Optimal control design of NMR and dynamic nuclear polarization experiments using monotonically convergent algorithms

Controlling NMR spin systems for quantum computation

Nuclear magnetic resonance is arguably both the best available quantum technology for implementing simple quantum computing experiments and the worst technology for building large scale quantum computers that has ever been seriously put forward. After a few years of rapid growth, leading to an implementation of Shor's quantum factoring algorithm in a seven-spin system, the field started to reach its natural limits and further progress became challenging. Rather than pursuing more complex algorithms on larger systems, interest has now largely moved into developing techniques for the precise and efficient manipulation of spin states with the aim of developing methods that can be applied in other more scalable technologies and within conventional NMR. However, the user friendliness of NMR implementations means that they remain popular for proof-of-principle demonstrations of simple quantum information protocols.

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.

Optimal control pulses for subspectral editing in low field NMR

Low field NMR is an inexpensive and low footprint technique to obtain physical, chemical, electronic and structural information on small molecules, but suffers from poor spectral dispersion, especially when applied to the analysis of mixtures. Subspectral editing employing optimal control pulses is a suitable approach to cope with the severe signal superpositions in complex mixture spectra at low field. In this contribution, the use of optimal control pulses is demonstrated to be feasible at a field strength as low as 0.5 T. The optimal control pulse shapes were calculated using the Krotov algorithm. Downsizing the complexity of the algorithm from exponential to polynomial is shown to be possible by using a system approach with each system corresponding to a (small) molecule. In this way compound selective excitation pulses can be calculated. The signals of substructures of the cyclopentenone molecule were excited using optimal control pulses calculated by the Krotov algorithm demonstrating the feasibility of subspectral editing. Likewise, for a mixture of benzoic acid and alanine, editing of the signals of either benzoic acid or alanine employing optimal control pulses was shown to be possible. The obtained results are very promising and can be extended to the targeted analysis of complex mixtures such as biofluids or metabolic samples at low field strengths opening access for benchtop NMR to point of care settings.

Spin-Scenario: A flexible scripting environment for realistic MR simulations

In this paper we present a new open source package, Spin-Scenario, aimed at developing an intuitive, flexible and unique scripting framework able to cover many aspects of simulations in both MR imaging and MR spectroscopy. For this purpose, we adopted the Liouville space model as the standard computing engine and let the consequent computational burden be afforded by parallel computing techniques. Benefitting from the powerful Lua scripting language, the pulse sequence programming syntax was specially designed to offer an extremely concise way of scripting. Moreover, the built-in dataflow graph based optimal control scheme enables an efficient optimization of shaped pulses or multiple cooperative pulses for real-life experiment evaluations. As the name states, the users are expected to be able to realize their creative ideas like a scenarist that creates a scenario script and looks at the spin actors acting accordingly. The validation of the framework was demonstrated with several examples within MR imaging and MR spectroscopy. Spin-Scenario is available for download at https://github.com/spin-scenario.

HR-MAS NMR Based Quantitative Metabolomics in Breast Cancer

High resolution magic-angle spinning (HR-MAS) nuclear magnetic resonance (NMR) spectroscopy is increasingly used for profiling of breast cancer tissue, delivering quantitative information for approximately 40 metabolites. One unique advantage of the method is that it can be used to analyse intact tissue, thereby requiring only minimal sample preparation. Importantly, since the method is non-destructive, it allows further investigations of the same specimen using for instance transcriptomics. Here, we discuss technical aspects critical for a successful analysis—including sample handling, measurement conditions, pulse sequences for one- and two dimensional analysis, and quantification methods—and summarize available studies, with a focus on significant associations of metabolite levels with clinically relevant parameters.

