The Effect of Fractional Derivatives on Thermo-Mechanical Interaction in Biological Tissues during Hyperthermia Treatment Using Eigenvalues Approach

Abstract

This article studies the effects of fractional time derivatives on thermo-mechanical interaction in living tissue during hyperthermia treatment by using the eigenvalues approach. A comprehensive understanding of the heat transfer mechanism and the related thermo-mechanical interactions with the patient’s living tissues is crucial for the effective implementation of thermal treatment procedures. The surface of living tissues is traction-free and is exposed to a pulse boundary heat flux that decays exponentially. The Laplace transforms and their associated techniques are applied to the generalized bio-thermo-elastic model, and analytical procedures are then implemented. The eigenvalue approach is utilized to obtain the solution of governing equations. Graphical representations are given for the temperature, the displacement, and the thermal stress results. Afterward, a parametric study was carried out to determine the best method for selecting crucial design parameters that can improve the precision of hyperthermia therapies.

1. Introduction

The temperature behavior of living tissue is still not fully understood due to the difficulty in accurately measuring it in vivo. This is because necropsy can alter tissue temperature characteristics and examining tissue outside of the body lacks perfusion effects. Various in vivo techniques exist to measure thermal behavior, but they yield varying results, and precise measurement of tissue temperature in vivo remains elusive. For effective treatment, it is critical to control the body’s heat transmission pathways. Thermal therapies aim to freeze or heat tumors without damaging surrounding healthy tissue. If doctors could predict how tissue would react thermally, they could plan therapy dosages and durations before surgery. Tissue and blood perfusion’s temperature response is associated with a range of diseases and injuries, including diabetes, skin grafts, and frostbites. The severity of these diseases is influenced by the extent to which blood can reach a certain location. If the thermal characteristics of damaged tissue can be precisely monitored before problems arise, appropriate and efficient therapy can be provided immediately. Heat transport analysis in living tissue is challenging due to its diverse internal structure, which includes perfusion via capillary tubes, convection between blood and tissues, thermal conduction between solid tissues and blood arteries, and heating generation through metabolism, evaporation, and other factors. Pennes’ [1] bioheat model, which is based on Fourier’s law of thermal conduction, is used to represent heat transmission in living tissues. Various thermotherapy procedures, such as laser tissue welding [2], laser operations [3], and hyperthermia [4], are frequently utilized in modern medicine. Skin tissue temperature distributions are influenced by complex factors such as metabolic heating generation and blood circulation, so researchers have expanded on basic relationships by incorporating a wide range of phenomenological mechanisms, such as metabolic heating production, thermal conduction, blood perfusions, radiation, and phase change. Biological tissues undergo diverse stages of change, which can take many different forms. Modified versions of Pennes’ bioheat models, which employ various numerical methods, are available in the relevant literature. Several techniques have been used to study heat transmission in living tissues, including the homotopy perturbation technique [5], Legendre wavelets Galerkin approaches [6,7], and the finite element approach [8]. When a person’s body temperature was unusually low, Esneault and Dillenseger [9] used finite difference approaches to investigate the time-dependent increase in temperature. During thermal therapy, Ghanmi and Abbas [10] performed an analytical investigation of the fractional time derivative in skin tissue. Marin et al. [11] utilized finite element analysis to study the nonlinear bio-heat model in skin tissue caused by external heating sources, while Hobiny and Abbas [12] explored the analytical solution for the fractional bioheat model in tissues with a spherical shape. On the other hand, Keangin and Phadungsak [13] conducted an analysis on the heat transport in porous liver during microwave ablation, specifically focusing on local thermal non-equilibrium. In their research, Keangin et al. [14] investigated the analysis of heat transfer in a deformed liver cancer model treated with a microwave coaxial antenna. Andreozzi et al. [15] studied the effects of a pulsating heat source on interstitial fluid transport in tumor tissues in which the effects of modulating-heat strategies to influence interstitial fluid transport in tissues were analyzed.

