Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 7249672, 7 pages

https://doi.org/10.1155/2017/7249672

## Effective Permittivity of Biological Tissue: Comparison of Theoretical Model and Experiment

Department of Biomedical Engineering, School of Electronic Information Engineering, Xi’an Technological University, Xi’an 710021, China

Correspondence should be addressed to Li Gun; nc.ude.utax@nugil

Received 12 March 2017; Revised 14 May 2017; Accepted 25 May 2017; Published 21 June 2017

Academic Editor: Marek Lefik

Copyright © 2017 Li Gun et al. 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

Permittivity of biological tissue is a critical issue for studying the biological effects of electromagnetic fields. Many theories and experiments were performed to measure or explain the permittivity characteristics in biological tissue. In this paper, we investigate the permittivity parameter in biological tissues via theoretical and experimental analysis. Firstly, we analyze the permittivity characteristic in tissue by using theories on composite material. Secondly, typical biological tissues, such as blood, fat, liver, and brain, are measured by* HP4275A* Multi-Frequency LCR Meter within 10 kHz to 10 MHz. Thirdly, experimental results are compared with the Bottcher-Bordewijk model, the Skipetrov equation, and the Maxwell-Gannett theory. From the theoretical perspective, blood and fat are regarded as the composition of liver and brain because of the high permittivity in blood and the opposite in fat. Volume fraction of blood in liver and brain is analyzed theoretically, and the applicability and the limitation of the models are also discussed. These results benefit further study on local biological effects of electromagnetic fields.

#### 1. Introduction

Biological effects induced by electromagnetic radiation depend on electric properties of biological tissue during exposure to electromagnetic radiation. Because of difficulties in measurement of biological tissue in vivo and permittivity of biological tissue which shows nonlinear characteristics in frequency domain, determining effective permittivity of biological tissue has received repeated attention in recent years [1–3]. Many techniques and principles of testing permittivity in biological tissue are similar to the traditional engineering materials. Capacitance of the tissue is measured first, and then permittivity can be calculated by the test results via relationship between the tissue morphology and the electromagnetic theory [4–6]. That is, the experimental study on permittivity of biological tissue is conducted via calculation from the test result (capacitance). The existing research has already reflected a lot of specific application. Otaki et al. suggested that dielectric blood coagulometry may be a useful method for measuring blood clotting and could provide the detailed assessment for the status of anticoagulant therapy [7]. The dielectric properties are not stable in biological system, which changes with external conditions such as temperature and external electric frequency. The dielectric properties of living tissue have its own features, and calculation of effective permittivity based on theory modeling can greatly reduce the workload in exploring them [8]. There are already many theoretical studies on simulation of effective permittivity in biological tissue by using perturbation expansion, effective medium approximation, the finite element method, and so on [9–12]. Among the studies on mathematical models of permittivity in tissue, most of them used the composite material model. Ashutosh Prasad et al. study experimental data yielded by different mixture equations in order to test the acceptability of dielectric mixture equations for high volume fractions of the inclusion material in the mixture [13]. Amooey considers the modification of volume fraction to improve mixing rules model on calculation of effective permittivity. Some mixing rules are compared for studying the permittivity of mixtures [14]. Zohdi et al. take for electromagnetic techniques which may provide an effective way to estimate the permittivity in blood, which can determine the volume fraction of blood cell in other tissue. They also attend to think that the deviation of the permittivity parameter can help characterize certain tissue disorders [15]. Prodan develops a theoretical framework for describing the dielectric response of live cells under low external electric fields by taking the presence of the cell’s membrane and its charge movement into account, and then the author also analyzed the effect of several other physical parameters on permittivity in cell [16]. Peón-Fernández et al. established a numerical model to estimate permittivity for periodically structured materials via the long wave approximation. When the system was subjected to a prescribed potential, Monte-Carlo random walk iterative method was used to estimate the numerical result, and the model can well evaluate field distribution. They also suggested that values of the effective permittivity for certain aggregate systems such as percolating simple cubic structures are important for the usage of biological tissue [17].

From the contents of the above-mentioned literatures, there is no study combining the experimental results and theory. This paper deals with the effective permittivity in biological system, especially in the blood, liver, fat, and brain. A theoretical analysis on several typical model of effective permittivity in materials is presented. We intend to explore the acceptability of the models in biotissues. So we take for the well feasibility of the volume fractions model such as the Bottcher-Bordewijk model, the Skipetrov equation, and the Maxwell-Wagner theory to analyze the acceptability of these models. Finally, experiment is conducted to validate the acceptability of the model mentioned.

