Improved 3D image plane parallel magnetic resonance imaging (pmri) method

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1 B. Wu, R. P. Millane, R. Watts, P. J. Bones, Improved 3D Image Plane Parallel Magnetic Resonance Imaging (pmri) Method, Proceedings of Image and Vision Computing New Zealand 2007, pp , Hamilton, New Zealand, December Improved 3D image plane parallel magnetic resonance imaging (pmri) method B. Wu 1, R. P. Millane 1, R. Watts 2 and P. J. Bones 1 1 Computational Imaging Group, Dept. Electrical & Computer Engineering, University of Canterbury, Christchurch, New Zealand. 2 Dept. Physics & Astronomy, University of Canterbury, Christchurch, New Zealand. bing.wu@elec.canterbury.ac.nz Abstract A new image plane parallel magnetic resonance imaging (pmri) method, Generalized Unaliasing Incorporating object Support constraint and sensitivity Encoding (GUISE), was recently introduced. It allows images to be recovered from arbitrary cartesian k-space (spatial frequency space) sampling patterns, restricted for efficiency to be those consisting of repeated blocks. Sensitivity encoding (SENSE) is seen as a special case of GUISE where a minimal repeating block size is used. An efficient sampling pattern design method for GUISE is proposed in order to reduce the overall noise level in the reconstructed image. Experimental results show that GUISE with this sampling pattern method provides better image reconstruction than SENSE. The improvement is attributed to better exploitation of the differences in spatial sensitivity across the set of receiver coils and the object support constraint. Keywords: 3D MRI, pmri, object support 1 INTRODUCTION 3D volumetric magnetic resonance imaging (MRI) allows images to be viewed in different planes independent of the order used for the data acquisition, and is widely used for breast imaging [1], brain imaging [2] and angiogram imaging [3]. However, clinical usage is limited by the long scan time. Recent development of parallel MRI (pmri) techniques have significantly reduced the necessary data acquisition time of 3D MRI examinations [3]. In general, pmri methods employ multiple receiver coils during data acquisition, and the different data sets gained from the coils, each of which has different spatial sensitivity variation, are exploited in post-processing to compensate for the skipped phase encoding (PE) steps. Among various pmri methods proposed to date, SENSitivity Encoding (SENSE) [4] has received much attention and has been implemented on commercial MR scanners because of its simplistic nature and high signal-tonoise ratio (SNR) performance. We have recently introduced a new pmri method, Generalized Unaliasing Incorporating object Support constraint and sensitivity Encoding (GUISE) [5], which generalizes the image plane pmri methods. In both SENSE and GUISE, a series of matrix inversions are to be performed to separate the folded voxels using knowledge of the coil sensitivity weighting for direct image recovery. The key difference is that SENSE results in fixed aliasing pattern whereas the aliasing pattern in GUISE is determined by the sampling pattern adopted. We postulate that this freedom in the aliasing pattern design allows more efficient exploitation of coil sensitivity and the object support region, which leads to improved SNR in the reconstructed image. In this paper, we present an overview of GUISE and propose a sampling pattern design method. We then compare the performance of GUISE with SENSE based on experimental data. 2 THEORY The term k-space is widely used in the MRI community to refer to the spatial frequency space in which data is acquired. In 3D MRI, the measured k-space data represent a 3D Fourier encoding of the magnetization signal. However, inverse discrete Fourier transformation along the fully sampled frequency encoding (FE) direction allows the 3D k-space to be decomposed to its 2D multi-slice equivalent. Each resulting image slice is thereby defined in terms of the two PE directions and reconstructions can be performed on individual slices to restore the entire 3D image. For ease of illustration and without loss of generality, we there- 311

