Influence of the Electromagnetic Energy Due to Cellular Devices in a Multi-Layer Human Head under Two-Temperature Heat Conduction Model

Abstract

In this study, a mathematical model of a human head with three layers of skin, bone, and brain for both children and adults was created. The phase-lag time definition was used to evaluate the internal thermal reaction that is created via the human head as a result of chemical processes. In the framework of the two-temperature heat conduction model, the governing equations were developed. The inversions were numerically derived using an approximation approach after applying Laplace transforms. When the human head is exposed to cellular devices that emit an electromagnetic wave, the dynamic and conductive temperature increments distributions have been calculated and discussed with various values of the two-temperature parameter, time, relaxation time, power transmission frequency, and power density. Each layer of the human cranium is affected by all of the characteristics analysed. The impacts of electromagnetic waves emitted by cellular devices were more detrimental to the bones of children’s heads than they are to the bones of adults’ heads.

1. Introduction

Low-level electromagnetic fields have been present in the environment for a very long time, and their biological effects have been well documented. Dosimetry of these field characteristics and their emissions from diverse mass usage sources have therefore been a matter of ongoing worry. The biological implications of mobile communications have also recently received attention. Children and adults alike utilise handheld mobile devices, such as cordless phones, in a variety of body postures. Additionally, the proliferation of mobile communication base stations has raised widespread concerns about the potential negative consequences of radiofrequency radiation on human health. There are two separate ways in which exposure to radiofrequency fields may harm one’s health. These are thermal side effects brought on by holding cell phones close to the body and having protracted talks. Second, both phones and base stations may have non-thermal impacts, and those effects may likewise build up with time [1,2]. The thermal and electromagnetic energy may lead to a thermal shock to the human head [3,4,5]. Now, the biggest concern for everyone is how to make safe the use of cellular devices. The radio frequencies and their power are very dangerous to our bodies with a high risk of using those devices. Many studies have indicated that short exposures to electromagnetic waves released by cellular devices can be hazardous, such as affecting brain cell proliferation when exposed to the energy emitted by these waves [6,7,8,9].

Many forms of study and scientific initiatives have recently been done to address the prospective health impacts of the human body’s exposure to electromagnetic wave radiation. It is critical to distinguish between biothermal and non-biothermal effects in this setting [2]. Lindholm et al. used many human bodies to test the biothermal effects of cell phone fields on youngsters [10].

The thermal wave models of bioheat transfer (TWMBT) are consist of the equation of the energy balance of temperature rise due to convection, conduction, metabolic heat response, and temperature increase via radiation due to the frequency of the cellular device. As illustrated in Figure 1, the governing equation of that model was used to depict the human head when a cellular radio frequency shock was administered and travelled through the three layers of the human skull [11,12].

Pennes developed a heat transfer model based on the classical Fourier rule, which is commonly used in biological tissue to estimate heat transmission [13]. Cattaneo [14] and Vernotte [15] proposed a different thermal wave model considering a finite speed of thermal propagation. Maillet proved that the thermal wave model does not give an accurate result according to some experimental results [16]. To combine microscopic influences in a macroscopic representation, Tzou proposed the dual-phase lag (DPL) model [17,18].

The value of the relaxation-time parameter is in the range of 15–30 s, according to certain experimental research and examinations of biological topics [2,11]. The time lag of the temperature gradient was analytically studied [2,10,11,12,17,18,19]. This sort of model has been used to estimate the impact of electromagnetic radiation on the skin [20,21,22], and others have used the dual-phase-lag model (DPL) for the same reasons [23,24]. Smirnov and Meshcherinov used a graphical approach to calculate the temperature increase of the various layers of the brain based on temperature measurements in the ear or the patient’s body temperature during craniocerebral cooling [25].

Zhihua et al. utilised thermal wave models of bioheat transfer to develop a face recognition computational model [26]. The mathematical derivations of nanoparticle-aided laser-induced interstitial thermotherapy on tumour tissue and cancer therapies were constructed using the same model by Xu et al. [27]. Other models have investigated the thermal wave models of bioheat transfer by applying them to the brain [1,28,29,30,31,32,33]. Youssef et al. [34] looked at how heat waves radiated by cellular devices affected a multi-layer human skull. Chen and Gurtin [3] created the heat conduction theory, which was based on two temperatures (conductive and dynamical) and Chen et al. [4,5]. Youssef devised and used the two-temperature generalized thermoelasticity theory, which contains a general uniqueness theorem, to a range of issues [35,36,37,38].

