Heteronuclear decoupling by optimal tracking

An improved algorithm for design of broadband excitation, inversion, and mixing pulse sequences by iterative optimization of phases: TOPS-2

This paper presents an improved iterative algorithm (TOPS-2) for the design of broadband inversion, excitation and coherent transfer mixing sequence (TOCSY) pulses. The evolution of the Bloch vector is presented as a sequence of small constant flip angle pulses with varying phases and constant amplitude. This paper describes an improved algorithm for iterative optimization of piece-wise constant phases as we incorporate the quadratic terms in the propagators. In our iterative optimization we obtain a closed-form expression for each phase, and these phases are optimized sequentially using the new improved algorithm. This paper compares the simulation results of the TOPS vs TOPS-2 and shows that TOPS-2 perform better. Experimental validation of excitation and inversion TOPS-2 pulse sequence is performed with .5% H2O in 99.5% D2O, and experimental validation of TOPS-2 mixing (TOCSY) pulse sequence is done with 0.1% of Ethylbenzene (EB) in CDCl3 solvent.

Flexible and efficient optimization of quantitative sequences using automatic differentiation of Bloch simulations

PURPOSE

To investigate a computationally efficient method for optimizing the Cramér-Rao Lower Bound (CRLB) of quantitative sequences without using approximations or an analytical expression of the signal.

METHODS

Automatic differentiation was applied to Bloch simulations and used to optimize several quantitative sequences without the need for approximations or an analytical expression. The results were validated with in vivo measurements and comparisons to prior art. Multi-echo spin echo and DESPO T 1 were used as benchmarks to verify the CRLB implementation. The CRLB of the Magnetic Resonance Fingerprinting (MRF) sequence, which has a complicated analytical formulation, was also optimized using automatic differentiation.

RESULTS

The sequence parameters obtained for multi-echo spin echo and DESPO T 1 matched results obtained using conventional methods. In vivo, MRF scans demonstrate that the CRLB optimization obtained with automatic differentiation can improve performance in presence of white noise. For MRF, the CRLB optimization converges in 1.1 CPU hours for N TR = 400 and has O ( N TR ) asymptotic runtime scaling for the calculation of the CRLB objective and gradient.

CONCLUSIONS

Automatic differentiation can be used to optimize the CRLB of quantitative sequences without using approximations or analytical expressions. For MRF, the runtime is computationally efficient and can be used to investigate confounding factors as well as MRF sequences with a greater number of repetitions.

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

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

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.

An optimal control approach to design entire relaxation dispersion experiments

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

Reaction monitoring using hyperpolarized NMR with scaling of heteronuclear couplings by optimal tracking

Off-resonance decoupling using the method of Scaling of Heteronuclear Couplings by Optimal Tracking (SHOT) enables determination of heteronuclear correlations of chemical shifts in single scan NMR spectra. Through modulation of J-coupling evolution by shaped radio frequency pulses, off resonance decoupling using SHOT pulses causes a user-defined dependence of the observed J-splitting, such as the splitting of ¹³C peaks, on the chemical shift offset of coupled nuclei, such as ¹H. Because a decoupling experiment requires only a single scan, this method is suitable for characterizing on-going chemical reactions using hyperpolarization by dissolution dynamic nuclear polarization (D-DNP). We demonstrate the calculation of [¹³C, ¹H] chemical shift correlations of the carbanionic active sites from hyperpolarized styrene polymerized using sodium naphthalene as an initiator. While off resonance decoupling by SHOT pulses does not enhance the resolution in the same way as a 2D NMR spectrum would, the ability to obtain the correlations in single scans makes this method ideal for determination of chemical shifts in on-going reactions on the second time scale. In addition, we present a novel SHOT pulse that allows to scale J-splittings 50% larger than the respective J-coupling constant. This feature can be used to enhance the resolution of the indirectly detected chemical shift and reduce peak overlap, as demonstrated in a model reaction between p-anisaldehyde and isobutylamine. For both pulses, the accuracy is evaluated under changing signal-to-noise ratios (SNR) of the peaks from reactants and reaction products, with an overall standard deviation of chemical shift differences compared to reference spectra of 0.02ppm when measured on a 400MHz NMR spectrometer. Notably, the appearance of decoupling side-bands, which scale with peak intensity, appears to be of secondary importance.

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.

Visualization and analysis of modulated pulses in magnetic resonance by joint time-frequency representations

We study the utility of joint time-frequency representations for the analysis of shaped or composite pulses for magnetic resonance. Such spectrograms are commonly used for the visualization of shaped laser pulses in optical spectroscopy. This intuitive representation provides additional insight compared to conventional approaches, which exclusively show either temporal or spectral information. We focus on the short-time Fourier transform, which provides not only amplitude but also phase information. The approach is illustrated for broadband inversion pulses, multiple quantum excitation and broadband heteronuclear decoupling. The physical interpretation and validity of the approach is discussed.

Further perspective on the theory of heteronuclear decoupling

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.

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.

Chemical shift correlations from hyperpolarized NMR using a single SHOT

A significant challenge in realizing the promise of the dissolution dynamic nuclear polarization technique for signal enhancement in high-resolution NMR lies in the nonrenewability of the hyperpolarized spin state. This property prevents the application of traditional two-dimensional correlation spectroscopy, which relies on regeneration of spin polarization before each successive increment of the indirect dimension. Since correlation spectroscopy is one of the most important approaches for the identification and structural characterization of molecules by NMR, it is important to find easily applicable methods that circumvent this problem. Here, we introduce the application of scaling of heteronuclear couplings by optimal tracking (SHOT) to achieve this goal. SHOT decoupling pulses have been numerically optimized on the basis of optimal control algorithms to obtain chemical shift correlations in C-H groups, either by acquiring a single one-dimensional (13)C spectrum with (1)H off-resonance decoupling or vice versa. Vanillin, which contains a number of functional groups, was used as a test molecule, allowing the demonstration of SHOT decoupling tailored toward simplified and accurate data analysis. This strategy was demonstrated for two cases: First, a linear response to chemical shift offset in the correlated dimension was optimized. Second, a pulse with alternating linear responses in the correlated dimension was chosen as a goal to increase the sensitivity of the decoupling response to the chemical shift offset. In these measurements, error ranges of ±0.03 ppm for the indirectly determined (1)H chemical shifts and of ±0.4 ppm for the indirectly determined (13)C chemical shifts were found. In all cases, we show that chemical shift correlations can be obtained from information contained in a single scan, which maximizes the ratio of signal to stochastic noise. Furthermore, a comprehensive discussion of the robustness of the method toward nonideal conditions is included based on experimental and simulated data. Unique features of this technique include the abilities to control the accuracy of chemical shift determination in spectral regions of interest and to acquire such chemical shift correlations rapidly-the latter being of interest for potential application in real-time spectroscopy.

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.

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.

Heteronuclear decoupling by multiple rotating frame technique

The paper describes the multiple rotating frame technique for designing modulated rf fields, that perform broadband heteronuclear decoupling in solution NMR spectroscopy. The decoupling method presented here is understood by performing a sequence of coordinate transformations, each of which demodulates a component of the rf field to a static component, that progressively averages the chemical shift and the dipolar interaction. We show that by increasing the number of modulations in the decoupling field, the ratio of dispersion in the chemical shift to the strength of the static component of the rf field is successively reduced in the progressive frames. The known decoupling methods like continuous wave decoupling, TPPM, etc., can be viewed as special cases of this method and their performance improves by adding additional modulations in the decoupling field. The technique is also expected to find use in design of broadband excitation, inversion and mixing sequences and broadband experiments in solid state NMR.