Interslice current change constrained B₀ shim optimization for accurate high-order dynamic shim updating with strongly reduced eddy currents

Abstract

Purpose:

To overcome existing challenges in dynamic B₀ shimming by implementing a shim optimization algorithm which limits shim current amplitudes and their temporal variation through the application of constraints and regularization terms.

Theory and Methods:

Spherical harmonic dynamic B₀ shimming is complicated by eddy currents, ill-posed optimizations, and the need for strong power supplies. Based on the fact that eddy current amplitudes are proportional to the magnitude of the shim current changes, and assuming a smoothness of the B₀ inhomogeneity variation in the slice direction, a novel algorithm was implemented to reduce eddy current generation by limiting interslice shim current changes. Shim degeneracy issues and resulting high current amplitudes are additionally addressed by penalizing high solution norms. Applicability of the proposed algorithm was validated in simulations and in phantom and in vivo measurements.

Results:

High-order dynamic shimming simulations and measurements have shown that absolute shim current amplitudes and their temporal variation can be substantially reduced with negligible loss in achievable B₀ homogeneity. Whereas conventional dynamic shim updating optimizations improve the B₀ homogeneity, on average, by a factor of 2.1 over second-order static solutions, our proposed routine reached a factor of 2.0, while simultaneously providing a 14-fold reduction of the average maximum shim current changes.

Conclusions:

The proposed algorithm substantially reduces the shim amplitudes and their temporal variation, while only marginally affecting the achievable B₀ homogeneity. As a result, it has the potential to mitigate the remaining challenges in dynamic B₀ shimming and help in making its application more readily available.

1 |. INTRODUCTION

Homogenization of the static magnetic field B₀ is a key requirement for obtaining high-quality data in most MR applications and is commonly referred to as B₀ shimming. Although B₀ is kept well under control during the manufacturing and installation of the bare magnet,^1,2 placement of tissues with varying magnetic susceptibilities into the magnetic field induces complex and high-order, subject-specific B₀ inhomogeneities.³ Standard MR systems contain sets of second- and sometimes third-order spherical harmonic (SH) shim coils⁴ that generate adjustable magnetic fields to cancel the apparent inhomogeneity terms. However, space limitations and shim coil efficiency considerations restrict the feasible number of shim coils and impose a natural limitation to this approach, leading to the inability to approximate higher-order terms. While already being problematic at clinical field strengths of 3T and below, the problem is exacerbated at higher field strengths and stands in contrast to the trend toward ultra-high-field MR systems, given that vendor-supplied means of B₀ shimming have not advanced to match demand.

This motivates the development of more-effective shim techniques, and recent approaches, such as static⁵ and dynamic⁶ multicoil shimming, integrated ΔB₀/Rx shim arrays^7,8 and very-high-order SH shimming,⁹ have already demonstrated remarkable improvements over standard static SH shimming. A further advanced approach is known as dynamic shim updating (DSU)^10,11 and has the advantage of making more-efficient use of the available SH shim coils. It exploits the fact that B₀ inhomogeneities can be better approximated over smaller volumes and thus updates shim coil currents during acquisition to be optimal for each subvolume. The initial first-order DSU experiments have been extended to also include second¹² and third¹³ orders and have proven to yield better B₀ homogeneity at ultrahigh fields¹⁴ and in both multivoxel and -slice applications,¹⁵ such as MR spectroscopy imaging¹⁶ and functional MRI.¹⁷

However, despite having been available for more than 2 decades, and notwithstanding its B₀ homogeneity improvement, DSU is not yet routine. Commonly reported complications pertaining to DSU implementation are the generation of eddy currents (ECs) through intra-acquisition shim current changes,¹⁸ an ill-posedness of the DSU optimization problem,¹² and a resulting need for stronger power supplies.¹⁹

We have found that these DSU-specific problems can be substantially reduced, if not eliminated, by appropriately constraining the DSU shim current calculation and by complementing the algorithm with a priori information about preferred solutions.²⁰ While reaching a similar B₀ homogenization performance as other state-of-the-art DSU techniques, the resulting strong shim current reduction limits potential ECs and addresses shim degeneracies at overall reduced hardware requirements. Following a detailed analysis of the DSU challenges, in this work we investigate the performance of our novel DSU shim current optimization and demonstrate its advantages in simulations and measurements.

2 |. THEORY

In contrast to conventional static shimming, DSU describes a method of changing shim currents during acquisition such as to optimally cancel residual B₀ inhomogeneities in distinct subvolumes. Calculation of DSU currents can be phrased as a minimization problem according to (Equation 1):

$$\underset{{\mathbf{x}}_k}{\min }{\Vert {\mathbf{A}}_k{\mathbf{x}}_k-{\mathbf{b}}_k\Vert }_2^2\mathrm{s.t.}\mathbf{lb}\le {\mathbf{x}}_k\le \mathbf{ub},$$ (1)

where ${\mathbf{x}}_k\in {\mathbb{R}}^{n\times 1}$ is the set of shim currents for n shim coils used to cancel residual B₀ inhomogeneities, ${\mathbf{b}}_k\in {\mathbb{R}}^{m\times 1}$, in the subvolume k, containing m mask voxels. The system matrix, ${\mathbf{A}}_k\in {\mathbb{R}}^{m\times n}$, describes the shim basis functions, and the lower and upper bounds, $\mathbf{lb},\mathbf{ub}\in {\mathbb{R}}^{n\times 1}$, are used to constrain the currents so as not to exceed the hardware limits.