Different methods can be used to solve the time-dependent heat transfer equation and model infinite thermal propagations based on classical Fourier heat conduction. Fractional computation has recently proved to be a successful method for modifying many physical models. Fractional derivatives have received significant attention, and various definitions and methods have been developed. The use of fractional time derivatives has enabled the successful modification of many physical models’ processes. Ezzat and colleagues introduced a novel bio-heat model based on the fractional heat conduction formulation, as described in their publications [16,17], while the investigation of transient heating in skin tissues caused by time-dependent thermal therapy, utilizing a heat transport law that incorporates memory, was carried out by Mondal and colleagues in reference [18]. Researchers have conducted several studies on the use of thermal transfer on living tissue [19] to improve treatment methods, develop more accurate temperature prediction technologies, and ultimately, find a cure for cancer. Over the past forty years, a wide range of researchers from various fields, including high-energy particle accelerators, continuum mechanics, acoustics, nuclear engineering, and aeronautics have expressed significant interest in generalized thermo-elastic models from both a technical and mathematical standpoint. The concepts of heat transfer and elasticity are linked in this model. Lord and Shulman [20] developed the thermo-elasticity hypothesis with multiple generalizations. Mondal et al. [21] employed the memory-dependent derivatives on a sliding interval within the framework of the Lord–Shulman model to investigate the heat transfer equation for this problem. Diaz et al. [22] used the finite element method to obtain the solutions of thermo-diffusion types present in living tissue to develop thermal damage. Zhu et al. [22] used the diffusion theory to consider rate process models for the results of thermal damages and the sedimentation of lighting energy in the tissue. When studying the actual phenomena of thermal transfer in finite media, the nonlinear and linear models of heating transfer are extended, and many authors have sought numerical or analytical solutions to the problems posed by these models [23,24,25,26,27,28,29,30,31,32,33]. Despite the growing popularity of laser, microwave, and other forms of thermal therapy in dermatology, thermo-mechanical interactions are seldom considered in current research, despite being central to the thermo-mechanically linked nature of these therapies. Generalized thermo-elastic models govern tissue thermo-elastic behaviors, which include the G-N model, the G-NII model, the DPL, the fractional model, and Li et al. [34,35,36], who further investigated the effects of heat-induced mechanical responses in skin tissues under temperature-dependent properties.

This research aims to create an analytical approach to examine the thermo-mechanical interactions with fractional time derivatives in living tissue that experiences instantaneous heating and has varying thermal and mechanical properties. A generalized thermo-elastic model is constructed that takes into account the tissue structure and variable thermal and mechanical characteristics within the bioheat transfer equation framework. The effect of fractional parameter in temperature, displacement, and thermal stress variations are shown in the graphics.

2. Materials and Methods

The field of bio-thermo-elasticity merges the principles of elasticity and bioheat conduction. The basic equations under fractional time derivatives in the living tissues can be given by [20,37,38]:

$$\left(\lambda +\mu \right)u_{j,ij}+\mu u_{i,jj}-\gamma T_{,i}=\rho \frac{{\partial }^2{u}_i}{\partial t^2},$$

(1)

$$k{\nabla }^{2}T=\left(1+\frac{{\tau }_o^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\frac{{\partial }^{\beta }}{\partial t^{\beta }}\right)\left(\rho c_e\frac{\partial T}{\partial t}+{\omega }_b{\rho }_b{c}_b\left(T-T_b\right)+\gamma T_o\frac{{\partial }^{2}u}{\partial t\partial x}-Q_m\right),0<\beta \le 1,$$

(2)

$${\sigma }_{ij}=\mu \left(u_{i,j}+u_{j,i}\right)+\left(\lambda u_{k,k}-\gamma \left(T-T_o\right)\right){\delta }_{ij},$$

(3)

where $t$ is the time,${\tau }_o$ is the thermal relaxation time, $T_b$ is the blood temperature, $c_e$ refers to the specific heat at constant strain, $\rho$ is the tissue mass density, $\gamma =\left(3\lambda +2\mu \right){\alpha }_t$, ${\alpha }_t$ refers to the linear thermal expansion coefficient, $\lambda , \mu$ refer to the Lame’s constants, $T$ is the tissue temperature, $k$ is the tissue thermal conductivity, ${\omega }_b$ is the blood perfusion rate, $Q_m$ is the metabolic heat generation in skin tissues, $e_{ij}$ are the strain components, ${\sigma }_{ij}$ are the stress components, $u_i$ are the displacement components, ${\delta }_{ij}$ is the Kronecker symbol and $c_b$ is the blood specific heat. The definition of the fractional order derivative is as follows:

$$\frac{{\partial }^{\beta }h\left(\mathit{r},t\right)}{\partial t^{\beta }}=\left\{\begin{array}{cc}h\left(\mathit{r},t\right)-h\left(\mathit{r},0\right),& \beta \to 0,\\ I^{\beta -1}\frac{\partial h\left(\mathit{r},t\right)}{\partial t},& 0<\beta <1,\\ \frac{\partial h\left(\mathit{r},t\right)}{\partial t},& \beta =1,\end{array}\right.,$$

