more uniform k-space gaps among d 1, d 2, and d can result in better reconstruction quality. In this article, G-SPEED was only demonstrated with a double-layer ghost signal model as shown in Eqs. [1] []. This is not a fundamental limitation. Signal model with more layers of ghosts can also be used for G-SPEED, with the rank of ghost phasor matrix as criterion to guide the choice of proper sampling schemes. However, exploring all the possibilities in this direction is beyond the scope of this article. Different regions of human body were tested for G-SPEED and all produced satisfactory results. However, the TRE value varied with different N (d 1, d 2, d ) combinations, for example, when N ¼ 6, the selection of combinations with minimum TRE value has some data dependence as shown in Figure 4b. How to select the optimized combinations N (d 1, d 2, d ) is a new challenge using G-SPEED. The data sparsity in different domain (28), the overlapping area, and the intensity of acquired k-space signal may contribute to the quality of final deghosted images. The reconstruction time for the gold standard DFT is about 0.16 s using MATLAB. The reconstruction time of G-SPEED increased with the N value as more (n 1, n 2 ) trials need to be tested in LSE solution for a larger N. Itis approximately ranged from 0.8 (N ¼ 4) to 4.5 (N ¼ 11) s for a image using MATLAB, which can be shortened if C programming language is used for coding. The algorithm of G-SPEED is based on analytical solution with limited number of trials, so it is attractive for efficient reconstruction in clinical applications. In this article, only straightforward examples are used to demonstrate the basic principles of G-SPEED. Many variations and applications of traditional SPEED are also possible for G-SPEED implementations. As an independent approach, G-SPEED can also be combined with existing fast imaging methods, such as partial Fourier (6 10), for further acceleration. The SPEED-array coil enhancement is a combination between SPEED and SENSE, where only one set of data with skip size N was acquired. For this variation, there is no prime number restriction for N, just like SENSE. It is only the singlecoil SPEED that has prime number restriction. In summary, this study revealed that although the prime number requirement for N is helpful, this restriction can be very much relaxed. Composite numbers combined with appropriately chosen PE shifts can also yield good reconstruction results. The more general and flexible sampling schemes offer considerably more freedom in practical implementations and applications of SPEED. REFERENCES 1. Mansfield P. Multi-planar image formation using NMR spin echoes. J Phys C: Solid State Phys 1977;10:L55 L Ahn CB, Kim JH, Cho ZH. High-speed spiral-scan echo planar NMR imaging. IEEE Trans Med Imaging 1986;MI-5:2 7.. Haase A, Frahm J, Matthaei D, Hanicke W, Merboldt KD. 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