However, successful application of the calculated shim currents may be compromised by DSU-specific problems. These are outlined in the following and are illustrated in Figure 1.

1. Intra-acquisition shim current changes can induce ECs in nearby conducting surfaces, which, in turn, generate time-varying magnetic fields.¹⁸ EC field amplitudes during DSU can be in the order of several hundred Hz/cmⁿ with time constants ranging up to seconds, which is high enough to make the B₀ homogeneity worse than for the static case.¹³

2. ECs are commonly handled by pre-emphasis modules, which superimpose correction currents to the shim outputs to cancel the EC fields.²¹ Therefore, sometimes up to 50% of the dynamic range of the power supply cannot be used for the purpose of shimming,²² and DSU optimization in its standard form can run into current limitations.

3. Given that they require a full 4D characterization of the ECs generated by each shim coil,²³ accurate pre-emphasis adjustments are time-consuming. Conventional MR sequence-based EC measurement techniques, for example, based on multiple projections²⁴ or full 4D EC maps,²⁵ can take up to a 1-hour measurement time per shim channel. Slight calibration inaccuracies can already translate into long-lived and uncompensated ECs with amplitudes that scale with the initial shim current change.

4. The shim basis functions are not necessarily orthogonal over the DSU volumes. This shim degeneracy degrades the linear independence of the system and renders the problem described by Equation 1 as being ill-conditioned, which can cause excessive DSU currents.²⁶

5. Shim optimization routines require system-specific shim calibrations, which relate the shim current through a shim coil with the magnitude of the resulting shim field.²⁷ Because of phase wrapping and intra-voxel dephasing artifacts induced by high shim offsets, the calibration data can often only be acquired at comparably low shim currents. Consequently, DSU shim fields calculated from conventional optimization routines, which therefore typically use higher shim currents,¹⁹ may be incorrectly modeled, when falling into a current regime not covered in the calibration procedure. The dynamic range of shim offsets that can be sampled during the calibration data acquisition depends on the efficiency of the shim coil and is generally narrower for the more-efficient lower-order shim coils.

FIGURE 1.

Challenges for DSU. A, A shim current change induces ECs in nearby conducting surfaces. Shim steps with high amplitudes S_H cause higher EC amplitudes EC_H (red) than shim steps of low amplitude (green). B, The amplitude of a pre-emphasis overshoot used to correct for shim-induced ECs is proportional to the initial change in shim current, thus high-amplitude shim steps require higher pre-emphasis amplitudes PE_H (red) than low-amplitude shim steps (green). C, Inaccuracies in the pre-emphasis adjustment cause uncompensated ECs. The offset of the correction signal (black dashed lines) scales with the original shim current change, and ECs of larger shim steps (red) will have stronger residuals than those of smaller shim steps (green). The exemplary data shown here demonstrate the effect of a 2% error in the adjustment of the amplitudes and time constants of a 3-term exponential pre-emphasis self-term correction of a B₀ coil. D, Image artefacts prohibit the acquisition of high shim offset data during the calibration of a shim system (magnitude data shown). As a consequence, the linear fit (blue line) derived from data sampled at relatively small offsets (green shaded area) might fail in modeling the shim system at high current offsets (red shaded area). For example, this can be attributed to small fitting errors (dotted) or slight power supply nonlinearities (dashed), which cannot be described by the linear model. The induced error scales with the magnitude of the calculated shim current. The exemplary data shown here qualitatively demonstrate this effect for the calibration data of a SH C2 coil of a 3T system. It also shows that, for the given acquisition, the maximum absolute offset current should be chosen to be <2 A

Being a severe source of artefacts on their own, these DSU problems are coupled and can accumulate in their negative effect on the MR signal. Nonetheless, because they originate from and are scaled by excessive dynamic shim currents and their temporal variation, their effects can be minimized if the absolute current amplitudes and the change between consecutive shim currents can be reduced substantially without affecting the achievable B₀ homogeneity.

In the following, it is hypothesized that for the general slice-wise DSU optimization problem described by Equation 1, the solution space is large enough to allow the feasible set of solutions to be strongly constrained while negligibly affecting the achievable B₀ homogeneity. In addition, it is assumed that for slice thicknesses on the order of a few millimeters, the B₀ inhomogeneities vary smoothly in slice direction and hence minor changes in DSU shim currents are sufficient to adapt to the new shim challenge as dictated by the slice acquisition order. Exploiting this knowledge by extending Equation 1 to include regularization terms and additional constraints provides a means of handling the DSU problems on the algorithmic side. By doing this, the general DSU process remains unaffected, while also reducing hardware requirements.

