On optimality of parallel MRI reconstruction in k -space

Estimation error bound for GRAPPA diffusion-weighted MRI

In recent years, diffusion weight magnetic resonance imaging (DW-MRI) has become one of the most important MRI imaging modalities. The importance of the DW-MRI grew thanks to the combination of parallel magnetic resonance imaging (pMRI) techniques with the echo-planar imaging (EPI), which minimize scan time and lead to reduced distortion, allowing the DW-MRI to become a routine clinical exam. Additionally, this has brought various new parameters that influence image quality and biomarkers used in DW-MRI. This work aims to investigate the effects of these parameters on the estimation quality, by using the Cramér-Rao bound tool, which gives analytical expressions of the lower limit on the estimation error variance of different DW-MRI variables when using the pMRI technique. In particular, these bounds will be used to study and optimize the impact of different factors of generalized autocalibrating partially parallel acquisition (GRAPPA) technique and system parameters on the estimation quality of the desired clinical metrics. Moreover, the obtained results of this study can be exploited and adapted in all human body DW-MRI clinical routines, further improving disease diagnosis, and tractography studies.

Improved Parallel Magnertic Resonance Imaging reconstruction with Complex Proximal Support Vector Regression

Generalized Auto-calibrating Partially Parallel Acquisitions (GRAPPA) has been widely used to reduce imaging time in Magnetic Resonance Imaging. GRAPPA synthesizes missing data by using a linear interpolation of neighboring measured data over all coils, thus accuracy of the interpolation weights fitting to the auto-calibrating signal data is crucial for the GRAPPA reconstruction. Conventional GRAPPA algorithms fitting the interpolation weights with a least squares solution are sensitive to interpolation window size. MKGRAPPA that estimates the interpolation weights with support vector machine reduces the sensitivity of the k-space reconstruction to interpolation window size, whereas it is computationally expensive. In this study, a robust GRAPPA reconstruction method is proposed that applies an extended proximal support vector regression (PSVR) to fit the interpolation weights with wavelet kernel mapping. Experimental results on in vivo MRI data show that the proposed PSVR-GRAPPA method visually improves overall quality compared to conventional GRAPPA methods, while it has faster reconstruction speed compared to MKGRAPPA.

On-the-Fly Adaptive ${k}$ -Space Sampling for Linear MRI Reconstruction Using Moment-Based Spectral Analysis

In high-dimensional magnetic resonance imaging applications, time-consuming, sequential acquisition of data samples in the spatial frequency domain ( -space) can often be accelerated by accounting for dependencies in linear reconstruction, at the cost of noise amplification that depends on the sampling pattern. Common examples are support-constrained, parallel, and dynamic MRI, and -space sampling strategies are primarily driven by image-domain metrics that are expensive to compute for arbitrary sampling patterns. It remains challenging to provide systematic and computationally efficient automatic designs of arbitrary multidimensional Cartesian sampling patterns that mitigate noise amplification, given the subspace to which the object is confined. To address this problem, this paper introduces a theoretical framework that describes local geometric properties of the sampling pattern and relates them to the spread in the eigenvalues of the information matrix described by its first two spectral moments. This new criterion is then used for very efficient optimization of complex multidimensional sampling patterns that does not require reconstructing images or explicitly mapping noise amplification. Experiments with in vivo data show strong agreement between this criterion and traditional, comprehensive image-domain- and -space-based metrics, indicating the potential of the approach for computationally efficient (on-the-fly), automatic, and adaptive design of sampling patterns.

