Field Camera versus Phantom-based Measurement of the Gradient System Transfer Function (GSTF) with dwell time compensation

Abstract

Purpose

The gradient system transfer function (GSTF) can be used to describe the dynamic gradient system and applied for trajectory correction in non-Cartesian MRI. This study compares the field camera and the phantom-based methods to measure the GSTF and implements a compensation for the difference in measurement dwell time.

Methods

The self-term GSTFs of a MR system were determined with two approaches: 1) using a dynamic field camera and 2) using a spherical phantom-based measurement with standard MR hardware. The phantom-based GSTF was convolved with a box function to compensate for the dwell time dependence of the measurement. The field camera and phantom-based GSTFs were used for trajectory prediction during retrospective image reconstruction of 3D wave-CAIPI phantom images.

Results

Differences in the GSTF magnitude response were observed between the two measurement methods. For the wave-CAIPI sequence, this led to deviations in the GSTF predicted trajectories of 4% compared to measured trajectories, and residual distortions in the reconstructed phantom images generated with the phantom-based GSTF. Following dwell-time compensation, deviations in the GSTF magnitudes, GSTF-predicted trajectories, and resulting image artifacts were eliminated (< 0.5% deviation in trajectories).

Conclusion

With dwell time compensation, both the field camera and the phantom-based GSTF self-terms show negligible deviations and lead to strong artifact reduction when they are used for trajectory correction in image reconstruction.

1. Introduction

Imperfections of the dynamic gradient system hardware lead to a loss in fidelity of the gradient waveforms during MR imaging. Imperfections can be caused by limitations in gradient amplifiers, coupling of gradient channels, vibration, and small timing inaccuracies, such that the nominal gradient waveform prescribed in the pulse sequence is not achieved on the scanner hardware.^1–6 Trapezoidal gradient waveforms used in standard Cartesian imaging are mainly affected by a constant time delay.²⁸ This time delay during the readout causes only a linear phase modulation in the reconstructed image.²⁹ In non-Cartesian imaging where the readout direction changes within or between excitations, delays or gradient waveform distortions from the desired gradients in general translate to misregistrations in the sampled data in k-space which result in image distortion.⁷ Thus, the impact of gradient waveform distortions is most problematic in non-Cartesian acquisition techniques like spiral, radial, wave-CAIPI, or single shot EPI-imaging.

Many methods have been implemented by vendors to correct k-space deviations by improving the scanner hardware and the built-in eddy current compensation.^8–10 Moreover, the MR community has developed strategies to compensate for residual gradient delays and trajectory imperfections.^11–13 The introduction of the gradient impulse response function (GIRF) or its Fourier transform, the gradient system transfer function (GSTF), describes the linear frequency transfer characterization of the dynamic gradient system. It serves as a gradient correction technique which can be applied for gradient prediction in image reconstruction^14–16 or as for gradient pre-emphasis^17–18. Using the GSTF-correction has been demonstrated to diminish trajectory distortions for many imaging sequences, like EPI^15,16-, spiral^15,16,18- or bSSFP^27– sequences. The main advantage of the GSTF is that it can be measured as a one-time calibration and can then be utilized to correct any arbitrary gradient waveform. Consequently, this is a simple technique that eliminates cumbersome and time-consuming trajectory measurement for each trajectory type, imaging parameters or slice orientation.^19–21

Typically, the acquisition of the GSTF is performed with either the scanner hardware and a spherical water phantom,^19–20,22 or using a dynamic field camera.^14–15 The latter technique uses additional hardware, i.e. field probes consisting of tiny vials mounted in small RF coils.

In this work, we compare the self-term field components, which play the dominant role^16,18 for trajectory correction, between the field camera and the phantom-based measurement technique and implement a dwell time compensation method. The resultant GTSFs are applied for trajectory correction of a 3D wave-CAIPI sequence²³.

