# Analytical Estimation of Temperature in Living Tissues Using the TPL Bioheat Model with Experimental Verification

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

## 3. Laplace Transforms

## 4. Numerical Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Gabay, I.; Abergel, A.; Vasilyev, T.; Rabi, Y.; Fliss, D.M.; Katzir, A. Temperature-controlled two-wavelength laser soldering of tissues. Lasers Surg. Med.
**2011**, 43, 907–913. [Google Scholar] [CrossRef] [PubMed] - Zhou, J.; Chen, J.; Zhang, Y. Dual-phase lag effects on thermal damage to biological tissues caused by laser irradiation. Comput. Biol. Med.
**2009**, 39, 286–293. [Google Scholar] [CrossRef] - Mahjoob, S.; Vafai, K. Analytical characterization of heat transport through biological media incorporating hyperthermia treatment. Int. J. Heat Mass Transf.
**2009**, 52, 1608–1618. [Google Scholar] [CrossRef] - Labonte, S. Numerical model for radio-frequency ablation of the endocardium and its experimental validation. IEEE Trans. Biomed. Eng.
**1994**, 41, 108–115. [Google Scholar] [CrossRef] [PubMed] - González-Suárez, A.; Berjano, E. Comparative analysis of different methods of modeling the thermal effect of circulating blood flow during RF cardiac ablation. IEEE Trans. Biomed. Eng.
**2015**, 63, 250–259. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Iasiello, M.; Andreozzi, A.; Bianco, N.; Vafai, K. The porous media theory applied to radiofrequency catheter ablation. Int. J. Numer. Methods Heat Fluid Flow
**2019**. [Google Scholar] [CrossRef] - Iasiello, M.; Vafai, K.; Andreozzi, A.; Bianco, N. Hypo-and hyperthermia effects on LDL deposition in a curved artery. Comput. Therm. Sci. Int. J.
**2019**, 11, 95–103. [Google Scholar] [CrossRef] - Egred, M.; Brilakis, E.S. Excimer laser coronary angioplasty (ELCA): Fundamentals, mechanism of action, and clinical applications. J. Invasive Cardiol.
**2020**, 32, E27–E35. [Google Scholar] - Ho, Y.-J.; Wu, C.-C.; Hsieh, Z.-H.; Fan, C.-H.; Yeh, C.-K. Thermal-sensitive acoustic droplets for dual-mode ultrasound imaging and drug delivery. J. Control. Release
**2018**, 291, 26–36. [Google Scholar] [CrossRef] - Andreozzi, A.; Iasiello, M.; Netti, P.A. A thermoporoelastic model for fluid transport in tumour tissues. J. R. Soc. Interface
**2019**, 16, 20190030. [Google Scholar] [CrossRef] [Green Version] - Pennes, H.H. Analysis of tissue and arterial blood temperatures in the resting human forearm. J. Appl. Physiol.
**1948**, 1, 93–122. [Google Scholar] [CrossRef] - Charny, C.K. Mathematical models of bioheat transfer. In Advances in Heat Transfer; Elsevier: Amsterdam, The Netherlands, 1992; Volume 22, pp. 19–155. [Google Scholar]
- Nakayama, A.; Kuwahara, F. A general bioheat transfer model based on the theory of porous media. Int. J. Heat Mass Transf.
**2008**, 51, 3190–3199. [Google Scholar] [CrossRef] [Green Version] - Andreozzi, A.; Brunese, L.; Iasiello, M.; Tucci, C.; Vanoli, G.P. Modeling heat transfer in tumors: A review of thermal therapies. Ann. Biomed. Eng.
**2019**, 47, 676–693. [Google Scholar] [CrossRef] [PubMed] - Tzou, D.Y. A Unified field approach for heat conduction from macro- to micro-scales. J. Heat Transf.
**1995**, 117, 8–16. [Google Scholar] [CrossRef] - Zhu, D.; Luo, Q.; Zhu, G.; Liu, W. Kinetic thermal response and damage in laser coagulation of tissue. Lasers Surg. Med.
