On the performance and accuracy of 2D navigator pulses.

The purpose of this study was to investigate and to optimize the performance of two-dimensional spatially selective excitation pulses used for navigator applications on a clinical scanner. The influence of gradient imperfections, off-resonance effects, and incomplete k-space covering on the pencil beam-shaped spatial excitation profile of the 2D RF pulse was studied. The studies involved experiments performed on phantoms and in vivo. In addition, simulations were carried out by numerical integration of the Bloch equations. The accuracy of positioning of the pencil beam was increased by a factor of three by employing a simple correction scheme for the compensation of gradient distortions. The spatial selectivity of the 2D RF pulse was improved by taking sampling density corrections into account. The 2D RF pulse performance was found to be sufficient to monitor the diaphragm motion even at moderate gradient strength. For applications, where a high spatial resolution is required or a less characteristic contrast is present a strong gradient system is recommended.     

Improving excitation and inversion accuracy by optimized RF pulse using genetic algorithm

In this study, a Genetic Algorithm (GA) is introduced to optimize the multidimensional spatial selective RF pulse to reduce the passband and stopband errors of excitation profile while limiting the transition width. This method is also used to diminish the nonlinearity effect of the Bloch equation for large tip angle excitation pulse design. The RF pulse is first designed by the k-space method and then coded into float strings to form an initial population. GA operators are then applied to this population to perform evolution, which is an optimization process. In this process, an evaluation function defined as the sum of the reciprocal of passband and stopband errors is used to assess the fitness value of each individual, so as to find the best individual in current generation. It is possible to optimize the RF pulse after a number of iterations. Simulation results of the Bloch equation show that in a 90 degrees excitation pulse design, compared with the k-space method, a GA-optimized RF pulse can reduce the passband and stopband error by 12% and 3%, respectively, while maintaining the transition width within 2 cm (about 12% of the whole 32 cm FOV). In a 180 degrees inversion pulse design, the passband error can be reduced by 43%, while the transition is also kept at 2 cm in a whole 32 cm FOV.     

On the use of steady-state signal equations for 2D TrueFISP imaging

To explain the signal behavior in 2D-TrueFISP imaging, a slice excitation profile should be considered that describes a variation of effective flip angles and magnetization phases after excitation. These parameters can be incorporated into steady-state equations to predict the final signal within a pixel. The use of steady-state equations assumes that excitation occurs instantaneously, although in reality this is a nonlinear process. In addition, often the flip angle variation within the slice excitation profile is solely considered when using steady-state equations, while TrueFISP is especially known for its sensitivity to phase variations. The purpose of this study was therefore to evaluate the precision of steady-state equations in calculating signal intensities in 2D TrueFISP imaging. To that end, steady-state slice profiles and corresponding signal intensities were calculated as function of flip angle, RF phase advance and pulse shape. More complex Bloch simulations were considered as a gold standard, which described every excitation within the sequence until steady state was reached. They were used to analyze two different methods based on steady-state equations. In addition, measurements on phantoms were done with corresponding imaging parameters. Although the Bloch simulations described the steady-state slice profile formation better than methods based on steady-state equations, the latter performed well in predicting the steady-state signal resulting from it. In certain cases the phase variation within the slice excitation profile did not even have to be taken into account.     

Partial angle inversion recovery (PAIR) MR imaging: spin-echo and snapshot implementation.

The effects of varying the inversion or excitation RF pulse flip angles on image contrast and imaging time have been investigated in IR imaging theoretically, with phantoms and with normal volunteers. Signal intensity in an IR pulse sequence as a function of excitation, inversion and refocusing pulse flip angles was calculated from the solution to the Bloch equations and was utilized to determine the contrast behavior of a lesion/liver model. Theoretical and experimental results were consistent with each other. With the TI chosen to suppress the fat signal, optimization of the excitation pulse flip angle results in an increase in lesion/liver contrast or allows reduction in imaging time which, in turn, can be traded for an increased number of averages. This, in normal volunteers, improved spleen/liver contrast-to-noise ratio (9.0 vs. 5.7, n = 8, p less than 0.01) and suppressed respiratory ghosts by 33% (p less than 0.01). Reducing or increasing the inversion pulse from 180 degrees results in shorter TI needed to null the signal from the tissue of interest. Although this decreases the contrast-to-noise ratio, it can substantially increase the number of sections which can be imaged per given TR in conventional IR imaging or during breathold in the snapshot IR (turboFLASH) technique. Thus, the optimization of RF pulses is useful in obtaining faster IR images, increasing the contrast and/or increasing the number of imaging planes.     

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.     

Design of self-refocused pulses under short relaxation times

The effect of using self-refocused RF pulses of comparable duration to relaxation times is studied in detail using numerical simulation. Transverse magnetization decay caused by short T2 and longitudinal component distortion due to short T1 are consistent with other studies. In order to design new pulses to combat short T1 and T2 the relaxation terms are directly inserted into the Bloch equations. These equations are inverted by searching the RF solution space using simulated annealing global optimization technique. A new T2-decay efficient excitation pulse is created (SDETR: single delayed excursion T2 resistive) which is also energy efficient. Inversion pulses which improve the inverted magnetization profile and achieve better suppression of the remaining transverse magnetization are also created even when both T1 and T2 are short. This is achieved, however, on the expense of a more complex B1 shape of larger energy content.     

Transient Decay of Longitudinal Magnetization in Heterogeneous Spin Systems under Selective Saturation. IV. Reformulation of the Spin-Bath-Model Equations by the Redfield-Provotorov Theory

Alternative formulations to the conventional Bloch equations for the RF saturation of the solid component in heterogeneous spin systems according to a spin-bath model are derived using the concept of spin temperature as suggested by Redfield and Provotorov. These formulations and the resulting equations derived by the projection-operator technique provide an analytical and explicit solution to the general problem of solid saturation under continuous RF irradiation. Using the Provotorov theory, a set of generalized (non-Markovian) equations of motions is derived. The solutions to these generalized equations approach those of the conventional Bloch formulation at one extreme when the applied RF is weak and the Redfield formulation at another when the applied RF is strong. In short, this development provides a simple alternative which removes the restriction of the lineshape function used to represent the solid component; the latter is well known to be non-Lorentzian, contrary to the tacit assumption made in the conventional Bloch formulation. Experimental verification of the generalized theory is provided by transient and steady-state longitudinal magnetization data acquired from cross-linked bovine serum albumin under selective saturation by continuous off-resonance RF irradiation.     

Phase control of the excitation of a two-level system with short laser pulses

The possibility of the phase control for the excitation of a quantum system in the discrete spectrum under the action of a short high-intensity laser pulse with a controlled phase is theoretically analyzed. The analysis is performed using the example of the simplest two-level system (TLS) employing a numerical solution to equations for the optical Bloch vector. It is demonstrated that the excitation probability and the mean dipole moment of the quantum system, which determine the normal luminescence and the superradiance in the TLS ensemble, respectively, substantially depend on the phase of the laser pulse for relatively short and powerful laser pulses.     

Dynamics of 23Na during completely balanced steady-state free precession

The dynamics of 23Na during completely balanced steady-state free precession (SSFP) have been studied in numerical simulations and experiments. Results from both agree well. It is shown that during SSFP multiple quantum coherences are excited and that their excitation affects the observable signal. The signal response to the sequence parameters (flip angle, TR, and RF pulse phase cycle) shows a structure which can not be described by the Bloch equations. Due to excitation of T31 (s,a), the amplitude ratio of the fast and slowly decaying components deviates from 3:2 and is a function of the sequence parameters. The results shown here represent a basis for the implementation and optimization of 23Na-SSFP imaging sequences.     

Generalized equation for describing the magnetization in spoiled gradient-echo imaging

The purpose of this study was to demonstrate a generalized equation for describing the magnetization in spoiled gradient-echo (SPGR) imaging in which the in-pulse relaxation and magnetization transfer (MT) effects are taken into account. First, the time-dependent Bloch equations for the two-pool exchange model with MT effect were reduced to an inhomogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation. Second, the equations describing the magnetization before and after the radiofrequency (RF) pulse were derived based on the above solution for the RF-pulse excitation and evolution phases. Finally, a generalized equation describing the steady-state magnetization was derived. The validity of this equation was investigated by comparing with the transverse magnetization obtained by the regular Ernst equation and analytical solution in which the in-pulse transverse relaxation is considered. When the same assumption was made in our method, there were good agreements between them, indicating the validity of our method. The in-pulse transverse and longitudinal relaxations decreased the transverse magnetization compared to the case in which these effects were neglected, whereas MT increased it. In conclusion, we derived a generalized equation for describing the magnetization in SPGR imaging. This equation will provide a suitable basis for understanding the signal intensity in SPGR imaging and/or T₁ measurement using an SPGR sequence in cases in which the effect of in-pulse relaxation and/or MT cannot be neglected.     

