# Computer Simulations of Dual-Antenna Microwave Ablation and Comparison to Experimental Measurements
## 2 Materials and Methods
### Finite Element Modeling  
The finite element simulation modeling was based on COMSOL Multiphysics (version. 5.5; COMSOL Inc., Stockholm, Sweden). The dimensions of the 2.45 GHz microwave antenna are shown in Table 1. As illustrated in Figure 2, the origin of the coordinate system was set at the emission point of one antenna. The initial and boundary conditions used in the simulation can be found in reference [24]. Two temperature measurement points P1 and P2 were arranged, where P1 is the midpoint of the two emission points and P2 is the point symmetrical to P1 about a microwave antenna. The positions of P1 and P2 in the model were varied with the dual-antenna array spacing. The dual-antenna MWA results were simulated by coupling the solution of two physical fields, including the electromagnetic field and biological heat transfer field. The Helmholtz harmonic Equation (1) was used to calculate the electromagnetic energy deposition [25], and the specific absorption rate (SAR) in the tissue was calculated by Equation (2) [26]. The Pennes bioheat transfer Equation (3) [27] was used to solve for transient heat transfer within the tissue. \[\
abla\times{\mu_{r}}^{-1}(\
abla\times\mathbf{E})-{k_{0}}^{2}\bigg{(}\varepsilon _{r}-\frac{j\sigma}{\omega\varepsilon_{0}}\bigg{)}\mathbf{E}=0 \tag{1}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Property** & **Value** \\ \hline Outer diameter of the antenna & 1.9 mm \\ Inner diameter of the antenna & 1.3 mm \\ Distance from emission point to tip of antenna & 11.5 mm \\ Total length & 191.5 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of the 2.45 GHz microwave antenna. Figure 2: Geometric model of dual-antenna in MWA simulation. \[\rho\cdot SAR=\frac{1}{2}\sigma|\textbf{E}|^{2} \tag{2}\]  
\[\rho\cdot C_{\text{p}}\cdot\frac{\partial T}{\partial t}+\
abla\cdot(-k\
abla T)= \rho\cdot SAR+(T_{\text{b}}-T)c_{\text{b}}\omega_{\text{b}}+Q_{\text{met}} \tag{3}\]  
where \(\varepsilon_{r}\) is the relative permittivity of the biological tissue; \(\varepsilon_{0}\) is the relative permittivity of the vacuum; \(\sigma\) is the electrical conductivity of the biological tissue (S/m); **E** is the electric field strength (V/m); \(\mu_{\text{r}}\) is the relative magnetic permeability (\(\mu_{\text{r}}=1\)); \(k_{0}\) denotes the free space wave number; \(\rho\) is the tissue density (kg/m\({}^{3}\)); \(SAR\) is the specific absorption rate (W/kg); \(T\) denotes the temperature (\({}^{\circ}\)C); \(t\) is the time (s); \(C_{\text{p}}\) denotes the specific heat capacity (J/(kg\(\cdot^{\circ}\)C)); \(c_{\text{b}}\) denotes the specific heat capacity of blood (J/(kg\(\cdot^{\circ}\)C)); \(k\) is the thermal conductivity (W/(m\(\cdot^{\circ}\)C)); \(\omega_{\text{b}}\) is the blood perfusion (kg/(m\({}^{3}\)-s)); \(Q_{\text{met}}\) is the heat generated by tissue metabolism (W/m\({}^{3}\)). The subscript \"b\" represents the related parameters of blood. In the FEM model, in order to describe the real variation in input parameters, the temperature-dependent electrical parameters, the specific heat capacity function based on water content variation, and the linear thermal conductivity function were used in this study. The specific heat capacity function based on the change in water content is as follows [28]:  
\[C_{\text{p}}(T)= \begin{cases}C_{\text{p25}\text{C}}+k_{\text{w}}\cdot(T-25)&T \leq 70\ ^{\circ}\text{C}\\ C_{\text{p70}\text{C}}-\frac{\alpha}{\rho}\cdot\frac{\partial W(T)}{\partial T} &T>70\ ^{\circ}\text{C}\end{cases}  \tag{4}\]  
\[W(T)= \begin{cases}0.778\cdot(1-\exp(\frac{T-106}{3.420}))&70\ ^{\circ}\text{C} \leq T<100\ ^{\circ}\text{C}\\ 7.053-0.0640\cdot T&100\ ^{\circ}\text{C}\leq T<104\ ^{\circ}\text{C}\\ 0.778\cdot\exp(-\frac{T-8.0}{34370})&104\ ^{\circ}\text{C}\leq T\end{cases}