MR systems often have 2^nd order, 3^rd order or even very high order spherical harmonic shim terms for B₀ shimming. However, most B₀ shimming methods make the assumption that the generated B₀ shim fields perfectly match spherical harmonic functions, while real B₀ shim fields can deviate greatly from the ideal ones.

In this work, we measured and modelled the generated real shim fields for each shim term (and at a range of amplitudes) and used these models to achieve a better B₀ shim in the human brain at 9.4T.

A 28 channel Resonance Research Inc (Billerica, MA) insert shim was used with a 9.4T Siemens (Erlangen, Germany) scanner. The insert shim consisted of complete 2^nd, 3^rd and 4^th as well as partial 5^th and 6^th order spherical harmonics.

The fields were measured on a 150mm spherical phantom. A 2D GRE sequence with the following parameters was used: resolution of 128x128; 200x200mm FOV; slice thickness of 3mm; 40 slices with a 20% distance factor; TR = 1200ms; TE=4.00/4.76 ms; read-out bandwidth of 1500Hz/px.

To account for possible amplitudes nonlinearities, the shim fields were measured at a range of field strengths. Each shim term field was measured at: -1.0A, -0.5A, -0.2A, -0.1A, 0.1A, 0.2A, 0.5A, 1.0A and decomposed. Therefore each term has 8 field measurements (at each of the different amplitudes) and each field measurement was characterised using a 6^th order decomposition.

The real shim field models were then incorporated into a custom-written MATLAB^TM shim algorithm resembling the one mentioned in [1] and tested by B₀ shimming a head-and-shoulder phantom. The shape of the volume-of-interest (VOI) could be arbitrary and was not only restricted to a rectangle. We used the following iterative algorithm to include the amplitude nonlinearities of each shim term:

1) Initialise field model coefficients using the field measured (at 100mA).

2) Calculate the shim values.

3) Replace the field model coefficients with an update rule.

4) If the shim currents do not change then STOP, otherwise go to step 2.

Three different update rules were used (see fig. 1); firstly, the amplitudes were assumed to be linear and field coefficients updated using a linearly regression model [2]; secondly, an iterative nearest-neighbour approach was used (the field model used was the one closest to the amplitude of the shim term); lastly, an iterative linear-piecewise approach for shimming; that is, the coefficients were interpolated between adjacent amplitude fields (i.e. between -1.0A to -0.5A, then between -0.5 to -0.2A and so forth).

The best suited real-field models were then used to shim a whole-brain VOI in vivo on 3 volunteers using 2^nd, 3^rd, 4^th and partial 5^th order. The resulting shim quality was thereafter compared to the shim results obtained when assuming perfect fields. The parameters for the B₀ mapping scans were the same as for the phantom (except with 0% distance factor).

All shim terms were found to behave linearly with respect to the amplitude (e.g. fig. 1) and so the linearly regressed model was used to characterise each term as input for the shim algorithm.

For 2^nd order shimming, both the perfect-field shims and real-field shims gave similar results indicating that the fields generated by the 2^nd order shim coils were almost pure (top images of fig. 2, 3). However, fig. 3 shows that assuming perfect-fields for 3^rd and higher order B₀ shimming was not reasonable and the real-field models need to be considered. Therefore for very high order B0 shimming, real-field characterisation is undoubtedly required.

Furthermore, from fig. 4 and 5 we can see a constant improvement in the B₀ homogeneity as we include higher order spherical harmonic shim fields (the figure shows results when the real-field models are considered). Fig. 5 shows a quantitative summary of the results from all 3 volunteers. Note that higher order shimming showed better B₀ homogeneity only when the real-field models for each B₀ shim term were used.

Higher-order B₀ shimming at 9.4T shows a noticeably better performance compared to low order B₀ shimming and consistently improves as more orders are used (as previously shown [3]).

We also showed that there is a need to model the real-fields produced by each shim term when shimming the B₀ field [4]. The fields were shown to be approximately linear with respect to the amplitude. Finally, the field models were confirmed with in vivo measurements.