Koki Matsuzawa^{1}, Katsumi Kose^{1}, and Yasuhiko Terada^{1}

Designing gradients coils with arbitrary geometries has been realized by matrix inversion optimization techniques. Use of a truncated singular value decomposition (SVD) is promising because magnetic field accuracies are controlled by choosing the appropriate SVD eigenmodes. However, in the SVD method, the gradient performances, such as inductance and power dissipation, cannot be optimized. Here we proposed a new strategy to optimize a desired coil performance. A key feature is the use of a genetic algorithm to optimize the appropriate combination of SVD eigenmodes. The concept is demonstrated for a biplanar geometry, and would be readily applicable to arbitrary geometries.

**INTRODUCTION **

**DESIGN METHOD**

Figure
1 shows a simplified depiction of the design pipeline. For the SVD module, the coil
current surface is discretized into a mesh of triangles and the stream function
of the current density, or the current potential, at each node $$$i$$$ is defined as $$$T_i$$$. The magnetic field at a point $$$j$$$ in the target area is expressed as $$$B_j = \sum_i^NA_{ij}T_i$$$. Here the matrix $$$A$$$ is calculated using the Bio-Savart
law. Then the matrix $$$A$$$ is decomposed using the SVD as $$$A=U \Lambda V^T$$$ where $$$U$$$ and $$$V$$$ are unitary matrices, and $$$\Lambda$$$ is a diagonal matrix containing the singular
values $$$\lambda _i , i=1,\cdots ,min(n,m)$$$. Using the columns of $$$U$$$, denoted $$$u_1,\cdots ,u_n$$$, and
those of $$$V$$$, denoted
$$$v_1,\cdots ,v_m$$$, $$$A=\sum_i^N \lambda _iu_iv_i^T$$$. In the
original SVD method, the solution, current potential matrix $$$T$$$, is calculated as $$$T=A^{-1}B=\sum_i^k \frac{v_iu_i^T}{\lambda _i}B$$$. Here $$$k$$$ is the truncation number which is
empirically determined depending on the given geometry.
Instead
in the proposed method, $$$B$$$ and $$$T$$$ are newly formulated as $$B=\sum_i^{k'} c_i\lambda _i u_i,$$ and
$$T=\sum_i^{k'} c_iv_i.$$ Then the
coefficients $$$\{c_i\}$$$ is optimized
by GA to maximize an objective function formulated to quantify the desired coil
performance. Here we generalize the objective function as $$f=\frac{\eta^a}{E^bL^cR^d}$$ where $$$\eta$$$ is the coil efficiency, $$$E$$$ is the gradient inhomogeneity, $$$L$$$ is the inductance, and $$$R$$$ is the resistance. The constants $$$a,b,c$$$ and $$$d$$$ are weighing factors to balance these
measures. For example, when $$$a=2, b=1,
c=1$$$, and $$$d=0$$$, $$$f$$$ corresponds to figure of merit proposed by
Turner^{4}.

The proposed method was tested to design transverse coils with a biplanar geometry (coil diameter = 320 mm; coil gap = 120 mm). The target region was set to be a 100 mm x 100 mm x 50 mm diameter ellipsoidal volume (Fig. 2). $$$k'$$$was set to 8 for simplicity. We used a real-coded GA algorithm (population size = 50; crossover rate = 75%; mutation rate = 1%) with BLX-$$$\alpha$$$ crossover $$$(\alpha =1)$$$ and an elite selection. We designed coils for different objective functions.

**RESULTS AND DISCUSSION**

**CONCLUSION**

^{1}Poole, M, et al., Concepts in Magnetic Resonance Part B: Magnetic Resonance
Engineering 31.3 (2007).

^{2}M. Abe et al., Phys. Plasmas, 10 (2003)

^{3}M. Abe, IEEE Trans. Magn., 49 (2013)

^{4}Turner, R., Journal of Physics E: Scientific Instruments 21.10
(1988).

Fig. 1: Pipeline of the proposed
method.

Fig. 2: Coil current surfaces and
target area. (a) xy projection. (b) xz
projection.

Table 1: Coil performances

Fig. 3: (a)-(c) Coil patterns
designed by the proposed method. (a) Efficiency-maximized
pattern (coil A). (b) Resistance-minimized pattern (coil B). (c) Balanced
pattern (coil C). (d) Coil pattern designed by the original SVD method (coil
D).

Fig. 4: (a)-(d) Gradient fields
calculated from the coil patterns shown in Figs. 3(a), (b), (c), and (d),
respectively.