Estimation of k-space trajectories in spiral MRI

Abstract

For non-2DFT data acquisition in MRI, k-space trajectory infidelity due to eddy current effects and other hardware imperfections will blur and distort the reconstructed images. Even with the shielded gradients and eddy current compensation techniques of current scanners, the deviation between the actual k-space trajectory and the requested trajectory remains a major reason for image artifacts in non-Cartesian MRI. It is often not practical to measure the k-space trajectory for each imaging slice. It has been reported that better image quality is achieved in radial scanning by correcting anisotropic delays on different physical gradient axes. In this paper, the delay model is applied in spiral k-space trajectory estimation to reduce image artifacts. Then a novel estimation method combining the anisotropic delay model and a simple convolution eddy current model further reduces the artifact level in spiral image reconstruction. The root mean square error (RMSE) and peak error in both phantom and in vivo images reconstructed using the estimated trajectories are reduced substantially compared to the results achieved by only tuning delays. After a one-time calibration, it is thus possible to get an accurate estimate of the spiral trajectory and a high-quality image reconstruction for an arbitrary scan plane.

INTRODUCTION

In magnetic resonance imaging (MRI), the theoretical k-space trajectory is proportional to the integral of the gradient current through each gradient coil. However, the actual k-space trajectory is always distorted by many undesired effects in spatial encoding such as eddy currents and anisotropic gradient amplifier delays. To reduce the effects of eddy currents, manufacturers have active shielding and pre-emphasis filters in current scanners to eliminate most of the errors. However, the residual error can still cause severe image artifacts, especially in non-Cartesian scanning such as radial and spiral imaging (1–3).

If the k-space trajectory is not distorted severely and the k-space center is sampled, we can use the actual k-space trajectory in the reconstruction and remove most of the artifacts. Many researchers have proposed methods to measure the actual k-space trajectory. Duyn et al. (4) and Zhang et al. (5) proposed using the phase difference of two measurements on the same thin slice per spatial encoding axis to measure the k-space trajectory. Spielman and Pauly measured the current through the gradient coil directly (6). Mason et al. placed a special phantom at different locations in the bore to measure the k-space trajectory (2). Takahashi et al. used the self encoding method to measured k-space trajectories to improve the multi-dimensional selective excitation (7). Alley et al. improved the self-encoding technique by extracting the k-space trajectory from the phase of the Fourier transform of the raw data (8). They also fit their k-space measurements to an impulse response model. Beaumont et al. (9) improved the accuracy at the k-space periphery by applying an additional dephasing gradient to the method proposed by Zhang et al. (5). With the measured k-space trajectory, the reconstructed image quality is improved greatly. However, it is not always practical to measure the trajectory for each imaging slice during scanning. Recently, Pruessmann et al. and Zanche et al. (10, 11) proposed magnetic field monitoring (MFM) during the MRI data acquisition using field probes. This method is very promising since it can remove undesired phase terms in each individual scan. One limitation is that the probes have to be aligned with each imaging slice.

Aside from uncompensated eddy current effects leading to distortions of the gradient waveform shape, another significant problem is small timing delay errors arising in the hardware. Peters et al. (12) measured the delays on different physical gradient axis in a calibration scan and corrected the miscentering of k-space on each projection angle using that information. Davies and Jezzard (13) proposed calibration and correction methods for gradient propagation delays for 2D RF pulses to improve the positional accuracy. Speier and Trautwein (14) parameterized the gradient delays for radial imaging applications. There are also frequency demodulation delays especially in off-axis imaging slices, as reported by Jung et al. (15).

In this work, we propose a k-space trajectory estimation method based on an anisotropic gradient delay model (16) and a simple eddy current model. Our goal is to estimate the actual k-space trajectory for spiral MRI in any arbitrary imaging slice and use the estimated trajectory in image reconstruction to remove artifacts. We applied our model to spiral sequences on three different scanners. When the estimated trajectory is adopted in the online reconstruction process, both phantom and in vivo images reconstructed using this method have a much smaller root mean square error (RMSE) and peak error compared with the results using delay only model.

