# Comparative Study of Minimally Invasive Microwave Ablation Applicators
## 2 Materials and Methods
### Simulation Framework and Methods for Applicator Analysis  
Microwave ablation (MWA) involves the absorption of electromagnetic energy by biological tissues, leading to localized heating. The electromagnetic field distribution is governed by [PERSON]'s equations:  
\[\
abla\mathrm{x}\overset{\rightarrow}{\mathrm{E}}=-\mathrm{j}\omega\mu \overset{\rightarrow}{\mathrm{H}} \tag{1}\]  
\[\
abla\mathrm{x}\overset{\rightarrow}{\mathrm{H}}=\overset{\rightarrow}{ \mathrm{J}}+\mathrm{j}\omega\varepsilon\overset{\rightarrow}{\mathrm{E}} \tag{2}\]  
where \(\overset{\rightarrow}{\mathrm{E}}\) and \(\overset{\rightarrow}{\mathrm{H}}\) are the electric and magnetic fields, \(\varepsilon\) is the electrical permittivity, \(\mu\) is the magnetic permeability, \(\omega\) is the angular frequency and \(\overset{\rightarrow}{\mathrm{J}}\) is the current density. These equations are coupled with tissue dielectric properties to calculate the power deposition in tissues, quantified as Specific Absorption Rate (SAR) [51]:  
\[\mathrm{SAR}=\frac{\sigma}{2\rho}\,|\,|\overset{\rightarrow}{\mathrm{E}}\,|\,|^ {2} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Design** & **Values of Parameters (mm)** & **Overall Diameter (mm)** & **Frequency (GHz)** & **Power (W)** \\ \hline Monopole & 1 = 140, \(\mathrm{lm}=12.7\) & 2.18 & 2.45 & 40 \\ \hline Capacitive capped monopole & L = 6.2, \(\mathrm{r}=1\), d = 1 & 2.797 & 5.8 & 10 \\ \hline Multi-slot & L1 = 4.8,L2 = 19,L3 = 18,dct = 1.1, N = 10, & 2.6 & 2.45 & 30 \\ \hline Dual-slot & L1 = 1, L2 = 9, L3 = 1.44, L4 = 4.5, L5 = 0.8 & 1 & 2.45 & NA \\ \hline Triaxial & If = 12.5, \(\mathrm{lm}=23.5\), inner and outer & 3.5 & 1.9 & 40 \\ \hline Choke dipole & \(\mathrm{lc}=30\), \(\mathrm{lf}=8\), \(\mathrm{lm}=23\) inner and outer & 3.5 & 1.9 & 40 \\  & diameter of choke = 2.5 and 3.2 & & & \\ \hline Balun-free base-fed monopole & 13 = 26, 12 = 35, 11 = 4, \(\mathrm{g}=4\), \(\mathrm{lh}=20\), N = 10 & & \\  & number of turns of helix, inner diameter of & 3.1 & 1.9 & 40 \\  & copper tube = 0.876 & & & \\ \hline FSD & hs = 23.5, \(\mathrm{hb}=10.4\), \(\mathrm{g}=2\), \(\mathrm{ha}=10.4\), inner & 3.5 & 1.9 & 42 \\  & and outer diameter of sleeve = 2.5 and 3.2 & & & \\ \hline Balun-free helical dipole & Im = 24, \(\mathrm{lh}=18\), N = 9.9 number of turns & 2.8 & 1.9 & 42 \\ \hline Helical open waveguide & \(\mathrm{lm}=7.61\), N = 4 number of spirals, & & & \\ \(\mathrm{hsp}=12\) mm, 1 = 56.56 mm, inner diameter & 7.96 & 0.915 & 25 \\  & of copper tube = 2.546 mm & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Details of the designs’ parameter values (all parameters listed in Table 2 correspond to magnitudes depicted in Figure 1 for the respective applicators). where \(\sigma\) is the tissue conductivity \(\overset{\rightarrow}{\mathrm{E}}\) is the electric field strength and \(\rho\) is the tissue density. The resultant heat diffusion in the tissue is governed by the [PERSON]'s bioheat equation:  
\[\rho\mathrm{c}\frac{\mathrm{d}\mathrm{T}}{\mathrm{d}t}=\
abla\cdot\mathrm{k} \
abla\mathrm{T}+\mathrm{Q}-\omega_{\mathrm{bl}}\mathrm{c}_{\mathrm{bl}}( \mathrm{T}-\mathrm{T}_{\mathrm{bl}})+\mathrm{Q}_{\mathrm{meta}} \tag{4}\]  
where \(\rho\) and \(\mathrm{c}\) are the tissue density and specific heat, \(\mathrm{T}\) is the temperature, \(\mathrm{k}\) is the thermal conductivity, \(\mathrm{Q}\) represents the heat generated by electromagnetic absorption (SAR), the perfusion term accounts for cooling by blood flow \(\omega_{\mathrm{bl}}\mathrm{c}_{\mathrm{bl}}(\mathrm{T}-\mathrm{T}_{\mathrm{ bl}})\), and \(\mathrm{Q}_{\mathrm{meta}}\) reflects tissue metabolism. In the electromagnetic simulations, boundary conditions of Perfectly Matched Layers (PMLs) are applied at the computational domain boundaries to absorb outgoing waves and prevent artificial reflections. For thermal boundary conditions, a constant temperature is assumed at the tissue surface, set to body temperature (37 \({}^{\circ}\)C).