Introduction

Optimal control radio frequency (RF) pulse design provides an efficient numerical framework for developing optimized RF pulses that show superior performance with respect to some user-defined target requirements, such as time-optimality, frequency selectivity and/or B₁-insensitive behavior^1-9. Producing highly uniform rotations that are immune to variation in the B₁-field is of particular importance when surface coils are used for acquisition. Adiabatic pulses can achieve close to a uniform magnetization profile over a wide range of frequencies¹⁰, however the high adiabatic threshold and consequent energy deposition can be a problem with high repetition rates, for example in experiments with ²H. ²H magnetic resonance spectroscopic imaging has been shown recently to be a viable technique for metabolic imaging in the clinic and numerous studies have demonstrated its potential as a diagnostic tool^11-17. Despite its low absolute sensitivity, the short relaxation time of deuterium allows for rapid signal averaging without saturation, thereby enabling metabolite detection. Rapid pulsing however makes the acquisition demanding both in terms of hardware requirements as well as specific absorption rate (SAR) constraints. Here we demonstrate that numerical optimal control-based pulse design could be used to design alternatives to the BIR4 pulse with similar performance but significantly lower adiabatic thresholds.

Methods

The optimization used the built‐in fminunc() function with the default quasi-newton solver in MATLAB (The Mathworks, Natick, Massachusetts) run on the amplitude and phase of each pulse point of the shaped pulse.The cost function contains Bloch simulations and the simulated magnetization profile is compared to the desired magnetization profile in the squared function L₂-norm on frequency-offset interval and B₁ values ranging from 0 to 5 relative to the nominal pulse amplitude. Since RF coils and amplifiers cannot produce arbitrarily high B₁ fields nor can they produce an instantaneous change in pulse amplitude the amplitudes of the shaped pulses were truncated to a maximum value of 1 G and then projected onto a set of waveforms with prescribed upper bounds on the first and second derivatives for each iteration of the optimization to yield a smooth amplitude variation¹⁸. Furthermore, unlike BIR4 pulses, the projected waveform starts and ends at 0 G, setting a more realistic task for the RF transmit chain. Five 2 ms long RF pulses were designed with flip-angles of 5, 15, 30, 68 and 90 degrees and starting the optimization from a linear-phase SLR pulse with 600 Hz wide pass-band. The results were compared to the corresponding 2 ms BIR4 pulses with respect to adiabatic threshold and overall performance over the and B1 grid. Relaxation was taken into account with a T₁ value of 140 ms, and a T₂ of 70 ms, typical of [2,3-²H₂] fumarate¹³.The simulation results for a 2 ms 90 BIR4 pulse and an equivalent optimized pulse were validated with measurements of the natural abundance HDO peak in vivo in a BALB/c Nude mouse (Charles River Laboratories). Experiments were performed at 7T (Agilent, Palo Alto, California) using a home-built 10 mm diameter single-loop surface coil for ²H transmit and receive, which was positioned over the abdominal area of the mouse. The acquisition was the sum of 100 transients acquired into 1024 spectral points with a bandwidth of 4006 Hz. TR was set to 700 ms to allow full recovery of the magnetization. The transmit gain was arrayed in 20 steps to trace the adiabatic threshold of the pulses. The experiment was carried out in compliance with project and personal licenses issued by the UK Home Office, and approved by a local Animal Welfare and Ethical Review Body.

Results

Simulation
The 2 ms long BIR4 pulses showed an approximately 1 G (~ 65%) higher adiabatic threshold when compared to the corresponding optimized pulses (Figure 1). A further advantage of the optimized design is the gradually decreasing flip-angle below the adiabatic threshold which is anticipated to mitigate edge effects when compared with the large variation observed with the BIR4 pulses.
In vivo experiment
The spectra of the HDO peak acquired in vivo showed good agreement with the simulated B¹ profile of the 90 excitation pulses. The adiabatic threshold is reached at a power level approximately 60% lower with the optimized design than with the BIR4 pulse (Figure 2). The BIR4 pulse also had a 1.32 times higher normalized area, which taking into account the higher adiabatic threshold gave an SAR estimated to be 3.67 times higher than that of the optimized pulse.

Conclusion

Here we have shown that pulses designed by optimal control can offer a better solution to the B¹-insensitive excitation problem than the adiabatic BIR4 pulses. A significant reduction in adiabatic threshold as well as SAR was observed and this superior performance was preserved in vivo.

Acknowledgements

VS is in receipt of a Cambridge European Scholarship from the Cambridge Trust.

References

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