2 fore present our work based on a single 2D plane and considerations for 3D space are given where needed. 2.1 Formulation of GUISE Consider the case of M receiver coils. The k-space data from the mth coil represents the 2D Fourier transform of the coil sensitivity weighted object magnetization. In the case of a full set of samples at Nyquist spacing, the system can be represented conveniently as F m = W diag(c m ) f, (1) where f, c m, and F m represent the stacked column vectors of the 2D object magnetization, the mth coil sensitivity map, and the acquired samples from the mth receiver coil, respectively. Diag(x) refers to a diagonal matrix with vector x on the main diagonal and zero elsewhere. W is the stacked Fourier matrix, left multiplication by which performs the 2D DFT. Under-sampling in k-space is equivalent to deleting the entries corresponding to skipped sample locations, which can be expressed as diag(h) F m = diag(h) W diag(c m ) f, (2) where H is a 2D k-space binary sampling mask stacked into a column vector. Inverse DFT of the under-sampled data set gives an image vector, f m, that contains aliasing artifacts: fm = W 1 diag(h) F m = W 1 diag(h) W diag(c m ) f = A diag(c m ) f, (3) where A is a matrix that maps the coil sensitivity weighted object magnetization to the aliased image, which we call a Fourier encoding matrix. The aliasing artifacts cannot be removed by inverting A directly because A is rank deficient. For multiple receiver coils, we can create a new hybrid encoding system, B, that maps the object magnetization to the aliased versions in different coils. Mathematically, this is achieved by stacking the aliased images from different coils and the corresponding encoding functions, i.e., B f = f, B = f = A diag(c 1 ) A diag(c 2 ) A diag(c M ) f1 f2 fm,. (4) B is invertible as long as the number of independent rows is greater than or equal to the number of independent columns. Due to distinct coil spatial sensitivity weightings, the simultaneously acquired samples in different coils are independent and the row rank increases by the number of phase encoding steps for every coil used. Thus an acceleration factor up to M is possible. Knowledge of the object support allows us to eliminate the regions outside the object from reconstruction, as they are known to generate zero signal. Only those entries in f that lie within the support region and the corresponding columns in B are retained. In the interests of brevity, we henceforth denote the reduced versions of f and B as f[ρ] and B[ρ], respectively, thus B[ρ] f[ρ] = f. (5) Denoting the ratio of the object support area to the total FOV area as φ, where φ 1, the maximum possible acceleration factor for recovering the 2D image is improved by the use of the support constraint to α max = M/φ. 2.2 System decomposition The system in (5) poses a problem for efficient matrix inversion as a result of its size, but this can be relieved by employing a periodic k-space sampling pattern that decomposes the overall system [6]. In a FOV of size N 1 N 2, if the k-space sampling pattern corresponding to H consists of a repeating pattern of size C 1 C 2 whose stacked column vector is G, then its point spread function has a sparse structure with nonzero values separated by L 1 = N 1 /C 1 and L 2 = N 2 /C 2 voxels in each respective direction. As a result, aliasing takes place in a well controlled manner, as depicted in Figure 1: only voxels that are separated by integer multiples of L 1 and L 2 in each direction respectively are aliased together, forming what we denote as a subsequence. Voxels within each such aliased subsequence are independent from voxels in all other sub-sequences [6]. Thus the overall system can be decomposed into L = L 1 L 2 sub-sequences in the above manner, and a sub-system matrix can be set up for the lth subsequence: B l [ρ l ] f l [ρ l ] = f l, fl = B l [ρ l ] = fl,1 fl,2 fl,m Â[ρ l ] diag(c l,1 [ρ l ]) Â[ρ l ] diag(c l,2 [ρ l ]) Â[ρ l ] diag(c l,m [ρ l ]),, (6) 312

3 2.3.1 Choosing the block size Figure 1: A periodic k-space sampling pattern leads to decomposition of the image plane system. On the left, the central region of the k-space sampling pattern, consisting of a 4 4 repeating block, is shown (bright and dark parts represent acquired and skipped samples, respectively). Aliasing in the resulting image plane is limited to occur within sub-sequences of voxels, one of which is depicted on the right as a regular set of white dots. For each such sub-sequence only those voxels which fall within the support region (outlined in white) need to be recovered. The the final 2D image is reconstructed by assembling all the recovered sub-sequences. Each image plane has its own support region. where  l [ρ l ] = (W 1 diag(g)w)[ρ l ], l = 0, 1, 2,..., L 1 L 2 1, and l is used to denote quantities related to the lth sub-sequence only. For the lth sub-sequence we describe the ratio of the number of voxels within the object support to the total number of sub-sequence voxels as the subsupport ratio, denoted φ l. The entire image can be recovered by inverting a series of sub-matrices followed by appropriate reordering. 