Application of optimal control theory to the design of broadband excitation pulses for high-resolution NMR

Hydrogen bond formation may enhance RDC-based discrimination of enantiomers

The distinction of enantiomers based on residual anisotropic parameters obtained by alignment in chiral poly-γ-benzyl-L-glutamate (PBLG) is among the strongest in high-resolution NMR spectroscopy. However, large variations in enantiodifferentiation among different solutes are frequently observed. One hypothesis is that the formation of hydrogen bonds between solute and PBLG is important for the distinction of enantiomers. With a small set of three almost spherical enantiomeric pairs, for which 1DCH residual dipolar couplings are measured, we address this issue in a systematic way: borneol contains a single functional group that can act as a hydrogen bond donor, camphor has a single group that may act as a hydrogen bond acceptor, and quinuclidinol can act as both hydrogen bond donor and acceptor. The results are unambiguous: although camphor shows low enantiodifferentiation with PBLG and alignment that can be predicted well by the purely steric TRAMITE approach, the distinction of enantiomers for the other enantiomeric pairs is significantly higher with alignment properties that must involve a specific interaction in addition to steric alignment.

Optimal control pulses for the 1.2-GHz (28.2-T) NMR spectrometers

The ability to measure nuclear magnetic resonance (NMR) spectra with a large sample volume is crucial for concentration-limited biological samples to attain adequate signal-to-noise (S/N) ratio. The possibility to measure with a 5-mm cryoprobe is currently absent at the 1.2-GHz NMR instruments due to the exceedingly high radio frequency power demands, which is four times compared to 600-MHz instruments. Here, we overcome the high-power demands by designing optimal control (OC) pulses with up to 20 times lower power requirements than currently necessary at a 1.2-GHz spectrometer. We show that multidimensional biomolecular NMR experiments constructed using these OC pulses can bestow improvement in the S/N ratio of up to 26%. With the expected power limitations of a 5-mm cryoprobe, we observe an enhancement in the S/N ratio of more than 240% using our OC sequences. This motivates the development of a cryoprobe with a larger volume than the current 3-mm cryoprobes.

Implementation and applications of shaped pulses in EPR

In this review, we describe the application of shaped pulses for EPR spectroscopy. Pulses generated by fast arbitrary waveform generators are mostly used in the field of EPR spectroscopy for broadband (200 MHz-1 GHz) excitation of paramagnetic species. The implementation and optimization of such broadband pulses in existing EPR spectrometers, often designed and optimized for short rectangular microwave pulses, is demanding. Therefore, a major part of this review will describe in detail the implementation, testing and optimization of shaped pulses in existing EPR spectrometers. Additionally, we review applications using such pulses for broadband inversion of longitudinal magnetization as well as for the creation and manipulation of transverse magnetization in the field of dipolar and hyperfine EPR spectroscopy. They demonstrate the great potential of shaped pulses to improve the performance of pulsed EPR experiments. We give a brief theoretical description of shaped pulses and their limitations, especially for adiabatic pulses, most often used in EPR. We believe that this review can on the one hand be of practical use to EPR groups starting to work with such pulses, and on the other hand give readers an overview of the state of the art of shaped pulse applications in EPR spectroscopy.

Simulation and design of shaped pulses beyond the piecewise-constant approximation

Response functions of resonant circuits create ringing artefacts if their input changes rapidly. When physical limits of electromagnetic spectroscopies are explored, this creates two types of problems. Firstly, simulation: the system must be propagated accurately through every response transient, this may be computationally expensive. Secondly, optimal control: circuit response must be taken into account; it may be advantageous to design pulses that are resilient to such distortions. At the root of both problems is the popular piecewise-constant approximation for control sequences in the rotating frame; in magnetic resonance it has persisted since the earliest days and has become entrenched in the commercially available hardware. In this paper, we report an implementation and benchmarks of recent Lie-group methods that can efficiently simulate and optimise smooth control sequences.

Distinguishing glutamate and glutamine in in vivo 1 H MRS based on nuclear spin singlet order filtering

PURPOSE

The signals of glutamate (Glu) and glutamine (Gln) are often significantly overlapped in routine ¹ H-MR spectra of human brain in vivo. Selectively probing the signals of Glu and Gln in vivo is very important for the study of the metabolisms in which Glu and Gln are involved.

METHODS

The Glu-/Gln- targeted pulse sequences are developed to selectively probe the signals of Glu and Gln. The core part of the Glu-/Gln- targeted pulse sequences lies on the preparation of the nuclear spin singlet orders (SSOs) of the five-spin systems of Glu and Gln. The optimal control method is used to prepare the SSOs of Glu and Gln with high efficiency.

RESULTS

The Glu-/Gln- targeted pulse sequences have been applied on phantoms to selectively probe the signals of Glu and Gln. Moreover, in the in vivo experiments, the signals of Glu and Gln in human brains of healthy subjects have been successfully probed separately.

