## 2 Materials and Methods
### Models of the Dynamics of Thermal and Radiation Fields  
The computational model should reproduce the process of measuring the brightness temperature \(T_{B}\) inside the tissue. The value of \(T_{B}\) is determined by the distributions of both the thermodynamic temperature and the electric field. Therefore, both of these quantities must be calculated self-consistently inside a multicomponent biological tissue with complex structure on very different scales from 10-20 cm to approximately 0.01 cm. Heat dynamics is determined by the heat conduction equation with different sources [33; 65; 88; 89; 90; 91]:  
\[\rho(\vec{r})C(\vec{r})\frac{\partial T(\vec{r},t)}{\partial t}=\
abla(\lambda (\vec{r})\
abla T)+Q^{(met)}(\vec{r},t)+Q^{(bl)}(\vec{r})+Q^{(can)}(\vec{r}), \tag{8}\]  
where \(\rho\) is the mass density, \(C\) is the heat capacity of tissue, \(T\) is the thermodynamic temperature, \(\lambda\) is the thermal conductivity coefficient of biological tissue, and \(\vec{r}=\{x,y,z\}\), \(\
abla\) is the nabla operator. We distinguish the following sources of heat source power density due to metabolic processes, produced by the metabolic processes in tissues (\(Q^{(met)}\)), the blood flows (\(Q^{(bl)}\)), and the cancerous tumors (\(Q^{(can)}\)), [\(Q\)] =W\(\cdot\)m\({}^{-3}\)[2; 92; 12; 93]. The distinctive feature of our approach is taking into account the spatial heterogeneity of physical parameters at different scales in the realistic multicomponent biological tissue, including the hierarchical circulatory system. This can significantly change the distributions of brightness and IR temperatures in our model compared to traditionally used multilayer models [72; 94]. The boundary conditions between the biological tissue (skin) and the environment are based on the continuity of the energy flux (Figure 8):  
\[\lambda(\vec{r})(\vec{n}\times\
abla T(\vec{r},t))=h\times(T_{air}-T(x,y,z,t)), \tag{9}\]where \(\vec{n}\) is the normal vector to the boundary of the interface \"biological tissue-environment\", \(h\) is the heat transfer coefficient (\(\text{W}/\text{m}^{2}\cdot K\)), and \(T_{air}\) is the ambient temperature. Measurements of internal and IR temperatures are carried out approximately 15 min after the start of the medical examination, when the patient gets used to the air temperature in the room [88]. The ambient temperature is a variable parameter when creating the dataset \(\left\langle\vec{G},\vec{F}\right\rangle\), so \(T_{air}\) is set similarly to other physical characteristics of biological tissues with \(T_{air0}=23\)\({}^{\circ}\)C, \(\delta T_{air\max}=2\)\({}^{\circ}\)C in accordance with the actual conditions of medical measurements. The second boundary condition is set for the pectoralis major muscle (see label 12 in Figure 5), for which the temperature \(T_{core}\) is fixed. The value of \(T_{core}\) is also variable when creating the vector of physical characteristics \(\mathcal{F}\), so that \(T_{core0}=37.5\)\({}^{\circ}\)C, \(\delta T_{air\max}=0.5\)\({}^{\circ}\)C (see (7)). Our numerical integration of the non-stationary Equation (8) together with (9) makes it possible to construct a quasi-stationary solution in a time interval of approximately 15 min, which is consistent with the medical examination technique. Calculations of the electric field distributions in the breast are based on the numerical integration of [PERSON]'s equations:  
\[\frac{\partial\vec{B}}{\partial t}+\
abla\times\vec{E}=0,\quad\frac{\partial \vec{D}}{\partial t}-\
abla\times\vec{H}=0,\quad\vec{B}=\mu\vec{H},\quad\vec{ D}=\varepsilon\vec{E}, \tag{10}\]  
where \(\vec{E}\) is the electric field, \(\vec{B}\) is the magnetic field, \(\varepsilon\) is the permittivity, and \(\mu\) is the magnetic permeability. The result of the numerical integration of Equation (10) is the stationary distribution of \(\vec{E}(\vec{r})\) inside the biological tissue, which is necessary to calculate the brightness temperature \(T_{B}\).