This article is about a mathematical model of the human skull, which is made up of three layers: skin, bone, and brain, with the fat layer left out. The metabolic responses caused by tissue chemical reactions have been addressed in the context of a phase-lag time definition to determine their influence on the human head’s three layers. The two-temperature heat conduction equation, a non-classical heat conduction equation, was used. When the human head was exposed to an electromagnetic wave, the temperature increment distribution was explored via the three layers of the human head with varied values of duration, relaxation time, frequency of power transmission, and power density.

2. Methods

The traditional heat conduction equation has been discussed and introduced in one dimension by Penne in Equations (1) and (2) as follows [32,33,34,37,39]:

$$q\left(x;t\right)=-k\nabla T\left(x;t\right)$$

(1)

and

$$k\frac{{\partial }^{2}T}{\partial x^2}-\rho C\frac{\partial T}{\partial t}=W_b{C}_b\left(T-T_b\right)-\left(Q_{met}+Q_{ext}\right)$$

(2)

where $C_b\left(\mathrm{J}/\mathrm{kg} °\mathrm{C}\right), W_b \left(\mathrm{mL}/\mathrm{C} \mathrm{m}\right)$ and $T_b\left(°\mathrm{C}\right)$ were the specific heat of the blood, the blood perfusion rate, and the blood temperature, respectively.$\rho \left({\mathrm{kg}/\mathrm{m}}^3\right), C\left(\mathrm{J}/\mathrm{kg} °\mathrm{C}\right)$ and $k\left(\mathrm{W}/\mathrm{m} °\mathrm{C}\right)$ were the density, the specific heat, and the thermal conductivity of the tissue, respectively. $T\left(°\mathrm{C}\right)$ was the absolute temperature, and $q\left({\mathrm{W}/\mathrm{m}}^3\right)$ was the heat flux. Through the living tissue, $Q_{met}\left({\mathrm{W}/\mathrm{m}}^3\right)$ was the metabolic heat generated due to the chemical reaction, and $Q_{ext}\left({\mathrm{W}/\mathrm{m}}^3\right)$ was denoted by the external heat source [34,37,40].

For simplicity, just a minor penetration depth was evaluated, and the curvature has been left out. Thus, in the x-direction, the mathematical model transformed into a one-dimensional flat surface and began at the surface of the tissue layer above the skull. Then, as seen in Figure 1, this path passed through various layers of skin, bone, and brain, while the destiny layer was ignored.

A non-Fourier model might be a more exact model since biological tissue has a micro-scale thermal reactivity and response [34,37].

Fourier’s law of heat conduction provided a relaxation time parameter for the heat flux to account for the lag time as follows [32,33,34]:

$$q\left(x,t+\tau \right)=-k\frac{\partial \phi }{\partial x}$$

(3)

and

$$\phi -T=\beta \frac{{\partial }^2\phi }{\partial x^2}$$

(4)

where $\beta \left(m^2\right)\ge 0$ was called the two-temperature parameter and $\tau \left(s\right)$ was given by the ratio of the thermal diffusivity ${\alpha }_o\left(m\right)$ over the speed of the thermal wave $C_t\left(m/s\right)$ in the biological materials $\left\{\tau \left(\mathrm{s}\right)={\alpha }_o\left(m\right)/C_t\left(m/s\right)\right\}$.

By applying the Taylor expansion on Equation (3) up to the first order, the heat conduction equation was in the form [34,40,41]:

$$\mathrm{x}k\frac{{\partial }^2\phi }{\partial x^2}-\rho C⟮1+\tau W_b{C}_b+\tau \frac{\partial }{\partial t}⟯\frac{\partial T}{\partial t}=W_b{C}_b\left(T-T_b\right)-⟮1+\tau \frac{\partial }{\partial t}⟯\left(Q_{met}+Q_{ext}\right)$$

(5)

In most materials, the value of the relaxation time was small, while in biological tissues, the relaxation time parameter was impacted. The term $W_b{C}_b\left(T-T_b\right)$ was the heat generated by convection inside the three layers of the human head per unit mass of tissue, and it was thought to be homogeneous within each layer [34,37].

