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.     

Reducing the duration of broadband excitation pulses using optimal control with limited RF amplitude

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.     

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.     

组合脉冲宽带激发2H-平均哈密顿理论计算研究

四级核回波实验通常需要射频脉冲能够激发谱宽超过100 kHz 的信号．在最近的研究中,作者发现组合脉冲COM-II (9018090135 45) 能够在氘核的四级核回波实验中实现宽带激发．此外,作者还结合了八步相位循环的方法,有效消除了由有限脉宽效应造成的谱图扭曲现象．利用了平均哈密顿原理,对该方法进行了理论计算研究．作者采用了自旋为1 的矩阵算符,通过计算解释了八步相位循环能够消除谱图扭曲的原因．     

J pulses for multiplet-selective NMR

Exact product operator solutions have been obtained for the evolution of weakly coupled spin-(1/2) I(m)S(n) systems during arbitrary RF irradiation of one spin. These solutions, which completely characterize the nature of J-coupling modulation during RF pulses, show that significant exchange occurs between single-spin magnetization and two-spin product operator states when the RF field strength is comparable to the coupling. In particular, a long (t(p) = [2J](-1) s), low-power (B(1) = J/2 Hz), constant amplitude pulse applied on resonance to one spin in an IS system completely interconverts the spinstates S(z) <--> 2S(x)I(z) and S(x) <--> 2S(z)I(z) when the RF is applied to the S spins, and interconverts S(x) <--> 2S(y)I(y) in 100% yield when the RF is applied to the I spins. Thus, these "J pulses," which select a bandwidth approximately equal to J Hz, may replace any combination of a (2J)(-1) delay period and a consecutive hard 90 degrees pulse in any polarization transfer or multiple quantum sequence. Although these rectangular pulses are highly frequency selective, in general they increase the replaced (2J)(-1) period by only a modest 40%, a time saving of a factor of 5 compared to existing pulses exhibiting the same selectivity. In favorable cases, there is no increase in duration of a pulse sequence using a particular type of J pulse, the 90(J) variety, which accomplishes the third spin state transformation listed above. J pulses will be advantageous for systems subject to rapid signal loss from relaxation and more generally for the enhanced operation of pulse sequences via the use of J modulation during RF irradiation.     

Application of optimal control to CPMG refocusing pulse design

We apply optimal control theory (OCT) to the design of refocusing pulses suitable for the CPMG sequence that are robust over a wide range of B(0) and B(1) offsets. We also introduce a model, based on recent progress in the analysis of unitary dynamics in the field of quantum information processing (QIP), that describes the multiple refocusing dynamics of the CPMG sequence as a dephasing Pauli channel. This model provides a compact characterization of the consequences and severity of residual pulse errors. We illustrate the methods by considering a specific example of designing and analyzing broadband OCT refocusing pulses of length 10t(180) that are constrained by the maximum instantaneous pulse power. We show that with this refocusing pulse, the CPMG sequence can refocus over 98% of magnetization for resonance offsets up to 3.2 times the maximum RF amplitude, even in the presence of ±10% RF inhomogeneity.     

相位可控的核四极矩共振激励脉冲发生器设计

在核四极矩共振（NQR）领域,射频激励脉冲信号的优劣对NQR响应信号有重要影响.针对常规方法中射频激励脉冲参数不可控的问题,本文基于32位闪存微型控制器STM32和直接数字频率合成（DDS）芯片AD9910设计了一种相位可控激励脉冲发生器.采用STM32控制AD9910产生波形参数（脉冲宽度、脉冲间隔、脉冲个数和共振频率等）可控的射频激励脉冲,利用LabVIEW软件平台设计脉冲参数设置界面,并建立计算机与微控制器通信,实现波形参数的精确优化控制.实验结果表明,该方法实现了相位可控的NQR激励脉冲序列,可为后续NQR信号检测提供有效激励源.     

Selective excitation of single lines of a multiplet for interpretation of spectra of complex multicomponent systems

The article discusses the most important aspects of the selective RF excitation of multiplet single lines, an approach that makes it possible to simplify the spectra of complex multicomponent systems, to measure the spin–spin coupling constant, and to determine molecular structure. Semi-selective and selective methods of recording spectra are considered and theoretically substantiated based on the product operator formalism. The results are verified using quantum-mechanical calculations within the framework of the density matrix formalism. The paper demonstrates that the proposed new method leads to a significant simplification of two-dimensional nuclear magnetic resonance spectra. This makes it possible to study chemical processes and spin state exchange by observing the dynamic behavior of the multiplet structure of bound nuclei.     

