Eric Van Reeth^{1}, Hélène Ratiney^{1}, Olivier Beuf^{1}, Soukaina Kanice^{1}, Steffen J Glaser^{2}, and Dominique Sugny^{3}

This study introduces a new family of broadband B1-robust excitation (90°) pulses for MRI with large enough bandwidth (+/- 1 kHz) to account for static field inhomogeneities, and minimal energy deposition. RF pulses are designed with a regularized optimal control algorithm, which is able to adapt the pulse B1-robustness range to fit the coil limits in terms of peak amplitude and energy. In vitro acquisitions using an endoluminal-shaped RF transmit coil show comparable excitation profiles than BIR4 pulses, although BEEEP pulses deposit 5.2 times less energy.

Introduction

Adiabatic excitation pulses are known for their high immunity to RF inhomogeneity. Once the adiabatic threshold is reached, such pulses guarantee a uniform flip of the magnetization [1]. Their performance is generally limited by the available peak pulse amplitude and/or maximal energy deposition (SAR) authorized by clinical systems. For this reason, robust and energy-efficient pulses have been investigated, in the context of NMR spectroscopy [2] and high-field MRI [3, 4].

This study introduces a new family of broadband B1-robust excitation (90°) pulses for MRI with large enough bandwidth (+/- 1 kHz) to account for static field inhomogeneities, and minimal energy deposition. The objective is to provide uniform excitation in the context of single loop surface RF transmission coil (e.g. with endoluminal coils), although they can be applied in any context willing to limit excitation energy deposition.

RF pulses are designed with a regularized optimal control algorithm, which is able to adapt the pulse B1-robustness range to fit the coil limits in terms of peak amplitude and energy. Optimal control is a well known pulse design tool in MRI [5]. It iteratively computes the pulse that optimizes a user-defined cost function, usually set to minimize the distance to a given magnetization target state. In this context, energy minimization can be performed by fixing hard constraints [2, 4], or penalty terms [3]. The cost function is regularized by a term proportional to the pulse energy ($$$||\omega||^2$$$):

$$C(\omega) = \frac{1}{I\times J}\sum_{i=1}^I \sum_{j=1}^J\Vert \overrightarrow{M_{i,j}}(t_f) - \overrightarrow{T_{i,j}}\Vert^2 + \lambda ||\omega||^2$$

where $$$i$$$ and $$$j$$$$ represent the indexes of the considered B0 and B1 inhomogeneity range, $$$\overrightarrow{T_{i,j}}$$$ the magnetization target state, and $$$\lambda$$$ a positive scalar. The optimal pulse thus balances the minimization of the average error norm between the magnetization state at the end of the pulse ($$$t_f$$$) and its target, and the energy deposition.

Our implementation is based on a second order GRAPE optimal control algorithm for cost minimization [6]. Taking advantage of the fact that symmetric pulses have shown to perform equally well as non-symmetric pulses in terms of robustness to B1 inhomogeneities [7], our optimization is performed only on 25 cosine Fourier series coefficients of the real and imaginary parts of the pulse, to ensure symmetry [8]. Therefore, this implies that isochromats with frequency offsets of opposite signs will have symmetric trajectories. As a result, only the positive resonance frequency offsets need to be considered in the optimization process which reduces the computation time.

The characteristics of the pulse were assessed on a pre-clinical 4.7T Bruker MRI on homogeneous agar phantoms. In a first study, a quadrature coil (inner diameter 40mm) was used to acquire the magnetization map with respect to B0 and B1 variations. Two pulses were compared, namely the optimized BEEEP pulse and a tanh/tan BIR4 pulse of the same duration, whose frequency modulation parameters were set to optimize the ratio between the adiabatic threshold and the targeted bandwidth ($$$\Delta \omega$$$ = 15kHz ; $$$\beta$$$ = 10 ; $$$\kappa$$$ = 1.5).

