Overview of Design Algorithm
The SLR algorithm reduces the problem of pulse design into that of designing two polynomials, An
) and Bn
), which may be reversibly transformed to yield the RF pulse (42
). The polynomials An
) and Bn
) may be represented by finite impulse response (FIR) filters and therefore may be generated using standard filter design algorithms. When designing the polynomials for a RF pulse, it is sufficient to first design a Bn
) polynomial with a frequency response that is equal to the desired spectral profile and then generate a matching minimum-phase An
) polynomial, which results in a minimum-energy RF pulse. (42
One property of adiabatic pulses is that they generate a spectral profile with quadratic phase. This is because the phase of the adiabatic RF pulse itself is a quadratic function, which produces a frequency sweep that satisfies the adiabatic condition. We found that when the Bn(z) polynomial is set to the response of a linear phase FIR filter modulated with a properly chosen amount of quadratic phase, the inverse SLR transform yields an RF pulse that is remarkably adiabatic. Not all quadratic phase pulses are adiabatic. However, if quadratic phase is added such that the frequency sweep varies slowly enough to satisfy the adiabatic condition, the pulse can be made adiabatic. Characteristics of the spectral profile such as BW, transition width, and ripple in the passband and stopband are set as inputs to the FIR filter design algorithm. In our case, this algorithm is a least-square algorithm, however an equiripple Parks–McClellan algorithm could be used as well. There may be some differences in the adiabatic behavior for pulses generated using these two algorithms.
In our design algorithm, we use a and b to denote the coefficients of the (n − 1)-order polynomials, An(z) and Bn(z), in the discrete time domain.
The discrete Fourier transforms of a and b are equivalent to the profiles of the polynomials evaluated along frequency.
A linear phase version of the desired spectral profile is found using the firls function in MATLAB (The Math-works, Natick, MA). firls designs a linear phase FIR filter that minimizes the weighted, integrated squared error between an ideal piecewise linear function and the magnitude response of the filter over a set of desired frequency bands. We begin by setting the sampling rate, fs, in Hz for the filter. fs is required to calculate the physical spectral BW of the pulse, a fundamental quantity that is set before application of quadratic phase. fs determines the sampling period of the pulse, ΔT, in seconds.
Similar to standard SLR pulse design, parameters such as the ripple in the passband and stopband, normalized spectral BW and transition band width are chosen and set as inputs to the firls function. An asymmetric profile may also be specified at this time. The vector inputs to the firls function are given by
is the number of samples. F
is a vector of frequency band edges given in the range [0, π] but normalized to [0,1] with ωp
denoting the passband and stopband edge angular frequencies. The vector A
specifies the desired amplitude of the frequency response of the resultant filter and, in this case, is set for a lowpass filter. W
contains the relative ripple amplitudes in the passband and stopband given by δp
, respectively. ωp
must be chosen so that the required spectral selectivity is achieved while keeping ripple in the passband and stopband within set limits (42
The normalized spectral BW for the pulse is given by
and the fractional transition width is
The physical spectral BW in Hz,
, is given by
The FIR filter produced by the firls function is zero padded to Ntotal samples and the inverse FT is taken to yield the linear phase spectral profile, Flp(ω). Zero padding is performed to increase the sampling resolution of Flp(ω).
Quadratic phase is overlaid on the linear phase spectral profile as given in Eq. 
, to obtain the final spectral profile, F
(ω). The constant, k
, determines the number of cycles of quadratic phase that are applied across the spectral profile.
The optimal value of k depends on the transition band width of Flp. The value of k determines the amount of quadratic phase that is applied and therefore the degree by which the RF energy is uniformly distributed over the pulse.
We found empirically that a k
value that satisfies Eq. 
results in sufficient phase applied across the transition bands of the spectral profile. A k
value that is significantly below this limit results in the distortion of transition bands at low B1
values, thus reducing the B1
range over which the pulse behaves adiabatically. Equation 9
provides an approximate value for k
and it is recommended that a few design iterations during the simulation stage be performed to arrive at the optimal k
value for a chosen set of parameters.
