# Numerical Study on the Application of Near-Infrared Temperature Distribution Measurement of HIFU
## 2 Theoretical Models
### Simulation  
To date, there is no analytical solution to the Westervelt equation. It can only be solved numerically. In this study, the COMSOL V6.1 finite element simulation software was used to solve the Westervelt acoustic wave propagation equation.  
Figure 1 illustrates a two-dimensional axisymmetric simulation model developed using COMSOL V6.1. Figure 1 represents biological tissues alone, while Figure 1 incorporates infrared glass. A focused ultrasound transducer with parameters, shown in Table 1, was used. The excitation signal was a sine wave with a frequency of 1 MHz. Multiple layers of human tissue, including skin, fat, muscle, and liver, were used with ultrasound beams focused 42 mm deep into the liver. The specific biological tissue parameters used in the model are shown in Table 2.  
Figure 1: Schematic diagram of the simulation model: (**a**) is a schematic of the geometric model without infrared glass and with only human tissue; (**b**) is a schematic of the geometrical model with infrared glass.  
When dividing the COMSOL V6.1 finite element model into grids, the two-dimensional axisymmetric model was discretized using a free triangular mesh. Reducing the grid size brings the numerical solution closer to the exact solution. However, it consumes significant computational resources. Therefore, when conducting temperature simulation calculations, it is necessary to determine the optimal grid size of the model. Considering the glass-free model depicted in Figure 1a, its temperature distribution along the acoustic axis direction was computed, as demonstrated in Figure 2. The model adopts four different grid sizes from four units per wavelength to seven units per wavelength for parameterized scanning. The grid sizes are shown in Table 3.  
Figure 2 shows that the temperature distribution waveforms of different grid size models are roughly the same, with some errors in the highest temperature. The maximum  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Frequency** & **Aperture** & **Focal Length** & **Power** & **Depth** \\
**(MHz)** & **Diameter (mm)** & **(mm)** & **(W)** & **(mm)** \\ \hline
1.12 & 95 & 150 & 20–100 & 42 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters for focused ultrasonic transducers.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Water** & **Liver** & **Skin** & **Muscle** & **Fat** & **Infrared Glass** \\ \hline Density (kg/m\({}^{3}\)) & 1000 & 1050 & 1109 & 1090 & 911 & 4890 \\ Speed of sound (m/s) & 1500 & 1587 & 1624 & 1588 & 1478 & 2692 \\ Thermal conductivity (W/m-K) & 0.6 & 0.48 & 0.3 & 0.46 & 0.2 & 11.72 \\ Thermal capacity (J/kg-K) & 4180 & 3400 & 3390 & 3500 & 2000 & 410 \\ Absorption coefficient (Np/m/MHz) & 0.028 & 7.728 & 7.952 & 23.804 & 4.883 & - \\ Non-linear coefficient & 5 & 7.3 & 7 & 7.1 & 10 & - \\ Thickness (mm) & 65 & 84 & 3 & 10 & 30 & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Organizational parameters used in the simulation.  
Figure 2: Axial temperature distribution corresponding to different grid sizes.  
axial temperature values of Grid1, Grid2, Grid3, and Grid4 are 51.7997, 49.4845, 48.4640, and 48.4003, respectively. The maximum error between Grid1 and Grid4 is 3%, while the maximum error between Grid3 and Grid4 is only 0.1%. Based on the temperature distribution curve results in this model, a maximum unit size of the grid at 0.2463 mm renders the numerical solution reliable. At this point, further grid refinement leads to marginal alterations in the solution accuracy. Considering the overall solution accuracy and calculation time, the grids for the models under different working conditions are segmented into six units per wavelength.