Spatiotemporal magnetic field fluctuations induced by physiological motion or scanner imperfections lead to artifacts in MR imaging^1–4. It has recently been shown that real-time field control⁵ enables stabilization of the field and reduction of related image artifacts. The system is based on multiple NMR field probes to measure spatiotemporal field changes^6,7 and a proportional-integral (PI) controller actuating a full 3^rd-order spherical harmonics shim system to counteract deviations from the desired field configuration.

Including shim pre-emphasis for self-terms and selected cross-terms to stabilize the controller output was shown to further enhance field control⁸. However, this implementation neglected the influence of several weaker cross-terms as well as cross-coupling induced by the applied cross-term correction itself.

To address these limitations, we include full 3^rd-order matrix pre-emphasis as an additional filter in the control loop, which directly corrects for all self- and cross-term distortions.

The ultimate goal of the feedback field control system (Fig. 1) is to produce a correction field ($$$b_c$$$) which counteracts disturbances ($$$b_d$$$) such that the total field ($$$b$$$) equals the reference field ($$$b_r$$$). This is achieved by a controller ($$$C$$$) that tries to minimize the error ($$$e$$$) between the reference and the measured total field, and a plant ($$$P$$$, representing the frequency response of a full 3rd-order shim system⁹) which produces the correction field ($$$b_c$$$).

To improve control performance, the controller was separated into a pure PI controller ($$$C$$$), and a filter for shaping and decoupling of the control channels by matrix pre-emphasis ($$$H$$$).
The controller was tuned for maximum control bandwidth while keeping noise amplification limited.

The pre-emphasis filter ($$$H(\omega)$$$) was designed as a combination of the desired target response

$$H_T(\omega)=e^{-(\omega/1300)^2} Id$$

and the inverse of the known plant response matrix $$$P(\omega)$$$ ¹⁰

$$H(\omega)=P(\omega)^{-1}\cdot{H_T(\omega)}$$

Sensitivity function
To evaluate the response of the control system to external disturbances, the sensitivity function¹¹

$$S(\omega)=(Id+P(\omega)H(\omega)C(\omega))^{-1}$$

$$b(\omega)=S(\omega)b_d(\omega)$$

was obtained by applying multisine¹³ input disturbances ($$$u_d$$$) and measuring the response with and without active field feedback:

$$b^{(feedback)}(\omega)=S(\omega)P(\omega)u_d(\omega)$$

$$b^{(no feedback)}(\omega)=P(\omega)u_d(\omega)$$

In order to calculate the full 16x16 sensitivity function matrix, 16 measurements of $$$b^{(feedback)}$$$ and $$$b^{(no feedback)}$$$ were combined into matrices $$$B^{(feedback)}$$$ and $$$B^{(no feedback)}$$$, respectively,:

$$S(\omega)=B^{(feedback)}(\omega)\cdot B^{(no feedback)}(\omega)^{-1}=[S(\omega)P(\omega)U_d(\omega)][P(\omega)U_d(\omega)]^{-1}$$

The measurement was repeated once with and once without active pre-emphasis in the control loop.

Step Response
To evaluate disturbance rejection efficiency, time domain responses to step disturbances in single channels were observed and the speed of the control was determined by the time needed to reject the disturbance to ≤ 3% of the original step amplitude.

Imaging
During a T2*-weighted gradient echo scan (TR=1sec, TE=25ms, resolution=(0.6mm)², slice thickness=1.5mm, 10 slices) a volunteer was instructed to move his arm up to his chin and down again every 20 seconds. Imaging results obtained with and without field control, as well as with and without active pre-emphasis in the control loop were evaluated.

Sensitivity
Measured sensitivity functions without pre-emphasis (Fig. 2, blue lines) showed varying self-term responses (diagonal elements) and some strongly coupling 3^rd-order channels amplifying the input disturbances by up to a factor of 5 (gray background). When using pre-emphasis in the control loop (red lines), the self-term responses were all the same and cross-terms were suppressed to below 0.12.

Step response
Without pre-emphasis, self-term responses exhibited undershoot and were generally slower than with pre-emphasis (Fig. 3 a&b). Applying pre-emphasis resulted in the actually applied correction steps ($$$\bf\it{b_c}$$$) matching the desired control field ($$$\bf\it{u_c}$$$) thus preventing undershoot. Strong cross-coupling induced fields of up to 220% the original disturbance amplitude (Fig. 3c) that led to disturbance rejection times of ≥ 2.5 seconds (Fig. 4). When pre-emphasis was applied, all cross-terms were suppressed to below 6.5% and disturbance rejection times were reduced to ≤ 1.2 seconds.

Imaging
Field control without pre-emphasis counteracted field distortions from hand motion but induced additional oscillations due to cross-coupling. Including pre-emphasis in the control loop successfully suppressed these oscillations and strongly improved the quality of the obtained T2*-weighted images (Fig. 5).

We showed that including pre-emphasis in the field control loop increases the achievable control bandwidth. This is due to all self-term responses being equal, hence optimal tuning of all channels can be achieved simultaneously. Additionally, counteracting coupling between shim channels directly decreases disturbance rejection times and noise amplification.

The proposed method enables stable field control also in demanding situations with disturbances of high spatial and temporal variability such as caused by limb motion of the subject.