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An investigation of biological tissue responses to thermal shock within the framework of fractional heat transfer theory

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Abstract

The present article addresses a novel mathematical model involving the Atangana-Baleanu (A-B) definition of fractional derivatives in time that offers a new interpretation of the thermo-mechanical effects inside skin tissue during thermal therapy. A Laplace transform mechanism is proposed to achieve closed-form solutions for prominent physical quantities, such as temperature, displacement, strain, and thermal stress. Computational results are obtained in time domains using an efficient numerical inversion algorithm of Laplace transform. The impact of the fractional parameter is investigated on the variations of the field quantities through the graphical results. The behavior of each physical field is speculated against the time parameter. The domain of influence of each field quantity is suppressed when the definition of the Atangana Baleanu fractional model is adopted, replicating that the waves under the A-B fractional model predict the finite nature of propagation compared to the conventional heat transport model. Further, we observe that the nature of the thermo-mechanical waves becomes stable earlier inside the tissue.

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Data Availability

No datasets were generated or analysed during the current study.

Abbreviations

\(T\) :

Temperature.

\(t\) :

Time variable.

\(\tau \) :

Phase-lag due to heat flux.

\(\rho _{b}\) :

Blood density.

\(c_{b}\) :

Specific heat of blood.

\(\omega _{b}\) :

Perfusion rate of blood.

\(c\) :

Specific heat.

\(Q_{m}\) :

Metabolic heat generation.

\(\alpha \) :

Atangana Baleanu fractional parameter.

\(c_{t}\) :

Heat capacity of a unit mass of the tissue.

\(\rho _{t}\) :

Mass density of the tissue.

\(\overrightarrow{u}\) :

Displacement vector.

\(div(\ \overrightarrow{u} )=e= e_{kk}\) :

Volumetric strain.

\(e_{ij}\) :

Strain tensor.

\(\gamma _{t}\) :

Thermal modulus.

\(\lambda _{t}\), \(\mu _{t}\) :

Lamé constants of the tissue.

\(\alpha _{t}\) :

Thermal expansion coefficient.

\(\overrightarrow{q}\) :

Heat flux vector.

\(T_{b}\) :

Arterial blood temperature.

\(k_{t}\) :

Thermal conductivity coefficient of the tissue.

\(\sigma \) :

Thermal stress inside tissue.

References

  • Andreozzi, A., Brunese, L., Iasiello, M., Tucci, C., Vanoli, G.P.: Modeling heat transfer in tumors: a review of thermal therapies. Ann. Biomed. Eng. 47, 676–693 (2019)

    Article  Google Scholar 

  • Aszad, M.I., Ikram, M.D., Akgul, A.: Analysis of MHD viscous fluid flow through porous medium with novel power law fractional differential operator. Phys. Scr. 95, 115209 (2020)

    Article  Google Scholar 

  • Aszad, M.I., Ali, R., Iqbal, A., Muhammad, T., Chu, Y.M.: Application of water based drilling clay-nanoparticles in heat transfer of fractional Maxwell fluid over an infinite flat surface. Sci. Rep. 11, 18833 (2021)

    Article  Google Scholar 

  • Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016). https://doi.org/10.2298/TSCI160111018A

    Article  Google Scholar 

  • Atangana, A., Koca, I.: Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Solitons Fractals 89, 447–454 (2016)

    Article  MathSciNet  Google Scholar 

  • Catteneo, C.: A form of heat-conduction equations which eliminates the paradox of instantaneous propagation. C. R. 247, 431–433 (1958)

    Google Scholar 

  • Chaudhary, R.K., Rai, K.N., Singh, J.: A study for multi-layer skin burn injuries based on DPL bioheat model. J. Therm. Anal. Calorim. 146, 1171–1189 (2021)

    Article  Google Scholar 

  • Chu, Y.M., Ikram, M.D., Aszad, M.I., Ahmadian, A., Ghaemi, F.: Influence of hybrid nano fluids and heat generation on coupled heat and mass transfer flow of a viscous fluid with novel fractional derivative. J. Therm. Anal. Calorim. 144, 2057–2077 (2021)

