Analytical solutions in the modeling of the endovenous laser ablation

Abstract

We modeled the operative treatment of incompetent truncal veins using endovenous laser ablation (EVLA). The main concern regarding the thermoablative technique is tissue damage, which is correlated with (1) the energy provided by the laser power and (2) temperature distribution. Our objective was to accurate the two functions, namely the fluence rate and the temperature, depending on the thermoablative technique and the endovenous laser treatment (ELT). First, we considered three differential equations: diffusion, heat, and bioheat equations in the endovenous-perivenous multidomain to describe the lumen, the vein wall, the tissue pad, and the skin. Second, we examined the power source according to the Beer-Lambert law in the incident beam irradiance and the heat source as the so-called absorbed optical power density. Third, we checked out the heat transfer at the skin boundary according to Newton’s law of cooling, which stands for a Robin boundary condition. For this new model, we proposed exact solutions: applying differential equations techniques, we solved (1) a diffusion approximation of the radiative transfer equation under the considered power source, and (2) the coupled heat and bioheat equations under the considered heat source accomplished with the Robin boundary condition. Then, we graphically illustrated the fluence rate profile and discussed its time dependence and steady state. Besides, we discuss thermal damage to the vein-tissue system and present open problems.