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Temperature Response in a Living Tissue with Different Heating Source at the Skin Surface Under Relaxation Time

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Abstract

In this paper we have analytically characterizes the temperature distribution in a living tissue for more effective thermal treatment of tumor cells based on the modified Pennes bioheat transfer equation subjected to constant, oscillatory, cosine heat flux as well as sinusoidally heating criteria. A thorough parametric study has been made and determined optimal values of the parameters that lead to reach the temperature above therapeutic value \(43^\circ \mathrm{C}\). We assumed that heat mainly propagates in the direction perpendicular to the skin surface. The analytical solutions of the thermal wave model of one-dimensional bioheat transfer equation with different heating conditions are obtained by using Laplace transform technique. The effects of thermal relaxation time and heat flux on the skin-surface, the blood perfusion rate, thermal conductivity and frequency of heating on temperature distribution are analyzed qualitatively as well as quantitatively. The study shows that the temperature distribution has an enhancing effect on the thermal relaxation time and heat flux at the skin-surface. Further, the blood perfusion rate has reducing effect on the temperature of the living tissues.

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References

  1. Roemer, R.B.: Engineering Aspects of Hyperthermia Therapy. Ann. Rev. Biomed. Eng. 1, 347–376 (1999)

    Article  Google Scholar 

  2. Minkowycz, W.J, Sprrow, E.M., Abraham, J.P.: Advances in numerical heat transfer, CRC Press,Boca Raton,USA, 3, (2009)

  3. Bischof, J.C.: Quantitative measurement and prediction of biophysical response during freezing in tissues. Ann. Rev. Biomed. Eng. 2, 257–288 (2000)

    Article  Google Scholar 

  4. Ng, E.Y.K., Chua, L.T.: Quick numerical assessment of skin burn injury with spreadsheet in PC. J. Mech. Med. Biol. 1, 1–10 (2001)

    Article  Google Scholar 

  5. Ng, E.Y.K., Chua, L.T.: Comparison of one and two dimensional programmes for predicting the state of skin burns. Burns 28, 27–34 (2002)

    Article  Google Scholar 

  6. Pennes, H.H.: Analysis of tissue and arterial blood temperature in the resting forearm. J. Appl. Physiol. 1, 93–122 (1948)

    Google Scholar 

  7. Wulff, W.: The energy conservation equation for living tissues. IEEE Trans.-Biomed. Eng. 21, 494–495 (1974)

    Article  Google Scholar 

  8. Klinger, H.G.: Heat transfer in perfused biological tissue. I. General theory. Bull. Math. Biol. 36, 403–415 (1974)

    MathSciNet  MATH  Google Scholar 

  9. Chen, M.M., Holmes, K.R.: Microvascular Contributions in Tissue Heat Transfer. Ann. N. Y. Acad. Sci. 335, 137–150 (1980)

    Article  Google Scholar 

  10. Chen, C., L.X, Xu: Tissue temperature oscillation in an isolated pig kidney during surface heating. Ann. Biomed. Eng. 30, 1162–1171 (2002)

    Article  MathSciNet  Google Scholar 

  11. Shih, T.C., Yuan, P., Li, W.L., Kou, H.S.: Analytical analysis of the Pennes bioheat transfer equation with sinusoidal heat flux condition on skin surface. Med. Eng. Phy. 29, 946–953 (2007)

    Article  Google Scholar 

  12. Deng, Z.S., Liu, J.: Analytical study on bioheat transfer problems with spatial or transient heat on skin surface or inside biological bodies. J. Biomed. Eng. 124, 638–649 (2002)

    Google Scholar 

  13. Jiang, S.C., Ma, N., Li, H.J., Zhang, X.X.: Effects of thermal properties and geometrical dimensions on skin burn injuries. Burns 28, 713–717 (2002)

    Article  Google Scholar 

  14. Diller, K.R.: Modeling of bioheat transfer processes at high and low temperatures. Adv. Heat Transfer. 22, 157–357 (1992)

    Article  Google Scholar 

  15. Cattaneo, C.: A form of heat conduction equation which eliminates the paradox of intantaneous propagation. Compte Rendus 247, 431–433 (1958)

    Google Scholar 

  16. Vernotte, P.: Some possible complications in the phenomena of thermal conduction. Compte Rendus 252, 2190–2191 (1961)

