Edward M Green^{1,2}, Yasmin Blunck^{1,2}, James C Korte^{3}, Bahman Tahayori^{4,5}, Peter M Farrell^{6}, and Leigh A Johnston^{1,2}

Inhomogeneous B1 excitation impedes image quality, particularly at high field. Adiabatic pulse modulation ameliorates this effect, however super-adiabatic properties can be exploited to further improve performance. Spin Lock Adiabatic Correction (SLAC) pulses can be applied to any adiabatic pulse shape, through reduction of flip angle inaccuracies induced by B1 variability. In this work, SLAC is derived for BIR4 pulse shapes, and the superior performance of SLAC-BIR4 is demonstrated in both simulation and phantom experiments at 7T. The SLAC procedure is an attractive analytical alternative to numerical optimisation of adiabatic pulses.

Consider an arbitrary adiabatic pulse defined by amplitude and phase functions, $$$B_{1,Ad}(t)$$$ and $$$\phi(t)$$$, respectively, with effective field, $$$\tilde{B}_{eff}$$$ and flip angle, $$$\alpha$$$ (Fig.1a). A second rotating frame aligned with $$$\tilde{B}_{eff}$$$ (Fig.1b) defines a cone with axis $$$\tilde{E} = \tilde{B}_{eff} + B_\perp$$$, where $$$B_\perp = \frac{d\alpha}{dt}$$$, and aperture, $$$\epsilon$$$. Cone aperture relates to flip angle accuracy.

The SLAC procedure reduces the cone aperture through introduction of an opposing spin lock component. SLAC is defined by the complex envelope function,

$$B_1(t) = \left(B_{1,Ad}(t) + i B_{1,SL}(t)\right)e^{i\phi (t)},$$

where $$$B_{1,SL}(t) = - \frac{1}{\gamma} \frac{d\alpha}{dt}$$$.

The effect of $$$B_{1,SL}$$$ is to null $$$B_\perp$$$. This occurs at the expense of increased $$$B_1$$$
amplitude. As shown below, the power can be matched to the adiabatic pulse power, however a small residual $$$B_\perp$$$ will remain unsupressed.

To demonstrate that
SLAC is applicable to arbitrary flip angles, it has been applied to a
45$$$^\circ$$$ 10ms tanh/tan BIR4 pulse with $$$\zeta=20$$$, $$$\kappa=\tan^{-1}(20)$$$, $$$\Delta\omega_{max}=$$$ 5kHz^{7}. As the power deposited by the analytically derived SLAC envelope is greater than for the BIR4, we implemented a SLAC envelope rescaled equal to BIR4 in terms of power deposition (Fig.2).

Simulation:

The Bloch equations were numerically simulated (MATLAB) to predict the transverse magnetisation following 45$$$^\circ$$$ BIR4 and SLAC-BIR4 pulses as a function of pulse duration and reference $$$\omega_1$$$ amplitude, defined as BIR4 $$$\omega_{1,max}$$$.

Experiment:

Data was acquired of a spherical phantom at 7T (Siemens Healthineers, Erlangen, Germany) using a single-channel TX-RX volume coil (QED, USA). Non-selective excitation pulses were followed by a custom-built 3D Cartesian GRE readout (TR=5000ms, TE=10ms, Bandwidth = 260Hz/px, Matrix size=256x128x64, FOV=200mm isotropic). An image volume was obtained for each of the following pulses: 45$$$^\circ$$$ flip angle, lengths of 0.5ms (block) and 10ms (BIR4 and SLAC-BIR4).

Simulation:

The comparative performance of the three pulses types, BIR4 (Fig.3a), power-matched SLAC-BIR4 (Fig.3b) and unmatched SLAC-BIR4 (Fig.3c), demonstrates the expected effects of longer and higher power pulses. More interesting, however, is that for most pulse durations and reference $$$\omega_1$$$ amplitudes, the power-matched SLAC-BIR4 outperforms the standard BIR4 (Fig.3d). The high power, low duration region in which BIR4 is better is a region in which a block pulse will outperform adiabatic excitation. For completeness, as expected, the mean transverse magnetisation following an unmatched SLAC-BIR4 exceeds the BIR4 everywhere (Fig.3e).

The resultant transverse magnetisation across $$$B_1$$$ strengths for 10ms, 45$$$^\circ$$$ BIR4 and SLAC-BIR4 pulses is depicted in Fig.4. The power-matched SLAC-BIR4 increases the signal for low $$$B_1$$$ regions compared with a standard BIR4, while all perform equally (except the block pulse) for higher $$$B_1$$$ powers.