(4)

$$I^{\nu }h\left(\mathit{r},t\right)={{\displaystyle \int }}_0^t\frac{{\left(t-s\right)}^{\nu }}{\mathsf{\Gamma }\left(\nu \right)}h\left(\mathit{r},s\right)ds, \nu >0,$$

(5)

$$\underset{\nu \to 1}{\lim }\frac{{\partial }^{\nu }h\left(\mathit{r},t\right)}{\partial t^{\nu }}=\frac{\partial h\left(\mathit{r},t\right)}{\partial t},$$

(6)

Equation (4) illustrates how the range of local thermal conduction can be characterized by two types of conductivity: standard thermal conduction and heat ballistic conduction. The fractional parameter is $\beta$, where $0<\beta \le 1$ is used to define these conductivities. For normal conductivity, $\beta =1$, while for low conductivity $0<\beta <1$. Here, we assume that the surface and the bottom boundary of a limited domain of tissues with thickness $L$. As a result, the displacement components can be expressed as follows:

$$u_x=u\left(x,t\right),u_y=0,u_z=0.$$

(7)

So that the model has the following form

$$\left(\lambda +2\mu \right)\frac{{\partial }^{2}u}{\partial x^2}-\gamma \frac{\partial T}{\partial x}=\rho \frac{{\partial }^{2}u}{\partial t^2},$$

(8)

$$k\frac{{\partial }^{2}T}{\partial x^2}=\left(1+\frac{{\tau }_o^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\frac{{\partial }^{\beta }}{\partial t^{\beta }}\right)\left(\rho c\frac{\partial T}{\partial t}+{\omega }_b{\rho }_b{c}_b\left(T-T_b\right)+\gamma T_o\frac{{\partial }^{2}u}{\partial t\partial x}-Q_m\right),$$

(9)

$${\sigma }_{xx}=\left(\lambda +2\mu \right)\frac{\partial u}{\partial x}-\gamma \left(T-T_o\right).$$

(10)

To obtain a solution for the equations, it is necessary to have two sets of initial and boundary conditions that align with the physical model description:

$$T\left(x,0\right)=T_b,\frac{\partial T\left(x,0\right)}{\partial t}=0,u\left(x,0\right)=0,\frac{\partial u\left(x,0\right)}{\partial t}=0,$$

(11)

$${\sigma }_{xx}\left(0,t\right)=0, {\sigma }_{xx}\left(L,t\right)=0,-k{\left.\frac{\partial T\left(x,t\right)}{\partial x}\right|}_{x=0}=q_o\frac{t^2{e}^{-\frac{t}{t_p}}}{16t_p^2},-k{\left.\frac{\partial T\left(x,t\right)}{\partial x}\right|}_{x=L}=0,$$

(12)

where $t_p$ points to the characteristic time of pulsing heat flux and $q_o$ is constant. Now, the dimensionless quantities may be utilized for ease by employing:

$$T^{\prime }=\frac{T-T_o}{T_o},\left(t^{\prime },{\tau }_o^{\prime },t_p^{\prime }\right)=\eta c^2\left(t,{\tau }_o,t_p\right), \left(x^{\prime },u^{\prime }\right)=\eta c\left(x,u\right),$$

$${\sigma }_{xx}^{\prime }=\frac{ {\sigma }_{xx}}{\lambda +2\mu }, q_o^{\prime }=\frac{q_o}{\eta c{T}_{o}K}, Q_m^{\prime }=\frac{Q_m}{T_{o}K{\eta }^2{c}^2},$$

(13)

where $\eta =\frac{\rho c_e}{K}$ and $c^2=\frac{\lambda +2\mu }{\mathsf{\rho }}$.

