Strongly modulating pulses for counteracting RF inhomogeneity at high fields

Authors


Abstract

A new pulse technique for counteracting RF inhomogeneity at high fields is reported. The pulses make use of the detailed knowledge of the voxels' B1 and B0 amplitude 2D histogram to generate, through an optimization procedure, gates where the flip angle is made uniform. Although most approaches to date require the use of parallel transmission, this method does not and therefore offers several advantages. The data necessary for the algorithm to determine an irradiation scheme requires only one transmit B1 along with a B0 inhomogeneity measurement. The use of a B1 and B0 amplitude 2D histogram instead of their spatial distribution also decreases substantially the complexity of the optimization problem, allowing the algorithm to find an RF solution in less than 30 s. Finally, the optimization procedure is based on an exact calculation and does not use any linear approximation. In this article, the theory behind the method in addition to spoiled gradient echo experimental data at 3T for 3D brain imaging are reported. The images obtained yield a reduction of the standard deviation of the sine of the flip angle by a factor of up to 15 around the desired value, compared to when a standard square pulse calibrated by the scanner is used. Magn Reson Med 60:701–708, 2008. © 2008 Wiley-Liss, Inc.

INTRODUCTION

Many applications in control of quantum systems involve the manipulation of a large ensemble by using the same control field. Often in practice, different members of the ensemble see variations in the experimental parameters that govern the dynamics. Magnetic resonance imaging (MRI) at high fields is an important case where the variation of an experimental parameter over the ensemble, mainly the radio-frequency (RF) field, has drastic consequences and raises challenging control problems. The irradiation of objects comparable or larger than the RF wavelength leads to destructive interferences and dielectric resonance effects which translate into RF inhomogeneity and thus dark zones on MRI images of human body parts such as the brain or pelvis(1, 2). Standard compensating pulse techniques such as adiabatic and composite pulses in general require too much energy to be implemented on a human subject(3). A simple composite pulse approach with a single coil(4) may however be implemented with a reasonable amount of energy, but still leaves room for some substantial improvement, and its performance quickly decreases as the spin resonance frequency deviates from the carrier frequency. Other B1 compensated 3D tailored RF pulses have also been reported in(5, 6), but rely on the small flip angle approximation or/and assume some particular B1 profile. Parallel transmission, namely Transmit SENSE(7–10) and RF shimming(3, 11), is another powerful method proposed for counteracting RF inhomogeneity at high fields. Although its feasibility has been verified experimentally, and is still the subject of stimulating and ongoing research, the method is costly, bulky, and time-consuming. It is costly mainly due to the fact that it requires (the purchase of) sophisticated hardware. It is bulky because it requires the designer to handle very large matrices, thus yielding long computation times. The time-consuming part, besides the computing part in the design process just mentioned above, arises from the N-fold transmit 3D B1 maps measurements (one for each channel) required for the algorithm to compute a solution, making an exam study even more lengthy for a patient. Finally, the nonlinearity of the Bloch equation tends to have an adverse effect on the performance of Transmit SENSE at large flip angles, despite recent improvements(12–15). As far as RF shimming is concerned, constraints of Maxwell's equations impose inevitable limits in performance when optimizing the RF field over a significant volume such as a human brain(3, 16) (but see(17) for an interesting combination of composite pulses with RF shimming).

In this article, we present an alternative method for uniformly exciting a volume of interest. The method is inspired from the strongly modulating pulses developed originally for nuclear magnetic resonance (NMR) quantum computing(18). Strongly modulating pulses initially fitted in the realm of “gate engineering,” where one is interested in finding an irradiation scheme that implements a target transformation with high fidelity no matter what the initial state of the system is. Soon after their creation, the design of these pulses was extended to account for RF inhomogeneity and their performance was measured and verified experimentally on numerous occasions(18, 19). As far as the external static B0 field inhomogeneity is concerned, it was simply too small to be a problem and the pulses turned out to be naturally robust with respect to small resonance frequency variations. This is not the case in magnetic resonance brain imaging because of the larger volumes involved and the varying magnetic susceptibilities, yielding a significant Larmor frequency dispersion that increases linearly with the external magnetic field. Although B1 and B0 inhomogeneities encountered in high-field MRI are much more severe than in liquid state NMR quantum computing, MRI often involves “state engineering” of a single spin (as opposed to gate engineering of coupled multispin systems). To achieve uniform excitation, indeed one must steer the equilibrium longitudinal state to a final one with a desired flip angle. As a result, the control problem is less demanding than in NMR quantum computing and hence relaxes some constraints on the optimization procedure, making the technique worth investigating for MRI needs.

