A Comparison of Optimization Algorithms for Localized in-vivo B0 Shimming
Sahar Nassirpour1,2, Paul Chang1,2, Ariane Fillmer3,4, and Anke Henning1,3

1Max Planck Institute For Biological Cybernetics, Tübingen, Germany, 2IMPRS for Cognitive and Systems Neuroscience, Eberhard Karls University of Tübingen, Tübingen, Germany, 3Institute for Biomedical Engineering, UZH and ETH Zürich, Zürich, Switzerland, 4Physikalisch-Technische Bundesanstalt, Berlin, Germany

Synopsis

This work presents a study on the performance of several least-squares optimization algorithms used for localized in-vivo B0 shimming. Seven different algorithms were tested in 4 different shim volumes in the brain: global shimming region, single slice, and single voxels in two different positions with 3rd order shimming at 7T. Each algorithm's robustness and convergence were tested against noisy inputs and different starting values. The results give an interesting overview of the properties of each algorithm and their applicability. The regularized iterative inversion algorithm proves to be the best algorithmic approach suited to this problem.

Introduction

Magnetic resonance spectroscopy and imaging applications benefit from an automated B0 shimming algorithm which is numerically stable, efficient and applicable to any arbitrary volume of interest. Recently, different approaches for solving this problem by 3D B0 mapping combined with least-squares optimization have been proposed [1-6]; yet there is still a need for a systematic comparison of these algorithms that would provide insights for choosing the best approach for B0 shimming applications. The aim of this work is to evaluate and compare the performance of 7 different optimization algorithms embedded into the same general software framework based on their convergence rate, numerical stability and solution optimality in a range of B0 shimming applications in human brain MRI at 7T.

Methods

The B0 shimming problem can be characterized by a system of linear equations, where each element of b is the value of the B0 field, A represents the sensitivity of each shim coil and x contains the applied currents. Thus each element of the matrix Ax gives the field generated by a given shim channel at a given position. The system of linear equations can then be cast as a linear least-squares optimization problem:

$$\min_x ||(Ax-b)||^2$$

The maximum shim fields can be included as box-constraints $$lb\leq x\leq ub$$

This problem is quadratic and hence convex. Therefore any solution is a global minimum. The convexity is still preserved even if A is ill-conditioned (which is often the case for single-slice and off-center single-voxel shimming volumes due to the lack of orthogonality of the shim fields at off-center positions), however, solving the problem becomes tricky because a small change in the data (e.g. noise in the acquired reference B0 map) can lead to radically different answers and optimal convergence is not guaranteed.

Seven different algorithms were implemented in MATLAB™ R2013b:

- nonlinear optimization using three methods: active-set (nonlin-as) [7], interior-point (nonlin-ip) [8] and sequential quadratic programming (sqp) [7] using MATLAB™’s fmincon;

- quadratic programming using the interior-point-convex method (qp-ip) [9] using MATLAB™’s quadprog ;

- iteratively inverting the system and truncating the values that exceed the limits (invcon);

- the same method except that the singular values of A are truncated to improve the conditioning (invcon-tsvd); and

- finally, for comparison purposes, unconstrained pseudo-inversion algorithm (pinv) using MATLAB™’s pinv, which always yields the best shim quality achievable (if the solver was not bound by constraints).

For this study, 33 in-vivo datasets from the brain of healthy volunteers obtained at a 7T Philips scanner were used. For each volunteer, the performance of all the algorithms for 3rd order shimming in 4 different shim volumes were studied (figure 1): a global brain region, a single slice positioned off-center, a single spectroscopy voxel located in the frontal cortex, and one in the visual cortex. The shim volumes were defined in a custom GUI application.

The methods were evaluated according to three criteria: convergence rate, starting-point sensitivity (i.e. how sensitive the result is based on the starting-point of the optimization algorithm), and the robustness of the solution against small perturbations to the input. For the starting-point sensitivity test, 1000 randomly initialized starting points were used. For the robustness test, 1000 different noisy B0 vectors generated by adding random white noise (with a maximum amplitude of 5% of the original signal) were considered. The amplitude constraints of the shim system are shown in figure 2.

Results/Discussion

Figure 3 shows the algorithms’ sensitivity to different starting values. Note that since the dedicated quadratic solvers do not require a starting value input, this test was not applicable for them. Among all the other algorithms, the interior-point method is the least sensitive.

