Optimal control design of excitation pulses that accommodate relaxation

Relaxational signal attenuation during soft refocusing pulses

Quantitative NMR is widely used, but the systematic errors introduced when signals are excited by anything other than a single hard pulse are not always well understood. One important source of error in experiments using soft pulses is the spin relaxation that takes place during pulses, which contains contributions from both spin-spin and spin-lattice relaxation. Here it is shown that relaxation on resonance during shaped soft 180° refocusing pulses in practical experiments can be well represented by biexponential decay, with rate constants R₂ and a shape-dependent linear combination of R₁ and R₂, where R₁ and R₂ are the inverses of the spin-lattice and spin-spin relaxation times T₁ and T₂. In principle this would allow correction for relaxational losses in experiments using on-resonance selective refocusing pulses.

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.

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.

Application of Optimal Control Theory to Fourier Transform Ion Cyclotron Resonance

We study the application of Optimal Control Theory to Ion Cyclotron Resonance. We test the validity and the efficiency of this approach for the robust excitation of an ensemble of ions with a wide range of cyclotron frequencies. Optimal analytical solutions are derived in the case without any pulse constraint. A gradient-based numerical optimization algorithm is proposed to take into account limitation in the control intensity. The efficiency of optimal pulses is investigated as a function of control time, maximum amplitude and range of excited frequencies. A comparison with adiabatic and SWIFT pulses is done. On the basis of recent results in Nuclear Magnetic Resonance, this study highlights the potential usefulness of optimal control in Ion Cyclotron Resonance.

Shaping EPR: Phase and amplitude modulated microwave pulses

The advent of fast arbitrary waveform generators in the sub-nanosecond time regime recently enabled new experimental developments in the field of pulsed EPR. In this article, the new possibilities of such fast phase/amplitude modulated microwave pulses are shortly described with respect to applications in pulsed dipolar spectroscopy. Some of the specific challenges of an accurate creation of such pulses in the field of EPR are outlined. Finally, a short outlook of potential applications is given and some specific experimental conditions are discussed, where shaped pulses might have an especially important impact in the future.

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.

Rotation relaxation splitting for optimizing parallel RF excitation pulses with T1- and T2-relaxations in MRI

Exact solutions of the Bloch equations with T₁- and T₂-relaxation terms for piecewise constant magnetic fields are numerically challenging. We therefore investigate an approximation for the achieved magnetization in which rotations and relaxations are split into separate operations. We develop an estimate for its accuracy and explicit first and second order derivatives with respect to the complex excitation radio frequency voltages. In practice, the deviation between an exact solution of the Bloch equations and this rotation relaxation splitting approximation seems negligible. Its computation times are similar to exact solutions without relaxation terms. We apply the developed theory to numerically optimize radio frequency excitation waveforms with T₁- and T₂-relaxations in several examples.

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.

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.

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.

Controlling open quantum systems: tools, achievements, and limitations

The advent of quantum devices, which exploit the two essential elements of quantum physics, coherence and entanglement, has sparked renewed interest in the control of open quantum systems. Successful implementations face the challenge of preserving relevant nonclassical features at the level of device operation. A major obstacle is decoherence, which is caused by interaction with the environment. Optimal control theory is a tool that can be used to identify control strategies in the presence of decoherence. Here we review recent advances in optimal control methodology that allow typical tasks in device operation for open quantum systems to be tackled and discuss examples of relaxation-optimized dynamics. Optimal control theory is also a useful tool to exploit the environment for control. We discuss examples and point out possible future extensions.

Sodium inversion recovery MRI on the knee joint at 7 T with an optimal control pulse

In the field of sodium magnetic resonance imaging (MRI), inversion recovery (IR) is a convenient and popular method to select sodium in different environments. For the knee joint, IR has been used to suppress the signal from synovial fluids, which improves the correlation between the sodium signal and the concentration of glycosaminoglycans (GAGs) in cartilage tissues. For the better inversion of the magnetization vector under the spatial variations of the B0 and B1 fields, the IR sequence usually employ adiabatic pulses as the inversion pulse. On the other hand, it has been shown that RF shapes robust against the variations of the B0 and B1 fields can be generated by numerical optimization based on optimal control theory. In this work, we compare the performance of fluid-suppressed sodium MRI on the knee joint in vivo, between one implemented with an adiabatic pulse in the IR sequence and the other with the adiabatic pulse replaced by an optimal-control shaped pulse. While the optimal-control pulse reduces the RF power deposited to the body by 58%, the quality of fluid suppression and the signal level of sodium within cartilage are similar between two implementations.

