Modeling of Photochemical and Photothermal Effects in Soft Tissue Subjected to Laser Irradiation
Abstract
The purpose of this study is to analyze the phenomena that occur in biological tissue during photodynamic therapy (PDT). Under the influence of the laser, triplet oxygen is transformed into singlet oxygen, which is cytotoxic to cancer tissue. The impact of the laser on the tissue may also be accompanied by changes in the thermophysical parameters, e.g., perfusion, which can affect the supply of oxygen to the tissue and, consequently, the outcome of the therapy. The proposed model uses the optical diffusion equation, the Pennes bioheat transfer equation, and reactions equations for PDT. The connection between bioheat transfer and PDT models is taken into account through the respective relationships between perfusion rate, capillary blood velocity, and the maximum oxygen supply rate. Furthermore, a method is proposed to model abnormal vascular patterns in the tumor subdomain. The boundary element method and the finite difference method were used in the numerical implementation stage.
Keywords:
bioheat transfer, optical diffusion equation, photodynamic therapy, boundary element method, finite difference methodReferences
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10. M. Paruch, Mathematical modeling of breast tumor destruction using fast heating during radiofrequency ablation, Materials, 13(1): 136, 2020, https://doi.org/10.3390/MA13010136
11. T.W. Secomb, J.P. Alberding, R. Hsu, M.W. Dewhirst, A.R. Pries, Angiogenesis: An adaptive dynamic biological patterning problem, PLoS Computational Biology, 9(3): e1002983, 2013, https://doi.org/10.1371/journal.pcbi.1002983
12. D. Goldman, Theoretical models of microvascular oxygen transport to tissue, Microcirculation, 15(8): 795–811, 2008, https://doi.org/10.1080/10739680801938289
13. L.A. Dombrovsky, D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, New York, 2010.
14. S.L. Jacques, B.W. Pogue, Tutorial on diffuse light transport, Journal of Biomedical Optics, 13(4): 041302, 2008, https://doi.org/10.1117/1.2967535
15. L.A. Dombrovsky, The use of transport approximation and diffusion-based models in radiative transfer calculations, Computational Thermal Sciences, 4(4): 297–315, 2012, https://doi.org/10.1615/ComputThermalScien.2012005050
16. L.A. Dombrovsky, J.H. Randrianalisoa, W. Lipinski, V. Timchenko, Simplified approaches to radiative transfer simulations in laser-induced hyperthermia of superficial tumors, Computational Thermal Sciences, 5(6): 521–530, 2013, https://doi.org/10.1615/Comput-ThermalScien.2013008157
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21. R.A. El-Nabulsi, Fractal Pennes and Cattaneo–Vernotte bioheat equations from productlike fractal geometry and their implications on cells in the presence of tumour growth, Journal of the Royal Society Interface, 18(182): 20210564, 2021, https://doi.org/10.1098/RSIF.2021.0564
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25. S. Hassanpour, A. Saboonchi, Modeling of heat transfer in a vascular tissue-like medium during an interstitial hyperthermia process, Journal of Thermal Biology, 62: 150–158, 2016, https://doi.org/10.1016/j.jtherbio.2016.06.022
26. E. Majchrzak, L. Turchan, M. Jasiński, Identification of laser intensity assuring the destruction of target region of biological tissue using the gradient method and generalized dual-phase lag equation, Iranian Journal of Science and Technology – Transactions of Mechanical Engineering, 43: 539–548, 2019, https://doi.org/10.1007/s40997-018-0225-2
