## 2 Mathematical Model and Theory  
### Acoustic Heating  
Acoustic waves are the propagation forms of sound, which are essentially the transmission of energy in the medium. When acoustic waves propagate, the classic case is to pass through a lossless fluid medium with homogeneous density that satisfies the simple conservation of momentum, mass and energy [31]. However, the reality is that the acoustic medium is heterogeneous, and its SOS and ambient density are different. At the same time, the propagation of acoustic waves cannot ignore the loss of acoustic energy due to random thermal motion. In addition, acoustic waves passing through the compressible medium will cause dynamic fluctuations in variables such as density, velocity of acoustic particle, pressure, temperature, etc. The vibration and friction between the acoustic waves and the medium will convert a part of the acoustic energy into thermal energy and generate acoustic absorption physically with dispersion [32]. Since we use a ring ultrasound transducer array in this study, acoustic heating is realized by the dynamic delayed emission of acoustic pulse signals for each ultrasound transducer element according to Equation (1):  
\[(x_{i}-o_{x0})^{2}+(y_{i}-o_{y0})^{2}=R^{2},\]  
\[l_{i}=\sqrt{(x_{i}-o_{x})^{2}+(y_{i}-o_{y})^{2}},\]  
\[t_{0}=\frac{l_{0}}{v_{s}},\]  
\[\Delta t_{i}=t_{0}-\frac{l_{i}}{v_{s}}, \tag{1}\]  
where \((x_{i},y_{i})\) is the coordinate of a single element on the ring ultrasound transducer array with the radius of R, which has 256 elements ( \(i\) ranges from 1 to 256). \((o_{x0},o_{y0})\) is both the center of the ring ultrasound transducer array and the center of the human thigh model whereas \((o_{x},o_{y})\) is both the focus position and the center of the tumor. In addition, \(v_{s}\) stands for the SOS of tissue and \(l_{i}\) stands for the distance from each element to the focus. We take the time \(t_{0}\) calculated from the farthest distance \(l_{0}\) as a benchmark, which can ensure that the delay value of each element\(\Delta t_{i}\) is greater than or equal to 0 when transmitting signals. When we add the corresponding delay \(\Delta t_{i}\) to each ultrasound transducer element, all the transmitted signals can reach the focus of the target position at the same time. The ring ultrasound transducer array model applied in this study is illustrated in Figure 1. Furthermore, the simulation model is immersed in the water bath at a constant temperature of 30 \({}^{\circ}\)C.  
Figure 1: Schematic illustration of the ring ultrasound transducer array. The green ring on the periphery stands for the ring ultrasound transducer array of 256 elements. The flesh-colored cylinder indicates human thigh whereas the orange area indicates the tumor. Furthermore, the red dot is the center of the thigh cross section and the yellow dot is the center of tumor. The heat conduction equation is an important partial differential equation in HIFU heating, which describes how the temperature in a region varies over time. Thermal diffusion is the main factor needing consideration during heating. In addition, [PERSON] bioheat equation [33] accounts for many aspects, such as heat deposition due to ultrasound absorption and advective heat loss due to tissue perfusion (blood flowing through the tissue). Therefore, the heat conduction equation including the factor of vascular heat perfusion is given by:  
\[\frac{dT}{dt}=\frac{k}{c\rho}\cdot\
abla^{2}T-\frac{B}{c\rho}(T-T_{a})+\frac{Q }{c\rho}, \tag{2}\]  
where \(k\) is thermal conductivity in W/m\({}^{\circ}\)C, \(c\) is specific heat capacity in J/kg\({}^{\circ}\)C, \(\rho\) is medium density in kg/m\({}^{3}\). \(Q\) is the volume rate of heat deposition in W/m\({}^{3}\), which can be treated as the initial heat source used for hyperthermia. Furthermore, \(B\) is the product of blood density (kg/m\({}^{3}\)), blood specific heat capacity (J/kg\({}^{\circ}\)C) and blood perfusion rate (\(s^{-1}\)) whereas \(T_{a}\) is the blood ambient temperature set to 37 \({}^{\circ}\)C by default.