We assumed that ${\phi }_{0s}=T_{0S}, {\phi }_{0B}=T_{0B}, {\phi }_{0R}=T_{0R}$ were the reference temperatures of the skin, bone, and brain, respectively, so then we obtained [37]:

$$T_S-T_b=\left(T-T_{0S}\right)-\left(T_b-T_{0S}\right)={\theta }_S-{\theta }_{0S}$$

(6)

$$T_B-T_b=\left(T-T_{0B}\right)-\left(T_b-T_{0B}\right)={\theta }_B-{\theta }_{0B}$$

(7)

$$T_R-T_b =\left(T-T_{0R}\right)-\left(T_b-T_{0R}\right)={\theta }_R-{\theta }_{0R}$$

(8)

Similarly,

$${\phi }_s=\phi -{\phi }_{0S}, {\phi }_B=\phi -{\phi }_{0B}, {\phi }_R=\phi -{\phi }_{0R}$$

(9)

The differential equations of the heat conduction in skin tissue took the forms

$$\begin{array}{l}{k}_S\frac{{\partial }^2{\phi }_S}{\partial x^2}-{\rho }_S{C}_S⟮1+{\tau }_S{W}_{bS}{C}_b+{\tau }_S\frac{\partial }{\partial t}⟯\frac{\partial {\theta }_S}{\partial t}=W_{bS}{C}_b\left({\theta }_S-{\theta }_{0S}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-⟮1+{\tau }_S\frac{\partial }{\partial t}⟯\left(Q_{met}+Q^S{}_{ext}\right), 0\le x\le x_S\end{array}$$

(10)

and

$${\phi }_S-{\theta }_S=\beta \frac{{\partial }^2{\phi }_S}{\partial x^2}, 0\le x\le x_S$$

(11)

The heat conduction equations of the bone took the forms

$$\begin{array}{l}{k}_B\frac{{\partial }^2{\phi }_B}{\partial x^2}-{\rho }_B{C}_B⟮1+{\tau }_B{W}_{bB}{C}_b+{\tau }_B\frac{\partial }{\partial t}⟯\frac{\partial {\theta }_B}{\partial t}=W_{bB}{C}_b\left({\theta }_B-{\theta }_{0B}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-⟮1+{\tau }_B\frac{\partial }{\partial t}⟯\left(Q_{met}+Q^B{}_{ext}\right),\hspace{1em} x_S\le x\le x_B\end{array}$$

(12)

and

$${\phi }_B-{\theta }_B=\beta \frac{{\partial }^2{\phi }_B}{\partial x^2},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_S\le x\le x_B$$

(13)

The heat conduction equations of the brain took the form:

$$\begin{array}{l}{k}_R\frac{{\partial }^2{\phi }_R}{\partial x^2}-{\rho }_R{C}_R⟮1+{\tau }_R{W}_{bR}{C}_b+{\tau }_R\frac{\partial }{\partial t}⟯\frac{\partial {\theta }_R}{\partial t}=W_{bR}{C}_b\left({\theta }_R-{\theta }_{0R}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -⟮1+{\tau }_R\frac{\partial }{\partial t}⟯\left(Q_{met}+Q^R{}_{ext}\right),\hspace{1em}\hspace{1em}\hspace{1em}{x}_B\le x\le x_R\end{array}$$

(14)

and

$${\phi }_R-{\theta }_R=\beta \frac{{\partial }^2{\phi }_R}{\partial x^2},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_B\le x\le x_R$$

(15)

We considered a unified form for simplicity as follows [34]:

$$\begin{array}{l}{k}_i\frac{{\partial }^2{\phi }_i}{\partial x^2}-{\rho }_i{C}_i⟮1+{\tau }_i{W}_{bi}{C}_b+{\tau }_i\frac{\partial }{\partial t}⟯\frac{\partial {\theta }_i}{\partial t}=W_{bi}{C}_b\left({\theta }_i-{\theta }_{0i}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-⟮1+{\tau }_i\frac{\partial }{\partial t}⟯\left(Q_{met}+Q^i{}_{ext}\right), i=S,B,R\end{array}$$

