Engineering Transactions, 71, 3, pp. 287–306, 2023
10.24423/EngTrans.3086.20230426

Modeling of the Influence of Elevated Temperature on Oxygen Distribution in Soft Tissue

Marek JASIŃSKI
ORCID ID 0000-0002-9475-9012
Silesian University of Technology
Poland

Maria ZADOŃ
ORCID ID 0000-0003-2833-8604
Silesian University of Technology
Poland

The purpose of the study was to analyze the combined model of bioheat transfer and oxygen distribution in tissue during exposition to the external heat impulse. The effect of temperature and thermal damage to the tissue on the values of its thermophysical parameters was taken into account. The variable value of the perfusion coefficient affects the blood velocity in the capillary and thus the distribution of the partial oxygen pressure in the tissue. Various models of the oxygen dissociation curves were also considered and a sensitivity analysis was performed for the parameters of the oxygen distribution model. In the  numerical realization stage, the finite difference method and the shooting method were used.

Keywords: bioheat transfer; tissue thermal damage; oxygen transport; Krogh cylinder; oxyhemoglobin dissociation curve; finite difference method; shooting method; sensitivity analysis
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References

Abraham J.P., Sparrow E.M., 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, doi: 10.1016/J.IJHEATMASSTRANSFER.2006.11.045.

Hassanpour S., Saboonchi A., Modeling of heat transfer in a vascular tissue-like medium during an interstitial hyperthermia process, Journal of Thermal Biology, 62, Part B: 150–158, 2016, doi: 10.1016/J.JTHERBIO.2016.06.022.

Afrin N., Zhou J., Zhang Y., Tzou D.Y., Chen J.K., 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, doi: 10.1080/10407782.2012.667648.

Jasiński M., 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, doi: 10.17512/JAMCM.2020.3.02.

He Y., Shirazaki M., Liu H., Himeno R., Sun Z., 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, doi: 10.1016/J.COMPBIOMED.2005.08.004.

Fry B.C., Roy T.K., Secomb T.W., Capillary recruitment in a theoretical model for blood flow regulation in heterogeneous microvessel networks, Physiological Reports, 1(3): e00050, 2013, doi: 10.1002/PHY2.50.

Goldman D., Theoretical models of microvascular oxygen transport to tissue, Microcirculation, 15(8): 795–811, 2008, doi: 10.1080/10739680801938289.

Secomb T.W., Alberding J.P., Hsu R., Dewhirst M.W., Pries A.R., Angiogenesis: An adaptive dynamic biological patterning problem, PLoS Computational Biology, 9(3): e1002983, 2013, doi: 10.1371/JOURNAL.PCBI.1002983.

Zhu T.C., Kim M.M., Liang X., Finlay J.C., Busch T.M., In-vivo singlet oxygen threshold doses for PDT, Photonics and Lasers in Medicine, 4(1): 59–71, 2015, doi: 10.1515/PLM-2014-0037.

Zhu T.C., Liu B., Penjweini R., Study of tissue oxygen supply rate in a macroscopic photodynamic therapy singlet oxygen model, Journal of Biomedical Optics, 20(3): 038001, 2015, doi: 10.1117/1.JBO.20.3.038001.

Hlastala M.P., Woodson R.D., Wranne B., Influence of temperature on hemoglobin-ligand interaction in whole blood, Journal of Applied Physiology, 43(3): 545–550, 1977, doi: 10.1152/JAPPL.1977.43.3.545.

Castaing M., Sinet M., Temperature and oxygenation of human blood at constant total CO2 content, Pflügers Archiv, European Journal of Physiology, 386(2): 135–140, 1980, doi: 10.1007/BF00584200.

Whiteley J.P., Gavaghan D.J., Hahn C.E.W., Mathematical modelling of oxygen transport to tissue, Journal of Mathematical Biology, 44: 503–522, 2002, doi: 10.1007/S002850200135.

Jasiński M., Numerical analysis of thermal damage and oxygen distribution in laser irradiated tissue, Journal of Applied Mathematics and Computational Mechanics, 21(2): 51–62, 2022, doi: 10.17512/JAMCM.2022.2.05.

El-Nabulsi R.A., Anukool W., Nonlocal thermal effects on biological tissues and tumors, Thermal Science and Engineering Progress, 34: 101424, 2022, doi: 10.1016/J.TSEP.2022.101424.

Paruch M., Mathematical modeling of breast tumor destruction using fast heating during radiofrequency ablation, Materials (Basel, Switzerland), 13(1): 136, 2020, doi: 10.3390/MA13010136.

Mochnacki B., Ciesielski M., 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, doi: 10.1515/BPASTS-2016-0066.

El-Nabulsi R.A., Fractal Pennes and Cattaneo–Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth, Journal of the Royal Society Interface, 18(182): 20210564, 2021, doi: 10.1098/RSIF.2021.0564.

Chaudhary R.K., Kumar D., Rai K.N., Singh J., 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, doi: 10.1142/S1793524521500406.