**2002**, 31, 313–321. [Google Scholar] [CrossRef] [PubMed] - Choudhuri, S.R. On a thermoelastic three-phase-lag model. J. Therm. Stresses
**2007**, 30, 231–238. [Google Scholar] [CrossRef] - Kumar, D.; Singh, S.; Rai, K. Analysis of classical Fourier, SPL and DPL heat transfer model in biological tissues in presence of metabolic and external heat source. Heat Mass Transf.
**2016**, 52, 1089–1107. [Google Scholar] [CrossRef] - Saeed, T.; Abbas, I. Finite element analyses of nonlinear DPL bioheat model in spherical tissues using experimental data. Mech. Based Des. Struct. Mach.
**2020**, 1–11. [Google Scholar] [CrossRef] - Hobiny, A.; Abbas, I. Analytical solutions of fractional bioheat model in a spherical tissue. Mech. Based Des. Struct. Mach.
**2019**, 1–10. [Google Scholar] [CrossRef] - Mondal, S.; Sur, A.; Kanoria, M. Transient heating within skin tissue due to time-dependent thermal therapy in the context of memory dependent heat transport law. Mech. Based Des. Struct. Mach.
**2019**, 1–15. [Google Scholar] [CrossRef] - Kumar, R.; Chawla, V. Reflection and refraction of plane wave at the interface between elastic and thermoelastic media with three-phase-lag model. Int. Commun. Heat Mass Transf.
**2013**, 48, 53–60. [Google Scholar] [CrossRef] - Hobiny, A.; Alzahrani, F.S.; Abbas, I. Three-phase lag model of thermo-elastic interaction in a 2D porous material due to pulse heat flux. Int. J. Numer. Methods Heat Fluid Flow
**2020**. [Google Scholar] [CrossRef] - Quintanilla, R.; Racke, R. A note on stability in three-phase-lag heat conduction. Int. J. Heat Mass Transf.
**2008**, 51, 24–29. [Google Scholar] [CrossRef] [Green Version] - Abbas, I.A.; Zenkour, A.M. Dual-phase-lag model on thermoelastic interactions in a semi-infinite medium subjected to a ramp-type heating. J. Comput. Theor. Nanosci.
**2014**, 11, 642–645. [Google Scholar] [CrossRef] - Abbas, I.A.; Youssef, H.M. Two-temperature generalized thermoelasticity under ramp-type heating by finite element method. Meccanica
**2013**, 48, 331–339. [Google Scholar] [CrossRef] - Abbas, I.A. Finite element analysis of the thermoelastic interactions in an unbounded body with a cavity. Forschung im Ingenieurwesen
**2007**, 71, 215–222. [Google Scholar] [CrossRef] - Abbas, I.A.; Abo-El-Nour, N.; Othman, M.I. Generalized magneto-thermoelasticity in a fiber-reinforced anisotropic half-space. Int. J. Thermophys.
**2011**, 32, 1071–1085. [Google Scholar] [CrossRef] - Zenkour, A.M.; Abbas, I.A. A generalized thermoelasticity problem of an annular cylinder with temperature-dependent density and material properties. Int. J. Mech. Sci.
**2014**, 84, 54–60. [Google Scholar] [CrossRef] - Marin, M. Cesaro means in thermoelasticity of dipolar bodies. Acta Mech.
**1997**, 122, 155–168. [Google Scholar] [CrossRef] - Marin, M.; Craciun, E. Uniqueness results for a boundary value problem in dipolar thermoelasticity to model composite materials. Compos. Part B Eng.
**2017**, 126, 27–37. [Google Scholar] [CrossRef] - Hassan, M.; Marin, M.; Ellahi, R.; Alamri, S.Z. Exploration of convective heat transfer and flow characteristics synthesis by Cu–Ag/water hybrid-nanofluids. Heat Transf. Res.
**2018**, 49, 1837–1848. [Google Scholar] [CrossRef] - Xu, F.; Seffen, K.; Lu, T. Non-Fourier analysis of skin biothermomechanics. Int. J. Heat Mass Transf.
