Introduction

In MRI gradient systems, feedback control based on measuring the output current eliminates the perturbations that arise from nonlinearities and time-varying parameters. These perturbations affect recently introduced feedforward controllers¹, which are preferable for gradient array and multi-coil designs^{2, 3}. In addition to the nonlinear modeling⁴, the feedforward model must consider time-varying parameters due to gradient heating. Here we extend our previous study that integrated the gradient coil resistance prediction into the feedforward controller⁵ and added a method for obtaining thermal and electrical model parameters for both the gradient coil and the gradient power amplifier with an improved thermal model that accounts both coil and the amplifier independently.

Method

The proposed system model (Figure 1) has three electrical and four thermal parameters. The electrical parameters R_L(t), R_SW(t) and L represent coil series resistance, lumped amplifier switch resistance, and coil inductance. The resistance variations are predicted using the thermal model with R_T and C_T parameters. This prediction is made separately for the lumped amplifier and coil resistance as they heat and cool at different rates. The thermal model is dependent on the real power loss on the gradient coil, while the power is dependent on the thermal model output. The time constant in the thermal model can be approximated as:
$$\tau\left(I\right)=\frac{R_TC_T}{1-\alpha_RR_0R_TI^2}$$
This equation implies that the system becomes unstable above a threshold. A calibration sequence is used to determine the model parameters, in which a known arbitrary voltage waveform is applied to the gradient coil via the gradient amplifiers for 6 minutes. Snapshots of the voltage and current waveforms are taken during this period. Since the electrical system model describes the voltage waveform as a linear combination of current and the time derivative of the current, the instantaneous R and L values can be calculated using the Gram matrix shown below.
$$\left[{\begin{array}{*{20}{c}}{\left\langle{I(t),I(t)}\right\rangle}&{\left\langle{I(t),\frac{{dI(t)}}{{dt}}}\right\rangle}\\{\left\langle{\frac{{dI(t)}}{{dt}},I(t)}\right\rangle}&{\left\langle{\frac{{dI(t)}}{{dt}},\frac{{dI(t)}}{{dt}}}\right\rangle}\end{array}}\right]\left[{\begin{array}{*{20}{c}}R\\L\end{array}}\right]=\left[{\begin{array}{*{20}{c}}{\left\langle{V(t),I(t)}\right\rangle}\\{\left\langle{V(t),\frac{{dI(t)}}{{dt}}}\right\rangle}\end{array}}\right]$$
For each voltage-current pair, the Gram matrix is solved to obtain R_L(t) and L values. Additionally, a single voltage waveform is measured without connecting the coil to find the voltage drop over the amplifier. This method allows us to obtain R_L(t)+R_SW(t) and L values from the Gram matrix. Since this process is carried out over a 6-minute period, the change in resistance caused by the self-heating of the resistive elements is visible. The thermal model parameters are fitted to the data using the Adam optimizer.
Because the thermal time constant is much larger than the electrical time constant, the thermal model operates independently and updates the resistance parameter of the feedforward model. The feedforward controller is implemented digitally on an Kintex Ultrascale KCU105 FPGA evaluation board. The images are obtained using a Siemens TimTrio 3T scanner. The z-direction is driven by a homemade gradient system composed of a gradient coil and an amplifier, while the scanner drives the x and y-directions. A trigger pulse is taken from the scanner, synchronizing our applied gradient with the scanner gradients. Since an external gradient coil is used, a homemade shielded Tx/Rx birdcage coil is used for RF transmission and reception inside the gradient coil. A photograph of the experimental setup is provided in Figure 2.
A multi-shot multi-slice EPI sequence (Matrix: 176x175; resolution: 0.86x0.86 mm²; FOV:150mm; slices: 6; slice spacing: 9mm; slice thickness: 6mm; flip angle: 16°; TR: 150ms; TE: 13ms; echo spacing: 3ms) is used for the imaging, which is developed on the open-source Pulseq framework⁶ (Figure 3). Number of scanned lines per TR is 7. The total scan time to acquire an image is 3.9 seconds. This results in a 25.3A RMS current while the entire sequence takes 6 minutes (92 images). For the phantom, a tomato is used as it is small enough to properly image while having an intricate internal shape that can expose image distortions caused by heat.

Results

The MRI images in Figure 4 demonstrate that the image shrinks along the readout direction when the gradient coil and power amplifier heat up. Due to the heating, the gradient current reduces during the scan (because of temperature-dependent resistances). As a result, the width of the image reduces gradually—the current and the image width show around an 18% reduction in 6 minutes. In Figure 5, the thermal model compensates for the change in resistance, and the feedforward controller can provide more reliable gradient currents. Therefore, the image size variations become minimal, with a 2% deviation in the current.

Discussion and Conclusion

In this work, we presented a method to extract the necessary thermal and electrical properties of a gradient system. The thermal model updates the electrical model's resistance parameter, which effectively compensates for image distortion caused by the heating of the gradient coil. An increase in resistance primarily affects the field of view since increased coil resistance decreases current and, consequently, the gradient field strength. Such effects are more noticeable with dense pulse sequences that heat the coils quickly, such as the ones used for fMRI imaging. Our method shows that it is possible to model the thermal behavior and compensate for the feedforward model accordingly. In order to further decrease the error, higher-order models or compensation with temperature measurements might be preferable.

Acknowledgements

No acknowledgement found.

References

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