THEORY

k-space trajectory infidelity

In reality, nothing is perfect. This is true of the gradient systems in MR scanners. While the deviation between the desired k-space trajectory and the actual trajectory is always there, the resulting artifacts are more obvious in non-Cartesian imaging. In Cartesian sampling, the deviation only causes some phase changes in the reconstructed image. In a modern scanner, the deviation between the actual k-space trajectory K_a and the theoretical k-space trajectory K_t is very small. However, the longer readout time in spiral imaging makes the problem worse and causes severe artifacts, especially in coronal, sagittal and oblique views. Eddy-current-induced k-space deviations can accumulate during a longer readout, leading to more deviation at the k-space periphery. Concomitant gradient field effects can add additional k-space errors for off-isocenter scans, as well as signal phase errors (17).

There are many causes for k-space trajectory infidelity, including eddy current effects, hardware imperfections and field inhomogeneity. In this study, we removed the phase from inhomogeneity and concomitant gradient terms in the k-space trajectory measurement. The residual difference between the actual k-space trajectory and theoretical trajectory is divided into two parts: anisotropic delay and eddy current induced. Then we can treat these two parts separately and combine them together to estimate the actual k-space trajectory.

Anisotropic delay model

Anisotropic timing delays exist between the application of a gradient waveform to the gradient amplifiers and the subsequent response of the gradient coils. When imaging an oblique or double oblique slice, the gradient delay on different physical axes might affect the same logical axis. So we need to measure the delay on each physical axis first and map them accordingly to each physical gradient.

Anisotropic delayed spiral gradient waveforms are illustrated in Fig. 1. For an arbitrary imaging slice, the logical gradient vector G_log= (g_ro g_ph g_s) can be transformed to the physical gradient vector G_phy = (g_x g_y g_z) by multiplying the rotation matrix R as in [1].

$${\mathbf{G}}_{\text{phy}}=\mathbf{R}{\mathbf{G}}_{\text{log}}$$ [1]

Then we can apply different delays on the resulting physical gradient vector G_phy to get the delayed physical gradient vector G_d. This can be expressed as follows:

$${\mathbf{G}}_{\mathrm{d}}=\mathrm{T}({\mathbf{G}}_{\text{phy}})=\left[\begin{array}{c}{g}_x(t-{\tau }_x)\\ g_y(t-{\tau }_y)\\ g_z(t-{\tau }_z)\end{array}\right]$$ [2]

The delayed physical k-space trajectory K_d is the integration of the delayed gradient vector. After we get the delayed physical gradient vector, we can use the rotation matrix to get the delayed gradient in the logical coordinate system:

$${\mathbf{G}}_{\text{dL}}={\mathbf{R}}^{\mathrm{T}}{\mathbf{G}}_{\mathrm{d}}={\mathbf{R}}^{\mathrm{T}}\mathrm{T}({\mathbf{G}}_{\text{phy}})$$ [3]

The delayed theoretical k-space trajectory K_dL is the integration of the delayed gradient vector. Since the sampling rate is different between the gradient system and the ADC system, we use spline interpolation to up sample the gradient signal first before we do the integration. We only need to measure the anisotropic delays on each physical gradient axis in order to estimate the delayed k-space trajectory.

Figure 1.

Illustration of anisotropic gradient delay on spiral gradients. This figure gives an exaggerated example of different delays on an oblique slice which causes image artifacts on reconstructed images.