CONCLUSION

The developed Glu-/Gln- targeted pulse sequences can be used to distinguish the ¹ H-MR signals of Glu and Gln in human brains in vivo. The optimal control method provides an effective way to prepare the SSO of a specific spin system with high efficiency and in turn selectively probe the signals of a targeted molecule.

Composite pulse combinations for chirp excitation

Composite pulses are the efficient method for broadband excitation to get control of the limitations of high field NMR, such as resonance offset effects with constraints on rf power that leads to signal intensity distortion. Phase-modulated chirp pulses are used as ordered composite pulse sequences in this paper as CHORUS sequence in a high-field NMR spectrometer (BRUKER 750 MHz) for broadband excitation. The composite pulse sequence applies chirp pulses with the forward and the reverse sweep mechanisms. A single excitation pulse combines adiabatic and non-adiabatic rotation, explained as a three-phase rotation, which leaves the magnetizing vectors to a non-uniform phase dispersion as a function of the offset frequency. One adiabatic refocusing pulse of the double sweep rate after the excitation pulse cannot satisfactorily compensate for the phase dispersion. Hence, composite self-refocussing CHORUS excitation pulse, with forward, reverse, and their combinations are used to remove the non-uniform phase dispersion generated due to offset resonance frequency. Four such combinations of composite pulses are produced with analytical explanation in this paper. MATLAB simulation results and experimental verification on the BRUKER 750 MHz NMR spectrometer of the composite pulses are also presented in this paper.

Adaptive optimal control of entangled qubits

Developing fast, robust, and accurate methods for optimal control of quantum systems comprising interacting particles is one of the most active areas of current science. Although a valuable repository of algorithms is available for numerical applications in quantum control, the high computational cost is somewhat overlooked. Here, we present a fast and accurate optimal control algorithm for systems of interacting qubits, QOALA (quantum optimal control by adaptive low-cost algorithm), which is predicted to offer [Formula: see text](M2) speedup for an M-qubit system, compared to the state-of-the-art exact methods, without compromising overall accuracy of the optimal solution. The method is general and compatible with diverse Hamiltonian structures. The proposed approach uses inexpensive low-accuracy approximations of propagators far from the optimum, adaptively switching to higher accuracy, higher-cost propagators when approaching the optimum. In addition, the utilization of analytical Lie algebraic derivatives that do not require computationally expensive matrix exponential brings even better performance.

Band-selective universal 90° and 180° rotation pulses covering the aliphatic carbon chemical shift range for triple resonance experiments on 1.2 GHz spectrometers

Biomolecular NMR spectroscopy requires large magnetic field strengths for high spectral resolution. Today's highest fields comprise proton Larmor frequencies of 1.2 GHz and even larger field strengths are to be expected in the future. In protein triple resonance experiments, various carbon bandwidths need to be excited by selective pulses including the large aliphatic chemical shift range. When the spectrometer field strength is increased, the length of these pulses has to be decreased by the same factor, resulting in higher rf-amplitudes being necessary in order to cover the required frequency region. Currently available band-selective pulses like Q3/Q5 excite a narrow bandwidth compared to the necessary rf-amplitude. Because the maximum rf-power allowed in probeheads is limited, none of the selective universal rotation pulses reported so far is able to cover the full [Formula: see text]C aliphatic region on 1.2 GHz spectrometers. In this work, we present band-selective 90° and 180° universal rotation pulses (SURBOP90 and SURBOP180) that have a higher ratio of selective bandwidth to maximum rf-amplitude than standard pulses. Simulations show that these pulses perform better than standard pulses, e. g. Q3/Q5, especially when rf-inhomogeneity is taken into account. The theoretical and experimental performance is demonstrated in offset profiles and by implementing the SURBOP pulses in an HNCACB experiment at 1.2 GHz.

Advances in nano- and microscale NMR spectroscopy using diamond quantum sensors

Quantum technologies have seen a rapid developmental surge over the last couple of years. Though often overshadowed by quantum computation, quantum sensors show tremendous potential for widespread applications in chemistry and biology. One system stands out in particular: the nitrogen-vacancy (NV) center in diamond, an atomic-sized sensor allowing the detection of nuclear magnetic resonance (NMR) signals at unprecedented length scales down to a single proton. In this article, we review the fundamentals of NV center-based quantum sensing and its distinct impact on nano- and microscale NMR spectroscopy. Furthermore, we highlight possible future applications of this novel technology ranging from energy research, materials science, to single-cell biology, and discuss the associated challenges of these rapidly developing NMR sensors.

Genetic algorithm-based optimization of pulse sequences

PURPOSE

The performance of pulse sequences in vivo can be limited by fast relaxation rates, magnetic field inhomogeneity, and nonuniform spin excitation. We describe here a method for pulse sequence optimization that uses a stochastic numerical solver that in principle is capable of finding a global optimum. The method provides a simple framework for incorporating any constraint and implementing arbitrarily complex cost functions. Efficient methods for simulating spin dynamics and incorporating frequency selectivity are also described.