The FIR filter given in 3 is of length N
+ 1. If no quadratic phase were applied, the resultant linear phase SLR pulse would be of duration (N
+ 1)ΔT (42
). The duration of the pulse is inversely related to the effective transition width.
The polynomial coefficients b
are given by the Fourier transform of the spectral profile:
A matching minimum-phase An
) polynomial is then calculated for the Bn
) polynomial with coefficients b
). Once An
) and Bn
) are specified, an RF waveform is found by applying the inverse SLR transform.
The pulse is truncated to the number of samples, Npulse, chosen such that
is the peak amplitude of the RF waveform. That is, the B1
amplitude at the end of the pulse should be less than 5% of the peak B1
value. Our simulations showed that truncating the pulse to a duration that is smaller than this value significantly degrades the adiabatic behavior. The pulse needs to be truncated because the final duration of the pulse, or equivalently the portion of the waveform generated by Eq. 
over which the RF energy is spread, is determined by the amount of applied quadratic phase. The final pulse duration, T
The complete method is summarized in the flow chart shown in .
FIG. 1 Flowchart illustrating main steps in the adiabatic SLR algorithm. The linear phase spectral profile is set to be the Fourier transform of a FIR filter produced by the firls function in MAT-LAB. Quadratic phase is overlaid on the linear phase profile. (more ...) Parameter Relationships
The peak B1
and time-bandwidth (TBW) product of the adiabatic SLR pulse are related to the peak B1
and TBW product of the 180° linear phase SLR pulse generated by the FIR filter, if no quadratic phase were applied. Using this algorithm, once the physical spectral BW is set, the initial values for peak B1
and TBW are those of the linear phase SLR pulse (42
). The application of quadratic phase increases the pulse duration, T
, and decreases the peak B1
value. The physical spectral BW and the selectivity of the profile remain the same. We found that the peak B1
is approximately inversely proportional to the square root of k
is approximately proportional to k
Variations on Pulse Design
Our simulations show that multiplying b
in Eq. 
with a constant scaling factor, 0 < c
≤ 1, generates an RF pulse with a slightly different shape and peak amplitude at adiabatic threshold. However, the spectral profile and B1
-insensitivity when operating above the adiabatic threshold are similar. This variant of b
is given in Eq. 
approaches 1, the RF pulse approaches the SLR design described in “Pulse Design” section. As c
is reduced below 1, the generated RF pulse becomes similar to a Fourier design. The Fourier design refers to setting the RF pulse equivalent to a scaled version of the polynomial coefficients b
in Eq. 
, without performing the SLR transform.
Above the adiabatic threshold, both the SLR and Fourier design result in B1-insensitive spectral profiles. The selectivity of the spectral profile produced by the SLR design is consistent with the parameter values used to generate fFIR. The Fourier design generates a spectral profile with marginally thinner transition bands and is therefore more selective. Below the adiabatic threshold, the spectral profile of the Fourier designed RF pulse develops winglike distortions at the edges of the passband that become more pronounced as the amplitude approaches adiabatic threshold. The ripple in the central portion of the passband is lower in the Fourier design when compared with the SLR design. The spectral profile of the SLR pulse retains its shape more faithfully when underdriven, with ripple values in the passband matching those specified for fFIR. Therefore, we have chosen to use the SLR design (i.e., set c = 1) for the pulses in this article. However, the option of using this scaled b design is available to the pulse designer.
An example adiabatic pulse was designed to validate the method outlined in “Pulse Design” section. The pulse was designed to have physical spectral BW of 9.8 kHz, pulse duration of 12 msec, and FTW of 0.21. fs was set to 119 kHz. Values used as inputs for the firls function are
We zero padded the output of the firls function to have Ntotal = 4, 000 samples. A k value of 1,150 was used and the RF waveform was truncated to 12 msec. shows plots of the amplitude and phase of the resultant polynomial coefficients b. The amplitude and phase of the Fourier transform of b, or equivalently, the profile of the Bn(z) polynomial evaluated along frequency are shown in . The magnitude, phase, and frequency waveforms for the adiabatic SLR pulse are shown in . The peak amplitude of the pulse is 17.2 μT. Frequency varies approximately linearly at a rate that satisfies the adiabatic condition for the duration of the pulse.