27. T. Saeed, I. Abbas, Finite element analyses of nonlinear DPL bioheat model in spherical tissues using experimental data, Mechanics Based Design of Structures and Machines, 50(4): 1287–1297, 2022, https://doi.org/10.1080/15397734.2020.1749068
28. F. Alzahrani, I. Abbas, A numerical solution of nonlinear DPL bioheat model in biological tissue due to laser irradiations, Indian Journal of Physics, 96(2): 377–383, 2022, https://doi.org/10.1007/s12648-020-01988-w
29. R.K. Chaudhary, D. Kumar, K.N. Rai, J. Singh, Analysis of thermal injuries using classical Fourier and DPL models for multi-layer of skin under different boundary conditions, International Journal of Biomathematics, 14(6): 2150040, 2021, https://doi.org/10.1142/S1793524521500406
30. R.K. Chaudhary, D. Kumar, K.N. Rai, J. Singh, Numerical simulation of the skin tissue subjected to hyperthermia treatment using a nonlinear DPL model, Thermal Science and Engineering Progress, 34: 101394, 2022, https://doi.org/10.1016/J.tsep.2022.101394
31. E. Majchrzak, Ł. Turchan, J. Dziatkiewicz, Modeling of skin tissue heating using the generalized dual phase-lag equation, Archives of Mechanics, 67(6): 417–437, 2015, https://doi.org/10.24423/aom.1777
32. E. Majchrzak, G. Kałuża, Sensitivity analysis of temperature in heated soft tissues with respect to time delays, Continuum Mechanics and Thermodynamics, 34(2): 587–599, 2021, https://doi.org/10.1007/s00161-021-01075-3
33. 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 Applications, Part A: Applications, 61(7): 483–501, 2012, https://doi.org/10.1080/10407782.2012.667648
34. E. Majchrzak, M. Stryczyński, Dual-phase lag model of heat transfer between blood vessel and biological tissue, Mathematical Biosciences and Engineering: MBE, 18(2): 1573–1589, 2021, https://doi.org/10.3934/MBE.2021081
35. T.N. Glenn, S. Rastegar, S.L. Jacques, Finite element analysis of temperature controlled coagulation in laser irradiated tissue, IEEE Transactions on Biomedical Engineering, 43(1): 79–87, 1996, https://doi.org/10.1109/10.477703
36. B.J. McGuire, T.W. Secomb, A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand, Journal of Applied Physiology, 91(5): 2255–2265, 2001, https://doi.org/10.1152/jappl.2001
37. M. Jasiński, Numerical analysis of the temperature impact to the oxygen distribution in the biological tissue, Journal of Applied Mathematics and Computational Mechanics, 19(3): 17–28, 2020, https://doi.org/10.17512/jamcm.2020.3.02
38. J.P. Whiteley, D.J. Gavaghan, C.E.W. Hahn, Mathematical modelling of oxygen transport to tissue, Journal of Mathematical Biology, 44(6): 503–522, 2002, https://doi.org/10.1007/s002850200135
39. B.C. Fry, T.K. Roy, T.W. Secomb, Capillary recruitment in a theoretical model for blood flow regulation in heterogeneous microvessel networks, Physiological Reports, 1(3): e00050, 2013, https://doi.org/10.1002/phy2.50
40. A. Korczak, M. Jasiński, Modelling of biological tissue damage process with application of interval arithmetic, Journal of Theoretical and Applied Mechanics, 57(1): 249–261, 2019, https://doi.org/10.15632/jtam-pl.57.1.249
41. B. Mochnacki, A. Piasecka Belkhayat, Numerical modeling of skin tissue heating using the interval finite difference method, Molecular & Cellular Biomechanics, 10(3): 233–244, 2013, https://doi.org/10.3970/mcb.2013.010.233
