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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Magn Reson Med. Author manuscript; available in PMC 2009 July 27.
Published in final edited form as:
PMCID: PMC2715966

Magnitude Least Squares Optimization for Parallel Radio Frequency Excitation Design Demonstrated at 7 Tesla With Eight Channels


Spatially tailored radio frequency (RF) excitations accelerated with parallel transmit systems provide the opportunity to create shaped volume excitations or mitigate inhomogeneous B1 excitation profiles with clinically relevant pulse lengths. While such excitations are often designed as a least-squares optimized approximation to a target magnitude and phase profile, adherence to the target phase profile is usually not important as long as the excitation phase is slowly varying compared with the voxel dimension. In this work, we demonstrate a method for a magnitude least squares optimization of the target magnetization profile for multichannel parallel excitation to improve the magnitude profile and reduce the RF power at the cost of a less uniform phase profile. The method enables the designer to trade off the allowed spatial phase variation for the improvement in magnitude profile and reduction in RF power. We validate the method with simulation studies and demonstrate its performance in fourfold accelerated two-dimensional spiral excitations, as well as for uniform in-plane slice selective parallel excitations using an eight-channel transmit array on a 7T human MRI scanner. The experimental results are in good agreement with the simulations, which show significant improvement in the magnitude profile and reductions in the required RF power while still maintaining negligible intravoxel phase variation.

Keywords: parallel excitation, acceleration, magnitude least squares, RF shimming

Parallel excitation offers a means of designing multidimensional radio frequency (RF) pulses using accelerated gradient trajectories resulting in a short pulse duration compared with single-channel excitation. Accelerations of four- to sixfold have been shown using an eight-channel transmit system (1), potentially enabling several important applications, including flexibly shaped excitation volumes, and mitigation of RF field inhomogeneity at high field. Various methods have been proposed for the design of such RF and gradient waveforms (25), primarily in the low flip domain (6), and successfully implemented on multichannel hardware (1,7).

In this work, we propose an extension to the spatial domain parallel excitation pulse design method introduced by Grissom et al. (5), where we apply magnitude least squares optimization to improve excitation magnitude profile and reduce the required RF power at a cost of increased phase variations in the excitation pattern. However, for many excitation applications, such as when magnitude images are recorded, low-order spatial phase variations do not impose a significant penalty. In fact, they can potentially decrease the dynamic range requirements of the imaging (which can be extensive for high field three-dimensional [3D] acquisitions) by reducing the amplitude at the center of k-space. We, therefore, developed a method for pulse calculation with an adjustable regularization parameter de-emphasizing the excitation phase profile and study the potential benefits in magnitude profile fidelity and SAR, which can accompany this relaxed constraint.

The idea of permitting phase variation in the excitation profile has previously been exploited in several applications, including the design of quadratic-phase RF pulses (8,9), RF shimming (1013), and frequency-sweep pulses (14), with benefits such as improved magnitude transition bands for saturation pulses, homogeneity for RF shimming, and reduced RF peak power for frequency-sweep pulses. In this work, we propose a method that allows us to take advantage of relaxed constraints on the phase profile for parallel excitation design. Our approach is based on a variant of a local optimization method used for solving general magnitude least square problems (15). We demonstrate the method using a 2D spiral excitation with R = fourfold acceleration over the Nyquist sampling and a uniform in-plane slice-selective spoke excitation (1,16) on an eight-channel excitation system at 7 Tesla (T).

Recently, at the ISMRM’07 conference, Katscher el al. (17) demonstrated the significant improvement gained in RF shimming when a magnitude least square approach is used. Kerr et al. (18) also proposed an approach for magnitude least square optimization for the design of parallel RF excitation and demonstrate the method using a spiral trajectory. The approach in this work differs in that it takes into account intravoxel dephasing and enables the designer to systematically trade off the allowed spatial phase variation for the improvement in magnitude profile and reduction in RF power. Furthermore, the work provides an extensive simulation and experimental study on the possible benefit of the algorithm on both spiral and spoke k-space trajectory excitations at high magnetic field strength.


Parallel Excitation

In this work, we based our parallel excitation formulation on the spatial-domain approach by Grissom et al. (5). Briefly, the small-tip multidimensional excitation approximation with R coils is written as

equation M1

Here, Sr and B1,r are the spatial sensitivity profiles and the RF waveforms for coils indexed by r, x is the spatial variable, m(x) is the target transverse magnetization after excitation, and k(t) is the excitation k-space trajectory, defined as equation M2,where γ is the gyromagnetic ratio, and G is the gradient, and T is the duration of the gradient waveform. After discretization of space and time, this expression can be written as a matrix equation, m = Ab, where the A-matrix incorporates the B1 coil profiles modulated by the Fourier kernel due to the k-space traversal, m is the target profile in space, and b contains the RF waveforms. With this formulation, the RF pulses can be designed by solving the following optimization problem by conventional least squares (LS):

equation M3

Here, the optimization is performed over the region of interest suggested by a weighting, w, and R(b) denotes a regularization term that may be use to control integrated and peak RF power. This minimization problem can be solved efficiently using conjugate-gradient (CG) methods.