2. Methods

2.1. GSTF theory

The dynamic gradient system can be treated as linear and time invariant and, thus, characterized by the GSTF. A detailed explanation of the GSTF measurement theory has been provided previously.^14,16 The nominal gradient input waveform g_in(t) is distorted to produce the output gradient waveform g_out(t):

$${\text{g}}_{\text{out}}(\text{t})={\int }_{-\infty }^{+\infty }{\text{g}}_{\text{in}}(\text{t})\cdot \text{h}(\text{t}-\tau )\text{dτ,}$$ [1]

where h(t) describes the impulse response function of the system. Or, in the frequency domain using the Fourier transforms G_in(f), G_out(f), H(f) of the respective time domain quantities:

$${\text{G}}_{\text{out}}(\text{f})={\text{G}}_{\text{in}}(\text{f})\cdot \text{H}(\text{f}).$$ [2]

According to equation 1 and 2, the gradient system impulse response function h(t) (GIRF) and the gradient system transfer function H(f) (GSTF) describe the system characteristics, including hardware imperfections.

In our implementation, twelve triangular input gradient waveforms g_in(t) with different pulse duration (100 – 320 μs, slew rate = 180 T/m/s) were applied, and the system’s response g_out (t) was measured¹⁴. The system transfer function of the dynamic gradient system was calculated for both acquisition techniques as follows:

$${\text{H}}_{\text{k},\text{l}}(\text{f})=\frac{{{\sum }_{\text{i}=1}^{\text{j}=12}{\text{G}}_{\text{in}}{}^{*}}_{\text{k}}^{\text{i}}(\text{f})\cdot {\text{G}}_{\text{out}}{}_{{}_{\text{k},\text{l}}}^{\text{i}}(\text{f})}{{\sum }_{\text{i}=1}^{\text{j}=12}{|{\text{G}}_{\text{in}}{}_{{}_{\text{k}}}^{\text{i}}(\text{f})|}^2},\begin{array}{c}\text{k}=\text{x},\text{y},\text{z}\\ \text{l}=\text{x},\text{y},\text{z}\\ \text{i}=1,2,3\dots 12\end{array}$$ [3]

where k represents the direction in which the input gradient is played out, l the direction in which the output gradient is measured. The index i is a particular triangular waveform. The self-terms (H_x,x(f), H_y,y(f) and H_z,z(f)) are given by k = l. For a simpler notation, we use GSTF_xx for H_x,x(f) etc.

2.2. Waveform measurement

All GSTF measurements were performed on a clinical 1.5T scanner (MAGNETOM Aera, Siemens Healthcare, Erlangen, Germany) with vendor built-in eddy current compensation enabled. The GSTF was determined for each gradient axis using a dynamic field camera (Skope Magnetic Resonance Technologies, Zurich, Switzerland) equipped with 16 ¹H NMR probes located around the surface of a spherical measurement head (diameter 20 cm). The gradient measurements were performed using the excitation and receiver chain of the dynamic field camera. For comparison, the gradients were also measured using phantom-based technique with a standard spherical water-phantom (diameter 17 cm) doped with NiSO₄.^14–15 For the phantom-based measurement, gradients were calculated from the difference in phase evolution along two parallel slices.¹⁹

The signal of the field camera was sampled with a fixed dwell time of τ = 1 μs (bandwidth = 1 MHz), and the dwell time τ of the phantom measurement was varied between 4.8 (bandwidth = 0.21 MHz) and 8.7 μs (bandwidth = 0.11 MHz). The acquisition duration was set to 40ms for the field camera and to 10ms for the phantom measurements. The number of points per read-out was limited for the phantom measurements and the desired acquisition time restricted the smallest dwell time which is achievable. In addition to twelve triangular input gradient waveforms, twelve reference acquisitions of the phase drift in the absence of gradients were measured for the phantom-based technique and subtracted from the phases of the triangular gradients. Forty measurements were performed to achieve the same total scan time for both techniques. For the phantom-based measurement, this is equivalent to 20 averages (for two slices measurements), and for the field camera-based measurement, this is equivalent to 40 averages. No filtering was applied to the raw data acquired using the field camera or MRI scanner to improve signal-to-noise ratio. Other relevant sequence parameters for the phantom-based technique were: TR = 1000 ms, slice thickness = 3 mm, slice positions = ±16.5 mm (parallel slices), flip-angle = 90°. The field camera was also used to measure the actual gradient waveforms for the wave-CAIPI test sequence. Figure 1 provides a schematic of the measurement methods and an example of the output phase data for one phantom slice and one single field probe.

FIGURE 1.

Schematic diagram of the acquisition process for the field camera and the phantom-based measurement technique containing a comparison of the magnitude and phase raw data signals for one slice (phantom) and a single field probe (field camera). The phase data is presented for one single triangular pulse.

