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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Recently, gradient performance and fidelity has become of increasing interest, as the fidelity of the magnetic resonance (MR) image is somewhat dependent on the fidelity of the gradient system. In particular, for high fidelity non-Cartesian imaging, due to non-fidelity of the gradient system, it becomes necessary to know the actual k-space trajectory as opposed to the requested trajectory. In this work we show that, by considering the gradient system as a linear time-invariant system, the gradient impulse response function (GIRF) can be reliably measured to a relatively high degree of accuracy with a simple setup, using a small phantom and a series of simple experiments. It is shown experimentally that the resulting GIRF is able to predict actual gradient performance with a high degree of accuracy. The method captures not only the frequency response but also gradient timing errors and artifacts due to mechanical vibrations of the gradient system. Some discussion is provided comparing the method presented here with other analogous methods, along with limitations of these methods.

In modern magnetic resonance imaging (MRI) instruments, the fidelity of the MR image is dependent upon the fidelity of the gradient system, such that the actual gradient outputs closely match the requested outputs. For conventional imaging, in which k-space points on a Cartesian grid are acquired, the image fidelity is somewhat immune to small gradient discrepancies, as long as the gradient areas are largely preserved. On the other hand, more efficient MRI data collection schemes that do not acquire the data on a Cartesian grid, such as spiral imaging, are sensitive to even small discrepancies in the gradient waveforms, which can give rise to significant blurring in the resulting images [

As there has been increased interest in recent years in more efficient data collection efficiencies for MRI, in which the data are not collected on a Cartesian grid, there has been a corresponding increased interest in methods for measuring the actual gradient output, with the purpose of providing corrections to improve the fidelity of the resulting image. For the most part, the gradient outputs are monitored by measuring the k-space locations actually acquired, as opposed to the k-space locations that would have been acquired by gradients of perfect fidelity.

There are multiple sources giving rise to gradient infidelity, including: non-linear gradient amplifier amplification and limited gradient amplifier frequency response, incomplete eddy current compensation, including gradient cross terms, the existence of a non-linear gradient field itself, and so-called concomitant gradients, which are higher order spatially varying magnetic fields which necessarily accompany the desired linear gradient fields [

For the most part, early approaches for measuring the gradient fields in MRI instruments (either to adjust eddy currents, or to measure k-space trajectories) involved measurements with small phantoms [

Instead of measuring the actual k-space trajectory, it is possible to consider the gradient system as a linear, time invariant system and measure the gradient impulse response function (GIRF). As discussed in detail in the Theoretical Background section, the actual gradient waveform is then predicted by applying the GIRF to the requested gradient waveform. The idea of considering the gradient system as a linear, time invariant system and obtaining a measurement of the GIRF to predict gradient response is not new, and has been presented in some detail by, for example, Alley

To alleviate the difficult susceptibility matched sample and probe construction utilized by the Pruessmann group, the Balcom group [

A number of articles have shown that, over limited gradient strengths and field of views, and within manufacturer prescribed slew rates and duty cycles, to a relatively high degree of accuracy the gradient response can be considered as a linear, time invariant system [

Taking the Fourier transform, such that:

The relationship may be expressed as:

Vannesjo

Our approach can be thought of as something in between the Pruessmann and Balcom approaches. Adopting some of the Balcom ideas, we use a single, small phantom, with its own transmit/receive coil. As the phantom size is several mm in diameter and not susceptibility matched, the construction is relatively straightforward. The phantom has its _{1} reduced to approximately 100 ms to enable fairly rapid pulsing on the sample. Unlike the Balcom approach, for the GIRF determination we do not apply RF pulses while the gradient is on, and more like the Pruessmann approach, we do multiple signal sampling during the free induction decay (FID) to follow the phase evolution of the sample signal. Overlapping acquisitions are used to enable sampling over the desired time period. Although Addy

Following a more complete description of our method, we demonstrate that our measured GIRFs are highly accurate and able to accurately predict measured gradient waveforms. Moreover, our GIRFs can be expected to provide important corrections to non-Cartesian gradient trajectories such as spiral gradient waveforms. The primary advantage of our method is that the ease of implementation of our approach should facilitate duplication by other MRI labs, and the experiments are readily implemented on a commercial MRI system.

