## 3 Results
### Influence of Tumor Spatial Location on the Brightness Temperature  
The brightness temperature \(T_{B}\) is an integral value and depends both on the thermodynamic temperature \(T\) and on the distribution of the electric field strength \(\vec{E}\) in the biological tissue. The factor of the spatial location of the tumor inside the breast significantly affects the results of temperature measurements and diagnosis. If two models (\(a\) and \(b\)) contain tumors at different depths \(L^{(cam)}\), then the deviations of the brightness temperature values in these models can be conveniently characterized by the formula  
\[\psi_{ab}=\frac{1}{N}\sum_{i=0}^{N=8}\frac{|T_{i}^{a}-T_{i}^{b}|}{T_{i}^{b}}, \tag{18}\]  
where the \(i\)th index denotes the position of the antenna on the surface of the breast according to the measurement technique (see Figure 10). Table 2 contains the results of calculations using Formula (18), which shows that the maximum deviations in the brightness temperature distribution can reach almost 6 percent, which is significant for medical diagnosis. Figure 19 shows the results of calculating the efficiency (15) of three machine learning methods for diagnosing the presence of a tumor. The dataset includes the results of modeling brightness and IR temperatures according to the classical measurement scheme (see Figure 10a). The efficiency exceeds 0.75 for a radius of 1 cm and larger, and the scatter for all three algorithms is within 5 percent. Even small tumors with radius of \(R=0.5\) cm allow for the correct definition of the class \"Cancer\" with probability of 62.5 percent. Support vector machine (SVM) gives a better result compared to other machine learning methods (KNN, NBC), which indicates that this method is better suited for this type of problem and for this training dataset structure. The gain of the SVM method with respect to NBC is 10 percent for the dataset with tumor \(R=0.5\) cm, which is significant for the \begin{table}
\begin{tabular}{c c c c c c} \hline \(\mathbf{L^{cam}}\) & **1 cm** & **2 cm** & **3 cm** & **4 cm** & **5 cm** \\ \hline
1 cm & 0 & 0.016 & 0.027 & 0.041 & 0.059 \\
2 cm & 0.016 & 0 & 0.011 & 0.026 & 0.043 \\
3 cm & 0.027 & 0.011 & 0 & 0.015 & 0.033 \\
4 cm & 0.041 & 0.026 & 0.015 & 0 & 0.019 \\
5 cm & 0.059 & 0.043 & 0.033 & 0.019 & 0 \\ \hline \end{tabular}
\end{table} 
Table 2: Relative brightness temperature deviations \(\psi_{ab}\) for two models with different tumor localizations in depth (\(L^{(cam)}\)). Figure 18: Thermodynamic temperature distributions at different depths from the nipple on flat sections: \(L^{(cam)}=2\) cm (**a**); \(L^{(cam)}=3\) cm (**b**); \(L^{(cam)}=4\) cm (**c**); \(L^{(cam)}=5\) cm (**d**). The tumor is in the vicinity of measurement point “8”. problem of medical diagnostics. The efficiency of the k-nearest neighbor method becomes unacceptable for small tumor radii (\(R\leq 0.5\) cm). Figure 20 demonstrates the integral result of evaluating the possibility of detecting a tumor depending on the size and specific energy release, which is determined by the tumor doubling time. Thick lines of different colors show the boundaries separating the parameter areas at which one or another method can detect a tumor. Heat release rates of \(Q^{(can)}\) = 30,000 W m\({}^{-3}\), which is typical of fast growing tumors doubling in 100 days or less, allow for detection of tumors up to 1 cm in diameter. The classification of even smaller tumors requires a transition to the analysis of feature spaces, an increase in the sample size, and the use of heuristics [22; 63; 64]. Feature spaces are built in the form of matrices of various dimensions, the elements of which are temperature differences at different measurement points (see Figure 10). Moreover, both brightness temperature and infrared temperature are used. We note study [94], in which a tumor with a radius of 0.5 cm is effectively detected by the MWR method when it is located at a depth of no more than 2.8 cm, which is consistent with our results.