Broadband excitation pulses with variable RF amplitude-dependent flip angle (RADFA)

Pulse sequences in NMR spectroscopy sometimes require the adjustment of effective flip angles with respect to experiment-specific or sample-specific parameters. Here, we present a quality factor for efficient optimization of offset-compensated broadband excitation pulses with RF amplitude-dependent effective flip angles (RADFA). After proof of principle, physical limits of RF amplitude-restricted and RF power-restricted broadband RADFA pulses are explored and corresponding pulse shapes and performances characterized in detail.

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.

Robust INEPT and refocused INEPT transfer with compensation of a wide range of couplings, offsets, and B1-field inhomogeneities (COB3)

Following the two-step optimization procedure previously introduced with the COB-INEPT (Ehni and Luy, 2012), a corresponding inphase-to-antiphase transfer element with close to optimal transfer efficiencies over a coupling range comprising approximately J-6J has been derived. The hard pulse sequence length is only 5.5 ms for coupling constants within 125-750 Hz. Robustness with respect to an offset range of 37.5 kHz on carbon (corresponding to 250 ppm on a 600 MHz spectrometer) and 10 kHz on protons (16.6 ppm at 600 MHz) is achieved with corresponding BUBI and BURBOP broadband pulses. As the sequence achieves a three times higher upper limit of J-compensation compared to the COB-INEPT, we name the transfer element COB3-INEPT. Next to the description of optimization and pulse sequence details, the performance of the resulting element is demonstrated on a test sample and partially aligned sample with actual total couplings in the range of 134 Hz⩽(1)TCH⩽391 Hz. The sequence can also be used for inphase-to-antiphase transfer starting from carbon, where the upper limit of J-compensation is 6J for CH-groups, 3J for CH2-groups, and slightly less than 2J for CH3. Theoretical transfers and experimental verification for the different multiplicities in an refocused INEPT are given.

In vivo measurement of apparent diffusion coefficients of hyperpolarized ¹³C-labeled metabolites

The combination of hyperpolarized MRS with diffusion weighting (dw) allows for determination of the apparent diffusion coefficient (ADC), which is indicative of the intra- or extracellular localization of the metabolite. Here, a slice-selective pulsed-gradient spin echo sequence was implemented to acquire a series of dw spectra from rat muscle in vivo to determine the ADCs of multiple metabolites after a single injection of hyperpolarized [1- ¹³C]pyruvate. An optimal control optimized universal-rotation pulse was used for refocusing to minimize signal loss caused by B1 imperfections. Non-dw spectra were acquired interleaved with the dw spectra and these were used to correct for signal decay during the acquisition as a result of T1 decay, pulse imperfections, flow etc. The data showed that the ADC values for [1- ¹³C]lactate (0.4-0.7 µm² /ms) and [1- ¹³C]alanine (0.4-0.9 µm² /ms) were about a factor of two lower than the ADC of [1- ¹³C]pyruvate (1.1-1.5 µm²/ms). This indicates a more restricted diffusion space for the former two metabolites consistent with lactate and alanine being intracellular. The higher ADC for pyruvate (similar to the proton ADC) reflected that the injected substance was not confined inside the muscle cells but also present extracellular.

Next-generation heteronuclear decoupling for high-field biomolecular NMR spectroscopy

Ultra-high-field NMR spectroscopy requires an increased bandwidth for heteronuclear decoupling, especially in biomolecular NMR applications. Composite pulse decoupling cannot provide sufficient bandwidth at practical power levels, and adiabatic pulse decoupling with sufficient bandwidth is compromised by sideband artifacts. A novel low-power, broadband heteronuclear decoupling pulse is presented that generates minimal, ultra-low sidebands. The pulse was derived using optimal control theory and represents a new generation of decoupling pulses free from the constraints of periodic and cyclic sequences. In comparison to currently available state-of-the-art methods this novel pulse provides greatly improved decoupling performance that satisfies the demands of high-field biomolecular NMR spectroscopy.