METHODS

Optimized pulse sequences for polarization transfer between protons and X-nuclei and excitation pulses that eliminate J-coupling modulation were evaluated experimentally using a surface coil on phantoms, and also the detection of hyperpolarized [2-13 C]lactate in vivo in the case of J-coupling modulation-free excitation.

RESULTS

The optimized polarization transfer pulses improved the SNR by ~50% with a more than twofold reduction in the B1 field, and J-coupling modulation-free excitation was achieved with a more than threefold reduction in pulse length.

CONCLUSION

This process could be used to optimize any pulse when there is a need to improve the uniformity and frequency selectivity of excitation as well as to design new pulses to steer the spin system to any desired achievable state.

SORDOR pulses: expansion of the Böhlen-Bodenhausen scheme for low-power broadband magnetic resonance

A novel type of efficient broadband pulse, called second-order phase dispersion by optimised rotation (SORDOR), has recently been introduced. In contrast to adiabatic excitation, SORDOR-90 pulses provide effective transverse 90∘ rotations throughout their bandwidth, with a quadratic offset dependence of the phase in the x , y plane. Together with phase-matched SORDOR-180 pulses, this enables the Böhlen-Bodenhausen broadband refocusing approach for linearly frequency-swept pulses to be extended to any type of 90∘ /180∘ pulse-delay sequence. Example pulse shapes are characterised in theory and experiment, and an example application is given with a 19 F -PROJECT experiment for measuring relaxation times with reduced distortions due to J -coupling evolution.

Concurrent J-evolving refocusing pulses

Conventional refocusing pulses are optimised for a single spin without considering any type of coupling. However, despite the fact that most couplings will result in undesired distortions, refocusing in delay-pulse-delay-type sequences with desired heteronuclear coherence transfer might be enhanced considerably by including coupling evolution into pulse design. We provide a proof of principle study for a Hydrogen-Carbon refocusing pulse sandwich with inherent J-evolution following the previously reported ICEBERG-principle with improved performance in terms of refocusing performance and/or overall effective coherence transfer time. Pulses are optimised using optimal control theory with a newly derived quality factor and z-controls as an efficient tool to speed up calculations. Pulses are characterised in theory and experiment and compared to conventional concurrent refocusing pulses, clearly showing an improvement for the J-evolving pulse sandwich. As a side-product, also efficient J-compensated resfocusing pulse sandwiches - termed BUBU pulses following the nomenclature of previous J-compensated BUBI and BEBE^tr pulse sandwiches - have been optimised.

Analytical models of probe dynamics effects on NMR measurements

This paper provides a detailed analysis of three common NMR probe circuits (untuned, tuned, and impedance-matched) and studies their effects on multi-pulse experiments, such as those based on the Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence. The magnitude of probe dynamics effects on broadband refocusing pulses are studied as a function of normalized RF bandwidth. Finally, the probe circuit models are integrated with spin dynamics simulations to design hardware-specific RF excitation and refocusing pulses for optimizing user-specified metrics such as signal-to-noise ratio (SNR) in grossly inhomogeneous fields. Preliminary experimental results on untuned probes are also presented.

Optimization of phase dispersion enables broadband excitation without homonuclear coupling artifacts

In NMR spectroscopy, many specialized shaped pulses are available for broadband excitation, beyond the bandwidth of conventional high-powered hard pulses. These shaped pulses typically have long duration. However, long-duration pulses are unsuitable for spectra containing significant homonuclear couplings, such as polyfluorinated compounds in 19F NMR. J-coupling evolution during the excitation pulse leads to spectral artifacts and incorrect peak integrals. Here, we report an approach to optimal control pulse design which significantly reduces the pulse length required to excite large bandwidths of chemical shift frequencies. The target state phase is not chosen beforehand but is instead only constrained to be linearly dependent on offset frequency. The first-order phase of the target state is then treated as a free-variable, to be optimized at the same time as the RF waveform itself. The resulting spectra are easily phased using standard NMR processing software. We observe that the required pulse length is significantly shorter than for currently available in-phase excitation schemes. Spectral artifacts from homonuclear couplings are avoided. We also demonstrate that pure in-phase excitation can be obtained over the same bandwidth by appending two inversion pulses, at the expense of increased overall duration.

Chirped ordered pulses for ultra-broadband ESR spectroscopy

Recently, applications of swept-frequency pulses proved to be a useful approach to circumvent the problem of limited excitation bandwidth in pulsed ESR posed by conventional pulses. Here, we present a chirped excitation sequence, CHirped ORdered pulses for Ultra-broadband Spectroscopy (CHORUS), for ultra-broadband ESR spectroscopy. It will be demonstrated that the application of this sequence can address the problems of excitation non-uniformity and sensitivity to instrumental instabilities to a greater extent compared to the current state of the art. This sequence is highly promising for finding applications beyond single excitation in many ESR experiments. Theoretical and experimental results for the proposed method are presented along with calibration strategies for experimental implementation.