3 |. METHODS

3.1 |. The proposed DSU current calculation algorithm

To minimize the problems discussed in section 0, the aims of the shim calculation algorithm are to reduce the shim amplitudes and their temporal variation while still optimally cancelling residual B₀ inhomogeneities. In the proposed algorithm, this is realized by complementing Equation 1 with 2 regularization terms as well as by tightening the constraints of the shim calculation according to (Equation 2):

$$\underset{{\mathrm{x}}_k}{\min }\underset{\text{Data Term}}{\underbrace{{\Vert {\mathbf{A}}_k{\mathbf{x}}_k-{\mathbf{b}}_k\Vert }_2^2}}+\underset{\text{Abs.Term}}{\underbrace{\lambda \cdot {\Vert {\mathbf{x}}_k\Vert }_2^2}}+\underset{\text{Diff. Term}}{\underbrace{\epsilon \cdot {\Vert {\mathbf{x}}_k-{\mathbf{x}}_{k-1}\Vert }_2^2}}\mathrm{s.t.}\underset{\text{Constraints}}{\underbrace{\{\begin{array}{c}{\mathbf{lb}}_k\le {\mathbf{x}}_k\le {\mathbf{ub}}_k\\ \mathbf{C}\cdot {\mathbf{x}}_k<\mathbf{d}\end{array}}}.$$ (2)

The inclusion of the Tikhonov regularization term ${\parallel {\mathbf{x}}_k\parallel }_2^2$ gives preference to solutions with small shim amplitudes, the inclusion of ${\parallel {\mathbf{x}}_k-{\mathbf{x}}_{k-1}\parallel }_2^2$ reduces the difference to the previous set of shim values, and the parameters λ and ε control the regularization weighting. Whereas the regularization terms drag the solution toward a preferred set of shim values, the lower and upper bounds, lb_k and ub_k, are updated for every slice. This is done to additionally set an explicit limit to the maximally permitted dynamic interslice shim current change. The added linear inequality constraints, given by C and d, are used to reduce the cumulative current output of a set of shims to user-selected values, which is a consequence of the hardware configuration used in this work (for more details, see Supporting Information). A table of used variables is given in Supporting Information Table S1.

To address through-slice dephasing, shim values for each slice are calculated in a moving boxcar mode by including both adjacent slices into the fit. This additionally reduces shim degeneracies and establishes data sharing between fits of consecutive slices, which is beneficial for keeping their individual DSU solutions similar.

Although Equation 2 can be used to loop over the slices to successively calculate all shim solutions, compliance with the explicit interslice current limit will not be guaranteed between the last and first slice. Therefore, an initial step is introduced in which the shim calculation for the outermost slices is coupled in a joint optimization. Consequently, the entire algorithm consists of 3 steps, which are detailed in the following and illustrated schematically in Figure 2.

FIGURE 2.

Processing steps of the DSU algorithm illustrated for 33 slices of 1 shim channel with ${\mathrm{I}}_{\max }^{\left(\mathrm{i}\right)}$ limited to 4 A and with a maximum interslice shim current limit of v⁽ⁱ⁾ = 0.5 A. (1) DSU currents are calculated for the end slices (blue circles) and constrained such that their difference is smaller than or equal to the chosen maximum interslice difference. (2) DSU currents are calculated for the starting slice (blue cross) with the upper and lower bounds tightened to ${\mathrm{I}}_{\max }^{\left(\mathrm{i}\right)}∕2$ in order to assure full flexibility for the calculation of the subsequent slices. (3) All other slice solutions (blue bars) are calculated successively under the guidance of the respective most stringent constraints as dictated by either the chosen interslice current limit, power supply limit, or end-slice solutions. The interslice limits are displayed as red boxes with magnitudes of ±v⁽ⁱ⁾, and the solid red line illustrates how the upper and lower bounds are tightened by ${\mathrm{lc}}_k^{\left(i\right)}$ and ${\mathrm{uc}}_k^{\left(i\right)}$ given by Equation 6 in order to maintain the interslice limits with respect to the end-slice solutions

3.1.1 |. Step 1: calculation of the end-slice solutions

Standard DSU solutions, ${\mathbf{x}}_k^s$, for the 2 end slices with indices k ∈ {1, N} are calculated individually from Equation 2, with the non-slice-specific power supply current limits, ±I_max, as the lower and upper bounds and by setting ε = 0. The scalar-valued norms, S_k, of these solutions follow from (Equation 3):

$${\mathrm{S}}_k={\Vert {\mathbf{A}}_k{\mathbf{x}}_k^s-{\mathbf{b}}_k\Vert }_2^2+\lambda \cdot {\Vert {\mathbf{x}}_k^s\Vert }_2^2$$ (3)

and yield quantitative parameters as a reference for the shim performance without application of any interslice current constraints. Now let v be a vector of the explicit interslice shim cur re nt limits given by ${\mathrm{v}}^{\left(i\right)}=p^{\left(i\right)}\cdot {\mathrm{I}}_{\max }^{\left(i\right)}$ with $\{p^{\left(i\right)}\in \mathbb{R}\mid 0<p^{\left(i\right)}\le 2\}$, i being the index of the shim channel, and let ${\mathbf{M}}_1,{\mathbf{M}}_2\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$ be the diagonal matrices (Equation 4):

$${\mathbf{M}}_1=\left[\begin{array}{ccc}+1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & +1\end{array}\right]\text{and}{\mathbf{M}}_2=\left[\begin{array}{ccc}-1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & -1\end{array}\right].$$ (4)