2.3. Dwell time compensation

We implemented a post-processing correction method to eliminate dwell time-dependent GSTF magnitude characteristics. The acquisition for a certain dwell time τ was modelled as a convolution with a box function of duration τ, so the influence of the dwell time can be compensated by dividing the GSTF by a sinc function, i.e. the Fourier transform of a box function. Mathematically the dwell time compensation can be expressed as:

$$|\text{H}(\text{f}){|}_{\text{compensated}}=\frac{|\text{H}(\text{f}){|}_{\text{measured}}}{\text{sinc}(\tau \cdot \text{f})}$$ [4]

2.4. Phantom non-Cartesian imaging experiments

GTSF-predicted trajectories were validated using a 3D wave-CAIPI sequence to image a structural phantom with an 18-channel body coil on the same MRI system. The wave-CAIPI pulse sequence used a 3D corkscrew trajectory²³ with a quasi-random sampling order (TR = 2.5 ms, TE = 1.1 ms, flip angle = 3°, slice thickness = 4mm, readout = 192×192×64). The wave-CAIPI sequence features sinusoidal gradient oscillations which were played out on the x- and y-axes with an amplitude of 4 mT/m and a frequency of 8 kHz. The GSTF-predicted gradient waveforms were compared to the nominal and measured gradient waveforms from the dynamic field camera. The wave-CAIPI sequence uses trajectories that are well-suited to image reconstruction with parallel imaging techniques.²³ Wave-CAIPI was chosen as a test sequence because it demands gradient system accuracy over a broad frequency range during data acquisition.

2.5. Image reconstructions

All phantom images were reconstructed in Matlab (The MathWorks, Natick, MA, USA) using a Nonuniform Fast Fourier Transform (NUFFT) with a Kaiser Bessel gridding kernel. The reconstructions were performed with the nominal k-space trajectories (no-correction) and with the GSTF-predicted trajectories. For the GSTF-prediction, three different GSTFs were applied for reconstruction: (a) the field camera-measured GSTF (GSTF_FC), (b) the standard (uncompensated) phantom-measured GSTF (GSTF_ST), (c) the dwell time compensated phantom-measured GSTF (GSTF_DT).

3. Results

3.1. GSTF measurement

Figure 2 shows the measured output gradients for a triangular nominal input gradient (Fig. 1) for the field camera and phantom-based measurements. A more detailed view of one single triangular output gradient (pulse width 300 μs) shows oscillations in the gradient time course and the noise level of the output signals. Because of the smaller dwell time the signal-to-noise ratio is lower for the field camera-based measurement.

FIGURE 2.

Measured gradient outputs for the x-gradient axis using the phantom-based (a) and field camera-based technique (b). The gradient outputs were averaged over all performed measurements (20 measurements for the phantom-based technique and 40 measurements for the field camera technique, respectively). In addition to the visualization of the 12 gradient outputs, the different noise level is presented for one output gradient (pulse duration 300μs) in a detailed view.

The GSTF magnitude was determined to be dwell time-dependent, as illustrated in Figure 3 showing the phantom-based measurement of the z-self-term. Longer dwell times result in a narrower magnitude GSTF response. That means that higher frequencies have a decreased transmission rate for low acquisition bandwidths. The GSTF phase transition is not dwell time-dependent.

FIGURE 3.

The dwell time dependence of the z-self-term GSTF. A 1μs dwell time was used for the field camera measurement and 4.8–8.7μs for phantom-based measurement. The magnitude response broadens for smaller dwell times (a), whereas the phase transitions show no dwell time dependent effects (b).

3.2. Dwell time compensation

With a dwell time of τ = 8.7 μs, the deviation between GSTF_ST and GSTF_FC is about 4% at the 3D wave-CAIPI frequency of 8kHz (Fig. 4 a–c). Following application of the proposed dwell time compensation (Eq. 4), the magnitude response of all GSTFs match within < 0.3%. In contrast to the magnitude response, the phase transition of the GSTF measurement techniques only shows negligible deviations of < 0.5% (Fig. 4 a–c) with and without dwell time compensation. Figure 4d shows the z-self-term for a limited frequency range of 5–8 kHz. The residual deviations in the GSTF magnitude and phase following dwell time compensation are on the order of the measurement noise.