(16)

and

$${\phi }_i-{\theta }_i=\beta \frac{{\partial }^2{\phi }_i}{\partial x^2}, i=S,B,R$$

(17)

We considered the following initial and boundary conditions:

$${\left.{\phi }_i\left(x,t\right)\right|}_{t=0}={\left.\frac{\partial {\phi }_i\left(x,t\right)}{\partial t}\right|}_{t=0}=0, i=S,B,R, {\left.{\phi }_S\left(x,t\right)\right|}_{x=0}=h\left(t\right), {\left.\frac{\partial {\phi }_R\left(x,t\right)}{\partial x}\right|}_{x=x_R}=0$$

(18)

where $h\left(t\right)$ was the thermal shock intensity function on the bounding surface of the head.

The temperature increments of the three layers satisfied the following continuity conditions:

$${\left.{\phi }_S\left(x,t\right)={\phi }_B\left(x,t\right)\right|}_{x=x_S}, {\left.{\phi }_B\left(x,t\right)={\phi }_R\left(x,t\right)\right|}_{x=x_B}$$

(19)

and

$${\left.k_s\frac{\partial {\phi }_S\left(x,t\right)}{\partial x}=k_B\frac{\partial {\phi }_B\left(x,t\right)}{\partial x}\right|}_{x=x_S}, {\left.k_B\frac{\partial {\phi }_B\left(x,t\right)}{\partial x}=k_R\frac{\partial {\phi }_R\left(x,t\right)}{\partial x}\right|}_{x=x_R}$$

(20)

Applying Laplace transform was defined as:

$$\overline{Z}(x,p)={\displaystyle \underset{0}{\overset{\infty }{\int }}Z(x,t) e^{-pt}dt} , p>0$$

(21)

Then, we had

$$\begin{array}{l}{k}_i\frac{d^2{\overline{\phi }}_i}{d{x}^2}=\left[{\rho }_i{C}_i\left(p+{\tau }_i{W}_{bi}{C}_{b}p+{\tau }_i{p}^2\right)+W_{bi}{C}_b\right]{\overline{\theta }}_i-\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} W_{bi}{C}_b{\theta }_{0i}-{\overline{Q}}_{met}-\left(1+{\tau }_{i}p\right){\overline{Q}}^i{}_{ext}\end{array}$$

(22)

and

$${\overline{\theta }}_i={\overline{\phi }}_i-\beta \frac{{\partial }^2{\overline{\phi }}_i}{\partial x^2}$$

(23)

Due to the chemical reactions, the metabolic heat source was considered, and it was assumed to be a constant, $Q_{met}=368.1 W/m^3$ which gave [32,33]:

$${\overline{Q}}_{met}=\frac{368.1}{p}$$

(24)

We considered that cellular devices were providing waves that generated electromagnetic energy absorbed by the tissue, and they worked as an external heat source $Q^i{}_{ext}$. The form of the electromagnetic heat source was given by [34]:

$$Q^i{}_{ext}=\frac{2I_{0}A}{{\delta }_i}{e}^{-\frac{2x_i}{{\delta }_i}}H\left(t\right)$$

(25)

where, $i= S,B, R$, $A\left(\mathrm{MHz}\right)$ was the power transmission coefficient between tissue and air, and $I_0$ was the power density of incident electromagnetic waves.

We considered the coefficient of power transmission at a frequency of A = 1800 MHz to take the values $f_s=0.4 A$, $f_B=0.8 A$, and $f_R=0.6 A$ for the skin, bone, and brain, respectively. The function $H\left(t\right)$ was the Heaviside unit step function. The parameter $\delta$ gave the penetration depth (the depth to which an electromagnetic wave may enter a certain material) where that value was determined as follows [32,33,34,37]:

$${\delta }_S=\frac{67.52}{\sqrt{{\epsilon }_S} f_S}, {\delta }_B=\frac{67.52}{\sqrt{{\epsilon }_B} f_B}, {\delta }_R=\frac{67.52}{\sqrt{{\epsilon }_R} f_R}$$

(26)

where ${\epsilon }_i$ were the relative permittivity modulus, as in

Table 1. Properties of the multi-layer human head [34,37].

