Dual-phase lag model of heat transfer between blood vessel and biological tissue

Ewa Majchrzak; Mikołaj Stryczyński; Ewa Majchrzak; Mikołaj Stryczyński

doi:10.3934/mbe.2021081

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 2: 1573-1589. doi: 10.3934/mbe.2021081

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Dual-phase lag model of heat transfer between blood vessel and biological tissue

Ewa Majchrzak ^,,
Mikołaj Stryczyński

Department of Computational Mechanics and Engineering, Silesian University of Technology Gliwice, Poland

Received: 15 December 2020 Accepted: 18 January 2021 Published: 01 February 2021

A single blood vessel surrounded by the biological tissue with a tumor is considered. The influence of the heating technique (e.g. ultrasound, microwave, etc.) is described by setting a fixed temperature for the tumor which is higher than the blood and tissue temperature. The temperature distribution for the blood sub-domain is described by the energy equation written in the dual-phase lag convention, the temperature distribution in the biological tissue with a tumor is described also by the dual-phase lag equation. The boundary condition on the contact surface between blood vessel and biological tissue and the Neumann condition are also formulated using the extended Fourier law. So far in the literature, the temperature distribution in a blood vessel has been described by the classical energy equation. It is not clear whether the Fourier's law applies to highly heated tissues in which a significant thermal blood vessel is distinguished, therefore, taking into account the heterogeneous inner structure of the blood, the dual-phase lag equation is proposed for this sub-domain. The problem is solved by means of the implicit scheme of the finite difference method. The computations were performed for various values of delay times, which were taken from the available literature, and the influence of these values on the obtained temperature distributions was discussed.
- bioheat transfer,
- thermal interaction between blood vessel and tissue,
- dual-phase lag model,
- finite difference method
Citation: Ewa Majchrzak, Mikołaj Stryczyński. Dual-phase lag model of heat transfer between blood vessel and biological tissue[J]. Mathematical Biosciences and Engineering, 2021, 18(2): 1573-1589. doi: 10.3934/mbe.2021081

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Abstract

A single blood vessel surrounded by the biological tissue with a tumor is considered. The influence of the heating technique (e.g. ultrasound, microwave, etc.) is described by setting a fixed temperature for the tumor which is higher than the blood and tissue temperature. The temperature distribution for the blood sub-domain is described by the energy equation written in the dual-phase lag convention, the temperature distribution in the biological tissue with a tumor is described also by the dual-phase lag equation. The boundary condition on the contact surface between blood vessel and biological tissue and the Neumann condition are also formulated using the extended Fourier law. So far in the literature, the temperature distribution in a blood vessel has been described by the classical energy equation. It is not clear whether the Fourier's law applies to highly heated tissues in which a significant thermal blood vessel is distinguished, therefore, taking into account the heterogeneous inner structure of the blood, the dual-phase lag equation is proposed for this sub-domain. The problem is solved by means of the implicit scheme of the finite difference method. The computations were performed for various values of delay times, which were taken from the available literature, and the influence of these values on the obtained temperature distributions was discussed.

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