International Journal of Antennas and Propagation

Volume 2018 (2018), Article ID 2969237, 9 pages

https://doi.org/10.1155/2018/2969237

## Modeling Human Body Using Four-Pole Debye Model in Piecewise Linear Recursive Convolution FDTD Method for the SAR Calculation in the Case of Vehicular Antenna

Harbin Institute of Technology, Xidazhi Street, Harbin, China

Correspondence should be addressed to Ammar Guellab; nc.ude.tih@004balleugramma and Qun Wu; nc.ude.tih@uwq

Received 16 September 2017; Revised 7 December 2017; Accepted 28 December 2017; Published 19 April 2018

Academic Editor: Rodolfo Araneo

Copyright © 2018 Ammar Guellab and Qun Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose an efficient finite difference time domain (FDTD) method based on the piecewise linear recursive convolution (PLRC) technique to evaluate the human body exposure to electromagnetic (EM) radiation. The source of radiation considered in this study is a high-power antenna, mounted on a military vehicle, covering a broad band of frequency (100 MHz–3 GHz). The simulation is carried out using a nonhomogeneous human body model which takes into consideration most of the internal body tissues. The human tissues are modeled by a four-pole Debye model which is derived from experimental data by using particle swarm optimization (PSO). The human exposure to EM radiation is evaluated by computing the local and whole-body average specific absorption rate (SAR) for each occupant. The higher in-tissue electric field intensity points are localized, and the SAR values are compared with the crew safety standard recommendations. The accuracy of the proposed PLRC-FDTD approach and the matching of the Debye model with the experimental data are verified in this study.

#### 1. Introduction

Nowadays, the battlefield is making more use of the electromagnetic spectrum to satisfy diverse operational needs that range from high-rate tactical links to broadband jammers. This calls for the use of vehicular antennas that transmit high powers, putting the life of crew personnel at stake through high electromagnetic (EM) exposure [1]. Such high-power radiation can occur in a broad frequency band (HF, VHF, and UHF). IEEE Technical Committee 95 (IEEE-TC95) proposed a standard for military workplaces whose purpose is to provide exposure limits to assure the personnel safety in a military workplace and provide protection against unfavorable effects of the electromagnetic radiation on the human body [2]. This standard expresses its recommendations as dosimetry reference limits (DRLs) which can be expressed by the within-tissue electric field strength or the specific absorption rate (SAR).

With the aim of protecting the crew against high-power EM radiation, much research is devoted to compute the level of the induced SAR from various EM sources. Accurate and efficient computational methods are sought after for this purpose [3–6]. This can be so challenging and complicated due to the human body and the vehicle structure complexity especially when a broad frequency band is considered.

A numerical simulation for the computation of the SAR in vehicle passengers due to an onboard 900 MHz transmission system is carried out in [3]. The FDTD method is used for the analysis of the SAR from a cell phone inside a vehicle in [4]. However, those studies are concerned with a single narrow band of frequency. Diao [5] uses the FDTD method for the estimation of the induced SAR from multiple communication devices, which occupy different frequencies inside the vehicle. In [6] the FDTD simulation of a complex structure vehicle is carried out to investigate the effect of the passenger’s number on the SAR level for many frequencies.

The previously cited works obtain a rigorous estimation for the SAR level induced by multiple sources. Nevertheless, they focus on isolated and narrow bands of frequency and use human body models that do not take into account the frequency dependency of electric characteristics of the human tissues.

A frequency-dependent model for the human tissue has been provided by Gabriel et al. [7], which is based on a four-pole Cole-Cole equation. This model leads to a highly precise evaluation of the complex electric characteristics (permittivity and conductivity) of the dispersive media constituting the human tissue. Nevertheless, the complexity of this model makes it disadvantageous for the implementation of time-domain numerical simulation.

Gabriel’s Cole-Cole model is reduced to a two-pole Debye model by using the least squares fitting technique in [8, 9]. The two-pole Debye tissue model is more suitable for the time domain methods but is less accurate than the Cole-Cole model. In [10], a fourth-order Debye model is used in the convolutional-based FDTD for the modeling of the electromagnetic waves propagation in the human head tissues. Based on this study, the fourth-order Debye model is more accurate than the two-pole Debye model and is simpler than the Cole-Cole model. Therefore, it combines the high accuracy of computations and the simplicity of implementation.

