**Kamila Aliane, Nasreddine Benahmed, Nadia Benabdallah, Abdelkader Benkaddour, and Fethi Tarik Bendimerad**

Magnetic resonance imaging (MRI) is widely used for noninvasive exploration inside the human body. It can provide clear images of organs and tissues, especially those with high water content like muscles and brain tissue. MRI systems operating at ultrahigh frequency (UHF) can benefit greatly from the use of a transverse electromagnetic (TEM) slotted elliptical tube resonator (SER), which can produce a uniform magnetic field. By working with electromagnetic (EM) software simulation tools based on the finite-element method (FEM) and the method of moments (MoM), the authors have successfully analyzed and designed a high-quality-factor (high-Q) quasi-TEM SER suitable for UHF MRI applications.

The fundamental principle of MRI is to receive nuclear magnetic resonance signals induced by radiating EM wave pulse to a human body, which is placed inside the high-intensity static magnetic field. MRI was developed by Lauterbur,^{1} Mansfield, and Grannell,^{2} and is based on applying a nuclear magnetic resonance (NMR) technique. Their work, which won the 2003 Nobel Prize in physiology or medicine, has since become a standard clinical method in modern medicine.

An MRI system is composed of various elements—including an RF coil, which plays an essential role in imaging. Several types of RF coils, such as a saddle coil,^{3} transverse electromagnetic (TEM) birdcage coil resonator (TEM BCR),^{4,5} TEM slotted tube resonator (TEM STR),^{6-8} and TEM slotted elliptical tube resonator (TEM SER),^{9-12} have been developed for a wide range of different applications. Among these RF coils, the TEM SER has gradually come to be most often employed since it produces an extremely uniform magnetic field and can effectively suppress the system’s electric field.

This article builds on work presented earlier in *Microwaves & RF*.^{12} In support of the analysis and design of a high-Q quasi-TEM resonator suitable for UHF-MRI applications based on loaded slotted elliptical tube resonator, the authors have adapted effective approaches based on the use of the finite-element method (FEM) and the method of moments (MoM).

For this type of quasi-TEM resonator, there are no numerical or experimental results in the scientific literature. Hence, the authors were obliged, for the same geometrical and physical parameters of our quasi-TEM elliptical resonator, to make simulations by using two numerical approaches (FEM and MoM). Modeling of this elliptical resonator consisted in analyzing the even- and odd-mode characteristic impedances (Z_{0e}, Z_{0o}), even- and odd-mode effective dielectric constants (e_{effe}, e_{effo}), and the primary inductive and capacitive matrices ([L], [C]), yielding the frequency response for the return loss, S_{11}, at the RF port of the designed inhomogeneous MRI probe using the transmission-line method (TLM).^{13}

As an application, the authors present the design results of a UHF-MRI probe loaded with a human head model^{14} of average relative dielectric constant of 64 and using the optimum configuration of the TEM SER. The probe with high quality factor (Q) operates at 340 MHz (proton imaging at 8 T) and has minimum reflection of -139.5 dB. The UHF-MRI probe using quasi-TEM SER is easy to construct, inexpensive, and simple to operate. Furthermore, the elliptical coil presented here may be constructed to work at different resonant frequencies.

Figure 1(a) shows a schematic depiction of the quasi-TEM slotted elliptical tube resonator. As shown in Fig. 1(b), this coil consists of two conductive bands containing a biological load (having a relative dielectric constant of e_{r}) with thickness, t, carrying opposite currents on each side of a cylinder. The two conductive bands can be mounted on the long (a) or short (b) axes of the ellipse. The conductive sheets are connected at the ends with capacitors to the cylindrical outer shield of radius r_{b} (Fig. 2). Figure 1(b) shows an elliptical cross section of the quasi-TEM SER. Angle θ is called the “window angle.” The quasi-TEM SER structure generally performs as well as inhomogeneous cylindrical birdcage coils, with the advantages of being easier to construct and operate.

For the analyzed TEM SER (i.e., unloaded resonator) of ref. 11 with a/b = 1.8 and r_{b}/a = 2.4, the optimum field homogeneity was obtained for a window angle of 72 deg. In ref. 12, the current authors presented the design results of a UHF-MRI probe with high Q, operating at 7 T (i.e., 300 MHz) and using the optimum configuration of the slotted elliptical tube-line TEM resonator. Unfortunately, changes introduced by human biological loads with high dielectric constants in the quasi-TEM resonator were not negligible, because of the nonhomogeneity of the structure. For this reason, the authors adapted the previous numerical tools based on FEM and MoM approaches from ref. 12 to analyze a slotted elliptical tube resonator loaded with biological material.