2.3 Noise propagation We have shown in [5] that the overall noise amplification level in the reconstruction can be written as: P 1 = trace(b[ρ]b[ρ] H ) +, (7) σ 2 i=1 i where σ denotes the singular value of B and P is the total number of voxels to be recovered. We refer this quantity as inverse trace metric (IT metric) and use it as a measurement tool to judge the noise immunity of different sampling strategies. In the case of using periodic sampling pattern, this quantity can be written as the sum of the IT metrics of individual sub-systems: trace(b[ρ]b[ρ] H ) + = L trace(b l [ρ l ]B l [ρ l ] H ) +. l=1 (8) Recall that the block has dimensions C 1 C 2. To achieve an acceleration factor of α, appropriate block sizes are restricted to those which make C 1 C 2 an integer multiple of α. This still leaves considerable freedom in choosing the block size. We now consider the factors in choosing the block parameters, first in terms of shape, then in terms of overall size. As can be seen in Figure 2, the values of C 1 and C 2 are directly reflected in the number of voxels being aliased in the corresponding direction. There is an advantage therefore in choosing the larger of C 1 or C 2 to correspond to the direction in which there is a greater variation in coil sensitivity. There is also an advantage to be gained by choosing the larger of C 1 or C 2 to correspond to the direction for which the out-of-support region is larger (i.e., for which the distance from the edge of the frame to the support is greater.) Assuming that the constraints above have been satisfied, we now consider the effects of choosing an overall large block size. As illustrated in Figure 2, larger k-space block sizes lead to a reduced number of sub-sequences in the spatial domain. Likewise the spatial extent of each sub-sequence is increased and sub-sequence elements are more closely packed. Based on the knowledge that coil sensitivities are smoothly varying spatial functions and object support areas are generally contiguous, we postulate that the use of large block sizes has the following influence: 1. The range of sensitivity variation within each sub-sequence is enhanced and finer. This can be exploited by using appropriate sampling patterns for improved reconstruction, however it also means that bad sampling patterns would lead to worse results compared to the case of using smaller block sizes. 2. Voxels within the object support are more evenly distributed among the resulting subsequences. Applying the support constraint therefore increases the degree of determinacy of the sub-systems, and the determinacy for each individual sub-system is determined by the sub-support ratio, φ l. Thus the distribution of φ l has a large impact on the noise profile and an uneven distribution of φ l values, as illustrated in Figure 2(a), leads to a nonuniform noise profile in the reconstruction. Furthermore, because the noise amplification level (IT metric) of a particular sub-system grows rapidly with increasing φ l, a narrow distribution of the φ l values is also a favorable condition for a lower overall noise level. 313

4 Figure 2: In an image plane of size , sub-systems are shown which correspond to k-space block sizes of (a) 2 4, (b) The histogram of all the φ l values indicates that using larger block size leads to an even distribution of the voxels within the support region (the central shaded circular region) and also a lower peak φ l. Thus expanding the k-space block size promotes the maximum acceleration factor achievable Sample selection The actual repeating pattern determines how the voxels are aliased and is a critical factor in the conditioning. Although periodic sampling reduces the problem size very significantly, an exhaustive search for the optimal pattern is still computationally intractable. Rather we propose to use the sequential forward selection (SFS) algorithm originally proposed by Gao et al [7], which sequentially selects sample positions based on minimizing a specified cost function in each step. However, the IT metric cost function involves a computationally expensive matrix inverse. We have shown previously [5] that the sum of singular values raised to the fourth power, i.e., P σi 4 = trace(b[ρ]b[ρ] H ) 2, (9) i=1 which we refer to as the squared trace metric (ST metric), exhibits similar general behavior to the IT metric under the process of minimization. Although the ST metric appears computationally expensive, very efficient calculation can be achieved in a sequential algorithm. 3 METHODS To compare the performance of GUISE and SENSE, a 3D brain scan of a healthy adult volunteer was performed on a GE 1.5T scanner using an 8-channel head coil. Informed consent was obtained prior to the study. The two PE directions were set to leftright (LR) and anterior-posterior (AP), which correspond to the two directions with significant coil sensitivity variation, and FE is set to the superiorinferior (SI) direction. T 1 weighted 3D Spoiled Gradient Recalled (SPGR) was used with the following sequence parameters: T R = 23ms, T E = 10ms, flip angle = 15, matrix = , resolution = 1.5mm 0.75mm 1.5mm. A full coronal plane data set was acquired to allow reconstructions with different sampling patterns to be performed in post-processing and the inverse DFT was performed along the FE direction prior to further reconstruction in individual axial slices. Image reconstructions using GUISE and SENSE methods were implemented in Matlab R (The Mathworks, Natick, MA, USA). 