Magnitude Least Squares

In the above formulation, RF waveforms are found that will reduce the deviations from the target profile in both the magnitude and phase. However, in many applications, such as conventional slice-selective excitation for structural imaging where only magnitude images are of interest, the primary metric of interest is the fidelity of the magnitude profile while the phase profile is relatively unimportant. In such cases, the following magnitude least square (MLS) optimization,

equation M4

may provide an improved magnitude profile, but at the cost of increased spatial phase variations. Here, m is specified as a real-valued vector. Unlike equation [2], the MLS optimization, [3], is unfortunately not convex and generally cannot be solved with a guarantee of global optimality. Kassakian (15) recently proposed two different methods (in the context of audio signal processing) for solving this MLS problem, but without any regard to the resulting phase profile. With these methods, superior magnitude profile can be achieved when compared with the solution from Equation [2]. For application of these methods in MRI RF excitation design, intravoxel dephasing can become an important constraint that limits the degree of allowable spatial phase variation. Thus, we extend Kassakian’s work on the so-called local-variable exchange solution, by imposing a mild constraint on the phase in the form of a smooth and slowly varying spatial phase profile.

We follow the notation of the local variable exchange method in (15). We start by rewriting Equation [3] as:

equation M5

where zi is an extra phase term and Ai is the ith row of A. From Equation [4], for a fixed z, the minimization of b can be solved as in Equation [2]. In turn, for a fixed b, the optimal vector, z, contains values zi with a modulus of 1 and phase equal to that of Aib. By starting with a feasible pair, b and z, and then keeping one variable fixed and solving for the other, we can only improve the objective function. Because the objective function is non-negative, by alternating this procedure a local minimum will eventually be reached.

As previously mentioned, the solution from the variable exchange method provides no guarantee that the phase behavior of the resulting excitation profile will be slowly varying compared with the pixel length. The phase profile will strongly depend on the solution to z, which contains the specified desired phase. To impose a smooth phase variation, we place a restriction on the evolution of phase of z during the optimization loop, as shown schematically in Figure 1, which is a flow diagram summarizing the modified local-variable exchange optimization. For each iteration step, rather than using the optimal z according to the original algorithm, a phase-smoothed version of the optimal z was calculated and used in the optimization. This modification forces the phase of z to evolve smoothly. However, there are natural tradeoffs in the choice of degree of phase smoothing for z, such that too much smoothing places overly conservative constraints on the optimization, resulting in a solution with an unnecessarily high cost. To avoid overconstraining the problem, the effect of the smoothing parameter was evaluated. For our designs, we chose a Gaussian smoothing kernel and empirically found that a good compromise of phase smoothing could be obtained by applying a 5 × 5-pixel Gaussian low-pass filter with σ = 2 pixels to smooth the phase profile variation to less than 8°/mm. This phase smoothing was only applied to non–zero-magnitude regions in the excitation target profile. As a final design step, after the completion of the optimization, a short “rewinder” gradient was appended to the excitation pulse to remove any residual linear component of the spatial phase variation.

FIG. 1
Flow diagram of the modified local-variable exchange method for magnitude least squares optimization. During the iterative optimization process, a smoothed version of the excitation phase from the previous iteration is used as the target excitation phase ...


Simulation Study

To compare the performance of the conventional LS with the MLS optimization, RF pulses were designed using these two methods, each for two types of excitation k-space trajectories: (i) a 2D spiral excitation with a fourfold acceleration, design resolution of 2 mm, and a field-of-view of 18 cm; (ii) a slice-selective four-spoke excitation with sinc subpulses’ time-bandwidth-product equal to 4. A square excitation target profile was used for the spiral excitation and a uniform slice-selective excitation of 1-cm-thickness target profile was used for the spoke. Maximum gradient amplitude of 25 mT/m and slew rate of 150 T/m/s were used for all designs. The predicted magnetization patterns were calculated using a Bloch equation simulation of the eight-channel parallel RF excitation, where the spin-domain representation of Cayley-Klein parameters was used (19). Spatial variation of the B1 excitation field was incorporated, and at each spatial location, the B1 excitation fields from all the coils were added as complex quantities, yielding the input for the Bloch equation simulation over space and time.