Finally, *in vitro* experiments
were performed using a simple prototype of endoluminal-shaped RF
transmit/receive coil [9], to
compare both pulse performances. The coil was sealed inside a tube,
and immersed into a nickel sulfate doped solution. A fast spin-echo
sequence is used with a turbo factor of 2,
TE = 11.7ms, TR = 1.5s. No gradient was applied during excitation,
meaning that the signal from the whole tube is integrated.

[1] Garwood M, DelaBarre L. The return of the frequency sweep: designing adiabatic pulses for contemporary NMR. Journal of magnetic resonance. 2001;153(2):155-177.

[2] Kobzar K, Skinner TE, Khaneja N, et al. Exploring the limits of broadband excitation and inversion: II. Rf-power optimized pulses. Journal of Magnetic Resonance. 2008;194(1):58-6.

[3] Boulant N, Mangin, JF and Amadon, A. Counteracting radio frequency inhomogeneity in the human brain at 7 Tesla using strongly modulating pulses. Magnetic resonance in medicine. 2009;61(5):1165-1172.

[4] Vinding MS, Guérin B, Vosegaard T, et al. Local SAR, global SAR, and power-constrained large-flip-angle pulses with optimal control and virtual observation points. Magnetic Resonance in Medicine. 2017;77(1):374-384.

[5] Van Reeth E, Ratiney H, Lapert M, et al. Optimal control theory for applications in Magnetic Resonance Imaging. Pacific Journal of Mathematics for Industry. 2017;9(1):1-9.

[6] de Fouquieres P, Schirmer SG, Glaser SJ, et al. Second order gradient ascent pulse engineering. Journal of Magnetic Resonance. 2011;212(2):412-417.

[7] Janich MA, McLean MA, Noeske R, et al. Slice-selective broadband refocusing pulses for the robust generation of crushed spin-echoes. Journal of Magnetic Resonance. 2012;223:129-137.

[8] Skinner TE, Gershenzon NI. Optimal control design of pulse shapes as analytic functions. Journal of Magnetic Resonance. 2010;204(2):248-255.

[9] Dorez H, Ratiney H, Canaple L, et al. In vivo MRS for the assessment of mouse colon using a dedicated endorectal coil: initial findings. NMR in Biomedicine. 2017;30.

Amplitude (a) and unwrapped phase (b) of the BEEEP pulse. It has a constant time-bandwidth product of 4.5 (unitless). This pulse is energy efficient because unlike BIR4 pulses, only the central part of the pulse has high amplitude coefficients. The central part features an adiabatic behavior with a quadratic phase pattern, which is completed by non-adiabatic patterns at the pulse extremities.

Comparison of acquired magnetization maps for
similar resonance offsets and B1 peak amplitude ranges. Dotted lines
illustrate the targeted bandwidth (+/- 1 kHz). Both pulses ensure an homogeneous excitation inside the targeted bandwidth. Notice how the BEEEP pulse is able to preserve the signal coherence at low B1 peak amplitudes.

Evolution of the pulse energy as a function of the B1 peak
amplitude. As expected, the BEEEP pulse induces several times (5.2) less
energy deposition than BIR4 for a given peak amplitude value.

Axial images acquired when RF pulses are transmitted by the
endoluminal coil. Images are shown on the same intensity scale.
Dashed green lines indicated the line along which the intensity
profiles are plotted (Figure 5). At high transmission gain (7dB),
BIR4 produces slightly more signal close to the coil, but presents a
faster decay. At low transmission gain (11dB), the adiabatic
threshold is quickly reached, which induces a quick signal drop,
while the BEEEP pulse preserves more signal away from the coil.

Intensity profiles derived from raw data shown in Figure 4. At higher pulse
amplitudes (amplifier attenuation of 7dB), BIR4 produces slightly
more signal close to the coil but slightly less further away. At low
transmission amplitudes (11dB), the BIR4 pulse induces a much quicker
signal drop than the BEEEP pulse. This could be anticipated from the
magnetization maps, which show that BEEEP pulses produce a much more
homogeneous signal at low peak amplitudes.