Efficient operator splitting algorithm for joint sparsity-regularized SPIRiT-based parallel MR imaging reconstruction

Self-consistent parallel imaging (SPIRiT) is an auto-calibrating model for the reconstruction of parallel magnetic resonance imaging, which can be formulated as a regularized SPIRiT problem. The Projection Over Convex Sets (POCS) method was used to solve the formulated regularized SPIRiT problem. However, the quality of the reconstructed image still needs to be improved. Though methods such as NonLinear Conjugate Gradients (NLCG) can achieve higher spatial resolution, these methods always demand very complex computation and converge slowly. In this paper, we propose a new algorithm to solve the formulated Cartesian SPIRiT problem with the JTV and JL1 regularization terms. The proposed algorithm uses the operator splitting (OS) technique to decompose the problem into a gradient problem and a denoising problem with two regularization terms, which is solved by our proposed split Bregman based denoising algorithm, and adopts the Barzilai and Borwein method to update step size. Simulation experiments on two in vivo data sets demonstrate that the proposed algorithm is 1.3 times faster than ADMM for datasets with 8 channels. Especially, our proposal is 2 times faster than ADMM for the dataset with 32 channels.

3D-accelerated, stack-of-spirals acquisitions and reconstruction of arterial spin labeling MRI

PURPOSE

The goal of this study was to develop a 3D acceleration and reconstruction method to improve image quality and resolution of background-suppressed arterial spin-labeled perfusion MRI.

METHODS

Accelerated acquisition was implemented in all three k-space dimensions in a stack-of-spirals readout using variable density spirals and partition undersampling. A single 3D self-consistent parallel imaging (SPIRiT) kernel was calibrated and iteratively applied to reconstruct each imaging volume. Whole-brain (including cerebellum) perfusion imaging was obtained at 3-mm isotropic resolution (nominal) using single- and 2-shot acquisitions and at 2-mm isotropic resolution (nominal) using four-shot acquisitions, achieving effective acceleration factors between 5.5 and 6.6. The signal-to-noise (SNR) performance of 3D SPIRiT was evaluated. The temporal SNR (tSNR) of the cerebral blood flow (CBF) maps and the gray/white matter CBF ratios were quantified.

RESULTS

The readout of the arterial spin labeling (ASL) sequence was significantly shortened with acceleration. The CBF values were consistent between accelerated and fully sampled ASL. With shorter spiral interleaves and shorter echo trains, the accelerated images demonstrated reduced blurring and signal dropout in regions with high susceptibility gradients, resulting in improved image quality and increased gray/white matter CBF ratios. The shortened readout was accompanied by a corresponding decrease in tSNR.

CONCLUSION

The 3D acceleration and reconstruction allow a rapid whole-brain readout that improved the quality of ASL perfusion imaging. Magn Reson Med 78:1405-1419, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

Accelerated dynamic cardiac MRI exploiting sparse-Kalman-smoother self-calibration and reconstruction (k - t SPARKS)

Accelerated dynamic MRI, which exploits spatiotemporal redundancies in k  -  t space and coil dimension, has been widely used to reduce the number of signal encoding and thus increase imaging efficiency with minimal loss of image quality. Nonetheless, particularly in cardiac MRI it still suffers from artifacts and amplified noise in the presence of time-drifting coil sensitivity due to relative motion between coil and subject (e.g. free breathing). Furthermore, a substantial number of additional calibrating signals is to be acquired to warrant accurate calibration of coil sensitivity. In this work, we propose a novel, accelerated dynamic cardiac MRI with sparse-Kalman-smoother self-calibration and reconstruction (k  -  t SPARKS), which is robust to time-varying coil sensitivity even with a small number of calibrating signals. The proposed k  -  t SPARKS incorporates Kalman-smoother self-calibration in k  -  t space and sparse signal recovery in x  -   f space into a single optimization problem, leading to iterative, joint estimation of time-varying convolution kernels and missing signals in k  -  t space. In the Kalman-smoother calibration, motion-induced uncertainties over the entire time frames were included in modeling state transition while a coil-dependent noise statistic in describing measurement process. The sparse signal recovery iteratively alternates with the self-calibration to tackle the ill-conditioning problem potentially resulting from insufficient calibrating signals. Simulations and experiments were performed using both the proposed and conventional methods for comparison, revealing that the proposed k  -  t SPARKS yields higher signal-to-error ratio and superior temporal fidelity in both breath-hold and free-breathing cardiac applications over all reduction factors.