However, the method reported in(18) is not directly applicable and needs important modifications. The first one is the cost function to minimize, where the gate engineering problem should be converted into the state engineering one, to achieve better performance. The goal here is to achieve a desired flip angle with minimum deviation with respect to the known B1 and B0 amplitudes. Hence in this work, the initial state is always assumed to be a z-magnetization state, limiting the demonstration to spoiled gradient echo sequences. Second, penalty functions are also adjusted to account for the different experimental limitations (time, power, frequency), and to minimize the specific absorption rate (SAR). Third, the search algorithm for the minimization procedure is now more sophisticated and combines a genetic algorithm (GA)(20) with the Nelder-Mead(21) direct search method. Although the work in(18, 19) uses the latter algorithm only, the use of a genetic algorithm increases the probability to find a global optimum, or at least generates a good start for the Nelder-Mead algorithm, thereby making the search faster. With the B1 and B0 inhomogeneity 2D histogram known, RF pulses whose durations are comparable or shorter than the transmit SENSE pulses(7–9) are returned in less than 30 s with a standard PC.

In the following section, we formulate the theory used to design the strongly modulating pulses. We use the exact Schrödinger, or Bloch, equation and account for B1 and B0 inhomogeneity by looping over their 2D histogram values. No magnetic field gradients during RF pulsing are assumed, making this method at the moment applicable only to nonselective 3D imaging. We proceed with some experimental results on two different human brains at 3 T. We then compare the strongly modulating pulses with adiabatic BIR4 pulses(22–24), also known to be robust against B1 and B0 inhomogeneity, in terms of performance and SAR. Finally, we discuss future work in prospect.

THEORY

In this section, we present the theory used to design the strongly modulating pulses. We use a spinor notation because of its compactness and to be consistent with(18). A Bloch formalism would work equally well. The goal is to find an RF pulse that rotates a magnetization initially along the z-axis with some specified tilt angle, given a B1 and B0 amplitude distribution determined via a preliminary measurement. Various strategies in NMR spectroscopy have been used for that purpose, including adiabatic and composite pulses(25–29). However in practice, the implementation of these techniques in high-field MRI would require unreasonably long pulse durations or large amounts of power(3). What the two techniques have in common though is the fact that the phases of the pulses are not constant over time. Noncommutativity is actually the key property that makes the design of compensating pulse sequences possible(30). Otherwise, at resonance, and for a constant phase in the rotating reference frame, the flip angle (FA) is always such that FA ∝ ∫math imageB1(t) dt, where τ is the duration of the pulse, so that a deviation in B1 in general necessarily yields a deviation of the flip angle. This result is no longer true if one is varying the phase of the pulses, or using off-resonance pulses. The parametrization of the strongly modulating pulses exploits both this result and the fact that for a phase varying linearly in time, analytical formulas are available.