Despite the fact that the high-order Debye model matches perfectly with the experimental dielectric characteristics, it leads to computational complexity and an accumulative numerical dispersion. In the literature, many works are devoted to the development of highly accurate methods for frequency-dependent media. In [10], a general recursive convolution FDTD is applied for the simulation of the fourth-order Debye model. However, this method is less accurate. In [11] the alternating direction implicit finite difference time domain (ADI-FDTD) method is extended to the simulation of the Debye dispersive media. The advantage of the ADI-FDTD is its unconditional stability. However, it suffers from the loss of precision. In [12], a simple trapezoidal recursive (TRC) technique is used in the FDTD analysis of frequency dependent media, where it is proven that the TRC method is more accurate than the RC technique. For more accuracy, a piecewise linear recursive convolution (PLRC) technique is used in [13]. The PLRC technique is an efficacious method for dealing with the dispersive models (Debye, Lorentz, and Drude) [13].

In this work, a highly accurate four-pole Debye model is developed, which perfectly fits with the experimental data [14] of the whole-body tissues in a broad band of frequency ranging from 100 MHz to 3 GHz. For this purpose, we applied the particle swarm optimization (PSO) algorithm for the optimization role. Then, we proposed an implementation of a three-dimension FDTD based on the piecewise linear recursive convolution (PLRC) technique for the simulation of the four-pole Debye model. The agreement of the Debye model with the experimental dielectric properties of each tissue of the human body and the accuracy of the proposed FDTD method are verified.

After that, a typical military vehicle, commonly known as (Humvee) is considered. It is equipped with a large antenna used for radio-transmission applications. A reduced complexity model is used for the vehicle. Human bodies are present inside where they are exposed to high-power electromagnetic radiation issued from the antenna. We applied the developed PLRC-FDTD method for the study of the electromagnetic waves’ behavior in this case and to compute the human bodies’ exposure to those radiations through the evaluation of the specific absorption rate (SAR).

#### 2. Formulations

##### 2.1. Four-Pole Debye Model

With the aim of modeling the response of the human body issues to the electromagnetic radiation at a broad band of microwave frequencies, the measured permittivity and conductivity data for each tissue [14] are approximated by a fourth-order Debye dispersive model, as expressed in the following equation:
where is the angular frequency, is the free space permittivity, and are the real part and the imaginary part of the complex relative permittivity () of the Debye dispersive media where denotes the electric conductivity, is the permittivity at the infinite frequency, *K* is the number of Debye modes, and are, respectively, the magnitude and the relaxation time of the *k*th Debye dispersion mode, and is the static conductivity.

##### 2.2. The Particle Swarm Optimization

In this paper, the Matlab toolbox based on the particle swarm optimization (PSO) algorithm is used to accurately fit the Debye parameterized model with a set of measured data [14], by minimizing the cost function *ζ* which is the mean of the relative errors between the data measurements and the Debye computed model at each measured frequency, as expressed in the following equation [10].
where , and are, respectively, the computed permittivity and conductivity using the Debye model. and are the measured permittivity and conductivity, respectively [14].

The application of the particle swarm algorithm implies the minimization of the cost function . The experimental data [14] of 19 tissues are fitted over the frequency ranging from 100 MHz to 3 GHz with the fourth-order Debye model.

The PSO algorithm’s flow involves first the creation of a random swarm of 2000 particles. Each particle has a position which is represented by a set of Debye parameters and a velocity which is the used to compute the next position. As a constraint, all the parameters are nonnegative numbers. At each iteration of the PSO algorithm, the cost function is evaluated at all particles to record the location of the best particle in the neighborhood of each particle and the best one over all particles. For each particle, the neighborhood is the first 500 nearest particles. The velocity of each particle is updated as a weighted sum of the previous velocity and the position of the neighborhood best particle and the global best particle. Then, the position of each particle is updated as a function of the previous position and the velocity. Over consecutive iterations, the swarm of particles converge toward the optimal solution with the lowest cost function. The maximum number of iterations is 10,000.

For the aim of comparison, the genetic algorithm is performed in such a manner to have the same resource consumption as the PSO algorithm. The population size is 2000. The maximum number of generations is 10,000.

Table 1 lists the four-pole Debye parameters of each tissue (due to the limited space, only a few of them are listed) and the evaluated errors of each model expressed by (2). The fitting results of some tissues are plotted in Figure 1.