Error decomposition for parallel imaging reconstruction using modulation-domain representation of undersampled data

This paper presents a quantitative approach to evaluating and optimizing parallel imaging reconstruction for a clinical requirement. By introducing a "modulation domain representation" for undersampled data, the presented approach decomposes parallel imaging reconstruction error into multiple error components that can be grouped into three categories: image fidelity error, residue aliasing artifacts, and amplified noise. It is experimentally found that these error components have different image-space patterns that compromise imaging quality in different fashions. An error function may be defined as the weighted summation of these error components. By choosing a set of weighting coefficients that can quantify desirable image quality, parallel imaging may be optimized for a clinical requirement. It is found that error decomposition model may improve clinical utility of parallel imaging, providing an application-oriented approach to clinical parallel imaging.

Pushing CT and MR imaging to the molecular level for studying the "omics": current challenges and advancements

During the past decade, medical imaging has made the transition from anatomical imaging to functional and even molecular imaging. Such transition provides a great opportunity to begin the integration of imaging data and various levels of biological data. In particular, the integration of imaging data and multiomics data such as genomics, metabolomics, proteomics, and pharmacogenomics may open new avenues for predictive, preventive, and personalized medicine. However, to promote imaging-omics integration, the practical challenge of imaging techniques should be addressed. In this paper, we describe key challenges in two imaging techniques: computed tomography (CT) and magnetic resonance imaging (MRI) and then review existing technological advancements. Despite the fact that CT and MRI have different principles of image formation, both imaging techniques can provide high-resolution anatomical images while playing a more and more important role in providing molecular information. Such imaging techniques that enable single modality to image both the detailed anatomy and function of tissues and organs of the body will be beneficial in the imaging-omics field.

k-t GRAPPA accelerated four-dimensional flow MRI in the aorta: effect on scan time, image quality, and quantification of flow and wall shear stress

PURPOSE

The purpose of this study was to evaluate the utility of k-t parallel imaging for accelerating aortic four-dimensional (4D)-flow MRI. The aim was to systematically investigate the impact of different acceleration factors and number of coil elements on acquisition time, image quality and quantification of hemodynamic parameters.

METHODS

k-t accelerated 4D-flow MRI (spatial/temporal resolution = 2.1 × 2.5 × 2.5 mm/40.0 ms) was acquired in 10 healthy volunteers with acceleration factors R = 3, 5, and 8 using 12- and 32-channel receiver coils. Results were compared with conventional parallel imaging (GRAPPA [generalized autocalibrating partial parallel acquisition], R = 2). Data analysis included radiological grading of three-dimensional blood flow visualization quality as well as quantification of blood flow, velocities and wall shear stress (WSS).

RESULTS

k-t GRAPPA significantly reduced scan time by 28%, 54%, and 68%, for R = 3, 5, and 8, respectively, while maintaining image quality as demonstrated by overall similar image quality grading. Significant differences in peak WSS (diff12ch = -5.9%, diff32ch = 18.5%) and mean WSS (diff32ch = 13.9%) were found at the descending aorta for both receiver coils for R = 5 (PWSS < 0.04). Peak velocity differed for R=8 at the aortic root (-7.4%) and descending aorta (-12%) with PpeakVelo < 0.03.

CONCLUSION

k-t GRAPPA acceleration with a 12- or 32-channel receiver coil and an acceleration of 3 or 5 can compete with a standard GRAPPA R = 2 acceleration.

