A numerical study of a 3D bioheat transfer problem with different spatial heating.

*(English)*Zbl 1062.92018Summary: We develop numerical methods for computer simulation and modeling of a three dimensional heat transfer problem in biological bodies. The technique is intended for temperature predications and parameter measurements in thermal medical practices and for studies of thermomechanical interaction of biological bodies at high temperature.

We examine a mathematical model based on the classical well-known Pennes equation for heat transfer in biological bodies. A finite difference discretization scheme is used to discretize the governing partial differential equation. A preconditioned iterative solver is employed to solve the resulting sparse linear system at each time step. Numerical results are obtained to demonstrate the efficacy of the proposed numerical methods.

We examine a mathematical model based on the classical well-known Pennes equation for heat transfer in biological bodies. A finite difference discretization scheme is used to discretize the governing partial differential equation. A preconditioned iterative solver is employed to solve the resulting sparse linear system at each time step. Numerical results are obtained to demonstrate the efficacy of the proposed numerical methods.

##### MSC:

92C30 | Physiology (general) |

80A20 | Heat and mass transfer, heat flow (MSC2010) |

65N06 | Finite difference methods for boundary value problems involving PDEs |

92C50 | Medical applications (general) |

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\textit{S. Karaa} et al., Math. Comput. Simul. 68, No. 4, 375--388 (2005; Zbl 1062.92018)

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##### References:

[1] | Anderson, G.T.; Burnside, G., A non-invasive technique to measure perfusion using a focused ultrasound heating source and a tissue surface temperature measurement, Proc. adv. meas. comput. temp. biomed., 147, 31-35, (1990) |

[2] | Chato, J.C., Fundamentals of bioheat transfer, (1989), Springer-Verlag Berlin |

[3] | Crank, J.; Nicolson, P., A practical method for numerical evaluation of solution of partial differential equations of the heat-conduction type, Proc. camb. philos. soc., 43, 50-67, (1947) · Zbl 0029.05901 |

[4] | Deng, Z.S.; Liu, J., Analytical study on bioheat transfer problems with spatial or transient heating on skin surface or inside biological bodies, ASME J. biomech. eng., 124, 638-649, (2002) |

[5] | Diller, K.R., Modeling of bioheat transfer processes at high and low temperatures, Adv. heat transfer, 22, 157-167, (1992) |

[6] | Gautherie, M., Clinical thermology, vol. 1-4, (1990), Springer-Verlag Heidelberg |

[7] | K.R. Holmes, Biological structures and heat transfer, in: Allerton Workshop on the Future of Biothermal Engineering, 1997. |

[8] | Killer, K.R.; Hayes, L.J., Analysis of tissue injury by burning: comparison of in situ and skin flap models, Int. J. heat mass transfer, 34, 1393-1406, (1991) |

[9] | Leonard, J.B.; Foster, K.B.; Athey, T.W., Thermal properties of tissue equivalent phantom materials, IEEE trans. biomed. eng., 31, 533-536, (1984) |

[10] | Liu, J., Uncertainty analysis for temperature prediction of biological bodies subject to randomly spatial heating, J. biomech., 34, 1637-1642, (2001) |

[11] | Liu, J.; Xu, L.X., Estimation of blood perfusion using phase shift in temperature response to sinusoidal heating at skin surface, IEEE trans. biomed. eng., 46, 1037-1043, (2001) |

[12] | Liu, P.; Liu, G., Action mechanisms of laser biology, (1989), Science Press Beijing, pp. 127-177 (in Chinese) |

[13] | Meijerink, J.A.; van der Vorst, H.A., An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. comput., 31, 148-162, (1977) · Zbl 0349.65020 |

[14] | Pennes, H.H., Analysis of tissue and arterial blood temperature in the resting human forearm, J. appl. physiol., 1, 93-122, (1948) |

[15] | Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Publishing New York, NY · Zbl 1002.65042 |

[16] | Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 856-869, (1986) · Zbl 0599.65018 |

[17] | Weinbaum, S.; Jiji, L.M.; Lemons, D.E., Theory and experiment for the effect of vascular microstructure on surface tissue heat transfer. part I. anatomical foundation and model conceptualization, ASME J. biomech. eng., 106, 321-330, (2001) |

[18] | J.J. Zhao, J. Zhang, N. Kang, F. Yang, A two-level finite difference scheme for one dimensional Pennesâ€™ bioheat equation, Appl. Math. Comput., in press (also as Technical Report No. 354-02, Department of Computer Science, University of Kentucky, Lexington, KY, 2002). · Zbl 1086.65090 |

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