### Analysis of the Bio-Thermoelasticity Response of Biological Tissues Subjected to Harmonic Heating Using a Refined Green–Lindsay Model

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt

3 Financial Mathematics and Actuarial Science (FMAS)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

4 Department of Mathematics, Faculty of Science and Arts in Almakhwah, Albaha University, Almakhwah 65311, Saudi Arabia

Abstract

This study focuses on the analysis of the bio-thermoelasticity response exhibited by biological tissues when their inner and outer surfaces are free from stress and exposing the outer surface of the skin to harmonic heating with heatlessness of the inner surface of the skin. The investigation employs a refined Green–Lindsay model for a comprehensive understanding of the phenomenon. A system of partial differential equations is written and the solution is obtained using the Laplace transform and numerical inverse Laplace. The current model's results for temperature, displacement, stress, and strain distributions are presented, and it is compared to various (coupled and uncoupled) models from previous literature. The relaxation times effect on the model with other models is clarified, the effect of time, and some vital parameters are also studied, and tabularly to illustrate the effect of blood perfusion on the four distributions.

Keywords

Main Subjects

[1]          H. H. Pennes, Analysis of Tissue and Arterial Blood Temperatures in the Resting Human Forearm, Journal of Applied Physiology, Vol. 85, No. 1, pp. 5-34, 1998.
[2]          C. Cattaneo, A Form of Heat-Conduction Equations Which Eliminates the Paradox of Instantaneous Propagation, Comptes Rendus, Vol. 247, pp. 431, 1958, 1958.
[3]          P. Vernotte, Les paradoxes de la theorie continue de l'equation de la chaleur, Compt. Rendu, Vol. 246, pp. 3154-3155, 1958, 1958.
[4]          D. Y. Tzou, The generalized lagging response in small-scale and high-rate heating, International Journal of Heat and Mass Transfer, Vol. 38, No. 17, pp. 3231-3240, 1995/11/01/, 1995.
[5]          S. Wahyudi, F. Gapsari, Analysis of Temperature Distribution of Human Skin Tissue in Various Environmental Temperature with the Finite Volume Method, International Journal of Mechanical Engineering and Robotics Research, pp. 99-105, 01/01, 2022.
[6]          S. K R, I. Ramarao, D. P.A, BIOHEAT TRANSFER EQUATION WITH TRANSIENT HEAT FLUX CONDITIONS TO SKIN TISSUE AN ANALYTICAL APPROACH, International Journal of Pure and Applied Mathematics, Vol. 120, pp. 871-879, 01/01, 2018.
[7]          I. Kaur, P. Lata, K. S. Handa, Effects of Memory Dependent Derivative of Bio-heat Model in Skin Tissue exposed to Laser Radiation, EAI Endorsed Transactions on Pervasive Health and Technology, Vol. 6, pp. 164589, 07/13, 2018.
[8]          M. B. Bera, M. Mondal, B. S. Mahapatra, G. Roymahapatra, P. Acharjya, Generalized theory of thermoelasticity in isotropic and homogenious thermoelastic solids, Turkish Journal of Computer and Mathematics Education (TURCOMAT), Vol. 11, No. 3, pp. 1877-1885, 2020.
[9]          G. Oguntala, Y. Hu, G. Sobamowo, 2021, Computational Modelling of Dual-Phase Lag Bioheat Process in Cooling of Cutaneous Tissue Exposed to High Heating for Burn Injury Prediction,
[10]        S. Ozen, S. Helhel, O. Cerezci, Heat analysis of biological tissue exposed to microwave by using thermal wave model of bio-heat transfer (TWMBT), Burns : journal of the International Society for Burn Injuries, Vol. 34, pp. 45-9, 03/01, 2008.
