Optimal control in a magnetization-prepared rapid acquisition gradient-echo sequence.

This article proposes a numerical framework to determine the optimal magnetization preparation in a three-dimensional magnetization-prepared rapid gradient-echo (MP-RAGE) sequence to obtain the best achievable contrast between target tissues based on differences in their relaxation times. The benefit lies in the adaptation of the algorithm of optimal control, GRAdient Ascent Pulse Engineering (GRAPE), to the optimization of magnetization preparation in a cyclic sequence without full recovery between each cycle. This numerical approach optimizes magnetization preparation of an arbitrary number of radio frequency pulses to enhance contrast, taking into account the establishment of a steady state in the longitudinal component of the magnetization. The optimal control preparation offers an optimized mixed T 1 / T 2 contrast in this traditional T 1 -weighted sequence. To show the versatility of the proposed method, numerical and in vitro results are described. Examples of contrasts acquired on brain regions of a healthy volunteer are presented for potential applications at 3 T.

KomaMRI.jl: An open-source framework for general MRI simulations with GPU acceleration.

PURPOSE: To develop an open-source, high-performance, easy-to-use, extensible, cross-platform, and general MRI simulation framework (Koma). METHODS: Koma was developed using the Julia programming language. Like other MRI simulators, it solves the Bloch equations with CPU and GPU parallelization. The inputs are the scanner parameters, the phantom, and the pulse sequence that is Pulseq-compatible. The raw data is stored in the ISMRMRD format. For the reconstruction, MRIReco.jl is used. A graphical user interface utilizing web technologies was also designed. Two types of experiments were performed: one to compare the quality of the results and the execution speed, and the second to compare its usability. Finally, the use of Koma in quantitative imaging was demonstrated by simulating Magnetic Resonance Fingerprinting (MRF) acquisitions. RESULTS: Koma was compared to two well-known open-source MRI simulators, JEMRIS and MRiLab. Highly accurate results (with mean absolute differences below 0.1% compared to JEMRIS) and better GPU performance than MRiLab were demonstrated. In an experiment with students, Koma was proved to be easy to use, eight times faster on personal computers than JEMRIS, and 65% of test subjects recommended it. The potential for designing acquisition and reconstruction techniques was also shown through the simulation of MRF acquisitions, with conclusions that agree with the literature. CONCLUSIONS: Koma's speed and flexibility have the potential to make simulations more accessible for education and research. Koma is expected to be used for designing and testing novel pulse sequences before implementing them in the scanner with Pulseq files, and for creating synthetic data to train machine learning models.

Quantitative MRI by nonlinear inversion of the Bloch equations.

PURPOSE: Development of a generic model-based reconstruction framework for multiparametric quantitative MRI that can be used with data from different pulse sequences. METHODS: Generic nonlinear model-based reconstruction for quantitative MRI estimates parametric maps directly from the acquired k-space by numerical optimization. This requires numerically accurate and efficient methods to solve the Bloch equations and their partial derivatives. In this work, we combine direct sensitivity analysis and pre-computed state-transition matrices into a generic framework for calibrationless model-based reconstruction that can be applied to different pulse sequences. As a proof-of-concept, the method is implemented and validated for quantitative T 1 $$ {\mathrm{T}}_1 $$ and T 2 $$ {\mathrm{T}}_2 $$ mapping with single-shot inversion-recovery (IR) FLASH and IR bSSFP sequences in simulations, phantoms, and the human brain. RESULTS: The direct sensitivity analysis enables a highly accurate and numerically stable calculation of the derivatives. The state-transition matrices efficiently exploit repeating patterns in pulse sequences, speeding up the calculation by a factor of 10 for the examples considered in this work, while preserving the accuracy of native ordinary differential equations solvers. The generic model-based method reproduces quantitative results of previous model-based reconstructions based on the known analytical solutions for radial IR FLASH. For IR bSFFP it produces accurate T 1 $$ {\mathrm{T}}_1 $$ and T 2 $$ {\mathrm{T}}_2 $$ maps for the National Insitute of Standards and Technology (NIST) phantom in numerical simulations and experiments. Feasibility is also shown for human brain, although results are affected by magnetization transfer effects. CONCLUSION: By developing efficient tools for numerical optimizations using the Bloch equations as forward model, this work enables generic model-based reconstruction for quantitative MRI.