Nearest-neighbor NMR spectroscopy: categorizing spectral peaks by their adjacent nuclei

Methyl-NMR enables atomic-resolution studies of structure and dynamics of large proteins in solution. However, resonance assignment remains challenging. The problem is to combine existing structural informational with sparse distance restraints and search for the most compatible assignment among the permutations. Prior classification of peaks as either from isoleucine, leucine, or valine reduces the search space by many orders of magnitude. However, this is hindered by overlapped leucine and valine frequencies. In contrast, the nearest-neighbor nuclei, coupled to the methyl carbons, resonate in distinct frequency bands. Here, we develop a framework to imprint additional information about passively coupled resonances onto the observed peaks. This depends on simultaneously orchestrating closely spaced bands of resonances along different magnetization trajectories, using principles from control theory. For methyl-NMR, the method is implemented as a modification to the standard fingerprint spectrum (the 2D-HMQC). The amino acid type is immediately apparent in the fingerprint spectrum. There is no additional relaxation loss or an increase in experimental time. The method is validated on biologically relevant proteins. The idea of generating new spectral information using passive, adjacent resonances is applicable to other contexts in NMR spectroscopy.

A novel spectrally selective fat saturation pulse design with robustness to B0 and B1 inhomogeneities: A demonstration on 3D T1-weighted breast MRI at 3 T

PURPOSE

Spectrally selective fat saturation (FatSat) sequence is commonly used to suppress signal from adipose tissue. Conventional SINC-shaped pulses are sensitive to B₀ off-resonance and B₁⁺ offset. Uniform fat saturation with large spatial coverage is especially challenging for the body and breast MRI. The aim of this study is to develop spectrally selective FatSat pulses that offer more immunity to B₀/B₁⁺ field inhomogeneities than SINC pulses and evaluate them in bilateral breast imaging at 3 T.

MATERIALS AND METHODS

Optimized composite pulses (OCP) were designed based on the optimal control theory with robustness to a targeted B₀/ B₁⁺ conditions. OCP pulses also allows flexible flip angles to meet different requirements. Comparisons with the vendor-provided SINC pulses were conducted by numerical simulation and in vivo scans using a 3D T₁-weighted (T₁w) gradient-echo (GRE) sequence with coverage of the whole-breast.

RESULTS

Simulation revealed that OCP pulses yielded almost half of the transition band and much less sensitivity to B₁⁺ inhomogeneity compared to SINC pulses with B₀ off-resonance within ±200 Hz and B₁⁺ scale error within ±0.3 (P < 0.001). Across five normal subjects, OCP FatSat pulses produced 25-41% lower residual fat signals (P < 0.05) with 27-36% less spatial variation (P < 0.05) than SINC.

CONCLUSION

In contrast to conventional SINC-shaped pulses, the newly designed OCP FatSat pulses mitigated challenges of wide range of B₀/ B₁⁺ field inhomogeneities and achieved more uniform fat suppression in bilateral breast T₁w imaging at 3 T.

Comprehensive and High-Throughput Exploration of Chemical Space Using Broadband 19 F NMR-Based Screening

Fragment-based lead discovery has become a fundamental approach to identify ligands that efficiently interact with disease-relevant targets. Among the numerous screening techniques, fluorine-detected NMR has gained popularity owing to its high sensitivity, robustness, and ease of use. To effectively explore chemical space, a universal NMR experiment, a rationally designed fragment library, and a sample composition optimized for a maximal number of compounds and minimal measurement time are required. Here, we introduce a comprehensive method that enabled the efficient assembly of a high-quality and diverse library containing nearly 4000 fragments and screening for target-specific binders within days. At the core of the approach is a novel broadband relaxation-edited NMR experiment that covers the entire chemical shift range of drug-like ¹⁹ F motifs in a single measurement. Our approach facilitates the identification of diverse binders and the fast ligandability assessment of new targets.

On the use of frequency-swept pulses and pulses designed with optimal control theory for the acquisition of ultra-wideline NMR spectra

Frequency-swept (FS) pulses, such as wideband uniform-rate smooth-truncation (WURST) pulses, have found much success for the acquisition of ultra-wideline (UW) solid-state NMR spectra. In this preliminary study, new pulses and pulse sequences are explored in simulation and experimentally for several nuclei exhibiting UWNMR powder patterns under static conditions, including ¹¹⁹Sn (I = 1/2), ¹⁹⁵Pt (I = 1/2), ²H (I = 1), and ⁷¹Ga (I = 3/2). First, hyperbolic secant (HS) and tanh/tan (THT) pulses are tested and implemented as excitation and refocusing pulses in spin-echo and Carr-Purcell/Meiboom Gill (CPMG)-type sequences, and shown to have comparable performances to analogous WURST pulses. Second, optimal control theory (OCT) is utilized for the design of new Optimal Control Theory Optimized Broadband Excitation and Refocusing (OCTOBER) pulses, using carefully parameterized WURST, THT, and HS pulses as starting points. Some of the new OCTOBER pulses used in spin-echo sequences are capable of efficient broadband excitation and refocusing, in some cases resulting in spectra with increased signal enhancements over those obtained in experiments using conventional FS pulses. Finally, careful consideration of the spin dynamics of several systems, by monitoring of the time evolution of the density matrix via the Liouville-von Neumann equation and analysis of the time-resolved Fourier transforms of the pulses, lends insight into the underlying mechanisms of the FS and OCTOBER pulses. This is crucial for understanding their performance in terms of generating uniformly excited patterns of high signal intensity, and for identifying trends that may offer pathways to generalized parameterization and/or new pulse shapes.

