Thermo-mechanical interaction on transient heating of skin tissue with variable thermal material properties
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 equations
The equations of motion
The general bio-heat transfer equation
The entropy equation
The DPL equation of bio-heat transferin the preceding equations, are
Linearization of governing equations
For the convenience of the following solutions, some non-dimensional variables are introduced as follows:
Substituting these non-dimensional variables into above governing equations and dropping the asterisks for convenience, we have
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)
- et al.
Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue
Int. Commun. Heat Mass Tran.
(2012) Importance of non-Fourier heat conduction in solid-phase reactions
Combust. Flame
(1998)Solution for non-Fourier dual phase heat conduction in a semi-infinite slab with surface heat flux
Int. J. Heat Mass Tran.
(1998)- et al.
Analytical study on the transient heating of a two-dimensional skin tissue using parabolic and hyperbolic bioheat transfer equations
Appl. Math. Model.
(2015) - et al.
Coupled thermoelastic analysis of a multi-layered hollow cylinder based on the C-T theory and its application on functionally graded materials
Compos. Struct.
(2015) - et al.
Theoretical analysis of thermal damages in skin tissue induced by intense moving heat source
Int. J. Heat Mass Tran.
(2018) - et al.
Verified non-linear DPL model with experimental data for analyzing heat transfer in tissue during thermal therapy
Int. J. Therm. Sci.
(2018) - et al.
Analytical study of transient thermo-mechanical response of dual-layer skin tissue with variable thermal material properties
Int. J. Therm. Sci.
(2018) - et al.
A modified fractional order generalized bio-thermoelastic theory with temperature-dependent thermal-material properties
Int. J. Therm. Sci.
(2018) - et al.
Investigation of transient thermo-mechanical response on the triple-layered skin tissue with temperature dependent blood perfusion rate
Int. J. Therm. Sci.
(2019)
Analytical solutions of non-Fourier bio-heat conductions for skin subjected to pulsed laser heating
Int. J. Therm. Sci.
Analysis for high-order effects in thermal lagging to thermal responses in biological tissue
Int. J. Heat Mass Tran.
Investigation for the dual phase lag behavior of bio-heat transfer
Int. J. Therm. Sci.
Analysis of heat transfer and burn damage in a laser irradiated living tissue with the generalized dual-phase-lag model
Int. J. Therm. Sci.
Short-pulse laser heating on metals
Int. J. Heat Mass Tran.
Asymptotic analysis of thermoelastic response in functionally graded thin plate subjected to a transient thermal shock
Compos. Struct.
Generalized solution of the thermoelastic problem for the axisymmetric structure with temperature-dependent properties
Eur. J. Mech. A Solid.
Skin biothermomechanics: modeling and experimental characterization
Adv. Appl. Mech.
Non-Fourier analysis of skin biothermomechanics
Int. J. Heat Mass Tran.
Biothermomechanics of skin tissues
J. Mech. Phys. Solid.
Cited by (17)
Legendre wavelet collocation method for investigating thermo-mechanical responses on biological tissue during laser irradiation
2024, Mathematics and Computers in SimulationOne-dimensional thermomechanical bio-heating analysis of viscoelastic tissue to laser radiation shapes
2024, International Journal of Heat and Mass TransferNon-local three phase lag bio thermal modeling of skin tissue and experimental evaluation
2023, International Communications in Heat and Mass TransferHeat transport across multi-layered skin tissue experiencing short-pulse laser irradiation: Case of temperature-dependent thermal physical parameters
2023, International Journal of Heat and Mass TransferStudy the effects of temperature and strain rates on transient thermomechanical responses on multilayer skin tissue
2023, European Journal of Mechanics, A/Solids