## 3 Results
### Results from Internal Analysis for Breast with Tumors and Healthy Tissue
#### 3.1.2 Horizontal Internal Path  
In the case of a horizontal path, Figure 6 shows the temperature distribution and its corresponding gradient for the same case as that detailed for the vertical path, a case of a tumor with a diameter of 5 mm. A horizontal path was drawn from one end to the other of the breast surface, and we ensured that it crossed the tumor center (Figure 3). At the initial point, P\({}_{0}\), located on the left side of the breast surface, the temperature was equal to 27.8 \({}^{\circ}\)C, increasing with depth over the path at an average rate of 0.11 \({}^{\circ}\)C/mm until the point corresponding to a coordinate x = 0 mm. At this point, the temperature was equal to 33.33 \({}^{\circ}\)C; this trajectory was followed over the healthy half of the breast. From this point (even a bit before) until the black point, a slight rise in the average rate was  
noticed due to the presence of tumor, reaching a value equal to 0.14 \({}^{\circ}\)C/mm. Nevertheless, between the black point and P\({}_{6}\) (which is the left boundary of tumor), the average rate tripled; it was equal to 0.41 \({}^{\circ}\)C/mm. In terms of gradient (punctual rate), the black point had a value equal to 24% of the maximum gradient, which was reached at point P\({}_{6}\). And in terms of temperature, this point compared to the same point in the healthy case presented a temperature increment of 2.3%. The distance between the black point and P\({}_{6}\) was equal to 2.52 mm. This was a value close to the 2.78 mm found on the vertical path and was defined as the length of a healthy surrounding ZWTG. For this reason, in the case of the horizontal path, the black point was also considered as the beginning of the healthy ZWTG. And taking the same distance (2.52 mm) from the right boundary of tumor, P\({}_{7}\), a blue point was marked to delimitate the warmed healthy zone in this orientation too.  
The area from point P\({}_{6}\) to point P\({}_{7}\) corresponds to the tumoral zone. The highest temperature in this region is 35.0 \({}^{\circ}\)C. This occurs at a coordinate x = 6.79 mm, almost at the tumor center. This point is marked with a green point, which corresponds, as expected, to a zero crossing in the temperature gradient graph.  
Points P\({}_{6}\) and P\({}_{7}\), corresponding to the left and the right boundaries of the tumor, coincided with the maximum positive (0.83 \({}^{\circ}\)C/mm) and the maximum negative (\(-\)0.89 \({}^{\circ}\)C/mm) values of the temperature gradient, respectively, and did so in the vertical case. The average increasing rate between P\({}_{6}\) and the green point was 0.16 \({}^{\circ}\)C/mm, and the average decreasing rate between green point and P\({}_{7}\) was \(-\)0.22 \({}^{\circ}\)C/mm.  
Figure 5: Temperature distributions (**a**) and corresponding gradients (**b**) along vertical path for cases with tumors of different sizes and a healthy case.  
Figure 6: Temperature distribution along horizontal path for the case of tumor with a diameter of 5 mm (**a**). Its corresponding temperature gradient (**b**).  
The average decrease rate between P\({}_{7}\) and the blue point was \(-0.56\)\({}^{\circ}\)C/mm; after this point, the average decrease rate diminished to a value of \(-0.23\)\({}^{\circ}\)C/mm until the other side of the breast surface, which presented a temperature of 27.8\({}^{\circ}\)C.  
In Table 5, the relevant values, analyzed previously for the breast with a tumor of 5 mm, are listed for all cases considering the horizontal path. In all tumor sizes, the black point is placed in the position corresponding to 24% of the maximum gradient value, which in all cases is reached at P\({}_{6}\). Regarding temperature, the black point corresponds to increments of 1.75% and 2.62% compared to the equivalent points in the corresponding healthy cases for tumors of 6 and 8 mm, respectively.  
Figure 7 shows temperature distributions along horizontal path and their respective gradients for all models, both cases with tumors and their respective healthy cases.