Liver cancer is a significant worldwide public health issue. The disease has a mortality rate of 100% at 5 years in untreated cases 1 and results in the deaths of more than one million people each year worldwide 2345. Although liver cancer can be treated successfully by surgical resection of the malignant tissue, approximately 90% of patients with the disease are ineligible for the procedure due to factors such as insufficient hepatic reserve and the close proximity of tumors to blood vessels 16. One promising alternative for these patients is hepatic microwave ablation (MWA), an experimental procedure in which an antenna is inserted percutaneously or during surgery 7 to induce cell necrosis through the heating of deep-seated tumors. Unlike other alternatives to resection such as RF ablation and cryoablation, MWA systems are able to produce large lesions in the presence of blood perfusion and are not restricted by tissue charring 57. MWA also has favorable one, two, and three year survival rates of 96%, 83%, and 73% 8.

The many perceived advantages of microwave ablation have driven researchers to develop innovative antennas to effectively treat deep-seated, nonresectable hepatic tumors. These designs have focused largely on thin, coaxial-based interstitial antennas 91011, which are minimally invasive and capable of delivering a large amount of electromagnetic power. These antennas can usually be classified as one of three types (dipole, slot, or monopole) based on their physical features and radiative properties.

To assist in antenna design for MWA, many researchers have employed the use of mathematical models rooted in computational electromagnetics (CEM), a discipline that employs numerical methods to describe propagation of electromagnetic waves. These methods center around the formulation of discrete solutions to the fundamental electromagnetic equations collectively referred to as Maxwell's equations. Three principal techniques exist within CEM. The first of these, the finite-difference time-domain (FDTD) method, is based on the Yee algorithm 12 and uses finite difference approximations of the time and space derivatives of Maxwell's curl equations to create a discrete three-dimensional representation of the electric and magnetic fields. This method has been widely used to numerically evaluate the electromagnetic radiation patterns of antennas in tissue 1314151617, although long computation times are generally required. Another commonly used CEM technique is method of moments (MoM) 1819, in which approximate numerical solutions to integral equations are formulated in the frequency domain to determine an unknown current distribution for an antenna. This distribution can then be subsequently extended to yield the antenna's radiation pattern.

An alternative to the two techniques above is the finite element method (FEM), which has been extensively used in simulations of cardiac and hepatic radio-frequency (RF) ablation 2021. FEM models can provide users with quick, accurate solutions to multiple systems of differential equations and as such, are well suited to heat transfer problems like ablation. In this study, we have adapted an existing axisymmetric electromagnetic FEM model that was implemented using FEMLAB™ 3.0 22 to demonstrate proper techniques for the design of an antenna for hepatic MWA.

The FEM model used in this study was adapted from a coaxial slot antenna general model, developed by COMSOL for microwave cancer therapy 22. In this model, the electric and magnetic fields associated with the time-varying transverse electromagnetic (TEM) wave generated by the microwave source propagating in a coaxial cable in the z-direction was expressed in 2D axially symmetric cylindrical coordinates as

with

where r_o and r₁are the outer and inner radii of the coaxial cable (m), P_in is the input power (W), ε_rd is the relative permittivity of the dielectric, is the wave impedance in the dielectric of the coaxial cable (Ω), is the intrinsic impedance (Ω), ε₀ = 8.854 × 10^-12 is the permittivity of free space (F/m), μ₀ = 4π × 10^-7 is the permeability of free space (H/m), ω = 2π f is the angular frequency (rad/s), f is the frequency (Hz), k = 2π/λ is the propagation constant (m^-1), and λ is the wavelength (m). For interstitial coaxial-based antennas during MWA, the magnetic field is purely azimuthal. The electric field is in the radial direction only inside the coaxial cable and in both radial and the axial direction inside the tissue. This allows for the antenna to be modeled using an axisymmetric transverse magnetic (TM) wave formulation, in which the source was modeled as a low reflection boundary

with excitation magnetic field H_φ0 = C/Zr. Such axisymmetric FEM models are highly desirable as they dramatically reduce computation time.

Careful examinations of both the antenna's specific absorption rate (SAR) pattern and frequency-dependent reflection coefficient in tissue are essential for the optimization of antennas for hepatic MWA. SAR represents the electromagnetic power deposited per unit mass in tissue (W/kg) and can be defined mathematically as

where σ is tissue conductivity (S/m) and ρ is tissue density (kg/m³) 23. For the treatment of deep-seated hepatic tumors, the SAR pattern of an interstitial antenna should be highly localized near the distal tip of the antenna. Antenna efficiency can be quantified using the frequency-dependent reflection coefficient, which can be expressed logarithmically as

where P_r indicates reflected power (W). The frequency where the reflection coefficient is minimum is commonly referred to as the resonant frequency and should be approximately the same as the operating frequency of the generator used. Antennas operating with high reflection coefficients (especially at higher power levels) can cause overheating of the feedline possibly leading to damage to the coaxial line or due to the thin outer conductor damage to the tissue 24.