    Article  Google Scholar 

  • Damor, R.S., Kumar, S., Shukla, A.K.: Numerical simulation of fractional bioheat equation in hyperthermia treatment. J. Mech. Med. Biol. 14, 1450018–1450033 (2014)

    Article  Google Scholar 

  • Díaz, S.H., Nelson, J.S., Wong, B.J.: Rate process analysis of thermal damage in cartilage. Phys. Med. Biol. 48, 19–29 (2002)

    Article  Google Scholar 

  • Ezzat, M.A.: Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer. Phys. Biol. 405, 4188–4194 (2010)

    Article  Google Scholar 

  • Ezzat, M.A.: Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer. Phys. Biol. 406, 30–35 (2011)

    Article  Google Scholar 

  • Ezzat, M.A.: Bio-thermo-mechanics behavior in living viscoelastic tissue under the fractional dual-phase lag theory. Arch. Appl. Mech. 91(9), 3903–3919 (2021)

    Article  Google Scholar 

  • Ezzat, M., Alsowayan, N., Al-Muhiameed, Z., Ezzat, S.: Fractional modelling of Pennes’ bioheat transfer equation. Heat Mass Transf. 50(7), 907–914 (2014)

    Article  Google Scholar 

  • Gabay, I., Abergel, A., Vasilyev, T., Rabi, Y., Fliss, D.M., Katzir, A.: Temperature-controlled two-wavelength laser soldering of tissues. Lasers Surg. Med. 43, 907–913 (2011)

    Article  Google Scholar 

  • Ghanmi, A., Abbas, I.A.: An analytical study on the fractional transient heating within the skin tissue during the thermal therapy. J. Therm. Biol. 82, 229–233 (2019)

    Article  Google Scholar 

  • Ghazizadeh, H.R., Maerefat, M., Azimi, A.: Modeling non-Fourier behavior of bioheat transfer by fractional single-phase-lag heat conduction constitutive model. In: Proc 4th IFAC Workshop on Fractional Differentiation and Its Applications. University of Extremadura, Badajoz (2010)

    Google Scholar 

  • Hobiny, A., Alzahrani, F., Abbas, I., Marin, M.: The effect of fractional time derivative of bioheat model in skin tissue induced to laser irradiation. Symmetry 12, 602 (2020)

    Article  Google Scholar 

  • Honig, G., Hirdes, U.: A method for the numerical inversion of the Laplace transform. J. Comput. Appl. Math. 10, 113–132 (1984)

    Article  MathSciNet  Google Scholar 

  • Hooshmand, P., Moradi, A., Khezry, B.: Bioheat transfer analysis of biological tissues induced by laser irradiation. Int. J. Therm. Sci. 90, 214–223 (2015)

    Article  Google Scholar 

  • Hu, Y., Zhang, X., Li, X.F.: Thermoelastic response of skin using time-fractional dual-phase-lag bioheat heat transfer equation. J. Therm. Stresses 45(7), 597–615 (2022)

    Article  Google Scholar 

  • Kumar, P., Kumar, D., Rai, K.N.: A mathematical model for hyperbolic space-fractional bioheat transfer during thermal therapy. Proc. Eng. 127, 56–62 (2015)

    Article  Google Scholar 

  • Kumar, R., Tiwari, R., Singhal, A., Mondal, S.: 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 Random Complex Media (2021). https://doi.org/10.1080/17455030.2021.1979273

    Article  Google Scholar 

  • Li, X., Qin, Q.H., Tian, X.: Thermomechanical response of porous biological tissue based on local thermal non-equilibrium. J. Therm. Stresses 42, 1481–1498 (2019)

    Article  Google Scholar 

  • Mahjoob, S., Vafai, K.: Analytical characterization of heat transport through biological media incorporating hyperthermia treatment. Int. J. Heat Mass Transf. 52, 1608–1618 (2009)

