Threedimensional biological tissue under highorder effect of twotemperature thermal lagging to thermal responses due to a laser irradiation
Hamdy M. Youssef^{1} , Najat A. Alghamdi^{2}
^{1, 2}Umm AlQura University, Makkah, Saudi Arabia
^{1}Corresponding author
Vibroengineering PROCEDIA, Vol. 22, 2019, p. 112117.
https://doi.org/10.21595/vp.2019.20521
Received 14 January 2019; accepted 21 January 2019; published 15 March 2019
36th International Conference on Vibroengineering in Dubai, United Arab Emirates, March 1517th, 2019
Using the lasers and heat transfer through biological skin tissue is essential for the processes of the medical treatment. Thus, twotemperature bioheat transfer equations were introduced and used to discuss the variation of temperature in a laserirradiated biological tissue. As a result, the Laplace and Fourier transform techniques have been applied, and the solutions of the conductive temperature have been represented in figures. The twotemperature parameter, the penetration depth parameter, the dimensions of the rectangular laser pulse, and the power density of laser irradiation parameter have significant effects on the thermal waves through the skin tissue.
Keywords: twotemperature, thermal lagging, biological tissue, laser irradiation.
1. Introduction
In the medical applications, the use of lasers has increased over the past halfcentury. For increasing the therapy quality, the observation of thermal damage is important. Effect of the laser irradiation on the biological tissues takes different types. The most common of those applications include the thermal effects. The thermal interaction is the most important since temperature rising due to thermal effects is a final factor to produce thermal damage. Due to lack of knowledge about the thermal reaction and response, the clinical doctors could not control the times of using the laser power [14]. To predict the thermal damage quantity, it is required to study the thermal response caused by the laser energy deposition. As a result, the analysis of thermal transport is essential to make an improvement to the treatment qualification and for more safety. The experiment is an essential method to know the medical treatment problems [4, 5]. The most famous researchers studied the bioheat transfer in the biological skin tissues are Pennes [6], Abramson [7]. The Pennes bioheat transfer equation is used to simulate and model the thermal behavior in the biological bodies. The review paper [8] represented many attempts to model the thermal behavior of the human body. Xu et al. [9, 10] presented a system discussion to the DPL effects on the biothermomechanical behavior of skin tissues. The linear DPL equation has been used in various theoretical studies [1113]. Youssef applies the twotemperature heat conduction in many applications [14, 15].
2. Materials and method
Consider homogeneous skin tissue in threedimensional fills the region $\mathrm{\Psi}$which is defined by $\mathrm{\Psi}=\left\{x,y,z:\mathrm{}0\le x\mathrm{\infty},\mathrm{}\mathrm{\infty}y\mathrm{\infty},\mathrm{\infty}z\mathrm{\infty}\right\}$ where the skin is initially at rest and has been thermally shocked by laser irradiation with a rectangular pulse at the bounding plane to the surface. Tzou proposed the DPL model in the classical heat flux model as [16]. Youssef proposed the twotemperature model to differentiate between the conductive temperature and the dynamical temperature as [17].
where $T$ is the temperature, $K$ the heat conductivity, $t$ is the time, ${\tau}_{q}$, ${\tau}_{T}$ are the phase lag of the heat flux and the phase lag of the temperature gradient, respectively, $\rho $ is the density, $C$ is the specific heat, ${C}_{b}$ and ${W}_{b}$ are the specific heat and perfusion rate of blood. ${Q}_{met}$ is the metabolic heat generation and ${Q}_{ext}$ is the heat source for spatial heating, ${T}_{b}$ is the arterial temperature, ${T}_{C}$ is the conductive temperature, and ${T}_{D}$ is the dynamical temperature:
where $\beta $ is a nonnegative parameter which is called twotemperature parameter.
The term ${W}_{b}{C}_{b}\left({T}_{D}{T}_{b}\right)$ expresses the heat caused by convection within the head per unit mass of the tissue and it is considered to be homogenous.
Assume the functions $\phi $ and $\theta $ is as follows:
From Eqs. (1) and (2), we obtain:
where ${\nabla}^{2}=\left(\frac{{\partial}^{2}}{\partial {x}^{2}}+\frac{{\partial}^{2}}{\partial {y}^{2}}+\frac{{\partial}^{2}}{\partial {z}^{2}}\right)$.