Using the solution norms of the standard solutions, the shim current calculation for both slices can be coupled in a joint optimization framework according to (Equation 5):

$$\underset{{\mathbf{x}}_1,{\mathbf{x}}_N}{\min }\sum _{k\in \{1,N\}}{({\Vert {\mathbf{A}}_k{\mathbf{x}}_k-{\mathbf{b}}_k\Vert }_2^2+\lambda \cdot {\Vert {\mathbf{x}}_k\Vert }_2^2-{\mathrm{S}}_k)}^2\mathrm{s.t.}\{\begin{array}{c}{\mathbf{lb}}_k\le {\mathbf{x}}_k\le {\mathbf{ub}}_k\\ \mathbf{C}\cdot {\mathbf{x}}_k<\mathbf{d}\\ \left[\begin{array}{cc}{\mathbf{M}}_1& {\mathbf{M}}_2\\ {\mathbf{M}}_2& {\mathbf{M}}_1\end{array}\right]\cdot \left[\begin{array}{c}{\mathbf{x}}_1\\ {\mathbf{x}}_N\end{array}\right]\le \left[\begin{array}{c}\mathbf{v}\\ \mathbf{v}\end{array}\right]\end{array}$$ (5)

Minimization of Equation 5 simultaneously calculates the solutions x₁ and x_N, which are ideally close to the B₀ shimming capabilities of the standard noncoupled solutions for these slices. The added linear inequality constraints limit the interslice difference between the outermost slices to be equal to or smaller than the values in v.

3.1.2 |. Step 2: calculation of a solution for a starting slice

Because all slice solutions depend on the shim values of their preceding slice, it is beneficial for the first processed slice not to have a solution that is close to the power supply limits. Hence, although the processing loop can start at any slice index, the first processed slice is chosen to be the center of the stack. To further emphasize the favored condition of having low currents for the first processed slice, the upper and lower bounds are reduced to ±I_max/2, and the starting slice solution follows from solving Equation 2 with a value of ε = 0.

3.1.3 |. Step 3: calculation of all slice solutions in acquisition order

Starting with the solution of step 2, the prior knowledge about the shim values of every preceding slice serves as an initial guess for the current slice solution. This information is also used to update the lower and upper bounds in Equation 2 so that shim current changes are limited to ±v. To ensure that all calculated solutions also maintain this limit with respect to the calculated end-slice solutions of step 1, an individual set of lower and upper current constraints is introduced and calculated from integer multiple values in v and for all slice indices k following (Equation 6):

$${\mathrm{lc}}_k^{(i)}=\max \left(\left\{{\mathrm{x}}_{k-1}^{(i)}-(N-k)\cdot {\mathrm{v}}^{(i)},{\mathrm{x}}_{k-1}^{(i)}-(k-1)\cdot {\mathrm{v}}^{(i)}\right\}\right)$$

and

$${\mathrm{uc}}_k^{(i)}=\min \left(\left\{{\mathrm{x}}_{k-1}^{(i)}+(N-k)\cdot {\mathrm{v}}^{(i)},{\mathrm{x}}_{k-1}^{(i)}+(k-1)\cdot {\mathrm{v}}^{(i)}\right\}\right).$$ (6)

The final lower and upper bounds are defined as the most stringent constraints and are dictated by either the interslice limits with respect to the preceding slice, the interslice limits with respect to one of the end slices, or the power supply limits. They then follow from (Equation 7):

$${\mathrm{lb}}_k^{(i)}=\max \left(\left\{{\mathrm{x}}_{k-1}^{(i)}-{\mathrm{v}}^{(i)},{\mathrm{lc}}_k^{(i)},-{\mathrm{I}}_{\max }^{(i)}\right\}\right)\text{and}{\mathrm{ub}}_k^{(i)}=\min \left(\left\{{\mathrm{x}}_{k-1}^{(i)}+{\mathrm{v}}^{(i)},{\mathrm{uc}}_k^{(i)},{\mathrm{I}}_{\max }^{(i)}\right\}\right)$$ (7)

and define the solution space boundaries for the entire data set.

3.2 |. Hardware

All measurements were performed on a 3T Tim Trio system using a 1Tx/8Rx-channel radiofrequency coil (Siemens Healthineers, Erlangen, Germany). The second-order shims of the scanner were extended by a 28-channel SH shim coil insert driven by a power supply with ±5 A of output current per channel and a DSU unit enabling dynamic shimming with EC compensation up to full fourth and partial fifth/sixth order (Resonance Research Inc., Billerica, MA). The shim insert was certified as an in-house product according to §12 of the German Act on Medical Devices.

Dynamic first-order shim updates were initiated from the sequence and applied through the scanner gradient system. The gradient dynamic interslice shim current change was limited in the algorithm to be 10% of the channel maximum output current. Dynamic shim updates for the Z0 and all higher-order terms were triggered from the sequence and applied through the shim insert. The maximum output currents of these channels were clipped to ±4 A, and the channels supplied from a common power supply subunit (cf. Supporting Information Table S2) were additionally constrained so as not to exceed a cumulative sum of ±8 A. Their maximum interslice shim current change was limited to ±0.5 A (p⁽ⁱ⁾ = 0.1), which implied that only non-negligible ECs from shim steps up to 0.5 A required a pre-emphasis compensation. These ECs were mapped with a variant of the method proposed by Bhogal et al²⁵ and compensated by adjusting the pre-emphasis modules of the DSU unit accordingly. Dynamic shim currents were updated 2 ms before slice excitation using a ramp function. Both shim systems were calibrated, and the results, as well as the applied constraints and the EC adjustments, are summarized in Supporting Information Table S2.