BEBE(tr) and BUBI: J-compensated concurrent shaped pulses for 1H-13C experiments

Shaped pulses designed for broadband excitation, inversion and refocusing are important tools in modern NMR spectroscopy to achieve robust pulse sequences especially in heteronuclear correlation experiments. A large variety of mostly computer-optimized pulse shapes exist for different desired bandwidths, available rf-field strengths, and tolerance to B1-inhomogeneity. They are usually derived for a single spin 1/2, neglecting evolution due to J-couplings. While pulses with constant resulting phase are selfcompensated for heteronuclear coupling evolution as long as they are applied exclusively on a single nucleus, the situation changes for concurrently applied pulse shapes. Using the example of a (1)H,(13)C two spin system, two J-compensated pulse pairs for the application in INEPT-type transfer elements were optimized: a point-to-point pulse sandwich called BEBE(tr), consisting of a broadband excitation and time-reversed excitation pulse, and a combined universal rotation and point-to-point pulse pair called BUBI, which acts as a refocusing pulse on (1)H and a corresponding inversion pulse on (13)C. After a derivation of quality factors and optimization protocols, a theoretical and experimental comparison with conventionally derived BEBOP, BIBOP, and BURBOP-180° pulses is given. While the overall transfer efficiency of a single pulse pair is only reduced by approximately 0.1%, resulting transfer to undesired coherences is reduced by several percent. In experiments this can lead to undesired phase distortions for pairs of uncompensated pulse shapes and even differences in signal intensities of 5-10% in HSQC and up to 68% in more complex COB-HSQC experiments.

Keeping PASE with WEFT: SHWEFT-PASE pulse sequences for 1H NMR spectra of highly paramagnetic molecules

Metalloproteins are a category of biomolecules in which the metal site is usually the locus of activity or function. In many cases, the metal ions are paramagnetic or have accessible paramagnetic states, many of which can be studied using NMR spectroscopy. Extracting useful information from (1)H NMR spectra of highly paramagnetic proteins can be difficult because the paramagnetism leads to large resonance shifts (~400 ppm), extremely broad lines, extreme baseline nonlinearity, and peak shape distortion. It is demonstrated that employing polychromatic and adiabatic shaped pulses in simple pulse sequences, then combining existing sequences, leads to significant spectral improvement for highly paramagnetic proteins. These sequences employ existing technology, with available hardware, and are of short duration to accommodate short nuclear T1 and T2. They are shown to display uniform excitation over large spectral widths (~75 kHz), accommodate high repetition rates, produce flat baselines over 75 kHz while maintaining peak shape fidelity, and can be used to reduce spectral dynamic range. High-spin (S = 5/2) metmyoglobin, a prototypical highly paramagnetic protein, was used as the test molecule. The resulting one-dimensional (1D) pulse sequences combine shaped pulses with super-water elimination Fourier transform, which can be further combined with paramagnetic spectroscopy to give shaped pulses with super-water elimination Fourier transform-paramagnetic spectroscopy. These sequences require, at most, direct current offset correction and minimal phasing. The performance of these sequences in simple (1)H 1D, 1D NOE, and two-dimensional NOESY experiments is demonstrated for metmyoglobin and Paracoccus denitrificans Co(2+)-amicyanin (S = 3/2), and employed to make new heme hyperfine resonance assignments for high-spin metBjFixLH(151-256), the heme sensing domain of Bradyrhizobium japonicum FixL.

The Fantastic Four: A plug 'n' play set of optimal control pulses for enhancing NMR spectroscopy

We present highly robust, optimal control-based shaped pulses designed to replace all 90° and 180° hard pulses in a given pulse sequence for improved performance. Special attention was devoted to ensuring that the pulses can be simply substituted in a one-to-one fashion for the original hard pulses without any additional modification of the existing sequence. The set of four pulses for each nucleus therefore consists of 90° and 180° point-to-point (PP) and universal rotation (UR) pulses of identical duration. These 1ms pulses provide uniform performance over resonance offsets of 20kHz ((1)H) and 35kHz ((13)C) and tolerate reasonably large radio frequency (RF) inhomogeneity/miscalibration of ±15% ((1)H) and ±10% ((13)C), making them especially suitable for NMR of small-to-medium-sized molecules (for which relaxation effects during the pulse are negligible) at an accessible and widely utilized spectrometer field strength of 600MHz. The experimental performance of conventional hard-pulse sequences is shown to be greatly improved by incorporating the new pulses, each set referred to as the Fantastic Four (Fanta4).

