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).

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.

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.

Optimal control design of excitation pulses that accommodate relaxation.

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.

Analyses, extensions and comparison of three experimental schemes for measuring ((n)J(CH)+D(CH))-couplings at natural abundance.

Three types of experiments for measuring (n)J(CH) heteronuclear long-range coupling constants are examined and extended with state-of-the-art pulse sequence building-blocks: The use of a HMBC with corresponding reference-HSQC for accurate coupling determination is combined with the constant time technique and the conversion of antiphase magnetization into ZQ/DQ-coherences; CPMG-based LR-CAHSQC and BIRD(r,X)-HSQMBC experiments are examined in detail with respect to their coherence transfer properties; finally, the HSQC-TOCSY-IPAP experiment is introduced, a sequence derived from previously published alpha and beta selective HSQC-TOCSYs using a different spin state selection technique and a recently developed ZQ-suppression method. The experiments are characterized with their advantages and disadvantages and compared using strychnine and menthol as standard molecules.

Spin state selective Hadamard encoding during transfer periods using multiple selective CW-HCP.

Hadamard spectroscopy today represents an alternative to conventional Fourier transform spectroscopy. The selective inversion of several narrow frequency bands is typically achieved by taylored inversion pulses in place of t1-evolution periods. However, band-selective inversion can also be achieved during coherence transfer steps, thereby shortening the period during which the magnetization is in the transverse plane. Using CW heteronuclear cross polarization (CW-HCP) as an example for highly selective coherence transfer, the implementation of Hadamard encoding within a transfer step is presented. Transfer characteristics, the preparation of multiple frequency selective CW-HCP and the possibility of acquiring spin state selective spectra are discussed in detail. The theoretical results are verified on two examples involving a cyclic pentapeptide and ubiquitin.

Optimal control design of constant amplitude phase-modulated pulses: application to calibration-free broadband excitation.

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.

Construction of universal rotations from point-to-point transformations.

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.

Orientational properties of stretched polystyrene gels in organic solvents and the suppression of their residual 1H NMR signals.

While residual dipolar couplings (RDCs) are an established method in high-resolution biomolecular NMR, their use for structure determination of small molecules in organic solvents is limited by the alignment media available. Only recently stretched polystyrene (PS) gels were introduced for the measurement of RDCs on small compounds that allowed urgently needed free scalability of the induced anisotropy. Here, the properties of such stretched PS gels in different organic solvents as well as for different magnetic field strengths and temperatures are studied and practical NMR-spectroscopic aspects are discussed.

Pattern pulses: design of arbitrary excitation profiles as a function of pulse amplitude and offset.

A novel class of pulses is presented which can be regarded as a generalization of both frequency-selective pulses and B1-selective pulses. The excitation profile of these pulses forms a pre-defined pattern in two dimensions, which are spanned by pulse offset and radio-frequency (RF) amplitude. The presented pulses were designed numerically based on principles of optimal control theory. For simple test patterns, we demonstrate the flexibility of this approach by simulations and experiments. This previously unknown flexibility may trigger novel applications in NMR spectroscopy and imaging. As a first practical application, we demonstrate a direct approach for calibrating RF pulses.

Exploring the limits of broadband excitation and inversion pulses.

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.