$\mathit{i}=\mathit{S},\mathit{B},\mathit{R}$	Skin	Bone	Brain	Units
$k_i$	0.2150	0.410	0.500	$\mathrm{W}/\mathrm{m} °\mathrm{C}$
${\rho }_i$	1000.0	1500.0	1050.0	$\mathrm{kG}/{\mathrm{m}}^3$
$C_i$	4000.0	2300.0	3700.0	$\mathrm{J}/\mathrm{kG} °\mathrm{C}$
$W_{bi}$	2.00	0.0	9.33	$\mathrm{mL}/\mathrm{C}\mathrm{m}$
$T_{0i}$	24.0	20.0	20.0	$°\mathrm{C}$
${\tau }_i$	20.0	10.0	20.0	s
${\epsilon }_i$	32.0	8.0	53.0	F/m
$x_i$	4.0	4.0	85.0	mm
${\delta }_i$	0.02	0.02	0.01	mm

.

Hence, we obtained

$${\overline{Q}}^S{}_{ext}=\frac{2I_{0}A}{{\delta }_{S}p}{e}^{- \frac{2x_S}{{\delta }_S}}, {\overline{Q}}^B{}_{ext}=\frac{2I_{0}A}{{\delta }_{B}p}{e}^{- \frac{2x_B}{{\delta }_B}}, {\overline{Q}}^R{}_{ext}=\frac{2I_{0}A}{{\delta }_{R}p}{e}^{- \frac{2x_R}{{\delta }_R}}$$

(27)

Eliminating ${\overline{\theta }}_i$ from Equations (22) and (23), we obtain:

$$\frac{d^2{\overline{\phi }}_i\left(x,p\right)}{d{x}^2}-{\lambda }_i^2{\overline{\phi }}_i\left(x,p\right)=-g_i\left(p\right), i=S,B,R$$

(28)

where

$${\lambda }_i^2\left(p\right)=\frac{\left[{\rho }_i{C}_i\left(p+{\tau }_i{W}_{bi}{C}_{b}p+{\tau }_i{p}^2\right)+W_{bi}{C}_b\right]}{\left[k_i+\beta \left[{\rho }_i{C}_i\left(p+{\tau }_i{W}_{bi}{C}_{b}p+{\tau }_i{p}^2\right)+W_{bi}{C}_b\right]\right]}, i=S,B,R$$

$$g_i\left(p\right)=\frac{W_{bi}{C}_b{\theta }_{0i}+{\overline{Q}}_{met}+\left(1+{\tau }_{i}p\right){\overline{Q}}^i{}_{ext} }{\left[k_i+\beta \left[{\rho }_i{C}_i\left(p+{\tau }_i{W}_{bi}{C}_{b}p+{\tau }_i{p}^2\right)+W_{bi}{C}_b\right]\right]}, i=S,B,R$$

Applying the Laplace transform on the boundary conditions (18), we obtained:

$${\left.{\overline{\phi }}_S\left(x,p\right)\right|}_{x=0}=\overline{h}\left(p\right), {\left.\frac{\partial {\overline{\phi }}_R\left(x,p\right)}{\partial x}\right|}_{x=x_R}=0$$

(29)

and using the Laplace transform on the continuity conditions (19) and (20), gives:

$${\left.{\left.{\overline{\phi }}_S\left(x,p\right)\right|}_{x=x_S}={\overline{\phi }}_B\left(x,p\right)\right|}_{x=x_S}, {\left.{\left.{\overline{\phi }}_B\left(x,p\right)\right|}_{x=x_B}={\overline{\phi }}_R\left(x,p\right)\right|}_{x=x_B}$$

(30)

and

$${\left.{\left.k_S\frac{\partial {\overline{\phi }}_S\left(x,p\right)}{\partial x}\right|}_{x=x_S}=k_B\frac{\partial {\overline{\phi }}_B\left(x,p\right)}{\partial x}\right|}_{x=x_S}, {\left.{\left.k_B\frac{\partial {\overline{\phi }}_B\left(x,p\right)}{\partial x}\right|}_{x=x_B}=k_R\frac{\partial {\overline{\phi }}_R\left(x,p\right)}{\partial x}\right|}_{x=x_B}$$

