Spectral-spatial radiofrequency (RF) pulses are selective in both frequency and space [10
]. This dual selectivity is accomplished by using an oscillating magnetic field gradient, with a shaped RF sub-pulse transmitted during each half-cycle of the gradient. As an approximation, the spatial profile is determined by the shape of each sub-pulse, and the spectral profile is determined by the envelope of the train of sub-pulses. Since the the envelope is discretized (by the sub-pulses), periodic replicas of the main spectral profile appear in frequency space.
For exciting a single resonance of the 13
C spectrum, the challenge is achieving a stopband of adequate width and attenuation between the central passband of the pulse and these distorted replicas in frequency space. The replicas are distorted because the trajectory through excitation k-space [11
] is a “zig-zag” pattern, as shown in . In addition to the imperfect replicas, the zig-zag trajectory also results in bipolar “ghosts” spaced every 1/(2τ
) from the main passband, where τ
is the period of the gradient waveform. These bipolar lobes are the result of imperfect cancellation between replicas from the positive and negative parts of the gradient waveform.
Figure 1 The zig-zag trajectory through excitation k-space. (a) A gradient waveform used with a typical spectral-spatial RF pulse. (b) The integral of the gradient shape gives the trajectory through excitation k-space. The “x” marks show that the (more ...)
To widen the stopband, the gradient oscillation frequency can be increased, which comes at the expense of a larger minimum slab width due to the decreased area under each gradient lobe. Also, one of the bipolar ghosts can be suppressed by using oscillating polarity in the RF sub-lobes, as introduced in [12
] and further explained in [13
]. However, for the high degree of attenuation needed to suppress the signal from the injected hyperpolarized compound, enabling imaging of a single metabolic product, a more advanced strategy that takes into account the zig-zag trajectory through excitation k-space must be used.
The zig-zag k-space trajectory used with a typical spectral-spatial pulse is shown in . The “x” marks in indicate that the temporal sample points are evenly distributed along the central line of the spatial k-space pattern, but are “staggered” away from the center. This staggered sampling pattern is the reason for the ghosts at 1/(2τ) from the main passband. The ghosts get worse further from the center of the frequency axis because the artifact caused by staggered sampling is worse for higher frequency signals. The reason that the technique described in the previous paragraph works is that staggered sampling has no effect on constant-amplitude (DC) signals, so adequate suppression can be achieved around 0 Hz despite the zig-zag pattern.
To further improve the attenuation in the stopband, the weighting function in excitation k-space can be designed for the specific zig-zag pattern traced out by the oscillating gradient waveform. This weighting function, defined in [11
] as B1
), is commonly used in the literature on multidimensional RF pulses and should not be confused with the weighting matrix W below. In conventional spectral-spatial RF pulse designs, the two-dimensional weighting function in excitation k-space is the outer product of the Fourier transforms of the desired spectral and spatial profiles [12
]. Thus, the shape of the weighting function along any line across the time dimension (i.e. the “x” marks in ) is the same. However, in this work, a different weighting function was used for each kz
position to compensate for the difference in sample spacing illustrated in .
To solve for r
, a vector containing the energy deposited by the RF pulse at each point along one of the non-uniformly sampled lines, we begin with discrete Fourier transform in matrix notation:
are column vectors and F
is a Fourier matrix with the elements:
is the number of points in s
. To account for the staggered sampling along each of the off-centre lines in , a Fourier matrix modified to include the sample delays along the particular line, 0 ≤ Δ ≤ 1, is used:
For example, the (−1)k
Δ values along the upper, staggered samples in might be [+0.25, −0.25, +0.25, −0.25…]. A standard weighted least squares formulation can then be applied to solve for the non-uniformly spaced r
values with a Fourier transform that most closely matches the desired frequency response H
W is a diagonal matrix of weights that specifies the important part of the spectrum to fit to. For this work, W was 1 for the central half of the matrix covering the stopband, and 0.01 elsewhere. H is a column vector containing the desired frequency response.
With only a single line of the spectrum excited using a spectral-spatial RF pulse, efficient imaging techniques with readout trajectories such as echo-planar methods can be used. This is the strategy taken in the experiments described below.