A systematic approach for optimizing the robustness of pulse sequence elements with respect to couplings, offsets, and B1-field inhomogeneities (COB)

Robust experiments that cover a wide range of chemical shift offsets and J-couplings are highly desirable for a multitude of applications in small molecule NMR spectroscopy. Many attempts to improve individual aspects of the robustness of pulse sequence elements based on rational and numerical design have been reported, but a general optimization strategy to cover all necessary aspects for a fully robust sequence is still lacking. In this article, a viable optimization strategy is introduced that covers a defined range of couplings, offsets, and B(1)-field inhomogeneities (COB) in a time-optimal way. Individual components of the optimization strategy can be optimized in any adequate way. As an example for the COB approach, we present the (1)H -(13)C-COB-INEPT with transfer of approximately 99% over the full carbon and proton bandwidth and (1)J(CH) -couplings in the range of 120-250 Hz, which have been optimized using efficient algorithms derived from optimal control theory. The theoretical performance is demonstrated in a number of corresponding COB-HSQC experiments.

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.

Tailored real-time scaling of heteronuclear couplings

Heteronuclear couplings are a valuable source of molecular information, which is measured from the multiplet splittings of an NMR spectrum. Radiofrequency irradiation on one coupled nuclear spin allows to modify the effective coupling constant, scaling down the multiplet splittings in the spectrum observed at the resonance frequency of the other nuclear spin. Such decoupling sequences are often used to collapse a multiplet into a singlet and can therefore simplify NMR spectra significantly. Continuous-wave (cw) decoupling has an intrinsic non-linear offset dependence of the scaling of the effective J-coupling constant. Using optimal control pulse optimization, we show that virtually arbitrary off-resonance scaling of the J-coupling constant can be achieved. The new class of tailored decoupling pulses is named SHOT (Scaling of Heteronuclear couplings by Optimal Tracking). Complementing cw irradiation, SHOT pulses offer an alternative approach of encoding chemical shift information indirectly through off-resonance decoupling, which however makes it possible for the first time to achieve linear J scaling as a function of offset frequency. For a simple mixture of eight aromatic compounds, it is demonstrated experimentally that a 1D-SHOT {(1)H}-(13)C experiment yields comparable information to a 2D-HSQC and can give full assignment of all coupled spins.

Designing optimal universal pulses using second-order, large-scale, non-linear optimization

Recently, RF pulse design using first-order and quasi-second-order pulses has been actively investigated. We present a full second-order design method capable of incorporating relaxation, inhomogeneity in B(0) and B(1). Our model is formulated as a generic optimization problem making it easy to incorporate diverse pulse sequence features. To tame the computational cost, we present a method of calculating second derivatives in at most a constant multiple of the first derivative calculation time, this is further accelerated by using symbolic solutions of the Bloch equations. We illustrate the relative merits and performance of quasi-Newton and full second-order optimization with a series of examples, showing that even a pulse already optimized using other methods can be visibly improved. To be useful in CPMG experiments, a universal refocusing pulse should be independent of the delay time and insensitive of the relaxation time and RF inhomogeneity. We design such a pulse and show that, using it, we can obtain reliable R(2) measurements for offsets within ±γB(1). Finally, we compare our optimal refocusing pulse with other published refocusing pulses by doing CPMG experiments.

Shaped optimal control pulses for increased excitation bandwidth in EPR

A 1 ns resolution pulse shaping unit has been developed for pulsed EPR spectroscopy to enable 14-bit amplitude and phase modulation. Shaped broadband excitation pulses designed using optimal control theory (OCT) have been tested with this device at X-band frequency (9 GHz). FT-EPR experiments on organic radicals in solution have been performed with the new pulses, designed for uniform excitation over a significantly increased bandwidth compared to a classical rectangular π/2 pulse of the same B(1) amplitude. The concept of a dead-time compensated prefocused pulse has been introduced to EPR with a self-refocusing of 200 ns after the end of the pulse. Echo-like refocused signals have been recorded and compared to the performance of a classical Hahn-echo sequence. The impulse response function of the microwave setup has been measured and incorporated into the algorithm for designing OCT pulses, resulting in further significant improvements in performance. Experimental limitations and potential new applications of OCT pulses in EPR spectroscopy will be discussed.