3.3 |. Data acquisition and processing parameters

The imaging protocol used to assess the achievable B₀ homogeneity included acquisition of both static and dynamic field maps using a 2D multiecho gradient-recalled echo sequence with monopolar readouts, TE = [4, 5, 6, 8, 12] ms and TR = 540 ms.²⁸ Field-of-view and image matrix sizes were adjusted to cover the region of interest (ROI) in 3-mm isotropic resolution at a total acquisition time of ~3:40 minutes.

The processing software and all graphical user interfaces were written in MATLAB (R2015b; The MathWorks, Inc., Natick, MA), and the minimization of Equation 2 was performed using the built-in fmincon solver with the sqp algorithm. Phase unwrapping for field map calculation was done with FSL PRELUDE,²⁹ and the ROI for the in vivo measurements was restricted to the brain area, which was segmented using FSL BET.³⁰

Phantom experiments using a water-filled sphere were performed to set up and calibrate the DSU hardware and test the correct functionality of the proposed algorithm. Statically shimmed in vivo field maps were acquired and used to simulate the effects of constraining the available shim current hardware. Dynamically shimmed in vivo field maps were acquired and used to evaluate the performance of the proposed algorithm and quantify the achievable B₀ homogeneity. All in vivo measurements were approved by the local ethics committee, and written informed consent was obtained from participants accordingly.

4 |. RESULTS

4.1 |. Feasibility of limiting shim amplitudes and their temporal variation

To evaluate the impact of limiting the DSU current amplitudes, simulations were performed for 8 second-order statically shimmed in vivo field maps. Standard DSU solutions were calculated from Equation 1 with values for the upper and lower bounds ranging from 100% to 1% of the maximum available current output. The result for 1 data set is illustrated in Figure 3 as a plot of the residual slice-wise standard deviation as a function of the applied power supply limit. A zoomed view of the slices with highest residual B₀ inhomogeneity reveals that by reducing the limits to 25% of the maximum current output, the standard deviation in the given slices increases by less than 0.5 Hz. For all data sets, the average increase in slice-wise standard deviation, as induced by decreasing the power supply limits by a factor of 4, is 0.1 Hz.

FIGURE 3.

Residual slice-wise standard deviation after static second-order shimming, after simulated 4th⁺-order static shimming (only resultant whole-brain standard deviation shown) and after simulated 4th⁺-order dynamic shimming with different power supply limits ranging from 100% to 1% of the channel’s maximum current outputs (= ±5 A to ±0.05 A). It can be seen in all slices on the plot, and in the zoomed view on the slices with highest B₀ inhomogeneity, that the slice-wise and the whole-brain standard deviation only increases slowly when cutting back large parts of the dynamic range of the power supply. It also shows that the addition of very-low-amplitude dynamic shim fields improves the B₀ homogeneity over the static second-order case already and that the application of the high-order static shims produces an intermediate solution

To evaluate the feasibility of penalizing high interslice shim current changes, DSU shim fields were simulated for all unconstrained DSU solutions and difference maps were calculated by subtracting the simulated field in each slice from that of every subsequent slice. Figure 4 illustrates the results for 1 data set and depicts the magnitude of the change in the shim fields for 6 exemplary slices. This approach approximates the actual change of the DSU shim fields and qualitatively demonstrates that the required changes in shim current amplitude are considerably lower than the absolute amplitudes required. For the 8 processed data sets, the respective maximum absolute field offset in the difference maps was determined and then averaged over all data sets. This average maximum absolute field offset, which can be seen as a measure for the required maximum interslice shim current change, was 34 Hz.

FIGURE 4.

Interslice shim field variation illustrated for 6 exemplary slices through the prefrontal cortex area. (1) The second-order statically shimmed field map slices show the residual B₀ inhomogeneities. (2) High-order dynamic shim fields can approximate large parts of these inhomogeneities. (3) The predicted shimmed slices show a resultant strong reduction of the B₀ inhomogeneities. (4) In contrast to the original shim fields (second row), the interslice changes in the shim field amplitudes, given by subtracting the simulated shim field in each slice from that of every subsequent slice, are low and demonstrate the applicability of interslice shim current constraints. MPRAGE = magnetization prepared rapid gradient echo

4.2 |. Standard vs. proposed DSU shim optimization framework

Constraining the shim optimization, as carried out in section 4.1, demonstrates that solutions for the optimization problem exist that reach similar homogeneities at lower overall current demands. To investigate the extent to which preference can be given to these solutions by applying regularization terms, simulations were performed and the performance of a standard DSU optimization was compared to our proposed routine.

Figure 5 shows the average power supply usage for the 8 analyzed subjects. These were derived from the nonregularized routine and from our proposed routine with different regularization weightings using the parameter values λ, ε = [0.001, 0.005, 0.01]. The plots show the mean and maximum current amplitudes, as well as the mean and maximum current changes between adjacent slices for each channel of the shim insert power supply. The mean absolute current amplitudes and the mean amplitude changes were averaged for each channel over all slices and then over all data sets. The maximum absolute current amplitudes and maximum amplitude changes were determined as the respective maximum value among all slices. This was then averaged over all data sets. While all plots show a significant reduction for all regularization weightings, the strongest relative reduction is achieved for the maximum interslice current changes, which is also the dominant EC-generating parameter. The average whole-brain standard deviation, obtained when applying the given solutions, was determined as 8.10 Hz (unconstrained) versus 8.29 Hz (λ, ε = 0.001), 8.54 Hz (λ, ε = 0.005), and 8.76 Hz (λ, ε = 0.01), as compared to 17.14 Hz for the static case. Depending on the chosen regularization parameter value, the slight degradation in whole-brain standard deviation for the regularized versus the standard optimization comes with the advantage of an average 6.3-, 14.2-, or even 23.0-fold reduction in the maximum DSU shim current change, and a 4.2-, 11.4-, or 18.2-fold reduction of the mean DSU shim current change.