The Hamiltonian of a spin 1/2 system under constant RF amplitude is given in the frame rotating at the carrier frequency by (we have set equation image for convenience)(31):

equation image

where equation image is the offset angular frequency which is spatially dependent because of B0 inhomogeneity, equation image is the spatially dependent nutation angular frequency, γ is the gyromagnetic ratio (in rad/T), σx,y,z are the Pauli spin matrices, and ϕ(t) is the time-dependent phase of the RF field. We will drop from now on the spatial dependence of Δω0 and ω1 for notational convenience. If ϕ(t) = ϕ0 + ωt, where ω and t are the RF offset frequency with respect to the carrier frequency and the time respectively, then jumping into the new frame rotating at ω, integrating Schrödinger equation and jumping back into the original frame yields for the propagator (the rotation matrix in Bloch's formalism):

equation image

where τ is the duration of the pulse. The evolution of a spin under this kind of phase modulated pulses therefore can be calculated in two steps. After applying equation image on the initial wavefunction, the calculation of the FA follows. If the phase ϕ(t) was not a linear function of time, then solving for the evolution in general would require a discretization in time and therefore more calculation steps. Although it was noticed earlier that in this case, the FA is not proportional to the integral of B1(t) with time, the parametrization so far does not allow to achieve a uniform FA when there are significant RF power variations. The design of the strongly modulating pulses then consists in applying a cascade of several evolution periods (like with composite pulses) as above, with different sets of four parameters. Likewise, the parameters for each period are the initial phase ϕk, the offset angular frequency ωk, the amplitude B1,k, and the duration τk. Note that there is in fact a position-dependent initial phase for the circularly polarized transmitted B1 field. But because we focus here on magnitude images, we ignore this dependence. For a series of N square pulses with respective parameter sets, the net propagator becomes:

equation image

where Umath imagek) = emath image is the inverse frame transformation for period k, and Heff = (Δω0 − ωkz/2 + ω1,kx cos(ϕk) + σy sin(ϕk)]/2 is the effective Hamiltonian in the new rotating frame corresponding to the same period. The parameter space thereby is 4N-dimensional. The nuclear spins' dynamics are complex enough to generate a gate robust against B1 and B0 inhomogeneity. And yet, it can be calculated quickly. Nevertheless, optimizing the spins' FA by calculating their evolution in each voxel would be a formidable task for a high resolution 3D image. Now if equation image and equation image are known, then their variations can be compactly incorporated in the optimization procedure by using their 2D histogram distribution. If B1,max is the maximum amplitude of the transmitted RF field over the volume of interest, let pm,n represent the fraction of spins over that volume which see an RF amplitude αmB1,max and a resonance frequency Δω0,n, with 0 < αm ≤ 1 and ∑m,npm,n = 1. The propagator equation image is replaced by Um,Δω0,n), i.e. a function of the attenuation of the maximum B1 amplitude and of the resonance frequency. The cost function in the minimization procedure can thus be symbolically calculated in the following way:

  • cost = 0

  • for each Δω0,n

  • for each αm

  • calculate Um,Δω0,n;

  • calculate FAm,n;

  • cost = cost + pm,n | (FAm,n − FAtarget/FAtarget|;

  • end for

  • end for

  • cost = cost + std(FAm,n)/〈FAm,n〉+…

  • penaltyPw + penaltyT + penaltyFq +

  • penaltySAR;

where “penaltyPw,” “penaltyT,” “penaltyFq,” and “penaltySAR” correspond to the penalty functions for the power, the duration, the frequency, and the SAR, respectively; while 〈·〉 and “std” denote the expectation value and the standard deviation over the {B0,B1} distribution. The penalty functions for the power, the time, and the SAR just ensure that unattainable powers, too long durations or too high energies are not returned. As far as the frequency penalty function is concerned, it prevents losing performance when discretizing the pulse, and avoids possible memory problems if too many points are required to sample accurately the waveform. All these functions are adjusted empirically and depend on the severity of the B1 and B0 dispersion, and on the hardware constraints. The algorithm therefore loops over a few tens of histogram values instead of hundreds of thousands if the joint spatial distribution of B1 and B0 was directly used. The procedure first starts with two periods. The genetic algorithm is used to find the region where a good solution is likely to be found. The solution returned by the GA then is taken as the initial guess for the Nelder-Mead simplex algorithm. This latter method is a direct search method and converges faster to the nearest local minimum of the cost function. If the value of the cost function is smaller than a given threshold, then the algorithm stops and returns the parameters of the RF pulse. If it is larger, it adds another period and continues the procedure until it matches the stopping criterion. It is worth adding that the cost function contains many local minima and that the full search algorithm is not deterministic. As a result, running the algorithm several times is likely to return different RF pulse solutions.