Maximizing MR signal for 2D UTE slice selection in the presence of rapid transverse relaxation

Ultrashort TE (UTE) sequences allow direct visualization of tissues with very short T2 relaxation times, such as tendons, ligaments, menisci, and cortical bone. In this work, theoretical calculations, simulations, and phantom studies, as well as in vivo imaging were performed to maximize signal-to-noise ratio (SNR) for slice selective RF excitation for 2D UTE sequences. The theoretical calculations and simulations were based on the Bloch equations, which lead to analytic expressions for the optimal RF pulse duration and amplitude to maximize magnetic resonance signal in the presence of rapid transverse relaxation. In steady state, it was found that the maximum signal amplitude was not obtained at the classical Ernst angle, but at an either lower or higher flip angle, depending on whether the RF pulse duration or amplitude was varied, respectively.     

Double-quantum excitation in the NMR of spinning solids by pulse-assisted rotational resonance

We describe a new technique for double-quantum excitation in magic-angle-spinning NMR of powdered solids. The technique is designed to efficiently excite double-quantum coherence in the vicinity of a rotational resonance condition. The offset from rotational resonance allows the double-quantum filtered signals to be observed with high resolution and sensitivity. The method uses rotational excitation of zero-quantum coherence, assisted by radiofrequency pulse cycles. The zero-quantum coherence is converted into double-quantum coherence by a frequency-selective inversion sequence. Experiments on [(13)C(2), (15)N]-glycine demonstrate a double-quantum filtering efficiency of approximately 41% at a sample rotation frequency of 8.300 kHz, which is 1.600 kHz away from the n = 1 rotational resonance. We achieve 32% double-quantum filtering efficiency at a spinning frequency of 9.250 kHz, which is 2.550 kHz away from rotational resonance.     

Fast fat-suppressed reduced field-of-view temperature mapping using 2DRF excitation pulses

The purpose of this study is to develop a fast and accurate temperature mapping method capable of both fat suppression and reduced field-of-view (rFOV) imaging, using a two-dimensional spatially-selective RF (2DRF) pulse. Temperature measurement errors caused by fat signals were assessed, through simulations. An 11×1140μs echo-planar 2DRF pulse was developed and incorporated into a gradient-echo sequence. Temperature measurements were obtained during focused ultrasound (FUS) heating of a fat-water phantom. Experiments both with and without the use of a 2DRF pulse were performed at 3T, and the accuracy of the resulting temperature measurements were compared over a range of TE values. Significant inconsistencies in terms of measured temperature values were observed when using a regular slice-selective RF excitation pulse. In contrast, the proposed 2DRF excitation pulse suppressed fat signals by more than 90%, allowing good temperature consistency regardless of TE settings. Temporal resolution was also improved, from 12 frames per minute (fpm) with the regular pulse to 28 frames per minute with the rFOV excitation. This technique appears promising toward the MR monitoring of temperature in moving adipose organs, during thermal therapies.     

Improved Broadband Inversion Performance for NMR in Liquids

New NMR broadband inversion pulses that compensate both for resonance offset and radiofrequency (RF) inhomogeneity are described. The approach described is a straightforward computer optimization of an initial digitized waveform generated from either a constant-amplitude frequency sweep or from an existing composite inversion pulse. Problems with convergence to local minima are alleviated by the way the optimization is carried out. For a given duration and maximum allowable RF field strength B₁ (but not necessarily given RMS power deposition), the resultant broadband inversion pulse (BIP) shows superior inversion compared to inversion pulses obtained from previous methods, including adiabatic inversion pulses. Any existing BIP can be systematically elaborated to build up longer inversion pulses that perform over larger and larger bandwidths. The resulting pulse need not be adiabatic throughout its duration or across the entire operational bandwidth.     

Fast amplitude-modulated pulse trains with frequency sweep (SW-FAM) in solid-state NMR of spin-7/2 nuclei

We here investigate the sensitivity enhancement of central-transition NMR spectra of quadrupolar nuclei with spin-7/2 in the solid state, generated by fast amplitude-modulated RF pulse trains with constant (FAM-I) and incremented pulse durations (SW-FAM). Considerable intensity is gained for the central-transition resonance of single-quantum spectra by means of spin population transfer from the satellite transitions, both under static and magic-angle-spinning (MAS) conditions. It is also shown that incorporation of a SW-FAM train into the excitation part of a 7QMAS sequence improves the efficiency of 7Q coherence generation, resulting in improved signal-to-noise ratio. The application of FAM-type pulse trains may thus facilitate faster spectra acquisition of spin-7/2 systems.     

