Steel and Composite Structures

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Analytical solutions of the temperature increment in skin tissues caused by moving heating sources

  • Hobiny, Aatef D. (Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Mathematics Department, King Abdulaziz University) ;
  • Abbas, Ibrahim A. (Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Mathematics Department, King Abdulaziz University)
  • Received : 2020.10.22
  • Accepted : 2021.07.12
  • Published : 2021.08.25
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Abstract

In this paper, mathematical bioheat transfer model in skin tissues in the bounded domain due to moving heat source are considered. The thermal damage to the tissues is totally evaluated by the denatured protein ranges by the Arrhenius formulation. The temporal complete solutions in Laplace time domain obtained by using the inversion scheme of the Laplace transform, to obtain the general solution (exact solution) for the increment of temperature. The numerical result of temperature and the thermal injurie are graphically demonstrated. In conclusions, parametric analysis are devoted to the identifications of appropriates procedures for choosing serious designs variables to reach the effectives thermal in hyperthermias treatments.

Keywords

Acknowledgement

The Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia funded this project, under grant no. (FP-142-42).

References

  1. Abbas, I. (2006), "Natural frequencies of a poroelastic hollow cylinder", Acta Mechanica, 186(1-4), 229-237. https://doi.org/10.1007/s00707-006-0314-y.
  2. Abbas, I.A. (2007), "Finite element analysis of the thermoelastic interactions in an unbounded body with a cavity", Forschung im Ingenieurwesen. 71(3-4), 215-222. https://doi.org/10.1007/s10010-007-0060-x.
  3. Abbas, I.A. (2014), "The effects of relaxation times and a moving heat source on a two-temperature generalized thermoelastic thin slim strip", Can. J. Phys., 93(5), 585-590. https://doi.org/10.1139/cjp-2014-0387.
  4. Abbas, I.A., Abo-El-Nour, N. and Othman, M.I. (2011), "Generalized magneto-thermoelasticity in a fiber-reinforced anisotropic half-space", Int. J. Thermophys., 32(5), 1071-1085. https://doi.org/10.1007/s10765-011-0957-3.
  5. Abbas, I.A. and Alzahrani, F.S. (2016), "Analytical solution of a two-dimensional thermoelastic problem subjected to laser pulse", Steel Compos. Struct., 21(4), 791-803. https://doi.org/10.12989/scs.2016.21.4.791.
  6. Abbas, I.A. and Kumar, R. (2016), "2D deformation in initially stressed thermoelastic half-space with voids", Steel Compos. Struct., 20(5), 1103-1117. https://doi.org/10.12989/scs.2016.20.5.1103.
  7. Abbas, I.A. and Youssef, H.M. (2013), "Two-temperature generalized thermoelasticity under ramp-type heating by finite element method", Meccanica, 48(2), 331-339. https://doi.org/10.1007/s11012-012-9604-8
  8. Abbas, I.A. and Zenkour, A.M. (2014), "Dual-phase-lag model on thermoelastic interactions in a semi-infinite medium subjected to a ramp-type heating", J. Comput. Theor. Nanosci., 11(3), 642-645. https://doi.org/10.1166/jctn.2014.3407.
  9. Ahmadikia, H., Fazlali, R. and Moradi, A. (2012), "Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue", Int. Commun. Heat Mass Transfer, 39(1), 121-130. https://doi.org/10.1016/j.icheatmasstransfer.2011.09.016.
