Thermo-mechanical interaction on transient heating of skin tissue with variable thermal material properties

https://doi.org/10.1016/j.euromechsol.2020.104173Get rights and content

Highlights

  • A generalized thermo-elastic model with multi-layered skin structure is proposed.

  • An analytical procedure dealing with variable material properties is proposed.

  • Thermo-mechanical interaction in skin tissue is explored qualitatively.

  • Effect of variable material properties on thermo-elastic response is studied.

  • The contribution of thermal stress on thermal pain of skin tissue is evaluated.

Abstract

Comprehension of the heat transfer process and associated thermo-mechanical interaction with skin tissue is the key issue on successful application of thermal treatment techniques. The purpose of this paper is to explore thermo-mechanical behavior taking place the instantaneously heated skin tissue via an analytical approach. The generalized thermo-elastic model involving dual-phase-lag model of bioheat transfer is proposed, in which a multi-layer skin structure is employed to represent variable thermal and mechanical properties with spatial location and temperature. Due to the Laplace transform and its limit theorem, an analytical procedure is then imposed on this generalized thermo-elastic model, in which the nonlinear terms involving variable properties are linearized prior to the solution of governing equations. The thermo-elastic response of skin tissue subjected to a sudden heating on its boundary is solved by this analytical approach. The exact solutions of each physical field can be obtained and its distributions are illustrated. The effects of phase lags, layered characteristics of thermal properties and temperature dependence on heat transport and evocative thermo-mechanical interaction are studied. The contribution of heat-induced stress on thermal pain is also evaluated.

Introduction

Heat transfer occurring in skin tissue is mainly a heat conduction process coupled to complicated physiological processes. The comprehension of this heat transfer process and related thermo-mechanical interaction is of great importance and can contribute to a variety of medical applications. Such as various thermal therapies, widely applied for diseased and injured skin tissue, its success depends on the precise prediction and control of temperature, damage and stress distributions in the tissue (Xu et al., 2008a, 2008b).

For knowing the thermal behavior, taking place in these thermal treatments, the experiment is the most essential method (Xu and Lu, 2009). Unfortunately, it is difficult to perform a complete experiment in vivo and vitro due to the variety of tissues and complexity of the physical and biochemical process. Therefore, as an alternative means, theoretical investigation with mathematical models has been used to explore the temperature distribution and related thermal damage in instantaneous heated tissues (Zhou et al., 2007; Liu, 2015; Liu and Chen, 2016). The Pennes model of bio-heat transfer (Pennes, 1948) based on the classical Fourier's law, is commonly used in early investigations for its simplicity. However, the nonphysical assumption that thermal disturbance or thermal wave propagates at an infinite speed through the medium limits its application on particular thermal conditions or heat conduction media (Qiu and Tien, 1992; Antaki, 1998a, 1998b).

The heat exchange between blood and tissues always happens at finite speed for its non-homogeneous inner structure, in which a non-Fourier feature such as thermal wave phenomenon can be observed and confirmed by some experiments (Roemer et al., 1985; Mitra et al., 1995). Attempt this truth, some modified models such as the CV model and dual-phase-lag model (DPL), are proposed to evaluate the thermal behavior taking place in living tissues. Here a finite propagation speed of thermal disturbance is predicted by the introduction of phase lag times (Antaki, 1998a, 1998b; Tzou, 1995a). Utilizing these modified models of bioheat transfer, a large number of investigations of thermal behavior within skin tissues have been conducted (Liu et al., 1995, 1999; Ahmadikia et al., 2012; Fazlali and Ahmadikia, 2013; Hobiny and Abbas, 2018; Dutta and Kundu, 2018; Liu and Chen, 2010, 2016; Lin and Li, 2016; Askarizadeh and Ahmadikia, 2014, 2015; Zhang et al., 2017; Zhang, 2009). Note that finite propagation speed of thermal disturbance is predicted in the CV model, however, some unusual physical solutions still exist in the CV model for some particular heat transfer progresses (Korner and Bergmann, 1998; Ahmadikia and Rismanian, 2011). Thereby the DPL model involving the microstructure interaction (Tzou, 1995a, 1995b), an effect absent in the CV model, is used more frequently in recent investigations.

Despite the widespread use of heating therapies in dermatology with the advances in laser, microwave, and similar technologies, the thermo-mechanical interaction is rarely involved in existing investigations, although it is equally important for the thermo-mechanical coupled essence in these thermal treatments. Treating the skin as a layered material, Xu et al. developed a theoretical framework for the coupled thermo-mechanical behavior of skin (Xu et al., 2008a, 2008b, 2008c), in which the heat-induced stress was solved by a ‘sequentially-coupled’ procedure for simplicity and stated that thermal stress may also contribute to thermal pain. Li et al. (2018a, 2018b, 2019) further explore the effect of temperature-dependent properties on the heat transfer and heat-induced mechanical response within the skin tissue, in which the generalized thermo-elastic theory involving the G-N model, fractional model, G-NII model, and DPL G-NII model are used separately to govern the thermo-elastic behavior taking place in skin tissue.

