## 2 Materials and Methods
### Mathematical Modelling
Limitations of Pennes model are:  
* It considers the entire tissue to contain a uniform blood perfusion rate and can not reflect local characteristics;
* The fact remains that it is challenging to confirm blood perfusion rate of tissue;
* Numerical simulation of large vessels is affected by blood perfusion rate, so tissue temperature has apparent deviations. In this work, the model is mainly applied to heat transfer analysis of muscle, fat, and bone. They are all uniform biological tissues and do not contain large vessels. So, a single blood perfusion rate can meet simulation requirements. Additionally, there are enough experiments to measure the blood perfusion rate of three types of tissues. Therefore, the Pennes model is appropriate for this work. Biological tissue properties are shown in Table 1. Theoretically, parameters such as thermal conductivity and electrical conductivity change with the increase of temperature, but there is no consensus on the changes in muscle, fat, and bone parameters. Therefore, fixed parameters are used for simulation in this work. To explore the law of parameter change, we used the trend of liver thermal conductivity with temperature to simulate the temperature and thermal damage distribution during muscle ablation. For related content, please see the supplementary material (Figures S1-S3). Compared with the standard transient heat transfer equation, the Pennes model adds metabolic heat source, blood perfusion, and microwave heating, which are defined in Equation (8). \[\rho C_{p}\frac{dT}{dt}=\
abla\cdot(k\
abla T)+Q_{mw}+Q_{met}+\rho_{b}C_{p,p} \omega_{b}(T_{b}-T) \tag{8}\]  
where, \(\rho\) is the density of biological tissue, and \(C_{p}\) is heat capacity, and therefore, \(\rho C_{p}\) presents volume-specific heat of tissue. \(k\) is tissue thermal conductivity, and \(Q_{met}\) and \(Q_{mw}\) are tissue metabolism heat production and external microwave heat source. So, \(\rho_{b}C_{p,b}\omega_{b}(T_{b}-T)\) is convective cooling of blood perfusion. \(\rho_{b}C_{p,b}\) is the volume-specific heat of blood. \(\omega_{b}\) reveals the blood perfusion rate, and \(T_{b}\) is blood temperature entering tissue. The external microwave heat source is the impedance heat generated by the electromagnetic field:  
\[Q_{mw}=\frac{1}{2}\text{Re}\bigg{(}(\sigma-j\omega e)\overset{\rightarrow}{E} \times\overset{\rightarrow}{E}^{*}\bigg{)} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Materials** & **Unit** & **Muscle** & **Fat** & **Bone** \\ \hline Conductivity & S/m & 0.439 & 0.0254 & 0.394 \\ Dielectric constant & & 60.0 & 12.0 & 11.4 \\ Thermal conductivity & W/mK & 0.49 & 0.21 & 0.32 \\ Constant pressure heat capacity & J/kgK & 3421 & 2348 & 1313 \\ Density & kg/m\({}^{3}\) & 1090 & 911 & 1908 \\ Metabolic thermogenesis & W/m\({}^{3}\) & 700 & 400 & 0 \\ Blood perfusion rate & 1/s & 0.0008 & 0.0002 & 0.000833 \\ Reference & & [18; 30] & [18; 31] & [32] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Electrical and thermo-physical properties of different biomedical tissue in MWA modeling. Biological tissue at \(r=0\) is set as an axisymmetric boundary, satisfying \(\overset{\rightarrow}{n}\times\left(K_{eff,t}\
abla T_{t}\right)=0\) and \(\overset{\rightarrow}{n}\times\left(K_{eff,b}\
abla T_{b}-\left(\rho C_{p} \right)_{b}\mu T_{b}\right)=0\). The contact surface of biological tissue, ablation antenna, and the outer surface of biological tissue are thermally insulated boundaries, which satisfy \(-\overset{\rightarrow}{n}\times\overset{\rightarrow}{q}=0\).