Optimal control design of band-selective excitation pulses that accommodate relaxation and RF inhomogeneity

Existing optimal control protocols for mitigating the effects of relaxation and/or RF inhomogeneity on broadband pulse performance are extended to the more difficult problem of designing robust, refocused, frequency selective excitation pulses. For the demanding case of T(1) and T(2) equal to the pulse length, anticipated signal losses can be significantly reduced while achieving nearly ideal frequency selectivity. Improvements in performance are the result of allowing residual unrefocused magnetization after applying relaxation-compensated selective excitation by optimized pulses (RC-SEBOPs). We demonstrate simple pulse sequence elements for eliminating this unwanted residual signal.

New strategies for designing robust universal rotation pulses: application to broadband refocusing at low power

Optimizing pulse performance often requires a compromise between maximizing signal amplitude and minimizing spectral phase errors. We consider methods for the de novo design of universal rotation pulses, applied specifically but not limited to refocusing pulses. Broadband inversion pulses that rotate all magnetization components 180° about a given fixed axis are necessary for refocusing and mixing in high-resolution NMR spectroscopy. The relative merits of various methodologies for generating pulses suitable for broadband refocusing are considered. The de novo design of 180° universal rotation pulses (180(UR)(°)) using optimal control can provide improved performance compared to schemes which construct refocusing pulses as composites of existing pulses. The advantages of broadband universal rotation by optimized pulses (BURBOP) are most evident for pulse design that includes tolerance to RF inhomogeneity or miscalibration. Nearly ideal refocusing is possible over a resonance offset range of ± 170% relative to the nominal pulse B(1) field, concurrent with tolerance to B(1) inhomogeneity/miscalibration of ± 33%. We present new modifications of the optimal control algorithm that incorporate symmetry principles (S-BURBOP) and relax conservative limits on peak RF pulse amplitude for short time periods that pose no threat to the probe. We apply them to generate a set of low-power 180(BURBOP)(°) pulses suitable for widespread use in (13)C spectroscopy on the majority of available probes. A quantitative measure for the reduced spectral phase error provided by these symmetry principles is also derived. For pulses designed according to this symmetry, refocusing phase errors are virtually eliminated upon application of EXORCYCLE or an equivalent G-180(S-BURBOP)(°)-G gradient sandwich, independent of resonance offset and RF inhomogeneity. The magnitude of the refocused component is not significantly compromised in achieving such ideal phase performance.

Robust slice-selective broadband refocusing pulses

Slice-selective broadband refocusing pulses are of great interest in localized MR spectroscopy for improving spatial selectivity, reducing chemical-shift displacement errors, and reducing anomalous J modulation. In practice the bandwidth of RF pulses is limited by the maximum available B1 amplitude. The goal of the present work is to design slice-selective and broadband refocusing pulses which are tolerant against B1 deviations. Pulse design is performed by numerical optimization based on optimal control theory. A comprehensive study of different cost functions and their effect on the optimization is given. The optimized slice-selective broadband refocusing pulses are compared to conventional Shinnar-Le Roux (SLR), broadband SLR, and hyperbolic secant pulses. In simulations and experiments optimized pulses were shown to fulfill broadband slice specifications over a range of ±20% B1 scalings. Experimental validation showed a reduction of chemical-shift displacement error by a factor of 3 compared to conventional SLR pulses.

Second order gradient ascent pulse engineering

We report some improvements to the gradient ascent pulse engineering (GRAPE) algorithm for optimal control of spin ensembles and other quantum systems. These include more accurate gradients, convergence acceleration using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm as well as faster control derivative calculation algorithms. In all test systems, the wall clock time and the convergence rates show a considerable improvement over the approximate gradient ascent.