Hybrid-state free precession in nuclear magnetic resonance

The dynamics of large spin-1/2 ensembles are commonly described by the Bloch equation, which is characterized by the magnetization's non-linear response to the driving magnetic field. Consequently, most magnetic field variations result in non-intuitive spin dynamics, which are sensitive to small calibration errors. Although simplistic field variations result in robust spin dynamics, they do not explore the richness of the system's phase space. Here, we identify adiabaticity conditions that span a large experiment design space with tractable dynamics. All dynamics are trapped in a one-dimensional subspace, namely in the magnetization's absolute value, which is in a transient state, while its direction adiabatically follows the steady state. In this hybrid state, the polar angle is the effective drive of the spin dynamics. As an example, we optimize this drive for robust and efficient quantification of spin relaxation times and utilize it for magnetic resonance imaging of the human brain.

Improved ultra-broadband chirp excitation

The design and application of ultra-broadband excitation pulses have been among the most interesting and timely areas in NMR and EPR methodology in recent years, due especially to advances in hardware design in EPR, the advent and popularity of high- and ultrahigh-field NMR, and the application of numerical methods like optimal control theory to the design and optimization of radiofrequency pulses and pulse sequences. In this communication, we present a short, robust, and flexible version of the CHORUS family of constant-phase, very broadband excitation sequences. We demonstrate that more than 0.5 MHz excitation with uniform amplitudes and phases can be achieved with this excitation sequence.

ASAP-HSQC-TOCSY for fast spin system identification and extraction of long-range couplings

Based on Ernst-angle-type excitation and Acceleration by Sharing Adjacent Polarization (ASAP), a fast HSQC-TOCSY experiment is introduced. In the approach, the DIPSI-2 isotropic mixing period of the ASAP-HSQC is simply shifted, which provides a TOCSY period without additional application of rf-energy. The ASAP-HSQC-TOCSY allows the acquisition of a conventional 2D in about 30 s. Alternatively, it allows the acquisition of highly carbon-resolved spectra (several Hz digital resolution) on the order of minutes. An ASAP-HSQC-TOCSY-IPAP variant, finally, allows the sign-sensitive extraction of heteronuclear long-range coupling constants from a pair of highly resolved spectra in less than an hour. Pulse sequences, several example spectra, and a discussion of results are given.

Assessing the potential of quantitative 2D HSQC NMR in 13C enriched living organisms

In vivo Nuclear Magnetic Resonance (NMR) spectroscopy has great potential to interpret the biochemical response of organisms to their environment, thus making it an essential tool in understanding toxic mechanisms. However, magnetic susceptibility distortions lead to 1D NMR spectra of living organisms with lines that are too broad to identify and quantify metabolites, necessitating the use of 2D ¹H-¹³C Heteronuclear Single Quantum Coherence (HSQC) as a primary tool. While quantitative 2D HSQC is well established, to our knowledge it has yet to be applied in vivo. This study represents a simple pilot study that compares two of the most popular quantitative 2D HSQC approaches to determine if quantitative results can be directly obtained in vivo in isotopically enriched Daphnia magna (water flea). The results show the perfect-HSQC experiment performs very well in vivo, but the decoupling scheme used is critical for accurate quantitation. An improved decoupling approach derived using optimal control theory is presented here that improves the accuracy of metabolite concentrations that can be extracted in vivo down to micromolar concentrations. When combined with 2D Electronic Reference To access In vivo Concentrations (ERETIC) protocols, the protocol allows for the direct extraction of in vivo metabolite concentrations without the use of internal standards that can be detrimental to living organisms. Extracting absolute metabolic concentrations in vivo is an important first step and should, for example, be important for the parameterization as well as the validation of metabolic flux models in the future.

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>B</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> -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.

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.

Boosting the NMR Assignment of Carbohydrates with Clean In-Phase Correlation Experiments

Novel CLIP-COSY based homo- and heteronuclear correlation experiments are reported for the rapid, semi-automated NMR assignment of small to medium-sized molecules. The homonuclear CLIP-COSY and corresponding relayed experiments employ the perfect-echo based mixing sequence for in-phase coherence transfer between directly and/or indirectly coupled proton spins. The combined analysis of the resulting CLIP-COSY and relayed spectra made it possible to easily track down, layer by layer, the proton-proton connectivity network. In larger molecules the narrow chemical shift range of protons may, however, compromise the efficacy of the homonuclear correlation based assignment strategy. To overcome this limitation, an HSQC variant of the CLIP-COSY experiment has now been devised. Combined treatment of HSQC-CLIP-COSY (relayed) and standard HSQC spectra facilitates simultaneous and semi-automatic assignment of ¹ H and ¹³ C resonances of medium-sized molecules, such as pentasaccharides. The recently introduced PSYCHE broadband homonuclear decoupling scheme has been also implemented into the devised homo- and heteronuclear CLIP-COSY based experiments, resulting in fully decoupled high-resolution pure-shift correlation spectra.