    Article  Google Scholar 

  • Mohammed, S., Zenkour, A.M.: Refined Lord–Shulman theory for 1D response of skin tissue under ramp-type heat. Mater. 15, 6292 (2022)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Qi, H., Guo, X.: Transient fractional heat conduction with generalized Cattaneo model. Int. J. Heat Mass Transf. 76, 535–539 (2014)

    Article  Google Scholar 

  • Riaz, M.B., Imran, M.A., Shabbir, K.: Analytic solutions of Oldroyd-B fluid with fractional derivatives in a circular duct that applies a constant couple. Alex. Eng. J. 55(4), 3267–3275 (2016)

    Article  Google Scholar 

  • Shah, N.A., Khan, I., Aleem, M., Imran, M.A.: Influence of magnetic field on double convection problem of fractional viscous fluid over an exponentially moving vertical plate: new trends of Caputo time-fractional derivative model. Adv. Mech. Eng. 11(7), 1–11 (2019)

    Article  Google Scholar 

  • Singh, S., Melnik, R.: Thermal ablation of biological tissues in disease treatment: a review of computational models and future directions. Electromagn. Biol. Med. 39(2), 49–88 (2020)

    Article  Google Scholar 

  • Singh, J., Gupta, P.K., Rai, K.N.: Solution of fractional bioheat equations by finite difference method and HPM. Math. Comput. Model. 54, 2316–2325 (2011)

    Article  MathSciNet  Google Scholar 

  • Sur, A., Mondal, S., Kanoria, M.: Influence of moving heat source on skin tissue in the context of two-temperature memorydependent heat transport law. J. Therm. Stresses 43, 55–71 (2020)

    Article  Google Scholar 

  • Vernotte, P.: Les paradoxes de la theorie continue de l’equation de la chaleur. C. R. Acad. Bulgare Sci. 246, 3154–3155 (1958)

    Google Scholar 

  • Xu, F., Lu, T.J., Seffen, K.A.: Bio-thermomechanical behavior of skin tissue. Acta Mech. Sin. 24(1), 1–23 (2008)

    Article  MathSciNet  Google Scholar 

  • Youssef, H.M., Alghamdi, N.A.: Characterization of thermal damage due to two-temperature high-order thermal lagging in a three-dimensional biological tissue subjected to a rectangular laser pulse. Polymers 12, 922 (2020a)

    Article  Google Scholar 

  • Youssef, H.M., Alghamdi, N.A.: The exact analytical solution of the dual-phase-lag two-temperature bioheat transfer of a skin tissue subjected to constant heat flux. Sci. Rep. 10, 15946 (2020b)

    Article  Google Scholar 

  • Zhang, Q., Sun, Y., Yang, J.: Bio-heat response of skin tissue based on three-phase-lag model. Sci. Rep. 10, 16421 (2020)

    Article  Google Scholar 

  • Zhang, Q., Sun, Y., Yang, J.: Bio-heat transfer analysis based on fractional derivative and memory-dependent derivative heat conduction models. Case Stud. Therm. Eng. 27, 101211 (2021)

    Article  Google Scholar 

  • Zhou, J., Chen, J., Zhang, Y.: Dual-phase lag effects on thermal damage to biological tissues caused by laser irradiation. Comput. Biol. Med. 39, 286–293 (2009)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Nitishwar College, A constituent unit of Babasaheb Bhimrao Ambedkar Bihar University, Bihar, India

    Rakhi Tiwari

  2. Department of Science and Liberal Arts, JK Lakshmipat University, Rajasthan, India

    Manushi Gupta

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  1. Rakhi Tiwari
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R.T. gave the idea of the manuscript and established mathematical model. M.G. prepared software.

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Correspondence to Rakhi Tiwari.

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Tiwari, R., Gupta, M. An investigation of biological tissue responses to thermal shock within the framework of fractional heat transfer theory. Mech Time-Depend Mater (2024). https://doi.org/10.1007/s11043-024-09700-9

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