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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.

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In this work, optimal control theory was used to design efficient excitation schemes in highly conductive materials, where both the radio frequency field strength and phase vary as a function of penetration depth. A pulse was designed to achieve phase alignment between signals at different depths within the conductor and thus to obtain higher signals from that region. In addition, an efficient suppression pulse was designed by insuring mutual suppression between the signals from various depths in the sample. The performance of the new approach was demonstrated experimentally for a bulk lithium sample for the excitation problem and for a biphasic metal/liquid sample for the selective suppression pulse.

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An exact general theory of heteronuclear decoupling is presented for spin-1/2 IS systems. RF irradiation applied to the I spins both modifies and generates additional couplings between states of the system. The recently derived equivalence between the dynamics of any N-level quantum system and a system of classical coupled harmonic oscillators makes explicit the exact physical couplings between states. Decoupling is thus more properly viewed as a complex intercoupling. The sign of antiphase magnetization plays a fundamental role in decoupling. A one-to-one correspondence is demonstrated between ±2SyIz and the sense of the S-spin coupling evolution. Magnetization Sx is refocused to obtain the desired decoupled state when ∫2SyIzdt=0. The exact instantaneous coupling at any time during the decoupling sequence is readily obtained in terms of the system states, showing that the creation of two-spin coherence is crucial for reducing the effective scalar coupling, as required for refocusing to occur. Representative examples from new aperiodic sequences as well as standard cyclic, periodic composite-pulse and adiabatic decoupling sequences illustrate the decoupling mechanism. The more general aperiodic sequences, obtained using optimal control, realize the potential inherent in the theory for significantly improved decoupling.

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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.

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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).

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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An optimal control algorithm for mitigating the effects of T(1) and T(2) relaxation during the application of long pulses is derived. The methodology is applied to obtain broadband excitation that is not only tolerant to RF inhomogeneity typical of high resolution probes, but is relatively insensitive to relaxation effects for T(1) and T(2) equal to the pulse length. The utility of designing pulses to produce specific phase in the final magnetization is also presented. The results regarding relaxation and optimized phase are quite general, with many potential applications beyond the specific examples presented here.

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An optimal control algorithm for generating purely phase-modulated pulses is derived. The methodology is applied to obtain broadband excitation with unprecedented tolerance to RF inhomogeneity. Design criteria were transformation of Iz-->Ix over resonance offsets of +/-25 kHz for constant RF amplitude anywhere in the range 10-20 kHz, with a pulse length of 1 ms. Simulations transform Iz to greater than 0.99 Ix over the targetted ranges of resonance offset and RF variability. Phase deviations in the final magnetization are less than 2-3 degrees over almost the entire range, with sporadic deviations of 6-9 degrees at a few offsets for the lowest RF (10 kHz) in the optimized range. Experimental performance of the new pulse is in excellent agreement with the simulations, and the robustness of the excitation pulse and a derived refocusing pulse are demonstrated by insertion into conventional HSQC and HMBC-type experiments.

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For a desired range of offsets, universal rotations of arbitrary flip angle can be constructed based on point-to-point rotations of I(y) with half the flip angle. This approach allows, for example, creation of broadband or bandselective refocusing pulses from broadband or bandselective excitation pulses. Furthermore, universal rotations about any axis can be obtained from point-to-point transformations that can easily be optimized using optimal control algorithms. The construction procedure is demonstrated on the examples of a broadband refocusing pulse, a broadband 120(x) degrees rotation and a z-rotation with offset pattern.

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The de facto standard cost function has been used heretofore to characterize the performance of pulses designed using optimal control theory. The freedom to choose new, creative quality factors designed for specific purposes is demonstrated. While the methodology has more general applicability, its utility is illustrated by comparison to a consistently chosen example--broadband excitation. The resulting pulses are limited to the same maximum RF amplitude used previously and tolerate the same variation in RF homogeneity deemed relevant for standard high-resolution NMR probes. Design criteria are unchanged: transformation of I(z)--> I(x) over resonance offsets of +/-20 kHz and RF variability of +/-5%, with a peak RF amplitude equal to 17.5 kHz. However, the new cost effectively trades a small increase in residual z magnetization for improved phase in the transverse plane. Compared to previous broadband excitation by optimized pulses (BEBOP), significantly shorter pulses are achievable, with only marginally reduced performance. Simulations transform I(z) to greater than 0.98 I(x), with phase deviations of the final magnetization less than 2 degrees, over the targeted ranges of resonance offset and RF variability. Experimental performance is in excellent agreement with the simulations.

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The design of broadband excitation and inversion pulses with compensation of B(1)-field inhomogeneity is a long standing goal in high resolution NMR spectroscopy. Most optimization procedures used so far have been restricted to particular pulse families to keep the scale of the problem within manageable limits. This restriction is unnecessary using efficient numerical algorithms based on optimal control theory. A systematic study of rf-limited broadband excitation by optimized pulses and broadband inversion by optimized pulses with respect to bandwidth and B(1)-field is presented. Upper limits on minimum pulse lengths are set for different degrees of pulse performance.

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Combining optimal control theory with a new RF limiting step produces pulses with significantly reduced duration and improved performance for a given maximum RF amplitude compared to previous broadband excitation by optimized pulses (BEBOP). The resulting pulses tolerate variations in RF homogeneity relevant for standard high-resolution NMR probes. Design criteria were transformation of Iz-->Ix over resonance offsets of +/-20kHz and RF variability of +/-5%, with a pulse length of 500 micros and peak RF amplitude equal to 17.5 kHz. Simulations transform Iz to greater than 0.995 Ix, with phase deviations of the final magnetization less than 2 degrees, over ranges of resonance offset and RF variability that exceed the design targets. Experimental performance of the pulse is in excellent agreement with the simulations. Performance tradeoffs for yet shorter pulses or pulses with decreased digitization are also investigated.

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