The EM properties of the quasi-TEM SER can be described in terms of its primary parameters [L], [C] and its secondary parameters: the even- and odd-mode characteristic impedances, Z_{0e} and Z_{0o}, the even- and odd-mode effective dielectric constants, e_{effe} and e_{effo}, and the loaded quality factor, Q, where the primary parameters can be found from:

The inductance matrix [L] contains the self-inductances of the sheets on the diagonal, and the mutual inductances between sheets in the off-diagonal terms. Matrix [C] accounts for the capacitative effects between the two conductive sheets, characterizing the electric field energy storage in the quasi-TEM SER.

The coefficients for these matrices are obtained by solving a two-dimensional static field problem using the FEM^{15,16} and MoM methods.^{17}

For the FEM approach and under the FreeFEM environment,^{18} the solution can be obtained by solving the Laplace equation as shown in Eq. 1 [Fig. 3(a)]:

where:

V = 1 V on the ith conductor’s surface, and

V = 0 V on all other conductors.

This solution represents the distribution of the potential, V, at the different mesh nodes of the structure [Fig. 3(b)].

When the potential V is known, it is possible to calculate the ith row of the [C] matrix from the electrical charge on each conductor, as in Eq. 2:

where:

V_{0} = 1 V;

q_{s} = e_{0}e_{r}E_{N};

1j represents the contour around the jth conductor; and

E_{N} = the normal component of the electric field.

In the high-frequency limit—i.e., the skin depth is sufficiently small such that current flow occurs only on the surface of the conductors—the inductance matrix [L] can be obtained from the matrix [C_{0}].^{8} The inductance matrix in terms of [C_{0}] calculated for e_{r} = 1 is:

For the MoM approach described in ref. 12, the numerical calculations of the EM-parameters of the studied resonator were carried out with LINPAR for Windows (Matrix Parameters for Multiconductor Transmission Lines), a two-dimensional (2D) software program for numerical evaluation of the quasistatic matrices for multiconductor transmission lines embedded in piecewise-homogeneous dielectrics.^{17} For the slotted elliptical tube-line quasi-TEM resonator, the authors were obliged to supply the cross section of the structure and all relevant dielectrics characteristics including the segmentation by using our programs in FORTRAN (Fig. 4).

When the EM-parameters are determined, it was possible to estimate the resonance spectrum (S_{11}) of the quasi-TEM resonator shown in Fig. 2 using programs based on the TLM or other numerical tools.

The UHF-MRI probe developed for this article consists of an SER resonator with length l, matching capacitor, C_{M}, and terminating capacitors, C_{Si} and C_{Li} (with I = 1, 2). The loaded Q of the quasi-TEM elliptical resonator can be estimated from the reflection-parameter (S_{11}) sweep with frequency^{12}:

where:

f_{r} = the resonant frequency of the circuit;

f_{u} = the 3-dB frequency above the resonant frequency; and

f_{1} = the 3-dB frequency below the resonant frequency.

To design a loaded UHF-MRI probe operating at 8 T (i.e., 340 MHz) and using the optimum configuration of the TEM SER (for θ = 72 deg.), the authors applied modified and coherent FEM- and MoM-based numerical modeling tools to the structure of Fig. 2 with the following set of features:

- A short b axis of 10 cm
- A long-to-short-axis ratio (a/b) of 1.8
- An outer radius-to-long-axis ratio (rb/a) of 2.4
- A sheet thickness-to-short-axis ratio (t/b) of 0.1; and
- A window angle (θ) of 72 deg

The numerical approaches make it possible to simulate the performance of a design and decide if a given set of constraints makes it possible to realize the UHF-probe.

The authors’ FEM and MoM approaches were employed as shown in Fig. 3 and Fig. 4 to determine the EM parameters of the quasi-TEM elliptical resonator. As discussed above, the integration of the normal flux over the conductor contours determines the per-unit-length parameter matrices. For instance, Table 1 lists the elements of the [L] and [C] matrices for e_{r} = 64. The table clearly shows good agreement between the results obtained by the two numerical approaches for inhomogeneous slotted elliptical tube-line resonator.

