The PILS algorithm is attractive because it does not require measurements of the coil sensitivity pattern, calibration of the coils, or complicated reconstructions. The PILS algorithm also offers better noise properties than SENSE and SMASH (7), with less artifacts than SMASH in many cases. Another strength of the PILS algorithm is that it enables truly parallel imaging through an acquisition and reconstruction scheme that is completely independent among different coils. Multiple coils can be placed at locations that are of interest and then images can be separately obtained in parallel. Since images from each coil can be reconstructed independently from images from other coils, the technique can also be applied to cases in which multiple coils are located at distant locations. Such cases would include, for example, simultaneous imaging of both breasts or both knees. The PILS method has the added advantage that unlike other parallel imaging techniques, such as SENSE and SMASH, the reconstruction remains computationally simple even when arbitrary k-space trajectories are used.
However, one concern with applying the PILS method is the fact that anti-aliasing filters are applied before the sampling stage. This poses a problem when the method is applied in a general sense. Here we define the problem and provide a solution.
During a scan it is often the case that the region of interest (ROI) is not at the center of the magnet (the so-called isocenter). In such cases, the FOV is assigned to be off-center. The difference between having the object of interest at the center of the magnet and having the same object off-center is a simple modulation. The signal received from an object off-center is a modulation of the signal that would be received from the same object if it were located at the center of the magnet. This is illustrated in Fig. 1. If the Fourier transform (FT) of an object m(x, y) is M(kx, ky) (Fig. 1a), the FT of the same object that is shifted off-center, m(x−xo, y−yo), is M(kx,ky)e (Fig. 1b).
Figure 1. Off-center imaging and data modulation. a: FT of an object m(x, y) located at isocenter is M(kx, ky). b: FT of the same object shifted, m(x−xo, y−yo) is M(kx,ky)e, a simple phase modulation of the FT of the object at the isocenter. Note that the k-space data are displayed as the real part to show the modulation effect.
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The implication of this relationship in multicoil imaging is very significant. When multiple coils are used to cover different ROIS at the same time, it is likely that the regions covered by each coil will be off-center. This means that signals from multiple coils will be modulated versions of the signals that would have been received if the ROIs were located at isocenter. As long as the data from each coil are demodulated properly, this is as if multiple images are acquired in parallel at the same time. This method was proposed in the original PILS paper (6) in a slightly different context.
In any given situation where analog signals are sampled and digitized, anti-aliasing filters are used. The main purpose of the anti-aliasing filter is obviously to make sure no aliasing (particularly from noise) occurs. The anti-aliasing is performed by filtering out any signal that is outside of the bandwidth supported by the real-time k-space sampling rate. The anti-aliasing filters are normally analog low-pass filters that suppress anything outside of the bandwidth that is supported by the sampling rate, assuming that the frequency content of the signal is centered around zero. Since the anti-aliasing filters are spatial filters in MRI, assuming that the signal frequency is around zero is equivalent to assuming that the object is at isocenter.
In the case of PILS imaging, the use of this anti-aliasing filter with a single demodulation hardware for all receive channels is problematic. Since the FOVs are located off-center with coils centered around each FOV location, unless the signals from all coils are demodulated separately so that their frequency range corresponds to being at isocenter, the anti-aliasing filter will end up filtering out signals of interest.
Figure 2 illustrates the problem with off-center imaging in the presence of anti-alias filtering for the case of 2DFT imaging. This figure demonstrates a case in which the readout is in the kx direction. Since the readout and anti-alias filtering are along a straight line in the kx direction, the filter supports a rectangular region in the x direction. When the FOV of interest is at isocenter (Fig. 2a), no problem exists. When the FOV of interest is off-center in the x direction, a problem arises because the spectral content of the signal of interest lies outside the bandwidth of the filter (Fig. 2b). Therefore, the signal must be demodulated prior to sampling so that its spectral content lies within the bandwidth of the anti-aliasing filter (Fig. 2c).
Figure 2. Anti-alias filtering and dynamic demodulation for 2DFT trajectories. a: When the object is located at isocenter, the anti-aliasing filter does not have any effect on the signal. b: If the object is located outside of the band region supported by the anti-aliasing filter, the signal of interest will be filtered out. c: Therefore, the signal received from the off-center object must be demodulated before it is anti-alias filtered and sampled.
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With time-varying gradients, such as in the case of spiral imaging, the readout signal bandwidth varies with the readout trajectory. Depending on the readout trajectory, signal bandwidth can be increased beyond the bandwidth of the signal in the Cartesian domain. However, this increase is generally difficult to quantify. Due to such an increase in bandwidth, with a k-space sampling rate that corresponds to the bandwidth in the Cartesian domain, the anti-aliasing filter will distort the signal. Therefore, sampling signals from such trajectories is not a trivial problem. Nevertheless, spiral imaging is often done assuming that such an effect is negligible. The analysis of the spiral imaging presented below is based on such assumptions.
For spiral imaging, the FOV supported by the sampling is circular. Since the anti-aliasing filter is applied in the readout direction along a spiral shaped trajectory, the anti-aliasing filter also supports a circular FOV. Similarly to the 2DFT imaging case, when the object is at isocenter (Fig. 3a), no problem exists. However, with the object outside the circular region supported by the anti-aliasing filter, the signal is distorted (Fig. 3b). This problem can be avoided by demodulating the signal before anti-alias filtering so that the signal frequency content is within the circular region supported by the bandwidth of the anti-aliasing filter (Fig. 3c).
Figure 3. Anti-alias filtering and dynamic demodulation for spiral trajectories. a: When the object is located at isocenter, the anti-aliasing filter does not have any effect on the signal. b: If the object is located off-center, the anti-aliasing filter will smear out the signal of interest. c: Therefore, the signal received from the off-center object must be demodulated before it is anti-alias filtered and sampled.
