Jan Ole Pedersen^{1,2,3}, Christian G. Hanson^{4}, Rong Xue^{5,6}, and Lars G. Hanson^{7,8}

Determining k-space trajectories inductively is conceptually simple, but rely on integration of the induced signal. Performing this integration digitally allow for higher degree of flexibility than analog integration, which is necessary to account for, e.g., refocusing RF pulses. Digital integration, however, require high bandwidth sampling of the induced signal as digitization error accumulate, making the overall approach less attractive. We show that the necessary bandwidth can be reduced by performing regularization using a gradient coil current measure.

The k-space location is given by

$$k_r(t) = \gamma\int_0^t G_r(\tau)\, d\tau$$,

where $$$ k_r(t) = 0$$$ at $$$ t=0$$$, and $$$G_r(t)$$$ is the gradient in direction $$$r$$$. An inductive gradient measure was obtained by placing a pick-up coil (20 windings, Ø3cm) in a gradient field and summing discrete samples of the generated voltage, $$$v_r(t)$$$:

$$G_r^v(t) = \sum_{i=1}^N b_1v_r(\tau_i) \Delta t + c$$.

In this sum over $$$N=(t-t_1)/\Delta t$$$ samples, $$$b_1$$$ is a constant depending on the position and geometry of the pick-up coil, $$$t_1$$$ is the time point of the first sample where $$$\Delta t$$$ is the sampling dwell time, $$$\tau_i = (t_1 + i \Delta t)$$$, and $$$c$$$ is a constant accounting for any gradient applied at $$$t=0$$$. A simple, regularized, inductive measure of the gradient field can then be obtained as

$$G_r^M(t) = \sum_{i=1}^N [b_1 v_r(\tau_1) \Delta t - R(i)]+ c$$,

with a regularization term:

$$R(i) = \eta[G_r^M(\tau_{i-1} - b_2 I(\tau_{i-1}) ]^2$$,

$$$b_2$$$ is a predetermined scaling factor, and $$$\eta$$$ ensures the correct sign of $$$R(i)$$$, so that $$$G_r^M(t)$$$ is relatively slowly pulled towards the current-based gradient measure,$$$b_2 I(t)$$$. A measure of $$$G_r^M(t)$$$ was in near real-time generated by custom open-source circuitry[3] that features ADCs sampling $$$v_r(t)$$$ and $$$I(t)$$$ (16-bit@200kS/s). The gradient measure was used for phase modulation of a carrier signal at the Larmor frequency, which was transmitted to a single receive channel of a 3T Achieva system (Philips, Best, The Netherlands). This allows for determining the k-space trajectory from the accumulated phase of the scanner-acquired signal, while the remaining receive channels concurrently acquire the MR signal.

[1] Senaj V et al. Inductive measurement of magnetic field gradients for magnetic resonance imaging. Rev. Sci. Instrum., 69(6):2400-2405 (1998)

[2] Spielman DM and Pauly JM. Spiral imaging on a small-bore system at 4.7T. Magn. Reson. Med., 34(4):580-585, (1995)

[3] Pedersen J et al. General purpose electronics for real-time processing and encoding of non-MR data in MR acquisitions. Concepts Magn. Reson. Part B Magn. Reson. Eng, e21385 (2018)

[4] Duyn J et al. Simple Correction Method for k-Space Trajectory Deviations in MRI, J. Magn. Reson., 132, 150-153 (1998)

Reconstructed EPI data using k-space trajectories based on [left] solely a digitally integrated inductive measure ($$$G_r^v(t)$$$), [mid] scaling of a gradient coil current measure ($$$b_2 I(t))$$$, and [right] a measure obtained by regularizing the inductive measure by the current-based measure ($$$G_r^M(t)$$$).

Equally scaled differences between the images presented in figure 1, and a reference image reconstructed using a k-space trajectory obtained using Duyn's method.

Reconstructed spiral data, using the [left] regularized, inductive k-space trajectory measure ($$$G_r^M(t)$$$), and [right] the nominal k-space trajectory.

Equally scaled differences between the images presented in figure 3, and a reference image reconstructed using a k-space trajectory obtained using Duyn's method.