Identification of boundary heat flux assuring the destruction of target region of biological tissue - application of generalized dual-phase lag model and gradient method

Abstract

In the paper an axially symmetrical biological tissue domain subjected to an external heat source is analyzed. The thermal processes occurring in the domain considered are described using the generalized dual-phase lag model supplemented by the Neumann boundary conditions and the appropriate initial conditions. The problem of tissue heating is solved using the implicit scheme of the finite difference method. The obtained solution allows one to determine the local and temporary values of the Arrhenius integral. Next, the inverse problem related to the identification of the boundary heat flux assuring the postulated destruction of the tissue target region is considered. The problem is solved using the gradient method. In the final part of the paper the results of computations and the conclusions are presented.

Keywords

Bioheat transfer, Generalized dual-phase lag model, Arrhenius integral, Inverse problem, Finite difference method, Gradient method,

References

[1] M.H. Niemz, Laser–Tissue Interaction: Fundamentals and Applications, Comptes Rendusdel’ Académiedes Sciences-Series I-Mathematics, Springer-Verlag, Berlin, Heidelberg, Germany, 2002, https://doi.org/10.1007/978-3-662-04717-0
[2] S.A. Sapareto, W.C. Dewey, Thermal dose determination in cancer therapy, International Journal of Radiation Oncology Biology Physics, 10(6): 787-800, 1984, https://doi.org/10.1016/0360-3016(84)90379-1
[3] M. Jasiński, Modeling of thermal damage in laser irradiated tissue, Journal of Applied Mathematics and Computational Mechanics, 14(4): 67-78, 2015, https://doi.org/10.17512/jamcm.2015.4.07
[4] H.H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forearm, Journal of Applied Physiology, l: 93-122, 1984, https://doi.org/10.1152/jappl.1948.1.2.93
[5] M. Jamil, E.Y.K. Ng, Ranking of parameters in bioheat transfer using Taguchi analysis, International Journal of Thermal Sciences, 63: 15-21, 2013, https://doi.org/10.1016/j.ijthermalsci.2012.07.002
[6] M. Ciesielski, B. Mochnacki, Application of the control volume method using the voronoi polygons for numerical modeling of bio-heat transfer processes, Journal of Theoretical And Applied Mechanics, 52(4): 927-935, 2014, https://doi.org/10.15632/jtam-pl.52.4.927
[7] B. Mochnacki, A. Piasecka-Belkhayat, Numerical modeling of skin tissue heating using the interval finite difference method, MCB: Molecular & Cellular Biomechanics, 10(3): 233-244, 2013,
http://dx.doi.org/10.3970/mcb.2013.010.233
[8] E. Majchrzak, B. Mochnacki, M. Dziewoński, M. Jasiński, Numerical modelling of hyperthermia and hypothermia processes, Advanced Materials Research, 268-270: 257-262, 2011, https://doi.org/10.4028/www/scientific.net/AMR.268-270.257
[9] Majchrzak, E., Mochnacki, B., Jasiński, M. (2003) Numerical modelling of bioheat transfer in multi-layer skin tissue domain subjected to a flash fire, Computational Fluid and Solid Mechanics, 1(2): 1766-1770,
https://doi.org/10.4028/www.scientific.net/AMR.268-270.257
[10] W. Kaminski, Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure, Journal of Heat Transfer, 112: 555-560, 1990, https://doi.org/10.1115/1.2910422
[11] K. Mitra, S. Kumar, A. Vedavarz, M.K. Moallemi, Experimental evidence of hyperbolic heat conduction in processed meat, Journal of Heat Transfer, 117: 568-573, 1995, https://doi.org/10.1115/1.2822615