Vibrational resonance in a driven two-level quantum system, linear and nonlinear response.

We consider a two-level quantum system interacting with two classical time-periodic electromagnetic fields. The frequency of one of the fields far exceeds that of the other. The effect of the high-frequency field can be averaged out of the dynamics to realize an effective transition frequency of the field-dressed two-level system. We examine the linear response, second harmonic response and Stokes and anti-Stokes Raman response of the dressed two-level system, to the weak frequency field. The vibrational resonance enhancement in each case is demonstrated for optimal strength of the high-frequency field. Our theoretical scheme is corroborated by full numerical simulation of the two-level, two-field dynamics governed by loss-free Bloch equations. We suggest that quantum optics can offer an interesting arena for the study of the vibrational resonance. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'.

A simplified framework to optimize MRI contrast preparation.

PURPOSE: This article proposes a rigorous optimal control framework for the design of preparation schemes that optimize MRI contrast based on relaxation time differences. METHODS: Compared to previous optimal contrast preparation schemes, a drastic reduction of the optimization parameter number is performed. The preparation scheme is defined as a combination of several block pulses whose flip angles, phase terms and inter-pulse delays are optimized to control the magnetization evolution. RESULTS: The proposed approach reduces the computation time of B 0 -robust preparation schemes to around a minute (whereas several hours were required with previous schemes), with negligible performance loss. The chosen parameterization allows to formulate the total preparation duration as a constraint, which improves the overall compromise between contrast performance and preparation time. Simulation, in vitro and in vivo results validate this improvement, illustrate the straightforward applicability of the proposed approach, and point out its flexibility in terms of achievable contrasts. Major improvement is especially achieved for short-T2 enhancement, as shown by the acquisition of a non-trivial contrast on a rat brain, where a short-T2 white matter structure (corpus callosum) is enhanced compared to surrounding gray matter tissues (hippocampus and neocortex). CONCLUSIONS: This approach proposes key advances for the design of optimal contrast preparation sequences, that emphasize their ability to generate non-standard contrasts, their potential benefit in a clinical context, and their straightforward applicability on any MR system.

Flow MRI simulation in complex 3D geometries: Application to the cerebral venous network.

PURPOSE: Develop and evaluate a complete tool to include 3D fluid flows in MRI simulation, leveraging from existing software. Simulation of MR spin flow motion is of high interest in the study of flow artifacts and angiography. However, at present, only a few simulators include this option and most are restricted to static tissue imaging. THEORY AND METHODS: An extension of JEMRIS, one of the most advanced high performance open-source simulation platforms to date, was developed. The implementation of a Lagrangian description of the flow allows simulating any MR experiment, including both static tissues and complex flow data from computational fluid dynamics. Simulations of simple flow models are compared with real experiments on a physical flow phantom. A realistic simulation of 3D flow MRI on the cerebral venous network is also carried out. RESULTS: Simulations and real experiments are in good agreement. The generality of the framework is illustrated in 2D and 3D with some common flow artifacts (misregistration and inflow enhancement) and with the three main angiographic techniques: phase contrast velocimetry (PC), time-of-flight, and contrast-enhanced imaging MRA. CONCLUSION: The framework provides a versatile and reusable tool for the simulation of any MRI experiment including physiological fluids and arbitrarily complex flow motion.

Fast myelin water fraction estimation using 2D multislice CPMG.

PURPOSE: T2 relaxometry based on multiexponential fitting to a single slice multiecho sequence has been the most common MRI technique for myelin water fraction mapping, where the short T2 is associated with myelin water. However, very long acquisition times and physically unrealistic models for T2 distribution are limitations of this approach. We present a novel framework for myelin imaging which substantially increases the imaging speed and myelin water fraction estimation accuracy. METHOD: We used the 2D multislice Carr-Purcell-Meiboom-Gill sequence to increase the volume coverage. To compensate for nonideal slice profiles, we numerically solved the Bloch equations for a range of T2 and B1 inhomogeneity scales to construct the bases for the estimation of the T2 distribution. We used a finite mixture of continuous parametric distributions to describe the complete T2 spectrum and used the constrained variable projection optimization algorithm to estimate myelin water fraction. To validate our model, synthetic, phantom, and in vivo brain experiments were conducted. RESULTS: Using the Bloch equations, we can model the slice profile and construct the forward model of the T2 curve. Our method estimated myelin water fraction with smaller error than the nonnegative least squares algorithm. CONCLUSIONS: The proposed framework can be used for reliable whole brain myelin imaging with a resolution of 2×2×4 mm3 in ≈17 min. Magn Reson Med 76:1301-1313, 2016. © 2015 Wiley Periodicals, Inc.

