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Articles

Fractional thermo-viscoelastic response of biological tissue with variable thermal material properties

Pages 1120-1137 | Received 02 Mar 2020, Accepted 12 May 2020, Published online: 08 Jun 2020

References

  • M.A. Meyers and K.K. Chawla, Mechanical Behavior of Materials, vol. 98. Upper Saddle River, NJ, USA: Prentice Hall, 1999, pp. 103.
  • B. Gross, Mathematical Structure of the Theories of Viscoelasticity. Paris, France: Hemann, 1953.
  • M.E. Gurtin and E. Sternberg, “On the linear theory of viscoelasticity,” Arch. Rational Mech. Anal., vol. 11, no. 1, pp. 291–367, 1962. DOI: 10.1007/BF00253942.
  • E. Sternberg, “On the analysis of thermal stresses in viscoelastic solids,” Brown Univ. Dir. Appl. Math. TR, vol. 19, pp. 213–219, 1963.
  • A.A. Ilioushin, “The approximation method of calculating the constructors by linear thermal viscoelastic theory,” Mekh. Polimerov Riga, vol. 2, pp. 168–178, 1968.
  • A.A. Ilioushin and B.E. Pobedria, Mathematical Theory of Thermal Viscoelasticity. Moscow, Russia: Nauka, 1970.
  • M. Caputo and F. Mainardi, “A new dissipation model based on memory mechanism,” PAGEOPH, vol. 91, no. 1, pp. 134–147, 1971. DOI: 10.1007/BF00879562.
  • M. Caputo and F. Mainardi, “Linear model of dissipation in an elastic solids,” Riv. Nuovo Cimento, vol. 1, no. 2, pp. 161–198, 1971. DOI: 10.1007/BF02820620.
  • M. Caputo, “Vibrations on an infinite viscoelastic layer with a dissipative memory,” J. Acoust. Soc. Am., vol. 56, no. 3, pp. 897–904, 1974. DOI: 10.1121/1.1903344.
  • K. Adolfsson and M. Enelund, “Fractional derivative viscoelasticity at large deformations,” Nonlinear Dyn., vol. 33, no. 3, pp. 301–321, 2003. DOI: 10.1007/s11071-004-3758-4.
  • K. Adolfsson, M. Enelund and S. Larsson, “Adaptive discretization of fractional order viscoelasticity using sparse time history,” Comput. Methods Appl. Mech. Eng., vol. 193, no. 42–44, pp. 4567–4590, 2004. DOI: 10.1016/j.cma.2004.03.006.
  • R.L. Bagley and P.J. Torvik, “Fractional calculus—a different approach to the analysis of viscoelastically damped structures,” AIAA J., vol. 21, no. 5, pp. 741–748, 1983. DOI: 10.2514/3.8142.
  • R.L. Bagley and P.J. Torvik, “On the fractional calculus model of viscoelastic behavior,” J. Rheol., vol. 30, no. 1, pp. 133–155, 1986. DOI: 10.1122/1.549887.
  • S.W.J. Welch, R.A.L. Rorrer and R.G. Duren Jr., “Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials,” Mech. Time-Depend. Mater., vol. 3, no. 3, pp. 279–303, 1999. [Mismatch] DOI: 10.1023/A:1009834317545.
  • M.A. Ezzat and A.A. El-Bary, “Fractional order theory to an infinite thermoviscoelastic body with a cylindrical cavity in the presence of an axial uniform magnetic field,” J. Electromagn. Waves Appl., vol. 31, no. 5, pp. 495–513, 2017. DOI: 10.1080/09205071.2017.1285728.
  • M.A. Ezzat, A.S. El-Karamany, A.A. El-Bary and M.A. Fayik, “Fractional calculus in one-dimensional isotropic thermo-viscoelasticity,” C. R., Mécc, vol. 341, no. 7, pp. 553–566, 2013. DOI: 10.1016/j.crme.2013.04.001.