Similar to the approach by Grissom et al. (5), Tikhonov regularization equation M6 was used for both conventional LS and MLS optimizations. For both spiral and spoke excitations, the RF pulses were designed for a set of Tikhonov regularization parameter values (β) to illustrate the clear tradeoff between the reduction in excitation error and the increased requirement in RF power. Deviations from the target profile were assessed using root magnitude mean square error (RMMSE) defined here as:

equation M7

To limit intravoxel dephasing in the MLS optimization design, the smoothing parameter of the Gaussian filter in the optimization was adjusted such that the resulting phase profile variation is less than approximately 8°/mm (after rewinding). In addition, a short rewinder gradient appended to the RF pulse was use to remove residual linear component of the spatial phase variation. The area of the rewinder was calculated based on a linear fit to the Bloch equation simulated spatial phase profile.

System Hardware

For experimental verification of the method, the RF pulses were tested on a Siemens prototype 7T Magnetom scanner (Erlangen, Germany), equipped with an eight-channel transmit system, with maximum gradient amplitude of 40 mT/m and slew rate of 200 T/m/s. An eight-channel strip-line coil array was used for transmit and receive (20). The RF array was built around a 28-cm diameter acrylic tube (Fig. 2a) and was driven through a transmit-receive switch at each of the eight rungs. The signal was received on eight receive channels from the same coils. All measurements were performed in a 17-cm diameter doped water phantom, containing 1.25 g/L of nickel sulfate and 5 g/L of sodium chloride. A cylindrical loading ring (filled with doped water containing manganese chloride to reduce its T1 to less than 3 ms), was used to reduce the amount of B1 inhomogeneity observed in the spherical water phantom to a similar level observed in human in vivo imaging. The bottom right of Figure 2a shows the transmit-received (TXRX) birdcage image for in vivo and for the water phantom with loading ring. The peak-to-trough signal ratio is ~9 and ~8.5 for in vivo and water phantom respectively. Without the loading ring, the water phantom’s peak-to-trough signal ratio would have been ~13 which is much higher than that observed in vivo with this particular setup.

FIG. 2
a: Eight-channel transmit/receive stripline coil array for 7 Tesla that was used for all experiments in this work. The insert on the right panel of panel a shows the estimated transmit (TX) and receive (RX) profiles and the transmit–received image ...

B1 Mapping

Excitation B1 profiles for the eight transmit coils were obtained by applying a slice-selective, low-flip-angle RF pulse to a single transmission channel at a time, and receiving through the birdcage reception mode of the array. In this reception mode, the complex data from the individual receive channels were phased with an incremental phase of 45 degrees and summed, to mimic reception with the traditional birdcage coil. The B1 profiles were estimated by recording the amplitude and phase of a low flip angle 2D gradient-recalled echo sequence with 128 × 128 pixels in x,y at 2 × 2 mm resolution, (TR/TE/BW = 100 ms, 5 ms, 260 pixel), and the transmit map was inferred by a division of the estimated reception profile (Fig. 2b).

At low B0 field, an estimation of the birdcage reception profile can be obtained by taking the square-root of the image intensity pattern created when the birdcage mode was used for both reception and transmission. However, because the transmit and receive patterns of a given coil use opposite circular polarizations that can have different profiles at high B0 field (21), the use of a quantitative coil profile mapping technique was required to estimate the birdcage reception profile. The technique used in this work, relies on transmitting and receiving with the birdcage mode at several transmission voltages, and fitting the resulting images to the appropriate image intensity equation (which can be derived from, e.g., Haake et al. (22), p. 454–455, noting that the TE decay is identical across all B1 excitation field mapping acquisitions and is ignored here):

equation M8

Where I is the image intensity, RX is the receive coil profile, TR is the sequence repetition time, T1 is the longitudinal relaxation time constant, and θ is the flip angle. A standard nonlinear search algorithm in Matlab was used to perform the fitting. From the fitting, the birdcage transmission profile (TX) can be approximated by [theta w/ tilde](x,y) and the reception profile (RX) by RX(x,y). The resulting profiles are shown in Figure 2a.

Parallel Excitation

For each of the parallel excitation designs, the eight-channel array was driven with eight independent transmit channels modulated in magnitude and phase by the calculated waveforms, and the data were received using the birdcage mode of the coil array. The excitation pattern was imaged with a 3D gradient-recalled echo sequence with matrix = 128 × 128 × 32, field of view = 256 mm × 256 mm × 160 mm, voxel resolution = 2 mm × 2 mm × 5mm, TR/TE/BW = 100 ms, 6 ms, 260 pixel. The excitation profile was inferred by dividing the reconstructed image by the estimated birdcage receive profile. The RF excitation pulses were adjusted so that all the experimental results are of the same target flip angle.