Cooperative broadband spin echoes through optimal control

The Hahn echo sequence is one of the most common building blocks in magnetic resonance, consisting of an excitation pulse and a refocusing pulse. Conventional approaches to improve the performance of echo experiments focused on the optimization of individual pulses, compensating their own imperfections. Here we present an approach to concurrently design both pulses such that they also compensate each others imperfections. The fact that for such cooperative pulses the individual pulses do not need to be perfect provides additional degrees of freedom, resulting in improved overall Hahn echo performance. Single-scan cooperative pulses are compared to conventional approaches by simulations as well as experiments.

Bang-bang optimal control of large spin systems: Enhancement of 13C-13C singlet-order at natural abundance

Using a bang-bang optimal control technique, we transfer polarization from a set of abundant high-γ nuclei directly to singlet order of a low-γ spin-pair. This approach is analogous to algorithmic cooling (AC) procedure used in quantum state purification. Specifically, we apply this method for enhancing the singlet order in a natural abundant ¹³C- ¹³C spin pair by exploiting nine equivalent protons of an 11-spin system. Compared to the standard method not involving polarization transfer, we find an enhancement of singlet order by about 3.4 times. In addition, since the singlet magnetization is contributed by the faster relaxing protons, the recycle delay is halved. Thus effectively we observe a reduction in the overall experimental time by a factor of 23. We also discuss a possible extension of AC, known as heat-bath algorithmic cooling (HBAC).

Broadband RF-amplitude-dependent flip angle pulses with linear phase slope

Pulse sequences in NMR spectroscopy sometimes require the application of pulses with effective flip angles different from 90° and 180°. Previously (Magn. Reson. Chem. 2015, 53, 886-893), offset-compensated broadband excitation pulses with RF-amplitude-dependent effective flip angles (RADFA) were introduced that are applicable in such cases. However, especially RF-amplitude-restricted RADFA pulses turned out to perform not as good as desired in terms of achievable bandwidths. Here, a class of RF-amplitude-restricted RADFA pulses with linear phase slope is introduced that allows excitation over much larger bandwidths with better performance. In this theoretical work, the basic principle of the pulse class is explained, their physical limits explored, and their properties, also compared with other pulse classes, discussed in detail. Copyright © 2017 John Wiley & Sons, Ltd.

An optimal control approach to design entire relaxation dispersion experiments

A general approach is introduced to optimize experiments for the analysis of spin systems in the presence of chemical exchange. Rather than optimizing individual pulse sequence elements, such as refocusing pulses, entire relaxation dispersion sequences are optimized in the form of a single shaped pulse. This is achieved by defining a performance index that is only based on the remaining signal after the relaxation dispersion sequence for a range of exchange, relaxation, offset, and rf inhomogeneity parameters. The approach is demonstrated by optimizing energy-limited broadband relaxation dispersion sequences that closely approach the overall effect of ideal CPMG sequences. As illustrated both theoretically and experimentally, significant improvements are found compared to standard amplitude or energy-limited CPMG sequences.

Chirp excitation

The paper describes the design of broadband chirp excitation pulses. We first develop a three stage model for understanding chirp excitation in NMR. We then show how a chirp π pulse can be used to refocus the phase of the chirp excitation pulse. The resulting magnetization still has some phase dispersion in it. We show how a combination of two chirp π pulses instead of one can be used to eliminate this dispersion, leaving behind a small residual phase dispersion. The excitation pulse sequence presented here allows exciting arbitrary large bandwidths without increasing the peak rf-amplitude. Experimental excitation profiles for the residual HDO signal in a sample of 99.5% D₂O are displayed as a function of resonance offset. Although methods presented in this paper have appeared elsewhere, we present complete analytical treatment that elucidates the working of these methods.

Whole-brain arteriography and venography: Using improved velocity-selective saturation pulse trains

PURPOSE

To develop velocity-selective (VS) MR angiography (MRA) protocols for arteriography and venography with whole-brain coverage.

METHODS

Tissue suppression using velocity-selective saturation (VSS) pulse trains is sensitive to radiofrequency field (B₁ +) inhomogeneity. To reduce its sensitivity, we replaced the low-flip-angle hard pulses in the VSS pulse train with optimal composite (OCP) pulses. Additionally, new pulse sequences for arteriography and venography were developed by placing spatially selective inversion pulses with a delay to null signals from either venous or arterial blood. The VS MRA techniques were compared to the time-of-flight (TOF) MRA in six healthy subjects and two patients at 3T.

