## 2 Materials and Methods
### Mathematical Modelling  
In coaxial cables, the propagation characteristic of electromagnetic waves is a transverse electromagnetic field (TEM). It is defined that a time-harmonic physical field with complex amplitude contains phase information, so electric and magnetic fields are Equations (1) and (2), respectively. \[E=e_{r}\frac{C}{r}e^{i(\omega t-kz)}k \tag{1}\]  
\[H=e_{\varphi}\frac{C}{r_{Z}}e^{i(\omega t-kz)} \tag{2}\]  
Time-averaged power flow in coaxial cable is Equation (3). \[P=\int_{r_{inner}}^{r_{outer}}\text{Re}\bigg{(}\frac{1}{2}E\times H^{*}\bigg{)} 2\pi rdr=e_{Z}\pi C^{2}/Z\ln(r_{outer}/r_{inner}) \tag{3}\]  
where \(r\), \(\varphi\) and \(z\) are the coordinate variables of the established cylindrical coordinates, and \(z\) is also the direction of wave propagation. \(\omega\) presents the angular frequency, \(k\) reveals propagation constant, and \(Z\) introduces wave impedance of dielectric. \(r_{inner}\) and \(r_{outer}\) are the inner radius and outer radius of the dielectric. In biological tissues, the electric field has a limited axial component, and the magnetic field has an omnidirectional angular direction. In a coaxial cable, the electric field is in the radial direction. While in biological tissues, it is in radial and axial directions. Therefore, an axisymmetric transverse magnetic field (TM) is used to model biological tissues and coaxial cables. Fluctuation direction is scalar \(H_{\varphi}\). TM is defined in Equation (4). \[\
abla\times\Big{(}(\varepsilon_{r}-j\sigma/\omega\varepsilon_{0})^{-1}\
abla \times H_{\varphi}\Big{)}-\mu k_{0}^{2}H_{\varphi}=0 \tag{4}\]  
where \(H\) is magnetic field strength, \(\varepsilon\) and \(\varepsilon_{0}\) are relative permittivity and free space permittivity, and \(\sigma\) reveals electrical conductivity. \(\omega\), \(\mu\), and \(k_{0}\) are angular frequency, relative permeability, and space wave number. In this work, \(\varepsilon\) of the conduit is 2.6, and \(\varepsilon\) of the dielectric is 2.03. To eliminate numerical error, the boundary condition of the inner conductor and outer conductor in the coaxial cable is set as an ideal electric conductor, which satisfies \(\overset{\rightarrow}{n}\times\overset{\rightarrow}{E}=0\). Boundaries of biological tissues and catheters are considered scattering boundary conditions to eliminate boundary reflections as defined in Equation (5). \[\overset{\rightarrow}{n}\times\big{(}\
abla\times\overset{\rightarrow}{E} \big{)}-jk\overset{\rightarrow}{n}\times\big{(}\overset{\rightarrow}{E} \times\overset{\rightarrow}{n}\big{)}=0 \tag{5}\]  
Central axis (r = 0) of coaxial cable adopts symmetrical boundary conditions:  
\[\left\{ \begin{array}{l}\overset{\rightarrow}{E}_{r}=0\\ \partial\overset{\rightarrow}{E}_{Z}/\partial r=0\end{array} \right. \tag{6}\]  
On the top of the dielectric (z = 40 mm, d = 0.135 mm\(\sim\)d = 0.470 mm), the boundary condition is set to the port boundary condition of the microwave power to model the feeding point. Input power is set to be consistent with the clinical experiment, so \(Pin=50\) W. This method is equivalent to the first-order low-reflection boundary condition using \(H_{\varphi 0}\) as an input:  
\[\overset{\rightarrow}{n}\times\sqrt{\varepsilon\overset{\rightarrow}{E}}- \sqrt{\mu}H_{\varphi}=-2\sqrt{\mu}H_{\varphi 0} \tag{7}\]  
where \(H_{\varphi 0}=\sqrt{P_{ap}Z/\pi r\ln(r_{outer}/r_{inner})}/r\). Unlike conventional solid and fluid heat transfer, biological tissues have blood flow, basal metabolism, and flow of body fluids due to vascular network system. Its heterogeneity and anisotropy make the heat transfer model of biological tissue more loaded, so it is also more difficult to simulate than ordinary materials. Some classic heat transfer models are the Weinbaum-Jiji model [22], the Klinger model [23], the Khaled-Vafai model [24], the Pennes model [25], and the Wulff model based on porous media theory [26]. The Pennes model has been widely applied in biological tissue models, and there have been many related studies [27; 28; 29].