Low-Flip-Angle Validation

In the report by Grissom et al. (5), it was shown that, with low Tikhonov regularization parameter value (β), that is, high RF power requirement, the excitation profile can be heavily degraded from the low flip angle approximation even at relatively low flip angle. For the current work, the applied excitation voltages were established to be well within the low-flip-angle regimen, although without a full quantitative B1 map to determine the exact excitation flip angle. For example, in the spiral-based pulse design, we made our calibration by exciting the design that required the highest power (the LS design with β = 0.3) with a sequence of increasing voltages and quantified the resulting profiles. For very low excitation voltages, the signal-to-noise ratio (SNR) is low, and at high excitation voltage, the low-flip-angle approximation fails. We chose the voltages for our experiments to be such that our data are acquired with ample SNR, but at the same time are low enough to not violate the low-flip-angle approximation. In our current setup, we found that this favorable operating range for the voltage extended from 30V to 100V for this design (β = 0.3).

To make a comparison between simulation and experimental results, the RMMSE for the simulation data is calculated assuming a target transverse magnetization value of mxy = 0.1. For the experimental results, the intensity is scaled so that the mean signal in the target excitation ROI matches that of the simulation, after which the RMMSE is then calculated.


Spiral Trajectory

The fourfold accelerated (4×) spiral excitation design yielded 3.51-ms-long pulse, with an additional 70 µs for the rewinder gradient in the MLS cases. Figure 3a) shows a comparison between the LS and MLS design for Tikhonov regularization parameter value of 1.5. Visually, the MLS magnitude profile shows improved target excitation and background noise suppression in areas outside the target square. For this particular β, the MLS optimization yields an improvement of 15.62% in RMMSE and 26% and 41% in integrated and peak RF power over the LS method. Figure 3b shows the plot of the normalized RMMSE versus the normalized RF voltage norm across all eight channels (‖b2) for four different settings of the Tikhonov regularization parameter (β). The dotted red and blue curves represent the simulation results for conventional LS and MLS, respectively. Similarly, the solid red and blue curves show the experimental tradeoff between mean square error and pulse energy at the different regularization parameters. The average drop in RMMSE over the four data points is 12% for simulation data and 10% for experimental data. The average integrated RF power drop is 5.7%, and the peak RF power drop is 9.3%.

FIG. 3
a: Experimental square-target excitation, imaged in a central axial section of a three-dimensional–encoded readout. The figure illustrates the excitation magnitude profile improvement with the magnitude least squares design when compared with ...

We note that the MLS computation time (3 GHz, 64-bit Xenon processor, Linux operating system), for the spiral-based designs ranged from 5–8 min for higher Tikhonov regularization parameter values, up to approximately 35 min for lower value (β = 0.3). Figure 4 illustrates the tradeoff between an increase in spatial phase variation for lower magnitude mean squared error and reduced RF power with MLS optimization compared conventional LS designs. The magnitude (left) and phase (right) profiles of the square target excitation with β = 1.5, for LS and MLS optimization are shown in row a and b, respectively. Similar to Figure 3a, the magnitude profile of the MLS design is visually better than the LS design. In Figure 4c, the increased in spatial phase variation across the target excitation area for MLS design can clearly be observed. This observed phase variation is shown to be very similar to the predicted variation.

FIG. 4
Experimental square-target excitation for a fixed value of Tikhonov regularization parameter (β = 1.5), magnitude (left) and phase (right) by, (a) conventional least squares optimization, and (b) by magnitude least squares. Larger variation of ...

Spoke Trajectory

The four-spoke pulse design yielded 2.86-ms-long pulses. Figure 5a,b shows a comparison between the best (lowest RMMSE) uniform in-plane slice selective profiles achieved using the LS (a) and the MLS (b) design. The magnitude profile resulting from the MLS design is significantly more uniform with a reduction in RMMSE of 51% compared with the LS design (the simulation predicted a 66% reduction). To further quantify the performance, we note that, with MLS design, 96.4% of the data deviate by less than 10% from the flat target profile and 100% deviate by less than 20%, compare with 82% and 97% with the LS design. Figure 5c shows the plot of the normalized RMMSE versus the normalized RF voltage norm across all eight channels (‖b2) for six different settings of the Tikhonov regularization parameter (β). The average drop in RMMSE over the six data points is 47% for simulation data and 34% for experimental data. The average integrated RF power drop is 49% and the peak RF power drop is 53%.