  • M.A. Ezzat, A.S. El-Karamany and A.A. El-Bary, “Thermo-viscoelastic materials with fractional relaxation operators,” Appl. Math. Model, vol. 39, no. 23–24, pp. 7499–7512, 2015. DOI: 10.1016/j.apm.2015.03.018.
  • Y.Z. Povstenko, “Fractional Cattaneo-type equations and generalized thermoelasticity,” J. Therm. Stress, vol. 34, no. 2, pp. 97–114, 2011. DOI: 10.1080/01495739.2010.511931.
  • H.H. Sherief, A.A. El-Said and A. Abd El-Latief, “Fractional order theory of thermo-elasticity,” Int. J. Solids Struct., vol. 47, no. 2, pp. 269–275, 2010. DOI: 10.1016/j.ijsolstr.2009.09.034.
  • F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,” J. Compu. Appli. Math., vol. 118, no. 1–2, pp. 283–299, 2000. DOI: 10.1016/S0377-0427(00)00294-6.
  • X.-J. Yang, “New rheological problems involving general fractional derivatives with nonsingular power-law kernels,” Proc. Roman Acad. Series A. Math. Phys. Tech. Sci. Inform. Sci., vol. 19, no. 1, pp. 45–52, 2018.
  • X.-J. Yang, “New general fractional-order rheological models with kernels of Mittag-Leffler functions,” Rom. Rep. Phys., vol. 69, no. 188, pp. 1–15, 2017.
  • X.-J. Yang, General Fractional Derivatives: Theory, Methods and Applications. Boca Raton, FL, USA: CRC Press, 2019.
  • X.-J. Yang, F. Gao and H.M. Srivastava, “New rheological models within local fractional derivative,” Rom. Rep. Phys., vol. 69, no. 113, pp. 1–12, 2017.
  • X.-J. Yang, M. Abdel-Aty and C. Cattany, “A new general fractional-order derivataive with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer,” Therm. Sci., vol. 23, no. 3 Part A, pp. 1677–1681, 2019. DOI: 10.2298/TSCI180320239Y.
  • X.-J. Yang, F. Gao and Y. Ju, General Fractional Derivatives with Applications in Viscoelasticity. New York, NY, USA: Academic Press, 2020.
  • N. Sasaki, Viscoelastic Properties of Biological Materials. London, UK: InTech Open, 2012.
  • W. Agnelli, C. Padra and C.V. Turner, “Shape optimization for tumor location,” J. Comput. Math. Appl., vol. 62, no. 11, pp. 4068–4081, 2011. DOI: 10.1016/j.camwa.2011.09.055.
  • J.N. Weinstein, R.L. Magin, R.L. Cysyk and D.S. Zaharko, “Treatment of solid L1210 murine tumors with local hyperthermia and temperature-sensitive liposomes containing methotrexate,” Cancer Res., vol. 40, no. 5, pp. 1388–1395, 1980.
  • H.H. Pennes, “Analysis of tissue and arterial blood temperatures in the resting human forearm,” J. Appl. Physiol., vol. 1, no. 2, pp. 93–122, 1948. DOI: 10.1152/jappl.1948.1.2.93.
  • M. Biot, “Thermoelasticity and irreversible thermodynamics,” J. Appl. Phys., vol. 27, no. 3, pp. 240–253, 1956. DOI: 10.1063/1.1722351.
  • A.V. Luikov, Analytical Heat Diffusion Theory. New York, NY, USA: Academic Press, 1968.
  • W. Kaminski, “Hyperbolic heat conduction equation for material with a non-homogenous inner structure,” ASME J. Heat. Transf., vol. 112, no. 3, pp. 555–560, 1990. DOI: 10.1115/1.2910422.
  • M.A. Ezzat, N. AlSowayan, Z.I. Muhiameed and S. Ezzat, “Fractional modelling of Pennes’ bioheat transfer equation,” Heat Mass Transfer, vol. 50, no. 7, pp. 907–914, 2014. DOI: 10.1007/s00231-014-1300-x.