METHODS

B1 and B0 Maps

Our measurements were performed on a Siemens 3 T Trio scanner (Siemens Medical Solutions, Erlangen, Germany) with a quadrature head coil used both for transmission and reception. The gradient amplitudes and slew rates available were 40 mT/m and 200 T/m/s, respectively. To extract the 3D B1 amplitude profile, we used the actual flip angle imaging (AFI) sequence reported in(32). By applying a 100 μs square pulse at resonance, the relationship between the flip angle and B1 being linear, the B1 values could be deduced. Errors in the calculation due to B0 variations are negligible in this case. The sequence parameters we used were TR1 = 20 ms, TR2 = 100 ms (n = 5), and nominal flip angle of π/4. We took a FOV of 210 × 210 × 192 mm3 and matrix 64 × 64 × 64. These conditions yielded a total acquisition time of 8 min. The B0 maps were obtained through fits of the phase evolution monitored using two additional echoes in TR1. We focused on brain imaging and therefore only used the brain-masked voxels to calculate the {B0,B1} 2D histogram and compute the pulses. We carried out experimental demonstrations in vivo on two volunteers' brains. Informed consent was obtained from both subjects in accordance with guidelines of our institutional review board.

Pulse Design at 3 T

The pulse design was coded using home-made C and Matlab code (The Mathworks, Natick, MA). We used for the GA parameters a population of 200 chromosomes, a mutation rate of 30%, a survival rate of 50% and a randomly picked single gene crossover (blending method)(20), over 500 generations. We empirically found in this case that the number of generations was a more important factor than the number of individuals. The GA then seeds its best solution to the Nelder-Mead simplex algorithm which converges quickly to the nearest local minimum. If the returned minimum value fulfills the convergence criteria, it stops. Otherwise, it increments the number of periods and starts the search again. On the first volunteer, only the B1 inhomogeneity profile was taken into account, while both B1 and B0 variations were used for the design of the pulses on the second volunteer. Choosing to do so allowed us to study the impact of using a 2D histogram instead of a 1D one, and to measure possible improvements. The number of histogram bins for the second volunteer thus was set equal to 30, i.e. 3 and 10 for the B0 and B1 directions, respectively, to find a trade-off between speed and accuracy. Typically, a pulse solution was returned by the algorithm in less than 30 sec using a standard PC.

We tested a series of three strongly modulating pulses (π/6, π/3, and π/2 pulses) on both volunteers. The pulses were designed on the fly, just after measurement and analysis of the 3D B1 and B0 maps of the volunteers' brains. To quantify the strongly modulating pulses' performances, we replaced the standard square pulses by the strongly modulating ones. We set their respective durations, according to the results returned by the algorithm, and voltages assuming a linear relationship with B1,max. We thus repeated the same FA measurement sequence(32) and deduced the sine of the FA (more directly related to the image intensity). We calculated the mean of the sine of the flip angle and its standard deviation over both brains for each pulse scenario. Finally, to numerically test the robustness of the pulses, we took the 2D histogram bins, multiplied their ordinate values by a random value between 0 and 2 and renormalized the distribution. For each of 1000 runs, we calculated the mean and standard deviation of the sine of the flip angle over the whole brain, and determined the robustness of the strongly modulating pulses accordingly.

