Influence of the modulation of the blood flow velocity in peripheral vessels on the temperature of the outer wall of the vessel: Finite element modeling of the adjoint problem

A finite element modelling of the process of the heat transfer from blood to the wall of an arterial vessel was carried out in order to solve a more general problem of determining the relationship between the amplitude-frequency characteristics of fluctuations in the volumetric blood flow velocity in peripheral vessels with temperature oscillations on the skin surface. A model was built in the ANSYS software with Fluid Flow CFX module which includes domains related to blood, the wall of a cylindrical vessel, and skin (bio-tissue). The model takes into account the convective heat transfer from blood to the vessel wall and thermal conductivity in the skin. The corresponding boundary value problem is posed which includes the Navier – Stokes equation and the Fourier heat equation. Dependences of the temperature oscillations of the vessel wall on the amplitude of fluctuations in the volumetric blood flow velocity in a wide frequency range of 0.01–1 Hz were obtained. The selected frequency range covers all currently known rhythms of hemodynamic fluctuations: endothelial, neurogenic, myogenic, respiratory and cardiac. A function is proposed that approximates the dependence of the amplitude of vessel wall temperature oscillations on the amplitude of the oscillations of the volumetric blood flow at various values of the blood flow velocity modulation frequency. The use of the introduced approximating function together with the solution of the heat equation for a thermal wave opens up the possibility of solving the inverse problem of determining the dynamics of volumetric blood flow in an arterial vessel based on the data on the temperature dynamics on the skin surface.