As indicated in the Background, the GIRF measurements involve comparing the gradient waveform actually produced with the requested waveform. The actual gradient waveform is obtained from differentiation of the sample signal at a known location. That is, as the frequency Ω is given by:

We have:

where γ is the gyromagnetic ratio,

While Addy

where

where

The calculations of the GIRFs were done as indicated by

Plots of magnitude GIRFs in the frequency domain are shown in _{0} terms. However, as these responses were all below 0.1%, they are not displayed here. Somewhat surprisingly, the GIRF

The phases of the GIRFs are shown in

Assuming the gradient systems to be linear and time invariant enables the actual gradient shape produced in the frequency domain to be predicted by

As outlined in the Introduction, quite a variety of imaging methods have been developed, for purposes of adjusting eddy currents, measuring k-space trajectories, or obtaining GIRFs. Some recent, rather sophisticated approaches include an elaborate imaging method for obtaining variable timing delays for improving spiral k-space trajectories [

Influenced in part by the reasoning and approach suggested by the Balcom group [

Had we taken only the 100 kHz data (See Experimental Section), the time to perform our GIRF measurements along one axis (with 400 averages and nine gradient waveforms) would have been one hour. Thus, the primary disadvantage of our approach is the time required for the measurements. This makes it difficult, for example, to see if the GIRFs are altered with gradient temperature, although such measurements could still be made by alternating GIRF measurements with MRI sequences to maintain the gradients at some elevated temperature.

Another potential disadvantage of our use of a larger phantom is that longer time duration gradient waveforms may require interleaved experiments to make experimental measurements of the waveform. On the other hand, the larger size phantom does improve the initial S/N of the experiment.

The GIRFs may be utilized in a variety of ways to improve image fidelity, particularly in non-Cartesian imaging. For example, Addy

Finally, the assumption of linearity and time invariance does have limitations. For example, gradient heating in gradient intensive MRI sequences may alter the GIRF, and the assumption of linearity means that concomitant gradients cannot be taken into account by the GIRF approach. In addition, the GIRF is not valid if the maximum slew rate is exceeded. Finally, the spatial non-linearity of the gradients must be accounted separately. For this reason, we used small displacements from isocenter (approximately 50 mm) to ensure we were well within the region of spatial linearity of the gradient system.

As a compromise between small size and ease of construction, we chose a Wilmad 529-A-8 spherical bulb insert, with a volume of approximately 110 μL, which fits inside an 8 mm NMR tube. A foil coil winding was formed around the NMR tube as shown in

The quarter wavelength cable used a Teflon dielectric, as previous experience showed significant contamination signal arose from use of a cable with polyethylene dielectric. Signal loss from use of the quarter wavelength cable was minimized by adjusting the coil and chip capacitor tuning close to the MRI resonance frequency. The components were mounted on a polystyrene foam board. The sample was doped with copper chloride solution to obtain a _{1} of around 100 ms. The signal-to-noise ratio (S/N) at a bandwidth of 100 kHz approached 1000. Using ξ as the sensitivity measure, defined as the S/N multiplied by the bandwidth (BW) [

yields a ξ of around 3 × 10^{5}. To connect the coil to the Siemens Skyra MRI system, an interface box and coil file (software file that Siemens requires as part of interfacing the coil to the MRI instrument) were purchased from Stark Contrast (See Acknowledgments).

For most of our measurements on our Siemens Skyra 3.0 T system with XQ gradients, we used a series of nine triangular gradients of slew rate 180 mT/m/ms, and rise and fall times T in increments of 10 μs from 30 μs to 110 μs as illustrated in

The data sets for the GIRF calculations were oversampled to 1 μs dwell times, and spliced together so that the initial data length of 20.48 ms was covered by the larger bandwidth signal, and the rest by the lower bandwidth signal. The rational for this was that the residual gradients at longer times arose primarily from incomplete cancellation of the longer time constant eddy currents, whose frequency content was adequately covered by the lower bandwidth acquisition.

