## 4 Discussion  
There are several tumor parameters to be predicted based on thermal information acquired from tumoral zones: depth, size, shape, malignancy grade, internal composition, metastasis stage, etc. However, first- or second-level sanitary attention would be desirable to know the most relevant malignancy probability and size, and if metastasis is present, although the last one can be deduced from the first two most of the times. In the analysis presented here, we tried to relate the temperature and gradient data, extracted from simulations that were nearly attached to real situations, to tumor characteristics. On the one hand, it was shown that internal and external temperature distributions are influenced by tumor size. In Figures 5a, 7a and 11a, the temperature behaviors along different paths (internal/vertical, internal/horizontal, and external arc, respectively) involving tumoral zone are presented and in all of them a direct relation between temperature increase and tumor size is observed: the smaller the tumor, the lesser the temperature increase. A Kruskal-Willis test was carried out to analyze the differences between the thermal distributions and healthy case. It was observed that a greater increase in temperature was presented with a tumor size of 8 mm with respect to tumor sizes of 5 mm and 6 mm (\(p<0.05\)). Nevertheless, between the tumors of 5 mm and 6 mm, a significant difference was not observed. It was even possible, based on data from the internal paths, to define the ZWTG, which is related with tumor size too: we proposed to mark the first zero-crossing point of the vertical gradient (called black point) as the beginning of this zone. This was mainly carried out due to this point representing the first change in the gradient's direction from a negative to positive one, which means a thermal tendency switch from tissue cooling (healthy tissue) to tissue heating (tumor). The distance between the black point and tumor \begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline \hline \multirow{3}{*}{**Tumor** **Diameter**} & **(A) Maximum Temperature Difference with Respect to Healthy Case** & **(B) Arc Length of Maximum Temperature Difference** & **(C) Arc Length of Zero Crossing in the Gradient Graph** & **(D) Arc Length of Intersection with Gradient of Healthy Case** \\ \hline (mm) & (\({}^{\circ}\)C) & (mm) & (mm) & (mm) \\ \hline
5 & 0.15 & 78.88 & 84.92 & 78.88 \\ \hline
6 & 0.18 & 78.64 & 85.55 & 78.64 \\ \hline
8 & 0.30 & 78.03 & 88.15 & 78.03 \\ \hline Healthy & – & – & 82.02 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters related to temperature and gradient behavior estimated on the arc length traced on breast surface. Figure 11: Temperature distribution over an arc length marked on breast surface (**a**) for breasts with different sizes of tumor and their corresponding temperature gradients (**b**). boundary was measured based on the vertical gradient; it was defined as the difference between the location of the black point and location of the maximum positive peak of the gradient that corresponded to the tumor beginning. To keep symmetry in the ZWTG, this distance was measured from the tumor end's to the healthy tissue after the tumor to mark the so-called blue point, which represents the end of the ZWTG. In Table 4, the distances between tumor boundaries and black and blue points are described for each tumor size in the footnotes. These distances added to tumor size results in the ZWTG length over the vertical path. Table 7 shows the ZWTG lengths measured on the vertical path for each tumor size based on the information extracted from the gradient. In all cases, ZWTG's vertical length doubles the tumor's diameter. For the horizontal path, a similar procedure was defined, and results close to those of the vertical case were obtained. In Table 8, the ZWTG horizontal lengths are listed for each tumor size; in this case, it also doubles the tumor diameter.