Comparison with the tanh/tan BIR4 Adiabatic Pulse

To check that the strongly modulating pulses potentially offer an energetic advantage compared to adiabatic pulses, we selected the following tanh/tan BIR4 pulse as suggested in(33):

equation image

for amplitude, and

equation image

for frequency modulation, and simulated the spins' dynamics. We took λ = 10 and tanβ = 10(33). Without imposing adiabaticity constraints, we varied the parameters ωmax and B1 to determine the smallest B1 value that allows to obtain the mean of the sine of the flip angle within 0.01 of the target value, and to achieve a standard deviation of less than 0.01 in absolute value (commonly reached by the strongly modulating pulses), using the measured {B0,B1} 2D histogram of the second volunteer's brain. Once found, we calculated the relative energy ratios EBIR4/ES.Mod. = ∫math imageBmath image(t)dt/∫math imageBmath image(t)dt to study the pulses' respective energy demands.

RESULTS

Figure 1 shows the B0 and B1 maps over a central sagittal slice of the second volunteer's brain and the corresponding whole brain-2D histogram. Roughly 3 and 2% of the spins were located in the bins centered at 55 Hz and −55 Hz, respectively; while the rest of the spins was contained in the 0 Hz bins, as expected. The maximum B1 value in the whole brain was measured to be 35.7 μT for a 100 μs square pulse with a nominal flip angle of π/4. We provide in Fig. 2 an example of a strongly modulating pulse shape, here a π/3, designed for the second volunteer using the measured B0 and B1 maps mentioned above.

Figure 1.

Measured B0 and B1 maps over the brain of the second volunteer and corresponding 2D histogram at 3T. Inset a: B0 map in Hz (with respect to the carrier frequency). Inset b: B1 map in μT. The maximum value measured in the whole brain was 35.7 μT for a 100 μs square pulse with nominal flip angle of π/4. Inset c: {B0,B1} 2D histogram over the whole brain. Roughly 3 and 2% of the spins are located in the bins centered at 55 Hz and −55 Hz, respectively; while the rest of the spins is contained in the 0 Hz bins. See text for the parameters used for the B1 and B0 mapping experiment.

Figure 2.

Pulse shape of a π/3 strongly modulating pulse. The real and imaginary parts of the pulse are shown. The pulse is made of 3 periods and lasts 2,311 μs.

The performance of the different pulses is illustrated in Figs. 3, 4, 5, 3–5 and again corresponds to the second volunteer's data. All figures are plotted on the same color scale. The figures clearly show that the strongly modulating pulses outperform the square pulses calibrated by the scanner and are in good agreement with the theoretical results (see Table 1 for a summary). Compared to the square pulses, not only the standard deviations corresponding to the strongly modulating pulses are significantly reduced (up to a factor of 15 for the π/2 pulse) but their mean values are also much closer to the target values. The discrepancy between theory and experiment can be explained by the finite number of histogram bins used for the calculation of the pulse performance, uncertainties of the FA measurements due to intrinsic noise (especially for the π/6 data where the measurement scheme is less sensitive(32)), possible transients in the pulse shape, nonlinearities in the amplifying electronic chain and the fact that T1 and T2 effects were not accounted for in the calculations.

Figure 3.

Measured sine of the flip angle for a π/6 square and strongly modulating pulse on three orthogonal slices. The mean of the sine of the flip angle over the whole brain for the square pulse and the strongly modulating pulse are respectively 0.445 and 0.490, the target value being 0.5. The respective standard deviations are 0.058 and 0.019. Three orthogonal views (from top to bottom: sagittal, axial, and coronal slice) are shown for each RF condition.

Figure 4.

Measured sine of the flip angle for a π/3 square and strongly modulating pulse on three orthogonal slices. The mean of the sine of the flip angle over the whole brain for the square pulse and the strongly modulating pulse are respectively 0.791 and 0.848, the target value being 0.866. The respective standard deviations are 0.080 and 0.012. Three orthogonal views (from top to bottom: sagittal, axial, and coronal slice) are shown for each RF condition.

Figure 5.

Measured sine of the flip angle for a π/2 square and strongly modulating pulse on three orthogonal slices. The mean of the sine of the flip angle over the whole brain for the square pulse and the strongly modulating pulse are respectively 0.965 and 0.999, the target value being 1. The respective standard deviations are 0.046 and 0.003. Three orthogonal views (from top to bottom: sagittal, axial, and coronal slice) are shown for each RF condition.