The Landau-Zener-Stückelberg-Majorana transition in the T2 ≪ T1 limit.

Landau-Zener-Stückelberg-Majorana (LZSM) transitions occur between quantum states when parameters in the system's Hamiltonian are varied continuously and rapidly. In magnetic resonance, losses in adiabatic rapid passage can be understood using the physics of LZSM transitions. Most treatments of LZSM transitions ignore the T2 dephasing of coherences, however. Motivated by ongoing work in magnetic resonance force microscopy, we employ the Bloch equations, coordinate transformation, and the Magnus expansion to derive expressions for the final magnetization following a rapid field sweep at fixed irradiation intensity that include T2 losses. Our derivation introduces an inversion-function, Fourier transform method for numerically evaluating highly oscillatory integrals. Expressions for the final magnetization are given for low and high irradiation intensity, valid in the T2≪T1 limit. Analytical results are compared to numerical simulations and nuclear magnetic resonance experiments. Our relatively straightforward calculation reproduces semiquantitatively the well known LZSM result in the T2→0 limit.

On the importance of electron-electron and electron-phonon scatterings and energy renormalizations during carrier relaxation in monolayer transition-metal dichalcogenides.

Anab initiobased fully microscopic many-body approach is used to study the carrier relaxation dynamics in monolayer transition-metal dichalcogenides. Bandstructures and wavefunctions as well as phonon energies and coupling matrix elements are calculated using density functional theory. The resulting dipole and Coulomb matrix elements are implemented in the Dirac-Bloch equations to calculate carrier-carrier and carrier-phonon scatterings throughout the whole Brillouin zone (BZ). It is shown that carrier scatterings lead to a relaxation into hot quasi-Fermi distributions on a single femtosecond timescale. Carrier cool down and inter-valley transitions are mediated by phonon scatterings on a picosecond timescale. Strong, density-dependent energy renormalizations are shown to be valley-dependent. For MoTe2, MoSe2and MoS2the change of energies with occupation is found to be about 50% stronger in the Σ and Λ side valleys than in theKandK' valleys. However, for realistic carrier densities, the materials always maintain their direct bandgap at theKpoints of the BZ.

Accuracy and performance analysis for Bloch and Bloch-McConnell simulation methods.

PURPOSE: To introduce new solution methods for the Bloch and Bloch-McConnell equations and compare them quantitatively to different known approaches. THEORY AND METHODS: A new exact solution per time step is derived by means of eigenvalues and generalized eigenvectors. Fast numerical solution methods based on asymmetric and symmetric operator splitting, which are already known for the Bloch equations, are extended to the Bloch-McConnell equations. Those methods are compared to other numerical methods including spin domain, one-step and multi-step methods, and matrix exponential. Error metrics are introduced based on the exact solution method, which allows to assess the accuracy of each solution method quantitatively for arbitrary example data. RESULTS: Accuracy and performance properties for nine different solution methods are analyzed and compared in extensive numerical experiments including various examples for non-selective and slice-selective MR imaging applications. The accuracy of the methods heavily varies, in particular for short relaxation times and long pulse durations. CONCLUSION: In absence of relaxation effects, the numerical results confirm the rotation matrices approach as accurate and computationally efficient Bloch solution method. Otherwise, as well as for the Bloch-McConnell equations, symmetric operator splitting methods are recommended due to their excellent numerical accuracy paired with efficient run time.

Simultaneous optimization of radio frequency and gradient waveforms with exact Hessians and slew rate constraints applied to kT-points excitation.