[11]        I. Abbas, A. Abdalla, F. Anwar, H. Sapoor, A numerical solution of 2-D single-phase-lag (SPL) bio-heat model using alternating direction implicit (ADI) finite difference method, Sohag Journal of Sciences, Vol. 7, No. 3, pp. 131-141, 2022.
[12]        M. Paruch, B. Mochnacki, Cattaneo-Vernotte bio-heat transfer equation. Identificaton of external heat flux and relaxation time in domain of heated skin tissue, Computer Assisted Mechanics and Engineering Sciences, Vol. 25, pp. 71-80, 2018.
[13]        S. K. Sharma, D. Kumar, A Study on Non-Linear DPL Model for Describing Heat Transfer in Skin Tissue during Hyperthermia Treatment, Entropy, Vol. 22, No. 4, pp. 481, 2020.
[14]        P. H. Ziaei, H. Moosavi, A. A. Moradi, Analysis of the dual phase lag bio-heat transfer equation with constant and time-dependent heat flux conditions on skin surface, Thermal Science, Vol. 20, pp. 1457-1472, 2016.
[15]        M. Ezzat, Bio-thermo-mechanics behavior in living viscoelastic tissue under the fractional dual-phase-lag theory, Archive of Applied Mechanics, Vol. 91, 09/01, 2021.
[16]        R. Kumar, R. Tiwari, A. Singhal, S. Mondal, Characterization of thermal damage of skin tissue subjected to moving heat source in the purview of dual phase lag theory with memory-dependent derivative, Waves in Random and Complex Media, 09/29, 2021.
[17]        Y. Hu, X. Zhang, X.-F. Li, Thermoelastic response of skin using time-fractional dual-phase-lag bioheat heat transfer equation, Journal of Thermal Stresses, Vol. 45, pp. 597-615, 06/06, 2022.
[18]        J. Zhou, J. K. Chen, Y. Zhang, Dual-Phase Lag Effects on Thermal Damage to Biological Tissues Caused By Laser Irradiation, Computers in biology and medicine, Vol. 39, pp. 286-93, 03/01, 2009.
[19]        D. Kumar, S. Singh, K. Rai, Analysis of Classical Fourier, SPL and DPL Heat Transfer Model in Biological Tissues in Presence of Metabolic and External heat source, Heat and Mass Transfer Springer, Vol. 52, 06/24, 2015.
[20]        R. Kumar, A. K. Vashisth, S. Ghangas, Analytical solution of bioheat transfer equation with variable thermal conductivity in skin, in Proceeding of.
[21]        R. Fazlali, H. Ahmadikia, Analytical solution of thermal wave models on skin tissue under arbitrary periodic boundary conditions, International Journal of Thermophysics, Vol. 34, pp. 139-159, 2013.
[22]        P. Forghani, H. Ahmadikia, A. Karimipour, Non-Fourier Boundary Conditions Effects on the Skin Tissue Temperature Response, Heat Transfer—Asian Research, Vol. 46, No. 1, pp. 29-48, 2017.
[23]        M. A. Biot, Thermoelasticity and irreversible thermodynamics, Journal of applied physics, Vol. 27, No. 3, pp. 240-253, 1956.
[24]        H. W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids, Vol. 15, No. 5, pp. 299-309, 1967.
[25]        A. E. Green, K. Lindsay, Thermoelasticity, Journal of elasticity, Vol. 2, No. 1, pp. 1-7, 1972.
[26]        A. Green, P. Naghdi, Thermoelasticity without energy dissipation, Journal of elasticity, Vol. 31, No. 3, pp. 189-208, 1993.
[27]        A. Bagri, M. R. Eslami, Analysis of Thermoelastic Waves in Functionally Graded Hollow Spheres Based on the Green-Lindsay Theory, Journal of Thermal Stresses, Vol. 30, No. 12, pp. 1175-1193, 2007/10/30, 2007.
[28]        A. Pramanik, S. Biswas, 2020, Eigenvalue approach to hyperbolic thermoelastic problem in porous orthotropic medium with Green- Lindsay model,
[29]        S. Abo-Dahab, Green Lindsay Model on Propagation of Surface Waves in Magneto-Thermoelastic Materials with Voids and Initial Stress, Journal of Computational and Theoretical Nanoscience, vol. 11, issue 3, pp. 763-771, Vol. 11, pp. 763-771, 03/01, 2014.