Table 1. Summary of the Pulses' Performances: Mean of the Sine of the Flip Angle ± the Standard Deviation
 π/6π/3π/2
  1. S. Mod. stands for strongly modulating pulse. The mean and standard deviations (of the sine of the FA) are over the whole brain of the second volunteer. The experimental results are in good agreement with the theory. For completeness, the duration and B1 maximum value are given for each strongly modulating pulse.

Experimental square0.445 ± 0.0580.791 ± 0.0800.965 ± 0.046
Experimental S. Mod.0.490 ± 0.0190.848 ± 0.0120.999 ± 0.003
Theoretical S. Mod.0.505 ± 0.0080.866 ± 0.0110.999 ± 0.002
S. Mod. duration (in μs)1,8902,3112,549
S. Mod. B1,max (in μT)18.516.614.2

For the sake of comparison, we provide in Fig. 6 data on the first and second volunteers. We remind the reader that B0 field inhomogeneity was not accounted for in the design of the pulses for the first volunteer. The region in the first volunteer's brain where the strongly modulating pulse did not perform well roughly corresponds to a −50 Hz deviation with respect to the carrier frequency. More quantitatively, for the two π/6 strongly modulating pulses (corresponding to Fig. 6), we measured a standard deviation of 0.055 over the whole brain of the first volunteer compared to 0.018 for the second one. It thus confirms the usefulness of including the B0 variation in the design of the pulses.

Figure 6.

Measured sine of the flip angle for the two different π/6 strongly modulating pulses on volunteers 1 and 2 (sagittal slices). The means of the sine of the flip angle are 0.504 and 0.490 for volunteers 1 and 2, respectively; while the standard deviations calculation yield 0.055 and 0.018. Once again, the quoted numbers correspond to measurements and calculations over the whole brains. It is visible that accounting for B0 inhomogeneity clearly allowed to converge towards a satisfying RF solution even in those regions where the spins have a different resonance frequency. Here on this figure, the region in the first volunteer's brain where the pulse did not perform well corresponds to a resonance frequency deviation of about −50 Hz from the carrier frequency.

To test the robustness of the pulses, we did the numerical study mentioned in the previous section. Over 1000 histogram variations, for the π/3 pulse at 3 T, we calculated that the mean of the sine of the flip angle never deviated by more than 0.4% of its mean value, while the maximum found in standard deviation was 0.015. This shows that the strongly modulating pulses are robust with respect to histogram deviations. In addition, still according to calculation, their performance should not be too much affected if the voltage value entered in the scanner differed from the optimal one by ±20%.

After the exam, for SAR considerations, we performed the energy ratio calculations mentioned in the previous section. We first took for the BIR4 pulse a duration T equal to 2.2 ms, i.e. roughly the average duration of the strongly modulating pulses reported here. We found for EBIR4/ES.Mod. values equal to 2.7, 2.9, and 1.4 respectively for the π/6, π/3, and π/2 pulses. One should remember however that the pulses used on the volunteers were designed on the fly, while they were in the scanner. Limited time was therefore available to work on a particular pulse optimization. After the exam, we were able to modify the penalty functions and find less energetic pulses which, with similar performances, yielded respective relative energy ratios with the adiabatic pulses equal to 3.2, 4.6, and 3. If as suggested in(33), we took T = 5 ms, then the energy ratios would be 2.1, 2.1, and 1 for the pulses designed on the fly, and 2.5, 3.3, and 2.1 for the reoptimized ones. With this particular BIR4 pulse, it therefore seems that for nearly equal performances on a human brain at 3 T, the strongly modulating pulses are less energetic, and the gain can sometimes be significant. However, it is worth adding that the authors make no claim here as far as the strongly modulating and adiabatic pulses' demands in energy are concerned. It is possible that other adiabatic pulses could perform better and require less RF power. But by the same token, there may also be other better local minima of the cost function so that less energetic strongly modulating pulses could be returned.