Eddy current model

Eddy currents are typically classified as B0 eddy currents or linear eddy currents based on their spatial dependence. B0 eddy currents cause unwanted phase modulation, while linear eddy currents distort the k-space trajectory. In a simple eddy current model (18), the eddy current g(t) on each physical gradient axis can be estimated by the convolution of the slew rate s(t) = −dG/dt of the desired/theoretical gradient waveform and the system impulse response function H(t) as the following.

$$H(t)=u(t){\displaystyle \sum _n{a}_n\text{exp}(-t/b_n)}$$ [4]

where u(t) is the unit step function. For each exponential term e(t) = a_nu(t)exp(−t/b_n) in H(t), if we only consider the first order Taylor expansion, i.e. e(t) ≈ a_nu(t)(1−t/b_n), the convolution can be simplified to the following expression.

$$\begin{array}{c}s(t)\otimes H(t)=-dG/dt\otimes {\displaystyle \sum _n{a}_{n}u(t)\text{exp}(-t/b_n)}\hfill \\ \hfill \approx -dG/dt\otimes {\displaystyle \sum _n{a}_{n}u(t)(1-t/b_n)}\hfill \\ ={\displaystyle \sum _n{a}_n[-{\displaystyle \underset{0}{\overset{t}{\int }}(dG/d\tau )d\tau +{\displaystyle \underset{0}{\overset{t}{\int }}(dG/d\tau )(t-\tau )/b_{n}d\tau ]}}}\hfill \\ =AG(t)+BG(t)t-B{\displaystyle \underset{0}{\overset{t}{\int }}(dG/d\tau )\tau d\tau }\hfill \end{array}$$ [5]

where $A=-{\displaystyle \sum _n{a}_n}\text{and}B={\displaystyle \sum _n(a_n/b_n)}$ are constants characterizing the impulse response function H(t). The eddy current induced k-space trajectory K_e is the integration of the above convolution result as in [6].

$$\begin{array}{c}{k}_e(t)={\displaystyle \underset{0}{\overset{t}{\int }}s(\mathit{t\text{'}})\otimes H(\mathit{t\text{'}})d\mathit{t\text{'}}}\hfill \\ \approx A{\displaystyle \underset{0}{\overset{t}{\int }}G(\mathit{t\text{'}})d\mathit{t\text{'}}+B{\displaystyle \underset{0}{\overset{t}{\int }}G(\mathit{t\text{'}})\mathit{t\text{'}}d\mathit{t\text{'}}-B{\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \underset{0}{\overset{\mathit{t\text{'}}}{\int }}(dG/d\tau )\tau d\tau }}d\mathit{t\text{'}}}}\hfill \end{array}$$ [6]

We can see that the first term is actually a scaling term for the theoretical k-space trajectory to correct for geometrical distortion. The other two terms are the system response to gradient switching. Once we know the system constants A and B, we can estimate a first order approximation of the eddy-current-induced k-space trajectory deviation on each physical axis.

METHODS

k-space trajectory measurement

First, we collect phase data on two thin slices offset symmetrically around isocenter with an even number of spiral interleaves for coronal, sagittal and transverse views. The excited slice should be thinner than the spatial resolution of the image and as thin as possible for accurate measurements. More averages can be used here to increase the signal to noise ratio. First, the phase from the baseline acquisition with no gradients applied is subtracted from the other measurements. Since the concomitant gradients are proportional to the square of the gradient amplitude and the B₀ eddy currents are linearly proportional to the gradient amplitude, we can remove the phase terms due to them separately (19). Dividing the corrected signal phase by the slice distance, we get the actual k-space trajectories K_a on each physical gradient axis in each view as follows for calibration purposes:

$${\mathbf{K}}_a=\left[(P_{+}(g_{+})-P_{+}(g_{-}))-(P_{-}(g_{+})-P_{-}(g_{-}))\right]/(4D)$$ [7]

where P_±() are the phase terms in two symmetrical slices; g_± are the gradient waveform to be tested and its inverted version; D is the distance between the offset slice and isocenter. Using these measured k-space trajectories, we perform gridding reconstruction to get the goal images I_g in the three normal views.

Anisotropic delay model estimation

There are two ways to find the delays on each physical gradient axis: one is to compare the delayed theoretical trajectory and the actual trajectory directly in the k-space domain and the other is to compare the reconstructed images using the delayed trajectory and the actual one. If the gradient pulse is very short as in radial imaging, the delay can be easily found in k-space using the first method. However, when the readout gradient is very long as in spiral imaging, the second method is more reliable. We have tested the two methods and found the results are very close if we only use the first few samples to measure the delay in the k-space domain. However, since the results from the k-space based method are dependent on the number of k-space samples used to calculate the error; the best delays would change if we change the number of readout samples per interleaf. Thus they should be combined with the image based method until consistent results are obtained.

