Ehsan Kazemivalipour^{1,2}, Alireza Sadeghi-Tarakameh^{1,2}, and Ergin Atalar^{1,2}

This abstract introduced a methodology to minimize the magnetic coupling between non-adjacent channels in a degenerate birdcage transmit array coil (DBC) by optimizing some of its physical parameters. The method is based on minimization of the mutual-inductances normalized by the self-inductances. Optimum radius, length in the z-direction, the width of end-rings, and the width of rungs are found for a 3T shielded twelve-channel head DBC with capacitive decoupled loops. Finite element based simulation results confirmed the validity of the results.

**Introduction**

**Methods**

Fig.1 shows the equivalent circuit models of an unloaded twelve-channel DBC enclosed by an RF shield. The effects of all self- and mutual-inductances originating from the coil and its shield are considered in the equivalent circuit model. The effect of the load and other coil losses are not considered in this model.By applying the Kirchhoff mesh current method, the matrix equation is obtained as:

$$(j\omega_{0}K+\frac{1}{j\omega_{0}}H)I=\frac{1}{j\omega_{0}c_{m}}i$$

where $$$\omega_{0}$$$ denotes the angular frequency, **I** denotes the vector of the loop currents and **i** denotes the vector of the input currents. $$${\bf{H}}=\begin{bmatrix}H_{p,q} \end{bmatrix}_{12\times12}$$$ represents capacitance matrix where $$$H_{p,q}$$$ is defined as:

$$H_{p,q}=\begin{cases}\frac{2}{c_{d}}+\frac{1}{c_{m}}+\frac{1}{c_{t}} & q = p\\\frac{-1}{c_{d}} & q-p\equiv\pm1(mode 12)\\0 & elsewhere\end{cases}$$

$$${\bf{K}}=\begin{bmatrix}K_{p,q} \end{bmatrix}_{12\times12}$$$ is an inductance matrix where $$$K_{p,q}$$$ denotes the mutual-inductance between the qth and pth loops. $$$K_{q,q}$$$ denotes the self-inductance of the qth loop. Using Neumann formula^{8} K-matrix elements which are dependent only on the coil’s geometry, can be calculated.

If the non-diagonal elements of $$${\bf{A}}=j\omega_{0}({\bf{K}}-\frac{1}{\omega_{0}^{2}}{\bf{H}})$$$ are nulled, the coupling will be zero because the current of the nth loop depend only on the nth input current $$$(I_{n}={\bf{A}}_{n,n}i_{n})$$$. By using the capacitors, represented by the H-matrix, just those elements of A which satisfy the condition of $$$q-p=\pm1$$$ (in mod 12) can be made zero. It means that only the coupling between the adjacent channels can be canceled.

Therefore, in the optimization of the physical parameters of the coil, only the mutual inductor between non-adjacent elements are considered. The coil symmetry enables us to consider just the one channel of the coil. Therefore, the error function, EF, for channel 1 can be obtained by normalizing these values with respect to the self-inductance value as $$$EF=||\frac{1}{K_{1,1}}\begin{bmatrix}K_{1,3}&K_{1,4}&...&K_{1,11}\end{bmatrix}||_{2}$$$.

**Results and discussion**

In order to appreciate the advantage of this optimization method, in a given range, the physical dimensions for a twelve-channel DBC with fix shield dimensions are calculated with two sets of design parameters that correspond to minimum and maximum error functions. The capacitor values for both of the DBCs (Fig.2) were found using the co-simulation strategy^{9} with the optimization criteria of minimum $$$\sum_{j=1}^{12}|S_{1j}|^{2}$$$.

The optimal dimensions reveal that the coil radius and end-rings widths should increase while the coil length and the rungs widths should decrease. The performance of the coils at 3T (123.2MHz) in terms of the error function (EF), the cost function which is defined as $$$CF=\sum_{j=1}^{12}|S_{1j}|^{2}$$$, reflection coefficient (S_{11}), maximum coupling (SdB_{max}), normalized power delivered to the load ($$$P_{load}\diagup{P_{accepted}}$$$), and transmit efficiency ($$$Tx_{eff}=B_1^+\diagup{\sqrt{P_{accepted}}}$$$) in circularly-polarized mode evaluated in Table 1.

Table 1 shows that optimizing the physical dimensions can improve the coupling level and as a result reduce CF. Decreasing $$$Tx_{eff}$$$ for the coil with the minimum EF in the CP mode showed that the close distance from the shield has a negative impact on the coil performance.

The performance of 8 twelve-channel DBC with different radiuses by minimization of the CF are assessed in Fig. 3. The capacitors were optimized to minimize CF. Decreasing R_{s}-R_{c} leading to a decrease in CF and maximum coupling. On the other hand, $$$Tx_{eff}$$$ in the CP mode is decreasing. Therefore the finding the optimal physical dimensions is a trade-off between the minimization of EF and increasing $$$R_{s}-R_{c}$$$. According to the results, $$$R_{c}$$$=180mm can be considered as the optimum value of the coil radius for the specific structure that we considered.

**Conclusion**

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Figure. 1 – Drawing of the shielded twelve-channel DBC and (b) its lumped equivalent circuit model. In the model, C_{d}, C_{m}, and C_{t} represent decoupling, matching, and tuning capacitors, respectively. The shield was modeled with an imaginary coil with the same physical shape except its radius is taken as, R_{i}=R_{s}^{2}/R_{c} where R_{c} and R_{s} are the coil and shield radii, respectively^{7}. Although both self- and mutual-inductances of the coil are considered in the equivalent circuit model, only the self-inductors (L_{ring} and L_{rung}) are shown in the model.

Figure. 2 – EM simulation models of two twelve-channel DBCs enclosed by a RF shield (diameter of 400mm and a length
of 400mm) and loaded by a uniform cylindrical phantom (diameter of 60mm, a
length of 250mm, a conductivity of 0.6s/m, and relative permittivity of 80) with
the minimum and maximum possible EF. The optimization was performed when the coil radius (R_{c})
ranging from 150mm to 190mm, the coil length (L_{c}) ranging from 260mm
to 300mm, the end-rings widths (W_{ring}) ranging from 10mm to 40mm, and
the rungs widths (W_{rung}) ranging from 10mm to 30mm. The EF_{max}/EF_{min}
is calculated as 2.245.

Table 1 – Analysis performance of two twelve-channel DBCs designed based on the maximum and minimum EF. In each coil, the capacitors optimized to minimize the CF factor. Applying the proposed method leading to decrease the coupling. Because of lower decoupling in the coil with optimal dimensions, we expected to observe an increase in P_{load}/P_{accepted} but it decreased. For the optimal dimension case, the distance of the coil from the shield is 10mm which seems a bit low. The reverse currents on the shield surface can have a destructive effect on the produced field by the coil.

Figure 3 – Analysis performance of 8 TxArrays with different R_{c} ranging from 140mm to 195mm when L_{c}=260mm, W_{ring},= 40mm, W_{rung}=10mm, and R_{s}=200mm. L_{c}, W_{ring}, and W_{rung} of all simulations have been kept same as the values obtained for the coil with EF_{min}.