#### 2. Theoretical Models and Their Applicability

The permittivity in biological system is related to its structure, compositions, nature, and so on. For a specific size object, permittivity can be calculated by experimentally detected results via

Here , , , and denote the permittivity, capacitance, length, and cross-sectional area of the object, respectively. Often, relative permittivity () should be used to express electrical properties of materials. Relative permittivity of tissue is the ratio of permittivity of tissue () and the permittivity of vacuum (); that is,

When blood is concerned, it can be easily shaped for detecting the capacitance. But elastic properties of other tissues may be influenced by attached proteins on biological membranes [18], and the detecting capacitance may be affected by the detection voltage of the measuring equipment [19]. Therefore, it is necessary to ensure good contact between the test sample and the electrode. For the biological system, the dielectric property is generally expressed via the multiphase permittivity. Many classical formulae were put forward for calculating permittivity of the two-phase composition. Establishment of tissue permittivity model for biological system would help us to simplify its complex constitution. A typical theory is Debye model, which is very famous and explains many compositions well and was put forward on the basis of empirical formula [20]. It is a widely used model for studying the dielectric properties of biological tissues in frequency domain. Binary mixtures’ model is usually used for investigating effective permittivity of mixture system for its significance in understanding the intermolecular interactions. There are many permittivity models on binary mixtures without considering the frequency domain, for instance, the Bottcher-Bordewijk model [21], Maxwell-Garnett formula [22], Bruggeman formula [23], and Hanai formula [24]. Each theory can only be successfully applied to a certain type of composition. Among them, the Bottcher-Bordewijk model suggests very well feasibility to predict the permittivity of a mixture [21]. Here the model is as follows:

Here denote the permittivity of the pure components and denote the volume fraction of each of the components, so the equation can be modified as follows for two-phase composition:

Cell structure is of a random nature with some predictable average properties such as cell size and density. It is can be modeled by an aggregate of randomly distributed spherical shells in (4). This model can be used to describe the two constituent materials. When the two-phase system is concerned, there is another famous model put forward by Skipetrov [25]. Skipetrov equation is as follows:

This model has been derived based on the assumption that here is much smaller than unity and that either correlation length or the particle diameter is well below the wavelength of electromagnetic waves used. The Skipetrov equation is original and more transparent than others and is assumed to give more correct results under tough situations. Skipetrov explored the effective permittivity of a random medium; the result showed that, for a two-component mixture, even volume fractions of one component , the effective permittivity is also calculated and discussed in Skipetrov’s work. Now, there is another famous model called Maxwell-Gannett model. Here we introduce the Maxwell-Gannett equation for the two-phase multisystem:

Here is the permittivity of the composite and and are permittivity of the doped phase and the matrix phase, respectively. So dielectric characteristics of tissues in human body are similar to the suspension system denoted by the Maxwell-Gannett conventional empirical models. Maxwell calculated permittivity of the composite media in the electrostatic field. With the in-depth study of the dielectric of the composite material, researchers are constantly looking for ways to model accurately the relationship between the various components of the composite material. When the dynamic electric field is further considered, Maxwell put out the permittivity model for complex biological tissue, which can be regarded as a small ball distributed in the continuum [22]. There are also other models, such as Bruggeman model [23], ideal mixture model [26], Peon-Iglesias model [27], and Kraszewski model [28]. All these models are been implicated to interpret the effective permittivity in two-phase composite material [29].

When biological tissue is concerned, uneven distribution of blood cells in tissues makes its representative dielectric properties [30]. Additionally, biological membranes are charged the dielectric properties, and in the vicinity of the membrane they are strongly influenced by orientational ordering of water dipoles which contributes to strong decrease of relative permittivity in the vicinity of the cell membranes and also partially within the membranes in their surface region. Therefore, accurately, the dielectric properties cannot be calculated from traditional models. In the following paper, we will discuss the applicability of these models for biological tissues from the perspective of experiment and theory, especially the deviation of these models from the experimental results.

#### 3. Experimental Results

Weight of male Wistar rats is g. Intraperitoneal injection of sodium pentobarbital is conducted with dose of 40 mg/kg. The chests of anesthetized rats were cut after being fixed on the disinfection board; 3–5mL blood was acupunctured from left apex of heart; and then the blood was put into the plexiglass tank (size: mm); all results are measured by* HP4275A* Multi-Frequency LCR Meter from 10 kHz to 10 MHz. Than brain, liver, and fat were measured in a similar way. Permittivity can be calculated via (1). Often, relative permittivity should be used to express electrical properties of materials, and relative permittivity of tissue is the ratio of permittivity of tissue and the permittivity of vacuum [31]; that is, . The relative permittivity of rat tissues is shown in Table 1. Here, of all tissues decline with the frequency; the main reason for the phenomena is that there is a period of time (relaxation time); different relaxation times characterize special responses of tissues to external electric fields.