So we conduct a 2D search for the minimum RMSE between the images reconstructed using different delays and the goal image for each view. With the optimum delays on each physical gradient axis, we have the first guess of the k-space trajectory K_d from the anisotropic delay gradient model.

Eddy current compensation

The k-space trajectory difference between the actual k-space trajectory and the delay model estimate K_r = K_a − K_d is small, but the images using this delay model have significant residual error. In order to reduce the deviation between the modeled k-space trajectory and the actual measured k-space trajectory, we introduce an additional eddy current compensation term on each physical gradient axis. In other words, we use the above mentioned eddy current model to approximate K_r.

Using the discretized version of equation [6], we can get the following matrix expression for the eddy current induced k-space trajectory K_e for each interleaf on each physical axis.

$${\mathbf{K}}_{\mathrm{e}}=[\begin{array}{ccc}A& B& -B\end{array}]{\mathbf{K}}_{\mathrm{b}}$$ [8]

where K_b = cumsum(G_e) is a 3 by N matrix for each physical gradient, ${\mathbf{G}}_{\mathrm{e}}={[\mathrm{G}({\mathrm{t}}_{\mathrm{n}});\mathrm{G}({\mathrm{t}}_{\mathrm{n}}){\mathrm{t}}_{\mathrm{n}};{\displaystyle \sum _{m=1}^{\mathrm{n}}}\mathrm{s}({\mathrm{t}}_{\mathrm{m}}){\mathrm{t}}_{\mathrm{m}}]}^{\mathrm{T}}$ and N is the number of k-space samples in one spiral interleaf. cumsum() is a Matlab function (The MathWorks) that calculates the cumulative sum of array elements. Then we can get an estimate of vector a = [A B − cB] by using weighted least squares (WLS) to fit the k-space difference K_r to K_e for each spiral interleaf on each physical gradient axis, where c is a constant related to the weighting used. In this study, c was set to 1. The mean value of a over all interleaves is then adopted as the final model parameter.

It is interesting to note that we have many choices here to use the fitting result for future calibration. The first choice would be to use an impulse function H(t) with only one exponential term, ignoring any additional exponential terms. However, when we get the exponential coefficient b_n = −A/B, we find it is sometimes negative and the first order Taylor approximation diverges quickly from the exponential function in that case as t increases. The second choice would be to combine the last two parameters together, since they are only different by a constant scale. The last option is to use is the full correction to give the system more freedom. We choose the last one here for implementation simplicity and robustness. So we are using the first order approximation for the impulse response. Thus no matter how many exponential terms are in H(t), we should be able to compensate most of the first order eddy current effects by applying these parameters on each physical axis. For an arbitrary scan slice, the logical k-space trajectory matrix K_L for image reconstruction can be derived as follows:

$${\mathbf{K}}_{\mathrm{L}}={\mathbf{R}}^{\mathrm{T}}({\mathbf{K}}_{\mathrm{d}}+{\mathbf{K}}_{\mathrm{e}})={\mathbf{K}}_{\text{dL}}+{\mathbf{R}}^{\mathrm{T}}\mathbf{A}{\mathbf{K}}_{\mathrm{B}}$$ [9]

where A is a 3 by 9 matrix with the following nonzero elements from the WLS results.

$$\begin{array}{c}\mathbf{A}(1,1:3)={\mathbf{a}}_x\\ \mathbf{A}(2,4:6)={\mathbf{a}}_y\\ \mathbf{A}(3,7:9)={\mathbf{a}}_z\end{array}$$ [10]

K_B is a 9 by N matrix with K_b on all three physical axes.

$${\mathbf{K}}_B=\left[\begin{array}{c}{\mathbf{K}}_{bx}\\ {\mathbf{K}}_{by}\\ {\mathbf{K}}_{bz}\end{array}\right]$$ [11]

The flow chart of the model estimation is shown in Fig. 2 and the flow chart for estimation of the k-space trajectory using the proposed model is shown in Fig. 3.