4 RESULTS Reconstructions of axial plane images using SENSE and GUISE at a sampling acceleration factor of 8 are shown in Figure 3. In SENSE, asymmetrical block sizes with a greater LR dimension than the AP dimension are chosen since there is more nonobject margin in the LR direction. In GUISE, a repeating block size of 8 8 is used and samples have been sequentially selected based on minimizing the overall ST metric. It is seen that in both cases GUISE method results in comparatively more uniform noise profile. The overall noise amplification levels are compared using the normalized root mean squared distance (NRMS), defined by 314

5 Figure 3: Comparing the reconstructed axial plane image slices using SENSE and GUISE at an overall acceleration factor of 8. Reconstructed axial slice of the brain region: (a) reconstruction from full data acquisition (b) using GUISE (c) using SENSE. Reconstructed axial slice through the eyeball: (d) reconstruction from full data acquisition (e) using GUISE (f) using SENSE. The NRMS for each accelerated reconstruction is shown in the bottom corner. NRMS = P i=1 ( Ireference i I reconstructed i ) 2 P, i=1 Ireference i 2 where I i is the ith voxel and P denotes the number of voxels recovered within the support region. The reconstructions gained from full sample acquisition were used as reference images (Figure 3(a) and Figure 3(d)). The NRMS values in the lower right of Figure 3(b), (c), (e) and (f) indicate that GUISE also out-performed SENSE in providing a lower overall noise level. Reconstructed brain slice images had better SNR compared to reconstructed eyeball slice images as the former features a smaller object support region and therefore lower φ value than the latter. 5 DISCUSSION GUISE treats the image plane aliasing due to undersampling in k-space as convolving the image with the point spread function of the sampling pattern. In general for pmri methods, the knowledge of image plane coil sensitivity weighting is used to remove the aliasing artifacts. GUISE explicitly incorporates the knowledge of the object s spatial support constraint in the image recovery process, which improves the determinacy of the linear system. By introducing periodicity in the k-space sampling pattern, the overall image plane system can be decomposed into independent sub-systems so that image recovery efficiency is greatly improved. The size of the repeating block in k-space sampling pattern determines the spatial extent of the resulting sub-systems and consequently their coil sensitivity weighting and sub-support ratios. Sufficient exploitation of the coil sensitivity distinctness and the finite object spatial extent is restricted by the small repeating block size used in SENSE. However, the repeating sampling block needs to welldesigned to keep the keep the sub-systems wellconditioned, and it is a difficult task to choose a suitable pattern for the block. The SFS method based on an approximated cost function achieves a satisfactory balance between efficiency and optimality. Since decreasing marginal gain is expected as the block size is further expanded in return of dramatically increased sample selection complexity, a moderate block size should be used in practice. 315

6 6 CONCLUSION GUISE is a new parallel MRI method with high flexibility in sampling pattern design. GUISE with appropriate sampling patterns can achieve better image reconstruction than the currently favoured method in clinical practice, SENSE, at high acceleration factors. References [1] K. Akazawa, Y. Tamaki, T. Taguchi, Y. Tanji, Y. Miyoshi, and S. Kim, Preoperative evaluation of residual tumor extent by threedimensional magnetic resonance imaging in breast cancer patients treated with neoadjuvant chemotherapy, The Breast Journal, vol. 12(2), pp , [2] C. Helmchen, H. Rambold, A. Sprenger, C. Erdmann, and F. Binkofski, Cerebellar activation in opsoclonus: an fmri study, Neurology, vol. 61(3), pp , [3] R. Muthupillai, G. Vick, S. Flamm, and T. Chung, Time-resolved contrast-enhanced magnetic resonance angiography in pediatric patients using sensitivity encoding, Journal of Magnetic Resonance Imaging, vol. 17, pp , [4] M. Weiger, K. Pruessmann, and P. Boesiger, 2D SENSE for faster 3D MRI, MAGMA, vol. 14, pp , [5] P. Bones, B. Wu, R. Millane, and R. Watts, Support constraint in 3-D magnetic resonance imaging, in Signal Recovery and Synthesis (Postdeadline Papers), The Optical Society of America, Washington DC, United States of America, April 2007, p. SMD6. [6] P. Bones, N. Alwesh, T. Connolly, and N. Blakeley, Recovery of limited-extent images aliased because of spectral undersampling, JOSA A, vol. 18, pp , [7] Y. Gao and S. Reeves, Optimal k-space sampling in MRSI for images with a limited region of support, IEEE Trans. Med. Imaging, vol. 12, pp ,