## 4 Materials and Methods
### The Modified Pennes Bioheat Transfer Model  
The most common model describing the heat exchange inside biological objects in macroscopic terms is the [PERSON] heat transfer model derived from [PERSON]'s heat conducting equation in 1948 [66]. The classic Pennes' equation took into account two biologically relevant heat sources reflecting the cooling effect of blood flowing in the arterial and venous blood vessels, as well as tissue metabolism. To estimate the thermal characteristics of female breast tissues in building the analyzed phantom, researchers successfully employed the modified [PERSON] bioheat equation in the following form [63, 67]:  
\[\rho\mathcal{C}\frac{\partial T}{\partial t}+\
abla\cdot(-KVT)=\rho_{b}C_{b} \rho\mathrm{HTR}(T_{b}-T)+\rho\mathrm{HGR}+\rho\mathrm{SAR} \tag{1}\]  
The first term of the above equation describes the phenomenon of heat accumulation in breast tissues during treatment time \(t\)[s], which is specified by such tissue parameters as the specific heat C (J/kg/K) and mass density \(\rho\) (kg/m\({}^{3}\)). The second element relates to the heat conduction phenomenon inside each tissue, which is significantly influenced by the tissue's thermal conductivity \(k\) (W/m/K). The third element describes the heat exchange between the individual tissues and arterial blood and determines the heat loss caused due to the cooling effect of blood flow through the tissue (commonly known as perfusion). These losses are proportional to the heat transfer rate HTR (mL/min/kg) and the temperature difference between arterial blood (\(T_{b}\)) and tissue (\(T\)) [66]. The last two elements of the modified Pennes' equation apply to heat generation due to the metabolic activity of tissue dependent on the heat generation rate (HGR; W/kg) as well as due to external EM sources (in our case, multi-element dipole antenna array) based on specific absorption rate (SAR; W/kg) defined in Equation (2). The last element \(Q=\rho\mathrm{SAR}\) (W/m\({}^{3}\)), commonly called external heat generation, relates to the coupling to the EM field, and from a theoretical point of view, it is a volumetric power density. Notably, the effective SAR parameter as a measure of the EM energy deposited by the unit mass of the biological objects a under unit period of time is written as follows [17]:  
\[\mathrm{SAR}=\frac{\mathrm{d}}{\mathrm{d}t}\bigg{(}\frac{\mathrm{d}W}{\mathrm{ d}m}\bigg{)}=\frac{\mathrm{d}}{\mathrm{d}t}\bigg{(}\frac{\mathrm{d}W}{\rho \mathrm{d}V}\bigg{)}=\frac{\sigma}{2\rho}|\mathbf{E}|^{2}=\frac{\sigma}{2\rho }\mathbf{E}\cdot\mathbf{E}^{*}\sim\frac{\mathrm{d}T}{\mathrm{d}t} \tag{2}\]  
Figure 14: The main frequency-dependent characteristics of the shortened 1-GHz half-wave dipole antenna: (**a**) the reflection coefficient (\(S_{11}=-16.11\) dB); (**b**) the input impedance (\(Z_{\mathrm{in}}=63.8\)\(\Omega\)). where \(W\) (J) refers to the energy absorbed by the breast tissue phantom with a specified volume \(V\) (m\({}^{3}\)), mass m (kg), and mass density \(\rho\) (kg/m\({}^{3}\)). Moreover, \(|\mathbf{E}|\) (V/m) corresponds to the magnitude of the total electric field produced by the antenna array as the superposition of each dipole, \(\sigma\) (S/m) stands for electrical conductivity of the body, and \(t\) (s) is the time of EMF exposure. Importantly, the SAR value is reflected in the increase in temperature distribution for the breast tissue \(T\) (K), as shown in the modified Pennes equation (Equation (1)). Therefore, the measurement and monitoring of this parameter may play an important role in terms of establishing EM hazards and human tissue safety [25; 68; 69; 70].