We consider an excitation pulse with piecewise constant gradient trajectories and radio frequency (RF) waveforms such that the solution of the Bloch equations without relaxation terms can be represented by rotations. Based on this analytic solution we formulate a non-linear program for finding sub-pulse durations, gradient strengths, and complex RF voltages which minimize the deviation between the achieved and desired magnetization. We develop explicit expressions for the first and second order derivatives of the objective function. We extend the non-linear program to precisely account for gradient slew rate constraints. Using an interior point solver we apply the developed theory to simultaneously optimize the positions of kT-points, their associated RF voltages and durations.

Characterization of the Observed Electric Field and Molecular Relaxation Times for Millimeter-Wave Chirped Pulse Instrumentation.

In a chirped pulse experiment, the strength of the signal level is proportional to the amplitude of the electric field, which is weaker in the millimeter-wave or submillimeter-wave region than in the microwave region. Experiments in the millimeter region thus require an optimization of the coupling between the source and the molecular system and a method to estimate the amplitude of the electric field as seen by the molecular system. We have developed an analytical model capable of reproducing the coherent transient signals obtained with a millimeter-wave chirped pulse setup operated in a monochromatic pulse mode. The fit of the model against the experimental data allowed access to the amplitude of the electric field and, as a byproduct, to the molecular relaxation times T 1 and T 2.

Prediction of flow effects in quantitative NMR measurements.

A method for the prediction of the magnetization in flow NMR experiments is presented, which can be applied to mixtures. It enables a quantitative evaluation of NMR spectra of flowing liquid samples even in cases in which the magnetization is limited by the flow. A transport model of the nuclei's magnetization, which is based on the Bloch-equations, is introduced into a computational fluid dynamics (CFD) code. This code predicts the velocity field and relative magnetization of different nuclei for any chosen flow cell geometry, fluid and flow rate. The prediction of relative magnetization is used to correct the observed reduction of signal intensity caused by incomplete premagnetization in fast flowing liquids. By means of the model, quantitative NMR measurements at high flow rates are possible. The method is predictive and enables calculating correction factors for any flow cell design and operating condition based on simple static T1 time measurements. This makes time-consuming calibration measurements for assessing the influence of flow effects obsolete, which otherwise would have to be carried out for each studied condition. The new method is especially interesting for flow measurements with compact medium field NMR spectrometers, which have small premagnetization volumes. In the present work, experiments with three different flow cells in a medium field NMR spectrometer were carried out. Acetonitrile, water, and mixtures of these components were used as model fluids. The experimental results for the magnetization were compared to the predictions from the CFD model and good agreement was observed.

Constant gradient elastography with optimal control RF pulses.

This article presents a new motion encoding strategy to perform magnetic resonance elastography (MRE). Instead of using standard motion encoding gradients, a tailored RF pulse is designed to simultaneously perform selective excitation and motion encoding in presence of a constant gradient. The RF pulse is designed with a numerical optimal control algorithm, in order to obtain a magnetization phase distribution that depends on the displacement characteristics inside each voxel. As a consequence, no post-excitation encoding gradients are required. This offers numerous advantages, such as reducing eddy current artifacts, and relaxing the constraint on the gradients maximum switch rate. It also allows to perform MRE with ultra-short TE acquisition schemes, which limits T2 decay and optimizes signal-to-noise ratio. The pulse design strategy is developed and analytically analyzed to clarify the encoding mechanism. Finally, simulations, phantom and ex vivo experiments show that phase-to-noise ratios are improved when compared to standard MRE encoding strategies.

Optimal control design of preparation pulses for contrast optimization in MRI.

This work investigates the use of MRI radio-frequency (RF) pulses designed within the framework of optimal control theory for image contrast optimization. The magnetization evolution is modeled with Bloch equations, which defines a dynamic system that can be controlled via the application of the Pontryagin Maximum Principle (PMP). This framework allows the computation of optimal RF pulses that bring the magnetization to a given state to obtain the desired contrast after acquisition. Creating contrast through the optimal manipulation of Bloch equations is a new way of handling contrast in MRI, which can explore the theoretical limits of the system. Simulation experiments carried out on-resonance quantify the contrast improvement when compared to standard T1 or T2 weighting strategies. The use of optimal pulses is also validated for the first time in both in vitro and in vivo experiments on a small-animal 4.7T MR system. Results demonstrate their robustness to static field inhomogeneities as well as the fact that they can be embedded in standard imaging sequences without affecting standard parameters such as slice selection or echo type. In vivo results on rat and mouse brains illustrate the ability of optimal contrast pulses to create non-trivial contrasts on well-studied structures (white matter versus gray matter).