42. C.A. Brebbia, J. Domínguez, Boundary Elements: An Introductory Course, 2nd ed., WIT Press, London, 1992.
43. E. Majchrzak, Ł. 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
44. K.K.H. Wang, J.C. Finlay, T.M. Busch, S.M. Hahn, T.C. Zhu, Explicit dosimetry for photodynamic therapy: Macroscopic singlet oxygen modeling, Journal of Biophotonics, 3(5–6): 304–318, 2010, https://doi.org/10.1002/jbio.200900101
45. E. Majchrzak, M. Jasiński, Ł. Turchan, Modeling of laser-soft tissue interactions using the dual-phase lag equation: Sensitivity analysis to selected tissue parameters, Defect and Diffusion Forum, 379: 108–123, 2017, https://doi.org/10.4028/www.scientific.net/DDF.379.108
2. M.H. Niemz, Laser-Tissue Interactions: Fundamentals and Applications, Springer-Verlag, Berlin, Heidelberg, 2007.
3. J.F. Algorri, M. Ochoa, P. Roldán-Varona, L. Rodríguez-Cobo, J.M. López-Higuera, Light technology for efficient and effective photodynamic therapy: A critical review, Cancers, 13(14): 3484, 2021, https://doi.org/10.3390/cancers13143484
4. T.C. Zhu, M.M. Kim, X. Liang, J.C. Finlay, T.M. Busch, In-vivo singlet oxygen threshold doses for PDT, Photonics and Lasers in Medicine, 4(1): 59–71, 2015, https://doi.org/10.1515/PLM-2014-0037
5. T.C. Zhu, B. Liu, R. Penjweini, Study of tissue oxygen supply rate in a macroscopic photodynamic therapy singlet oxygen model, Journal of Biomedical Optics, 20(3): 038001, 2015, https://doi.org/10.1117/1.jbo.20.3.038001
6. M.M. Kim, A. Darafsheh, Light sources and dosimetry techniques for photodynamic therapy, Photochemistry and Photobiology, 96(2): 280–294, 2020, https://doi.org/10.1111/php.13219
7. T. Sheng, Y. Ong, W. Guo, T.C. Zhu, Reactive oxygen species explicit dosimetry to predict tumor growth for benzoporphyrin derivative-mediated vascular photodynamic therapy, Journal of Biomedical Optics, 25(6): 063805, 2020, https://doi.org/10.1117/1.jbo.25.6.063805
8. M. Jasiński, M. Zadoń, Mathematical modeling of the phenomena that occur in a biological tissue containing a photosensitizer, Journal of Applied Mathematics and Computational Mechanics, 21(4): 40–51, 2022, https://doi.org/10.17512/jamcm.2022.4.04
9. J.P. Abraham, E.M. Sparrow, A thermal-ablation bioheat model including liquid-to-vapor phase change, pressure- and necrosis-dependent perfusion, and moisture-dependent properties, International Journal of Heat and Mass Transfer, 50(13–14): 2537–2544, 2007, https://doi.org/10.1016/j.ijheatmasstransfer.2006.11.045
10. M. Paruch, Mathematical modeling of breast tumor destruction using fast heating during radiofrequency ablation, Materials, 13(1): 136, 2020, https://doi.org/10.3390/MA13010136
11. T.W. Secomb, J.P. Alberding, R. Hsu, M.W. Dewhirst, A.R. Pries, Angiogenesis: An adaptive dynamic biological patterning problem, PLoS Computational Biology, 9(3): e1002983, 2013, https://doi.org/10.1371/journal.pcbi.1002983
12. D. Goldman, Theoretical models of microvascular oxygen transport to tissue, Microcirculation, 15(8): 795–811, 2008, https://doi.org/10.1080/10739680801938289