The primary equations can be expressed in a non-dimensional form by removing the dashes and introducing appropriate parameters, as shown in Equation (13):

$$\frac{{\partial }^{2}u}{\partial x^2}-{\epsilon }_1\frac{\partial T}{\partial x}=\frac{{\partial }^{2}u}{\partial t^2},$$

(14)

$$\frac{{\partial }^{2}T}{\partial x^2}=\left(1+\frac{{\tau }_o^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\frac{{\partial }^{\beta }}{\partial t^{\beta }}\right)\left(\frac{\partial T}{\partial t}+{\epsilon }_{2}T+{\epsilon }_3\frac{{\partial }^{2}u}{\partial t\partial x}-Q_m\right),$$

(15)

$${\sigma }_{xx}=\frac{\partial u}{\partial x}-{\epsilon }_{1}T,$$

(16)

$$u\left(x,0\right)=0,\frac{\partial u\left(x,0\right)}{\partial t}=0,T\left(x,0\right)=0,\frac{\partial T\left(x,0\right)}{\partial t}=0,$$

(17)

$${\left.{\sigma }_{xx}\left(0,t\right)=0,{\sigma }_{xx}\left(L,t\right)=0,\frac{\partial T\left(x,t\right)}{\partial x}\right|}_{x=0}=-q_o\frac{t^2{e}^{-\frac{t}{t_p}}}{16t_p^2},{\left.\frac{\partial T\left(x,t\right)}{\partial x}\right|}_{x=L}=0,$$

(18)

where ${\epsilon }_1=\frac{T_o{\gamma }_e}{\left({\lambda }_e+2{\mu }_e\right)},{\epsilon }_2=\frac{{\rho }_b{c}_b{\omega }_b}{K{\eta }^2{c}^2},{\epsilon }_3=\frac{{\gamma }_e}{\rho c_e}$.

3. Analytical Solutions in the Transform Domain

Equations (14)–(18) can be transformed using Laplace transforms, as in

$$\overline{f}\left(x,s\right)=L\left[f\left(x,t\right)\right]={{\displaystyle \int }}_0^{\infty }f\left(x,t\right)e^{-st}dt.$$

(19)

Therefore, the following equations can be obtained

$$\frac{d^2\overline{u}}{d{x}^2}=s^2\overline{u}+{\epsilon }_1\frac{d\overline{T}}{dx},$$

(20)

$$\frac{d^2\overline{T}}{d{x}^2}=\left(1+\frac{s^{\beta }{\tau }_o^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\right)\left(\left(s+{\epsilon }_2\right)\overline{T}+s{\epsilon }_3\frac{d\overline{u}}{dx}\right)-\frac{Q_m}{s},$$

(21)

$${\overline{\sigma }}_{xx}=\frac{d\overline{u}}{dx}-{\epsilon }_1\overline{T},$$

(22)

$${\left.{\overline{\sigma }}_{xx}\left(0,s\right)=0, {\overline{\sigma }}_{xx}\left(L,s\right)=0,\frac{d\overline{T}\left(x,s\right)}{dx}\right|}_{x=0}=\frac{-q_o{t}_p}{8{\left(s{t}_p+1\right)}^3}, {\left.\frac{d\overline{T}\left(x,s\right)}{dx}\right|}_{x=L}=0.$$

(23)

Equations (20) and (21) can be used to represent the vector–matrix differential equation as follows:

$$\frac{d\mathrm{M}}{dx}=BM-f,$$

(24)

where $M=\left(\begin{array}{c}\overline{u}\\ \overline{T}\\ \frac{d\overline{u}}{dx}\\ \frac{d\overline{T}}{dx}\end{array}\right)$, $B=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ s^2& 0& 0& {\epsilon }_1\\ 0& \left(1+\frac{s^{\beta }{\tau }_o^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\right)\left(s+{\epsilon }_2\right)& \left(1+\frac{s^{\beta }{\tau }_o^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\right)s{\epsilon }_3& 0\end{array}\right)$, $f=\left(\begin{array}{c}0\\ 0\\ 0\\ \frac{Q_m}{s}\end{array}\right)$.

To solve Equation (24) using the eigenvalue techniques described in [39,40,41,42,43,44,45,46,47,48], Matrix $B$’s characteristic equation is expressed as:

$${\alpha }^4-\left(s^2+\left(1+\frac{s^{\beta }{\tau }_o^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\right)\left(s+{\epsilon }_2\right)+\left(1+\frac{s^{\beta }{\tau }_o^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\right)s{\epsilon }_3{\epsilon }_1\right){\alpha }^2+\left(1+\frac{s^{\beta }{\tau }_o^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\right)\left(s+{\epsilon }_2\right)s^2=0,$$

(25)