    Google Scholar 

  17. Ozisik, M.N., Tzou, D.Y.: On the wave theory in heat conduction. ASME J. Heat Transf. 116, 526–535 (1994)

    Article  Google Scholar 

  18. Weymann, H.D.: Finite speed of propagation in heat conduction, diffusion, and viscous shear motion. Am. J. Phys. 35, 488–496 (1967)

    Article  Google Scholar 

  19. Shit, G.C., Bera, A.: Effect of thermal relaxation time on heat transfer in a two layer composite system of living tissues. Int. Commun. Heat Mass Transf. 61, 96–101 (2015)

    Article  Google Scholar 

  20. Kaminski, W.: Hyperbolic heat conduction equation for material with a nonhomogenous inner structure. ASME J. Heat Transfer. 112, 555–560 (1990)

    Article  Google Scholar 

  21. Mitra, K., Kumar, S., Vedavarz, A., Moallemi, M.K.: Experimental evidence of hyperbolic heat conduction in processed meat. ASME. J. Heat Transfer. 117, 568–573 (1995)

    Article  Google Scholar 

  22. Liu, K.C.: Thermal propagation analysis for living tissue with surface heating. Int. J. Therm. Sci. 47, 507–513 (2008)

    Article  Google Scholar 

  23. Liu, J., Chen, X., Xu, L.X.: New thermal wave aspects on burn evaluation of skin subjected to instantaneous heating. IEEE Trans. Biomed. Eng. 46, 420–428 (1999)

    Article  Google Scholar 

  24. 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. Comm. Heat Mass Transf. 39, 121–130 (2012)

    Article  Google Scholar 

  25. Ströher, G.R., Ströher, G.L.: Numerical thermal analysis of skin tissue using parabolic and hyperbolic approaches. Int. Commun. Heat Mass Transf. 57, 193–199 (2014)

    Article  Google Scholar 

  26. Afrin, N., Zhou, J., Zhang, Y., Tzou, D.Y., Chen, J.K.: Numerical simulation of thermal damage to living biological tissues induced by laser irradiation based on a generalised dual phase lag model. Numer. Heat Transf. 61, 483–501 (2012)

    Article  Google Scholar 

  27. Das, K., Mishra, S.C.: Estimation of tumor characteristics in a breast tissue with known skin surface temperature. J. Therm. Biol. 38, 311–317 (2013)

    Article  Google Scholar 

  28. Das, K., Mishra, S.C.: Non-invasive estimation of size and location of a tumor in a human breast using a curve fitting technique. Int. Commu. Heat Mass Transf. 56, 63–70 (2014)

    Article  Google Scholar 

  29. Damor, R.S., Kumar, S., Shukla, A.K.: Numerical simulation of fractional bioheat equation in hyperthermia treatment. J. Mech. Med. Biol. 14, 1450018 (2014)

    Article  Google Scholar 

  30. Singh, S., Kumar, S.: Numerical study on triple layer skin tissue freezing using dual phase lag bio-heat model. Int. J. Therm. Sci. 86, 12–20 (2014)

    Article  Google Scholar 

  31. Shih, T.C., Yuan, P., Lin, W.L., Kou, H.S.: Analytical analysis of the Pennes bioheat transfer equation with sinusoidal heat flux condition on skin surface. Med. Eng. Phys. 29, 946–953 (2007)

    Article  Google Scholar 

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Acknowledgments

The authors wish to convey their sincere thanks to all the esteemed reviewers for their comments and suggestions based upon which the present version of the manuscript has been revised. One of the authors (G. C. Shit) is thankful to the Institute of Mathematical Sciences, Chennai for providing sufficient research facility to carry out this investigation during his stay at IMSc., Chennai as Visiting Associate.

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  1. Department of Mathematics, Jadavpur University, Kolkata, 700032, India

    G. C. Shit & Amal Bera

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  1. G. C. Shit
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  2. Amal Bera
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Correspondence to G. C. Shit.

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Shit, G.C., Bera, A. Temperature Response in a Living Tissue with Different Heating Source at the Skin Surface Under Relaxation Time. Int. J. Appl. Comput. Math 3, 381–394 (2017). https://doi.org/10.1007/s40819-015-0120-0

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  • DOI: https://doi.org/10.1007/s40819-015-0120-0

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