  10. Andreozzi, A., Brunese, L., Iasiello, M., Tucci, C. and Vanoli, G.P. (2019), "Modeling heat transfer in tumors: a review of thermal therapies", Annal. Bio. Eng., 47(3), 676-693. https://doi.org/10.1007/s10439-018-02177-x.
  11. Andreozzi, A., Iasiello, M. and Netti, P.A. (2019), "A thermoporoelastic model for fluid transport in tumour tissues", J. Roy. Soc. Interface, 16(154), 20190030. https://doi.org/10.1098/rsif.2019.0030.
  12. Biot, M.A. (1941), "General theory of three-dimensional consolidation", J. Appl. Phys., 12(2), 155-164. http://dx.doi.org/10.1063/1.1712886.
  13. Biot, M.A. (1956), "Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range", J. Acoust. Soc. Am., 28(2), 179-191. https://doi.org/10.1121/1.1908241.
  14. Charny, C.K. (1992), Mathematical models of bioheat transfer, Elsevier
  15. Debnath, L. and Bhatta, D. (2014), Integral transforms and their applications, Chapman and Hall/CRC.
  16. Eftekhari, S.A. (2018), "A coupled ritz-finite element method for free vibration of rectangular thin and thick plates with general boundary conditions", Steel Compos. Struct., 28(6), 655-670. http://dx.doi.org/10.12989/scs.2018.28.6.655.
  17. Egred, M. and Brilakis, E.S. (2020), "Excimer laser coronary angioplasty (ELCA): fundamentals, mechanism of action, and clinical applications", J. Invasive Cardiol, 32(2), 27-35.
  18. Ezzat, M. and El-Bary, A. (2017), "Fractional magneto-thermoelastic materials with phase-lag Green-Naghdi theories", Steel Compos. Struct., 24(3), 297-307. http://dx.doi.org/10.12989/scs.2017.24.3.297.
  19. Ezzat, M.A. and El-Bary, A.A. (2017), "A functionally graded magneto-thermoelastic half space with memory-dependent derivatives heat transfer", Steel Compos. Struct., 25(2), 177-186. http://dx.doi.org/10.12989/scs.2017.25.2.177
  20. Gabay, I., Abergel, A., Vasilyev, T., Rabi, Y., Fliss, D.M. and Katzir, A. (2011), "Temperature-controlled two-wavelength laser soldering of tissues", Laser. Surg. Med.. 43(9), 907-913. http://dx.doi.org/10.1002/lsm.21123.
  21. Gonzalez-Suarez, A. and Berjano, E. (2015), "Comparative analysis of different methods of modeling the thermal effect of circulating blood flow during RF cardiac ablation", IEEE T. Bio. Eng., 63(2), 250-259. https://dx.doi.org/10.1109/TBME.2015.2451178.
  22. Green, A. and Naghdi, P. (1993), "Thermoelasticity without energy dissipation", J. Elasticity, 31(3), 189-208. https://doi.org/10.1007/BF00044969.
  23. Green, A.E. and Naghdi, P.M. (1991), "A re-examination of the basic postulates of thermomechanics", Proc. Roy. Soc. London. Series A: Math. Phys. Sci., 432(1885), 171-194. https://doi.org/10.1098/rspa.1991.0012.
  24. Hassan, M., Marin, M., Ellahi, R. and Alamri, S.Z. (2018), "Exploration of convective heat transfer and flow characteristics synthesis by Cu-Ag/water hybrid-nanofluids", Heat Transfer Res., 49(18). https://dx.doi.org/10.1615/HeatTransRes.2018025569.
  25. Henriques Jr, F. and Moritz, A. (1947), "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. Pathology. 23(4), 530.
  26. Ho, Y.J., Wu, C.C., Hsieh, Z.H., Fan, C.H. and Yeh, C.K. (2018), "Thermal-sensitive acoustic droplets for dual-mode ultrasound imaging and drug delivery", J. Controlled Release, 291 26-36. https://doi.org/10.1016/j.jconrel.2018.10.016.
  27. Hobiny, A. and Abbas, I. (2019), "Analytical solutions of fractional bioheat model in a spherical tissue", Mech. Based Des. Struct. Mach., 1-10. https://doi.org/10.1080/15397734.2019.1702055.
