# The Skin Tissue of the Human Head Subjected to Thermal Diffusion

Authors: Eman A. N. Al-Lehaibi
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

We consider a problem of skin tissue of the human head subjected to thermal diffusion with relaxation time. The bounding surface of the tissue of the human head is subjected to a thermal shock. The chemical potential is also assumed to be a known function of time on the bounding plane. Laplace transform techniques are used and the results are obtained numerically using a numerical method for the inversion of the Laplace transform based on Tzou method. The temperature distribution and the concentration distribution are obtained and represented in figures. According to the results, the values of the time and the distance have significant effects on the temperature increment and the concentration. The values of the temperature increment and the concentration increase when the time increases and decrease when the distance increases.


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## Body

## 1. Introduction

Biomechanics is the area of science where the laws, principles, and methods of mechanics are applied to the structure and function of the human body. Mechanics can be divided into two categories: statics, which is the study of stationary objects, and dynamics, which is the study of moving objects [1].Some human trial studies have expressed that, in the short term, being subjected to cellular phones frequencies can cause such problems like exciting the brain cells when subjected to a wave [1], the possible occurrence of cataracts with continuous exposure weakening the neural behavioral and cognitive functioning, and possibly a disruption in the functionality of the nervous system [1]. Lindholm et al. [2] conducted a human trial to see the effects of short term radio frequency when the electromagnetic fields are subjected to preadolescents.Ferreri et al. studied the mobile phone emissions and human brain excitability [3] and Keetley et al. studied the neuropsychological squeal of digital mobile phone exposure in humans [4], while Lin discussed the cataracts and personal communication radiation [5]. The effects of electromagnetic radiation of mobile phones on the central nervous system have been studied by Hossmann and Hermann [6] and the thermal effects of radiation from cellular telephones have been studied by Wainwright [7].The most important work regarding specific absorption rate and temperature distributions in human head subjected to mobile phone radiation at different frequencies was introduced by Wessapan et al. [8]. Elwassif et al. constructed the bioheat transfer model of deep brain stimulation-induced temperature changes [9]. Ley and Bayazitoglu analyzed the brain temperature distribution during deep hypothermic circulatory arrest in humans [10] and got the effect of physiology on the temperature distribution of a layered head with external convection [1].The thermal wave model of bioheat transfer (TWMBT) equation is an energy balance equation that includes the heat caused by conduction, convection, metabolic heat generation, and the heat produced by radiation due to the cellular frequency [11]. The model incorporates a relaxation time to account for the lag time it takes for the radio frequency wave to respond to the heat penetrating through the tissue, that is, heat flux associated with the tissues [12, 13]. Biological experimental studies have shown that this value can range from 15 to 30 s [12, 14–20]. An additional relaxation time accommodating the phase lag of the temperature gradient is analytically studied.Morimoto et al. [21] studied the time constants of the temperature elevations in human head and body models exposed to simulated radiation from dipole antennas, electromagnetic beams, and plane waves within frequency range considered to be from 1 to 30 GHz. Tang et al. discussed the thermal and vibrational characterization of human skin to find suitable and reproducible biomarkers for further clinical studies on intrinsic and extrinsic aging of dermis [22].


