## 2 Thermophysical Model of Endovasal Laser Treatment of a Vein with TOTFC  
The technology of TOTFC manufacture is described in detail in the references [12,13]. TOTFC is known to consist of titanium dioxide (TiO\({}_{2}\)) microspheres with a diameter of \(\sim\)1.2 \(\upmu\)m surrounded by silica (SiO\({}_{2}\)), and to be located at the distal end of an optical quartz fiber. The TOTFC model used in this research is shown in Figure 1a. The microspheres are evenly distributed over the volume of the converter, and the volume fraction of the TiO\({}_{2}\) (k) microspheres is 0.22. The converter has a strong mechanical connection with the optical quartz fiber with the diameter of 440 \(\upmu\)m. The scheme describing the EVLT model used in the current study is shown in Figure 1b. The vein is presented in the form of a cylindrical tube with the length of 200 mm, the internal diameter of 5 mm, and the wall thickness of 1 mm. An optical quartz fiber with TOTFC is located on the axis of the vein. It can be moved along the axis of the vein using a fiber moving device. The traction speed ranges from 1 mm/s to 7 mm/s. Laser radiation passing through the beamsplitter is fed to the input of the optical quartz fiber. The laser radiation has the wavelength of 980 nm. In studies previously performed for TOTFC with k = 0.22, the efficiency of converting laser radiation with the wavelength of 980 nm into heat was 83.2% [15,16]. Laser radiation propagating through the optical quartz fiber reaches TOTFC and heats the latter and the vein wall. The thermal radiation (IR signal) from the laser heated TOTFC propagating along with the optical quartz fiber in the direction opposite to the direction of propagation of the laser radiation reaches the beamsplitter and falls on the photodetector (PD). The PD signal enters the laser control system for subsequent processing and making a decision on the change in the average laser power required for optimal EVLT (feedback). Computer simulation of radiation and thermal processes occurring during EVLT using TOTFC was carried out in the COMSOL Multiphysics 5.4 software package (\"COMSOL Inc.\", USA). After heating TOTFC by laser radiation with the wavelength of 980 nm, there is heat exchange of the converter with the environment by natural convection and thermal radiation. A thermal insulation condition was assigned at the entrance of the optical quartz fiber and at the ends of the vein. In the COMSOL Multiphysics 5.4 software package (COMSOL Inc., Burlington, MA, USA), the thermal conductivity equation was solved by the finite element method using the COMSOL PDE toolkit, which is described in two-dimensional space as follows:  
\[\rho C_{p}\frac{\partial T}{\partial t}=\kappa\left(\frac{\partial^{2}T}{ \partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}\right)+Q_{0} \tag{1}\]  
where \(C_{p}\left[\text{J}/\left(\text{kg}\cdot\text{K}\right)\right]\) is specific heat capacity; \(\kappa\left[\text{W}/\left(\text{m}\cdot\text{K}\right)\right]\) is thermal conductivity; \(\rho\left[\text{kg}/\text{m}^{3}\right]\) is physical density; \(Q_{0}\left[\text{W}/\text{m}^{3}\right]\) is specific energy of the heat source. The following boundary and initial conditions were set: (i) optical quartz fiber with TOTFC at the distal end is located horizontally on the axis of the vein; (ii) the medium surrounding the fiber with TOTFC is blood; (iii) the initial temperature of the fiber, converter, and blood is equal to the human body's normal temperature (36.6 \({}^{\circ}\)C); (iv) the blood, the vein, and the vein wall are stationary, and (v) the optical quartz fiber with TOTFC can move progressively from left to right (Figure 1b). When simulating, the distribution of heat sources in TOTFC was uniform. Boiling of the blood around the converter was not simulated, however, it was taken into account by 200 times increase in the thermal conductivity coefficient in the area where the temperature exceeded the threshold of 95 \({}^{\circ}\)C [21].