(31)

By solving the Equation (28), we obtain:

$${\overline{\phi }}_S\left(x,p\right)=A_1\left(p\right)e^{{\lambda }_{s}x}+A_2\left(p\right)e^{-{\lambda }_{s}x}+\frac{g_S\left(p\right)}{{\lambda }_S^2}, 0\le x\le x_S$$

(32)

$${\overline{\phi }}_B\left(x,p\right)=B_1\left(p\right)e^{{\lambda }_{B}x}+B_2\left(p\right)e^{-{\lambda }_{B}x}+\frac{g_B\left(p\right)}{{\lambda }_B^2}, x_S\le x\le x_B$$

(33)

and

$${\overline{\phi }}_R\left(x,p\right)=C_1\left(p\right)e^{{\lambda }_{R}x}+C_2\left(p\right)e^{-{\lambda }_{R}x}+\frac{g_R\left(p\right)}{{\lambda }_R^2}, x_B\le x\le x_R$$

(34)

To calculate the parameters, $A_{1,}{A}_{2,}{B}_1, B_2,C_1,C_2$ we apply the boundary conditions (29). Hence, we get:

$$A_1\left(p\right)+A_2\left(p\right)=\overline{h}\left(p\right)-\frac{g_S\left(p\right)}{{\lambda }_S^2}$$

(35)

and

$$C_1\left(p\right)e^{{\lambda }_R{x}_3}-C_2\left(p\right)e^{-{\lambda }_R{x}_3}=0$$

(36)

Moreover, applying the continuity conditions (30) and (31), we obtain the following system:

$$A_1\left(p\right)e^{{\lambda }_s{x}_S}+A_2\left(p\right)e^{-{\lambda }_s{x}_S}-B_1\left(p\right)e^{{\lambda }_B{x}_S}-B_2\left(p\right)e^{-{\lambda }_B{x}_S}=\frac{g_B\left(p\right)}{{\lambda }_B^2}-\frac{g_S\left(p\right)}{{\lambda }_S^2}$$

(37)

$$C_1\left(p\right)e^{{\lambda }_R{x}_B}+C_2\left(p\right)e^{-{\lambda }_R{x}_B}-B_1\left(p\right)e^{{\lambda }_B{x}_B}-B_2\left(p\right)e^{-{\lambda }_B{x}_B}=\frac{g_B\left(p\right)}{{\lambda }_B^2}-\frac{g_R\left(p\right)}{{\lambda }_R^2}$$

(38)

$$k_s{\lambda }_s\left(A_1\left(p\right)e^{{\lambda }_s{x}_S}-A_2\left(p\right)e^{-{\lambda }_s{x}_S}\right)-k_B{\lambda }_B\left(B_1\left(p\right)e^{{\lambda }_B{x}_S}-B_2\left(p\right)e^{-{\lambda }_B{x}_S}\right)=0$$

(39)

$$k_R{\lambda }_R\left(C_1\left(p\right)e^{{\lambda }_R{x}_B}-C_2\left(p\right)e^{-{\lambda }_R{x}_B}\right)-k_B{\lambda }_B\left(B_1\left(p\right)e^{{\lambda }_B{x}_B}-B_2\left(p\right)e^{-{\lambda }_B{x}_B}\right)=0$$

(40)

Now, we consider the thermal shock intensity function $h\left(t\right)$ as follows:

$$h\left(t\right)=h_{0}H\left(t\right)$$

(41)

where $h_0$ is constant, and it is the intensity of the thermal shock.

Applying Laplace transform, we obtained

$$\overline{h}\left(p\right)=\frac{h_0}{p}$$

(42)

By solving the system of algebraic linear Equations (37)–(40), we get the solutions of the dynamical and conductive temperature increment in the Laplace transform domain.