Figure 2.

Flowchart of model estimation using calibration data from coronal, sagittal and transverse views.

Figure 3.

Flowchart of k-space trajectory estimation for an arbitrary imaging slice using the proposed model.

A natural extension of the above model is to divide the residual error K_r into multiple time segments to get a closer approximation of the impulse response H(t) during each shortened time period. For example, we could divide 8192 samples to eight segments with 1024 samples in each segment and get eight sets of Taylor expansion parameters. We tried as many as 32 segments for 8192 k-space samples, but found the improvement is very small when using more than eight segments. There is a tradeoff between more time segments and the variance of the system parameters over all interleaves. The variance of the parameters should be kept as small as possible in order to get a stable result. We thus used eight segments on 1.5T scanners and four segments on the 7T scanner in this work. Since we only use three variables to describe H(t) for one time segment on each axis, there is only a tiny increase of memory load while the computation time is the same for using more segments in the extended model.

Sequence Parameters

The k-space trajectories for spiral sequences were measured on two whole-body Siemens 1.5T scanners (Avanto and Sonata) and a 7T Siemens/Bruker Clinscan small animal. On the 1.5 T scanners, the readout for each slice was done with 14 interleaved spirals with 8192 data samples and a duration of 16.38 ms. A low resolution field map was estimated using two single-shot spiral readouts at the beginning of each scan. Linear B0 inhomogeneity correction was performed using the low resolution field map (20). The reconstructed image matrix was 512 by 512. The nominal in plane resolution was 0.9 mm. The distance between the excited slice and the isocenter in the k-space trajectory measurement was 50 mm and the field of view (FOV) was 280 mm. The slice thickness was 0.9 mm for k-space trajectory measurement and 5 mm for imaging. The maximum gradient amplitude was 40 mT/m and the maximum slew rate was 170 mT/m/ms. On the 7T scanner, the spiral sequence had 42 interleaves with 2048 data points and a readout duration of 4.096 ms. The FOV was 40 mm with slice thickness 0.1 mm in k-space trajectory measurement. The distance between the thin slice and the isocenter was 6 mm. The FOV was 30 mm with slice thickness of 0.5 mm for imaging. The reconstructed image matrix was 256 by 256. The maximum gradient amplitude was 290 mT/m and the maximum slew rate was 1160 mT/m/ms.

RESULTS

We observed that the actual spiral trajectories were slightly distorted on three different scanners. As shown in Fig. 4a, the actual k-space trajectory K_a and the theoretical k-space trajectory K_t are almost on top of each other and their difference in Fig. 4b is very small. However, the difference between these two trajectories is big enough to cause severe artifacts as shown in Fig. 5. The reconstructed objects have severe artifacts on the edge and are geometrically stretched. The artifacts are more obvious in the coronal and sagittal views than in the transverse view because the physical gradient on z axis is much different from the other two gradient axes. We adopt the corrected trajectory retrospectively to the raw data in gridding reconstruction since the k-space center is guaranteed to be sampled.

Figure 4.

Actual k-space trajectory and theoretical trajectory (a) and their difference (b). The trajectory distortion is small, but still produces significant artifacts.

Figure 5.

Coronal results using a single delay on both axes (a) and the difference image (b) between the uncorrected image and the goal image (RMSE = 21.142, NRMSE = 1.0 and NMaxE = 1.0).