[12] P.J. Antaki, New interpretation of non-Fourier heat conduction in processed meat, Journal of Heat Transfer, 127: 189-193, 2005,
https://doi.org/10.1115/1.1844540
[13] J. Zhou, J.K. Chen, Y. Zhang, Dual phase lag effects on thermal damage to biological tissues caused by laser irradiation, Computers in Biology and Medicine, 39: 286-293, 2009, https://doi.org/10.1016/j.compbiomed.2009.01.002
[14] J. Zhou, Y. Zhang, J.K. Chen, An axisymmetric dual-phase lag bio-heat model for laser heating of living tissues, International Journal of Thermal Sciences, 48(8): 1477-1485, 2009, https://doi.org/10.1016/j.ijthermalsci.2008.12.012
[15] K.C. Liu, H.T. Chen, Investigation for the dual phase lag behavior of bioheat transfer, International Journal of Thermal Sciences, 49: 1138-1146, 2010, https://doi.org/10.1016/j.ijthermalsci.2010.02.007
[16] E. Majchrzak, Application of different variants of the BEM in numerical modeling of bioheat transfer processes, MCB: Molecular & Cellular Biomechanics, 10(3): 201-232, 2013, http://dx.doi.org/10.3970/mcb.2013.010.201
[17] E. Majchrzak, L. Turchan, The general boundary element method for 3D dual-phase lag model of bioheat transfer, Engineering Analysis with Boundary Elements, 50: 76-82, 2015, https://doi.org/10.1016/j.enganabound.2014.07.012
[18] B. Mochnacki, E. Majchrzak, Numerical model of thermal interactions between cylindrical cryoprobe and biological tissue using the dual-phase lag equation, International Journal of Heat and Mass Transfer, 108: 1-10, 2017, https://doi.org/10.1016/j.ijheatmasstransfer.2016.11.103
[19] Y. Zhang, Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues, International Journal of Heat and Mass Transfer, 52: 4829-4834, 2009,
https://doi.org/10.1016/j.ijheatmasstransfer.2009.06.007
[20] N. Afrin, J. Zhou, Y. Zhang, D.Y. Tzou, J.K. Chen, Numerical simulation of thermal damage to living biological tissues induced by laser irradiation based on a generalized dual phase lag model, Numerical Heat Transfer, Part A: Applications, 61(7): 483-501, 2012, https://doi.org/10.1080/10407782.2012.667648
[21] E. Majchrzak, L. Turchan, J. Dziatkiewicz, Modeling of skin tissue heating using the generalized dual-phase lag equation, Archives of Mechanics, 67(6): 417-437, 2015.
[22] M. Jasiński, E. Majchrzak, L. Turchan, Numerical analysis of the interactions between laser and soft tissues using dual-phase lag model, Applied Mathematical Modeling, 40(2): 750-762, 2016, https://doi.org/10.1016/j.apm.2015.10.025
[23] A.R.A. Khaled, K. Vafai, The role of porous media in modeling of flow and heat transfer in biological tissues, International Journal of Heat and Mass Transfer, 46: 4989-5003, 2003, https://doi.org/10.1016/S0017-9310(03)00301-6
[24] A. Nakayama, F. Kuwahara, A general bioheat transfer model based on the theory of porous media, International Journal of Heat and Mass Transfer, 51: 3190-3199, 2008, https://doi.org/10.1016/j.ijheatmasstransfer.2007.05.030
[25] E. Majchrzak, Ł. Turchan, Numerical analysis of tissue heating using the bioheat transfer porous model, Computer Assisted Methods in Engineering and Science, 20: 123-111, 2013.
[26] K. Kurpisz, A.J. Nowak, Inverse thermal problems, Computational Mechanics Publications, Southampton, Boston 1995.
[27] M. Paruch, Identification of the cancer ablation parameters during RF hyperthermia using gradient, evolutionary and hybrid algorithms, International Journal of Numerical Methods for Heat & Fluid Flow, 27(3): 674-697, https://doi.org/10.1108/HFF-03-2016-0114
[28] E. Majchrzak, B. Mochnacki, Sensitivity analysis and inverse problems in bio-heat transfer modelling, Computer Assisted Mechanics and Engineering Sciences, 13: 85-108, 2006.