Experiment:

Block pulse excitation (Fig.5a) shows higher intensity in the high $$$B_1$$$ regions and lower intensity in the low $$$B_1$$$ regions than BIR4 (Fig.5b) and SLAC-BIR4 (Fig.5c). Areas of low $$$B_1$$$ in the SLAC-BIR4 image contain on the order of 10% more signal than the BIR4 image (Fig.5d), while in areas of high $$$B_1$$$ strength they are equivalent. Both of these observations agree well with simulation predictions, that SLAC reduces the sensitivity to B1 inhomogeneity without a concomitant increase in power deposition.

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4. Ugurbil K, Garwood M, Bendall MR. Amplitude- and frequency-modulated pulses to achieve 90° plane rotations with inhomogeneous B1 fields. JMR 1987;72:177-85.

5. Ugurbil K, Garwood M, Rath AR, Bendall MR. Amplitude- and frequency/phase-modulated refocusing pulses that induce plane rotations even in the presence of inhomogeneous B1 fields. JMR 1988 Jul 1;78(3):472-97.

6. Green EM, Korte JC, Tahayori B, Farrell PM, Johnston LA. Spin Lock Adiabatic Correction (SLAC) Excitation. In Proceedings of the 26th Annual Meeting of the ISMRM, Paris, France. 2018.

7. Garwood M, Ke Y. Symmetric pulses to induce arbitrary flip angles with compensation for RF inhomogeneity and resonance offsets. JMR 1991;94(3):511-525.

Figure 1. The
effective field, $$$\tilde{B}_{eff}$$$, and its components in rotating frames of
reference. (a) The $$$\tilde{B}_{eff}$$$ frame $$$x^{\prime\prime}$$$, $$$y^{\prime\prime}$$$, $$$z^{\prime\prime}$$$ is related to the frequency
modulated frame $$$x^\prime$$$, $$$y^\prime$$$, $$$z^\prime$$$, by $$$\alpha$$$, the angle that $$$\tilde{B}_{eff}$$$ makes with
the $$$z^\prime$$$ axis of the FM frame. (b) Effective field components during a BIR4 pulse. (c)
Effective field components during a SLAC-BIR4
pulse, demonstrating the additional Spin Lock component, $$$B_{1,SL}$$$, that opposes and acts to reduce the conical aperture.

Figure 2. Comparison of pulse envelope shapes. (a) $$$B_1$$$ magnitude and (b) phase for BIR4 (black), power-matched SLAC-BIR4 (blue) and unmatched SLAC-BIR4 (red, directly underneath blue) pulses. (c) $$$\epsilon$$$ over the course of BIR4 (black), power-matched SLAC-BIR4 (blue) and unmatched SLAC-BIR4 (red) pulses.

Figure 3. Performance
heatmaps: Mean transverse magnetisation across variation in reference $$$\omega_1$$$ and pulse duration, for isochromats in an inhomogeneous $$$B_1$$$ field (0.5-1.5 x
reference $$$B_1$$$ strength) following an on-resonance (a) BIR4 pulse, (b) power-matched SLAC-BIR4
pulse, and (c) unmatched SLAC-BIR4 pulse. Difference heatmaps: (d) power-matched SLAC-BIR4 –
BIR4 performance heatmap difference, (e) unmatched SLAC-BIR4 – BIR4
performance heatmap.

Figure 4. Simulated
transverse magnetisation vs. $$$B_1$$$ strength following an on-resonance 10ms 45$$$^\circ$$$ BIR4 pulse with $$$\omega_{1,max}$$$= 200Hz for $$$B_{1,ref}$$$ (black), 10ms 45$$$^\circ$$$ power-matched SLAC-BIR4 (blue), 10ms 45$$$^\circ$$$ unmatched SLAC-BIR4 (red), a 45$$$^\circ$$$ 0.5ms block pulse (purple). $$$\sin\left(45^\circ\right)$$$ (orange dashed line).

Figure 5. Spherical spectroscopy phantom. (a) Image acquired using 0.5ms 45$$$^\circ$$$
block pulse, (b) 10ms 45$$$^\circ$$$ BIR4, (c) 10ms 45$$$^\circ$$$ power-matched SLAC-BIR4.
(d) Ratio image: SLAC-BIR4 image/BIR4 image. (e) $$$B_1$$$ map of spherical phantom.