Reducing SAR in parallel excitation using variable-density spirals: a simulation-based study

Parallel excitation using multiple transmit channels has emerged as an effective method to shorten multidimensional spatially selective radiofrequency (RF) pulses, which have a number of important applications, including B1 field inhomogeneity correction in high-field MRI. The specific absorption rate (SAR) is a primary concern in high-field MRI, where wavelength effects can lead to local peaks in SAR. In parallel excitation, the subjects are exposed to RF pulses from multiple coils, which makes the SAR problem more complex to analyze, yet potentially enables greater freedom in designing RF pulses with lower SAR. Parallel-excitation techniques typically employ either Cartesian or constant-density (CD) spiral trajectories. In this article, variable-density (VD) spiral trajectories are explored as a means for SAR reduction in parallel-excitation pulse design. Numerical simulations were conducted to study the effects of CD and VD spirals on parallel excitation. Specifically, the electromagnetic fields of a four-channel transmit head coil with a three-dimensional head model at 4.7 T were simulated using a finite-difference time domain method. The parallel RF pulses were designed and the resulting excitation patterns were generated using a Bloch simulator. The SAR distributions due to CD and VD spirals were evaluated quantitatively. The simulation results show that, for the same pulse duration, parallel excitation with VD spirals can achieve a lower SAR compared to CD spirals for parallel excitation. VD spirals also resulted in reduced artifact power in the excitation patterns. This gain came with slight, but noticeable, degrading of the spatial resolution of the resulting excitation patterns.     

Spectral Resolution and Inversion of the Bloch Equations with Relaxation

It is demonstrated that the linear Bloch equations, describing near-resonant excitation of two-level media with relaxation, can be resolved into a 3n-dimensional nonlinear system associated with a special spectral problem, generalizing the classical Zakharov–Shabat spectral problem. Remarkably, for n = 1 it is the well-known Lorenz system, and for n > 1 several such systems coupled with each other in a manner dependant on the excitation pulse. The unstable manifold of a saddle equilibrium point in this ensemble characterizes possible excitations of the spins from the initial equilibrium state. This enables us to get a straightforward geometric extension of the inverse scattering method to the damped Bloch equations and hence invert them, i.e., design frequency selective pulses automatically compensated for the effect of relaxation. The latter are essential, for example, in nuclear magnetic resonance and extreme nonlinear optics.      

特形脉冲的设计与计算机模拟

根据LiouvillevonNeumann方程从理论上对特形脉冲做了全面的描述,提出了一种具体的调幅特形脉冲设计方案:首先将待设计的脉冲展成一个有限Fourier级数,然后根据Bloch方程的解析解准确计算出各阶正弦、余弦波的频谱,再将这些频谱组合后与该脉冲的理想频谱进行比较构成误差函数,最后运用鲍威尔-模拟退火组合优化算法计算出全局最优Fourier系数,即可得到所需脉冲的表达式.应用此设计方案,得到了体系处于热平衡态时的特形激励脉冲和反转脉冲的具体表达式.计算机模拟表明,所得脉冲的频谱具有较好的选择性 关键词： 核磁共振 特形脉冲 Bloch方程 鲍威尔模拟退火组合优化算法     

An integrated program for amplitude-modulated RF pulse generation and re-mapping with shaped gradients

Efficient generation of amplitude modulated, frequency selective RF pulses has been demonstrated by the Shinnar-Le Roux (SLR) algorithm. In the present article, we provide an overview of a relatively comprehensive computer program that includes a version of the SLR algorithm and also incorporates an algorithm for re-mapping a selective RF pulse onto a new dwell time with modulated gradients. The re-mapping may be used to reduce SAR, or to shorten the RF pulse time by increasing the gradient and RF strength in regions where the original RF pulse amplitude was low. The program includes additional useful features including a Bloch equations algorithm, and pulse scaling, to enable examination of pulse profiles under a variety of conditions such as RF inhomogeneity and even nuclear relaxation. The program, MATPULSE, was developed with the MATLAB for Windows programming language and makes extensive use of the MATLAB graphical user interface (GUI) features to generate a userfriendly interface. A number of examples are provided to illustrate the capabilities of the MATPULSE program.     

A chemical-shift-insensitive slice-selective RF pulse (CHISS)

In NMR imaging and in vivo spectroscopy, slice selection is usually achieved by applying a frequency-selective RF pulse in the presence of a magnetic field gradient. A serious limitation of this method of slice selection is that, in a system with many different chemical shifts, the selected slice is offset in space for each chemically shifted resonance. In the present study, a composite RF pulse that is insensitive to chemical-shift differences has been developed. The pulse involves applying a RF pulse of desired shape in the presence of an alternating magnetic field gradient, together with hard 180° pulses at each gradient transition. Calculations are presented to show that excitation with the proposed pulse averages the chemical-shift term to zero. An exact calculation for a rectangular RF excitation shape verifies this. Experiments based on observing the RF excitation profiles have been performed to demonstrate the validity of the proposed pulse.     