Simple analysis of scattering data with the Ornstein-Zernike equation

In this paper we propose and explore a method of analysis of the scattering experimental data for uniform liquidlike systems. In our pragmatic approach we are not trying to introduce by hands an artificial small parameter to work out a perturbation theory with respect to the known results, e.g., for hard spheres or sticky hard spheres (all the more that in the agreement with the notorious Landau statement, there is no physical small parameter for liquids). Instead of it being guided by the experimental data we are solving the Ornstein-Zernike equation with a trial (variational) form of the interparticle interaction potential. To find all needed correlation functions this variational input is iterated numerically to satisfy the Ornstein-Zernike equation supplemented by a closure relation. Our method is developed for spherically symmetric scattering objects, and our numeric code is written for such a case. However, it can be extended (at the expense of more involved computations and a larger amount of required experimental input information) for nonspherical particles. What is important for our approach is that it is sufficient to know experimental data in a relatively narrow range of the scattering wave vectors (q) to compute the static structure factor in a much broader range of q. We illustrate by a few model and real experimental examples of the x-ray and neutron scattering data how the approach works.

Local SAR, global SAR, and power-constrained large-flip-angle pulses with optimal control and virtual observation points

PURPOSE

To present a constrained optimal-control (OC) framework for designing large-flip-angle parallel-transmit (pTx) pulses satisfying hardware peak-power as well as regulatory local and global specific-absorption-rate (SAR) limits. The application is 2D and 3D spatial-selective 90° and 180° pulses.

THEORY AND METHODS

The OC gradient-ascent-pulse-engineering method with exact gradients and the limited-memory Broyden-Fletcher-Goldfarb-Shanno method is proposed. Local SAR is constrained by the virtual-observation-points method. Two numerical models facilitated the optimizations, a torso at 3 T and a head at 7 T, both in eight-channel pTx coils and acceleration-factors up to 4.

RESULTS

The proposed approach yielded excellent flip-angle distributions. Enforcing the local-SAR constraint, as opposed to peak power alone, reduced the local SAR 7 and 5-fold with the 2D torso excitation and inversion pulse, respectively. The root-mean-square errors of the magnetization profiles increased less than 5% with the acceleration factor of 4.

CONCLUSION

A local and global SAR, and peak-power constrained OC large-flip-angle pTx pulse design was presented, and numerically validated for 2D and 3D spatial-selective 90° and 180° pulses at 3 T and 7 T. Magn Reson Med 77:374-384, 2017. © 2015 Wiley Periodicals, Inc.

Application of the limited-memory quasi-Newton algorithm for multi-dimensional, large flip-angle RF pulses at 7T

OBJECTIVE

Ultrahigh field MRI provides great opportunities for medical diagnostics and research. However, ultrahigh field MRI also brings challenges, such as larger magnetic susceptibility induced field changes. Parallel-transmit radio-frequency pulses can ameliorate these complications while performing advanced tasks in routine applications. To address one class of such pulses, we propose an optimal-control algorithm as a tool for designing advanced multi-dimensional, large flip-angle, radio-frequency pulses. We contrast initial conditions, constraints, and field correction abilities against increasing pulse trajectory acceleration factors.

MATERIALS AND METHODS

On an 8-channel 7T system, we demonstrate the quasi-Newton algorithm with pulse designs for reduced field-of-view imaging with an oil phantom and in vivo with scans of the human brain stem. We used echo-planar imaging with 2D spatial-selective pulses. Pulses are computed sufficiently rapid for routine applications.

RESULTS

Our dataset was quantitatively analyzed with the conventional mean-square-error metric and the structural-similarity index from image processing. Analysis of both full and reduced field-of-view scans benefit from utilizing both complementary measures.

CONCLUSION

We obtained excellent outer-volume suppression with our proposed method, thus enabling reduced field-of-view imaging using pulse trajectory acceleration factors up to 4.