To improve the S/N, particularly in the high frequency regime of the GIRF, Savitzky-Golay filtering was done, with greater smoothing performed on the higher frequency region of the GIRF. For a variety of reasons, the GIRF calculations became unreliable beyond approximately 30 kHz, and even showed an artificial upswing in the very high frequency regime of the GIRF. To avoid this artificial upswing, a two term polynomial was fitted to a logarithm of the GIRF over the 16 kHz to 27 kHz region, and used to extrapolate the GIRF to zero. The same procedure was performed on the magnitude and real GIRFs, and the imaginary GIRFs calculated to be consistent with the difference between magnitude and real components. As actual requested gradients in general do not have significant frequency component at these high frequencies (above 20 kHz), this did not result in significant errors in application of the GIRFs to calculate resulting gradients from requested waveforms. In addition, some of the calculated GIRFs exhibited a small blip at zero frequency, presumably to slight instrument drifts that were not completely cancelled out. These blips at zero frequency were removed to provide more realistic estimate of the GIRFs.

Finally, the phantom was positioned at approximately 50 mm from isocenter along the axis being measured for the GIRF measurements (and at isocenter for B_{0} measurements). The initial estimates of the phantom position were obtained by measuring the frequency in the presence of a 0.5 mT/m gradient. However, due to somewhat asymmetric lineshapes, there remained some small uncertainty in the exact position. Therefore, a slight adjustment was done on the phantom positions so that the GIRF

In addition to the GIRF measurements, gradient measurements of additional trapezoidal gradient waveforms with equal rise, top, and fall times were taken to compare with the GIRF predictions of these waveforms. The gradient shapes were measured using the identical setup as used for the GIRF calculations, with the phantom positioned approximately 50 mm from isocenter. As done with the GIRF calculations, repetitions were taken with alternating gradient and background acquisitions. As before, the background data were collected as the average of the gradient off data prior to and following the gradient on data, and the actual gradient calculated by

Some additional experiments were performed at still higher bandwidths (500 kHz), but these acquisitions were not found to improve the GIRF estimations. In addition, some low bandwidth acquisitions (at 25 kHz) were done with a sample with longer _{1} and _{2}^{*} to obtain a longer time acquisition, as the sample with 100 ms _{1} (and somewhat shorter _{2}^{*}) did not produce reliable results beyond approximately 50 ms. However, these longer time acquisitions also did not improve the GIRF estimations, and there was no evidence of persistence of gradient effects beyond several ms after turnoff of the gradients. Thus, it appears that acquisitions using just the 100 kHz bandwidth acquisitions would have given reasonable GIRF estimates. For this reason, in the Discussion section we discuss the time for the experiments as if we had just taken the 100 kHz bandwidth measurements.

We have presented a simple approach for obtaining GIRFs with relatively high precision, and shown that this approach does provide accurate predictions of actual gradient waveforms. Our measured GIRFs capture both the frequency response of the gradient amplifier, as well as deviations due to mechanical vibrations (and potentially other sources as well). Our approach utilizes a rather simple sample and coil construction, and simple MRI experiments with triangular gradient shapes to enable GIRF calculations. Other than the restrictions imposed by the assumption of linearity and time invariance, the primary disadvantage of our approach is the time required to make the measurements.

The authors thank Helmult Stark of Stark Contrast in Erlangen, Germany, for the interface unit and coil file that enabled our coil to be connected to the Siemens Skyra MRI. Grant sponsor: National Center for Research Resources and the National Institute of Biomedical Imaging and Bioengineering, National Institutes of Health; Grant number: P41 RR 023953; Grant sponsor: Veterans Affairs Medical Center.

The authors declare no conflict of interest.

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Plots of gradient impulse response function (GIRF) magnitude and phase for the three gradient directions. (

Time domain plots of the GIRFs in the three gradient directions. The plots are offset in the vertical direction for clarity. All plots show amplitudes prior to zero time. The higher frequency response of the

Requested (green), predicted (red) and experimental trapezoidal gradient waveforms for a

(

The series of triangular gradient waveforms use for the GIRF calculations. The rise and fall times varied from 30 μs to 110 μs in increments of 10 μs, and all waveforms used a slew rate of 180 mT/m.