Application of optimal control theory to the design of broadband excitation pulses for high-resolution NMR

Optimal control theory is considered as a methodology for pulse sequence design in NMR. It provides the flexibility for systematically imposing desirable constraints on spin system evolution and therefore has a wealth of applications. We have chosen an elementary example to illustrate the capabilities of the optimal control formalism: broadband, constant phase excitation which tolerates miscalibration of RF power and variations in RF homogeneity relevant for standard high-resolution probes. The chosen design criteria were transformation of I(z)-->I(x) over resonance offsets of +/- 20 kHz and RF variability of +/-5%, with a pulse length of 2 ms. Simulations of the resulting pulse transform I(z)-->0.995I(x) over the target ranges in resonance offset and RF variability. Acceptably uniform excitation is obtained over a much larger range of RF variability (approximately 45%) than the strict design limits. The pulse performs well in simulations that include homonuclear and heteronuclear J-couplings. Experimental spectra obtained from 100% 13C-labeled lysine show only minimal coupling effects, in excellent agreement with the simulations. By increasing pulse power and reducing pulse length, we demonstrate experimental excitation of 1H over +/-32 kHz, with phase variations in the spectra <8 degrees and peak amplitudes >93% of maximum. Further improvements in broadband excitation by optimized pulses (BEBOP) may be possible by applying more sophisticated implementations of the optimal control formalism.     

Novel methods for characterizing a decoupler channel using "undetectable" quantum coherences.

Exact solutions for the effect of time-independent RF pulses on any initial configuration of an IS J-coupled system demonstrate that on-resonance CW decoupling yields signals whose frequency depends on RF field strength and homogeneity. These signals are enhanced starting with "undetectable" antiphase and multiple quantum coherences, which can also produce centerband intensity to mimic the signal from decoupled Sx. Conversely, these coherences can be generated from Sx using a low-power pulse, B1 = J/2, of length (2J)-1, dubbed a "90J pulse" since it is the selective equivalent of {(2J)-1-90[I]}. Utilizing 90J pulses, new characterization-of-decoupler (COD) pulse sequences can determine the performance of an insensitive I-spin channel by observing large signals from either antiphase or multiple quantum coherences with the S-spin channel, allowing, in minutes rather than hours: (i) frequency calibration to an accuracy of 0.1 Hz; (ii) measurement of RF amplitudes over a 500-fold variation; and (iii) mapping of RF homogeneity along the sample axis with a single 1D B1 spectrum. These 90J coherence transfer pulses are of potential general use for selective spectroscopy.     

Tailoring the optimal control cost function to a desired output: application to minimizing phase errors in short broadband excitation pulses

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.     

Methylene spectral editing in solid-state 13C NMR by three-spin coherence selection

A robust new solid-state nuclear magnetic resonance (NMR) method for selecting CH2 signals in magic-angle spinning (MAS) 13C NMR spectra is presented. Heteronuclear dipolar evolution for a duration of 0.043 ms, under MREV-8 homonuclear proton decoupling, converts 13C magnetization of CH2 groups into two- and three-spin coherences. The CH2 selection in the SIJ (C H H) spin system is based on the three-spin coherence S(x)I(z)J(z), which is distinguished from 13C magnetization (S(x)) by a 1H 0 degrees/90 degrees pulse consisting of two 45 degrees pulses. The two-spin coherences of the type S(y)I(z) are removed by a 13C 90 degrees x-pulse. The three-spin coherence is reconverted into magnetization during the remainder of the rotation period, still under MREV-8 decoupling. The required elimination of 13C chemical-shift precession is achieved by a prefocusing 180 degrees pulse bracketed by two rotation periods. The selection of the desired three-spin coherence has an efficiency of 13% theoretically and of 8% experimentally relative to the standard CP/MAS spectrum. However, long-range couplings also produce some three-spin coherences of methine (CH) carbons. Therefore, the length of the 13C pulse flipping the two-spin coherences is increased by 12% to slightly invert the CH signals arising from two-spin coherences and thus cancel the signal from long-range three-spin coherences. The signal intensity in this cleaner spectrum is 6% relative to the regular CP/TOSS spectrum. The only residual signal is from methyl groups, which are suppressed at least sixfold relative to the CH2 peaks. The experiment is demonstrated on cholesteryl acetate and applied to two humic acids.     

