Theoretical Analysis of Transient Bioheat Transfer in Multi-Layer Tissue

Heat conduction in skin tissue is a problem of significant technological importance. A theoretical understanding of such a problem is essential as it may lead to design potential therapeutic measures for needed cancer therapy or novel medical devices for various applications including hyperthermia. To understand the physical phenomenon of energy transport in biological systems a transient model is chosen for this study. The most common transport equation to estimate temperature distribution in humans was developed by H.H. Pennes and is popularly known as the Pennes bioheat transfer equation. A generalized Pennes bioheat transfer equation accounts for the effect of various physical phenomena such as conduction, advection, volumetric heat generation, etc. are considered. In this paper, a general transient form of the Pennes bioheat transfer equation is solved analytically for a multilayer domain. The boundary value problem considers the core of the tissue is maintained at uniform temperature of 37°C, convective cooling is applied to the external surface of the skin and the sidewalls are adiabatic. The computation of transient temperature in multidimensional and multilayer bodies offers unique features. Due to the presence of blood perfusion in the tissue, the reaction term in the Pennes governing equation is modeled similar to a fin term. The eigenvalues may become imaginary, producing eigenfunctions with imaginary arguments. In addition the spacing between the eigenvalues between zero and maximum value varies for different cases; therefore the values need to be determined with precision using second order Newton’s method. A detailed derivation of the temperature solution using the technique of separation of variables is presented in this study. In addition a proof of orthogonality theorem for eigenfunctions with imaginary eigenvalues is also presented. The analytical model is used to study the thermal response of skin tissue to different parameters with the aid of some numerical examples. Results shown in this paper are expected to facilitate a better understand of bioheat transfer in layered tissue such as skin.