  • C. Cattaneo, “Sur une forme de l’équation de la Chaleur éliminant le paradoxe d’une propagation instantaneée,” C.R. Acad. Sci. Paris, vol. 247, no. 3, pp. 431–433, 1958.
  • J. Liu, “Preliminary survey on the mechanisms of the wavelike behaviors of heat transfer in living tissues,” Forsch. Im Ingen., vol. 66, no. 1, pp. 1–10, 2000. DOI: 10.1007/s100100000031.
  • P.K. Gupta, J. Singh and K.N. Rai, “Numerical simulation for heat transfer in tissues during thermal therapy,” J. Therm. Biol., vol. 35, no. 6, pp. 295–301, 2010. DOI: 10.1016/j.jtherbio.2010.06.007.
  • P.K. Gupta, et al., “Solution of the heat transfer problem in tissues during hyperthermia by finite difference-decomposition method,” Appl. Math. Comput., vol. 219, no. 12, pp. 6882–6892, 2013. DOI: 10.1016/j.amc.2013.01.020.
  • J. Liu, Y.-X. Zhou and Z.-S. Deng, “Sinusoidal heating method to noninvasively measure tissue perfusion,” IEEE Trans. Biomed. Eng., vol. 49, no. 8, pp. 867–877, 2002. DOI: 10.1109/TBME.2002.800769.
  • D. Zhu, Q. Luo, G. Zhu and W. Liu, “Kinetic thermal response and damage in laser coagulation of tissue,” Lasers Surg. Med., vol. 31, no. 5, pp. 313–321, 2002. DOI: 10.1002/lsm.10108.
  • N. Fox, “Generalized thermoelasticity,” Int. J. Eng. Sci., vol. 7, no. 4, pp. 437–445, 1969. DOI: 10.1016/0020-7225(69)90077-9.
  • H.W. Lord and Y.A. Shulman, “Generalized dynamical theory of thermoelasticity,” J. Mech. Phys. Solids, vol. 15, no. 5, pp. 299–309, 1967. DOI: 10.1016/0022-5096(67)90024-5.
  • A.E. Green and K.A. Lindsay, “Thermoelasticity,” J. Elasticity, vol. 2, no. 1, pp. 1–7, 1972. DOI: 10.1007/BF00045689.
  • M.A. Ezzat and A.S. El-Karamany, “The uniqueness and reciprocity theorems for generalized thermoviscoelasticity with two relaxation times,” Int. J. Eng. Sci., vol. 40, no. 11, pp. 1275–1284, 2002. DOI: 10.1016/S0020-7225(01)00099-4.
  • A.S. El-Karamany and M.A. Ezzat, “Discontinuities in generalized thermo-viscoelasticity under four theories,” J. Therm. Stress, vol. 27, no. 12, pp. 1187–1212, 2004. DOI: 10.1080/014957390523598.
  • M.A. Ezzat, A.S. El-Karamany and A.A. Samaan, “State–space formulation to generalized thermoviscoelasticity with thermal relaxation,” J. Therm. Stress, vol. 24, no. 9, pp. 823–846, 2001. DOI: 10.1080/014957301750379612.
  • M.A. Ezzat, M.I. Othman and A.S. El-Karamany, “State space approach to generalized thermo-viscoelasticity with two relaxation times,” Int. J. Eng. Sci., vol. 40, no. 3, pp. 283–302, 2002. DOI: 10.1016/S0020-7225(01)00045-3.
  • M.I. Othman, M.A. Ezzat, S.A. Zaki and A.S. El-Karamany, “Generalized thermo-viscoelastic plane waves with two relaxation times,” Int. J. Eng. Sci., vol. 40, no. 12, pp. 1329–1347, 2002. DOI: 10.1016/S0020-7225(02)00023-X.
  • H.H. Sherief, F.A. Hamza and A.M. Abd El-Latief, “2D problem for a half-space in the generalized theory of thermo-viscoelasticity,” Mech. Time-Depend Mater., vol. 19, no. 4, pp. 557–568, 2015. DOI: 10.1007/s11043-015-9278-4.