Product Operator Theory for Spin‐5/2 Nuclei: Application to 2D J‐Resolved NMR Spectroscopy

Product operator theory is a simple quantum mechanical method that has often been used to analytically describe multi‐pulse NMR experiments for weakly coupled spin systems. Considering the existence of 2D‐J resolved NMR spectra of aqueous solutions containing S = 5/2 nuclear spins, the product operator formalism has been extended to the weakly coupled IS (I = 1/2, S = 5/2) spin system. The evolution of I_x, I_y, I_xS_z and I_yS_z product operators under spin–spin coupling Hamiltonian are given here. The analytical results obtained are applied to the well‐known gated decoupler pulse sequence for heteronuclear 2D‐J resolved NMR spectroscopy.     

Lifetimes of the singlet-states under coherent off-resonance irradiation in NMR spectroscopy

Singlet-states |S=(|alphabeta> - |betaalpha>)/sq.rt.2 can be excited in pairs of coupled spins I and S, first by preparing either a non-vanishing zero-quantum coherence I(+)S(-) or a state of longitudinal two-spin order I(z)S(z) and then by applying a coherent radio-frequency (RF) irradiation with a carrier frequency omega(rf) = (Omega(I) + Omega(S))/2 that lies half-way between the chemical shifts of the two spins involved. The life-times T(S) can be much longer than the spin-lattice relaxation time T(1) of longitudinal magnetization, but singlet-states are ultimately relaxed, not only by dipolar interactions between the active spins or with the external spins, but also as a result of a non-vanishing offset Deltaomega = omega(rf) - (Omega(I) + Omega(S))/2 or an insufficient amplitude of the RF irradiation that fails to fulfill the condition omega(1) > DeltaOmega = (Omega(I) - Omega(S)). In this work, the effect of off-resonance irradiation is explored and an approximate formula for the effective relaxation rate of the singlet population is provided on the basis of perturbation theory. The qualitative features of the dependence of the relaxation rate of the singlet population on the offset Deltaomega and on the difference DeltaOmega of the chemical shifts of the two spins are illustrated by comparison with numerical simulations.     

Population and coherence transfer induced by double frequency sweeps in half-integer quadrupolar spin systems

We have recently shown that the sensitivity of single- and multiple-quantum NMR experiments of half-integer (N/2) quadrupolar nuclei can be increased significantly by introducing so-called double frequency sweeps (DFS) in various pulse schemes. These sweeps consist of two sidebands generated by an amplitude modulation of the RF carrier. Using a time-dependent amplitude modulation the sidebands can be swept through a certain frequency range. Inspired by the work of Vega and Naor (J. Chem. Phys. 75, 75 (1981)), this is used to manipulate +/-(m - 1) <--> +/-m (3/2 < or = m < or = N/2) satellite transitions in half-integer spin systems simultaneously. For (23)Na (I = 3/2) and (27)Al (I = 5/2) spins in single crystals it proved possible to transfer the populations of the outer +/-m spin levels to the inner +/-1/2 spin levels. A detailed analysis shows that the efficiency of this process is a function of the adiabaticity with which the various spin transitions are passed during the sweep. In powders these sweep parameters have to be optimized to satisfy the appropriate conditions for a maximum of spins in the powder distribution. The effects of sweep rate, sweep range, and RF field strength are investigated both numerically and experimentally. Using a DFS as a preparation period leads to significantly enhanced central transition powder spectra under both static and MAS conditions, compared to single pulse excitation. DFSs prove to be very efficient tools not only for population transfer, but also for coherence transfer. This can be exploited for the multiple- to single-quantum transfer in MQMAS experiments. It is demonstrated, theoretically and experimentally, that DFSs are capable of transferring both quintuple-quantum and triple-quantum coherence into single-quantum coherence in I = 5/2 spin systems. This leads to a significant enhancement in signal-to-noise ratio and strongly reduces the RF power requirement compared to pulsed MQMAS experiments, thus extending their applicability. This is demonstrated by (27)Al 3QMAS experiments on 9Al(2)O(3). 2B(2)O(3) and the mineral andalusite. In the latter compound, Al experiences a quadrupolar-coupling constant of 15.3 MHz in one of the sites. Finally a 5QMAS spectrum on 9Al(2)O(3). 2B(2)O(3) demonstrates the sensitivity enhancement of this experiment using a double frequency sweep.     