Sparsity-promoting calibration for GRAPPA accelerated parallel MRI reconstruction

The amount of calibration data needed to produce images of adequate quality can prevent auto-calibrating parallel imaging reconstruction methods like generalized autocalibrating partially parallel acquisitions (GRAPPA) from achieving a high total acceleration factor. To improve the quality of calibration when the number of auto-calibration signal (ACS) lines is restricted, we propose a sparsity-promoting regularized calibration method that finds a GRAPPA kernel consistent with the ACS fit equations that yields jointly sparse reconstructed coil channel images. Several experiments evaluate the performance of the proposed method relative to unregularized and existing regularized calibration methods for both low-quality and underdetermined fits from the ACS lines. These experiments demonstrate that the proposed method, like other regularization methods, is capable of mitigating noise amplification, and in addition, the proposed method is particularly effective at minimizing coherent aliasing artifacts caused by poor kernel calibration in real data. Using the proposed method, we can increase the total achievable acceleration while reducing degradation of the reconstructed image better than existing regularized calibration methods.

Parallel MR imaging

Parallel imaging is a robust method for accelerating the acquisition of magnetic resonance imaging (MRI) data, and has made possible many new applications of MR imaging. Parallel imaging works by acquiring a reduced amount of k-space data with an array of receiver coils. These undersampled data can be acquired more quickly, but the undersampling leads to aliased images. One of several parallel imaging algorithms can then be used to reconstruct artifact-free images from either the aliased images (SENSE-type reconstruction) or from the undersampled data (GRAPPA-type reconstruction). The advantages of parallel imaging in a clinical setting include faster image acquisition, which can be used, for instance, to shorten breath-hold times resulting in fewer motion-corrupted examinations. In this article the basic concepts behind parallel imaging are introduced. The relationship between undersampling and aliasing is discussed and two commonly used parallel imaging methods, SENSE and GRAPPA, are explained in detail. Examples of artifacts arising from parallel imaging are shown and ways to detect and mitigate these artifacts are described. Finally, several current applications of parallel imaging are presented and recent advancements and promising research in parallel imaging are briefly reviewed.

K-t sparse GROWL: sequential combination of partially parallel imaging and compressed sensing in k-t space using flexible virtual coil

Because dynamic MR images are often sparse in x-f domain, k-t space compressed sensing (k-t CS) has been proposed for highly accelerated dynamic MRI. When a multichannel coil is used for acquisition, the combination of partially parallel imaging and k-t CS can improve the accuracy of reconstruction. In this work, an efficient combination method is presented, which is called k-t sparse Generalized GRAPPA fOr Wider readout Line. One fundamental aspect of this work is to apply partially parallel imaging and k-t CS sequentially. A partially parallel imaging technique using a Generalized GRAPPA fOr Wider readout Line operator is adopted before k-t CS reconstruction to decrease the reduction factor in a computationally efficient way while preserving temporal resolution. Channel combination and relative sensitivity maps are used in the flexible virtual coil scheme to alleviate the k-t CS computational load with increasing number of channels. Using k-t FOCUSS as a specific example of k-t CS, the experiments with Cartesian and radial data sets demonstrate that k-t sparse Generalized GRAPPA fOr Wider readout Line can produce results with two times lower root-mean-square error than conventional channel-by-channel k-t CS while consuming up to seven times less computational cost.

K-space reconstruction with anisotropic kernel support (KARAOKE) for ultrafast partially parallel imaging

PURPOSE

Partially parallel imaging (PPI) greatly accelerates MR imaging by using surface coil arrays and under-sampling k-space. However, the reduction factor (R) in PPI is theoretically constrained by the number of coils (N(C)). A symmetrically shaped kernel is typically used, but this often prevents even the theoretically possible R from being achieved. Here, the authors propose a kernel design method to accelerate PPI faster than R = N(C).

METHODS

K-space data demonstrates an anisotropic pattern that is correlated with the object itself and to the asymmetry of the coil sensitivity profile, which is caused by coil placement and B(1) inhomogeneity. From spatial analysis theory, reconstruction of such pattern is best achieved by a signal-dependent anisotropic shape kernel. As a result, the authors propose the use of asymmetric kernels to improve k-space reconstruction. The authors fit a bivariate Gaussian function to the local signal magnitude of each coil, then threshold this function to extract the kernel elements. A perceptual difference model (Case-PDM) was employed to quantitatively evaluate image quality.

