## Theory
### Dynamic Tissue Properties  
The acoustic and thermal parameters of tissue were strongly dependent on tissue temperature, and many experimental data had been obtained [10; 11; 12; 13]. The data for acoustic absorption coefficient and sound speed were derived from measurements in liver tissue by [PERSON] [10] and [PERSON] [11], respectively. The polynomials to fit the acoustic absorption coefficient and sound speed in liver tissue to experimental data are [15]:  
\[\begin{array}{c}\alpha_{liver}=5.5367-2.9950\times 10^{-1}T+3.3357\times 10^{-2}T ^{2}-1.6058\times 10^{-3}T^{3}+3.4382\times 10^{-5}T^{4}\\ -3.2486\times 10^{-7}T^{5}+1.1181\times 10^{-9}T^{6}\qquad\qquad 30\ ^{\circ} \text{C}\leq\text{T}\leq 90\ ^{\circ}\text{C}.\end{array} \tag{8}\]\[\begin{array}{c}c_{\mathit{linear}}=1529.3+1.6856T+6.1131\times 10^{-2}T^{2}-2.2967 \times 10^{-3}T^{3}\\ +2.2657\times 10^{-5}T^{4}-7.1795\times 10^{-8}T^{5}\qquad\qquad 30\ ^{\circ}\mathrm{C} \leq\mathrm{T}\leq 90\ ^{\circ}\mathrm{C}\end{array} \tag{9}\]  
In this study, the experimental data for the change in the non-linearity parameter with temperature in the liver tissue are derived from measurements by [PERSON] (1988), and the experimental data for the changes in the specific heat capacity, thermal conductivity, and density with temperature in liver tissue are derived from measurements by [PERSON] (1988). We obtain the expressions of the temperature-dependent non-linearity parameter, specific heat capacity, thermal conductivity, and density respectively by the least squares polynomial fitting their experimental data in liver tissue:  
\[\begin{array}{c}\left(\frac{B}{A}\right)_{\mathit{linear}}=6.68-0.41448\ T+0.03364\ T^{2}-0.00101\ T^{3}+1.34407\times 10^{- 5}T^{4}\\ -6.35346\times 10^{-8}T^{5}\qquad\qquad 30\ ^{\circ}\mathrm{C}\leq\mathrm{T} \leq 75\ ^{\circ}\mathrm{C}\end{array} \tag{10}\]  
\[\begin{array}{c}C_{\mathit{linear}}=3600+53.55552T-3.96009T^{2}+0.10084T^{3}- 0.00106T^{4}\\ +4.01666\times 10^{-6}T^{5}\qquad\qquad 20\ ^{\circ}\mathrm{C}\leq\mathrm{T} \leq 90\ ^{\circ}\mathrm{C}\end{array} \tag{11}\]  
\[\begin{array}{c}K_{\mathit{linear}}=0.84691-0.02094T+3.89971\times 10^{-4}T^{2}-5.47451\times 10^{-7}T^{3}\\ -4.14455\times 10^{-8}T^{4}\ +2.97188\times 10^{-10}T^{5}\qquad 20\ ^{\circ} \mathrm{C}\leq\mathrm{T}\leq\ 90\ ^{\circ}\mathrm{C}\end{array} \tag{12}\]  
\[\begin{array}{c}\rho_{\mathit{linear}}=1084.09352-2.97434T+0.0042T^{2}+0.0029 3T^{3}-6.14447\times 10^{-5}T^{4}\\ +3.33019\times 10^{-7}T^{5}\qquad\qquad 20\ ^{\circ}\mathrm{C}\leq T \leq 90\ ^{\circ}\mathrm{C}\end{array} \tag{13}\]  
The polynomials above are shown in Figure 1, which have the validity in their own temperature range. In this study, their use is also strictly restricted to their respective temperature ranges.  
The variation of the blood perfusion rate with temperature and thermal damage can be described by:  
\[W_{b,\mathit{linear}}(T,\Omega)=W_{b,0}f_{T}f_{u} \tag{14}\]  
Figure 1: Temperature-dependent tissue properties in liver tissue.  
where \(W_{b,0}\) is the constitutive blood perfusion rate, 18.2 Kg m\({}^{-3}\) s\({}^{-1}\) for liver, and \(f_{T}\) is a dimensionless function that accounts for vessel dilation at slightly elevated temperatures, which can be approximated as [23; 24]:  
\[f_{T}=\left\{\begin{array}{c}4+0.6\;(T-42)\;37\;^{\circ}\mathrm{C}\leq T\leq 4 2\;^{\circ}\mathrm{C}\\ 4T\geq 42\;^{\circ}\mathrm{C}\end{array}\right. \tag{15}\]  
The blood perfusion rate increases as the temperature rises, but as tissue coagulation develops, it is decreased to zero due to the factor of thermal damage [25].  
In order to study the effects of dynamic tissue properties on HIFU hyperthermia, we compare the simulation using dynamic tissue properties with the conventional method using tissue properties with constant values. In this study, the constant values of tissue properties were obtained from the values of the above fitting formula at 37 \({}^{\circ}\mathrm{C}\). The values of the acoustic and thermal parameters are listed in Tables 1 and 2, respectively.