The Observation and Dynamics of 1H NMR Spin Noise in Methanol

The observation of ¹H spin noise in relation to prior established magnetization and radiation damping has revealed a correlated dynamics. The spin noise of methyl satellites in ¹³C-enriched methanol was observed in the presence of an antiphase magnetization, created by the combination of ¹H–¹³C J coupling evolution and radiofrequency (RF) pulses. A gradient pulse was applied to remove residue spin coherence coming from the RF pulses, and as a result spin noise phenomena were uncovered. While magnetization was in an inverted metastable state, the spin–spin relaxation time was shortened to prevent a super radiation burst. The relation between magnetization, radiation damping, and absorption or emission of the spin noise of methyl satellites has been studied. In relation to magnetization and radiation damping, spin noise bump and dip have been observed simultaneously in the same molecule. Both can be created through a proper inversion of magnetization. The revealed spin noise dynamics of spin system coupling to the probe circuit via radiation damping allows performance of a transformation from dip into bump by proper application of pulses combined with ¹H–¹³C J coupling evolution.     

Heteronuclear Correlation Spectroscopy with RF Gradients

The applicability of RF gradients for suppressing the resonances of uncoupled spins in inverse-detected heteronuclear spectroscopy has been investigated. Pulse sequences were designed which incorporate RF gradients in heteronuclear single-quantum-correlation experiments (HSQC), and they can be divided into three categories based on how the RF gradients are used. In the first type of experiment, the desired coherences are spin locked in the RF-gradient held, while unwanted terms, placed perpendicular to the direction of the RF-gradient held, are dephased. In a second type of experiment, the dephasing action of the gradient and the coherence-transfer RF pulses are combined into a single RF-gradient pulse. A second RF-gradient pulse is then used to rephase the desired spin terms. The third type of experiment uses a period of longitudinal storage of the heteronuclear magnetization, during which time the magnetization of the uncoupled spins is destroyed by an RF-gradient pulse. Experimental results are shown from all three techniques, and the techniques are compared.     

Determination of quadrupole parameters with a composite pulse for spurious signal suppression

Spurious signals such as the piezoelectric signal from a ferroelectric crystal or the ringing signal from the NMR probe head tuned for low gyromagnetic ratio nuclei are often observed in pulsed NMR. Both signals are cancelled using the Hahn echo sequence with appropriate phase cyclings. The present paper applies a composite-pulse sequence to cancel the ringing signal. The main advantage of this sequence over the Hahn echo sequence is in the simplicity of optimizing the line intensity: the optimization of only one pulse duration for this sequence but of two pulse durations and the interpulse delay for the Hahn echo sequence. We are interested in half-integer quadrupole spins (I = 3/2, 5/2, 7/2, and 9/2), which means that we must consider the first-order quadrupole interaction during the pulses. For simplicity, we deal mainly with spin I = 3/2 nuclei. Since the central-line intensity depends on the ratio of the quadrupole coupling constant (QCC) to the amplitude of the RF pulse, we can determine the QCC from a featureless lineshape by fitting the variation of the experimental central-line intensity for increasing pulse duration with theoretical results. Contrary to the one-pulse sequence where the central-line intensity is proportional to the pulse duration if the latter is short, there is no such condition with the composite-pulse sequence. In other words, this sequence does not allow us to quantify the relative spin populations in powders. The size of the sample must be much smaller than that of the RF coil in order for the RF magnetic field to become homogeneous for the sample. We used (87)Rb (I = 3/2) in an aqueous solution of RbCl and in RbNb2O5F powder, (131)Xe (I = 3/2) of xenon gas physisorbed in Na-Y zeolite, and (23)Na (I = 3/2) in two well-known powders (NaNO3 and NaNO2) to support our theoretical result.     

