## 2 Materials and Methods
### Finite Element Simulation
#### 2.2.2 Mathematical Model  
The ablation zone is formed by the deposition of electromagnetic wave energy around the tissue, which causes the temperature of the tissue to rise. Therefore, the electromagnetic field effect, biochemical effect and tissue specific absorption rate need to be considered in the simulation model.  
The simulation model of this study uses two physical fields of electromagnetic wave and biological heat transfer, and the temperature distribution of lung tissue after microwave ablation is calculated through the coupling of these two physical fields.  
Solving the electromagnetic field simulation uses the wave equation of the plane transverse magnetic field wave [26,27]. The formula is as follows:  
\[\
abla\times\left[\left(\varepsilon_{\mathrm{r}}-\frac{\mathrm{j}\sigma}{ \omega\varepsilon_{0}}\right)^{-1}\
abla\times\mathrm{H}_{\varphi}\right]- \mu_{\mathrm{r}}\mathrm{k}_{0}^{2}\mathrm{H}_{\varphi}=0 \tag{1}\]  
In this formula, \(\varepsilon_{\mathrm{r}}\) is the relative permittivity of lung tissue, \(\sigma\) is the electrical conductivity of the tissue (S/m), \(\omega\) is the angular frequency of the microwave, \(\varepsilon_{0}\) is the relative permittivity of vacuum (\(8.85\times 10^{-12}\) F/m), \(\mathrm{H}_{\varphi}\) is the magnetic field strength (A/m), \(\mu_{\mathrm{r}}\) is the relative permeability (\(\mu_{\mathrm{r}}=1\)), \(\mathrm{k}_{0}\) is the wave number of free space (rad/m).  
\[\mathrm{SAR}=\frac{\sigma_{\mathrm{lung}}}{2\rho}\left|\frac{\mathrm{j}}{ \mathrm{E}}\right|^{2} \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Property** & **Value** \\ \hline Diameter of the central conductor & 0.29 mm \\ Inner diameter of the outer conductor & 0.94 mm \\ \hline Outer diameter of the outer conductor & 1.19 mm \\ \hline Diameter of catheter & 1.79 mm \\ \hline \hline \end{tabular}
\end{table}
Table 2: Size of microwave antenna.  
Figure 4: Two-dimensional lung tissue model.  
SAR is the specific absorption rate of lung tissue to microwave, where \(\sigma_{\rm lung}\) is the electrical conductivity of the lung tissue, \(\rho\) is the density of tissue and E is the electric field strength.  
The temperature rise of the tissue is caused by the deposition of microwave electromagnetic energy in the tissue, so the heat transfer of biological tissue under microwave radiation can be described by the Pennes equation [28]. The equation is as follows:  
\[\rho\rm c\frac{\partial T}{\partial t}=\
abla\cdot(k\
abla T)+\omega_{b}C_{b}(T _{b}-T)+q_{m}+q_{r} \tag{3}\]  
In the formula, T is the temperature (\({}^{\circ}\)C), \(\rho\) is the density (kg/m\({}^{3}\)), C is the specific heat (J/kg \({}^{\circ}\)C) and k is the heat conductivity of tissue (W/m \({}^{\circ}\)C), \(\omega_{b}\) is the blood perfusion rate (kg/m\({}^{3}\) s), \(C_{b}\) is the blood specific heat, \(T_{b}\) is the blood temperature, \(q_{m}\) is the heat production rate of tissue metabolism, and \(q_{r}\) is the external heat source. The relationship between \(q_{r}\) and SAR can be expressed as follows:  
\[q_{r}=\rho\rm SAR=\frac{1}{2}\sigma_{\rm lung}\bigg{|}_{E}\bigg{|}^{2} \tag{4}\]  
Therefore, the simulation results of microwave ablation can be calculated by solving the electromagnetic wave and biological heat conduction model equations, respectively.