We can get the dynamical temperature increment by substituting the Equations (32)–(34) into the Equation (23). Then, we obtained:

$${\overline{\theta }}_S\left(x,p\right)=\left[A_1\left(p\right)-\beta {\lambda }_S^2\right]e^{{\lambda }_{s}x}+\left[A_2\left(p\right)-\beta {\lambda }_S^2\right]e^{-{\lambda }_{s}x}+\frac{g_S\left(p\right)}{{\lambda }_S^2}, 0\le x\le x_S$$

(43)

$${\overline{\theta }}_B\left(x,p\right)=\left[B_1\left(p\right)-\beta {\lambda }_B^2\right]e^{{\lambda }_{B}x}+\left[B_2\left(p\right)-\beta {\lambda }_B^2\right]e^{-{\lambda }_{B}x}+\frac{g_B\left(p\right)}{{\lambda }_B^2}, x_S\le x\le x_B$$

(44)

$${\overline{\theta }}_R\left(x,p\right)=\left[C_1\left(p\right)-\beta {\lambda }_R^2\right]e^{{\lambda }_{R}x}+\left[C_2\left(p\right)-\beta {\lambda }_R^2\right]e^{-{\lambda }_{R}x}+\frac{g_R\left(p\right)}{{\lambda }_R^2}, x_B\le x\le x_R$$

(45)

3. Results

We will use the Riemann-sum approach to obtain numerical data for the temperature distribution for each layer, which allows any function in the Laplace domain to be inverted to the time domain as [18]:

$$Z(x,t)=\frac{e^{\kappa t}}{t}\left[\frac{1}{2}\overline{Z}\left(x,\kappa \right)+\mathrm{Re}{\displaystyle \sum _{n=1}^N{\left(-1\right)}^n\overline{Z}⟮x,\kappa +\frac{i n\pi }{t}⟯}\right]$$

(46)

We obtained $“i”$, the well-known imaginary complex number. Moreover “Re” gave the real part. Many numerical tests proved that the value $\kappa$ meets the relation $\kappa t\approx 4.7$ for quick convergence [18]. The thermal material properties of the skin, the bone, and the brain are given in Table 1.

$C_b=3800 \mathrm{J}/\mathrm{kg} °\mathrm{C}$ was the specific heat of the blood and $T_b=37 °\mathrm{C}$ [32,33,37]. We assumed the intensity of the thermal shock $h_0=0.1 °\mathrm{C}$ and $\beta =1.0\left({10}^{-6}\right) {\mathrm{m}}^2$ (assumed the same value for the three layers).

4. Discussion

For all the figures, we considered the depth of the adult head and took the range $x\left(0\le x\le 0.018 \mathrm{m}\right)$, and the child’s head took the range $x\left(0\le x\le 0.014 \mathrm{m}\right)$.

Figure 2 and Figure 3 show the conductive and dynamical temperature increment distributions for the three layers of the adult and child head, respectively, when the power transmission coefficient $A=1800 \mathrm{MHz}$, the power density $I_o=300 {\mathrm{mW}/\mathrm{cm}}^2$, and at time $t=100 \mathrm{s}$. The curves in the two figures had the same behaviour with different values. The conductive temperature increment distribution was continuous, which agrees with the continuity conditions between the internal boundaries of the layers. The dynamical temperature increment was a discontinuous distribution. The peak point of the dynamical temperature increment occurred at the end of the skin layer for the adult $\theta \left(0.004\right)=0.33 °\mathrm{C}$ and the child $\theta \left(0.002\right)=0.43 °\mathrm{C}$. The peak point of the conductive temperature increment occurred at the skin layer $\phi \left(0.0026\right)=0.18 °\mathrm{C}$ for the adult’s head and $\phi \left(0.002\right)=0.21 °\mathrm{C}$ for child’s head. Almost all the values of the dynamical temperature increment were higher than the values of the conductive temperature increment in three layers for adult and child heads. For the adult head, the difference value at the first discontinuous point of the dynamical temperature increment is ${\left.\theta \left(0.004\right)\right|}_{skin}-{\left.\theta \left(0.004\right)\right|}_{bone}=0.01 °\mathrm{C}$, while at the second discontinuous point was ${\left.\theta \left(0.008\right)\right|}_{bone}-{\left.\theta \left(0.008\right)\right|}_{brain}=0.05 °\mathrm{C}$. For the child’s head, the difference value at the first discontinuous point of the dynamical temperature increment of the adult head was ${\left.\theta \left(0.002\right)\right|}_{skin}-{\left.\theta \left(0.002\right)\right|}_{bone}=0.07 °\mathrm{C}$, while at the second discontinuous point was ${\left.\theta \left(0.004\right)\right|}_{bone}-{\left.\theta \left(0.004\right)\right|}_{brain}=0.08 °\mathrm{C}$. The differences value of the dynamical temperature increment between the bone and the brain layers was higher than the difference value of the skin and bone layers for the adult and child. For the two discontinuous points of the dynamical temperature increment in the internal boundaries, the values of the difference within the child’s head were more significant than its value within the adult head, which made the child feel tired from using cellular phones before the adults.