Free induction decay with radiation damping II. Possible limitation of NMR thermometers at very low temperatures

Solution of Bloch equations with a term including the radiation damping for the pulse excitation of a two-level spin system shows, that the shape of the envelope of the free induction decay can be, under certain conditions, temperature dependent. This result is quantitatively exact for spin systems obeying these equations. Qualitatively, we show that this fact can give a limitation to the widespread use of NMR thermometers at very low temperatures (in the region of and below 1 [mK]).     

Excitation of a two-level system by a chirped laser pulse

The excitation of a two-level system by a chirped laser pulse is analyzed using an analytical approach (in the perturbation-theory limit) that involves the modified rotating-wave approximation and the numerical solution to the Bloch equation. The dependence of the population of the upper energy level on the chirp is studied for various radiation intensities and pulse durations. An exact solution is compared with the result obtained using the modified rotating-wave approximation and the perturbation theory. It is demonstrated that, for a certain range of parameters, the excitation probability of a two-level system can be effectively controlled via a variation in the chirp.     

Saturation in hadamard NMR spectroscopy and its description by a correlation expansion

In broadband NMR spectroscopy excitation with pseudorandom binary amplitude or phase modulation permits the distribution of the excitation power over the entire data acquisition time while peak power requirements are kept low. For sufficiently low excitation power, the magnetization is the linear response of the spin system to its input. The transfer function of the linearly driven system is recovered with the fast Hadamard transform. It is identical to the FID signal in FT NMR. Increasing excitation levels produce distorted lineshapes resulting from linear processing of a nonlinear spin response. Spectra measured for different degrees of saturation are reproduced faithfully by a numerical solution of the Bloch equations including relaxation during excitation. The origin of the lineshape distortions is discussed on the basis of an expansion of the nonlinear response in terms of the linear response. This expansion is in good agreement with the Bloch equations for limited excitation levels. Its nonlinear response terms are generalized to account for connectivities in coupled spin systems.     

Generalized RF Pulse Train with Insensitivity to B 1 Inhomogeneity

Adiabatic inversion recovery radiofrequency (RF) pulse techniques are used to address B ₁ inhomogeneity; however, the specific absorption rates of these techniques are significantly higher than that of non-adiabatic RF pulse techniques. In addition, time efficiency is poorer because of the required longer inversion recovery time. Therefore, an RF pulse train with three subpulses was previously developed and reported. The purpose of this article was to generalize the RF pulse train for tissues with different T ₁ relaxation times and in a different application. The RF pulse train B ₁ insensitivities and frequency responses were calculated with different T ₁ relaxation times and different subpulse durations using the Bloch equation. The previously reported optimal flip angle (FA) combination was used. When using the optimal FA combination, the RF pulse train B ₁ insensitivity did not change even if the T ₁ relaxation times and the subpulse durations did change. In other words, the optimal FA combination does not require adjustments according to the T ₁ and subpulse duration. The RF pulse train frequency responses with these subpulses can be dramatically improved even if the inherent subpulse frequency response is poor. This finding will facilitate RF pulse train technique implementation on magnetic resonance imaging scanners.     

T1rho-weighted MRI using a surface coil to transmit spin-lock pulses

T1rho-weighted MRI is a novel basis for generating tissue contrast. However, it suffers from sensitivity to B1 inhomogeneity. First, excitation with a spatially varying B1 causes flip-angle artifacts and second, spin locking with an inhomogeneous B1 results in non-uniform T1rho contrast. In this study, we overcome the former complication with a specially designed spin-locking pulse sequence and we successfully obtain T1rho-weighted images with a surface coil. In this pulse sequence, the spin-lock pulse was divided into segments of equal duration and alternating phase. This "self-compensating" T1rho-preparatory pulse sequence was analyzed and the effect of an inhomogeneous B1 field was simulated using the Bloch equations. T1rho-weighted MR images of a phantom and a human knee joint in vivo were obtained on a clinical scanner with a surface coil to demonstrate the utility of the pulse sequence. The self-compensating T1rho-prepared pulses sequence resulted in substantially reduced image artifacts compared to the conventional, single-phase spin-lock pulse.