RESULTS

More uniform suppression of stationary tissue was observed when the hard pulses were replaced by OCP pulses in the VSS pulse trains, which improved contrast ratios between blood vessels and tissue background for both arteries (0.87 vs. 0.77) and veins (0.80 vs. 0.59). Both arteriograms and venograms depicted all major cervical and intracranial arteries and veins, respectively. Compared to TOF MRA, VS MRA not only offers larger spatial coverage but also depicts more small vessels. Initial clinical feasibility was shown in two patients with comparisons to TOF protocols.

CONCLUSION

Noncontrast-enhanced whole-brain arteriography and venography can be obtained without losing sensitivity to small vessel detection. Magn Reson Med 79:2014-2023, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Improvements, extensions, and practical aspects of rapid ASAP-HSQC and ALSOFAST-HSQC pulse sequences for studying small molecules at natural abundance

Previously we introduced two novel NMR experiments for small molecules, the so-called ASAP-HSQC and ALSOFAST-HSQC (Schulze-Sünninghausen et al., 2014), which allow the detection of heteronuclear one-bond correlations in less than 30s at natural abundance. We propose an improved symmetrized pulse scheme of the basic experiment to minimize artifact intensities and the combination with non-uniform sampling to enable the acquisition of conventional HSQC spectra in as short as a couple of seconds and extremely ¹³C-resolved spectra in less than ten minutes. Based on steady state investigations, a first estimate to relative achievable signal intensities with respect to conventional, ASAP-, and ALSOFAST-HSQC experiments is given. In addition, we describe several extensions to the basic pulse schemes, like a multiplicity-edited version, a revised symmetrized CLIP-ASAP-HSQC, an ASAP-/ALSOFAST-HSQC sequence with broadband BIRD-based ¹H,¹H decoupling, and a symmetrized sequence optimized for water suppression. Finally, RF-power considerations with respect to the high duty cycle of the experiments are given.

Active control of the spatial MRI phase distribution with optimal control theory

This paper investigates the use of Optimal Control (OC) theory to design Radio-Frequency (RF) pulses that actively control the spatial distribution of the MRI magnetization phase. The RF pulses are generated through the application of the Pontryagin Maximum Principle and optimized so that the resulting transverse magnetization reproduces various non-trivial and spatial phase patterns. Two different phase patterns are defined and the resulting optimal pulses are tested both numerically with the ODIN MRI simulator and experimentally with an agar gel phantom on a 4.7T small-animal MR scanner. Phase images obtained in simulations and experiments are both consistent with the defined phase patterns. A practical application of phase control with OC-designed pulses is also presented, with the generation of RF pulses adapted for a Magnetic Resonance Elastography experiment. This study demonstrates the possibility to use OC-designed RF pulses to encode information in the magnetization phase and could have applications in MRI sequences using phase images.

Rapid convergence of optimal control in NMR using numerically-constructed toggling frames

We present a numerical method for rapidly solving the Bloch equation for an arbitrary time-varying spin-1/2 Hamiltonian. The method relies on fast, vectorized computations such as summation and quaternion multiplication, rather than slow computations such as matrix exponentiation. A toggling frame is constructed in which the Hamiltonian is time-invariant, and therefore has a simple analytical solution. The key insight is that constructing this frame is faster than solving the system dynamics in the original frame. Rapidly solving the Bloch equations for an arbitrary Hamiltonian is particularly useful in the context of NMR optimal control. Optimal control theory can be used to design pulse shapes for a range of tasks in NMR spectroscopy. However, it requires multiple simulations of the Bloch equations at each stage of the algorithm, and for each relevant set of parameters (e.g. chemical shift frequencies). This is typically time consuming. We demonstrate that by working in an appropriate toggling frame, optimal control pulses can be generated much faster. We present a new alternative to the well-known GRAPE algorithm to continuously update the toggling-frame as the optimal pulse is generated, and demonstrate that this approach is extremely fast. The use and benefit of rapid optimal pulse generation is demonstrated for 19F fragment screening experiments.

Perspectives of shaped pulses for EPR spectroscopy

This article describes current uses of shaped pulses, generated by an arbitrary waveform generator, in the field of EPR spectroscopy. We show applications of sech/tanh and WURST pulses to dipolar spectroscopy, including new pulse schemes and procedures, and discuss the more general concept of optimum-control-based pulses for applications in EPR spectroscopy. The article also describes a procedure to correct for experimental imperfections, mostly introduced by the microwave resonator, and discusses further potential applications and limitations of such pulses.

Multiple-Quantum Transitions and Charge-Induced Decoherence of Donor Nuclear Spins in Silicon

We study single- and multiquantum transitions of the nuclear spins of an ensemble of ionized arsenic donors in silicon and find quadrupolar effects on the coherence times, which we link to fluctuating electrical field gradients present after the application of light and bias voltage pulses. To determine the coherence times of superpositions of all orders in the 4-dimensional Hilbert space, we use a phase-cycling technique and find that, when electrical effects were allowed to decay, these times scale as expected for a fieldlike decoherence mechanism such as the interaction with surrounding ^{29}Si nuclear spins.

