## 3 Result and Discussion
### Dynamic Acoustic Absorption Coefficient  
Simulations are carried out that only consider the change of the acoustic absorption coefficient with temperature independently. To get more accurate simulation results, the acoustic absorption coefficient and acoustic field are updated here every 0.2 s. Figure 4a depicts the axial profile of the acoustic absorption coefficient during 3 s of HIFU irradiation. The acoustic absorption coefficient near ultrasonic focus increases with the time of HIFU irradiation. At time t = 3 s, Figure 4b illustrates the axial distribution of peak acoustic pressure. Clearly, the peak acoustic pressures are almost the same between dynamic \(\alpha_{\mathit{liter}}\) and constant \(\alpha_{\mathit{liter}}\). It can be explained that the temperature only has a great effect on the acoustic absorption coefficient near the ultrasonic focus, as shown in Figure 4a. In Figure 4c, the maximum value of \(Q_{\mathit{ext}}\) is \(8.938\times 10^{7}\mathrm{W/m^{2}}\) for constant \(\alpha_{\mathit{liter}}\), and \(2.179\times 10^{8}\mathrm{W/m^{2}}\) for dynamic \(\alpha_{\mathit{liter}}\). The features can be explained that \(Q_{\mathit{ext}}\) is proportional to the acoustic absorption coefficient, according to Formula (5). Figure 4d contrasts the evolution of the focal temperature with time for dynamic and constant \(\alpha_{\mathit{liter}}\). Before t = 1 s, the rate of the focal temperature rise is almost the same for simulations using dynamic \(\alpha_{\mathit{liter}}\) and constant \(\alpha_{\mathit{liter}}\). This may be due to the small change of the acoustic absorption coefficient during the early HIFU irradiation stage, as shown in Figure 4a. After t = 1 s, the focal temperature for dynamic \(\alpha_{\mathit{liter}}\) rises much faster than that for constant \(\alpha_{\mathit{liter}}\). When t = 3 s, the focal temperature is 65.94 \({}^{\circ}\)C for constant \(\alpha_{\mathit{liter}}\) and 85.53\({}^{\circ}\)C for dynamic \(\alpha_{\mathit{liter}}\). Figure 4e plots the shape of the thermal damage zone, representing the heated region for more than 240 min equivalent time at 43 \({}^{\circ}\)C. The thermal damage zone is an ellipse of 0.51 cm \(\times\) 0.12 cm size for constant \(\alpha_{\mathit{liter}}\), and an ellipse of 0.6 cm \(\times\) 0.16 cm size for dynamic \(\alpha_{\mathit{liter}}\). These phenomena indicate that dynamic \(\alpha_{\mathit{liter}}\) has a greater effect on the focal temperature and thermal damage zone as the  
Figure 3: Flowchart of the iterative method for coupling acoustic pressure and temperature calculation.  
HIFU irradiation time increases, compared with constant \(\alpha_{\mathit{liver}}\), which can be explained by the greater acoustic absorption coefficient being related to a greater the value of \(Q_{\mathit{ext}}\), higher focal temperature, and larger thermal damage zone. Figure 4f describes the axial profile of the thermal dose, and the black dotted line denotes the value \(\log_{10}(240)\) min. The axial length AB of the thermal damage zone is 0.51 cm for constant \(\alpha_{\mathit{liver}}\), and the axial length CD of the thermal damage zone is 0.6 cm for dynamic \(\alpha_{\mathit{liver}}\), which are consistent with the axial length of thermal damage zone in Figure 4e. Meanwhile, the thermal dose of ultrasonic focus for dynamic \(\alpha_{\mathit{liver}}\) is much greater than that for constant \(\alpha_{\mathit{liver}}\).