Computation of Exact g-Factor Maps in 3D GRAPPA Reconstructions (Submitted to Magnetic Resonance in Medicine

Purpose: To characterize the noise distributions in 3D-MRI accelerated acquisitions reconstructed with GRAPPA using an exact noise propagation analysis that operates directly in k–space. Theory and Methods: We exploit the extensive symmetries and separability in the reconstruction steps to account for the correlation between all the acquired k–space samples. Monte Carlo simulations and multirepetition phantom experiments were conducted to test both the accuracy and feasibility of the proposed method; an in–vivo experiment was performed to assess the applicability of our method to clinical scenarios. Results: Our theoretical derivation shows that the direct k–space analysis renders an exact noise characterization under the assumptions of stationarity and uncorrelation in the original k–space. Simulations and phantom experiments provide empirical support to the theoretical proof. Finally, the in–vivo experiment demonstrates the ability of the proposed method to assess the impact of the sub-sampling pattern on the overall noise behavior. Conclusions: By operating directly in the k–space, the proposed method is able to provide an exact characterization of noise for any Cartesian pattern sub-sampled along the two phase–encoding directions. Exploitation of the symmetries and separability into independent blocks through the image reconstruction procedure allows us to overcome the computational challenges related to the very large size of the covariance matrices involved. INTRODUCTION Characterization of noise statistics in MR images is essential for multiple applications including quality assurance (1), (2), protocol optimization (3), (4), and tailoring of subsequent post– processing steps (5), (6), (7). For fully–sampled acquisitions, the noise distribution is spatially homogeneous, and is well modelled by a zero–mean spatially uncorrelated independent and identically distributed (IID) complex Gaussian process for each coil . However, in the presence of parallel MRI (pMRI) acceleration, the noise distribution (e.g. the pixel-wise noise standard deviation) generally has a strong spatial dependence (8), (9). For these reasons, there is a clinical and research need for accurate estimation of the pixel-wise maps of noise standard deviation (i.e. ‘noise maps’), that is widely applicable to common pMRI methods. In particular, commonly used pMRI methods where the reconstruction takes place in the k– space, such as generalized autocalibrating partially parallel acquisitions (GRAPPA) (10), introduce correlations in the k–space that propagate into the x–space. For GRAPPA reconstructions, a If the data are acquired by several receiving coils, the multi-coil noise can be characterized by its covariance matrix 4 direct noise propagation analysis requires operating with very large covariance matrices (11) and renders this analysis challenging. In recent work, we have shown that, by exploiting the extensive symmetries and separability in each step of the reconstruction, a computationally efficient and exact noise analysis can be obtained for 2D Cartesian acquisitions (12), where acceleration is applied along a single (phase-encoding) dimension. However, no feasible and exact noise analysis has been reported for 3D Cartesian acquisitions, which are commonly obtained with acceleration in two dimensions. Indeed, 3D acquisitions are commonplace in clinical and research MRI exams, both in brain and body imaging applications. Compared to 2D imaging, 3D acquisitions enable high (eg: isotropic) spatial resolution with high signal-to-noise ratio (SNR) efficiency. Further, pMRI techniques can be combined with 3D imaging by sub-sampling the k–space along both phase–encoding directions (13), resulting in highly flexible sub-sampling schemes. However, GRAPPA reconstruction in 3D imaging introduces correlations in k–space across the three dimensions, notably increasing the size of the matrices involved in the noise propagation analysis. For this reason, noise characterization in 3D imaging is substantially more challenging than in 2D imaging. Two previously proposed approaches to estimate the noise maps are given by the Monte–Carlo (11) and by the x–space methods (14). The Monte–Carlo method is based on repeatedly corrupting the acquired data with synthetic noise properly scaled and correlated, and assessing the empirical statistics of the GRAPPA reconstructions resulting from these noisy data. Unfortunately, this method is inherently time consuming and is only able to provide a noisy estimate of the noise–maps due to the limited number of Monte–Carlo realizations. In contrast, x–space methods reformulate the GRAPPA reconstruction as a pixel-wise multiplication in x–space, providing computationally efficient noise characterization by avoiding the extensive k–space correlations. However, as shown in Refs. (15), (16), the x–space reconstruction is not exactly equivalent to a GRAPPA reconstruction, rendering systematic errors in the estimated noise–maps (12). Further, errors in x–space methods are generally associated with the boundaries between k–space regions with different different sampling patterns (12), which is troublesome in 3D acquisitions obtained with 5 highly non–uniform sub-sampling patterns. In this work, we propose a method for exact noise characterization for 3D GRAPPA reconstructions under the assumptions of stationarity and uncorrelation in the original k–space sub-sampled acquisition. The proposed method extends a recently reported 2D approach (12) to the more challenging analysis of 3D imaging, presenting an exact and computationally efficient solution by exploting the extensive symmetries and separability in each step of the reconstruction. We demonstrate the accuracy of the proposed method under multiple sub-sampling scenarios, including various sub-sampling patterns (Rectangular-type, CAIPIRINHA-type or Random), autocalibration region shapes (rectangular or ellipsoidal) and sampling density (uniform or variable across the k–space), on both synthetic and real phantoms. Finally, an in–vivo experiment is performed to illustrate the applicability of the proposed noise characterization method. THEORY Overview of the problem Parallel MRI techniques achieve acceleration by sub-sampling the k–space data acquisition. In 3D Cartesian acquisitions with parallel imaging, the 3D data-set is sub-sampled along two phase encoding directions in order to exploit the sensitivity variations of the receiver coils in these two spatial dimensions. The acquired data across the L coils ~s(k) = [s1(k) · · · sL(k)] can be modeled as in (17), (3): ~s(k) = ~a(k) + ~n(k;Γa,Ca), [1] where ~a(k) = [a1(k) · · · aL(k)] is the noiseless signal in the k–space and ~n(k;Γa,Ca) is the acquisition noise, which is assumed to follow a L-variate complex zero-mean normal distribution (see (18)) for each k–space point. This noise distribution is fully characterized by its covariance matrices Γa = E{~n · ~nH} and Ca = E{~n · ~nT}, where the operators T and H denote the transpose and Hermitian operator of a matrix, respectively. Furthermore, we also assume the acquisition noise in MRI is stationary (17), i.e. Γa and Ca do not depend on k. GRAPPA reconstructs the k–space for each coil by filling the missing points with a linear com6 bination of the acquired points within its neighborhood η(k) across all the coils (10): sRl (k) = L ∑