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The anti-alias filtering effects for spiral trajectories (8) are shown in Fig. 4. The anti-aliasing filter gives rise to a space-variant response. Therefore, to appreciate the filter response, the impulse response is plotted for several different locations. Since the filtering effects are radially symmetric for spiral trajectories, four different impulses along the x-axis were used for the demonstration. The input impulse location is shown in Fig. 4a. The FOV region is marked with the circle.
Figure 4. Impulse response of anti-aliasing filters for spiral trajectories. a: To characterize the shift variant response of a spiral trajectory's anti-aliasing filter, the impulse response is plotted for impulses at four different locations. The four dots along the x-axis show the four impulse locations. b: The cross sections of the four impulse responses along the x-axis are shown. The amplitude decrease as the impulse moves away from the FOV region is clearly demonstrated. c: The cross section along the y-axis at each impulse location is overlaid. The amplitude decrease and the widening of the impulse are observed as the impulse moves away from the FOV region.
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For the simulation of the anti-aliasing filter effects, a 40×40 cm2 FOV and 1×1 mm2 resolution spiral trajectory were used. The spiral trajectory chosen for the experiment consisted of 32 interleaves and 3968 readout points (15.9 ms readout time). The samples were taken from an analytical FT of the impulses. To simulate the anti-alias filtering effect, a digital filter of length 101 and cutoff frequency 0.25 π rad (bandwidth 0.5 π rad) was applied to the data. This emulates an anti-aliasing filter that corresponds to supporting a 10 cm FOV about the isocenter. Then the data were undersampled by a factor of 4 in both the readout and interleave directions, resulting in an eight-interleave, 992 readout point spiral trajectory. This spiral trajectory supports a 10×10 cm2 FOV and 1×1 mm2 resolution.
Figure 4b shows the cross section of the four impulse responses along the x-axis. It is clear that the response of impulses outside of the FOV region is attenuated due to the filtering. Such attenuation is also observed in Fig. 4c, where the cross section of the four impulse responses along the y-axis are overlapped. In the cross section along the y-axis, widening of the impulse response is also observed as the impulse moves away from the FOV. This is due to the decrease in curvature of the spiral trajectory in the outer portions of k-space. Because the impulses were shifted along the x-axis, the signal variation in k-space occurs along the kx direction. The decrease in curvature of spiral trajectories in the outer part of k-space results in more filtering in the direction of the variation (kx) in the outer part of k-space in the ky direction. This results in a weighting in the ky direction that ends up being a convolution in the y direction. Since the weighting becomes narrower as the impulse moves further in the x direction, the impulse response in the y direction correspondingly becomes wider.
In addition to readout functions, gradient performance in terms of strength and speed are driven by the need for short, strong encoding pulses (e.g., for diffusion imaging) and shorter-duration, slice-selective RF pulses at any given slice thickness. The use of ever more powerful amplifiers is one way to increase performance. However, such performance boosts are not in lock step with the A/D conversion rate. Therefore, if the strongest possible gradient is used for readout, the receiver anti-aliasing filter bandwidth supports a small FOV at the isocenter. For example, with a 40 mT/m maximum gradient strength and 4 μs minimum sample interval digital receiver, the supported FOV is only 14.68 cm. Therefore, when the FOV of interest is greater than 14.68 cm, the maximum gradient strength cannot be utilized for readout. Using smaller readout gradients leads to longer readout times, which result in longer minimum TRs and greater off-resonance artifacts.
When the maximum gradient strength jumps ahead of the A/D conversion rate in a scanner that has multiple receiver channels (N), multiple demodulators allow N units of bandwidth to be arbitrarily located in the image domain. These bandwidths can be spliced together into one bandwidth piece that can be the same size as the bandwidth of each piece, N times bigger, or anything in between. This is because the center of the anti-alias filter-supported FOV region can be moved around by means of dynamic analog demodulation. In other words, the demodulation lets one shift the “stuck at isocenter filter bandwidth” to anywhere one desires. By having multiple such demodulators, the center location of the bandwidth can be customized to each coil's center location, which minimizes the need for higher A/D conversion rate.
Therefore, for PILS, centering units of bandwidth on each of the separate coils is a natural approach. By demodulating the signals from each coil separately based on the location of the coil, the k-space sampling rate in the readout direction will not be limited by the FOV size measured from the isocenter. Compared to the case in which PILS imaging is done without such multiple-demodulation hardware (8), readout time can be significantly reduced. This can also lead to a shorter scan time when the scan time is limited by the minimum TR. For k-space trajectories, such as in EPI or spiral imaging, where the readout length is limited by the off-resonance artifacts, for a given readout length the number of excitations can be decreased with new trajectories that can utilize faster gradient speeds.
A block diagram of the proposed multiple-demodulation hardware configuration is shown in Fig. 5. Dynamic demodulation is applied to each coil separately depending on each coil's location, before the anti-alias filtering is applied. For the signal from a coil with the sensitivity region centered at (xi, yi, zi), the demodulation factor is e. Note that the demodulation factor is dependent on the center location of the coil as well as the k-space sampling locations, which can be time-varying within a single readout.
Figure 5. Multiple dynamic demodulation hardware configuration. Dynamic demodulation must be done separately for each coil depending on each coil's location before the anti-aliasing filter is applied. f0 is the Larmor frequency and (xi, yi, zi) (i=1, …, n) indicate the center location of each coil's sensitivity region. Note that the demodulation factor is dependent on the k-space trajectory, which can also be time-varying.
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