自旋置换对称体系多脉冲及二维核磁共振实验的密度算符描述(Ⅱ)——多量子积算符方法

在对称化乘积算符(简称SAPO)方法基础上提出了多量子积算符(简称MQCPO)方法。改进的密度算符理论对I_nS(I=1/2,S=1/2;n为任意正整数)自旋体系多脉冲及二维核磁共振实验的描述普遍适用。MQCPO与SAPO从不同角度反映了自旋体系的对称性,故它们之间存在简单线性关系。文中给出I_n(I=1/2,n=2,3)自旋体系MQCPO的SAPO表示。MQCPO有利于自由演化过程的描述,而脉冲作用的描述则是SAPO为佳;利用MQCPO与SAPO的线性关系及SAPO笛卡儿分量的坐标轮换性质,“z”表象下脉冲作用的描述变得简单而直观。对异核谱剪辑及自旋拓扑滤波(spin topology filtration)等实验脉冲序列的分析,该方法是方便的。 关键词：     

14N核四极共振的偏共振效应

用虚拟１／２自旋算符讨论了核四极共振（ＮＱＲ）中自旋Ｉ＝１的自旋系统对激发脉冲宽度和频率偏置的响应。用单脉冲和双脉冲来观测的核四极共振信号与理论预期相符合。此外还证明，若只考虑射频场在分子电场梯度（ＥＰＧ）张量主轴坐标系（ＰＡＳ）中的一个轴上的分量（即有效射频场分量）的作用，就可用ＮＭＲ矢量模型来处理Ｉ＝１的核自旋系统。 关键词：     

Improved pulse schemes for separated local field spectroscopy for static and spinning samples

An improved pulse sequence for SLF experiments based on the magic sandwich (MS) scheme for homo-nuclear dipolar decoupling is proposed. The sequence incorporates a double MS, both on I and S spins and has been named as EXE-MS2. The proposed scheme which has a scaling factor of 1 is observed to be free from low intensity artifacts and provides better line-widths particularly for S spins labeled at multiple sites. The pulse sequence which has been applied on static oriented samples incorporates the EXE scheme where direct polarization of the S spin in the B(0) field is utilized in the place of polarization inversion and is observed to perform well without any loss of sensitivity while ensuring considerable reduction in rf power input into the sample. The EXE scheme has also been tested for solid samples under MAS.     

Observing in-phase single-quantum 15N multiplets for NH2/NH3+ groups with two-dimensional heteronuclear correlation spectroscopy

Two-dimensional (2D) F1-(1)H-coupled HSQC experiments provide 3:1:1:3 and 1:0:1 multiplets for AX(3) and AX(2) spin systems, respectively. These multiplets occur because, in addition to the 2S(y)H(z)(a)-->2S(y)H(z)(a) process, the coherence transfers such as 2S(y)H(z)(a)-->2S(y)H(z)(b) occurring in t(1) period provide detectable magnetization during the t(2) period. Here, we present a 2D F1-(1)H-coupled (1)H-(15)N heteronuclear correlation experiment that provides a 1:3:3:1 quartet for AX(3) spin system and a 1:2:1 triplet for AX(2). The experiment is a derivative of 2D HISQC experiment [J. Iwahara, Y.S. Jung, G.M. Clore, Heteronuclear NMR spectroscopy for lysine NH(3) groups in proteins: unique effect of water exchange on (15)N transverse relaxation. J. Am. Chem. Soc. 129 (2007) 2971-2980] and contains a scheme that kills anti-phase single-quantum terms generated in the t(1) period. The purge scheme is essential to observe in-phase single-quantum multiplets. Applications to the NH(2) and NH(3)(+) groups in proteins are demonstrated.     

