FDTD Analysis of the Electromagnetic Fields Produced by a Four-channel RF Applicator

Abstract:

A four-channel radio-frequency (RF) electromagnetic (EM) phased array applicator has been designed for hyperthermia treatments of locally advanced breast cancer. Simulations of the electric field produced inside and outside of the tank are performed by the finite difference time domain (FDTD) method, which is an efficient approach for computating of electromagnetic wave propagation within complex structures. In these 3D FDTD simulations, the absorbing boundary condition (ABC) is implemented as a perfectly matched layer (PML) on the outer surface of the computational domain. The tank is modeled with the staircase approximation, and to reduce the numerical error introduced by the sloped surface of the tank, a cell spacing of 2.4mm is selected in all three dimensions. The end-loaded dipole antennas are modeled by the Dey-Mittra technique for structures containing perfect electric conductors (PECs), which is the result of the Yee algorithm derived from the integral form of Faraday's Law and Ampere's Law instead of the differential form of these expressions. The size of the computational domain is 1m by 0.72m by 0.7m, and the source frequency is 140MHz. Simulation results obtained with the FDTD method demonstrate that the phased array applicator generates standing waves within the water tank, and constructive interference ultimately produces a focus in the center of the tank.

Applicator 3D models for EM simulation:

Four end-loaded dipole antennas^[^1] are mounted on a water tank such that each antenna points in the direction of the intended target. The lexan water tank is filled with deionized water, which is depicted in figure 1. The geometric focus is in the upper central region of the applicator. The source frequency is 140MHz in all simulations.

Figure 1 : RF phased array applicator with 4 antennas

Methods :

The EM field inside the applicator is simulated with the Finite-Difference Time-Domain (FDTD) method. Berenger’s perfectly matched layer (PML) is chosen as the absorbing boundary condition (ABC)[3][5]. The simulation grid consists of cubic FDTD cells that are 2.4mm on a side, which is about 1/100 of the wavelength in water. The lexan tank is modeled with staircase cells and all cells containing antennas are modeled with the Dey-Mittra technique for PEC structures^[3]

Results :

The size of tank is 30cm(x) by 55cm(y) by 27cm(z). The FDTD simulation results are evaluated in deionized water, in the lexan enclosure, and in the air surrounding the tank. Fig. 2 contains two orthogonal 2D slices of a simulated 3D electric field (E field) evaluated for the 4 antenna breast applicator depicted in Fig. 1. This result shows that the E field at (0, 0, -3cm) is maximized with a sinusoidal 140MHz excitation. The upper figure shows the E field in the xy plane located at z = -3 cm while the lower figure shows the E field in the yz plane which is located at x = 0cm. All of the results are shown in dB.

Fig. 2: E field evaluated in the xy and yz planes

Figure 3a demonstrates simulated and measured electric field values along the Z axis from about 2.5 cm above the water surface to 14cm below the surface. The FDTD simulation correctly predicts the locations of the measured minimum and maximum values for the 4-antenna applicator. The maximum measured and simulated values are observed 3cm below the water surface, whereas the minimum measured and simulated values occur 12cm below the water surface. Fig 3b and 3c also shows a broad peak along the X and Y axes for both the measured and simulated electric fields. In Figure 3, the measurements confirm the overall trend of the simulated electric field in each direction.

Fig. 3a: Measured and simulated E field evaluated along the Z axis (X=0, Y=0)

Fig. 3b: Measured and simulated E field evaluated along the X axis (Y=0, Z=-3cm)

Fig. 3c: Measured and simulated E field evaluated along the Y axis (X=0, Z=-3cm)

After validating the FDTD algorithm on this model, a plastic cup is inserted into the tank to fix the position of breast, as shown in Fig. 1, the cup inner diameter is 10cm, the thickness is 3mm, and the height is 12 cm. Next, a breast phantom is placed in the center of the applicator. The breast phantom conforms to the shape of the cup, the outer 5mm of the phantom material is modeled as mammary tissue, and the remaining phantom material is modeled as tumor, as shown in Fig 4. All material parameters are listed in Table 1.

Fig. 4: Breast model

	Material/tissue	Permittivity	Conductivity σ (S m-1)	Density (kg m-3)
Patient	Mammary	10	0.06	880
	Muscle	75	0.75	1050
	Malignant mammary	65	0.74	1050
Applicator	Deionized water	76.5	0.001	1000
	metal	1	107	7900
	Lexan	2.9	0	1190

Table 1: Material and tissue properties for FDTD modeling^[1],[2]

Fig. 5 shows the simulated E field distribution when the cup is included in the model. Fig. 6 shows the E field distribution when a breast phantom is placed into the applicator. All results show the same yz plane located at x = 0 and xy plane located.

Fig. 5: E field distribution for the cup model Fig.6: E field distribution for a phantom breast

evaluated in the xy and yz planes placed in the applicator with an additional cup.

Summary & Future work:

Comparisons between simulated and measured electric fields show that FDTD results successfully predict the measured E field as demonstrated in the figures above. FDTD simulations in a breast phantom show that this applicator can heat a tumor surrounded by a thin layer of mammary tissue. More detailed human models, based on MR and/or CT images, will be included in future FDTD simulation results.

References:

W. T. Joines, Y. Zhang, C Li, and R. L. Jirtle, “The Measured Eletrical Properties of Normal and Malignant Human tissues from 50 to 900 MHz” Med. Phys. 21(4), 547-550 (1994).
H. Kroeze, J. B. Van de Kamer, A. A. C. De Leeuw and J. J. W. Lagendijk. “Regional hyperthermia applicator design using FDTD modeling”. Phys. Med. Biol. 46 No 7 1919-1935 (2001).
”Computational Electrodynamics: The Finite-Difference Time-Domain Method”, Second edition. by A. Taflove, S. C. Hagness. Artech House (2000).
S. M. Foroughipoul and K. P. Esselle, “The theory of a Singularity-Enhanced FDTD Method of Diagonal Metal Edges”, IEEE Transactions on Antennas and Propagation Vol. 51, No.2, 312-321 (2003).
J. P. Berenger, “Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves”, Journal of Computational Physics 127, Article No. 0181 363–379 (1996)

Author:

Ruihua Ding

Contact:

Links: