This work describes a method for measuring the 0th and 1st order gradient impulse response functions (GIRF) using existing intrinsic MR methods and for applying these response functions to accurately estimate the k-space trajectory and B0 phase errors in the presence of eddy currents and channel delays. Measurement of the GIRF is rapid (< 3 minutes) and needs only be done once for a given gradient system. This method will be shown to reduce gradient related artifacts in the context of 2D spiral imaging.

Two-dimensional multi-slice Spiral and Cartesian data were acquired on a healthy volunteer. For the spiral acquisition, previously determined channel delays and gradient corrections were disabled to highlight the proposed method (eddy current correction was not disabled). The process illustrated in figure 1 was used to measure the B0 eddy current response, H₀, and the linear gradient field response, H₁. An impulse response measurement framework was used as set forth by Vannesjo et al¹, but the actual measurement was accomplished with a birdcage coil and a uniform spherical phantom using the B0 eddy current measurement method of Brodsky et al².

The phase produced by B0 eddy currents can be predicted according to

$$\hat \theta_{B0}(\omega) = G_x(\omega) H_{0x}(\omega) + G_y(\omega) H_{0y}(\omega) + G_z(\omega) H_{0z}(\omega),$$

and the actual k-space trajectory can be predicted according to

$$\hat k_l(\omega) = k_l(\omega) H_{1l}(\omega),$$

where $$$\hat \theta_{B0}$$$ is the predicted B0 eddy current induced phase, $$$H_{0l}$$$ and $$$H_{1l}$$$ are the measured 0th and 1st order GIRFs and $$$l$$$ is one of the three physical gradient axes $$$l \in \{x, y, z\}$$$, $$$G_l$$$ are the theoretical gradient waveforms, and $$$\hat k_l$$$ and $$$k_l$$$ are the predicted and theoretical k-space coordinates. Artifact reduction was achieved by applying the conjugate of the predicted B0 phase to the spiral data and by reconstructing this data using the predicted k-space coordinates.

To validate the predicted k-space coordinates, the actual k-space trajectory was independently measured using the self-encoding gradient method of Alley et al³. This method requires several minutes to measure all interleaves of a 2D spiral trajectory but is very accurate. Similarly, the B0 eddy current induced phase from the spiral trajectory was measured independently using the method of Brodsky et al². All data processing and reconstruction was performed in GPI⁴.

Spiral data reconstructed with and without correction are shown in figure 2. Cartesian data are also shown as a reference. As is evident from the figure, the B0 eddy current phase and k-space trajectory predicted using the measured H₀ and H₁ GIRFs provide substantial improvements in image quality. The predicted k-space trajectory is compared in figure 3 to the designed theoretical spiral trajectory and the independently measured spiral trajectory. As can be seen from the figure, the predicted trajectory very closely follows the measured trajectory, and both differ from the theoretical trajectory. Figure 4 compares the independently measured B0 eddy current induced phase to the predicted phase. The predicted phase follows the shape of the measured phase.

This work has presented a framework for predicting the effects of eddy currents, channel delays, and other gradient errors using a fast, one-time measurement. It should be noted that the k-space trajectory measurements extracted using this method also benefit in accuracy from the removal of the phase associated with B0 eddy currents. To our knowledge, the effects of B0 eddy current induced phase have not been taken into account in any of the previously published methods that measure k-space using a thin-slice excitation.

The predicted B0 eddy current phase and k-space trajectory follow the measured trajectories very closely. The match is less good for the B0 eddy current phase; however, the match is close enough that image quality is similar whether the measured or predicted phase is used to correct the spiral data.

While the application here was spiral MRI, the measured GIRFs can be applied with equal efficacy to projection reconstruction MRI, EPI, or any other acquisition as long as the theoretical gradient waveforms are known.