## 2. Formulation of the Problem

We consider a small bounded cross section of the skin tissue of the human head occupies the region0≤x≤l subjected to thermal diffusion (such as a gas) in the upper plane of the skin tissue x=0 as in Figure 1, and this tissue is initially quiescent.Figure 1
The human head.The heat transfer in living tissue is accompanied by blood perfusion and metabolic heat generation; the bioheat equation takes the following form [4]:(1)K∂2Ts∂x2=∂∂t+τ1∂2∂t2ρceTs+αC+1+τ1∂∂tρpωpcpTs-Tb-Qmet-Qout,where Ts=Tsx,t is the temperature of the tissue, Tb is the blood temperature (constant), K is the thermal conductivity, ρ is the density of the skin tissue, ρp is the blood mass density, cp is the specific heat of the blood, ce is the specific heat of the skin tissue, ωp is the blood perfusion rate, Qout is the external heat source, Qmet is the heat source due to metabolic heating, α is a measure of thermodiffusion effect, τ1 is the relaxation time of the skin tissue, and C=Cx,t is the concentration of the diffusive material in the skin tissue.The diffusion equations take the form(2)D∂2P∂x2=∂∂t+τ2∂2∂t2C,where τ2 is the diffusion relaxation time, D is the diffusion coefficient, and P=Px,t is the chemical potential which satisfies(3)P=βC-αTs-Tp,where β is a measure of diffusive effect.To simplify the equations in (1)–(3), we will use the following nondimension variables:(4)x′=ωpρpcpKx,t′,τ1′,τ2′=ωpρpcpρct,τ1,τ2,P′=PαTp,C′=βαTpC,θ=Ts-TpTp,Qmet′,Qout′=Qmet,QoutωpρpcpTp.After applying the dimensionless variables, (1)–(3) take the forms(5)∂2θ∂x2=∂∂t+τ1∂2∂t2θ+a1C+1+τ1∂∂tθ-Qmet-Qout,(6)∂2P∂x2=a2∂∂t+τ2∂2∂t2C,(7)P=C-θ,where a1=α2/ρβc and a2=K/ρβcD.We dropped the prime for simplicity.We will consider the tissue has no external heat sourceQout=0 and the heat source due to metabolic heating Qmet is constant; then we get(8)∂2θ∂x2=∂∂t+τ1∂2∂t2θ+a1C+1+τ1∂∂tθ-Qmet.By eliminating P between (6) and (7), we obtain(9)∂2C∂x2-∂2θ∂x2=a2∂∂t+τ2∂2∂t2C.We will apply Laplace transform on (6) and (8) which is defined as(10)f-x,s=∫0∞fx,te-stdt,s>0,where Cx,tt=0=θx,tt=0=∂C(x,t)/∂tt=0=∂θx,t/∂tt=0=0.Hence, we get(11)∂2θ-∂x2=1+τ1s1+sθ-+a1s+τ1s2C--Qmets,∂2C-∂x2-∂2θ-∂x2=a2s+τ2s2C-.By eliminating θ- between the equations in (11), we obtain(12)∂4∂x4-α1+α2+α3∂2∂x2+α1α3C-=0,and by eliminating C- between the equations in (11), we get(13)∂4∂x4-α1+α2+α3∂2∂x2+α1α3θ-=α4,where α1=1+τ1s1+s,α2=a1s+τ1s2,α3=a2s+τ2s2,α4=α3Qmet/s.We can write the general solutions of (12) and (13) in the forms(14)θ-x,s=γ1eλ1x+γ2eλ2x+β1e-λ1x+β2e-λ2x+α4M,C-x,s=λ12-α3λ12γ1el1x+λ22-α3λ22γ2eλ2x+λ12-α3λ12β1e-λ1x+λ22-α3λ22β2e-λ2x,where γ1=γ1s,γ2=γ2s,β1=β1s,β2=β2s are parameters and ±λ1s, ±λ2s are the roots of the characteristic equation(15)λ4-Lλ2+M=0,where L=α1+α2+α3,M=α1α3.To get the parametersγ1,γ2,β1,β2, we will consider the upper plane of the tissue x=0 is subjected to thermal and chemical potential shock which depends on time only as follows:(16)θ0,t=ft,θl,t=0,P0,t=gt,Pl,t=0.Applying Laplace transform on conditions (16), we get(17)θ-0,s=f-s,P-0,s=g-s.By using (7) and (17), we get(18)C-0,s=f-s+g-s.Applying the conditions in (17) and (18) on (14), we obtain the following system of linear equations:(19)γ1+γ2+β1+β2=f-s-α4M,λ12-α3λ12λ1+λ22-α3λ22γ2+λ12-α3λ12β1+λ22-α3λ22β2=f-s+g-s,λ1γ1eλ1l+λ2γ2eλ2l-λ1β1e-λ1l-λ2β2e-λ2l=0,λ12-α3λ1γ1eλ1l+λ22-α3λ2γ2eλ2l-λ12-α3λ1β1e-λ1l-λ22-α3λ2β2e-λ2l=0.By solving the system above, we get(20)γ1=λ12α3α4s-f-sM-λ22α4s+g-sMα3Mλ22-λ121+e2lλ1,γ2=λ22α3α4s-f-sM-α12α4s+g-sMα3Mλ12-λ221+e2lλ2,β1=λ12e2lλ1α3α4s-f-sM-α22α4s+g-sMα3Mλ22-λ121+e2hλ1,β2=λ22e2lλ2α3α4s-f-sM-λ12α4s+g-sMα3Mλ12-λ221+e2lλ2.Thus, we have the solution of the problem in the Laplace transform domain in the form(21)θ-x,s=λ22α3α4s-f-sM-α12α4s+g-sMα3Mλ12-λ221+e2lλ2eλ1x+λ22α3α4s-f-sM-α12α4s+g-sMα3Mλ12-λ221+e2lλ2eλ2x+λ12e2lλ1α3α4s-f-sM-α22α4s+g-sMα3Mλ22-λ121+e2hλ1e-λ1x+λ22e2lλ2α3α4s-f-sM-λ12α4s+g-sMα3Mλ12-λ221+e2lλ2e-λ2x+α4M,C-x,s=λ12-α3λ12λ12α3α4s-f-sM-λ22α4s+g-sMα3Mλ22-λ121+e2lλ1eλ1x+λ22-α3λ22λ22α3α4s-f-sM-α12α4s+g-sMα3Mλ12-λ221+e2lλ2eλ2x+λ12-α3λ12λ12e2lλ1α3α4s-f-sM-α22α4s+g-sMα3Mλ22-λ121+e2hλ1e-λ1x+λ22-α3λ22λ22e2lλ2α3α4s-f-sM-λ12α4s+g-sMα3Mλ12-λ221+e2lλ2e-λ2x.