A peripheral component interconnect express-based scalable and highly integrated pulsed spectrometer for solution state dynamic nuclear polarization

High sensitivity, high data rates, fast pulses, and accurate synchronization all represent challenges for modern nuclear magnetic resonance spectrometers, which make any expansion or adaptation of these devices to new techniques and experiments difficult. Here, we present a Peripheral Component Interconnect Express (PCIe)-based highly integrated distributed digital architecture pulsed spectrometer that is implemented with electron and nucleus double resonances and is scalable specifically for broad dynamic nuclear polarization (DNP) enhancement applications, including DNP-magnetic resonance spectroscopy/imaging (DNP-MRS/MRI). The distributed modularized architecture can implement more transceiver channels flexibly to meet a variety of MRS/MRI instrumentation needs. The proposed PCIe bus with high data rates can significantly improve data transmission efficiency and communication reliability and allow precise control of pulse sequences. An external high speed double data rate memory chip is used to store acquired data and pulse sequence elements, which greatly accelerates the execution of the pulse sequence, reduces the TR (time of repetition) interval, and improves the accuracy of TR in imaging sequences. Using clock phase-shift technology, we can produce digital pulses accurately with high timing resolution of 1 ns and narrow widths of 4 ns to control the microwave pulses required by pulsed DNP and ensure overall system synchronization. The proposed spectrometer is proved to be both feasible and reliable by observation of a maximum signal enhancement factor of approximately -170 for (1)H, and a high quality water image was successfully obtained by DNP-enhanced spin-echo (1)H MRI at 0.35 T.

Real-time 2D spatially selective MRI experiments: Comparative analysis of optimal control design methods

There is an increasing need for development of advanced radio-frequency (RF) pulse techniques in modern magnetic resonance imaging (MRI) systems driven by recent advancements in ultra-high magnetic field systems, new parallel transmit/receive coil designs, and accessible powerful computational facilities. 2D spatially selective RF pulses are an example of advanced pulses that have many applications of clinical relevance, e.g., reduced field of view imaging, and MR spectroscopy. The 2D spatially selective RF pulses are mostly generated and optimised with numerical methods that can handle vast controls and multiple constraints. With this study we aim at demonstrating that numerical, optimal control (OC) algorithms are efficient for the design of 2D spatially selective MRI experiments, when robustness towards e.g. field inhomogeneity is in focus. We have chosen three popular OC algorithms; two which are gradient-based, concurrent methods using first- and second-order derivatives, respectively; and a third that belongs to the sequential, monotonically convergent family. We used two experimental models: a water phantom, and an in vivo human head. Taking into consideration the challenging experimental setup, our analysis suggests the use of the sequential, monotonic approach and the second-order gradient-based approach as computational speed, experimental robustness, and image quality is key. All algorithms used in this work were implemented in the MATLAB environment and are freely available to the MRI community.

Computer-intensive simulation of solid-state NMR experiments using SIMPSON

Conducting large-scale solid-state NMR simulations requires fast computer software potentially in combination with efficient computational resources to complete within a reasonable time frame. Such simulations may involve large spin systems, multiple-parameter fitting of experimental spectra, or multiple-pulse experiment design using parameter scan, non-linear optimization, or optimal control procedures. To efficiently accommodate such simulations, we here present an improved version of the widely distributed open-source SIMPSON NMR simulation software package adapted to contemporary high performance hardware setups. The software is optimized for fast performance on standard stand-alone computers, multi-core processors, and large clusters of identical nodes. We describe the novel features for fast computation including internal matrix manipulations, propagator setups and acquisition strategies. For efficient calculation of powder averages, we implemented interpolation method of Alderman, Solum, and Grant, as well as recently introduced fast Wigner transform interpolation technique. The potential of the optimal control toolbox is greatly enhanced by higher precision gradients in combination with the efficient optimization algorithm known as limited memory Broyden-Fletcher-Goldfarb-Shanno. In addition, advanced parallelization can be used in all types of calculations, providing significant time reductions. SIMPSON is thus reflecting current knowledge in the field of numerical simulations of solid-state NMR experiments. The efficiency and novel features are demonstrated on the representative simulations.

Optimized multiple-quantum filter for robust selective excitation of metabolite signals

The selective excitation of metabolite signals in vivo requires the use of specially adapted pulse techniques, in particular when the signals are weak and the resonances overlap with those of unwanted molecules. Several pulse sequences have been proposed for this spectral editing task. However, their performance is strongly degraded by unavoidable experimental imperfections. Here, we show that optimal control theory can be used to generate pulses and sequences that perform almost ideally over a range of rf field strengths and frequency offsets that can be chosen according to the specifics of the spectrometer or scanner being used. We demonstrate this scheme by applying it to lactate editing. In addition to the robust excitation, we also have designed the pulses to minimize the signal of unwanted molecular species.