  • I.A. Abbas and R. Kumar, “2D deformation in initially stressed thermoelastic half-space with voids,” Steel Compos. Struct., vol. 20, no. 5, pp. 1103–1117, 2016. DOI: 10.12989/scs.2016.20.5.1103.
  • B. Kumari and S. Mukhopadhyay, “Fundamental solutions of thermoelasticity with a recent heat conduction model with a single delay term,” J. Therm. Stress, vol. 40, no. 7, pp. 866–878, 2017. DOI: 10.1080/01495739.2017.1281715.
  • H.H. Sherief and E.M. Hussein, “Fundamental solution of thermoelasticity with two relaxation times for an infinite spherically symmetric space,” Z. Angew. Math. Phys., vol. 68, no. 2, pp. 50–63, 2017. DOI: 10.1007/s00033-017-0794-8.
  • M.A. Ezzat and M.Z. Abd-Elaal, “State space approach to viscoelastic fluid flow of hydromagnetic fluctuating boundary-layer through a porous medium,” Z. Angew. Math. Mech., vol. 77, no. 3, pp. 197–207, 1997. DOI: 10.1002/zamm.19970770307.
  • M.A. Ezzat and M.Z. Abd-Elaal, “Free convection effects on a viscoelastic boundary layer flow with one relaxation time through a porous medium,” J. Frank. Inst., vol. 334, no. 4, pp. 685–706, 1997. DOI: 10.1016/S0016-0032(96)00095-6.
  • M. Ghassemi and A. Shahidian, “Bioheat transfer, biosystems heat and mass transfer,” in Nano and Bio Heat Transfer and Fluid Flow, A. Shahidian and M. Ghassemi, Eds. New York, NY, USA: Elsevier, 2017, pp. 31–56.
  • K.C. Liu and H.T. Chen, “Investigation for the dual phase lag behavior of bio-heat transfer,” Int. J. Therm. Sci., vol. 49, no. 7, pp. 1138–1146, 2010. DOI: 10.1016/j.ijthermalsci.2010.02.007.
  • L. Cao, Q.H. Qin and N. Zhao, “An RBF–MFS model for analysing thermal behaviour of skin tissues,” Int. J. Heat Mass Tranf., vol. 53, no. 7–8, pp. 1298–1307, 2010. DOI: 10.1016/j.ijheatmasstransfer.2009.12.036.
  • M.A. Ezzat, A.A. El-Bary and N.S. Al-Sowayan, “Tissue responses to fractional transient heating with sinusoidal heat flux condition on skin surface,” Anim. Sci. J, vol. 87, no. 10, pp. 1304–1311, 2016. DOI: 10.1111/asj.12568.
  • A. Bawadekji, M. Amin and M.A. Ezzat, “Skin tissue responses to transient heating with memory-dependent derivative,” J. Therm. Biol, vol. 86, no. 12, pp. 102427, 2019. DOI: 10.1016/j.jtherbio.2019.102427.
  • Z. Liu and L. Bilston, “On the viscoelastic character of liver tissue: Experiments and modeling of the linear behavior,” Biorheology, vol. 37, no. 3, pp. 191–201, 2000.
  • S. Ocal, et al., “Effect of preservation period on the viscoelastic material properties of soft tissues with implications for liver transplantation,” J. Biomech. Eng., vol. 132, no. 10, pp. 101007, 2010. DOI: 10.1115/1.4002489.
  • M.M. Attar, et al., “Thermo-mechanical analysis of soft tissue in local hyperthermia treatment,” J. Mech. Sci. Technol., vol. 30, no. 3, pp. 1459–1469, 2016. DOI: 10.1007/s12206-015-1053-6.
  • G. Honig and U. Hirdes, “A method for the numerical inversion of the Laplace transform,” J. Comp. Appl. Math, vol. 10, no. 1, pp. 113–132, 1984. DOI: 10.1016/0377-0427(84)90075-X.