FIG. 5
Acquired data and simulation results for a four-spoke excitation by (a) conventional least-squares and (b) magnitude least squares for the Tikhonov regularization parameter pair that resulted in the lowest experimental RMMSE. In each panel, the top row ...

Because we only need to calculate the spokes coefficient during the RF design, the MLS computation time (3 GHz, 64-bit Xenon processor, Linux operating system), for the spoke-based designs is much less compared with the spiral- based design, with calculation time of less than 1 min for any of the Tikhonov regularization parameter values.

Similar to Figure 4, Figure 6 illustrates the tradeoff between an increase in spatial phase variation for lower magnitude mean squared error and reduced RF power with MLS optimization compared conventional LS designs for the spoke design. Again in row c), the observed phase closely resembles the simulation; with small and smoothly varying variation, resulting in negligible intravoxel dephasing.

FIG. 6
Experimental magnitude (left) and phase (right) results for a four-spokes excitation by (a) conventional least squares and (b) magnitude least squares for the Tikhonov regularization parameter pair that resulted in the lowest experimental root magnitude ...


We have successfully developed, implemented, and demonstrated a design algorithm for multichannel parallel excitation that minimizes the deviation in the magnitude of the transverse magnetization profile from its target and reduces the pulse power required. This improved performance is achieved at the cost of increased spatial phase variation, under the control of the designer, and resulted here in negligible intravoxel dephasing. Experimental results at 7T matched well with the simulation; showing significantly improved performance compared with conventional least-squares design where the phase target is explicitly constrained.

For the 2D spiral design, the experimental RMMSE matched well with the predicted values from the simulated data for both LS and MLS methods (Fig. 3b). However, at smaller Tikhonov regularization parameter values (β), the experimental RMMSE starts to deviate more from the prediction. We explain this phenomenon by observing that for low β values more cancellations among profiles are required to achieve a better excitation (as can be seen from the increased in the required RF power). This large amount of required cancellations can result in an amplification of noise in the excitation profile from, for example, noise in the B1 maps, causing a larger RMMSE than expected. This was observed for both the LS and MLS methods where the experimental excitation profiles at β = 0.3 contains more edge artifacts, when compare with the excitation profiles of higher β values.

For the spoke design, in Figure 5c the experimental RMMSE follows a similar trend to that of the simulation results, although with higher error values. In our simulation studies, we found that the spoke design is much more susceptible to B0 off-resonance and eddy current effects when compared with the spiral design. This vulnerability is likely to be part of the cause for the increased error. Our future work includes the exact quantification of the source of this error and possibly the design for a correction method.

Also in Figure 5c, an increase in error at lower Tikhonov regularization parameter value (β) can be observed in the LS experimental result. In the simulation result over the same β values range, a very small reduction in error for a much higher required RF power can be observed. This finding suggests that, at lower β values, a significantly larger amount of profile cancellation is required during the excitation. In the experiment, this causes large noise amplification, which resulted in the increased error.

In comparing the spiral and the spoke excitations, the improvement gained from using the MLS design in the spoke case is much more prominent. This is to be expected, because the spokes trajectory, which is just a few delta points in kx-ky, allows for much less control of the in-plane excitation profile when compared with the spiral trajectory. Therefore, the gain from relaxing the in-plane excitation phase constrain for the spoke design should be much larger.

This work is a novel study of the possible benefits from relaxing the phase constraint in parallel excitation design. To relax the phase constraint, the design problem was mapped to an MLS optimization. Because of the nonconvex nature of this optimization, currently no method solves the problem with a guaranteed global optimality of the solution. We chose to extend the local variable exchange method (15) to solve the problem, because it allows us to impose the smoothly varying phase constraint. Other methods exist for solving MLS problems, including the relaxed semidefinite programming method (15). Future work includes an exploration of alternate means to solve the posed MLS problem for parallel RF excitation.

This work demonstrated the benefit of MLS optimization in parallel excitation. A water phantom with large B1 profile variation, similar to that observed in human imaging at high field, was used in our experimental set up. Given the significant benefit achieved with the MLS optimization, particularly for the mitigation of the inhomogeneous B1 profile through the use of the spoke trajectory excitation, we expect the MLS optimization to play an important role in parallel excitation design for human imaging at high field.


Grant sponsor: Siemens Medical Solutions; Grant sponsor: National Institutes of Health NCRR; Grant number: P41RR14075; Grant number: R01EB006847; Grant sponsor: R.J. Shillman Career Development Award; Grant sponsor: the MIND Institute.


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