Dynamic nuclear polarization using frequency modulation at 3.34 T

During dynamic nuclear polarization (DNP) experiments polarization is transferred from unpaired electrons to their neighboring nuclear spins, resulting in dramatic enhancement of the NMR signals. While in most cases this is achieved by continuous wave (cw) irradiation applied to samples in fixed external magnetic fields, here we show that DNP enhancement of static samples can improve by modulating the microwave (MW) frequency at a constant field of 3.34 T. The efficiency of triangular shaped modulation is explored by monitoring the (1)H signal enhancement in frozen solutions containing different TEMPOL radical concentrations at different temperatures. The optimal modulation parameters are examined experimentally and under the most favorable conditions a threefold enhancement is obtained with respect to constant frequency DNP in samples with low radical concentrations. The results are interpreted using numerical simulations on small spin systems. In particular, it is shown experimentally and explained theoretically that: (i) The optimal modulation frequency is higher than the electron spin-lattice relaxation rate. (ii) The optimal modulation amplitude must be smaller than the nuclear Larmor frequency and the EPR line-width, as expected. (iii) The MW frequencies corresponding to the enhancement maxima and minima are shifted away from one another when using frequency modulation, relative to the constant frequency experiments.

Designing dipolar recoupling and decoupling experiments for biological solid-state NMR using interleaved continuous wave and RF pulse irradiation

Rapid developments in solid-state NMR methodology have boosted this technique into a highly versatile tool for structural biology. The invention of increasingly advanced rf pulse sequences that take advantage of better hardware and sample preparation have played an important part in these advances. In the development of these new pulse sequences, researchers have taken advantage of analytical tools, such as average Hamiltonian theory or lately numerical methods based on optimal control theory. In this Account, we focus on the interplay between these strategies in the systematic development of simple pulse sequences that combines continuous wave (CW) irradiation with short pulses to obtain improved rf pulse, recoupling, sampling, and decoupling performance. Our initial work on this problem focused on the challenges associated with the increasing use of fully or partly deuterated proteins to obtain high-resolution, liquid-state-like solid-state NMR spectra. Here we exploit the overwhelming presence of (2)H in such samples as a source of polarization and to gain structural information. The (2)H nuclei possess dominant quadrupolar couplings which complicate even the simplest operations, such as rf pulses and polarization transfer to surrounding nuclei. Using optimal control and easy analytical adaptations, we demonstrate that a series of rotor synchronized short pulses may form the basis for essentially ideal rf pulse performance. Using similar approaches, we design (2)H to (13)C polarization transfer experiments that increase the efficiency by one order of magnitude over standard cross polarization experiments. We demonstrate how we can translate advanced optimal control waveforms into simple interleaved CW and rf pulse methods that form a new cross polarization experiment. This experiment significantly improves (1)H-(15)N and (15)N-(13)C transfers, which are key elements in the vast majority of biological solid-state NMR experiments. In addition, we demonstrate how interleaved sampling of spectra exploiting polarization from (1)H and (2)H nuclei can substantially enhance the sensitivity of such experiments. Finally, we present systematic development of (1)H decoupling methods where CW irradiation of moderate amplitude is interleaved with strong rotor-synchronized refocusing pulses. We show that these sequences remove residual cross terms between dipolar coupling and chemical shielding anisotropy more effectively and improve the spectral resolution over that observed in current state-of-the-art methods.

Genetic algorithms and solid state NMR pulse sequences

The use of genetic algorithms for the optimisation of magic angle spinning NMR pulse sequences is discussed. The discussion uses as an example the optimisation of the C7(2)(1) dipolar recoupling pulse sequence, aiming to achieve improved efficiency for spin systems characterised by large chemical shielding anisotropies and/or small dipolar coupling interactions. The optimised pulse sequence is found to be robust over a wide range of parameters, requires only minimal a priori knowledge of the spin system for experimental implementations with buildup rates being solely determined by the magnitude of the dipolar coupling interaction, but is found to be less broadbanded than the original C7(2)(1) pulse sequence. The optimised pulse sequence breaks the synchronicity between r.f. pulses and sample spinning.