**2008**, 51, 2237–2259. [Google Scholar] [CrossRef] - Ahmadikia, H.; Fazlali, R.; Moradi, A. Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue. Int. Commun. Heat Mass Transf.
**2012**, 39, 121–130. [Google Scholar] [CrossRef] - Kumar, D.; Rai, K. Three-phase-lag bioheat transfer model and its validation with experimental data. Mech. Based Des. Struct. Mach.
**2020**, 1–15. [Google Scholar] [CrossRef] - Mitchell, J.W.; Galvez, T.L.; Hengle, J.; Myers, G.E.; Siebecker, K.L. Thermal response of human legs during cooling. J. Appl. Physiol.
**1970**, 29, 859–865. [Google Scholar] [CrossRef] - Gardner, C.M.; Jacques, S.L.; Welch, A. Light transport in tissue: Accurate expressions for one-dimensional fluence rate and escape function based upon Monte Carlo simulation. Lasers Surg. Med. Off. J. Am. Soc. Laser Med. Surg.
**1996**, 18, 129–138. [Google Scholar] [CrossRef] - Hobiny, A.D.; Abbas, I.A. Theoretical analysis of thermal damages in skin tissue induced by intense moving heat source. Int. J. Heat Mass Transf.
**2018**, 124, 1011–1014. [Google Scholar] [CrossRef] - Tzou, D.Y. Macro-to Micro-Scale Heat Transfer: The Lagging Behavior; CRC Press: Boca Raton, FL, USA, 1996. [Google Scholar]
- Noroozi, M.J.; Goodarzi, M. Nonlinear analysis of a non-Fourier heat conduction problem in a living tissue heated by laser source. Int. J. Biomath.
**2017**, 10, 1750107. [Google Scholar] [CrossRef] - Henriques, F., Jr.; Moritz, A. Studies of thermal injury: I. The conduction of heat to and through skin and the temperatures attained therein. A theoretical and an experimental investigation. Am. J. Pathol.
**1947**, 23, 530. [Google Scholar] - Moritz, A.R.; Henriques, F., Jr. Studies of thermal injury: II. The relative importance of time and surface temperature in the causation of cutaneous burns. Am. J. Pathol.
**1947**, 23, 695–720. [Google Scholar] - Museux, N.; Perez, L.; Autrique, L.; Agay, D. Skin burns after laser exposure: Histological analysis and predictive simulation. Burns
**2012**, 38, 658–667. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 5.**Variations in temperature versus the distance under the TPL model for several values of the rate of blood perfusion ${\omega}_{b}$.

**Figure 6.**Temperature under the TPL model for several values of the rate of blood perfusion ${\omega}_{b}$ at the skin surface in relation to time.

**Figure 7.**Variations in thermal damage under the TPL model for several values of the rate of blood perfusion ${\omega}_{b}$ at the skin surface $x=0$.

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**MDPI and ACS Style**

Hobiny, A.; Alzahrani, F.; Abbas, I.
Analytical Estimation of Temperature in Living Tissues Using the TPL Bioheat Model with Experimental Verification. *Mathematics* **2020**, *8*, 1188.
https://doi.org/10.3390/math8071188

**AMA Style**

Hobiny A, Alzahrani F, Abbas I.
Analytical Estimation of Temperature in Living Tissues Using the TPL Bioheat Model with Experimental Verification. *Mathematics*. 2020; 8(7):1188.
https://doi.org/10.3390/math8071188

**Chicago/Turabian Style**

Hobiny, Aatef, Faris Alzahrani, and Ibrahim Abbas.
2020. "Analytical Estimation of Temperature in Living Tissues Using the TPL Bioheat Model with Experimental Verification" *Mathematics* 8, no. 7: 1188.
https://doi.org/10.3390/math8071188