FIGURE 5.

Shim current amplitude evaluation for simulated solutions obtained from and averaged over 8 in vivo data sets. The current values are shown for the channels of the different SH orders of the shim insert system. A, Mean absolute DSU amplitude of each channel averaged over all slices and then over all data sets. B, Highest absolute DSU amplitude of each channel calculated as the maximum absolute current among all slices, which was then averaged over all data sets. C, Mean absolute DSU current difference between adjacent slices for each channel averaged over all slices and then over all data sets. D, Highest absolute DSU current difference between adjacent slices for each channel calculated as the maximum absolute DSU current difference among all slices, which was then averaged over all data sets

Figure 6 visualizes the difference in temporal shim current variation between the solutions derived from the standard approach versus our proposed algorithm, with a regularization weighting of λ, ε = 0.01. The shim current amplitudes from the given subject are plotted for 1 exemplary term from each SH shim order greater than n = 1. It can be seen that our proposed routine reduces the absolute current amplitudes and imposes a smoothness on the temporal variation, thus supporting the findings displayed in Figure 5.

FIGURE 6.

Comparison between the results obtained from a standard DSU routine versus our proposed routine with λ, ε = 0.01. A, Temporal shim current variation derived from our proposed routine (top row) and from the standard routine (bottom row) shown for 1 exemplary term of each SH shim order greater than n = 1. The magnitude of the change in shim current is color-coded. B, Acquired second-order static field map slices and predicted DSU field map slices for the unconstrained case and for our proposed routine. The regularized solution shows a significant improvement over the static shim solution, and only minor differences exist to the unconstrained solution. For display purposes, only every fourth slice of the data set is shown

Additionally, Figure 6 shows simulated field maps and illustrates how the regularized solutions can be used to improve the field homogeneity over the second-order static case. The majority of the field homogenization capabilities and the most significant improvements over the static solution are already provided by a very stringent regularization weightings of λ, ε = 0.01. Barely any difference can be seen to the unconstrained solutions.

4.3 |. Phantom experiments

A phantom experiment was performed to test the pre-emphasis adjustments and the accuracy of the applied DSU shim fields. A second-order statically shimmed, in vivo field map was acquired and a high-order DSU solution with λ, ε = 0.01 was calculated. The slice locations of the in vivo data were copied and the calculated DSU shim currents were applied in a phantom. The subtraction of a static phantom field map from the acquired dynamic phantom field map yielded the effectively applied DSU shim fields as determined for the in vivo shim set and allowed comparison to the predicted shim fields without any subject influences.

The results for 3 neighboring slices from 3 different brain regions are illustrated in Figure 7 and demonstrate the match between the predicted and acquired shim fields. Despite the applied regularization terms and the added constraints, versatile field shaping capabilities, as provided by the shim set, can be seen from the maps.

FIGURE 7.

Dynamic shimming phantom results. A, DSU currents were calculated from an in vivo data set and applied in a measurement using a spherical water phantom. The magnitude images of both data sets show their volumetric overlap. B, A common ROI is generated from both data sets and used to mask the in vivo and phantom images for better comparison. C, Three exemplary neighboring slices from the 3 different brain regions, which are marked on the sagittal magnitude image, show the match between the predicted shim field for the brain slices and the real shim fields as measured in the phantom

4.4 |. In vivo experiments

Unshimmed field maps, followed by second-order whole-brain statically shimmed field maps, were acquired from 3 volunteers. These were then used in the proposed DSU optimization process with λ, ε = 0.01, and the results were applied to acquire dynamically shimmed field maps. The static and dynamic shim calculation was performed with our fully automated custom shim software and the resulting field maps were acquired noniteratively.

Figure 8 illustrates the results of 1 data set and shows the homogeneity gain provided by the dynamic shim. The sagittal views show the elimination of residual B₀ inhomogeneities in large parts of the brain and a strong reduction of the most prominent offsets in the prefrontal cortex area. A simulated 4th⁺ static shim solution is shown for comparison, to estimate the gain provided by switching the shim set dynamically instead of statically.

FIGURE 8.

High-order dynamic in vivo shimming. The sagittal views show the B₀ inhomogeneity distribution after measured static second-order shimming and simulated static 4th⁺-order shimming. Moreover, the match between the simulated and the acquired high-order dynamic shimming field maps can be seen from the sagittal views. The homogeneity improvements when switching the high-order shims dynamically can be seen from the comparison between the acquired second-order static and the acquired 4th⁺-order dynamic axial field map slices. 4th⁺ indicates the usage of full fourth- and partial fifth- and sixth-order harmonics

Dynamic shimming results from all 3 subjects are summarized in Figure 9. The slice-wise standard deviation of the residual B₀ inhomogeneity is plotted for the static and the dynamic field mapping results and is in agreement with the simulations from section 4.2 using the same regularization parameters. To further quantitatively demonstrate that the number of high-offset voxels can be reduced through the addition of dynamic shimming, the 90th-percentile-range plots indicate for each slice the frequency offset range that 90% of all voxels lie within.