MR imaging: deconstructing timing diagrams and demystifying k-space.

Magnetic resonance imaging (MRI) works on the principle that hydrogen molecules, which are abundant in organic tissue, have a magnetic moment arising from the spin of the protons in the nucleus. All atoms consist of a nucleus made of protons and neutrons. When a sample is put in a large magnet field, the hydrogen atoms become magnetized resulting in a bulk magnetization of the sample. Each of these hydrogen atoms acts like a bar magnet, spinning at a frequency about the applied main magnetic field. The frequency of spin is proportional to the applied main field and hence to encode position, we apply an additive field that increases linearly with position in a given direction. Hence, the spins in that direction will precess at a linearly increasing frequency and can be resolved by matching each resolvable frequency bin to a given position. This allows one direction to be resolved. By repeating the same procedure for the other dimension, a 2D image can be resolved by averaging over the third dimension.

Rabi resonance in spin systems: theory and experiment.

The response of a magnetic resonance spin system is predicted and experimentally verified for the particular case of a continuous wave amplitude modulated radiofrequency excitation. The experimental results demonstrate phenomena not previously observed in magnetic resonance systems, including a secondary resonance condition when the amplitude of the excitation equals the modulation frequency. This secondary resonance produces a relatively large steady state magnetisation with Fourier components at harmonics of the modulation frequency. Experiments are in excellent agreement with the theoretical prediction derived from the Bloch equations, which provides a sound theoretical framework for future developments in NMR spectroscopy and imaging.

Bloch equations for anisotropic paramagnetic centers with spin of 1/2.

The Bloch equations are an invaluable tool in magnetic resonance for describing the dynamics of isotropic spin systems. However, when the Bloch equations are reformulated for anisotropic spin systems, much of their utility is lost because the spin evolution they describe is not physically observable. A set of Bloch-like equations are derived for these anisotropic systems in terms of the magnetic moment which is the physical property measured in magnetic resonance and other experiments. The equations describe the dynamics of the magnetic moment including relaxation and only contain parameters that are experimentally measurable.

A theoretical and numerical consideration of the longitudinal and transverse relaxations in the rotating frame.

We previously derived a simple equation for solving time-dependent Bloch equations by a matrix operation. The purpose of this study was to present a theoretical and numerical consideration of the longitudinal (R1ρ=1/T1ρ) and transverse relaxation rates in the rotating frame (R2ρ=1/T2ρ), based on this method. First, we derived an equation describing the time evolution of the magnetization vector (M(t)) by expanding the matrix exponential into the eigenvalues and the corresponding eigenvectors using diagonalization. Second, we obtained the longitudinal magnetization vector in the rotating frame (M1ρ(t)) by taking the inner product of M(t) and the eigenvector with the smallest eigenvalue in modulus, and then we obtained the transverse magnetization vector in the rotating frame (M2ρ(t)) by subtracting M1ρ(t) from M(t). For comparison, we also computed the spin-locked magnetization vector. We derived the exact solutions for R1ρ and R2ρ from the eigenvalues, and compared them with those obtained numerically from M1ρ(t) and M2ρ(t), respectively. There was excellent agreement between them. From the exact solutions for R1ρ and R2ρ, R2ρ was found to be given by R2ρ=(2R2+R1)/2-R1ρ/2, where R1 and R2 denote the conventional longitudinal and transverse relaxation rates, respectively. We also derived M1ρ(t) and M2ρ(t) for bulk water protons, in which the effect of chemical exchange was taken into account using a 2-pool chemical exchange model, and we compared the R1ρ and R2ρ values obtained from the eigenvalues and those obtained numerically from M1ρ(t) and M2ρ(t). There was also excellent agreement between them. In conclusion, this study will be useful for better understanding of the longitudinal and transverse relaxations in the rotating frame and for analyzing the contrast mechanisms in T1ρ- and T2ρ-weighted MRI.