[29] M. Kleiber, Parameter Sensitivity in Non-linear Mechanics, J. Willey & Sons, London, 1997.
[30] K. Dems, B. Rousselet, Sensitivity analysis for transient heat conduction in a solid body – Part I: External boundary modification, Structural Optimization, 17: 36-45, 1999, https://doi.org/10.1007/BF01197711
[31] G. Kaluza, E. Majchrzak, L. Turchan, Sensitivity analysis of temperature field in the heated soft tissue with respect to the perturbations of porosity, Applied Mathematical Modelling, 49: 498-513, 2017,
https://doi.org/10.1016/j.apm.2017.05.011
[32] L. Turchan, E. Majchrzak, Identification of Neumann boundary condition assuring the destruction of target region of biological tissue, Proceedings of the International Conference of Computational Methods, 5: 835-846, 2018.
[33] M. Paruch, L. Turchan, Mathematical Modelling of the Destruction Degree of Cancer Under the Influence of a RF Hyperthermia, AIP Conference Proceedings 1922, Lublin, Poland, 2017, https://doi.org/10.1063/1.5019064
[34] C. Junmeng, H. Fang, Y. Weiming, Y. Fusheng, A new formula approximating the Arrhenius integral to perform the nonisothermal kinetics, Chemical Engineering Journal, 124: 15–18, 2006, https://doi.org/10.1016/j.cej.2006.08.003
[35] A. Szasz, N. Szasz, O. Szasz, Oncothermia: Principles and Practices, Springer, Heidelberg, Germany, 2011, https://doi.org/10.1007/978-90-481-9498-8
[36] H.W. Huang, Z.P. Chen, R.B. Roemer, A counter current vascular network model of heat transfer in tissues, Journal of Biomechanical Engineering, 118: 120-129, 1996, https://doi.org/10.1115/1.2795937
[37] S. Hassanpour, A.Saboonchi, Validation of local thermal equilibrium assumption in a vascular tissue during interstitial hyperthermia treatment, Journal of Mechanics in Medicine and Biology, 17(5): 1750087 (26 pages), 2017, https://doi.org/10.1142/S0219519417500877
[38] T. Burczyński, W. Kus, A. Dlugosz, P. Orantek, Optimization and defect identification using distributed evolutionary algorithms, Engineering Applications of Artificial Intelligence, 17(4): 337-344, 2004, https://doi.org/10.1016/j.engappai.2004.04.007
[39] H. Li, An evolutionary algorithm for multi-criteria inverse optimal value problems using a bilevel optimization model, Applied Soft Computing, 23: 308-318, 2014, https://doi.org/10.1016/j.asoc.2014.06.044
[40] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, Berlin, 1996, https://doi.org/10.1007/978-3-662-03315-9
[41] M. Paruch, Identification of the degree of tumor destruction on the basis of the Arrhenius integral using the evolutionary algorithm, International Journal of Thermal Sciences, 130: 507-517, 2018,
https://doi.org/10.1016/j.ijthermalsci.2018.05.015
[42] E.G. Baquela, A.C. Olivera, A novel hybrid multi-objective metamodel-based evolutionary optimization algorithm, Operations Research Perspectives, 6: 100098, 2019, https://doi.org/10.1016/j.orp.2019.100098
[43] F. Han, Q. Liu, A diversity-guided hybrid particle swarm optimization based on gradient search, Neurocomputing, 137(5): 234-240, 2014,
https://doi.org/10.1016/j.neucom.2013.03.074
How to Cite
TURCHAN, Lukasz; MAJCHRZAK, Ewa. Identification of boundary heat flux assuring the destruction of target region of biological tissue - application of generalized dual-phase lag model and gradient method. Computer Assisted Methods in Engineering and Science, [S.l.], v. 26, n. 1, p. 21-34, aug. 2019. ISSN 2299-3649. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/247>. Date accessed: 11 may 2021.
Section
Articles