RESULTS

A MR phantom experiment showed that k-space anisotropy increased as a function of magnetic field strength. The authors tested a K-spAce Reconstruction with AnisOtropic KErnel support ("KARAOKE") algorithm with both MR phantom and in vivo data sets, and compared the reconstructions to those produced by GRAPPA, a popular PPI reconstruction method. By exploiting k-space anisotropy, KARAOKE was able to better preserve edges, which is particularly useful for cardiac imaging and motion correction, while GRAPPA failed at a high R near or exceeding N(C). KARAOKE performed comparably to GRAPPA at low Rs.

CONCLUSIONS

As a rule of thumb, KARAOKE reconstruction should always be used for higher quality k-space reconstruction, particularly when PPI data is acquired at high Rs and/or high field strength.

Parallel magnetic resonance imaging using localized receive arrays with sinc interpolation (PILARS)

Large arrays with localized coil sensitivity make it possible to use parallel imaging to significantly accelerate MR imaging speed. However, the need for auto calibration signals limits the actual acceleration factors achievable with large arrays. This paper presents a novel method for parallel imaging with large arrays. The method uses Sinc kernels for k-space data interpolation that only requires one phase parameter to be estimated using a small size of calibration signals. Simulations based on synthetic array data and phantom experiments show that the new method can achieve higher actual acceleration factors with comparable reconstruction quality.

A radial self-calibrated (RASCAL) generalized autocalibrating partially parallel acquisition (GRAPPA) method using weight interpolation

A generalized autocalibrating partially parallel acquisition (GRAPPA) method for radial k-space sampling is presented that calculates GRAPPA weights without synthesized or acquired calibration data. Instead, GRAPPA weights are fitted to the undersampled data as if they were the calibration data. Because the relative k-space shifts associated with these GRAPPA weights vary for a radial trajectory, new GRAPPA weights can be resampled for arbitrary shifts through interpolation, which are then used to generate missing projections between the acquired projections. The method is demonstrated in phantoms and in abdominal and brain imaging. Image quality is similar to radial GRAPPA using fully sampled calibration data, and improved relative to a previously described self-calibrated radial GRAPPA technique.

On the undersampling strategies to accelerate time-resolved 3D imaging using k-t-GRAPPA

The purpose of this study was to explore how to optimally undersample and reconstruct time-resolved 3D data using a k-t-space-based GRAPPA technique. The performance of different reconstruction strategies was evaluated using data sets with different ratios of phase (N(y)) and partition (N(z)) encoding lines (N(y) × N(z) = 64-128 × 40-64) acquired in a moving phantom. Image reconstruction was performed for different kernel configurations and different reduction factors (R = 5, 6, 8, and 10) and was evaluated using regional error quantification and SNR analysis. To analyze the temporal fidelity of the different kernel configurations in vivo, time-resolved 3D phase contrast data were acquired in the thoracic aorta of two healthy volunteers. Results demonstrated that kernel configurations with a small kernel extension yielded superior results especially for more asymmetric data matrices as typically used in clinical applications. The application of k-t-GRAPPA to in vivo data demonstrated the feasibility of undersampling of time-resolved 3D phase contrast data set with a nominal reduction factors of up to R(net) = 8, while maintaining the temporal fidelity of the measured velocity field. Extended GRAPPA-based parallel imaging with optimized multidimensional reconstruction kernels has the potential to substantially accelerate data acquisitions in time-resolved 3D MRI.

SPIRiT: Iterative self-consistent parallel imaging reconstruction from arbitrary k-space

A new approach to autocalibrating, coil-by-coil parallel imaging reconstruction, is presented. It is a generalized reconstruction framework based on self-consistency. The reconstruction problem is formulated as an optimization that yields the most consistent solution with the calibration and acquisition data. The approach is general and can accurately reconstruct images from arbitrary k-space sampling patterns. The formulation can flexibly incorporate additional image priors such as off-resonance correction and regularization terms that appear in compressed sensing. Several iterative strategies to solve the posed reconstruction problem in both image and k-space domain are presented. These are based on a projection over convex sets and conjugate gradient algorithms. Phantom and in vivo studies demonstrate efficient reconstructions from undersampled Cartesian and spiral trajectories. Reconstructions that include off-resonance correction and nonlinear l(1)-wavelet regularization are also demonstrated.