DISCUSSION

This article demonstrates a new and efficient method that counteracts RF inhomogeneity in high-field MRI. The method uses the exact Schrödinger equation, except for T1 and T2 dependencies, and a 2D B1 and B0 amplitude histogram corresponding to a volume of interest. The method is fast and can find an irradiation scheme in less than 30 s, once the {B0,B1} histogram is known. The pulse being applied everywhere, without any magnetic field gradient, 3D imaging techniques need to be used at this time. Although more demanding, if several spins with distinct resonance frequencies were encoded in the problem, then slice selection could also be studied. In fact, results reported in(34) show that there exists some control fields that can steer an ensemble of spins that see RF power and resonance frequency variations to any desired set of states with arbitrary accuracy. Unfortunately, in high-field MRI, the RF inhomogeneity is too severe and its maximum value too small to make the recipe in(34) implementable in a reasonable time. However, it is worth pointing that the problem in(34) was posed as a gate engineering problem. By relaxing some constraints on the target, i.e. again by switching from gate to state engineering, likewise tools borrowed from optimal control theory could lead to sequences achieving this great level of coherent control.

Also in this article, an initial longitudinal magnetization state was assumed. The pulses designed in this way would not perform well on other states, e.g. on transverse magnetization states. One could eventually design strongly modulating pulses to work on arbitrary states, as was done in(18, 19), to provide good control of the spins during a spin echo sequence for instance. We intend to study in the future the relationship between the B1 spread and the ability to perform rotations on arbitrary states using this technique.

In contrast to all other previous methods, the method presented here does not exploit parallel transmission and saves the long premeasurement time of the inhomogeneity profile of each coil. However in more extreme high field conditions, if the strongly modulating pulses required too much energy to be implemented on a human subject, combining them with parallel transmission could allow one to find accurate and less energetic pulses. For instance, one could imagine doing some RF-shimming first in order to optimize the B1 histogram and then keep the corresponding array amplitude and phase configuration to compute a strongly modulating pulse. The RF-shimming simulations reported in(3) actually show that it could indeed be a good starting point at fields up to 14 T, for further pulse optimization. This is also the subject of some future work.

Finally, it is interesting to note that B1 and B0 measurements from one patient to another may not be necessary if the respective statistical distributions of such values are somewhat similar. While we know that locally the B1 field value is patient-dependent, it is not clear if it is the case for the histogram, which may be a much more robust quantity. In Fig. 7 we provide the two different B1/B1,max 1D histograms corresponding to the two different subjects we performed our experiments on, in addition to their first five moments calculation. The two distributions are relatively similar and according to calculation, a strongly modulating pulse designed for one of the two subjects should still perform very well on the other one (assuming also a somewhat similar B0 distribution). If with further investigation we could confirm such robustness, this would imply that a pulse designed for a given FA would need to be computed only once and could be applied on any patient, at least with roughly the same head size, and that only a scaling in the applied voltage based on the scanner's RF calibration would be required. We intend to study this in some future work as well.

Figure 7.

B1/B1,max 1D histograms for two human brains and first 5 moments calculation. The two histograms are somewhat similar so that a strongly modulating pulse designed using one subject's data should perform well on the other subject.

In conclusion, we have developed a new control scheme to counteract RF inhomogeneity at high fields for 3D brain imaging. We have successfully conducted a first series of experiments at 3 T which validate the performance and the feasibility of the method. Calculations tend to show smaller demands in energy than for the specific tanh/tan BIR4 adiabatic pulses. The algorithm is fast and can find an irradiation scheme in less than 30 s. In addition to the greater simplicity of the method compared to parallel transmission schemes, one possible advantage is the fact that B1 profile measurements and pulse (re)computations may not be necessary to achieve uniform excitation across a population of patients, thus making the method particularly attractive for in vivo applications.

Acknowledgements

The authors thank A. Vignaud from Siemens France for technical assistance.

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