Analytical analysis of the Pennes bioheat transfer equation with sinusoidal heat flux condition on skin surface

This study focuses on the effect of the temperature response of a semi-infinite biological tissue due to a sinusoidal heat flux at the skin. The Pennes bioheat transfer equation such as ρtct(∂T/∂t) + Wbcb(T − Ta) = k∂2T/∂x2 with the oscillatory heat flux boundary condition such as q(0,t) = q0eiωt was investigated. By using the Laplace transform, the analytical solution of the Pennes bioheat transfer equation with surface sinusoidal heating condition is found. This analytical expression is suitable for describing the transient temperature response of tissue for the whole time domain from the starting periodic oscillation to the final steady periodic oscillation. The results show that the temperature oscillation due to the sinusoidal heating on the skin surface is unstable in the initial period. Further, it is unavailable to predict the blood perfusion rate via the phase shifting between the surface heat flux and the surface temperature. Moreover, the lower frequency of sinusoidal heat flux on the skin surface induces a more sensitive phase shift response to the blood perfusion rate change, but extends the beginning time of sampling because of the avoidance of the unavailable first cyclic oscillation.