Design and application of robust rf pulses for toroid cavity NMR spectroscopy

We present robust radio frequency (rf) pulses that tolerate a factor of six inhomogeneity in the B₁ field, significantly enhancing the potential of toroid cavity resonators for NMR spectroscopic applications. Both point-to-point (PP) and unitary rotation (UR) pulses were optimized for excitation, inversion, and refocusing using the gradient ascent pulse engineering (GRAPE) algorithm based on optimal control theory. In addition, the optimized parameterization (OP) algorithm applied to the adiabatic BIR-4 UR pulse scheme enabled ultra-short (50 μs) pulses with acceptable performance compared to standard implementations. OP also discovered a new class of non-adiabatic pulse shapes with improved performance within the BIR-4 framework. However, none of the OP-BIR4 pulses are competitive with the more generally optimized UR pulses. The advantages of the new pulses are demonstrated in simulations and experiments. In particular, the DQF COSY result presented here represents the first implementation of 2D NMR spectroscopy using a toroid probe.

Cooperative pulses

We introduce the concept of cooperative (COOP) pulses which are designed to compensate each other's imperfections. In multi-scan experiments, COOP pulses can cancel undesired signal contributions, complementing and generalizing phase cycles. COOP pulses can be efficiently optimized using an extended version of the optimal-control-based gradient ascent pulse engineering (GRAPE) algorithm. The advantage of the COOP approach is experimentally demonstrated for broadband and band-selective pulses.

Exact solution to the Bloch equations and application to the Hahn echo

The exact symbolic solution of the Bloch equations is given in the Lagrange form and illustrated with R2 experiments using a Hahn echo. Two different methods are also applied to approximately solve the Bloch equations, we find that splittings with effective-field interpretations are very substantially better than other approximations by comparing the errors. Estimates of transverse relaxation, R2, from Hahn echos are effected by frequency offset and field inhomogeneity. We use exact solutions of the Bloch equations and simulations to quantify both effects, and find that even in the presence of expected B0 inhomogeneity, off-resonance effects can be removed from R2 measurements, when∥ω∥⩽0.5γB1, by fitting the exact solutions of the Bloch equations. Further, the experiments and simulations show that the fitting models with the exact solutions of the Bloch equations do not depend on the sampling density and delay times.

Optimal control design of pulse shapes as analytic functions

Representing NMR pulse shapes by analytic functions is widely employed in procedures for optimizing performance. Insights concerning pulse dynamics can be applied to the choice of appropriate functions that target specific performance criteria, focusing the solution search and reducing the space of possible pulse shapes that must be considered to a manageable level. Optimal control theory can accommodate significantly larger parameter spaces and has been able to tackle problems of much larger scope than more traditional optimization methods. However, its numerically generated pulses, as currently constructed, do not readily incorporate the capabilities of particular functional forms, and the pulses are not guaranteed to vary smoothly in time, which can be a problem for faithful implementation on older hardware. An optimal control methodology is derived for generating pulse shapes as simple parameterized functions. It combines the benefits of analytic and numerical protocols in a single powerful algorithm that both complements and enhances existing optimization strategies.

Optimal excitation of (23)Na nuclear spins in the presence of residual quadrupolar coupling and quadrupolar relaxation

Optimal control theory is applied for designing pulse sequences to optimally excite a spin-3/2 system with residual quadrupolar coupling in the presence of quadrupolar relaxation. A homogeneous form of the master equation is constructed to simulate the dynamics of the spin system, and a general optimization procedure with a homogeneous form of the equation of motion is described. The optimized pulses are tested with (23)Na NMR, and their performance is compared with that of pulses optimized in the absence of relaxation.

Heteronuclear decoupling by optimal tracking

The problem to design efficient heteronuclear decoupling sequences is studied using optimal control methods. A generalized version of the gradient ascent engineering (GRAPE) algorithm is presented that makes it possible to design complex non-periodic decoupling sequences which are characterized by tens of thousands of pulse sequence parameters. In contrast to conventional approaches based on average Hamiltonian theory, the concept of optimal tracking is used: a pulse sequence is designed that steers the evolution of an ensemble of spin systems such that at a series of time points, a specified trajectory of the density operator is tracked as closely as possible. The approach is demonstrated for the case of low-power heteronuclear decoupling in the liquid state for in vivo applications. Compared to conventional sequences, significant gains in decoupling efficiency and robustness with respect to offset and inhomogeneity of the radio-frequency field were found in simulations and experiments.