We have applied the proposed method to three scanners. More than two datasets were collected on each scanner with the time interval between scans longer than one month. The first dataset was used for calibration to estimate the anisotropic delays and eddy current model parameters. Then the proposed model was used to estimate the k-space trajectory for gridding reconstruction on the second dataset. To quantify the artifact level, we used the normalized RMSE (NRMSE) and normalized absolute peak error defined in the following equations.

$$\text{NRMSE}=\text{RMSE(}{\mathrm{I}}_{\text{proposed}},{\mathrm{I}}_{\mathrm{g}})/\text{RMSE(}{\mathrm{I}}_{\text{bench}},{\mathrm{I}}_{\mathrm{g}})$$ [12]

where I_proposed is the image using the proposed k-space trajectory, I_g is the image using the measured k-space trajectory and I_bench is the image using the same delay on all axes. The normalized absolute peak error is defined in similar sense.

$$\text{NMaxE}=\underset{\text{all pixels}}{\text{max}}(\text{abs}({\mathrm{I}}_{\text{proposed}}-{\mathrm{I}}_{\mathrm{g}}))/\underset{\text{all pixels}}{\text{max}}(\text{abs}({\mathrm{I}}_{\text{bench}}-{\mathrm{I}}_{\mathrm{g}}))$$ [13]

Both NRMSE and NMaxE are 0 for the reconstructed images using the measured trajectories and 1 for the uncorrected images. The benchmark is the most common way to do online spiral reconstruction. We just use a linear field map and use the same delays on all the axes to get the k-space trajectories. Gridding is then used followed by an inverse FFT. We choose the same delay as our benchmark for two reasons. First, although groups have used different ways to measure delays and one group studied the effects of anisotropic delays on 2D spiral gradients (16), there is no report of characterizing anisotropic delays and combining them with a rotation matrix to compensate for them in spiral image reconstruction. So the anisotropic delay method is part of our proposed method. Furthermore, using the same delay method as the benchmark, we can quantify the improvements from both the delay-only method and the combined model.

In Fig. 5a, we show the benchmark image in the coronal view using the same delay (9.5us) on all axes. We can see the shape of the sphere phantom is distorted and there are bright and dark artifacts around the edge. The edges around the north and south poles are very bright while the edges around the left and right are very dark. This is mainly because of the anisotropic delays on two physical axes. The residual error in Fig. 5b is surprisingly high and has to be corrected.

Table 1 gives the anisotropic delays measured using both image based and k-space based methods on three scanners. The results from two methods are almost the same on two 1.5T scanners with 2us sampling time. Though the results from two methods for the 7T scanners are different, we found if we only use the first 50 k-space samples they are much closer to each other. We can see the delays on physical X and Y axes are very close to each other and the delay on Z axis is always the largest. This is consistent with the physical properties of the gradient coils in the body coils since the gradient coil in the axis direction is different from the other two coils. Furthermore, the eddy current model parameters are very close on physical X and Y axes while the parameters on Z are more different.

Table 1.

Anisotropic delays (in us) on different physical gradient axis obtained using image domain based method (row 2–4) and k-space domain based method using the first 100 samples (row 5–6).

Scanners		Avanto	Sonata	7T
Methods and axis
Image based	Physical X	8.50	14.75	5.50
	Physical Y	9.75	15.00	5.50
	Physical Z	11.50	17.00	7.50
K-space based	Physical X	8.50	14.50	6.75
	Physical Y	9.50	15.00	6.25
	Physical Z	11.50	17.00	10.00

Table 2 gives the NRMSE in the k-space defined in Eq. [14] below by using different methods. The NRMSE for three physical axes are listed for all three scanners. Here K_bench is the k-space trajectory with the same delays on all physical gradient axes. Thus an NRMSE value less/larger than one means the deviation in k-space domain is reduced/increased. The RMSE of the proposed method is always lower than the uncorrected and delay model. The RMSE is actually reduced to less than 20 percent in all cases except on the z axis of the Avanto.

$$\text{NRMSE}=\text{RMSE}({\mathrm{K}}_{\text{proposed}},{\mathrm{K}}_{\mathrm{a}})/\text{RMSE}({\mathrm{K}}_{\text{bench}},{\mathrm{K}}_{\mathrm{a}})$$ [14]

Table 2.

NRMSE in the k-space domain on three scanners. We used eight segments with 1024 samples in each segment for 1.5T scanners and four segments for the 7T scanner. Here 1.0 means the error is the same as using the same delay on all physical gradient axes.