Figure 4 and Figure 5 represent the conductive temperature increment of one- and two-temperature heat conduction models with various values of power transmission$A=\left(800 ,1200 ,1800\right)\mathrm{MHz}$ and power density $I_o=300 {\mathrm{mW}/\mathrm{cm}}^2$. The importance of electricity transmission A had a tremendous impact on the thermal wave as it travelled through the three layers of the human skull. For the two types of heat conduction models, the temperature increment values grew as the value of A increased. Peak points existed in the skin area of the adult head, whereas they appeared in the bone area of the child’s head.

Figure 6 and Figure 7 show the conductive temperature increment of one- and two-temperature models with various values of power density$I_o=\left(100 ,600\right){\mathrm{mW}/\mathrm{cm}}^2$, and $A=1800 \mathrm{MHz}$. The value of the power density $I_o$ had a significant impact on the conductive temperature increment across the three layers of both the adult’s and child’s heads. The conductive temperature increment increased when $I_o$ increased in the context of the two models of heat conduction. The peak points occurred in the skin area within the adult’s head while it happened in the bone area of the child’s head. The effect of the two-temperature parameter was more significant when the value of the power density increased.

Figure 8 and Figure 9 show the temperature increment distribution of one- and two-temperature theorems with various values of relaxation time ${\tau }_S={\tau }_B={\tau }_R=0.0 s$ and ${\tau }_S\ne {\tau }_B\ne {\tau }_R\ne 0.0 s$ when $A=1800 \mathrm{MHz}$ and the power density. The values of the relaxation times had significant effects on the temperature increment through the three layers of the human head. When the relaxation times parameters increased, the values of the temperature increment increased in the skin and bone areas while it was vise versa in the brain area.

Figure 10 and Figure 11 show the temperature increment distribution of one- and two-temperature models with various values of the time $t=\left(100,150\right) \mathrm{s}$ and $A=1800 \mathrm{MHz}$, and power density $I_o=300 {\mathrm{mW}/\mathrm{cm}}^2$. The value of the time had significant effects on the conductive temperature increment through the three layers of the head—the values of conductive temperature increment increased when the value of the time increased. The two-temperature parameter had a significant effect on the conductive temperature increment distribution at any value of time.

Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19 represent the dynamical temperature increment distributions with various values of power transmission $A=\left(100 ,1800\right)\mathrm{MHz}$, the power density $I_o=\left(100 ,600\right){\mathrm{mW}/\mathrm{cm}}^2$, relaxation times ${\tau }_i=0 s \mathrm{and} {\tau }_i\ne 0s$, and time $t=\left(100 ,150\right)\mathrm{s}$, for adult and child heads, respectively. Those figures confirmed that all the parameters mentioned above had significant effects on the dynamical temperature increment distributions.

When the values of power transmission, power density, relaxation time, and time increased, the value of the dynamical temperature increment increased. Moreover, the dynamical temperature increment had a discontinuous point in each inside layer of the head.

The results of this paper agree with the results in [25,34]. Moreover, some results by Van Leeuwen et al. [42] agree with the current results.

5. Conclusions

Based on the two-temperature model of heat conduction, this study presented the thermal analysis of a single lag model to depict the three layers of an adult’s and child’s human skull, which was heated by thermal shock and electromagnetic radiation.

The conductive and dynamic temperature increases in a multi-layer human skull were influenced by power transfer, power density, relaxation period, and time.

The discontinuous spots of the dynamical temperature increment distributions of the three layers of the human skull were influenced by relaxation time, power density, power transmission, and time.

The two-temperature parameter had significant effects on the dynamical and conductive temperature increment.

Cellular phones and devices were more harmful to children than adults. The effects of the electromagnetic energy due to cellular phones reached the bone of the head significantly.