Phase-constrained parallel MR image reconstruction

Phase-Constrained Parallel MR Image Reconstruction: Using Symmetry to Increase Acceleration and Improve Image Quality J. D. Willig-Onwuachi1, E. N. Yeh2, A. K. Grant1, M. A. Ohliger2, C. A. McKenzie1, D. K. Sodickson1 1

Beth Israel Deaconess Medical Center and Harvard Medical School, Boston, MA, United States, 2Harvard-MIT Division of Health Sciences and Technology, Boston, MA, United States Synopsis: A generalized method for phase-constrained parallel magnetic resonance image reconstruction is presented. This method can be used to reconstruct traditional partial-Fourier data as well as trajectories not simply handled by partial-Fourier approaches, including variable-density (self-calibrating) trajectories for parallel imaging. For full-Fourier applications, use of data symmetry results in reduced noise multiplication and higher achievable acceleration factors. In some circumstances, this new method allows a near doubling of achievable acceleration, even enabling acceleration factors larger than the number of array elements. At fixed acceleration factors, this phase-constrained method shows improved g-factor and signal-to-noise ratio compared with standard generalized encoding matrix methods. Introduction: Parallel MRI accelerates imaging using a reduced data set. Data points omitted from the acquisition are filled in during reconstruction by exploiting the information from multiple radiofrequency coils. Other techniques prior to the advent of parallel MRI have also used a reduced data set, including phase-constrained (partial-Fourier) methods such as the Margosian method (1,2) or POCS (3). These constrained reconstructions (4) use a priori knowledge of data symmetry to fill in portions of k-space not acquired during the scan. In this abstract we summarize one method of combining parallel MRI and phase-constraint concepts in a general formulation that permits the use of k-space trajectories not compatible with standard parallel or partial Fourier MRI. Theory: In its discretized form, the demodulated MR signal induced in a bar ρ (rj ) = exp(iϕ m (rj )) ρ (rj ), Pjj ≡ exp(iϕ m (r j )), Pj , j ' ≠ j ≡ 0 [1] coil from the spin excitation in an arbitrary image plane or volume can be written as S=Bρ, as in the generalized encoding matrix (GEM) (5) or generalized SENSE (6) formulations. Here S and ρ are vectors containing  Re S   Re( BP )  bar bar bar bar = ρ or S =B ρ [2] the measured signal data in all coils and the magnetization density,  Im S   Im( BP )  respectively, and B is an encoding matrix containing the coil sensitivity Cl and gradient coil modulations: Bp(k,l),j=Cl(rj)exp(-ik.r j). If we represent ρ as σj bar bar bar bar † some underlying magnetization density modulated by a measurable phase g = , σ 2j = Binv Binv , σ 2j ,full = l ,l ′ Clj* Ψ ll−1′ Cl ′j [3] jj full ϕm(rj) common to all coils, we can then rewrite the signal equation as σj R S=BPρbar where P is a diagonal matrix containing the common phases (Eq. bar [1]). If we can measure or estimate the phase ϕm(rj) with sufficient accuracy, so that ρ is real, then the signal equation simplifies and can be written as in Eq. [2]. Inverting the extended encoding matrix Bbar and multiplying by the extended signal vector Sbar reconstructs the target magnetization density ρbar. Maximal SNR is achieved using a modified Moore-Penrose pseudoinverse (5,6). The g-factor can be written as in Eq. [3]. The noise variance σ2 in any voxel j of the reconstructed image is also shown in Eq. [3] and involves a transformed noise correlation matrix, Ψbar, which includes separate correlations of real and imaginary noise. In the current notation, σ2j,full can be rewritten for Cartesian trajectories in terms of the sensitivities and the standard noise resistance matrix, Ψ.

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Methods: In this abstract we present data from simulations, phantom scans, and in vivo scans. For simulations, a real-valued image multiplied by coil sensitivities derived from a Biot-Savart calculation was Fourier transformed and decimated to yield simulated signal data. Gaussian noise with σ = |Smax|/100 was added to the signal. Scan data were acquired on a Siemens Symphony magnet with quantum gradients [Siemens Medical Systems, Erlangen, Germany]. Combined coil sensitivity and phase information (BP in Eq. [2]) was derived from the same reference images obtained either from a separate acquisition or from internal calibration lines. Both simulations and experimental scans used a four-element linear coil array geometry with the image plane oriented parallel to the coil array. All reconstructions were performed by first forming either the traditional GEM B matrix or the expanded Bbar matrix as described in Eq. [2]. This matrix was inverted using line-by-line block diagonalization as described in (5) or by an iterative conjugate gradient approach similar to that described in (7). Maps of g-factor were calculated using Eq. [3]. Results: Fig.1 shows cross sectional plots of simulated g-factor maps for 2-fold (2x) and 3-fold (3x) undersampling using standard (top) and constrained (bottom) GEM. Trajectories with paired (e.g. …-2,0,2…) and unpaired (e.g. …-4,-2,0,1,3… or …-5,-2,1,4…) sets of conjugate k-space lines were tested, including: 1a) 2x paired, 1b) 2x unpaired, and 1c) 3x unpaired. Fig.2 compares standard (top) and constrained (bottom) GEM reconstructions of gradient echo data with acceleration factors ranging from 1 to 5 (4-coil array). A separate fully sampled reference image was used for extracting coil sensitivities and phases. Fig.3 shows a self-calibrated in vivo image acquired using a variable density HASTE sequence (half-Fourier with 32 fully sampled central lines and an outer reduction factor of 2).

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Discussion: We have presented data to illustrate specific benefits of the phase-constrained GEM reconstruction, including reduced noise amplification and increased acceleration factors. As a consequence of the phase constraint in constrained GEM, the solution vector is smaller than in standard GEM, and the inversion is more overdetermined. This results in more degrees of freedom to minimize noise multiplication. Alternatively, the additional degrees of freedom result from an effective separation of the real and imaginary sensitivities, doubling the number of independent sensitivities for spatial encoding. Aside from helping to reduce noise amplification, this also allows the extension of achievable acceleration factor beyond the number of coils. Note that high acceleration factors, for example R=5 in Fig.2, could in some cases be achieved by using a lower reduction factor and partial-Fourier methods with standard GEM. Constrained GEM, however, is a generalized method that allows these accelerations even for full-Fourier trajectories. In fact, the general formulation enables comparison of different undersampling schemes to determine those trajectories that give the lowest g-factor. Phase encoding schemes that use lines placed asymmetrically about k=0 (unpaired) benefit the most from constrained GEM (e.g. Fig.1b,c), since they contain the largest quantity of distinct information, but benefits are also seen for paired trajectories (e.g. Fig.1a & Fig.2). The intrinsic rephasing of the image in constrained GEM also performs an automatic segregation of the real channel, eliminating noise from the imaginary channel which, in the absence of explicit correction, enters into unconstrained magnitude images. This feature yields improved SNR even for unaccelerated images (Fig 2, 1x). Of course, use of constrained GEM requires accurate representations of the true phase variations present in the imaged object. Tissue interfaces with large susceptibility discontinuities can pose problems especially for low-resolution calibrations, since sharp changes in phase require high resolution phase information. References: 1. Margosian P, et al. Health Care Instrum 1986; 1: 195-197. 2. Noll DC, et al. IEEE Trans Med Imag 1991; MI-10: 154-163. 3. Cuppen JJ, Van Est A. Top. Conf. Fast MRI Tech, Cleveland, Ohio 1987.

Proc. Intl. Soc. Mag. Reson. Med. 11 (2003)

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