Synopsis

Root-flipped multiband pulses have peculiar spin-echo behaviour due to their non-linear phase profile. In spin-echo sequences, different slices will typically have different relaxation weightings. This work investigates the typical time-delays of the spin-echoes such pulses, and proposes a novel root-flipping method to minimize differences in relaxation weighting.

Purpose

Root-flipping is a recently proposed method for designing multiband band refocusing pulses (1). It produces shorter pulses than the phase-optimized (2) and time-shifted (3) methods because RF energy is more evenly distributed across the duration of the pulse. Consequently, the spin-echo formation in different slices develops differently. The two quantities of interest are the echo arrival time ($$$T_{arr}$$$) for each slice, and the time that magnetization in each slice spends in the transverse plane ($$$T_{xy}$$$) (see Figure 1). For a conventional sequence these terms are equal to the echo time, but this is not the case for root-flipped designs. Here we use $$$\Delta$$$$$$T_{xy}$$$ and $$$\Delta$$$$$$T_{arr}$$$ to describe the largest spread in these parameters across the multiple slices. Previous work (3,1) has identified three terms to classify this effect:

1) Aligned-echo excitation: Matching the bandwidth of the excitation pulse to that of the refocusing pulse. Echoes arrive simultaneously but with different $$$T_{2}$$$-weighting. At readout, $$$\Delta$$$$$$T_{arr}=0$$$ and $$$\Delta$$$$$$T_{xy}\neq0$$$.

2) Align-TE excitation: Matching the duration of the excitation pulse to that of the refocusing pulse. Echoes arrive at different times with equal $$$T_2$$$-weighting. At readout, $$$\Delta$$$$$$T_{arr}\neq 0$$$ and $$$\Delta$$$$$$T_{xy}=0$$$

3) Minimum-duration excitation: Matching the peak-amplitude of the excitation pulse to that of the refocusing pulse. Echoes arrive at different times with different $$$T_2$$$-weighting. At readout, $$$\Delta$$$$$$T_{arr}\neq0$$$ and $$$\Delta$$$$$$T_{xy}\neq0$$$.

In this work, we propose a novel root-flipping algorithm that aligns the echoes such that $$$\Delta$$$$$$T_{arr}\approx0$$$ and $$$\Delta$$$$$$T_{xy}\approx0$$$.

Theory

In the root-flipping method the multi-slice frequency profile is represented as roots along the unit circle on the complex plane. The stopband frequencies have roots on the unit circle and the passband frequencies have roots away from this. Flipping roots inside or outside the unit circle redistributes the pulse energy in the time-domain. In the original method from Sharma et al. (1), the root-pattern across each passband was unconstrained. Having an unequal number of roots inside and outside the unit circle across a passband leads to temporal displacement, which leads to a difference in spin-echo development for different slices.

Ensuring an equal number of roots inside and outside the unit circle for each passband restricts temporal displacement, which results in $$$\Delta$$$$$$T_{arr}\approx0$$$ regardless of the excitation method. This arrangement approximates the root pattern from a linear-phase design typically found with phase-optimized pulses (2). Using this in conjunction with the “align-TE” excitation method also gives the desired property, $$$\Delta$$$$$$T_{xy}\approx0$$$. We refer to this as the “align-all” approach.

Methods

The proposed root-flip pattern requires pulses with an even number of passband roots. This is related to the time-bandwidth product (TBP) of the target design. For example, TBP=4 gives two roots per passband; all pulses in this work were designed this way. The root-flipping method searches through root-flipping patterns to obtain the solution with the lowest RF peak-amplitude. The Genetic Algorithm in Matlab was used to perform this search. The original flipping pattern was used (code made available by Sharma et al. http://www.vuiis.vanderbilt.edu/~grissowa/) alongside a modified version which selects an even number of roots per passband closest to the passband-center and flips them evenly across the unit circle (Figure 2).

The signal evolution for transverse magnetization in the presence of dephasing was simulated using Bloch simulations with a secondary frequency axis. The predicted signal was estimated by Lorentzian weighted-averaging along this axis for $$$T_{2}^{*}$$$=45ms and $$$T_{2}$$$=80ms, which correspond to human white matter values at 3T (4). The nominal echo time of this sequence was 90ms.

Results

Figure 3 shows a multiband-5 example spin-echo simulation. As expected, either $$$T_{arr}$$$ or $$$T_{xy}$$$ differs through the group of five simultaneously excited slices for the existing methods. The proposed approach results in nearly complete alignment of the echoes. Figure 4 shows that the result is stable across a range of designs. Figure 5 shows that on average the align-all root-flipped pulses are 19.6% longer in duration than those from the original method. This is expected, since they are more constrained designs.

Discussion and Conclusion

Acknowledgements

References

1. Sharma A, Lustig M, Grissom WA. Root-flipped multiband refocusing pulses. Magn. Reson. Med. 2016;75:227–237. doi: 10.1002/mrm.25629.

2. Wong E. Optimized phase schedules for minimizing peak RF power in simultaneous multi-slice RF excitation pulses. Proc. Intl. Soc. Mag. Reson. Med. 20 2012;20:2209.

3. Auerbach EJ, Xu J, Yacoub E, Moeller S, Ugurbil K. Multiband accelerated spin-echo echo planar imaging with reduced peak RF power using time-shifted RF pulses. Magn. Reson. Med. 2013;69:1261–7. doi: 10.1002/mrm.24719.

4. Wansapura JP, Holland SK, Dunn RS, Ball WS. NMR relaxation times in the human brain at 3.0 Tesla. J. Magn. Reson. Imaging 1999;9:531–538. doi: 10.1002/(SICI)1522-2586(199904)9:4<531::AID-JMRI4>3.0.CO;2-L.