[30]        A. Kumar, O. N. Shivay, S. Mukhopadhyay, Infinite speed behavior of two-temperature Green–Lindsay thermoelasticity theory under temperature-dependent thermal conductivity, Zeitschrift für angewandte Mathematik und Physik, Vol. 70, 01/02, 2019.
[31]        N. Sarkar, S. De, N. Sarkar, Modified Green–Lindsay model on the reflection and propagation of thermoelastic plane waves at an isothermal stress-free surface, Indian Journal of Physics, Vol. 94, pp. 1215-1225, 08/08, 2020.
[32]        S. R. Choudhuri, On a thermoelastic three-phase-lag model, Journal of Thermal Stresses, Vol. 30, No. 3, pp. 231-238, 2007.
[33]        A. Hobiny, F. Alzahrani, I. Abbas, Analytical Estimation of Temperature in Living Tissues Using the TPL Bioheat Model with Experimental Verification, Mathematics, Vol. 8, pp. 1188, 07/19, 2020.
[34]        A. M. Zenkour, T. Saeed, A. A. Al-Raezah, A 1D thermoelastic response of skin tissue due to ramp-type heating via a fractional-order Lord–Shulman model, Journal of Computational Applied Mechanics, Vol. 54, No. 3, pp. 365-377, 2023.
[35]        M. Sobhy, A. Zenkour, Refined Lord–Shulman Theory for 1D Response of Skin Tissue under Ramp-Type Heat, Materials, Vol. 15, pp. 6292, 09/10, 2022.
[36]        A. Zenkour, T. Saeed, K. Alnefaie, Refined Green–Lindsay Model for the Response of Skin Tissue under a Ramp-Type Heating, Mathematics, Vol. 11, pp. 1437, 03/16, 2023.
[37]        A. Zenkour, T. Saeed, A. Aati, Refined Dual-Phase-Lag Theory for the 1D Behavior of Skin Tissue under Ramp-Type Heating, Materials, Vol. 16, pp. 2421, 03/17, 2023.
[38]        A. M. Zenkour, T. Saeed, A. M. Aati, Analyzing the Thermoelastic Responses of Biological Tissue Exposed to Thermal Shock Utilizing a Three-Phase Lag Theory, Journal of Computational Applied Mechanics, 2023.
[39]        M. Aljadani, A. Zenkour, A Modified Two-Relaxation Thermoelastic Model for a Thermal Shock of Rotating Infinite Medium, Materials, Vol. 15, pp. 9056, 12/18, 2022.
[40]        A. M. Zenkour, Exact coupled solution for photothermal semiconducting beams using a refined multi-phase-lag theory, Optics & Laser Technology, Vol. 128, pp. 106233, 2020/08/01/, 2020.
[41]        A. M. Zenkour, H. F. El-Mekawy, On a multi-phase-lag model of coupled thermoelasticity, International Communications in Heat and Mass Transfer, Vol. 116, pp. 104722, 2020/07/01/, 2020.
[42]        A. M. Zenkour, On Generalized Three-Phase-Lag Models in Photo-Thermoelasticity, International Journal of Applied Mechanics, Vol. 14, No. 02, pp. 2250005, 2022.
[43]        A. M. Zenkour, Thermal diffusion of an unbounded solid with a spherical cavity via refined three-phase-lag Green–Naghdi models, Indian Journal of Physics, Vol. 96, No. 4, pp. 1087-1104, 2022/03/01, 2022.
[44]        D. Y. Tzou, Experimental support for the lagging behavior in heat propagation, Journal of Thermophysics and Heat Transfer, Vol. 9, No. 4, pp. 686-693, 1995.
[45]        G. Honig, U. Hirdes, A method for the numerical inversion of Laplace transforms, Journal of Computational and Applied Mathematics, Vol. 10, No. 1, pp. 113-132, 1984.
[46]        Q. Zhang, Y. Sun, J. Yang, Bio-heat response of skin tissue based on three-phase-lag model, Scientific Reports, Vol. 10, No. 1, pp. 16421, 2020.
[47]        T. Okabe, T. Fujimura, J. Okajima, S. Aiba, S. Maruyama, Non-invasive measurement of effective thermal conductivity of human skin with a guard-heated thermistor probe, International Journal of Heat and Mass Transfer, Vol. 126, pp. 625-635, 11/01, 2018.
###### Volume 54, Issue 4December 2023Pages 588-606
• Receive Date: 10 October 2023
• Revise Date: 30 December 2023
• Accept Date: 10 October 2023