Email this article and Applied Mechanics
57, 1, pp. 249-261, Warsaw 2019
DOI: 10.15632/jtam-pl.57.1.249
Modelling of biological tissue damage process with application of interval arithmetic
tissue subjected to laser irradiation is presented. The transient heat transfer is described
by the bioheat transfer equation in Pennes formulation. The internal heat source resulting
from the laser-tissue interaction based on the solution of the diffusion equation is taken
into account. Thermophysical and optical parameters of the tissue are assumed as directed
intervals numbers. At the stage of numerical realization. the interval finite difference method
has been applied. In the final part of the paper, the results obtained are shown.
References
Abraham J.P., Sparrow E.M., 2007, 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
Banerjee S., Sharma S.K., 2010, Use of Monte Carlo simulations for propagation of light in
biomedical tissues, Applied Optics, 49, 4152-4159
Dawood H., 2011, Theories of Interval Arithmetic: Mathematical Foundations and Applications,
LAP LAMBERT Academic Publishing GmbH & Co., Germany
Dombrovsky L.A., Baillis D., 2010, Thermal Radiation in Disperse Systems: An Engineering
Approach, Begell House, New York
Dombrovsky L.A., Randrianalisoa J.H., Lipinski W., Timchenko V., 2013, Simplified
approaches to radiative transfer simulations in laser induced hyperthermia of superficial tumors,
Computational Thermal Sciences, 5, 6, 521-530
Fasano A., H¨omberg D., Naumov D., 2010, On a mathematical model for laser-induced thermotherapy,
Applied Mathematical Modelling, 34, 12, 3831-3840
Gajda K., Marciniak A., Szyszka B., 2000, Three- and four-stage implicit interval methods
of Runge-Kutta type, Computational Methods in Science and Technology, 6, 41-59
Hansen E., Walster G.W., 2004, Global Optimization Using Interval Analysis, Marcell Dekker,
New York
Henriques F.C., 1947, Studies of thermal injuries, V. The predictability and the significance of
thermally induced rate process leading to irreversible epidermal injury, Journal of Pathology, 23, 489-502
Jacques S.L., Pogue B.W., 2008, Tutorial on diffuse light transport, Journal of Biomedical
Optics, 13, 4, 1-19
Jankowska M.A., Sypniewska-Kaminska G., 2012, Interval finite difference method for bioheat
transfer problem given by the Pennes equation with uncertain parameters, Mechanics and
Control, 31, 2, 77-84
Jasiński M., 2014, Modelling of tissue thermal injury formation process with application of direct
sensitivity method, Journal of Theoretical and Applied Mechanics, 52, 947-957
Jasiński M., 2015, Modelling of thermal damage in laser irradiated tissue, Journal of Applied
Mathematics and Computational Mechanics, 14, 67-78
Jasiński M., 2018, Numerical analysis of soft tissue damage process caused by laser action, AIP
Conference Proceedings, 060002, 1922
Jasiński M., Majchrzak E., Turchan L., 2016, Numerical analysis of the interactions between
laser and soft tissues using generalized dual-phase lag model, Applied Mathematic Modelling, 40, 2, 750-762
Kałuża G., Majchrzak E., Turchan L., 2017, Sensitivity analysis of temperature field in the
heated soft tissue with respect to the perturbations of porosity, Applied Mathematical Modelling, 49, 498-513
Di Lizia P., Armellin R., Bernelli-Zazzera F., Berz M., 2014, High order optimal control
of space trajectories with uncertain boundary conditions, Acta Astronautica, 93, 217-229
Majchrzak E., Mochnacki B., 2016, 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, 89-96
Majchrzak E., Mochnacki B., 2017, Implicit scheme of the finite difference method for 1D
dual-phase lag equation, Journal of Applied Mathematics and Computational Mechanics, 16, 3, 37-46
Majchrzak E., Turchan L., Dziatkiewicz J., 2015, Modeling of skin tissue heating using the
generalized dual-phase lag equation, Archives of Mechanics, 67, 6, 417-437
Markov S.M., 1995, On directed interval arithmetic and its applications, Journal of Universal
Computer Science, 1, 514-526
Mochnacki B., Ciesielski M., 2016, 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
Mochnacki B., Piasecka-Belkhayat A., 2013, Numerical modeling of skin tissue heating using
the interval finite difference method, Molecular and Cellular Biomechanics, 10, 3, 233-244
Mochnacki B., Suchy J.S., 1995, Numerical Methods in Computations of Foundry Processes,
PFTA, Cracow
Moore R.E., 1966, Interval Analysis, Prentice-Hall, Englewood Cliffs
Nakao M., 2017, On the initial-boundary value problem for some quasilinear parabolic equations
of divergence form, Journal of Differential Equations, 263, 8565-8580
Paruch M., 2014, Hyperthermia process control induced by the electric field in order to destroy
cancer, Acta of Bioengineering and Biomechanics, 16, 4, 123-130
Piasecka-Belkhayat A., 2011, Interval Boundary Element Method for Imprecisely Defined
Unsteady Heat Transfer Problems (in Polish), Silesian University of Technology, Gliwice
Piasecka-Belkhayat A., Jasiński M., 2011, Modelling of UV laser irradiation of anterior part
of human eye with interval optic, [In:] Evolutionary and Deterministic Methods for Design, Opti-
mization and Control. Applications, Burczyński T., P´eriaux J. (Eds.), CIMNE, Barcelona, 316-321
Piasecka-Belkhayat A., Korczak A., 2016, Numerical modelling of the transient heat transport
in a thin gold film using the fuzzy lattice Boltzmann method with alpha-cuts, Journal of
Applied Mathematics and Computational Mechanics, 15, 1, 123-135
Piasecka-Belkhayat A., Korczak A., 2017, Modeling of thermal processes proceeding in a 1D
domain of crystalline solids using the lattice Boltzmann method with an interval source function,
Journal of Theoretical and Applied Mechanics, 55, 1, 167-175
Popova E.D., 1994, Extended interval arithmetic in IEEE floating-point environment, Interval
Computations, 4, 100-129
Popova E.D., 2011, Multiplication distributivity of proper and improper intervals, Reliable Com-
puting, 7, 129-140
Welch A.J., 2011, Optical-Thermal Response of Laser Irradiated Tissue, M.J.C. van Gemert
(Eds.), 2nd edit., Springer