Critical to the development of electromagnetic models of hepatic MWA is an accurate knowledge of the dielectric properties of liver tissue. Fig. 1 shows the dielectric properties of fresh bovine liver that was obtained from a local slaughterhouse and measured approximately 1 h postslaughter. Measurements were obtained using the dielectric spectroscopy procedure described by Popovic et al 25, using a custom designed, stainless steel and glass borosilicate open-ended coaxial probe. These measurements were subsequently converted to complex permittivity using a rational function model 2627.

Fig. 3(a) shows the axisymmetric finite element mesh, which was generated by FEMLAB™ using the mesh parameters in Table 1 and the maximum element sizes determined by the convergence study. This mesh consists of 8920 triangular elements. Fig. 3(b) shows the corresponding two-dimensional normalized SAR distribution produced by FEMLAB™ for a double slot choked antenna. Calculation time for a single simulation on a personal computer with 2.8 GHz Pentium™ 4 processor and 1.5 GB memory required approximately 8 s. This same simulation required 75 s on a laptop computer with 1.6 GHz Pentium™ 4 processor and 512 MB memory.

Fig. 4 shows the simulated antenna reflection coefficient in liver expressed logarithmically as a function of frequency with the dashed line indicating the commonly used MWA frequency of 2.45 GHz. This frequency response was calculated using the procedure described above and required approximately 30 min to complete. Fig. 5 shows the double slot choked antenna that was built to validate these results. Fig. 6 shows the simulated and measured reflection coefficient of this constructed antenna in saline.

Fig. 3(b) shows that the double slot choked antenna is able to provide a degree of power localization similar to those produced by other recently designed antennas. Volumes for 50% and 90% energy dissipation are 7.55 cm³ and 99.3 cm³ respectively. This initial analysis also shows that it may be capable of providing more localized power deposition than the earlier double slot antenna and standard cap-choke antenna, and should therefore be considered as a viable design for future research. Fig. 4 shows that the antenna is resonant at 2.45 GHz with a low reflection coefficient of approximately -22 dB. Reducing catheter thickness by 0.1 mm slightly improved localization but decreased the resonant frequency of the antenna by approximately 400 MHz. Such effects were expected because the catheter is an important part of the antenna structure. However more studies are apparently necessary to learn the exact effects of the catheter and the antenna geometry with catheter should be optimized together for the desired resonant frequency and SAR localization.

Several factors should be considered when designing and simulating antennas for hepatic MWA. Due to phase considerations, both antenna dimensions and the thickness of the catheter should be chosen and optimized based on the effective wavelength in tissue. As this quantity is highly dependent on the relative permittivity of the medium, it is also essential that electromagnetic models of antennas for hepatic MWA employ accurate dielectric properties of liver tissue. Great care should be taken when measuring these properties or using previously published data, as the dielectric properties of liver tissue are highly dispersive and change with tissue type 35, water content 36, and temperature 37.

In addition, tradeoffs that are not well understood exist between performance metrics such as SAR and the reflection coefficient. From our experience, it is usually more desirable to achieve a localized power deposition than a highly resonant antenna since external impedance matching networks can be used to improve antenna efficiency 38. However, our additional experiments have shown that due to thermal conduction, a localized SAR pattern will only result in controlled tissue heating when treatment duration is minimized. Not only through the tissue, heat can also be effectively conducted by the antenna metal body and result in uncontrolled tissue heating along the antenna body for long treatment durations even with localized SAR pattern of the antenna.

Compared with other CEM software packages such as XFDTD™ and FEKO™, FEMLAB™ provides a user-friendly graphical interface for quick and reliable antenna design using axisymmetric electromagnetic models that can be integrated with MATLAB™ for increased functionality. Also, as the software was designed for multiphysics applications, it should also be ideal for future thermal simulations of hepatic MWA that are based on the Pennes bio-heat equation 39 and are dependent on both the electromagnetic and thermal properties of tissue. Initial work has also shown that electromagnetic models implemented using FEMLAB™ can be used in genetic algorithms 40 to better optimize antennas.

An antenna for hepatic MWA was quickly and accurately simulated using an axisymmetric electromagnetic model implemented in FEMLAB™ 3.0. This model is ideal for analyzing the SAR patterns and reflection coefficient of antennas used in hepatic MWA, two metrics which are commonly used for the evaluation of such antennas. Using this model, a double slot choked antenna with performance comparable to current antennas was designed and evaluated.

JB performed the research. All authors revised, read and approved the final manuscript.