Tailored sinc pulses for uniform excitation and artifact-free radio frequency time-domain EPR imaging

A method to generate shaped radiofrequency pulses for uniform excitation of electron spins in time-domain radio frequency (RF) electron paramagnetic resonance (EPR) imaging is presented. A commercial waveform generator was integrated with the transmit arm of the existing time-domain RF-EPR spectrometer to generate tailored excitation pulses with sub-nano second resolution for excitation with a 90 degrees flip-angle. A truncated sinc [sin(x)/x] pulse, tailored to compensate for the Q-profile (RF frequency response) of the resonator, was shown to yield images from phantom objects as well as in vivo images, with minimal distortion. These studies point to the advantages in using shaped sinc pulses to achieve improved uniform excitation over a relatively wide bandwidth region in time-domain RF-EPR imaging (RF-FT-EPRI).     

Relaxation dispersion in MRI induced by fictitious magnetic fields

A new method entitled Relaxation Along a Fictitious Field (RAFF) was recently introduced for investigating relaxations in rotating frames of rank ≥ 2. RAFF generates a fictitious field (E) by applying frequency-swept pulses with sine and cosine amplitude and frequency modulation operating in a sub-adiabatic regime. In the present work, MRI contrast is created by varying the orientation of E, i.e. the angle ε between E and the z″ axis of the second rotating frame. When ε > 45°, the amplitude of the fictitious field E generated during RAFF is significantly larger than the RF field amplitude used for transmitting the sine/cosine pulses. Relaxation during RAFF was investigated using an invariant-trajectory approach and the Bloch-McConnell formalism. Dipole-dipole interactions between identical (like) spins and anisochronous exchange (e.g., exchange between spins with different chemical shifts) in the fast exchange regime were considered. Experimental verifications were performed in vivo in human and mouse brain. Theoretical and experimental results demonstrated that changes in ε induced a dispersion of the relaxation rate constants. The fastest relaxation was achieved at ε ≈ 56°, where the averaged contributions from transverse components during the pulse are maximal and the contribution from longitudinal components are minimal. RAFF relaxation dispersion was compared with the relaxation dispersion achieved with off-resonance spin lock T(?ρ) experiments. As compared with the off-resonance spin lock T(?ρ) method, a slower rotating frame relaxation rate was observed with RAFF, which under certain experimental conditions is desirable.     

Slice profile optimization in arterial spin labeling using presaturation and optimized RF pulses

OBJECTIVE: An important source of error in arterial spin labeling (ASL) is incomplete static tissue subtraction due to imperfect slice profiles. This effect can be reduced by saturating the spins in the imaging area prior to labeling. In this study, the use of optimized presaturation is compared with the use of optimized RF pulses for minimizing this error. MATERIALS AND METHODS: Different methods for optimizing presaturation were simulated by numerical solution of the Bloch equation. Presaturation was optimized by changing the number of presaturation pulses, their position in the pulse sequence and the area of the crusher gradients following each saturation pulse. It was also investigated whether the use of optimized presaturation could reduce the tag gap needed to ensure complete static tissue subtraction. Simulation results were verified using phantom and in vivo studies at 3T. RESULTS: In proximal inversion with control for off-resonance effects, optimized presaturation could reduce the necessary tag gap to 15% of the imaging slab for experiments using standard RF pulses, while c-FOCI RF pulses could reduce it to 11%. In flow-sensitive alternating inversion recovery, a single presaturation pulse could reduce the inversion width to 122% of the imaging slab and neither multiple presaturation pulses nor optimized RF pulses could reduce it further. CONCLUSION: Optimized presaturation can reduce the necessary inversion width to the same level as if using optimized RF pulses and can, therefore, be used to optimize ASL sensitivity. Furthermore, optimized presaturation can reduce the B(1)-dependent sensitivity in static tissue subtraction.