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

We propose that a transmit coil can be optimized for certain modes of operation. It is well-known that when the number of channels of a degenerate birdcage coil increases coupling between channels becomes a significant problem. The high total reflection when only one of its channels is used as a transmitter caused a small portion of the power delivered to the body. However, when it is properly designed, in some modes of operations such as circularly-polarized mode, the total reflection become negligibly small. In this work, we demonstrate this effect on various two-row degenerate birdcage-coils together with simulations and experiments.

**Introduction**

Transmit array (TxArray) coils are being widely investigated for ultra-high field systems in order to obtain a homogeneous excitation. These coils can be also very useful for the conventional scanners since, without any increase in the cost, they will provide degrees of freedom to the designer of pulse sequencer. This freedom may be used, for example, to increase the power efficiency^{1-3} or to implement implant-friendly mode^{4} in a very efficient way.

In TxArrays, however, the coupling between channels is a significant problem. This problem gets more difficult to solve as the number of channels increases^{5}. It should be noted that the power efficiency of the TxArray coil is a function of the phase and amplitude of the input power. The reflected power can be written as $$$P_{reflected}=\frac{1}{2}\frac{|V^{-}|^{2}}{Z_{0}}$$$ where $$$V^{-}$$$ denotes the reflected voltage vector given in the phasor form. $$$V^{-}$$$ can be written as $$$V^{-}={\bf{S}}V^{+}$$$ where $$$\bf{S}$$$ denotes the scattering matrix and $$$V^{+}$$$ denotes the incident voltage. As can be noted in this formulation, in order to obtain zero reflection for all incident voltages, $$$\bf{S}$$$ has to be zero. However, for a non-zero $$$\bf{S}$$$-matrix, zero reflected power can still be obtained for certain input voltages.

In this study, we claim that is possible to design an array coil such that its total reflected power is zero or low in some important modes of operation. For this purpose, we designed a double-row degenerate birdcage coil that operated in the circularly-polarized (CP) mode with very low reflection while its single-channel (SC) operation mode has an acceptable reflection level. Note that if the degenerate birdcage coil is used in the CP mode, its excitation profile will be similar to the profile of a conventional birdcage coil^{6-9}.

**Methods**

We simulated 4 shielded degenerate birdcage head TxArrays with varying number of channels and the same overall dimensions loaded with a uniform cylindrical phantom shown in Fig.1. Transmit loops distributed in both the circumferential and the z-direction and could be decoupled by adjustment of decoupling capacitors placed between nearest neighbors. Exciting both rows separately in the CP mode, while the currents of the upper and lower rows cancel each other in their mutual ring segment, provides the ability of the CP mode excitation by this type of TxArrays same as conventional birdcage coils^{6-9}.

Fig.2 depicts an arbitrary N-port microwave network. The normalized-TR for the nth port of this network, when the characteristic impedance is real, can be defined as^{10}:

$$TR_n=\frac{P_{reflected_{n}}}{P_{incident_{n}}}=|\frac{V_n^-}{V_n^+}|^{2}$$

By adjusting capacitor values, **S**-matrix can be optimized to minimize the TR value such that CP mode of operation is obtained with very low reflection. The optimization problem is a non-convex problem and it also gives us the opportunity to minimize the TR in additional modes of operations as well. For this, we chose the SC mode. In this mode, the TR can be expressed as $$$\sum_{n=1}^N|S_{mn}|^{2}$$$. To evaluate the performance of this idea, the coils capacitors are found using the co-simulation strategy^{11} with the criteria to minimize TR in both CP and SC modes.

**Results and discussion**

_{}We assessed the performance of the TxArrays at 123.2MHz shown in Fig.3 in the SC mode and two different CP modes. In CP1 mode, both rows individually excited with identical power and linearly increasing phase, whereas the lower-row channels excited with a 180° phase shift relative to axially adjacent upper-row channels, while in the CP2 mode, both rows are in the same phase. In both CP1 and CP2 modes, TR is less than 1% for all coils. In the SC mode of operation, however, the TR increases with the increased number of channels.

Fig.4 shows the surface current density on 2x4-channels DBC, and the B_{1}^{+}-maps inside of the phantom for the SC, CP1, and CP2 modes. As it is obvious, the current on the TxArray’s middle-ring is almost zero in the CP1 mode which provides the possibility of having a uniform B_{1}^{+}-profile in the iso-center. Additionally, for the CP2 mode, the B_{1}^{+}-field is almost zero in the central slice while the profiles in the other slices are homogeneous which make this coil to use for some safety applications and multi-slice imaging.

As
proof of the method, the structure of 2×4-channels TxArray constructed (Fig.5(a)). The reflection coefficient,
maximum coupling, TR, B_{1}^{+}-profile of each individual
channel, and GRE images of the CP1 and CP2 modes are shown in Fig.5(b-d).