Figure 4 shows the robustness of the algorithms against a small noise perturbation in the input data and the invcon-tsvd algorithm proves to be the most robust.

Figure 5 shows the overall performance of these algorithms in all 4 shimming areas. As can be seen, the invcon-tsvd method consistently provides the best shim quality even in hard-to-shim areas where the problem becomes most ill-conditioned.

Finally in figure 6, a comparison of the convergence rate and run-time of these algorithms is shown. The invcon-tsvd algorithm was found to be the fastest.

Conclusion

For localized in-vivo B0 shimming the use of a dedicated quadratic programming solver instead of a generic nonlinear least-squares solver is highly recommended. The reason being that the quadratic solvers do not depend on a starting value and also converge much faster. Among the quadratic solvers, the regularized iterative inversion method (invcon-tsvd) was found to be both fastest and the most robust.

Acknowledgements

No acknowledgement found.

References

[1] Gruetter R, et al, “Fast, non-iterative shimming of spatially localized signals”, JMRI 96, 323-334, 1992.

[2] Kim D, et al, “Regularized Higher order in-vivo shimming”, MRM 48:715-722 2002.

[3] Hetherington H. et al, “Robust fully automated shimming of the human brain for high-field 1H spectroscopic imaging”, MRM 56:26-33, 2006.

[4] M. Weiger et al, “Gradient shimming with spectrum optimization”, Journal of Magnetic Resonance Volume 182, Issue 1, September 2006.

[5] Juchem et al, “Dynamic shimming of the human brain at 7 Tesla”, Concepts of Magnetic Resonance Part B MR Eng.: 116-128, 2010.

[6] Fillmer A, et al, “Constrained Image-BasedB0Shimming Accounting for “Local Minimum Traps” in the Optimization and Field inhomogeneity outside the Region of Interest”, MRM 2014 Epub.

[7] Fletcher, R., "Practical Methods of Optimization," John Wiley and Sons, 1987.

[8] Byrd, R.H., J. C. Gilbert, and J. Nocedal, "A Trust Region Method Based on Interior Point Techniques for Nonlinear Programming," Mathematical Programming, Vol 89, No. 1, pp. 149–185, 2000.

[9] Gill, P.E., W. Murray, and M.H. Wright, "Numerical Linear Algebra and Optimization", Vol. 1, Addison Wesley, 1991.

Figures

Figure 1 - the maximum available field amplitudes in mT/m2

Figure 2 - Representative slice overlaid with different shimming volumes in red: a) single slice b) single spectroscopy voxel positioned in the visual cortex c) single spectroscopy voxel positioned in the frontal cortex. Note that this slice is selected from a 20-slice dataset. Each slice is 2.5mm in thickness. The spectroscopy voxel dimensions are 2x2x2cm3.

Figure 3 – Starting-value sensitivity tests for 3rd-order shimming in a single voxel located in the visual cortex. The change in the standard deviation (std) of the shimmed B0 map was used as a measure of robustness. (a) Each box contains the results of one algorithm applied to all 33 in-vivo datasets for 1000 different starting values. The minimum achievable std using pinv is shown for comparison. (b) Results of the same test in one representative in-vivo dataset (150 of 1000 test cases).


Figure 4 - Robustness against a small noise perturbation of the input for 3rd-order shimming in a single voxel located in the visual cortex. Change in the standard deviation of the shimmed B0 map was used as a robustness measure. (a) Each box contains the results of one algorithm applied to all 33 in-vivo datasets for 1000 different noisy inputs each. (b) Results of the same test in one representative in-vivo dataset (150 of 1000 test cases).

Figure 5 - The performance of the different shim algorithms for 3rd-order shimming in 4 different shim volumes. The change in the standard deviation of the shimmed B0 map has been used as a measure of the shim quality. Each box contains the results of one algorithm (labeled at the bottom) applied to all 33 in-vivo datasets.

Figure 6 - The number of iterations and run time (s) of the different algorithms averaged over 33 in-vivo datasets in a global shimming test case. Note that since the pinv, invcon and invcon-tsvd algorithms essentially cast the problem at each step as an unconstrained problem and perform a simple matrix inversion instead of an elaborate optimization, the number of iterations was not applicable to them and hence was not reported. The run times are based on a 2.3 GHz 6-core Intel CPU.



Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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