Since the exact solutions of these bio-heat transfer models and generalized bio-thermo-elastic models are more effective to reveal the thermal and related thermo-mechanical responses, many analytical attempts have been performed in previous investigations (Liu, 2015; Liu and Chen, 2016; Antaki, 1998a, 1998b; Liu et al., 1999; Ahmadikia et al., 2012; Fazlali and Ahmadikia, 2013; Lin and Li, 2016; Askarizadeh and Ahmadikia, 2014; Askarizadeh and Ahmadikia, 2015; Li et al., 2018a, 2018b). Though exact solutions are obtained successfully in some of the attempts (Liu et al., 1999; Ahmadikia et al., 2012; Fazlali and Ahmadikia, 2013; Askarizadeh and Ahmadikia, 2014; Askarizadeh and Ahmadikia, 2015), however, all these exact solutions are derived in the constant properties case and focus on the thermal response. So far, few attempts are performed to solve analytically thermo-mechanical behavior within skin tissue involving variable material properties, although this is very important to explore the thermo-mechanical interaction within various thermal treatments.

The purpose of this work is to seek an analytical procedure to explore thermo-mechanical interaction occurring in instantaneously heated skin tissue with variable thermal and mechanical properties. Treating the skin as a layered structure, a generalized thermo-elastic model with variable thermal and mechanical properties is developed in the context of the DPL equation of bioheat transfer. Due to the Laplace transform and its limit theorem, an analytical approach, validated for instantaneous heating process in the works of Balla (1991) and Wang et al. (2012, 2016, 2017, 2019), is employed to solve this generalized model, in which the nonlinear terms involving in governing equations are prior to linearized via a layered means (Fu et al., 2015; Wang et al., 2016, 2019). The thermo-elastic response of instantaneously heated skin tissue is solved by this analytical approach, and the exact solutions of each physical field are obtained. The distributions of the temperature, deformation, and thermal stress are illustrated and compared with the constant properties case. The effects of variable thermal and mechanical properties as well as the contribution of thermal stress on thermal pain are also discussed.

Section snippets

Formulation of the problem

As well known thermoelasticity is a fusion of the theories of elasticity and heat conduction. Its fundamental equations involving the DPL model of bio-heat transfer (Tzou, 1995a, 1995b) can be taken as

The constitutive equationsσij=λγkkδij+2μγijβθδij

The equations of motionρu¨i=σij,jfi

The general bio-heat transfer equationqi,i=ρT0S˙+wbρbcb(TbT)+qmet+qext

The entropy equationρT0S=ρcθ+T0βγ˙kk

The DPL equation of bio-heat transferq(ri,t+τq)=kT(ri,t+τT),iin the preceding equations, σij are

Linearization of governing equations

For the convenience of the following solutions, some non-dimensional variables are introduced as follows:x=a1ve1x,L=a1ve1L,t=a1(ve1)2t,τq=a1(ve1)2τq,τT=a1(ve1)2τT,(Ti)=(TiT0)/T0,Tb=(TbT0)/T0,(ui)=a1ve1λ1+2μ1T0β1ui,(σxxi)=σxxi/(T0β1),(qmeti)=qmeti/(ρ1c1T0a1(ve1)2)

Substituting these non-dimensional variables into above governing equations and dropping the asterisks for convenience, we haveσxxi=fλ+2μiuixfβiθiσyyi=σzzi=fλiuixfβiθifρi2uit2=fλ+2μi2uix2fβiθix(1+τqt)[fρifciTi

Verification of the analytical procedure

It is very necessary to validate present analytical approach to following thermo-mechancial analysis by means of these exact solutions (42)–(44). Since lack of experimental data or closed-form solutions on thermo-mechanical response of similar problems involving bio-heat transfer, hence the same validation procedure frequently used by previous investigation (Li et al., 2018a, 2018b; Askarizadeh and Ahmadikia, 2014, 2015) is employed in present work. Firstly if we ignore the thermo-mechanical

Conclusions

The heat transfer process and related thermo-mechanical interaction with skin tissue, induced by the heat input on its outer boundary, are studied in present work. A multi-layer model involving various thermal and mechanical material parameters and corresponding analytical procedure are proposed in the context of the generalized thermo-elastic model with DPL heat transfer. The thermo-elastic response induced by instantaneous temperature rise on its outer boundary is solved and discussed, which

CrediT author statement

Wang Yingze: Conceptualization, Methodology, Software. Li Meijun: Data curation, Writing-Original Draft. Liu Dong: Writing-Review & Editing, Supervision.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Science Foundation of China [grant numbers 51676086, 51575247].

References (41)

Cited by (17)

  • Non-local three phase lag bio thermal modeling of skin tissue and experimental evaluation

    2023, International Communications in Heat and Mass Transfer
View all citing articles on Scopus
View full text