First, a UHF-MRI probe was designed using an unloaded slotted elliptical tube-line TEM resonator with the following features: resonator length, l (with respect to the wavelength of free space, λ_{0}), of 20 cm (l ˜ λ_{0}/4); a matching capacitor, C_{M}, with value of 22.24 pF; and source and load trimming capacitors, C_{S} and C_{L}, respectively, both with value of 1.44 pF. The simulated S_{11} responses at the RF port for the designed unloaded MRI probe are shown in Fig. 5 for both TLM programs and for MATPAR software.^{19}

In practice, for UHF-MRI proton imaging at 8 T, such results remain valid when the quasi-TEM SER is filled by an inhomogeneous biological load (such as a human head). In ref. 14, 18 tissue types (in addition to air) were identified in the images given in Fig. 6 in order to obtain a detailed human head structure. These included blood, bone-cancellous material, bone-cortical material, cartilage, cerebellum, cornea, cerebro spinal fluid (CSF), dura, fat, gray-matter (GM), mucosa, muscle, nerve, skin, tongue, vitreous-humor, white-matter (WM), and mixed-GM-WM.

For the same length (i.e., l = 20 cm) of the unloaded elliptical MRI probe, the authors introduced a biological load having a relative dielectric constant (e_{r})^{20} into the MRI resonator and numerically tuned matching capacitor C_{M} and terminating capacitors C_{Si} and C_{Li} until achieving resonance. At 340 MHz, the values obtained for these capacitors are shown in Table 2, along with the EM parameters of the elliptical resonator, for each biological load. From Table 2, the value of the matching capacitor varies between 1 and 2.06 pF for MRI use when applying the optimum configuration of the quasi-TEM SER. The values shown in Table 1 (for the element of matrix [L]) and Table 2 are essential for designing inhomogeneous elliptical UHF-MRI probes operating at 8 T. Table 2 provides the key parameter values for the wide range of load types in the MRI probe at 340 MHz, including blood, bone, cartilage, cornea, muscle, nerve, and skin materials.

Considering that the average relative dielectric constant of the human head is 64,^{14} the wavelength inside the head is approximately 11 cm. As a result, the EM parameters of the quasi-TEM SER loaded with the human head model obtained from the authors’ MoM analyses include even- and odd-mode characteristic impedances, Z_{0e} and Z_{0o}, of 134.6 O and 8.8 O, respectively, and effective dielectric constants, e_{effe} and (e_{effo}, of 1.066 and 35.36, respectively, and primary inductive and capacitive matrices, [L] and [C], respectively, as follows:

Figure 7 shows the simulated frequency responses of S_{11} at the RF port for the designed UHF-MRI probe using quasi-TEM SER loaded with the human head model, using both of the authors’ TLM programs and commercial MATPAR software. From this figure, it appears that the biological load introduced into the SER improves the value of the reverse transmission, indicated by the response of parameter S_{11}, at 340 MHz. For matching capacitor C_{M} with value of 20.83 pF and terminating capacitors C_{Si} and C_{Li}, both with capacitance value of 1.31 pF, the loaded UHF-probe operates at 340 MHz (proton imaging at 8 T) and has -139.5 dB minimum reflections. Using Eq. 4, Q was estimated to be very superior to 500.

This report has presented the analysis and the design of a UHF-MRI probe with high Q, operating at 8 T (i.e., 340 MHz) and using the optimum configuration of the slotted elliptical tube-line quasi-TEM resonator. The EM parameters of the elliptical quasi-TEM resonator were characterized using modified FEM and MoM programs. When the EM parameters were determined, it was possible to simulate the frequency response of S_{11} at the RF port of the designed quasi-TEM resonator loaded with any biological element having any combination of relative dielectric constants. The high-Q quasi-TEM UHF-MRI probe that was designed operated at 340 MHz. Using a SER loaded with a human head model having average relative dielectric constant of 64, minimum reflections of -139.5 dB were measured. The UHF-MRI probe described in this report is inexpensive, easy to construct, simple to operate, and can be easily modified to work at different resonant frequencies. It can be effectively applied to research on organs and tissues with high water content, including muscles and brain tissues.

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