Kazuya Sakaguchi^{1} and Yasuhiko Terada^{1}

^{1}Institute of Applied Physics, University of Tsukuba, Tsukuba, Japan

### Synopsis

Most of gradients coils are
cylindrical or planar, but deformed gradients that fit an object shape should
have superior performance. A truncated singular value decomposition
(SVD) is a design method for arbitrary shaped gradients, but the design
process is time-consuming, especially for shielded gradients. Moreover, coil performance,
such as coil inductance and power dissipation cannot be optimized. In this
study, we proposed a new strategy that combines the SVD method with a
meta-heuristic method for automatically optimizing desired performance for shielded
gradients.

### INTRODUCTION

Gradients
coils with arbitrary geometries are promising because the coil performance can
be optimized to suit an object shape. They are often designed by a combination
of matrix inversion optimization techniques and discrete representations of
current surface, which is solved using a truncated singular value decomposition
(SVD) [1,2]. However,
the selection method of the SVD eigenmodes is empirical, and hence the gradient
performances, such as coil inductance and power
dissipation, cannot be optimized. Moreover, design for shielded gradients using
the SVD requires time and effort, because there is a trade-off between pattern
complexity and field accuracy, and hence main and shield coils should be
alternatively calculated with many iterations [3]. To address these issues, we proposed
a method for designing shielded gradients with optimized performance to be
desired. The proposed method combines the SVD method with a meta-heuristic
approach. We designed a shielded gradients with power efficiency optimization,
and demonstrated that the proposed method outperformed the original SVD method
without meta- heuristic approach.### DESIGN METHODS

In
our previous study, we used the similar method to optimize given performance
for an unshielded coil [4]. Here, we added the following improvements to the
previous algorithm: (1) We modified the genetic algorithm to an artificial bee
colony algorithm (ABC) to improve the computational efficiency, and (2) added a
constraint of the maximum leakage field.

Figure
1 shows ^{}a flowchart of the proposed method. In the SVD module, the current
potential **T** that generated the
target magnetic field **B** was calculated
back by the SVD calculation for each of main and shield coils. The current
surface was represented by triangular meshes, and the current distribution was obtained
from the contour lines of the current potential **T**_{i} of each node i. The required magnetic field was expressed
by **B**_{j} = ∑_{i}**AT**_{i} at each evaluation point j, where **A** was calculated from the Biosavart’s
law. By calculating the inverse matrix of **A**
with SVD, **T** was expressed as the sum
of orthonormal basis **T**^{k}(mode). After
determining the number of truncation modes k, the alternative SVD calculations
of the main and shield coils were repeated twice to correct the error caused by
the mutual interference of the magnetic fields, which gave an initial solution.

In
the ABC optimization module, **T**^{k} was multiplied by the weight coefficient c_{k} and redefined as **T** = ∑_{k}c_{k}**T**^{k}, and a
combination of c_{k} that formed optimal coil was automatically
searched by the ABC algorithm. The optimization was performed to maximize an objective
function formulated according to the target performance. In this study, only
the main coil was optimized to simplify the calculation.

The SVD-ABC
method was tested for a cylindrical shielded gradient coils. In order to verify
the superiority of this method, we compared two methods: the SVD method alone and
the SVD-ABC module method. Fig. 2 shows the geometric conditions of the
designed coil. The main coil (truncation number = 20, number of windings = 36)
and shield coil (k = 392, number of windings = 16) were arranged coaxially. The
diameter of each conductor was 0.5 mm. The gradient magnetic field was formed
in a 26 mm sphere, and the shield area was a cylindrical surface coaxial with
the main coil.

In the ABC algorithm, the objective function was
set as follows to optimize the power efficiency (population = 50, generation =
1200, and employ bee rate = 0.5),

$$f=\frac {\eta^2}{RN^2\max{(|B_{leak}|)}}$$.

Here,
$$$\eta$$$ was the gradient field efficiency, $$$R$$$ was the resistance, $$$N$$$ was the number of windings, and $$$\max{(|B_{leak}|)}$$$ was the maximum strength of the leakage
magnetic field on the shield surface. We set a limiting condition that the
nonlinearity of the gradient field and the leakage field strength were 10 % and
10 µT or less.

Figure of merit (FOM) was defined as $$$\eta^2/R\sqrt \delta$$$ [4], where $$$\delta$$$ was the RMS error of the gradient magnetic
field. The power efficiency was calculated as $$$P = \eta^2/R$$$.### RESULT AND DISCUSSION

Fig. 3 shows the winding pattern
of the designed shielded gradient coil. Table 1 summarizes the coil performance. The SVD-ABC yielded a coil with a higher current density
at the center of the winding pattern because the curve was steeper there. As a
result, the efficiency of the gradient field in the center was improved (Table
1), the overall resistance was reduced, and the power efficiency was improved
by 35%, while maintaining the maximum leakage within the given constraint. This
indicates that the automatic optimization by the ABC algorithm worked
effectively.### CONCLUSION

We
proposed a new design method for arbitrarily shaped shielded gradient coils. We
demonstrated that the combination the SVD method with the ABC algorithm enables
the automatic design of shielded gradient coils optimized for the desired
performance.### Acknowledgements

No acknowledgement found.### References

[1]
M. Abe et al., Phys. Plasmas, 10 (2003) 1022
-1033. [2] M. Abe, IEEE Trans. Magn., 49
(2013) 5645 -5655. [3] M. Ave et al., IEEE Trans. Magn., 50(2015)
5100911. [4] K. Matsuzawa, ISMRM 4336 (2017).