Optimizing k-space sampling trajectories is a promising yet challenging topic for fast magnetic resonance imaging (MRI). This work proposes to optimize a reconstruction method and sampling trajectories jointly concerning image reconstruction quality in a supervised learning manner. We parameterize trajectories with quadratic B-spline kernels to reduce the number of parameters and apply multi-scale optimization, which may help to avoid sub-optimal local minima. The algorithm includes an efficient non-Cartesian unrolled neural network-based reconstruction and an accurate approximation for backpropagation through the non-uniform fast Fourier transform (NUFFT) operator to accurately reconstruct and back-propagate multi-coil non-Cartesian data. Penalties on slew rate and gradient amplitude enforce hardware constraints. Sampling and reconstruction are trained jointly using large public datasets. To correct for possible eddy-current effects introduced by the curved trajectory, we use a pencil-beam trajectory mapping technique. In both simulations and in- vivo experiments, the learned trajectory demonstrates significantly improved image quality compared to previous model-based and learning-based trajectory optimization methods for 10× acceleration factors. Though trained with neural network-based reconstruction, the proposed trajectory also leads to improved image quality with compressed sensing-based reconstruction.

This paper proposes a new method for joint design of radiofrequency (RF) and gradient waveforms in Magnetic Resonance Imaging (MRI), and applies it to the design of 3D spatially tailored saturation and inversion pulses. The joint design of both waveforms is characterized by the ODE Bloch equations, to which there is no known direct solution. Existing approaches therefore typically rely on simplified problem formulations based on, e.g., the small-tip approximation or constraining the gradient waveforms to particular shapes, and often apply only to specific objective functions for a narrow set of design goals (e.g., ignoring hardware constraints). This paper develops and exploits an auto-differentiable Bloch simulator to directly compute Jacobians of the (Bloch-simulated) excitation pattern with respect to RF and gradient waveforms. This approach is compatible with arbitrary sub-differentiable loss functions, and optimizes the RF and gradients directly without restricting the waveform shapes. For computational efficiency, we derive and implement explicit Bloch simulator Jacobians (approximately halving computation time and memory usage). To enforce hardware limits (peak RF, gradient, and slew rate), we use a change of variables that makes the 3D pulse design problem effectively unconstrained; we then optimize the resulting problem directly using the proposed auto-differentiation framework. We demonstrate our approach with two kinds of 3D excitation pulses that cannot be easily designed with conventional approaches: Outer-volume saturation (90° flip angle), and inner-volume inversion.

This paper introduces a fast, general method for dictionary-free parameter estimation in quantitative magnetic resonance imaging (QMRI) parameter estimation via regression with kernels (PERK). PERK first uses prior distributions and the nonlinear MR signal model to simulate many parameter-measurement pairs. Inspired by machine learning, PERK then takes these parameter-measurement pairs as labeled training points and learns from them a nonlinear regression function using kernel functions and convex optimization. PERK admits a simple implementation as per-voxel nonlinear lifting of MRI measurements followed by linear minimum mean-squared error regression. We demonstrate PERK for $ {\textit {T}_{1}}, {\textit {T}_{2}}$ estimation, a well-studied application where it is simple to compare PERK estimates against dictionary-based grid search estimates and iterative optimization estimates. Numerical simulations as well as single-slice phantom and in vivo experiments demonstrate that PERK and other tested methods produce comparable $ {\textit {T}_{1}}, {\textit {T}_{2}}$ estimates in white and gray matter, but PERK is consistently at least $140\times $ faster. This acceleration factor may increase by several orders of magnitude for full-volume QMRI estimation problems involving more latent parameters per voxel.