Dynamic nuclear polarization and optimal control spatial-selective 13C MRI and MRS

Aimed at (13)C metabolic magnetic resonance imaging (MRI) and spectroscopy (MRS) applications, we demonstrate that dynamic nuclear polarization (DNP) may be combined with optimal control 2D spatial selection to simultaneously obtain high sensitivity and well-defined spatial restriction. This is achieved through the development of spatial-selective single-shot spiral-readout MRI and MRS experiments combined with dynamic nuclear polarization hyperpolarized [1-(13)C]pyruvate on a 4.7 T pre-clinical MR scanner. The method stands out from related techniques by facilitating anatomic shaped region-of-interest (ROI) single metabolite signals available for higher image resolution or single-peak spectra. The 2D spatial-selective rf pulses were designed using a novel Krotov-based optimal control approach capable of iteratively fast providing successful pulse sequences in the absence of qualified initial guesses. The technique may be important for early detection of abnormal metabolism, monitoring disease progression, and drug research.

Fast numerical design of spatial-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods

The use of increasingly strong magnetic fields in magnetic resonance imaging (MRI) improves sensitivity, susceptibility contrast, and spatial or spectral resolution for functional and localized spectroscopic imaging applications. However, along with these benefits come the challenges of increasing static field (B(0)) and rf field (B(1)) inhomogeneities induced by radial field susceptibility differences and poorer dielectric properties of objects in the scanner. Increasing fields also impose the need for rf irradiation at higher frequencies which may lead to elevated patient energy absorption, eventually posing a safety risk. These reasons have motivated the use of multidimensional rf pulses and parallel rf transmission, and their combination with tailoring of rf pulses for fast and low-power rf performance. For the latter application, analytical and approximate solutions are well-established in linear regimes, however, with increasing nonlinearities and constraints on the rf pulses, numerical iterative methods become attractive. Among such procedures, optimal control methods have recently demonstrated great potential. Here, we present a Krotov-based optimal control approach which as compared to earlier approaches provides very fast, monotonic convergence even without educated initial guesses. This is essential for in vivo MRI applications. The method is compared to a second-order gradient ascent method relying on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method, and a hybrid scheme Krotov-BFGS is also introduced in this study. These optimal control approaches are demonstrated by the design of a 2D spatial selective rf pulse exciting the letters "JCP" in a water phantom.

Exploring the limits of broadband 90° and 180° universal rotation pulses

90° and 180° universal rotation (UR) pulses are two of the most important classes of pulses in modern NMR spectroscopy. This article presents a systematic study characterizing the achievable performance of these pulses as functions of bandwidth, pulse length, and tolerance to B(1)-field inhomogeneity/miscalibration. After an evaluation of different quality factors employed in pulse design algorithms based on optimal control theory, resulting pulses are discussed in detail with a special focus on pulse symmetry. The vast majority of resulting BURBOP (broadband universal rotations by optimal control) pulses are either fully symmetric or have one symmetric and one antisymmetric Cartesian rf component, where the importance of the first symmetry has not been demonstrated yet and the latter one matches the symmetry that results from a previously derived construction principle of universal rotation pulses out of point-to-point pulses [3]. Optimized BURBOP pulses are shown to perform better than previously reported UR pulses, resulting in shorter pulse durations for the same quality of broadband rotations. From a comparison of qualities of effective universal rotations, we find that the application of a single optimal refocusing pulse matches or improves the performance of two consecutive inversion pulses in INEPT-like pulse sequence elements of the same total duration.

Recent advances in nuclear magnetic resonance quantum information processing

Quantum information processors have the potential to drastically change the way we communicate and process information. Nuclear magnetic resonance (NMR) has been one of the first experimental implementations of quantum information processing (QIP) and continues to be an excellent testbed to develop new QIP techniques. We review the recent progress made in NMR QIP, focusing on decoupling, pulse engineering and indirect nuclear control. These advances have enhanced the capabilities of NMR QIP, and have useful applications in both traditional NMR and other QIP architectures.