Scanners		Avanto	Sonata	7T
Methods and axis
Delay only	Physical X	0.57	0.79	1.43
	Physical Y	0.94	1.00	1.25
	Physical Z	3.62	2.26	0.68
Proposed	Physical X	0.09	0.08	0.16
	Physical Y	0.10	0.12	0.13
	Physical Z	0.25	0.15	0.13

In Fig. 6, the differences between the benchmark and images reconstructed using both models are shown in three different views from the calibration data set collected on the Avanto scanner. The artifacts around the edge are mostly removed in Fig. 6(a–c). The residual error is removed almost completely in the difference images in Fig. 6(d–f). The NRMSE and NMaxE are consistent with the artifact level.

Figure 6.

Difference images of the sphere phantom between benchmark and reconstruction using theoretical k-space trajectory (a–c), using delay model (d–f) and the proposed method (g–i). (a) Coronal view: NRMSE = 1..000, NMaxE = 1.000. (b) Sagittal view: NRMSE = 1.000, NMaxE = 1.000. (c) Transverse view: NRMSE = 1.000, NMaxE = 1.000. (d) Coronal view: NRMSE = 0.457, NMaxE = 0.840. (e) Sagittal view: NRMSE = 0.542, NMaxE = 0.855. (f) Transverse view: NRMSE = 0.785, NMaxE = 0.725. (g) Coronal view: NRMSE = 0.279, NMaxE = 0.202. (h) Sagittal view: NRMSE = 0.372, NMaxE = 0.194. (i) Transverse view: NRMSE = 0.496, NMaxE = 0.170.

In Fig. 7, reconstructed images from a resolution phantom are shown in (a–c) and the absolute difference between (a–c) and the goal image are shown in (d–f). The data was collected on an oblique slice (C > T 20 degrees) nine months after the calibration dataset was obtained on Avanto using the body coil. The FOV is 250mm and there are 18 interleaves. The uncorrected image in Fig. 7a has artifacts around the edge and its shape is distorted. Note that the delay only model actually has worse NMaxE in Fig 7b while the proposed method can remove most artifacts consistently as on the calibration dataset. The improvements can be easily seen in the zoomed images in (g–h) in the areas indicated by the arrows.

Figure 7.

Reconstructed images of an oblique (C > T 20 degrees) slice in the resolution phantom and their absolute difference images to the goal image. The difference images are brightened to the same scale for more details. (a) Uncorrected: NRMSE = 1.000, NMaxE = 1.000. (b) Delay only: NRMSE = 0.803, NMaxE = 1.162. (c) Proposed: NRMSE = 0.343, NMaxE = 0.274. (d) Difference between (a) and the goal image. (e) Difference between the (b) and goal image. (f) Difference between the (c) and goal image. (g) Zoomed in image of the uncorrected image. (h) Zoomed in image of delay only result. (i) Zoomed in image of the proposed combined model. The arrows indicate regions of improvement.

In Fig. 8, in vivo images from a healthy human brain in the coronal view are shown. The data was collected using a twelve channel head coil on a Siemens Avanto scanner. The FOV is 280mm and the slice thickness is 5mm. Compared with the uncorrected method results in Fig. 8(a) and Fig. 8(d), the delay only method improved the image quality as shown in Fig. 8(b) and Fig. 8(e). In the zoomed in images, we can see that the central part of the image using our proposed method is better than the uncorrected image and the delay only method, particularly in the regions indicated by the arrows.

Figure 8.

In vivo coronal images of a healthy human brain (a–c). Zoomed images of the central parts of (a–c) are shown in (d–f). (a, d) Uncorrected. (b, e) Delay only. (c, f) Proposed method. The proposed method produces the sharpest image. The arrows indicate regions of improvement.

DISCUSSION

k-space trajectory infidelity is one of the main obstacles to a reliable spiral scanning implementation. There exist methods to measure the k-space trajectory, but it is not practical to do that for every imaging slice. The method proposed here measures the anisotropic delay on each different physical gradient axis and minimizes the k-space trajectory deviation by introducing a simple eddy current model to compensate the residual error. Using the estimated spiral trajectory in reconstruction, we can eliminate most of the artifacts around the edge and remove the shading and shape distortion partially in the phantom and in vivo images.

