Fast switching of shims and gradients, as used for example in dynamic shimming applications, relies on settling of shim fields within milliseconds. This is generally inhibited due to long living eddy currents induced at shim switching. Pre-emphasis can be applied to counteract these distortions either using pure exponential terms ^1,2 or designed from measured impulse responses ³. However, these approaches are limited when considering self- and cross-terms terms individually, which disregards the perturbations that pre-emphasis itself causes via cross-term. By regarding the shim system as one multiple-input / multiple-output system, a full matrix description can be employed which permits pre-emphasis of all self- and cross-terms at once and in a fully mutually compensated fashion.

A MR shim system can be regarded as a linear time invariant system: the output is, in the frequency domain, the multiplication of the input and the shim’s impulse response function H. For a multiple input / multiple output system, $$$H(\omega)$$$ is a matrix of frequency responses including self-terms on the diagonal and cross-terms on the off-diagonal.

The frequency response of a full 3rd order shim system was measured as detailed in Vannesjö et al⁴ using frequency swept pulses on each channel successively. The shims were controlled via digital-to-analog converters (25kS/s, National Instrument) connected to the shim amplifiers (Resonance Research Inc., Billerica, USA). The output was measured with a field camera (16 NMR field probes) connected to a standalone spectrometer. ⁵

To determine $$$H(\omega)$$$ experimentally, it is necessary that the number of linearly independent measurements equals the number of degrees of freedom of the system. The shim system has 16 degrees of freedom, therefore a full characterization of the system requires 16 linearly independent measurements and yield to a 16x16 matrix for the impulse response function $$$H(\omega)$$$.

A desired output of the shim system $$$O(\omega)$$$ can be achieved by filtering the input $$$I(\omega)$$$ with a pre-emphasis filter $$$P(\omega)$$$ such that $$$O(\omega) = H(\omega)P(\omega)I(\omega)$$$. Ideally, $$$P(\omega)=H(\omega)^{-1}$$$ such that the output exactly matches the input. This is, however, not feasible due to hardware limitations and hence a target system response $$$H_T(w)$$$ is defined and $$$P(\omega)=H(\omega)^{-1}H_T(\omega)$$$. In our case $$$H_T(\omega)=e^{-(\omega /\omega _0)^2} – 1$$$ where $$$\omega_0$$$ is 1kHz which correspond to a shim settling time of 1ms and $$$1$$$ is the unity matrix. $$$HP(\omega) = H(\omega)P(\omega)$$$ is referred to as matrix pre-emphasis.

$$$HP(\omega)$$$ was measured using the same method as for the shim system response measurement with the difference that the filter $$$P(\omega)$$$ was digitally applied to the input frequency sweeps.

Matrix pre-emphasis was used for dynamic shimming in vivo: 5 volunteers were measured (32 channel receive array, NOVA Medical). A B0 map of the volunteer covering the imaging volume and matching the targeted geometry (off-centre & angulation) was first acquired (50 transversal slices, 3.5x3.5x2mm3, TE 3ms, dTE 1ms, FOV 240x240mm2). Based on 16 measured shims field maps, slice-wise optimal shim values were calculated using a least-square minimization constrained such as to remain within the hardware limitation. The fit included the slice itself plus 2 slices above and 2 slices below it (Fig.1a). A nominal waveform was computed for slice-wise shim update (Fig.1b). The update was done 5ms prior to next slice excitation. The nominal waveform was filtered with $$$HP(\omega)$$$ to yield a pre-emphasized waveform.

EPI were acquired (50 transversal slices, 1.25x1.25x2mm3, slice TR 50ms, FOV 220x180mm2, SENSE factor 4) first using the nominal waveform for dynamic shim update, second, using the pre-emphasized waveform, with the global shim settings as determined by the host system and without shims. Images were reconstructed at the scanner. Magnitude data underlying the B0 map were used to outline the contour of the anatomy of the volunteer’s brain on the figures.

Fig.1 shows the measured SIRFs without pre-emphasis $$$H(\omega)$$$ (up) and with matrix pre-emphasis $$$HP(\omega)$$$ (bottom). Without pre-emphasis, the shapes of the self-term and the cross-terms vary substantially. With pre-emphasis, the self-terms are close to the targeted response and the cross-terms are reduced to noise level.

Fig.3 shows in vivo data. Globally shimmed data (2nd column) exhibit distortions (slices 1-3) and dropouts (slices 32&36) which get mostly corrected when using matrix pre-emphasis dynamic shimming. Data acquired with dynamic shimming but without matrix PE (3rd column) show pronounced distortions especially in slices 48&44.

In this work we showed that full matrix pre-emphasis is feasible. It requires knowledge of the entire matrix pre-emphasis and full digital control of the shim inputs. It is the key to fast shim switching and was successfully used for dynamic shimming of EPIs.