Chaudhary R.K., Kumar D., Rai K.N., Singh J., Numerical simulation of the skin tissue subjected to hyperthermia treatment using a nonlinear DPL model, Thermal Science and Engineering Progress, 34: 101394, 2022, doi: 10.1016/J.TSEP.2022.101394.

Majchrzak E., Turchan Ł., Dziatkiewicz J., Modeling of skin tissue heating using the generalized dual phase-lag equation, Archives of Mechanics, 67(6): 417 – 437, 2015, doi: 10.24423/AOM.1777.

Saeed T., Abbas I., 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, doi: 10.1080/15397734.2020.1749068.

Alzahrani F., Abbas I., A numerical solution of nonlinear DPL bioheat model in biological tissue due to laser irradiations, Indian Journal of Physics, 96(2): 377–383, 2022, doi: 10.1007/s12648-020-01988-w.

Akula S.C., Maniyeri R., 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, doi: 10.1007/s40430-019-2132-x.

Majchrzak E., Stryczyński M., Dual-phase lag model of heat transfer between blood vessel and biological tissue, Mathematical Biosciences and Engineering: MBE, 18(2): 1573–1589, 2021, doi: 10.3934/MBE.2021081.

Dombrovsky L.A., Laser-induced thermal treatment of superficial human tumors: An advanced heating strategy and non-Arrhenius law for living tissues, Frontiers in Thermal Engineering, 1: 807083, 2022, doi: 10.3389/FTHER.2021.807083.

Oden J.T., Diller K.R., Bajaj C., Browne J.C., Hazle J., Babuška I., Bass J., Biduat L., Demkowicz L., Elliott A., Feng Y., Fuentes D., Prudhomme S., Rylander M.N., Stafford R.J., Zhang Y., Dynamic data-driven finite element models for laser treatment of cancer, Numerical Methods for Partial Differential Equations, 23(4): 904–922, 2007, doi: 10.1002/NUM.20251.

Hamilton G., Investigations of the Thermal Properties of Human and Animal Tissues, Ph. D. Thesis, University of Glasgow, UK, 1998.

McGuire B.J., Secomb T.W., A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand, Journal of Applied Physiology, 91(5): 2255–2265, 2001, doi: 10.1152/jappl.2001.91.5.2255.

Majchrzak E., Kałuża G., Sensitivity analysis of temperature in heated soft tissues with respect to time delays, Continuum Mechanics and Thermodynamics, 34(2): 587–599, 2021, doi: 10.1007/s00161-021-01075-3.

Majchrzak E., Turchan L., Jasiński M., 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, doi: 10.1007/s40997-018-0225-2.

Jasiński M., Modelling of tissue thermal injury formation process with application of direct sensitivity method, Journal of Theoretical and Applied Mechanics, 52(4): 947–957, 2014, doi: 10.15632/JTAM-PL.52.4.947.

Davies C.R., Saidel G.M., Harasaki H., Sensitivity analysis of one-dimensional heat transfer in tissue with temperature-dependent perfusion, Journal of Biomechanical Engineering, 119(1): 77–80, 1997, doi: 10.1115/1.2796068.

Majchrzak E., Kałuża G., Sensitivity analysis of biological tissue freezing process with respect to the radius of spherical cryoprobe, Journal of Theoretical and Applied Mechanics, 44(2): 381–392, 2006.

Mochnacki B., Piasecka Belkhayat A., Numerical modeling of skin tissue heating using the interval finite difference method, Molecular & Cellular Biomechanics, 10(3): 233–244, 2013, doi: 10.3970/MCB.2013.010.233.

Majchrzak E., Mochnacki B., Dual-phase lag equation. Stability conditions of a numerical algorithm based on the explicit scheme of the finite difference method, Journal of Applied Mathematics and Computational Mechanics, 15(3): 89–96, 2016, doi: 10.17512/JAMCM.2016.3.09.

Keskin A.Ü., Boundary Value Problems for Engineers, Springer International Publishing, Heidelberg, 2019, doi: 10.1007/978-3-030-21080-9.

Attili B.S., Syam M.I., Efficient shooting method for solving two point boundary value problems, Chaos, Solitons & Fractals, 35(5): 895–903, 2008, doi: 10.1016/J.CHAOS.2006.05.094.

Ha S.N., A nonlinear shooting method for two-point boundary value problems, Computers & Mathematics with Applications, 42(10–11): 1411–1420, 2001, doi: 10.1016/S0898-1221(01)00250-4.

Korczak A., Jasiński M., Modelling of biological tissue damage process with application of interval arithmetic, Journal of Theoretical and Applied Mechanics, 57(1): 249–261, 2019, doi: 10.15632/JTAM-PL.57.1.249.

Jasiński M., Zadoń M., 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, doi: 10.17512/jamcm.2022.4.04.




DOI: 10.24423/EngTrans.3086.20230426