## 3 Results
### Results from Internal Analysis for Breast with Tumors and Healthy Tissue  
#### 3.1.1 Vertical Internal Path  
To illustrate temperature and gradient behaviors, the case of a breast model containing a tumor with a diameter of 5 mm was selected. In Figure 4, the temperature distribution and its corresponding gradient are plotted along a vertical path crossing the tumor. Analyzing the temperature distribution and the gradient it produced from the chest wall to the breast surface, it was noticed that a temperature of 37 \({}^{\circ}\)C was established at the thoracic region (P\({}_{0}\)). This slowly decreased (at an average rate of \(-\)0.06 \({}^{\circ}\)C/mm) to a minimum value of 34.42 \({}^{\circ}\)C at a point (black point) located 2.78 mm before P\({}_{3}\), which corresponded to the fat-tumor interface (Figure 4a). In the temperature gradient graph (Figure 4b), as expected, this black point corresponded to a zero crossing. This point was defined as the beginning of the zone warmed by tumor growth (ZWTG). From this point until P\({}_{3}\), the temperature rose at an average rate of 0.19 \({}^{\circ}\)C/mm, producing an abnormal thermal distribution in the healthy zone surrounding the tumor.  
The maximum rate was reached at P\({}_{3}\), which corresponded to the tumor boundary, and was equal to 0.55 \({}^{\circ}\)C/mm. After P\({}_{3}\) and until P\({}_{4}\), the temperature continued increasing, but the average rate presented a slight decrease; it was equal to 0.13 \({}^{\circ}\)C/mm. The warmest point in the tumor zone was in P\({}_{4}\), with a temperature of 35.13 \({}^{\circ}\)C. From this point, the temperature decreased at an average rate equal to \(-\)0.17 \({}^{\circ}\)C/mm until P\({}_{5}\), a point that corresponded to the interface between a galactophore duct and the tumor. Near to the tumor-fat boundary, P\({}_{6}\), there was a zone inside the tumor (with a depth of 1.03 mm measured from the boundary) with the largest average decreasing rate (\(-\)0.52 \({}^{\circ}\)C/mm), reaching a maximum value of \(-\)1.14 \({}^{\circ}\)C/mm at exactly the boundary. Following the path into the healthy fatty tissue, the average decreasing rate in the next 2.78 mm (assessed to consider a symmetrical heating profile before and after tumor formation) was equal to \(-\)0.77 \({}^{\circ}\)C/mm. The end point of the 2.78 mm segment is marked with a blue point in the figure. Thus, the ZWTG is considered to be the zone between black and blue points. The final segment, from the blue point to the breast surface (P\({}_{f}\)), was composed of fat and skin tissues, where temperature continued decreasing at an average rate equal to \(-\)0.38 \({}^{\circ}\)C/mm. The temperature on the breast surface was 27.67 \({}^{\circ}\)C.  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Tumor Size** & \(P_{0}\) & \(P_{1}\) & \(P_{2}\) & \(P_{3}\) & \(P_{4}\) & \(P_{5}\) & \(P_{6}\) & \(P_{7}\) & \(P_{8}\) & _Pf_ \\
**(Diameter** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** \\ \hline
5 & \(-\)36.23 & \(-\)34.10 & \(-\)9.34 & \(-\)6.76 & \(-\)4.2 & \(-\)2.14 & 4.5 & 9.5 & 34.1 & 36.23 \\ \hline
6 & \(-\)36.66 & \(-\)34.54 & \(-\)9.62 & \(-\)6.88 & \(-\)4.56 & \(-\)2.14 & 4 & 10 & 34.54 & 36.66 \\ \hline
8 & \(-\)37.47 & \(-\)35.40 & \(-\)10.05 & \(-\)7.25 & \(-\)4.76 & \(-\)2.06 & 3 & 11 & 35.40 & 37.48 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The X coordinates of the points of interest for tumors with a diameter of 5, 6, and 8 mm (Horizontal path).  
In Table 4, the relevant values of temperature and gradients for all tumor sizes along vertical path are summarized. In all cases, the maximum positive and negative gradient values were identified at the posterior (P\({}_{3}\)) and anterior (P\({}_{6}\)) boundaries of the tumor, respectively. Temperature distributions along vertical paths for all cases, with tumor and healthy, and their respective gradients are displayed in Figure 5.