Selective detection of ordered sodium signals by a jump-and-return pulse sequence

A simple pulse sequence, derived from the shaped pulse optimally exciting the central transition of a spin 3/2, can be used to selectively detect ordered sodium with a given quadrupolar coupling. The pulse sequence consists of two pulses with opposite phases and separated by a delay, called a quadrupolar jump-and-return (QJR) sequence. This QJR sequence is tested with a phantom made of sodium ions in bacteriophage and in aqueous solution and its feasibility for contrast modification based on the quadrupolar coupling is demonstrated.

RF pulses for in vivo spectroscopy at high field designed under conditions of limited power using optimal control

Localized in vivo spectroscopy at high magnetic field strength (>3T) is susceptible to localization artifacts such as the chemical shift artifact and the spatial interference artifact for J-coupled spins. This latter artifact results in regions of anomalous phase for J-coupled spins. These artifacts are exacerbated at high magnetic field due to the increased frequency dispersion, coupled with the limited RF pulse bandwidths used for localization. Approaches to minimize these artifacts include increasing the bandwidth of the frequency selective excitation pulses, and the use of frequency selective saturation pulses to suppress the signals in the regions with anomalous phase. The goal of this article is to demonstrate the efficacy of optimal control methods to provide broader bandwidth frequency selective pulses for in vivo spectroscopy in the presence of limited RF power. It is demonstrated by examples that the use of optimal control methods enable the generation of (i) improved bandwidth selective excitation pulses, (ii) more efficient selective inversion pulses to be used for generation of spin echoes, and (iii) improved frequency selective saturation pulses. While optimal control also allows for the generation of frequency selective spin echo pulses, it is argued that it is more efficient to use dual inversion pulses for broadband generation of spin echoes. Finally, the optimal control routines and example RF pulses are made available for downloading.

Optimal nuclear magnetic resonance excitation schemes for the central transition of a spin 3/2 in the presence of residual quadrupolar coupling

Optimal control theory is applied for enhancing the intensity of the central peak of a spin 3/2 signal in the presence of a residual quadrupolar coupling. While a maximum enhancement is always possible in the regime omega(rf) << omega(Q) via the use of modulated and shaped pulses, the intermediate rf-power regime omega(rf)-omega(Q) does not admit simple solutions based on intuition. In this work we present optimized shaped pulses that have been derived using an optimization algorithm based on optimal control and test these with (23)Na NMR in this regime. In addition to enhancing the intensity of the central transition signal, the satellite peaks can be effectively suppressed, which is a useful feature for the implementation in (23)Na imaging sequences.

Exploring the limits of broadband excitation and inversion: II. Rf-power optimized pulses

In [K. Kobzar, T.E. Skinner, N. Khaneja, S.J. Glaser, B. Luy, Exploring the limits of broadband excitation and inversion, J. Magn. Reson. 170 (2004) 236-243], optimal control theory was employed in a systematic study to establish physical limits for the minimum rf-amplitudes required in broadband excitation and inversion pulses. In a number of cases, however, experimental schemes are not limited by rf-amplitudes, but by the overall rf-power applied to a sample. We therefore conducted a second systematic study of excitation and inversion pulses of varying pulse durations with respect to bandwidth and rf-tolerances, but this time using a modified algorithm involving restricted rf-power. The resulting pulses display a variety of pulse shapes with highly modulated rf-amplitudes and generally show better performance than corresponding pulses with identical pulse length and rf-power, but limited rf-amplitude. A detailed description of pulse shapes and their performance is given for the so-called power-BEBOP and power-BIBOP pulses.

Linear phase slope in pulse design: application to coherence transfer

Using optimal control methods, robust broadband excitation pulses can be designed with a defined linear phase dispersion. Applications include increased bandwidth for a given pulse length compared to equivalent pulses requiring no phase correction, selective pulses, and pulses that mitigate the effects of relaxation. This also makes it possible to create pulses that are equivalent to ideal hard pulses followed by an effective evolution period. For example, in applications, where the excitation pulse is followed by a constant delay, e.g. for the evolution of heteronuclear couplings, part of the pulse duration can be absorbed in existing delays, significantly reducing the time overhead of long, highly robust pulses. We refer to the class of such excitation pulses with a defined linear phase dispersion as ICEBERG pulses (Inherent Coherence Evolution optimized Broadband Excitation Resulting in constant phase Gradients). A systematic study of the dependence of the excitation efficiency on the phase dispersion of the excitation pulses is presented, which reveals surprising opportunities for improved pulse sequence performance.