## 2 Female Breast Carcinoma
### Modelling
The most accurate dielectric and thermo-physical properties of female breast tissues, including the tumor tissue considered for this in silico simulation, are collected in Table 1. They were taken from the IT'S material properties database [61] for exciting an EM frequency for the dipole antenna array equal to \(f\) = 1 GHz. It should be emphasized that the values of all key electro-thermal parameters from Table 1, discussed in Section 3, are different for various types of human tissues (e.g., blood, fat, gland, skin, muscle, tumor) and determined experimentally in [62; 63]. It was assumed that the breast tumor had breast muscle tissue properties with different heat transfer rates (HTRs). The temperature-dependent HTR parameter in a tumor is due to the specific dense vascularity and blood perfusion of breast carcinoma [52; 64]. It is worth adding that the electrical parameters such as relative permittivity (\(\varepsilon_{\mathrm{r}}\)) and electrical conductivity (\(\sigma\)) strongly depend on the frequency of the external EM field produced by the matrix of the dipole antennas. This is due to the dispersion properties of human tissues and tissue relaxation processes occurring under the influence of an alternating EM field [62]. In our case, \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Tissue** & \(\varepsilon_{\mathrm{r}}\) & \(\sigma\) & \(\rho\) & \(C\) & \(k\) & HTR * & \(\rho_{\mathrm{b}}\)C\({}_{\mathrm{r}}\)\(\rho\)HTR & **HGR ** \\  & **(S/m)** & **(kg/m\({}^{3}\))** & **(J/kg/K)** & **(W/m/K)** & **(mL/min/kg)** & **(W/m\({}^{3}\)/K)** & **(W/kg)** \\ \hline Blood & 61.06 & 1.583 & 1050 & 3617 & 0.517 & 10,000 & 6.646\(\cdot\)10\({}^{5}\) & 0 \\ Breast & 5.41 & 0.053 & 911 & 2348 & 0.209 & 47 & 2710 & 0.728 \\ fat & 59.47 & 1.079 & 1041 & 2960 & 0.334 & 150 & 9884 & 2.323 \\ gland & 11.29 & 0.116 & 911 & 2348 & 0.211 & 33 & 1903 & 0.507 \\ Muscle & 54.81 & 0.978 & 1090 & 3421 & 0.495 & 37 & 2553 & 0.906 \\ Skin & 40.94 & 0.900 & 1109 & 3391 & 0.372 & 106 & 7441 & 1.648 \\ Tumor ** & 54.81 & 0.978 & 1090 & 3421 & 0.495 & see Equation (3) & see Equation (3) & 0.906 \\ \hline \hline \end{tabular}  
* heat transfer rate, ** heat generation rate, *** tumor was modelled using the muscle tissue parameters.  
\end{table} 
Table 1: Dielectric and thermal tissue parameters employed in a female breast phantom for dipole antennas working at a frequency \(f\) = 1 GHz [61]. Figure 3: Axial cross-section (_z_ = 0) through an MRI-derived breast phantom including the main breast tissues: skin (red), fat (green), breast fat (yellow), breast gland (blue), muscle (orange), tumor (pink). the temperature-dependence of breast parameters was omitted excluding the tumor blood perfusion defined in Equation (3) as a non-linear quantity. The entire model, including the female breast tissues with embedded tumor and dipole antenna array, consisted of \(265\times 379\times 259\) voxels of a volume 1 mm\({}^{3}\) each. The considered antennas matrix included eight dipole antennas forming an array with diameter of 10 cm surrounding the female breast model, as presented in Figure 4. All the dipole applicators placed around the breast were powered at the same frequency of 1 GHz, but each dipole could be individually tuned for amplitude and phase so that the appropriate power deposition could be focused on the treated tumor region or ROI, avoiding healthy tissues. The problem was solved using the Cartesian coordinate system (\(x,y,z\)), and the center of the dipole array was taken as a reference point (0, 0, 0). The [PERSON]'s equations [17] together with the modified [PERSON] bioheat model (Equation (1)), described in detail in Section 3 which follows, are supplemented by suitable boundary and initial conditions. The dipole antennas were considered to be perfect electric conductors (PECs). The computational domain was bounded by six perfectly matched layers (PMLs)--two in the direction of each axis, which leads to undistorted penetration of the EM field beyond the computing area.