Nonlinear phase RF pulses have been proposed and commonly used for broadband saturation and inversion with reduced peak B₁ amplitude, including quadratic-phase pulse ^1,2 and root-flip optimized pulse ³. However, practical comparison between them is not well known. In this work, the two classes of pulses were studied in terms of spatial selectivity, peak B₁ requirement, pulse energy, and B₁ sensitivity, when used for outer volume suppression (OVS) applications.

To achieve high spatial selectivity for OVS a high time-bandwidth-product (TBW) is necessary, typically by adjusting bandwidth and fixing pulse duration. A broadband pulse can also reduce B₀ sensitivity and displacement from chemical shifts. Nonlinear phase pulses distribute RF energy more evenly over the pulse duration, which helps keep peak B₁ within system limits.

Both RF pulses are designed based on the Shinnar-Le Roux (SLR) algorithm ⁴, which addresses the nonlinearity of the Bloch equation at higher flip angles and achieves good agreement with target pulse profile, by reversibly transforming the RF pulse design to the well-established finite impulse response (FIR) filter design. The FIR filters are designed and optimized within the convex optimization framework ⁵.

(A) Quadratic-phase pulse design. The quadratic phase Beta polynomial is designed by specifying a quadratic phase in the desired frequency response, and solved as a second order cone programming. Different from the equal-ripple designs in ², an equal-ripple-passband and decaying-ripple-stopband pulse is designed to reduce perturbation into the imaging volume. The amount of quadratic phase is set with a few design iterations.

(B) Root-flip pulse design. A minimum phase Beta polynomial that has the minimal transition bandwidth by bisection search ⁵ was first designed, then its passband roots are individually flipped with respect to the unit circle, to find a configuration with minimum peak B₁ amplitude through all possible combinations. The monte-carlo flip pattern and advanced algorithm for calculating roots introduced in ⁶ was used to reduce numerical error.

One example of design of the quadratic-phase pulse and the root-flip pulse was shown in Fig. 1, with the same pulse duration of 4ms, number of samples 400 (2x upsampling), passband/stopband ripple 0.006/10^-4, and transition width 6.25mm. Bloch simulation of selection profile was shown in Fig. 2, notice the quadratic-phase pulse does have a quadratic Mxy phase profile. Though optimized for peak B₁, the root-flip pulse also shows an approximate quadratic Mxy phase profile. The pulse performance is summarized in the table in Fig. 2.

Simulated saturation profile with B₁ deviation was shown in Fig. 3. Saturation profile measured on a whole body 1.5T clinical scanner (AllTech EchoStar, AllTech Medical Systems, Chengdu, China) with a sphere water phantom is depicted in Fig. 4. Volunteer L-spine image examples with the quadratic-phase OVS pulses are in Fig. 5 to demonstrate the practical effectiveness.

Both root-flip and quadratic-phase pulses can achieve high spatial selectivity while keeping peak B₁ within typical system limits in practical applications. Root-flip pulses spread energy more evenly over time, thus has smaller peak B₁ demands compared to quadratic-phase pulse, especially at high TBW. Root-flip pulses also have sharper transitions, because only the magnitude frequency response target is considered during the FIR filter design. Thus, root-flip pulses tend to have lower peak B₁ and power given the same transition, or sharper transition given the same peak B₁.

With B₁ deviations, the actual flip angles change for both pulses because they are not adiabatic. Nevertheless, we found the quadratic-phase pulses maintain the target bandwidth and profile shape much better than the root-flip pulses, as the selection profiles of the latter are rapidly distorted, especially over passband (saturation band), as demonstrated in simulation and imaging experiments. This is because quadratic-phase pulse with quadratic phase modulation in the time domain, as shown in Fig. 1, behaves similar as an adiabatic pulse with offset-independent adiabaticity, but scaled down to the non-adiabatic regime ⁷. Adiabaticity is not a design target in this work, as B₁ insensitivity can also be achieved by a train of OVS pulse, which contributes to T₁ insensitivity as well ⁸. Retaining similar flip angles over the saturation band with B₁ variations makes quadratic-phase pulse more robust when repeated multiple times in OVS. However, a non-uniform saturation profile with B₁ deviations may be less problematic for some other applications, like nonlinear phase refocusing pulse, where signal is averaged over the slice ⁹.

This comparison provides practical insights to pulse designers in choosing one of the two classes of pulses in nonlinear phase OVS pulse design.