## 4 Materials and Methods
### The Electric Field from the Dipole Antenna  
The electro-thermal interaction of female breast tissues affected by the tumor with microwaves produced by a circular array of dipole antennas was solved using the finite-difference time-domain (FDTD) engine relying on the Maxwell's equations in a time domain, as described in [17]. Details of the FDTD paradigm based on the cubic cells can be found in several critical studies [20; 25; 65]. A matrix of 8 dipole antennas arranged symmetrically around the female breast phantom was used as the source of EM excitation (see Figure 4). Due to limited space around the analyzed breast tissue, the shortened-length, half-wave dipole antennas with a shortening factor established at value \(k_{\mathrm{s}}\) = 0.67 were employed in the model. Therefore, each antenna of the array operated at a working frequency of 1 GHz and consisted of two symmetrical dipole arms of length \(L_{\mathrm{d}}\) = 50 mm and radius \(r_{\mathrm{d}}\) = 1 mm, as well as an AC voltage source \(e_{i}(t)=A_{\mathrm{m}}\mathrm{cos}(2\pi\mathrm{f}t+\phi_{\mathrm{i}})\) supplying the dipole, as shown in Figure 12. Importantly, a dipole source induces currents flowing in opposite directions on the surface of the metal dipole arms. In turn, these currents cause the antenna radiation and the \(E\)-field distribution around the dipole (see Figure 13). Figure 11: The transient temperature distributions inside the tumors subjected to the eight-dipole antenna array after the optimization procedure. Figure 12: View of the dipole antenna including dimensions (\(d\) = 1 mm, \(L_{\mathrm{d}}\) = 50 mm, \(r_{\mathrm{d}}\) = 1 mm) and the sinusoidal voltage dipole source. In the popular FEM- and FDTD-based engines, the dipole antenna is modelled as a two-armed metal antenna with a perfect electric conductor (PEC). Thus, the zero value of the tangent component of the electric field strength (\(E_{x}\) = 0) is assumed on the surface of each dipole arm. Generally, a symmetrical transmission line with characteristic impedance of 50 \(\Omega\) supplies a dipole antenna. For numerical calculation purposes, modeling of the feeding coaxial cable is omitted. It is sufficient that each dipole is fed by a point-dipole sinusoidal voltage source with an internal load of 50 \(\Omega\), located between the dipole arms. In the FDTD-based software Sim4 Life [36], this was modelled using the edge port of length \(d\) = 1 mm (\(d<<\lambda\)) with a time-harmonic excitation signal defined by the amplitude \(A_{\text{m}}\) and phase \(\phi_{i}\) of the dipole source. It was assumed that the direction of this source was in opposition (i.e., in the \(x\)-direction) to the resulting vector of electric field strength \(\mathbf{E}=E_{x}\mathbf{e}_{x}\) between the antenna arms, and that its amplitude was equal to \(A_{\text{m}}\) = 1 V. Moreover, the initial input power values for the dipole sources were set at \(P_{i}\) = 1 W, and thus we were freely able to set the crucial parameters of the dipole antennas \(P_{i}\) and \(\phi_{i}\), while the amplitude \(A_{\text{m}}\) and the frequency \(f\) of dipole sources did not change. By changing of the antenna frequency \(f\), and thus the wavelength \(\lambda\) = _cf_, where \(c\) = 3\(\cdot\)10\({}^{8}\) m/s is the speed of light, we were able to influence the shape of the dipole antennas' radiation characteristics. Obviously, a higher frequency is related to a shorter wavelength and thus a smaller size of dipole antenna. In the case of the analyzed shortened 1-GHz-half-wave applicator, the \(E\)-field characteristics around a single dipole are depicted in Figure 13.  
As presented in Figure 13, the greatest values for the \(E\)-field appear around the excitation source and the ends of the dipole antenna arms. For better visibility, the near field dipole distribution scale has been limited to the value of \(E\) = 100 V/m.