Optimal pulse design in quantum control: a unified computational method

Many key aspects of control of quantum systems involve manipulating a large quantum ensemble exhibiting variation in the value of parameters characterizing the system dynamics. Developing electromagnetic pulses to produce a desired evolution in the presence of such variation is a fundamental and challenging problem in this research area. We present such robust pulse designs as an optimal control problem of a continuum of bilinear systems with a common control function. We map this control problem of infinite dimension to a problem of polynomial approximation employing tools from geometric control theory. We then adopt this new notion and develop a unified computational method for optimal pulse design using ideas from pseudospectral approximations, by which a continuous-time optimal control problem of pulse design can be discretized to a constrained optimization problem with spectral accuracy. Furthermore, this is a highly flexible and efficient numerical method that requires low order of discretization and yields inherently smooth solutions. We demonstrate this method by designing effective broadband π/2 and π pulses with reduced rf energy and pulse duration, which show significant sensitivity enhancement at the edge of the spectrum over conventional pulses in 1D and 2D NMR spectroscopy experiments.

Guiding the time-evolution of a molecule: optical control by computer

The theory and computation of optical control has been developed over the last 25 years and is now a mature field of research. Initial work provided pictures of how control using light fields in simple systems may be achieved, for example using multiple excitation pathways or pulse sequences. The development of optimal control theory then provided a general method for guiding a system to its target using a shaped laser pulse. Combined with quantum dynamics simulations this has become a widely used tool, and has been applied to a range of systems to show what can be controlled. The present challenge is to gain more insight into the mechanism of control. In addition, methods need to be extended to reach the size of system of interest to technology. In this perspective article we shall give a brief overview of present capabilities and some of the recent developments in quantum dynamics and control simulations.

A kilowatt pulsed 94 GHz electron paramagnetic resonance spectrometer with high concentration sensitivity, high instantaneous bandwidth, and low dead time

We describe a quasioptical 94 GHz kW pulsed electron paramagnetic resonance spectrometer featuring pi/2 pulses as short as 5 ns and an instantaneous bandwidth of 1 GHz in nonresonant sample holders operating in induction mode and at low temperatures. Low power pulses can be as short as 200 ps and kilowatt pulses as short as 1.5 ns with timing resolution of a few hundred picoseconds. Phase and frequency can be changed on nanosecond time scales and complex high power pulse sequences can be run at repetition rates up to 80 kHz with low dead time. We demonstrate that the combination of high power pulses at high frequencies and nonresonant cavities can offer excellent concentration sensitivity for orientation selective pulsed electron double resonance (double electron-electron resonance), where we demonstrate measurements at 1 microM concentration levels.

Optimal control in NMR spectroscopy: numerical implementation in SIMPSON

We present the implementation of optimal control into the open source simulation package SIMPSON for development and optimization of nuclear magnetic resonance experiments for a wide range of applications, including liquid- and solid-state NMR, magnetic resonance imaging, quantum computation, and combinations between NMR and other spectroscopies. Optimal control enables efficient optimization of NMR experiments in terms of amplitudes, phases, offsets etc. for hundreds-to-thousands of pulses to fully exploit the experimentally available high degree of freedom in pulse sequences to combat variations/limitations in experimental or spin system parameters or design experiments with specific properties typically not covered as easily by standard design procedures. This facilitates straightforward optimization of experiments under consideration of rf and static field inhomogeneities, limitations in available or desired rf field strengths (e.g., for reduction of sample heating), spread in resonance offsets or coupling parameters, variations in spin systems etc. to meet the actual experimental conditions as close as possible. The paper provides a brief account on the relevant theory and in particular the computational interface relevant for optimization of state-to-state transfer (on the density operator level) and the effective Hamiltonian on the level of propagators along with several representative examples within liquid- and solid-state NMR spectroscopy.