磁共振现代射频脉冲理论在非均匀场成像中的应用

在磁共振非均匀场成像中，传统的射频脉冲导致回波信号的衰减.为了减小和消除这种磁共振信号的衰减，在讨论了经典理论的基础上根据非线性动力学中的逆散射理论和Shinnar-Le Roux方法导出了用于非均匀场成像的射频脉冲设计方法.模拟结果表明，采用逆散射理论和Shinnar-Le Roux方法优化的脉冲序列可以明显提高信号的信噪比. 关键词： 磁共振成像 射频脉冲 非线性系统     

Broadband Excitation by Method of Double Sweep

The paper describes the design of broadband excitation pulses in high resolution nuclear magnetic resonance (NMR) by method of double sweep. We first show the design of a pulse sequence that produces broadband excitation to the equator of Bloch sphere with phase linearly dispersed as frequency. We show how this linear dispersion can then be refocused by nesting free evolution between two adiabatic inversions (sweeps). We then show how this construction can be generalized to exciting arbitrary large bandwidths without increasing the peak rf-amplitude and by incorporating more adiabatic sweeps. Finally, we show how the basic design can then be modified to give a broadband x rotation over arbitrary large bandwidth and with limited rf-amplitude. Experimental excitation profiles for the residual HDO signal in a sample of \(99.5\%\) D\(_2\)0 are displayed as a function of resonance offset. Application of the excitation is shown for \(^{13}\)C excitation in a labelled sample of alanine.     

Adjustable, broadband, selective excitation with uniform phase

An advance in the problem of achieving broadband, selective, and uniform-phase excitation in NMR spectroscopy of liquids is outlined. Broadband means that, neglecting relaxation, any frequency bandwidth may be excited even when the available radiofrequency (RF) field strength is strictly limited. Selective means that sharp transition edges can be created between pure-phase excitation and no excitation at all. Uniform phase means that, neglecting spin-spin coupling, all resonance lines have nearly the same phase. Conventional uniform-phase excitation pulses (e.g., E-BURP), mostly based on amplitude modulation of the RF field, are not broadband: they have an achievable bandwidth that is strictly limited by the peak power available. Other compensated pulses based on adiabatic half-passage, like BIR-4, are not selective. By contrast, inversion pulses based on adiabatic fast passage can be broadband (and selective) in the sense above. The advance outlined is a way to reformulate these frequency modulated (FM) pulses for excitation, rather than just inversion.     

用于同时多层MRI的选择性射频脉冲的优化设计

近年来,为提高磁共振成像（MRI）信号信噪比（SNR）、缩短成像时间,同时多层成像技术受到了极大的关注.为了实现同时多层的选择性激发,现有的多层成像序列大多使用组合射频（RF）脉冲,该脉冲可包含多个独立的幅值相同相位不同的简单脉冲,由于其采用简单的线性叠加方法,该类脉冲射频功率随脉冲数量呈现平方增长,因而应用受限.针对这一问题,基于自旋动力学和优化控制原理,本文提出了一种针对同时多层MRI的选择性射频脉冲的数值优化方法,该方法充分运用射频脉冲的调控机制,获得优化脉冲,并配合层选梯度,可实现任意层厚、层间距、层数的同时高效选择性激发.最后,通过数字模体的同时多层模拟成像实验验证了优化脉冲的有效性.     

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).     

Optimizing delays in the MBOB,broadband HMBC,and broadband XLOC NMR pulse sequences

The MBOB, broadband HMBC, and broadband XLOC NMR pulse sequences (A. Meissner and O. W. S?rensen (2000, Magn. Reson. Chem. 38, 981-984; 2001, 39, 49-52)) were introduced as a means of obtaining heteronuclear long-range correlation spectra with broadband excitation over an interval of heteronuclear long-range J coupling constants. However, it is not trivial what combination of delays to choose for a given purpose, particularly if one-bond and long-range correlation spectra are obtained simultaneously as in MBOB. This paper presents a way to determine sets of delays for MBOB, broadband HMBC, and broadband XLOC resolving the problem. The results tabulated suit various ranges of J coupling constants and transverse relaxation times.