Rapid, reliable quantification of MR relaxation parameters T1and T2is desirable for many clinical applications. Steady-state sequences such as Spoiled Gradient-Recalled Echo (SPGR) and Dual-Echo Steady-State (DESS) are fast and well-suited for relaxometry because the signals they produce are quite sensitive to T1and T2variation. However, T1, T2estimation with these sequences typically requires multiple scans with varied sets of acquisition parameters. This paper describes a systematic framework for selecting scan types (e.g., combinations of SPGR and DESS scans) and optimizing their respective parameters (e.g., flip angles and repetition times). The method is based on a Cramér-Rao Bound (CRB)-inspired min-max optimization that finds scan parameter combinations that robustly enable precise object parameter estimation. We apply this technique to optimize combinations of SPGR and DESS scans for T1, T2relaxometry in white matter (WM) and grey matter (GM) regions of the human brain at 3T field strength. Phantom accuracy experiments show that SPGR/DESS scan combinations are in excellent agreement with reference measurements. Phantom precision experiments show that trends in T1,T2pooled sample standard deviations reflect CRB-based predictions. In vivo experiments show that in WM and GM, T1and T2estimates from a pair of optimized DESS scans exhibit precision (but not necessarily accuracy) comparable to that of optimized combinations of SPGR and DESS scans. To our knowledge, T1maps from DESS acquisitions alone are new. This example application illustrates that scan optimization may help reveal new parameter mapping techniques from combinations of established pulse sequences.

We propose a new method for the joint design of k-space trajectory and RF pulse in 3D small-tip tailored excitation. Designing time-varying RF and gradient waveforms for a desired 3D target excitation pattern in MRI poses a non-linear, non-convex, constrained optimization problem with relatively large problem size that is difficult to solve directly. Existing joint pulse design approaches are therefore typically restricted to predefined trajectory types such as EPI or stack-of-spirals that intrinsically satisfy the gradient maximum and slew rate constraints and reduce the problem size (dimensionality) dramatically, but lead to suboptimal excitation accuracy for a given pulse duration. Here we use a 2nd-order B-spline basis that can be fitted to an arbitrary k-space trajectory, and allows the gradient constraints to be implemented efficiently. We show that this allows the joint optimization problem to be solved with quite general k-space trajectories. Starting from an arbitrary initial trajectory, we first approximate the trajectory using B-spline basis, and then optimize the corresponding coefficients. We evaluate our method in simulation using four different k-space initializations: stack-of-spirals, SPINS, KT-points, and a new method based on KT-points. In all cases, our approach leads to substantial improvement in excitation accuracy for a given pulse duration. We also validated our method for inner-volume excitation using phantom experiments. The computation is fast enough for online applications.

Fast and accurate quantification of spin-spin relaxation parameter T2is of importance for clinical MRI applications. Classical spin echo (SE) sequences yield straightforward T2estimates, but require undesirably long scans. By contrast, steady-state sequences such as the Dual-Echo Steady-State (DESS) sequence are considerably faster, but produce signals that depend on more complex functions of both desired and nuisance parameters. Conventional method-of-moments estimators exhibit systematic error because of the approximations used to bypass nuisance parameter estimation. To improve T2mapping accuracy, we propose a novel, model-based approach to this nonlinear estimation problem. We use a fast scan to estimate nuisance parameters M0*and T1, and then use the exact DESS signal model for regularized T2estimation from DESS data, with minimal approximations. MR brain simulation results show that the proposed approach substantially improves T2estimation accuracy and precision, compared to conventional method-of-moments estimators.

Magnetic resonance image (MRI) reconstruction from undersampled k-space data requires regularization to reduce noise and aliasing artifacts. Proper application of regularization however requires appropriate selection of associated regularization parameters. In this work, we develop a data-driven regularization parameter adjustment scheme that minimizes an estimate [based on the principle of Stein's unbiased risk estimate (SURE)] of a suitable weighted squared-error measure in k-space. To compute this SURE-type estimate, we propose a Monte-Carlo scheme that extends our previous approach to inverse problems (e.g., MRI reconstruction) involving complex-valued images. Our approach depends only on the output of a given reconstruction algorithm and does not require knowledge of its internal workings, so it is capable of tackling a wide variety of reconstruction algorithms and nonquadratic regularizers including total variation and those based on the l1-norm. Experiments with simulated and real MR data indicate that the proposed approach is capable of providing near mean squared-error optimal regularization parameters for single-coil undersampled non-Cartesian MRI reconstruction.