  • W. Shen, J. Zhang and F. Yang, “Modeling and numerical simulation of bioheat transfer and biomechanics in soft tissue,” Math. Comput. Model, vol. 41, no. 11–12, pp. 1251–1265, 2005. DOI: 10.1016/j.mcm.2004.09.006.
  • Y.C. Fung, Foundation of Solid Mechanics. Upper Saddle River, NJ, USA: Prentice Hall, 1968.
  • M.A. Ezzat, “The relaxation effects of the volume properties of electrically conducting viscoelastic material,” J. Mater. Sci. Eng. B., vol. 130, no. 1–3, pp. 11–23, 2006. DOI: 10.1016/j.mseb.2006.01.020.
  • M.A. Ezzat and A.A. El-Bary, “Effects of variable thermal conductivity on Stokes' flow of a thermoelectric fluid with fractional order of heat transfer,” Int. J. Therm Sci., vol. 100, no. 2, pp. 305–315, 2016. DOI: 10.1016/j.ijthermalsci.2015.10.008.
  • M.A. Ezzat and A.A. El-Bary, “Effects of variable thermal conductivity and fractional order of heat transfer on a perfect conducting infinitely long hollow cylinder,” Int. J. Therm. Sci., vol. 108, no. 10, pp. 62–69, 2016. DOI: 10.1016/j.ijthermalsci.2016.04.020.
  • R.B. Hetnarski, Thermal Stresses I, 2nd Series, vol. 1. North-Holland, the Netherlands: Taylor & Francis Group, LLC, 1986.
  • L.Y. Bahar and R. Hetnarski, “State space approach to thermoelasticity,” J. Therm. Stress, vol. 1, no. 1, pp. 135–145, 1978. DOI: 10.1080/01495737808926936.
  • H.H. Sherief, “State space approach to thermoelasticity with two relaxation times,” Int. J. Eng. Sci., vol. 31, no. 8, pp. 1177–1189, 1993. DOI: 10.1016/0020-7225(93)90091-8.
  • M.A. Ezzat, “State space approach to generalized magneto-thermoelasticity with two relaxation times in a medium of perfect conductivity,” Int. J. Eng. Sci., vol. 35, no. 8, pp. 741–752, 1997. DOI: 10.1016/S0020-7225(96)00112-7.
  • F. Durbin, “Numerical inversion of Laplace transforms: An effective improvement of Dubner and Abate's method,” Comput. J., vol. 17, no. 4, pp. 371–376, 1974. DOI: 10.1093/comjnl/17.4.371.
  • M. Mohajer, M.B. Ayani and H.B. Tabrizi, “Numerical study of non-Fourier heat conduction in a biolayer spherical living tissue during hyperthermia,” J. Therm. Biol, vol. 62, no. Pt B, pp. 181–188, 2016. DOI: 10.1016/j.jtherbio.2016.06.019.
  • Y. Sun, D. Fang, M. Saka and A.K. Soh, “Laser-induced vibrations of micro-beams under different boundary conditions,” Int. J. Solids Struct., vol. 45, no. 7–8, pp. 1993–2013, 2008. DOI: 10.1016/j.ijsolstr.2007.11.006.
  • M.A. Ezzat and A.A. El-Bary, “On thermo-viscoelastic infinitely long hollow cylinder with variable thermal conductivity,” Microsyst. Technol., vol. 23, no. 8, pp. 3263–3270, 2017. DOI: 10.1007/s00542-016-3101-2.
  • M.A. Ezzat and A.S. El-Karamany, “Fractional thermoelectric viscoelastic materials,” J. Appl. Polym. Sci., vol. 124, no. 3, pp. 2187–2199, 2012. DOI: 10.1002/app.35243.
  • M.A. Ezzat, A.S. El-Karamany and A.A. El-Bary, “On thermo-viscoelasticity with variable thermal conductivity and fractional-order heat transfer,” Int. J. Thermophys., vol. 36, no. 7, pp. 1684–1697, 2015. DOI: 10.1007/s10765-015-1873-8..
  • A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Tables of Integral Transforms, Vol. 1. New York, NY, USA: McGraw-Hill, 1954.

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