## 4 Materials and Methods
### Field from Annular Dipole Antenna Array and Optimization Procedure  
The considered hyperthermia system is formed using a circular array of 8 dipole antennas, each with a radius of \(R=100\) mm surrounding an anatomically correct female breast phantom as schematically viewed in Figure 15. The authors chose the antenna array radius arbitrarily to accommodate female breasts of different sizes inside the applicator. The dipoles were evenly distributed around the breast model on the \(yz\) plane and their sources were localized in the \(x=0\) plane and numbered. The antenna array position was chosen with regard to the antennas feeding points and the tumor center in order to achieve symmetrical tumor exposition. Before the optimization procedure, all dipole sources were running at the same frequency of \(f=1\) GHz, unit amplitudes of \(A_{\text{m}}=1\) V, and zero phases \(\phi_{i}\). Moreover, initially all applicators worked with the same input power level, set at \(P_{i}=1\) W, and the whole eight-element dipole array operated with a correspondingly higher MW power. The limitation of the total input power of the antenna array to the sum \(P_{i}=8\) W was derived from our earlier studies [17] as the best option to obtain healing temperature levels for hyperthermia, i.e., 42 \({}^{\circ}\)C.  
The drawings compared in Figure 16a,b depict, respectively, the electric and magnetic field distributions generated by the analyzed antenna matrix before optimization. To better illustrate the EM field in the ROI, we normalized the \(E\)- and \(H\)-field distributions to the maximum value and were given on a decibel scale. As expected, the same initial set points for the dipole sources resulted in symmetrical field distribution around the dipoles and homogeneous field distribution inside the antenna array. Only proper parameter optimization of the dipole antenna array will allow for EM energy deposition in the target tissue and thereby heat the breast cancer to healing temperatures. The EM energy deposition in the analyzed breast tissues with an immersed tumor is determined numerically for each dipole antenna applicator. The electric field \(\mathbf{E}\) (V/m) generated by a single dipole antenna supplied by voltage with a unit amplitude and no phase shift, at a specific location in the space given by a distance vector \(\mathbf{r}=x\mathbf{e}_{x}+y\mathbf{e}_{y}+z\mathbf{e}_{z}\), is expressed by the following formula [19]:  
\[\mathbf{E}_{n}(\mathbf{r})=\sum_{i}^{x,y,z}E_{n,i}(\mathbf{r})\exp[i\Phi_{n,i} (\mathbf{r})]\mathbf{e}_{i} \tag{5}\]  
where \(E_{n,l}\) (V/m) and \(\Phi_{n,l}\) (rad) are, respectively, the amplitude and the phase shift of the resulting electric field strength of the \(n\)-th applicator for each \(i\)-th orthogonal component of the Cartesian  
Figure 16: Normalized electric and magnetic field distributions from the antenna array when each dipole is working at \(f=1\) GHz and \(P_{i}=1\) W before the optimization procedure: (**a**) \(E\)-field; (**b**) \(H\)-field. Figure 15: Arrangement of dipole antennas around the breast with a tumor of radius \(r=2\) mm centered at (0, \(-12\),12) mm: (**a**): schematic sectional view; (**b**): dipole sources positions at \(x=0\) plane (antenna array radius \(R=100\) mm). coordinate system (\(x\),\(y\),\(z\)) and given location. Moreover, \(\mathbf{e}_{i}\) represents the unit vectors toward the individual axis of the rectangular coordinate system and \(\mathrm{j}=\sqrt{(-1)}\) is the imaginary unit.