  28. Hobiny, A., Alzahrani, F.S. and Abbas, I. (2020), "Three-phase lag model of thermo-elastic interaction in a 2D porous material due to pulse heat flux", Int. J. Numer. Method. Heat Fluid Fl., https://doi.org/10.1108/HFF-03-2020-0122.
  29. Iasiello, M., Andreozzi, A., Bianco, N. and Vafai, K. (2019), "The porous media theory applied to radiofrequency catheter ablation", Int. J. Numer. Method. Heat Fluid Fl., https://doi.org/10.1108/HFF-11-2018-0707.
  30. Iasiello, M., Vafai, K., Andreozzi, A. and Bianco, N. (2019), "Hypo-and hyperthermia effects on LDL deposition in a curved artery", Comput. Therm. Sci., 11(1-2). https://dx.doi.org/10.1615/ComputThermalScien.2018024754
  31. Kahya, V. and Turan, M. (2018), "Vibration and buckling of laminated beams by a multi-layer finite element model", Steel Compos. Struct., 28(4), 415-426. https://doi.org/10.12989/scs.2018.28.4.415.
  32. Karageorghis, A., Lesnic, D. and Marin, L. (2014), "A moving pseudo-boundary MFS for void detection in two-dimensional thermoelasticity", Int. J. Mech. Sci., 88, 276-288. https://doi.org/10.1016/j.ijmecsci.2014.05.015.
  33. Kaur, H. and Lata, P. (2020), "Effect of thermal conductivity on isotropic modified couple stress thermoelastic medium with two temperatures", Steel Compos. Struct., 34(2), 309-319. https://doi.org/10.12989/scs.2020.34.2.309.
  34. Kumar, C.S. and Mohammad, F. (2011), "Magnetic nanomaterials for hyperthermia-based therapy and controlled drug delivery", Adv. Drug Delivery Reviews, 63(9), 789-808. https://doi.org/10.1016/j.addr.2011.03.008.
  35. Kumar, D. and Rai, K. (2020), "Three-phase-lag bioheat transfer model and its validation with experimental data", Mech. Based Des. Struct. Mach., 1-15. https://doi.org/10.1080/15397734.2020.1779741.
  36. Kumar, R. and Chawla, V. (2013), "Reflection and refraction of plane wave at the interface between elastic and thermoelastic media with three-phase-lag model", Int. Commun. Heat Mass Transfer, 48, 53-60. https://doi.org/10.1016/j.icheatmasstransfer.2013.08.013.
  37. Kumar, R., Sharma, N. and Lata, P. (2016), "Thermomechanical interactions due to hall current in transversely isotropic thermoelastic with and without energy dissipation with two temperatures and rotation", J. Solid Mech., 8(4), 840-858.
  38. Labonte, S. (1994), "Numerical model for radio-frequency ablation of the endocardium and its experimental validation", IEEE T. Biomed. Eng., 41(2), 108-115. https://doi.org/10.1109/10.284921.
  39. Lata, P. (2018), "Effect of energy dissipation on plane waves in sandwiched layered thermoelastic medium", Steel Compos. Struct., 27(4), 439-451. http://dx.doi.org/10.12989/scs.2018.27.4.439.
  40. Lata, P. and Kaur, I. (2019), "Thermomechanical interactions in transversely isotropic magneto thermoelastic solid with two temperatures and without energy dissipation", Steel Compos. Struct., 32(6), 779-793. http://dx.doi.org/10.12989/scs.2019.32.6.779.
  41. Lata, P., Kumar, R. and Sharma, N. (2016), "Plane waves in an anisotropic thermoelastic", Steel Compos. Struct., 22(3), 567-587. https://doi.org/10.12989/scs.2016.22.3.567.
  42. Mahjoob, S. and Vafai, K. (2009), "Analytical characterization of heat transport through biological media incorporating hyperthermia treatment", Int. J. Heat Mass Transfer., 52(5-6), 1608-1618. https://doi.org/10.1016/j.ijheatmasstransfer.2008.07.038.
  43. Marin, M. (1997), "Cesaro means in thermoelasticity of dipolar bodies", Acta mechanica. 122(1-4), 155-168. https://doi.org/10.1007/BF01181996.
