A typical B-S pulse is a large tip pulse (> 1200° if it were applied on resonance); however by virtue of being sufficiently off-resonant, the axis of the effective B
field is nearly parallel to B0
, i.e. nearly parallel to the longitudinal magnetization direction. As there is minimal perturbation of the longitudinal magnetization we assume the small-tip approximation (?
), meaning that simple Fourier analysis holds within the resonant band. In order to maximize ANR for a given pulse width, we are looking for the B-S pulse which generates the maximum B-S phase shift. Maximizing B-S phase shift is achieved by maximizing B-S pulse energy and minimizing B-S frequency offset (?). A key aspect of our pulse design methodology is the use of quadratic programming to design an equiripple high-stop filter that maximizes pulse power for defined values of pulse width, peak B1;max
, stopband ripple and stopband edge, described as follows:
are the N scaled samples of the fixed-duration RF pulse, I is the identity matrix, W is a DFT matrix evaluating the stopband performance of the pulse, δ
is the stopband ripple, B1;max
is the peak
expected to be used, Δt
is the duration of each RF sample so that total pulse width T
, and γ
is the gyromagnetic ratio. The minimization problem is solved using the quadratic programming solver provided by MATLAB (The Mathworks, Natick, USA). As the problem is not strictly convex, local minima are avoided by seeding the program with a Fermi window as an initial solution (?). With the above quadratic programming tool in place, the design strategy is to find the globally optimum B-S pulse, defined as the one that generates maximum B-S phase shift subject to the constraints. We use an exhaustive search strategy, described in the following set of steps:
- Define values for T (pulse width), δ (stopband ripple threshold), B1;max
- Start of design loop: set value of stopband edge (initial value: 1 kHz)
- Using quadratic programming, design high-stop filter with stopband edge set in step 2, subject to constraints set in step 1
- Perform Bloch simulation to find true stopband edge accounting for minor inaccuracy in small-tip approximation
- Modulate the pulse off-resonance by an amount equal to true stopband edge + 600 Hz assuming on-resonance band of interest is ±600 Hz
- Perform second Bloch simulation on frequency-offset pulse to evaluate B-S shift
- Record value of B-S shift
- Cycle back to step 2, increase stopband edge by 10 Hz, and repeat steps 2–8 until complete nominal stopband edge range has been spanned (1–4 kHz)
- Search the complete nominal stopband edge range for ”globally optimal” B-S pulse, defined as the one with maximum B-S phase shift.
Optional: Repeat entire cycle (steps 1–9) for different value of T (and possibly also different δ or B1;max values)
shows a schematic diagram showing the time- and frequency-domain representations of low, mid and high value stopband edge pulses, and how they differ in terms of B-S shift. To compare different pulses with different pulse widths, we have to consider two factors. First, in a given TR, SAR will be mostly proportional to the B-S pulse energy assuming that the excitation pulse is small and its contribution to SAR can be neglected compared to the B-S pulse. Therefore, we define the intrinsic B-S pulse efficiency as B-S phase shift divided by B-S pulse energy. This efficiency metric allows us to compare B-S pulses of different pulse widths under the condition of constant SAR. The second factor is the additional echo time caused by the insertion of B-S pulse into the pulse sequence. To account for the magnitude signal loss during the B-S pulse, and therefore the loss in
map ANR, we add a multiplicative factor of
, where pw
is the B-S pulse width, into the definition of total B-S pulse efficiency. With the above definition of total B-S pulse efficiency, we ran simulations with different B-S pulse widths and
values relevant to human tissue at 7T, and sought to optimize this efficiency figure of merit. All simulations and B-S pulse design were done on a 2.66 GHz Intel Core i7 Macbook Pro laptop (Apple, Cupertino, USA) with 4 GB of RAM.
Figure 1 B-S pulse design steps. (a) Using quadratic programming, a 4 ms B-S pulse is designed with a maximum of 1% (−40 dB) on-resonance excitation over the stopband, when the stopband edge is set to 1000 Hz (top row), 1120 Hz (middle row) and 1500 Hz (more ...)
The B-S pulse was integrated into a conventional gradient echo pulse sequence (?) and is shown in . To minimize artifacts due to on-resonant excitation by the B-S pulse, crushers were added before and after B-S pulse, thus spoiling any signal excited by the B-S pulse. This was implemented by moving the de-phasing part of the readout gradient before the Bloch-Siegert pulse. In addition the slice-select re-phaser is moved after the Bloch-Siegert pulse for the same purpose. The amplitude of B-S pulse was also chopped, i.e. inverted on each new phase encode (producing a B-S phase offset alternation of π radians with respect to excitation) so that any artifact would appear as an N/2 ghost.
(a) B-S pulse sequence using 4 ms off-resonance Quad pulse. (b) B-S phase dependence on B1;max (B0 = 0 Hz), and (c) B-S phase dependence on B0 (B1;max = 20 μT) for both 4 ms Quad (solid line) & 6 ms Fermi (dashed line) pulses.
Using this modified gradient echo sequence, a 4 ms optimized B-S pulse with frequency offset of 1.96 kHz and a 6 ms Fermi pulse with frequency offset of 4 kHz were compared for
mapping on a 7T scanner (GE Healthcare, Waukesha, WI). The 4 kHz offset 6 ms Fermi pulse was chosen as a reference since it was used in the original description of the B-S
mapping method (?). Phantom studies, using a 1% copper sulfate head-neck phantom, were acquired using the following parameters: TR=50 ms, flip angle=40 deg, FOV=24 cm, thickness = 5 mm and 64 × 64 matrix. The minimum TE was 7.1 ms for the optimized pulse and 9.1 ms for the Fermi pulse. The
mapping sequence was repeated 20 times for each pulse from which dataset mean
and angle-to-noise ratio maps were computed with noise defined as the standard deviation computed across the 20 repeats. The nominally symmetric off-resonant pulses were then both shifted up or down by 600 Hz to simulate B0
off-resonance effects. The two Bloch-Siegert pulses were also compared on a volunteer with the same sequence with the exception of a longer TR (TR=150 ms). The volunteer was scanned in accordance with institutional review board guidelines for in vivo research, and provided informed consent.