3D highly heterogeneous thermal model of pineal gland in-vitro study for electromagnetic exposure using finite volume method

In this paper, the relationship between electromagnetic power absorption and temperature distributionsinside highlyheterogeneous biologicalsamples was accuratelydeter-minated using ﬁnite volume method. An in-vitro study on pineal gland that is responsible for physiological activities was for the ﬁrst time simulated to illustrate effectiveness of the proposed method. © 2017 Author(s). All article content, except where oth-erwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4991464]


I. INTRODUCTION
In recent years, wireless telephones have been widely used in our life. There is an increasing public concern about the health hazards resulting from exposure to cell phone radiofrequency (RF) radiation. The International Agency for Research on Cancer (IARC) classified RF radiation as a possible human carcinogen. 1 An interesting finding of the U.S. National Toxicology Program (NTP) support the IARC conclusions regarding the possible carcinogenic potential of RF radiation and reports that "low incidences of tumors in the brains and hearts of male rats, but not in female rats." 2 Temperature elevation from RF energy absorption (usually expressed in terms of the specific absorption rate SAR) is known to be a dominant factor inducing adverse health effects. The difficulty existing in the analysis of the temperature rise in biological bodies caused by electromagnetic fields is the highly complex computation and the lack of preciseness.
The bio-heat transfer equation for homogeneous material model can be easily calculated by using second order finite difference approximation to discretize the spatial derivatives. However, for highly heterogeneous thermal model, the spatial derivative ∇(K ·∇T ) is difficult to be discretized using FDTD. For a large object, such as human head or human body, the second order finite difference expressions might be suitable enough, but in practice, the in-vitro studies on cells or isolated tiny tissues are often desirable which require accurate simulation. To overcome the disadvantages of the FDTD method, the Finite Volume Method (FVM) 3 has been investigated by researchers. The method is based on integral conservation and therefore has distinct advantages for being readily applicable to multidimensional problems involving variable mesh and physical properties. We propose numerical methods that use FVM method to analyse the fundamental problem of bio-heat transfer caused by electromagnetic (EM) exposure for inhomogeneous materials models. To the best of our knowledge, the method has not been utilized to calculate bio-thermal response of pineal gland caused by electromagnetic exposure. Melatonin which is secreted by the pineal gland in the brain is a major regulator of core a body temperature. We recently conducted an in-vitro study on pineal gland that is responsible for physiological activities.

II. METHODS
Biological media are highly heterogeneous, thermal conductivity K varies from tissue to tissue. The bio-heat transfer equation is expressed as: 4 with the boundary condition where K is the thermal conductivity [J/(s · m · • C)]; A 0 is volumetric heat sources [J/(s · m 3 )] due to metabolic processes; the specific absorption rate (SAR) is the input EM heating source into the bio-heat equation; B is a parameter proportional to the blood perfusion [J/(s · m 3 · • C)]; T a is the ambient temperature; T b is the blood temperature; h is the convective heat-transfer coefficient [W /m 2 · • C]; and C p [J/(kg · • C)] and ρ [kg/m 3 ] are the tissue specific heat and density, respectively. Many tasks require the in-vitro studies on cells or isolated tiny tissues. If the solution is not sufficiently smooth, the order of accuracy will be reduced. This can happen if the coefficients are not smooth, e.g., due to the thermal conductivity jump across the interface of different tissues. We propose in this context the FVM method which is fully compatible with the FDTD method and allows the use of highly inhomogeneous objects. FVM is developed from integral interpolation method of FD. As the name implies, a finite-volume formulation involves integrating equation over a control volume (CV). The method has been described previously. 5 Briefly, using the notations in field theory, Eq. (1) can be rewritten as: The integral on the left-hand side of Eq. (4) can be removed by means of the mean value theorem for integrals. Similarly, the integral over S is removed. The volume integral over the divergence of the heat flux vector is transformed to a surface integral by means of the divergence theorem Eq. (4) can be expressed as: Using half-grid central differences, Eq. (5) can be simplified to yield Temperature updating equation: In order to ensure the numerical stability, the maximum time step is chosen to satisfy the stability criterion δ t ≤ 2ρc p δ min 2 12k+Bδ min 2 which was derived from Von Neumann's condition. 6 By expanding in its finite volume approximation, Eq. (2) can be written as

III. NUMERICAL EXAMPLE
Equations (6) and (7) can be easily implemented on a computer. To illustrate effectiveness of the proposed method, the in-vitro study on pineal gland shown in Figure 1 in a sector of the radial waveguide exposure system 7 was conducted.
It is calculated for 6 different materials, including the air surrounding the model, glass, rubber, pineal gland, attached skull, and Krebs-Ringer buffer. Table I lists the thermal properties of all materials. Figure 2-4 show the electric field distributions plotted across the centre of the model due to excitation of 1800MHz. Determination of the electromagnetic heating source 8 will not be discussed in this paper.  The specific absorption rate distribution is evaluated as: where σ(i, j, k) and ρ(i, j, k) are the conductivity and density of the tissue filling the (i, j, k) cell. Figure 5-7 show the SAR distributions inside this pineal gland in-vitro model. The temperature rise of the target model after 5 min exposure was calculated at 800 mW/kg average SAR in the small pineal organ. The initial temperature began from 37 • C.  phone use; 10 The research by MJ Schoemaker suggests that there is no substantial risk of acoustic neuroma in the first decade after starting mobile phone use; 11 The result of F Malek et al.
indicates that no negative health effect can be associated with electromagnetic field (EMF) exposure; 12 The study by C Johansen provides no support for an association between mobile phones and ocular. 13 Byeon and Back discovered that melatonin level increased as temperature increased when rice seedlings were exposed to various temperatures for 1 hr. 14 Melatonin which is secreted by the pineal gland in the brain is a major regulator of core body temperature. Hence, temperature rise due to EM exposure may cause body's melatonin level to increase. Many experiments do show increased melatonin levels after exposure to magnetic or electromagnetic fields. [15][16][17] So hypothetically, the results of temperature rise in the pineal gland might indicate that melatonin could be a good mechanistic candidate to explain potentially deleterious or salubrious effects of cell phone RF radiation. Melatonin is used to treat insomnia. Cohen et al proposed the hypothesis that a decrease in melatonin levels might promote the development of breast cancer in humans. 18 Melatonin has also been reported to exert additional functions in other organs. [19][20][21][22][23]

IV. CONCLUSION
Simulation of temperature and electromagnetic field inside pineal gland cell was for the first time performed to illustrate effectiveness of the FVM method. The results exclude the possibility of thermally induced health hazards from cell phone radiofrequency exposure. In future study, we will apply the method implicitly 24 to 3D multi-material model with non-uniform grids.

ACKNOWLEDGMENTS
The paper was supported by the National Natural Science Foundation of China (61370185).