**Conclusion**

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[2] Kozlov M, Turner R. Analysis of RF transmit performance for a multi-row multi-channel MRI loop array at 300 and 400 MHz. 2011. IEEE. p 1190-1193.

[3] Guérin B, Gebhardt M, Serano P, Adalsteinsson E, Hamm M, Pfeuffer J, Nistler J, Wald LL. Comparison of simulated parallel transmit body arrays at 3 T using excitation uniformity, global SAR, local SAR, and power efficiency metrics. Magn Reson Med 2015; 73(3):1137-1150.

[4] Eryaman Y1, Guerin B, Akgun C, Herraiz JL, Martin A, Torrado-Carvajal A, Malpica N, Hernandez-Tamames JA, Schiavi E, Adalsteinsson E, Wald LL. Parallel transmit pulse design for patients with deep brain stimulation implants, Magn Reson Med. 2015 May;73(5):1896-903.

[5] Mahmood Z, Guerin B, Adalsteinsson E, Wald L, Daniel L. An Automated Framework to Decouple pTx Arrays with Many Channels. 2013:2722.

[6] Alagappan V, Nistler J, Adalsteinsson E, Setsompop K, Fontius U, Zelinski A, Vester M, Wiggins GC, Hebrank F, Renz W. Degenerate mode bandāpass birdcage coil for accelerated parallel excitation. Magn Reson Med 2007; 57(6):1148-1158.

[7] Sadeghi-Tarakameh A, Kazemivalipour E, Demir T, Gundogdu U, Atalar E. Design of a Degenerate Birdcage Radiofrequency Transmit Array Coil for the Magnetic Resonance Imaging Using Equivalent Circuit Model. In Proceeding of the 34th Annual Scientific Meeting of ESMRMB, Barcelona, Spain 2017:300-301.

[8] Kazemivalipour E, Sadeghi-Tarakameh A, Yilmaz U, Acikel V, Sen B, Atalar E. A 12-Channel Degenerate Birdcage Body Transmit Array Coil for 1.5T MRI Scanners. In Proceeding of the 26th Joint Annual Meeting ISMRM-ESMRMB, Paris, France 2018: Abstract 1708.

[9] Sadeghi-Tarakameh A, Kazemivalipour E, Demir T, Gundogdu U, Atalar E. Accelerating the Co-Simulation Method for Fast Design of Transmit Array Coils: An Example Study on a Degenerate Birdcage Coil. In Proceeding of the 26th Joint Annual Meeting ISMRM-ESMRMB, Paris, France 2018: Abstract 4404.

[10] Pozar DM. Microwave engineering: John Wiley & Sons, 2009.

[11] Kozlov M, Turner R. Fast MRI coil analysis based on 3-D electromagnetic and RF circuit co-simulation. Journal of Magnetic Resonance. 2009; 200(1):147–152.

Figure 1 – EM simulation models of 4 TxArrays loaded by a cylindrical phantom (diameter of *80mm*, a length of *350mm*, a conductivity of *0.6s/m*, and relative permittivity of *80*). All TxArrays had the same overall dimensions (diameter of *315mm*, length of *270mm*, shield diameter of *408.9mm*, and shield length of *420mm*). All rings and legs were *15mm* wide copper strips. Each loop was broken by 1 matching, 2 tuning, and 2 decoupling capacitors in the circumferential direction and 2 decoupling capacitors in the z-direction placed along its circumference. 2 series capacitors in the input port are used for matching improvement.

Figure 2 – An arbitrary *N*-port microwave network where *$$$V_n^+$$$* and *$$$V_n^-$$$* are defined as equivalent voltages for the incident and reflected waves, respectively^{10}. At the terminal plane, the total voltage of the nth port is given by $$$V_{n}={V_n^+}+{V_n^-}$$$. The vector of reflected ($$$V^{-}$$$) voltages can be obtained by multiplying $$$\bf{S}$$$-matrix by the vector of incident (*$$$V^{+}$$$*) voltages as $$$V^{-}={\bf{S}}V^{+}$$$.

Figure 3 – (a) Analysis performance of TxArrays in terms of *TR*, reflection coefficient, maximum coupling, power delivered to the load, and TxArray efficiency as performance metrics within three different transverse planes (b) in the load for SC, CP1, and CP2 modes. Although the *TR* in SC mode is rising with increasing the number of channels, the *TR* and TxArray efficiency in the CP1 and CP2 modes remain constant. As the number of channels increased, the coupling will increase in the SC mode but the $$$Tx_{eff-max}$$$ and $$${P_{load}}\diagup{P_{incident}}$$$ for the CP1 and CP2 modes remained constant.

Figure 4 – The surface current density on the *2×4*-channels TxArray, and the $$$B_1^+$$$-maps in three transverse slices inside of the load for the SC, CP1, and CP2 modes. The results obtained for a constant total incident power of *20mW* for all structures.