FIG. 2 (a) Magnitude and (b) phase of the polynomial coefficients b created by overlaying quadratic phase on the filter that is output by the firls function. (c) Magnitude and (d) phase of the Fourier transform of b. The profile in (c) is the desired spectral (more ...)
(a) Amplitude, (b) phase, and (c) frequency waveforms for adiabatic SLR pulse with optimal quadratic phase and spectral bandwidth of 9.8 kHz. Frequency varies approximately linearly at a rate that does not violate the adiabatic condition.
Simulations were performed in MATLAB by explicitly multiplying the rotation matrices to obtain the spectral profile of the pulse. In the presence of a slice-select gradient, this is equivalent to the slice profile. T1
effects were neglected. The spin-echo and inversion profile were simulated for the exemplary RF pulse shown in . The magnitude and phase of the spin-echo profile and the magnitude of the inversion profile were plotted. When used as a single pulse with crusher gradients to produce a spin-echo, the spin-echo profile, Mse
, is the spectral profile that is multiplied by the transverse magnetization. When used as an inversion pulse, the inversion profile, Minv
, is the spectral profile of the longitudinal magnetization (42
). The relationship between the two profiles is given by Minv
= 1 − 2|Mse
|. Given these definitions, the spectral BW for the two profiles are equivalent.
To study the effects of varying applied quadratic phase, RF pulses were generated using the inputs in Eq. 
values ranging from 450 to 1,150. Spectral profiles for a range of B1
overdrive factors above adiabatic threshold were simulated for each pulse and compared. The B1
overdrive factor is equal to the amplitude at which we operate the pulse divided by the amplitude of the pulse at adiabatic threshold. Therefore, a B1
overdrive factor of 1.5 would mean a 50% increase of pulse amplitude above the adiabatic threshold. Spectral profiles were simulated for B1
overdrive factors from 0.5 to 5 to show behavior of the pulse below and above adiabatic threshold.
A HS and adiabatic SLR pulse with the same physical spectral BW were generated and the peak RF amplitudes of the pulses were compared.
Phantom experiments were conducted using a standard transmit/receive birdcage head coil at 3 T (Whole-body magnet; GE Healthcare, Waukesha, WI) to validate the pulse performance. Because we were limited by the maximum output of our RF coil/amplifier combination, we used a pulse designed to have a physical spectral BW of 5 kHz and consequently reduced peak RF amplitude of 12.8 μT when compared with the 9.8 kHz pulse shown in . This way, it was possible to overdrive the pulse to check a larger range of adiabatic behavior and better illustrate the pulse performance experimentally. The passband and stop-band edge angular frequencies set as inputs to the firls function were δp = 0.03π and δs = 0.058π, respectively. The k value used was 2,000, which was chosen to be greater than the minimum k required to generate an undistorted profile to uniformly spread the RF energy over the targeted 12 msec pulse duration.
The pulse was added to a standard gradient recalled echo (GRE) sequence as an inversion pulse before the 90° excitation pulse. The amplitude and phase waveforms for the pulse are shown in the pulse sequence diagram in . A slice-selective gradient was played in conjunction with the pulse to invert a 14-mm slice. Frequency and phase encoding dimensions for the GRE sequence were both set to be orthogonal to the excited slice to provide an image of the inversion slice profile. A spherical water phantom was imaged. Several images were obtained with the inversion pulse amplitude set to the nominal B1 (at approximately the adiabatic threshold) and at 25, 50, 75, and 100% above the nominal B1 value. These were subtracted from an image obtained without the adiabatic inversion pulse. Because the objective was to measure the inversion profile for the pulse, a long repetition time (TR) of 5,500 msec was used to allow for full T1 relaxation between pulse applications so that the signal-to-noise ratio of the measurement could be maximized. Acquisition parameters were: TE/TR= 9/5,500 msec, matrix size = 256 × 128, F0V = 24 × 24 cm2 and scan time = 11 : 44 min.
FIG. 4 RF, phase, and gradient waveforms for the pulse sequence which uses the adiabatic SLR pulse for inversion. The inversion pulse is selective in the z dimension; the 90° excitation pulse is selective in x dimension; the readout gradient is applied (more ...)