Regularization is an effective means of reducing noise and artifacts in MR image reconstruction from undersampled k-space data. Proper application of regularization demands appropriate selection of the associated regularization parameter. Generalized cross-validation (GCV) is a popular parameter tuning technique especially for linear reconstruction methods, but its application to nonlinear iterative MRI reconstruction is more involved as it demands the evaluation of the Jacobian matrix of the reconstruction algorithm with respect to complex-valued data. We derive analytical expressions for recursively updating this Jacobian matrix for an iterative reweighted least-squares reconstruction algorithm. Our method can also be used to calculate a predicted risk estimate (PSURE) for MRI based on Stein's principle. We demonstrate with simulations and experiments with real data that regularization parameter selection based on GCV and PSURE provides near-MSE-optimal results for nonlinear MRI reconstruction from undersampled k-space data using ℓ1-regularization.

Regularized iterative reconstruction algorithms for imaging inverse problems require selection of appropriate regularization parameter values. We focus on the challenging problem of tuning regularization parameters for nonlinear algorithms for the case of additive (possibly complex) Gaussian noise. Generalized cross-validation (GCV) and (weighted) mean-squared error (MSE) approaches (based on Steinfs Unbiased Risk Estimate. SURE) need the Jacobian matrix of the nonlinear reconstruction operator (representative of the iterative algorithm) with respect to the data. We derive the desired Jacobian matrix for two types of nonlinear iterative algorithms: a fast variant of the standard iterative reweighted least-squares method and the contemporary split-Bregman algorithm, both of which can accommodate a wide variety of analysis- and synthesis-type regularizers. The proposed approach iteratively computes two weighted SURE-type measures: Predicted-SURE and Projected-SURE (that require knowledge of noise variance Ð2), and GCV (that does not need Ð2) for these algorithms. We apply the methods to image restoration and to magnetic resonance image (MRI) reconstruction using total variation (TV) and an analysis-type .1-regularization. We demonstrate through simulations and experiments with real data that minimizing Predicted-SURE and Projected-SURE consistently lead to near-MSE-optimal reconstructions. We also observed that minimizing GCV yields reconstruction results that are near-MSE-optimal for image restoration and slightly suboptimal for MRI. Theoretical derivations in this work related to Jacobian matrix evaluations can be extended, in principle, to other types of regularizers and reconstruction algorithms.

Compressed sensing (CS) has been used for accelerating magnetic resonance imaging acquisitions, but its use in applications with rapid spatial phase variations is challenging, e.g., proton resonance frequency shift (PRF-shift) thermometry and velocity mapping. Previously, an iterative MRI reconstruction with separate magnitude and phase regularization was proposed for applications where magnitude and phase maps are both of interest, but it requires fully sampled data and unwrapped phase maps. In this paper, CS is combined into this framework to reconstruct magnitude and phase images accurately from undersampled data. Moreover, new phase regularization terms are proposed to accommodate phase wrapping and to reconstruct images with encoded phase variations, e.g., PRF-shift thermometry and velocity mapping. The proposed method is demonstrated with simulated thermometry data and in vivo velocity mapping data and compared to conventional phase corrected CS.

The temporal resolution of dynamic magnetic resonance imaging (MRI) can be increased by sampling a fraction of k-space in an interleaved fashion, which introduces spatial and temporal aliasing. We describe algebraically and graphically the aliasing process caused by dynamic undersampled spiral imaging within 3-D xyf space (the Fourier transform of k(x)k(y)t space) and formulate the unaliasing problem as a set of independent linear inversions. Since each linear system is numerically underdetermined, the use of prior knowledge in the form of bounded support regions is proposed. To overcome the excessive memory requirements for handling large matrices, a fast implementation of the conjugate gradient (CG) method is used. Numerical simulation and in vivo experiments using spiral twofold undersampling demonstrate reduced motion artifacts and the improved depiction of fine cardiac structures. The achieved reduction of motion artifacts and motion blur is comparable to simple filtering, which is computationally more efficient, while the proposed algebraic framework offers greater flexibility to incorporate additional algebraic acceleration techniques and to handle arbitrary sampling schemes.