Relation (25) has four roots, which are the eigenvalues of matrix $B$, which are defined by $\pm {\alpha }_1, \pm {\alpha }_2$. The general solutions to the nonhomogeneous system (24) can be obtained by adding the complementary solution of the corresponding homogeneous system to a particular solution of the nonhomogeneous system. To find the particular solution to the nonhomogeneous Equation (24), we need to consider that the inhomogeneous terms in (24) contain functions of the Laplace parameter s. As a result, the particular solution can be expressed as:

$$\mathrm{M}\left(x,s\right)=A_1{X}_1{e}^{-{\alpha }_{1}x}+A_2{X}_2{e}^{{\alpha }_{1}x}+A_3{X}_3{e}^{-{\alpha }_{2}x}+A_4{X}_4{e}^{{\alpha }_{2}x}+\left(\begin{array}{c}0\\ R\\ 0\\ 0\end{array}\right).$$

(26)

where $R=\frac{Q_m}{s\left(s+{\epsilon }_2\right)\left(1+\frac{s^{\beta }{\tau }_o^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\right)}$. Hence, in the Laplace domain, the general solutions of displacement, temperature, and stress can be given by:

$$\overline{u}\left(x,s\right)=A_1{U}_1{e}^{-{\alpha }_{1}x}+A_2{U}_2{e}^{{\alpha }_{1}x}+A_3{U}_3{e}^{-{\alpha }_{2}x}+A_4{U}_4{e}^{{\alpha }_{2}x}.$$

(27)

$$\overline{T}\left(x,s\right)=A_1{T}_1{e}^{-{\alpha }_{1}x}+A_2{T}_2{e}^{{\alpha }_{1}x}+A_3{T}_3{e}^{-{\alpha }_{2}x}+A_4{T}_4{e}^{{\alpha }_{2}x}+R.$$

(28)

$$\begin{array}{c}{\overline{\sigma }}_{xx}=A_1\left(-{\alpha }_1{U}_1-{\epsilon }_1{T}_1\right)e^{-{\alpha }_{1}x}+A_2\left({\alpha }_1{U}_2-{\epsilon }_1{T}_2\right)e^{{\alpha }_{1}x}+A_3(-{\alpha }_2{U}_3-\\ {\epsilon }_1{T}_3)e^{-{\alpha }_{2}x}+A_4\left({\alpha }_2{U}_4-{\epsilon }_1{T}_4\right)e^{{\alpha }_{2}x}+R,\end{array}$$

(29)

where $U_i,T_i$ refer to the eigenvectors of displacement and temperature, respectively. The problem boundary conditions can be utilized to determine the values of $A_1, A_2,A_3,$ and $A_4$. To obtain the final solutions for the displacement, temperature, and stress distributions, the Stehfest approach [49] can be used, which has an inverse function $f\left(x,t\right)$ defined by a specific formulation:

$$f\left(x,t\right)=\frac{ln2}{t}{{\displaystyle \sum }}_{j=1}^M{V}_j\overline{f}\left(x,j\frac{ln2}{t}\right),$$

(30)

where $V_j$ can be given by

$$V_j={\left(-1\right)}^{\frac{M}{2}+1}{{\displaystyle \sum }}_{k=\frac{i+1}{2}}^{min\left(i,\frac{M}{2}\right)}\frac{k^{\frac{M}{2}+1}\left(2k\right)!}{\left(\frac{M}{2}-k\right)!k!\left(i-k\right)!\left(2k-1\right)!}$$

4. Numerical Outcomes and Discussion

To demonstrate the theoretical results discussed in the above sections, the numerical values of the physical parameters are presented. The material parameters for living tissues at the reference temperature, used in the following calculation, are denoted as follows [35]:

$$\lambda =8.27\times {10}^8\left(N\right)\left(m^{-2}\right),{\alpha }_t=1\times {10}^{-4}\left(k^{-1}\right),T_o=310\left(k\right), t_p=0.15,$$

$$\mu =3.446\times {10}^7\left(N\right) \left(m^{-2}\right),c_b=3770 \left(J\right)\left(k{g}^{-1}\right)\left(k^{-1}\right), {\rho }_b=1060\left(kg\right)\left(m^{-3}\right),$$

$$\begin{gathered} {\tau }_o=0.05, K=0.235 \left(W\right)\left(m^{-1}\right)\left(k^{-1}\right), Q_m=1.19\times {10}^3\left(W\right)\left(m^{-3}\right), \rho = \\ 1190\left(kg\right)\left(m^{-3}\right),c_e=3600 \left(J\right)\left(k{g}^{-1}\right)\left(k^{-1}\right) \end{gathered}$$