FIGURE 9.

Dynamic shimming results from 3 subjects. For each data set, the slice-wise residual standard deviation is given alongside with the whole-brain standard deviation for the acquired second-order static, the acquired 4th⁺-order dynamic, and a simulated 4th⁺-order static shim. The 90th percentile range plots delineate the frequency range that 90% of all field map voxels lie within and proves that the number of very-high-offset voxels can be substantially reduced with added high-order dynamic shimming. 4th⁺ indicates the usage of full fourth- and partial fifth- and sixth-order harmonics

4.5 |. Extension to other acquisition strategies

The analyses so far assumed sequential acquisition strategies. Many multislice MR imaging applications, however, use different slice-ordering schemes, such as slice-interleaving or simultaneous multislice imaging. Slice-interleaving increases the distance between consecutively fitted slices and introduces a discontinuity between the last odd and the first even slice. Simultaneous multislice imaging requires sets of distant slices to be jointly optimized. Given that the acquisition order is critical for the proposed interslice current constraints, both scenarios were evaluated in simulations.

For a comprehensive comparison, 4 acquisition strategies for SH shimming were analyzed: Dynamic shimming for sequential acquisition modes was simulated for increasing SH shim orders. Additionally, 4th⁺ order results for slice-interleaved dynamic shimming acquisitions and for simultaneous multislice acquisitions with acceleration factors 2, 3, 4, and 6 were simulated. To be able to set all results into context with the achievable B₀ homogeneity for conventional B₀ shimming, second-, third-, and 4th⁺-order static shim solutions were calculated. In Figure 10, the average whole-brain standard deviations from 3 subjects for each of these acquisition strategies are displayed. Moreover, animations can be found in Supporting Information Video S1, which show the increase in field shaping capabilities, when higher-order terms are added to the dynamic shim optimization.

FIGURE 10.

Comparison of the effectiveness of spherical harmonic shimming applied to different acquisition schemes. The plot shows the simulated whole-brain standard deviation averaged over 3 subjects when applying static, sequential dynamic, interleaved dynamic, or simultaneous multislice dynamic shimming. Conventional static whole-brain shimming improves according to expectations with the order of the included SH. The same holds true for DSU in sequential slice acquisition mode. Furthermore, the plot indicates that the proposed algorithm can equally well be applied to slice interleaved acquisition (Slice-IL). The simultaneous multislice dynamic shim setting for a 2-fold acceleration is comparable to the single-slice dynamic shim solutions and converges toward the static solution of the applied shim set for increasing accelerations. The dashed line indicates the achievable whole-brain standard deviation when using the 4th⁺-order shim set in static mode. The dashed box highlights the comparable B₀ homogenization performance for sequential, slice-interleaved, and simultaneous multislice acquisitions of equal shim order

The static and sequential dynamic shimming results show an expected B₀ homogeneity improvement when including higher shim orders. The altered slice acquisition orders for the interleaved and for the multiband acquisition schemes entail an increase in achievable B₀ homogeneity of 0.07 Hz (interleaved) and 0.40 Hz (simultaneous multislice), respectively. Merely by increasing the multiband acceleration, the increase in achievable B₀ homogeneity becomes more noticeable and converges toward a corresponding static solution. However, this degradation in shim quality is attributed to the requirement to optimize the shim over more regions simultaneously, rather than the applied current constraints.

5 |. DISCUSSION

In this work, high-order dynamic shimming hardware was installed and calibrated and the required acquisition and processing software was implemented. Resulting B₀ shimming improvements were analyzed in simulations and validated in phantom and in vivo measurements. Our novel shim optimization framework has proven to substantially reduce the shim amplitudes and their temporal variation, and thus provides the means for better control over shim-induced eddy currents. The simulations for the slice-interleaved and the simultaneous multislice acquisition schemes indicate that an extension to these more prevalent acquisition types is possible and introduces no significant loss in achievable B₀ homogeneity. Both the simulated and acquired results for the sequential acquisition strategies, as well as the simulated results for these alternative acquisition strategies, are in accord with simulated DSU performance reported in the literature³¹ and show an approximate 2-fold homogeneity gain, as compared to standard static second-order SH shimming. It is shown that the loss in achievable B₀ homogeneity, as introduced by the applied constraints and regularization terms, is negligibly small and outweighed by the benefits of superior eddy current handling, lower DSU hardware requirements, and increased accuracy of the shim simulations. As a result, the addition of dynamic shimming capabilities to existing static shimming routines is facilitated and has the potential to make its application in 2D multislice or 3D slab acquisitions more readily available. The source code of the proposed algorithm can be shared upon request.