13. L.A. Dombrovsky, D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, New York, 2010.
14. S.L. Jacques, B.W. Pogue, Tutorial on diffuse light transport, Journal of Biomedical Optics, 13(4): 041302, 2008, https://doi.org/10.1117/1.2967535
15. L.A. Dombrovsky, The use of transport approximation and diffusion-based models in radiative transfer calculations, Computational Thermal Sciences, 4(4): 297–315, 2012, https://doi.org/10.1615/ComputThermalScien.2012005050
16. L.A. Dombrovsky, J.H. Randrianalisoa, W. Lipinski, V. Timchenko, Simplified approaches to radiative transfer simulations in laser-induced hyperthermia of superficial tumors, Computational Thermal Sciences, 5(6): 521–530, 2013, https://doi.org/10.1615/Comput-ThermalScien.2013008157
17. M. Friebel, A. Roggan, G.J. Müller, M.C. Meinke, Determination of optical properties of human blood in the spectral range 250 to 1100 nm using Monte Carlo simulations with hematocrit-dependent effective scattering phase functions, Journal of Biomedical Optics, 11(3): 034021, 2006, https://doi.org/10.1117/1.2203659
18. A. Paul, A. Paul, Computational study of photo-thermal ablation of large blood vessel embedded tumor using localized injection of gold nanoshells, Journal of Thermal Biology, 78: 329–342, 2018, https://doi.org/10.1016/j.jtherbio.2018.10.021
19. Y. He, M. Shirazaki, H. Liu, R. Himeno, Z. Sun, A numerical coupling model to analyze the blood flow, temperature, and oxygen transport in human breast tumor under laser irradiation, Computers in Biology and Medicine, 36(12): 1336–1350, 2006, https://doi.org/10.1016/j.compbiomed.2005.08.004
20. M. Jasiński, Numerical analysis of thermal damage and oxygen distribution in laser irradiated tissue, Journal of Applied Mathematics and Computational Mechanics, 21(2): 51–62, 2022, https://doi.org/10.17512/jamcm.2022.2.05
21. R.A. El-Nabulsi, Fractal Pennes and Cattaneo–Vernotte bioheat equations from productlike fractal geometry and their implications on cells in the presence of tumour growth, Journal of the Royal Society Interface, 18(182): 20210564, 2021, https://doi.org/10.1098/RSIF.2021.0564
22. R.A. El-Nabulsi, W. Anukool, Nonlocal thermal effects on biological tissues and tumors, Thermal Science and Engineering Progress, 34: 101424, 2022, https://doi.org/10.1016/j.tsep.2022.101424
23. B. Mochnacki, M. Ciesielski, Sensitivity of transient temperature field in domain of forearm insulated by protective clothing with respect to perturbations of external boundary heat flux, Bulletin of the Polish Academy of Sciences: Technical Sciences, 64(3): 591–598, 2016, https://doi.org/10.1515/bpasts-2016-0066
24. S.C. Akula, R. Maniyeri, Numerical simulation of bioheat transfer: a comparative study on hyperbolic and parabolic heat conduction, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42(62): 1–13, 2020, https://doi.org/10.1007/s40430-019-2132-x
25. S. Hassanpour, A. Saboonchi, Modeling of heat transfer in a vascular tissue-like medium during an interstitial hyperthermia process, Journal of Thermal Biology, 62: 150–158, 2016, https://doi.org/10.1016/j.jtherbio.2016.06.022
26. E. Majchrzak, L. Turchan, M. Jasiński, Identification of laser intensity assuring the destruction of target region of biological tissue using the gradient method and generalized dual-phase lag equation, Iranian Journal of Science and Technology – Transactions of Mechanical Engineering, 43: 539–548, 2019, https://doi.org/10.1007/s40997-018-0225-2
27. T. Saeed, I. Abbas, Finite element analyses of nonlinear DPL bioheat model in spherical tissues using experimental data, Mechanics Based Design of Structures and Machines, 50(4): 1287–1297, 2022, https://doi.org/10.1080/15397734.2020.1749068