The numerical values of the calculated physical quantities under the fractional biothermo-elastic model, considering one thermal relaxation time and using the previous parameters, are displayed in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. Numerical calculations have been performed at time t = 0.2 to determine the temperature variations, the variation of displacement, and stress variation along the distance $x$ under different values of the studied parameters as in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. Figure 1, Figure 4, Figure 7 and Figure 10 display the temperature variation along the distance $x$. It can be observed from the figures that the temperature initially peaks at the tissue surface $\left(x=0\right)$ due to the exponentially decaying pulse boundary heat flux. As the distance x increases, the temperature steadily decreases. Figure 2, Figure 5, Figure 8 and Figure 11 display the variations of displacement along the distance $x$. Observing the figures, it can be noted that the displacement reaches its highest negative values at the tissue surface $\left(x=0\right)$. Subsequently, it progressively increases towards peak values near the surface before decreasing back to zero. Figure 3, Figure 6, Figure 9 and Figure 12 show the variation of stress ${\sigma }_{xx}$ along the distance $x$. It can be observed that the stress ${\sigma }_{xx}$ starts from zero and ends at zero to comply with the boundary condition of the problem. Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 can be classified into four distinct groups.

In the first group, Figure 1, Figure 2 and Figure 3 show the variations of temperature, displacement, and stress under various models. In Figure 1, Figure 2 and Figure 3, the solid line (—) refers to the Pennes model (Pennes model) without thermal relaxation time (${\tau }_o=0$) and without fractional time derivative ($\beta =1$), the dashed line (---) points to the single-phase lag model (SPL model) with one thermal relaxation time (${\tau }_o=0.02$) and without fractional time derivative ($\beta =1$), while the dotted line (…) refers to the single-phase lag model under fractional time derivative (FSPL model) with one thermal relaxation time (${\tau }_o=0.02$) and with fractional time derivative ($\beta =0.5$). The significant effects on the quantities under consideration are evident due to the various models.

In the second group, Figure 4, Figure 5 and Figure 6 display the variation of temperature, displacement, and stress under different values of the fractional parameter ($\beta =1,0.5,0.1$) when (${\tau }_o=0.02$). It is evident that the fractional time derivatives are responsible for the evident significant impacts on the quantities under consideration. A decrease in fractional time derivatives weakens the effect of thermo-mechanical propagation, as evidenced by a decrease in the maximum amplitude of temperature, displacement, and stress.

In the third group, Figure 7, Figure 8 and Figure 9 show the variation of temperature, displacement, and stress under various values of thermal relaxation time (${\tau }_o=0, 0.02, 0.2$) under fractional time derivative when ($\beta =0.5$). It is evident that the thermal relaxation time has significant effects on the quantities under consideration. The maximum amplitude of the temperature, displacement, and stress decreases with the increase the thermal relaxation time, which means that the thermal relaxation time is apt to weaken the effect of thermo-mechanical propagation.

In the fourth group, Figure 10, Figure 11 and Figure 12 show the variations of temperature, displacement, and stress under various of the characteristic time of pulsing heat flux ($t_p=0.1, 0.15, 0.2$) with one thermal relaxation time (${\tau }_o=0.02$) under fractional time derivative when ($\beta =0.5$). It is observed that the characteristic time of pulsing heat flux has a significant effect on the quantities under consideration. An increase in the characteristic time of pulsing heat flux weakens the effect of thermo-mechanical propagation, as evidenced by a decrease in the maximum amplitude of temperature, displacement, and stress.

5. Conclusions

The aim of this research paper was to examine how biological tissues react to a sudden pulsing heat flux load by employing generalized thermo-elasticity under the fractional time derivatives framework, which incorporates one thermal relaxation time.

• This study specifically focused on the impacts of fractional parameter, thermal relaxation time, and the pulsing heat flux characteristic time on bio-thermo-elastic behaviors.
• A comparative analysis was conducted between the fractional single-phase lag model (FSPL model) and previous single-phase lag (SPL model) and Pennes (Pennes model) models.
• The findings of this study, which presented a modified thermo-elasticity approach, offered a fresh perspective on the propagation of thermal waves, representing the first attempts in this area.
• These results significantly contribute to enhancing our understanding of thermo-elastic behavior in living tissue.