Given that the DSU-specific problems are coupled and additive, the benefits of the proposed solution are manifold and have a cascading effect. The average 6- to 23-fold reduction of the maximally occurring shim steps, as induced by the 3 different regularization weightings chosen in this work, directly translates into a reduction of the maximum ECs by the same amount. Additionally, in all the DSU volumes for which the regularization parameter ε is chosen too conservatively, the added box constraints will still guarantee that the current steps do not exceed the user-selected interslice current limit. As a result of both EC handling strategies, the number of channels that still require a pre-emphasis correction can be potentially decreased and counteract the projected EC compensation calibration effort for very-high-order DSU systems.¹³ When shim current steps were limited to 0.5 A in magnitude, the shim system used in this work required a pre-emphasis correction for only 5 channels. Furthermore, because the maximum occurring current steps are dictated by the applied box constraints, and are consequently known beforehand, saving large parts of the dynamic range of the power supply for arbitrarily large pre-emphasis overshoots is no longer required. Moreover, because residual ECs after pre-emphasis correction scale with the applied shim step, it follows that when targeting equal EC suppression levels as in standard DSU implementations, the pre-emphasis adjustments are more robust to calibration inaccuracies.

By additionally regularizing the amplitudes of the absolute shim currents and controlling their impact by the parameter λ, the ill-posedness of the optimization problem is intrinsically addressed in a similar way to Kim et al.³² Furthermore, a shim degeneracy analysis²⁶ or the exclusion of certain SH shim terms¹² is no longer needed. In contrast to the notion that optimal B₀ homogeneity for dynamically shimmed multislice applications can only be achieved by high-power shim supplies,¹⁹ our results indicate that an equally good shim performance can be achieved at substantially lower current amplitudes. Another direct effect is that the calculated currents will be closer to the current regime that can be covered in the static shim calibration procedure, thus making the simulations more accurate. Also, it is to be noted that high current amplitudes are only penalized and not suppressed. This assures that the valuable shim currents that contribute significantly to cancelling residual inhomogeneities are accepted and only those with marginal improvements are omitted.

In addition to the implementation of data sharing between adjacent DSU volumes, which is achieved by fitting stacks of 3 slices in a moving boxcar mode, the use of previous slice solutions as an initial guess for current iterations supports the regularization terms and helps the algorithm in finding solutions with smooth current variations. This is also true at the transition between the end slices, and despite the application of a joint optimization framework to calculate the solutions for these slices simultaneously, no noticeable degradation in the achievable shim quality was found.

The high-order shim insert used in this work is of a smaller diameter than the scanner shim coils and thus has a larger distance to the cryoshield and other eddy current surfaces of the MR scanner. It is to be expected that this setup is, therefore, intrinsically less prone to generating ECs than the scanner’s in-built shim coils. However, it can be stated that when switching the larger host shims dynamically, our proposed routine is likely to be even more beneficial.

For the proof of principle and the results obtained for this work, no regularization parameter optimization of λ and ε was performed and this leaves room for improvements. Moreover, given that the higher-order SH shim coils become increasingly less efficient, the regularization weighting can be individually adapted for each channel. The same holds true for the maximum interslice shim current steps, which can be less restrictive for the higher-order terms. In general, the regularization terms and constraints can be reformulated to be based on the given coil efficiencies, rather than on the applied currents. Furthermore, it can be investigated to what extent similar results with respect to the shim current reduction can be achieved when optimizing all slice solutions simultaneously and including the current variation into the optimization problem as a further parameter to be minimized.

In summary, it can be stated that dynamic shimming for 2D multislice applications makes optimal use of the available shim hardware. The added benefits of our proposed optimization framework facilitates its successful implementation and allows its incorporation into future studies. It was shown that the remaining inhomogeneities are not caused by the introduced constraints and regularization terms, but rather they are beyond the field shaping capabilities of the shim set.

To date, none of the existing shimming techniques can fully mitigate all B₀ inhomogeneities. This is true for 2D approaches, and even more so for 3D and multiband applications. Consequently, future work will concentrate on further hardware developments that are capable of more accurately approximating the complex B₀ inhomogeneities induced by the subject-dependent susceptibility distributions.

Supplementary Material

video1

VIDEO S1 Simulations of the proposed dynamic shimming optimization. Each animation file corresponds to the dynamic shim optimization of a given in vivo field map data set. The top row shows the dynamic shim current amplitudes for each channel that was included in the optimization. The blue solid line corresponds to the currently present global static shim current for each shim channel. The green bars correspond to the additional dynamic shim current amplitudes for the given slice. The horizontal red line shows for each shim channel the shim current value of the preceding slice, and the light red boxes around the line correspond to the applied ±0.5 A interslice shim value limit, that the current slice needs to stay within. The black solid lines indicate the applied upper and lower bounds for the maximum absolute shim current for each channel. The bottom row displays the acquired field map (second-order static shim) and the predicted dynamic shim solutions as well as the predicted shim fields when including higher spherical harmonic orders into the fit. The field maps are overlaid to the associated magnitude images to provide for an anatomical reference. Note that because of the applied hardware, only 4 fifth- and 2 sixth-order terms were included into the simulation. The resultant slice-wise standard deviation is given for each spherical harmonic shim order

suppl_1

TABLE S1 Table of variables used in the algorithm

TABLE S2 Spherical harmonic shim coils of the shim insert system. The coil efficiencies relate the effects of the applied current constraints to the resulting maximum achievable B0 field offsets. The amplifier bank indices denote which channels are collectively constrained by the linear inequality constraints given in Equation 2. EC self- and cross-term compensation adjustments were performed for shims that generated non-negligible ECs following a shim step of 0.5 A. Amplitude and time constants are given in DAC and ms, and the conversion factor between DAC and A is 1.53e-4