Quantum Ensemble Classification: A Sampling-Based Learning Control Approach

Quantum ensemble classification (QEC) has significant applications in discrimination of atoms (or molecules), separation of isotopes, and quantum information extraction. However, quantum mechanics forbids deterministic discrimination among nonorthogonal states. The classification of inhomogeneous quantum ensembles is very challenging, since there exist variations in the parameters characterizing the members within different classes. In this paper, we recast QEC as a supervised quantum learning problem. A systematic classification methodology is presented by using a sampling-based learning control (SLC) approach for quantum discrimination. The classification task is accomplished via simultaneously steering members belonging to different classes to their corresponding target states (e.g., mutually orthogonal states). First, a new discrimination method is proposed for two similar quantum systems. Then, an SLC method is presented for QEC. Numerical results demonstrate the effectiveness of the proposed approach for the binary classification of two-level quantum ensembles and the multiclass classification of multilevel quantum ensembles.

Arbitrary waveform modulated pulse EPR at 200GHz

We report here on the implementation of arbitrary waveform generation (AWG) capabilities at ∼200GHz into an Electron Paramagnetic Resonance (EPR) and Dynamic Nuclear Polarization (DNP) instrument platform operating at 7T. This is achieved with the integration of a 1GHz, 2 channel, digital to analog converter (DAC) board that enables the generation of coherent arbitrary waveforms at K_u-band frequencies with 1ns resolution into an existing architecture of a solid state amplifier multiplier chain (AMC). This allows for the generation of arbitrary phase- and amplitude-modulated waveforms at 200GHz with >150mW power. We find that the non-linearity of the AMC poses significant difficulties in generating amplitude-modulated pulses at 200GHz. We demonstrate that in the power-limited regime of ω₁<1MHz phase-modulated pulses were sufficient to achieve significant improvements in broadband (>10MHz) spin manipulation in incoherent (inversion), as well as coherent (echo formation) experiments. Highlights include the improvement by one order of magnitude in inversion bandwidth compared to that of conventional rectangular pulses, as well as a factor of two in improvement in the refocused echo intensity at 200GHz.

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).

Application of spin echoes in the regime of weak dephasing to T1 -mapping of the lung

PURPOSE

This work presents an approach to mapping the entire lung's proton density and T₁ within a single breath-hold and analyzes the apparent T₁ when exciting with a spin echo generating pulse in comparison to a standard gradient echo acquisition.

METHODS

An inversion-recovery SNAPSHOT-FLASH sequence with a stack-of-stars k-space readout with a golden angle increment was modified to use a spin echo generating radiofrequency-pulse for excitation. Data of five volunteers were acquired on a 3T scanner and image reconstruction was performed by an iterative algorithm adopted from MR-Fingerprinting.

RESULTS

The feasibility of acquiring quantitative maps of the entire lung with a resolution of 5 × 5 × 10 mm within 7.5 s is demonstrated. It is shown that the proposed spin echo forming radiofrequency-pulse increases the apparent proton density compared to a rectangular pulse. Further, the apparent T₁ is reduced in the spin echo case compared to the gradient echo sequence.

CONCLUSION

The proposed spin echo based method results in T₁ maps that are comparable to the ones that were acquired with ultra-short echo time sequences elsewhere. The T₁ shortening is believed to originate from increased signal contributions of the extra vascular compartment, which has a short T2∗ and T₁ . Magn Reson Med 79:960-967, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Learning robust pulses for generating universal quantum gates

Constructing a set of universal quantum gates is a fundamental task for quantum computation. The existence of noises, disturbances and fluctuations is unavoidable during the process of implementing quantum gates for most practical quantum systems. This paper employs a sampling-based learning method to find robust control pulses for generating a set of universal quantum gates. Numerical results show that the learned robust control fields are insensitive to disturbances, uncertainties and fluctuations during the process of realizing universal quantum gates.

Ultra broadband NMR spectroscopy using multiple rotating frame technique

The paper describes the design of broadband excitation and inversion pulses by method of multiple rotating frame technique. The ideal situation for perfect excitation and inversion is to have no chemical shift offset and thereby everything on resonance. However, when chemical shifts span a wide range, this condition is not realized. We achieve this condition using a multiply modulated rf-field, whose effect can be understood by progressing into multiple frames. As we progress through the frames, the ratio of chemical shift dispersion to strength of static rf-field decreases, resulting in a final frame, where there is negligible chemical shift as compared to the effective rf-field and we get good excitation and inversion. Increasing the number of frames, increases excitation bandwidth and the ratio of bandwidth to rms excitation amplitude. We show, in principle, it is possible to excite arbitrary large bandwidth for a given rms rf-amplitude by increasing the number of frames. The time of excitation increases linearly with the bandwidth when we keep the rms rf-amplitude constant. Experimental demonstration of proposed method is presented on (1)H excitation over a bandwidth of 52 kHz with a rms amplitude of 10 kHz. Increasing the frames increases excitation bandwidth for same rms amplitude of 10 kHz. Experimental spectra obtained from 100%(13)C labeled arginine shows uniform excitation over the entire carbon spectra, obtained with a 8-frame pulse sequence.