FIGURE 4.

The magnitude and phase of the GSTF measured for x, y and z self-terms is shown in (a-c) for the field camera (1μs dwell time) and phantom technique (8.7μs dwell time). In all axes, the phantom-based GSTF magnitude responses show apparent deviations to the field camera GSTFs. The sinc function-based dwell time- compensation reduces the deviation between phantom and field camera GSTFs. Due to the small dwell time of the field camera (1μs), the deviations between the standard and compensated field camera GSTFs are negligible. The differences in the phase transitions are small. A detailed view for the limited frequency ranges from 5 to 8kHz is illustrated in (d) for the z-self-term.

3.3. Trajectory comparison

The GSTF-predicted x- and y-gradient waveforms using the phantom-based measurement were also dwell time-dependent. GSTF_ST vs GSTF_FC produced < 4% deviation for the wave-CAIPI sequence (Fig. 5). After applying dwell time compensation, the deviations were reduced to < 0.5% (GSTF_DT vs GSTF_FC). Comparisons to the field camera measured gradient waveforms, generated < 1.5% deviations from GSTF_FC-predicted and GSTF_DT-predicted gradients (Fig. 5). All GSTF-predicted gradients showed significant deviation from the nominal input gradient waveforms from the pulse sequence program (mean deviation = 11%), as did the measured gradient waveform. The nominal gradient waveform is not achieved due to gradient hardware imperfections, and the executed waveform is measured by the field camera. By applying GSTF-based gradient correction to the nominal gradient, the GSTF-predicted gradient waveforms, especially GSTF_FC and GSTF_DT, approaches the measured gradients.

FIGURE 5.

Comparison of x, y and z gradient waveforms between nominal, measured, phantom GSTF-predicted (standard and compensated) and field camera GSTF-predicted (a, b, c). The deviations between the measured, and the GSTF-predicted gradient waveforms are small for the wave-CAIPI sequence. The dwell time compensation results in increased similarity between GSTF_DT-predicted and GSTF_FC-predicted gradient waveforms.

3.4. Reconstructed image comparison

Figure 6 shows a comparison of the reconstructed images for the wave-CAIPI sequence using the nominal trajectory (a) and the predicted trajectories after applying GSTF_FC (b), GSTF_ST (c) and GSTF_DT (d) during reconstruction. Image artifacts are highlighted with a red rectangle. All GSTF corrected images show diminished artifact level (Figure 6b–d). The difference image between the uncorrected image (Fig. 6a) and the GSTF_FC corrected image (Fig. 6b) is shown in Fig. 6e, where significant deviations in signal intensity are observed throughout the phantom. The differences between GSTF_FC and GSTF_ST are apparent (Fig. 5f), with a ~9% maximum difference in signal intensity between the two reconstructions (see yellow rectangles in the intensity profile, Fig. 6i). Fig. 6g shows that the deviations between the GSTF_FC and GSTF_DT are negligible. This is highlighted in the intensity profiles (Fig. 6h,j) with green rectangles, where differences are at the level of the noise.

FIGURE 6.

Phantom images acquired using a 3D wave-CAIPI sequence. The images were reconstructed with the nominal (a), the GSTF_FC-predicted (b), the GSTF_ST-predicted (c) and GSTF_DT-predicted (d) k-space trajectories. Artifacts are highlighted by a red rectangle. (e) shows the difference image between a reconstruction with the nominal and the field camera corrected trajectory. The difference between the 3D wave-CAIPI images using trajectories corrected by the GSTF determined by the field camera and by the standard/compensated phantom technique in (f, g). The corresponding intensity profiles for the difference images (dashed line in e-g) are shown in (h-j). (g) shows the deviations between the images corrected by GSTF_FC and GSTF_DT are negligible (green rectangles in h and j), whereas the image differences between the GSTF_FC-corrected and the GSTF_ST-corrected trajectories are apparent (f) (yellow rectangles in h and i). The intensities in the difference images (e-g) were normalized to the maximum intensity of (a).

4. Discussion and Conclusion

Here we compared the self-terms of the gradient system transfer function (GSTF) measured with two different acquisitions techniques that have been used previously for distortion correction of non-Cartesian images. We showed that the magnitude of the phantom-based GSTF measurement is dwell time-dependent, but dwell time compensation can be applied such that GSTF measurement method had negligible impact on the trajectories and the reconstructed 3D wave-CAIPI images.

