Jose EC Serralles^{1}, Elfar Adalsteinsson^{1}, Lawrence L Wald^{2}, and Luca Daniel^{1}

^{1}Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, United States, ^{2}A. A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, MA, United States

In this paper we efficiently combine a coil optimization process with an electromagnetic field solver and show preliminary proof of concept on simple systems leading to 22% to 45% improvement in Signal to Noise Ratios.

$$ \underset{\mu}{\text{max}} \frac{S}{P} $$

where

$$S=\sum_{v\in\text{ROI}}\left|H_v^+\right|^2$$

and

$$P=\sum_{v\in V} \sigma_v \left|E_v\right|^2\Delta V\text{.}$$

$$$H^+$$$ denotes the right-handed component of the magnetic field; $\sigma$ denotes the electric conductivity; $$$E$$$ denotes the electric field; and $\Delta V$ denotes the volume of each voxel. The decision variables that we consider are the radius of each loop and its distance from the origin.

We achieve tuning and matching by attaching capacitors to distributed ports along each loop and constraining them to be

$$C=\text{argmin}_C \|S\|_{\rm F}^2\text{ s.t. }C_l \leq C_i \leq C_u, \; i = 1, \ldots, N_c$$

where $$$S$$$ is the scattering parameter matrix and the norm is the Frobenius norm. The scattering parameter matrix is given by $$$S = \left(Z_p+z_0I\right)^{-1}\left(Z_p-z_0I\right)$$$ where $$$Z_p$$$ is the impedance parameter matrix. We shim by maximizing the SNR over the excitation applied to each coil while ensuring unit magnitude or smaller:

$$\alpha = \text{argmax}_\alpha \frac{S}{P}\text{ s.t. }|\alpha_i|^2 \leq 1, \; i = 1, \ldots, N_p$$

In our second simulation, we maximized the SNR on hypothalamus region. We used two loops, adjusting independently radii and distances of each loop. We started with 6 cm radii and 15 cm distances, yielding an SNR of 121. After merely 4 iterations (i.e. 85 minutes), we converged to an SNR of 148, with radii 13.9 cm and 15.8 cm and with distances 18.0 cm and 18.4 cm, respectively.

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First simulation experiment involving optimization of one loop's radius and distance from the origin. Top row depicts the (a) initial coil design in front of the head, (b) the initial electric field along a transverse slice, and (c) the initial magnetic field along the same transverse slice. Bottom row depicts the (d) final coil design, (e) the final electric field along a transverse slice, and (f) the final magnetic field along a transverse slice.

Second simulation experiment involving
optimization of two loops' radii and distances from the origin. Top row
depicts the (a) initial coil design around the head, (b) the
initial electric field along a transverse slice, and (c) the initial
magnetic field along the same transverse slice. Bottom row depicts the
(d) final coil design, (e) the final electric field along a transverse
slice, and (f) the final magnetic field along a transverse slice.