## 3. Inverse Laplace Transforms

To get solutions in the time domain, the Riemann-sum approximation method is used to obtain the numerical results. In this method, any function in Laplace domain can be inverted to the time domain as follows:(22)ht=eκtt12h-κ+Re⁡∑n=1N-1nh-κ+inπt,where Re is the real part and i is the imaginary number unit. For faster convergence, many numerical experiments have shown that the value of κ satisfies the relation κt≈4.7.We considered that the functions on the boundary in (16) will be defined as a shock in the following forms:(23)ft=gt=ωHt,where ω is constant (the intensity of the shock) and H(t) is the Heaviside unit step function.By using Laplace transform, (23) takes the form(24)f-s=g-s=ωs.


## 4. The Numerical Values

The following properties of a skin tissue are used in the numerical analysis [1–5]:(25)ρ=1190kg/m3,K=0.235W/mK,ce=3600J/kg K,ρp=1060kg/m3,cp=3770J/kg Kωp=0.5kg/m3s,Qm=33800W/m3,qo=5000W/m3,Tp=310K,l=0.002m,D=0.85×10-8,α=1.2×104,β=0.9×106,Qmet=5.46×10-5,τ1=0.002,τ2=0.02,a1=32.098,a2=7.171×10-6.Figures2 and 3 represent the temperature increment distribution and the concentration distribution, respectively, with a wide range of the thickness of the layer of the tissue when 0≤x≤3 and at different values of time t=0.2,0.4,0.7. Those figures show that the value of the time has a significant effect on the temperature increment and the concentration. The values of the temperature increment and the concentration increase when the time increases and decrease when the distance increases. The thermal and the concentration waves have finite speeds of wave propagation which agrees with the real physical behavior.Figure 2
The temperature increment distribution at different values of time.Figure 3
The concentration distribution at different values of time.Figures4 and 5 represent the temperature increment distribution and the concentration distribution, respectively, with a wide range of the time when 0≤t≤1.0 and at different values of distance x=0.2,0.4,0.6. Those figures show that the value of the distance has a significant effect on the temperature increment and the concentration. The values of the temperature increment and the concentration increase when the distance decreases and go to the steady state position when the time increases. Moreover, the steady state positions start first in the nearest distance. The steady states of the thermal wave and the concentration wave occur almost at the same instance.Figure 4
The temperature increment distribution at different values of distance.Figure 5
The concentration distribution at different values of distance.



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*Source: 8781950-2018-01-31.xml*