Due to the chemical reactions within the tissues, the metabolic heat source is valid, and it is assumed to be a constant ${Q}_{met}=$ 368.1 W/m^{3}. When the laser beam shocked the skin surface, the laser energy is absorbed, scattered, and transmitted. Lambert expresses the laser power intensity along the tissue depthBeer’s law, as follows [2]:
Heat generation due to scattering is assumed to be negligible; therefore, the specific absorption rate in the target zone can be expressed as follows:
In the Eq. (6), ${I}_{0}$ (W/m^{2}) represents the power density of laser irradiation and $H\left(t\right)$ is the Heaviside unit step function. The parameter $\delta $ represents the penetration depth. It means, the rectangular thermal pulse of the laser irradiation acts on a band of width 2$a$ centered on the $y$axis and 2$b$ centered around the $z$axis on the surface of the halfspace $x=\text{0}$ and is zero elsewhere:
$\bullet \left(\rho C\frac{\partial \theta}{\partial t}+{W}_{b}{C}_{b}{\rho}_{p}\theta \left(\mathcal{l}{I}_{0}{e}^{\delta x}H\left(t\right)H\left(a\lefty\right\right)H\left(b\leftz\right\right)+{Q}_{met}\right)\right).$
Applying the Laplace transform for Eqs. (5), (7) and (8) defined as:
where:
Thus, we get:
where ${h}_{T}=\left(1+s{\tau}_{T}\frac{{s}^{2}{\tau}_{T}^{2}}{2}\right)$, ${h}_{q}=\left(1+s{\tau}_{q}\frac{{s}^{2}{\tau}_{q}^{2}}{2}\right)$.
3. The governing equations in the Fourier transform domain
Using double Fourier transform for any function $f\left(x,y\right)$ which is defined as follows:
where the inversion transform of the double Fourier transform takes the form:
Hence, we have:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\frac{2\delta {I}_{0}{h}_{q}\mathrm{s}\mathrm{i}\mathrm{n}\left(pa\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(qb\right)}{\pi \mathrm{}s\mathrm{}pqK\mathrm{}{h}_{T}}{e}^{\delta x}\frac{{Q}_{met}\mathrm{}{h}_{q}\delta \left(p\right)\delta \left(q\right)}{sK\mathrm{}{h}_{T}},$
By eliminating $\stackrel{~}{\stackrel{}{\theta}}$ from Eqs. (15) and (16), we get the following differential equation:
where:
The general solution of the differential Eq. (17) takes the form:
where ${c}_{1}$ and ${c}_{2}$ are some parameters which have to be determined.
Apply the boundary conditions:
Hence, we obtain:
where:
4. Results
The values of the relevant thermal parameters used in the present calculations are as follows in Table 1 [4, 9, 13, 18].
Table 1. The material properties of the skin tissue
Parameter  Unit  Skin tissue  Parameter  Unit  Skin tissue 
$K$  W/m°C  0.628  ${W}_{b}$  ml/Cm  0.00187 
$\rho $  kg/m^{3}  1000  ${T}_{b}$  °C  37 
${\rho}_{b}$  kg/m^{3}  1060  ${\tau}_{T}$  s  10 
$C$  J/kg°C  4187  ${\tau}_{q}$  s  15 
${C}_{b}$  J/kg°C  3860  $L$  m  0.05 
5. Discussion
Fig. 1 shows the conductive temperature increment with a range of $x$axis and various positions of $y$ and $z$. The values of both $y$ and $z$ have significant effects. When the values of $y$ and $z$ are increasing, the value of the conductive temperature increment is decreasing. Fig. 2 shows the conductive temperature increment with a range of $x$axis and various value of $\beta $ and $y=z=$ 0.0. The value of $\beta $ has a significant effect. When the value of $\beta $ is increasing, the value of the conductive temperature is increasing. Fig. 3 shows the conductive temperature distribution with a range of $x$axis and various value of $\delta $ when $y=z=$ 0.0. The value of $\delta $ has a significant effect. When $\delta $ is decreasing, the value of the conductive temperature is decreasing. Fig. 4 shows the conductive temperature distribution with a range of $x$axis and various $a$, $b$ and $y=z=$ 0.0. The values of $a$ and $\mathrm{}b$ have significant effects. When the values of $a$ and $\mathrm{}b$ are increasing, the value of the conductive temperature is increasing. Fig. 5 shows the conductive temperature distribution with a range of $x$axis and various value of ${I}_{0}$ when $y=z=$ 0.0. The value of ${I}_{0}$ has a significant effect. When the value of ${I}_{0}$ is increasing, the value of the conductive temperature is increasing.
Fig. 1. The conductive temperature increment with various of $y$, $z$
Fig. 2. The conductive temperature increment distribution with various values of $\beta $
Fig. 3. The conductive temperature increment distribution with various values of $\delta $
Fig. 4. The conductive temperature increment distribution with various of $a$ and $b$
Fig. 5. The conductive temperature increment with various of ${I}_{0}$
6. Conclusions
The twotemperature parameter, the dimensions of the rectangular laser pulse, the penetration depth parameter, and the power density of laser irradiation have significant effects on the conductive temperature.
Acknowledgements
The authors are grateful for the support of this work as a part of the project provided by the “LongTerm Comprehensive National Plan for Science, Technology, and Innovation” through STU of Umm AlQura University by Grant Number 14MAT16210.
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