Previous work by Graedel et al.²⁴ also compared the two GSTF measurement methods for a 3D EPI (TURBINE blade) sequence. In their study, only small differences in the GSTFs were found and the trajectory prediction for a TURBINE blade was slightly better using the field camera predicted GSTF. This was attributed by the authors to the inclusion of cross terms, increased frequency resolution and higher SNR of field-camera based GSTF. In their study, the SNR of the field camera technique was higher because of the higher MR field strength (7T) and because of the thinner slices (0.8 mm) used for the phantom-based approach.

Here we showed for 3D wave-CAIPI that the time resolution of the phantom-based GSTF acquisition affects the correction process and requires the application of dwell time compensation to generate the equivalent artifact reduction compared with an image corrected with the field camera-measured GSTF. Since the dwell time dependence effects solely the magnitude GSTF response and is negligible in the phase GSTF response, the artifact improvement may be less prominent for spiral and EPI imaging methods. We chose to use a wave-CAIPI sequence because the magnitude GSTF response is dominant for trajectory correction, and by comparison, the correction of spiral imaging sequences is dominated by the phase GSTF response. All measurements were performed with eddy-current compensation enabled, to replicate the clinical application of this sequence.

It was not possible to perform a direct comparison of the transfer functions for both GSTF acquisition techniques with identical dwell times. With the vendor provided hard- and software configuration of the scanner it is not possible to achieve dwell times around of 1 μs for acquisition times of at least 30 ms, as it was used for field camera acquisitions. Moreover, the dwell time of the field camera is fixed to 1 μs and cannot be changed by the user. However, the application of dwell time compensation eliminates the dwell time dependency and makes both approaches comparable. One potential advantage of the field camera is the higher frequency resolution that enables better resolution of mechanical resonances in the transfer functions. The phantom-measured GSTFs were determined with a frequency resolution of 100 Hz as opposed to 25 Hz with the field camera. The higher frequency resolution of the field camera-based GSTFs facilitates to resolve sharp resonances better than the phantom-based technique with the chosen parameters. The maximum possible acquisition duration of 100 ms was limited to 40 ms for the field camera measurements to limit T2 delay of the probes and gain acceptable SNR values. The GSTF data in this study was averaged with 20 repetitions for the phantom-based technique (2 slice measurement) and 40 repetitions for the field-camera to allow the same acquisition duration (30 min acquisition per self-term). With the measurement parameters used in this study, the phantom-based technique provides higher SNR than the field camera measurement. Filter operations to reduce signal noise were not applied to the measured field camera and phantom raw data. In the relevant frequency range of 0–8 kHz, the noise of the field camera and phantom-based approach can be quantified by the standard errors SE < 1.97% and SE < 0.23%, respectively. For the chosen GSTF-acquisition parameters, the measurement noise does not affect image reconstruction with the GSTF_FC-corrected and GSTF_DT-corrected trajectory. Similar SNR for both techniques can be achieved by increasing the number of measurements for the field camera technique. Nevertheless, we found that both techniques performed trajectory correction in a wave-CAIPI sequence equivalently when dwell time compensation is applied independent of the differences in frequency resolution and SNR.

Another feature of the field camera measurement is the intrinsic acquisition of additional GSTF terms, including cross terms, higher order terms and 0^th-order spatial field components.^14–15 The dynamic field camera allows measurement of all spatial field components up to the 3^rd order, whereas the phantom-based measurement approach we used enables to determine the 0^th and 1^st order components.^16,22,25 Adding additional phase encoding gradients in two directions makes it also feasible, but time-consuming, to gain 2^nd and 3^rd order field components using a standard phantom approach.²⁶ Cross-terms, 0^th order and higher order (2^nd and 3^rd order) field components weren’t investigated in this study due to their minor impact on trajectory distortions and resultant image quality for this wave-CAIPI sequence.

In conclusion, we have shown that the GTSF-based trajectory correction is not affected by the measurement technique used for acquiring the transfer functions when dwell time compensation is applied. With the same measurement time, the field camera-based approach offers a higher frequency resolution but also lower SNR at 1.5T, compared with the phantom-based approach.