Data consistency criterion for selecting parameters for k-space-based reconstruction in parallel imaging

k-space-based reconstruction in parallel imaging depends on the reconstruction kernel setting, including its support. An optimal choice of the kernel depends on the calibration data, coil geometry and signal-to-noise ratio, as well as the criterion used. In this work, data consistency, imposed by the shift invariance requirement of the kernel, is introduced as a goodness measure of k-space-based reconstruction in parallel imaging and demonstrated. Data consistency error (DCE) is calculated as the sum of squared difference between the acquired signals and their estimates obtained based on the interpolation of the estimated missing data. A resemblance between DCE and the mean square error in the reconstructed image was found, demonstrating DCE's potential as a metric for comparing or choosing reconstructions. When used for selecting the kernel support for generalized autocalibrating partially parallel acquisition (GRAPPA) reconstruction and the set of frames for calibration as well as the kernel support in temporal GRAPPA reconstruction, DCE led to improved images over existing methods. Data consistency error is efficient to evaluate, robust for selecting reconstruction parameters and suitable for characterizing and optimizing k-space-based reconstruction in parallel imaging.

Using the GRAPPA operator and the generalized sampling theorem to reconstruct undersampled non-Cartesian data

As expected from the generalized sampling theorem of Papoulis, the use of a bunched sampling acquisition scheme in conjunction with a conjugate gradient (CG) reconstruction algorithm can decrease scan time by reducing the number of phase-encoding lines needed to generate an unaliased image at a given resolution. However, the acquisition of such bunched data requires both modified pulse sequences and high gradient performance. A novel method of generating the "bunched" data using self-calibrating GRAPPA operator gridding (GROG), a parallel imaging method that shifts data points by small distances in k-space (with Deltak usually less than 1.0, depending on the receiver coil) using the GRAPPA operator, is presented here. With the CG reconstruction method, these additional "bunched" points can then be used to reconstruct an image with reduced artifacts from undersampled data. This method is referred to as GROG-facilitated bunched phase encoding (BPE), or GROG-BPE. To better understand how the patterns of bunched points, maximal blip size, and number of bunched points affect the reconstruction quality, a number of simulations were performed using the GROG-BPE approach. Finally, to demonstrate that this method can be combined with a variety of trajectories, examples of images with reduced artifacts reconstructed from undersampled in vivo radial, spiral, and rosette data are shown.

Regionally optimized reconstruction for partially parallel imaging in MRI applications

Based on the conventional SENSE and GRAPPA, a regionally optimized reconstruction method is developed for reduced noise and artifact level in partially parallel imaging. In this regionally optimized reconstruction, the field-of-view (FOV) is divided into a number of small regions. Over every small region, the noise amplification and data fitting error can be balanced and minimized locally by taking advantage of spatial correlation of neighboring pixels in reconstruction. The full FOV image can be obtained by "region-by-region" reconstruction. Compared with the conventional SENSE, this method gives better performance in the regions where there are pixels with high SENSE g-factors. Compared with GRAPPA, it is better in the regions where all the pixels have low SENSE g-factors. In this work, we applied the regionally optimized reconstruction in four important imaging experiments: brain, spine, breast, and cardiac. It was demonstrated in these experiments that the overall image quality using this regionally optimized reconstruction is better than that using the conventional SENSE or GRAPPA.

Block circulant quasi-band matrix property for the SENSE unfolding in k-space and justification for GRAPPA

The k-space unfolding matrix for parallel imaging in MRI was examined and was found to have the structure of block circulant quasi-band, providing a rigorous mathematical justification for the interpolation kernel in the commonly used GRAPPA method. The optimal GRAPPA kernel size and dimension were closely related to sensitivity spectrum.