Measured k-space trajectories have been adopted to improve image quality by many researchers. In this work, we measured the anisotropic delays and calculated the resulting k-space trajectory using the rotation matrix to compensate for their effects in spiral image reconstruction. After adopting the delay only model, we found the residual error required further attention. In order to quantify the improvements, we adopted the same delays on all gradients axes as the benchmark for a comparison between the delay only model and the proposed combined model.

It is interesting to notice that the anisotropic delay model can sometimes increase the error in k-space trajectory and image domain as well. This is one reason we should include the eddy current model to account for the remaining error in k-space trajectory. The proposed model has consistent performance compared with the delay only model.

In some cases the k-space NRMSE increased after applying the anisotropic delay model, even though the overall image quality is improved. The reason for this is that the NRMSE is based on the entire k-space trajectory, while the image quality is more dependent on the k-space samples in the center. The relatively high NMRSE also explains the remaining edge artifacts when using the anisotropic delay model, because there are residual deviations in the outer part of k-space, which contains the high frequency information of the object.

We found the gradient delays based on the NRMSE in the k-space domain are dependent on the number of samples used to calculate the NRMSE; i.e., the estimated delay is not a constant over the entire spiral by the minimum NRMSE standard in k-space. This is different from radial imaging, which has a very short readout gradient. The eddy current model can compensate most of the changing delay effect as the spiral extends to the outer k-space. Thus it reduces the deviation between the estimated spiral and the actual spiral.

In fact, we can divide the residual error in one spiral interleaf into many time segments to get different Taylor expansion parameters for each segment. Since the first order Taylor expansion is a linear approximation, it will give a more accurate estimation in a smaller time span. Thus we can get a better estimate of the impulse response function. The tradeoff is that the parameter variance over different interleaves increases when more time segments are used.

The first order Taylor approximation gives a reasonable estimation for the k-space residual error. The model parameters are nearly the same on each spiral interleaf. However, when we try to use higher order approximation, we find the coefficients are more dependent on the spiral interleaf number; i.e. they are angularly dependent in k-space. So it would be beneficial to calibrate with as many interleaves as possible and save the coefficients in a table for future reference if higher order error is of concern.

There are three terms in the first order Taylor expansion for the simple eddy current model. The first one is actually a scaled physical gradient term that compensates for scaling differences between the gradient axes. The second and third term are coupled together to account for the different slew rate response. The contribution from the first term is commensurate as the other two terms.

The k-space trajectory measurement method adopted here could also be used to measure the cross terms induced by coupling. We measured the effects of the cross terms and found they were nearly negligible. The image quality was not significantly affected by the cross terms, so we did not include those terms in our model.

The delays on each physical gradient axis are dependent on the receiver bandwidth as previously mentioned by Speier and Trautwein in (14). In their work, they measured gradient delays on five different Magnetom systems (Siemens AG, Erlangen) and found the delays were nearly isotropic, although the physical z axis had the largest delay. They also introduced a linear approximation for the delays as a function of bandwidth on different systems. In this study, we kept the bandwidth fixed, but a more general model could incorporate the delay variations caused by bandwidth changes. This could be done either by using a linear approximation or by saving the parameters corresponding to a finite number of sampling rates.

CONCLUSIONS

We measured significant gradient delay asymmetry causing severe image artifacts as in Fig. 2. In order to reduce the artifacts from k-space trajectory infidelity, we proposed an estimation method that combines tuning the delays on different physical gradient axes and an additional eddy current compensation model. We tested the method on phantoms and in vivo on three scanners. The proposed method improved image quality on all cases. The RMSE and peak difference have been reduced substantially. Furthermore, there is little time penalty for using this method after the one-time system calibration. We have demonstrated the benefits of the combined model in spiral imaging; however, it should be fairly easy to extend it to other non-Cartesian trajectories.