  44. Marin, M. (1999), "An evolutionary equation in thermoelasticity of dipolar bodies", J. Math. Phys., 40(3), 1391-1399. https://doi.org/10.1063/1.532809.
  45. Marin, M. and Craciun, E. (2017), "Uniqueness results for a boundary value problem in dipolar thermoelasticity to model composite materials", Compos. Part B: Eng., 126, 27-37. https://doi.org/10.1016/j.compositesb.2017.05.063.
  46. Marin, M. and Ochsner, A. (2017), "The effect of a dipolar structure on the Holder stability in Green-Naghdi thermoelasticity", Continuum Mech. Therm., 29(6), 1365-1374. https://doi.org/10.1007/s00161-017-0585-7.
  47. Marin, M., Vlase, S. and Paun, M. (2015), "Considerations on double porosity structure for micropolar bodies", Aip Adv., 5(3), 037113. https://doi.org/10.1063/1.4914912.
  48. Mitchell, J.W., Galvez, T.L., Hengle, J., Myers, G.E. and Siebecker, K.L. (1970), "Thermal response of human legs during cooling", J. Appl. Physiology, 29(6), 859-865. https://doi.org/10.1152/jappl.1970.29.6.859.
  49. Mondal, S., Sur, A. and Kanoria, M. (2019), "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., 1-15. https://doi.org/10.1080/15397734.2019.1686992.
  50. Moritz, A.R. and Henriques Jr, F. (1947), "Studies of thermal injury: II. The relative importance of time and surface temperature in the causation of cutaneous burns", Am. J. Pathology, 23(5), 695.
  51. Noroozi, M.J. and Goodarzi, M. (2017), "Nonlinear analysis of a non-Fourier heat conduction problem in a living tissue heated by laser source", Int. J. Biomathematics, 10(8), 1750107. https://doi.org/10.1142/S1793524517501078.
  52. Othman, M.I. and Abbas, I.A. (2012), "Generalized thermoelasticity of thermal-shock problem in a nonhomogeneous isotropic hollow cylinder with energy dissipation", Int. J. Thermophys., 33(5), 913-923. https://doi.org/10.1007/s10765-012-1202-4.
  53. Othman, M.I. and Marin, M. (2017), "Effect of thermal loading due to laser pulse on thermoelastic porous medium under GN theory", Results in Phys., 7, 3863-3872. https://doi.org/10.1016/j.rinp.2017.10.012.
  54. Pennes, H.H. (1948), "Analysis of tissue and arterial blood temperatures in the resting human forearm", J. Appl. Physiology, 1(2), 93-122. https://doi.org/10.1152/jappl.1948.1.2.93.
  55. Quintanilla, R. and Racke, R. (2008), "A note on stability in three-phase-lag heat conduction", Int. J. Heat Mass Transfer., 51(1-2), 24-29. https://doi.org/10.1016/j.ijheatmasstransfer.2007.04.045.
  56. Saeed, T. and Abbas, I. (2020), "Finite element analyses of nonlinear DPL bioheat model in spherical tissues using experimental data", Mech. Based Des. Struct. Mach., 1-11. https://doi.org/10.1080/15397734.2020.1749068.
  57. Stehfest, H. (1970), "Algorithm 368: Numerical inversion of Laplace transforms [D5]", Commun. ACM. 13(1), 47-49. https://doi.org/10.1145/361953.361969.
  58. Xu, F., Seffen, K. and Lu, T. (2008), "Non-Fourier analysis of skin biothermomechanics", Int. J. Heat Mass Transfer., 51(9), 2237-2259. https://doi.org/10.1016/j.ijheatmasstransfer.2007.10.024.
  59. Zenkour, A.M. and Abbas, I.A. (2014), "A generalized thermoelasticity problem of an annular cylinder with temperature-dependent density and material properties", Int. J. Mech. Sci., 84, 54-60. https://doi.org/10.1016/j.ijmecsci.2014.03.016.
  60. Zhou, J., Chen, J. and Zhang, Y. (2009), "Dual-phase lag effects on thermal damage to biological tissues caused by laser irradiation", Comput. Biology Medicine, 39(3), 286-293. https://doi.org/10.1016/j.compbiomed.2009.01.002.