28. F. Alzahrani, I. Abbas, A numerical solution of nonlinear DPL bioheat model in biological tissue due to laser irradiations, Indian Journal of Physics, 96(2): 377–383, 2022, https://doi.org/10.1007/s12648-020-01988-w
29. R.K. Chaudhary, D. Kumar, K.N. Rai, J. Singh, Analysis of thermal injuries using classical Fourier and DPL models for multi-layer of skin under different boundary conditions, International Journal of Biomathematics, 14(6): 2150040, 2021, https://doi.org/10.1142/S1793524521500406
30. R.K. Chaudhary, D. Kumar, K.N. Rai, J. Singh, Numerical simulation of the skin tissue subjected to hyperthermia treatment using a nonlinear DPL model, Thermal Science and Engineering Progress, 34: 101394, 2022, https://doi.org/10.1016/J.tsep.2022.101394
31. E. Majchrzak, Ł. Turchan, J. Dziatkiewicz, Modeling of skin tissue heating using the generalized dual phase-lag equation, Archives of Mechanics, 67(6): 417–437, 2015, https://doi.org/10.24423/aom.1777
32. E. Majchrzak, G. Kałuża, Sensitivity analysis of temperature in heated soft tissues with respect to time delays, Continuum Mechanics and Thermodynamics, 34(2): 587–599, 2021, https://doi.org/10.1007/s00161-021-01075-3
33. 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 Applications, Part A: Applications, 61(7): 483–501, 2012, https://doi.org/10.1080/10407782.2012.667648
34. E. Majchrzak, M. Stryczyński, Dual-phase lag model of heat transfer between blood vessel and biological tissue, Mathematical Biosciences and Engineering: MBE, 18(2): 1573–1589, 2021, https://doi.org/10.3934/MBE.2021081
35. T.N. Glenn, S. Rastegar, S.L. Jacques, Finite element analysis of temperature controlled coagulation in laser irradiated tissue, IEEE Transactions on Biomedical Engineering, 43(1): 79–87, 1996, https://doi.org/10.1109/10.477703
36. B.J. McGuire, T.W. Secomb, A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand, Journal of Applied Physiology, 91(5): 2255–2265, 2001, https://doi.org/10.1152/jappl.2001
37. M. Jasiński, Numerical analysis of the temperature impact to the oxygen distribution in the biological tissue, Journal of Applied Mathematics and Computational Mechanics, 19(3): 17–28, 2020, https://doi.org/10.17512/jamcm.2020.3.02
38. J.P. Whiteley, D.J. Gavaghan, C.E.W. Hahn, Mathematical modelling of oxygen transport to tissue, Journal of Mathematical Biology, 44(6): 503–522, 2002, https://doi.org/10.1007/s002850200135
39. B.C. Fry, T.K. Roy, T.W. Secomb, Capillary recruitment in a theoretical model for blood flow regulation in heterogeneous microvessel networks, Physiological Reports, 1(3): e00050, 2013, https://doi.org/10.1002/phy2.50
40. A. Korczak, M. Jasiński, Modelling of biological tissue damage process with application of interval arithmetic, Journal of Theoretical and Applied Mechanics, 57(1): 249–261, 2019, https://doi.org/10.15632/jtam-pl.57.1.249
41. B. Mochnacki, A. Piasecka Belkhayat, Numerical modeling of skin tissue heating using the interval finite difference method, Molecular & Cellular Biomechanics, 10(3): 233–244, 2013, https://doi.org/10.3970/mcb.2013.010.233
42. C.A. Brebbia, J. Domínguez, Boundary Elements: An Introductory Course, 2nd ed., WIT Press, London, 1992.
43. E. Majchrzak, Ł. 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
44. K.K.H. Wang, J.C. Finlay, T.M. Busch, S.M. Hahn, T.C. Zhu, Explicit dosimetry for photodynamic therapy: Macroscopic singlet oxygen modeling, Journal of Biophotonics, 3(5–6): 304–318, 2010, https://doi.org/10.1002/jbio.200900101
45. E. Majchrzak, M. Jasiński, Ł. Turchan, Modeling of laser-soft tissue interactions using the dual-phase lag equation: Sensitivity analysis to selected tissue parameters, Defect and Diffusion Forum, 379: 108–123, 2017, https://doi.org/10.4028/www.scientific.net/DDF.379.108
