## 4 Discussion
Nowadays, MRI thermal mapping based on PRF can provide in vivo temperature measurement with a precision of about 1 \({}^{\circ}\)C; hence, there is a possibility that detecting small tumors with a high Q\({}_{m}\), generally associated with malignant early-stage lesions, based on absolute temperature differences (compared to healthy tissue) could be a reality. However, it is important to remark that, despite the outstanding advances in technologies used to estimate internal temperature distributions, the reached resolution with most of techniques remains a challenge, not to mention the difficulties estimate the temperature of adipose tissue such as the breast [40; 43]. The use of PRF-based MRI with specific contrast agents based on paramagnetic nanoparticles tends to improve this precision to a tenth of a degree Celsius. In the cases discussed here, the temperature difference between a tumor with a Q\({}_{m}\) = 120,800 W/m\({}^{3}\) (Diam = 5 mm) and another with Q\({}_{m}\) = 60,830 W/m\({}^{3}\) (Diam = 6 mm) was only 0.04 \({}^{\circ}\)C, which was indistinguishable between them even with the most sophisticated equipment. In this key issue, the relevance of working with gradient data could be notable: the maximum gradient difference between the tumors mentioned \begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline \hline
**A.** & **B1.** & **B2.** & **B3.** & **C.** & **D.** & **E.** & **F.** & **G.** & **H.** \\ \hline \multirow{3}{*}{**Tumor Diam**} & **Location of Tumor Projected on** & **Arc Length of Maximum** & **Projection of (C) on** & **Dist. of (C) on** & **Arc Length from (D)** & **Arc Length of Maximum** & **Projection of (F) on \(x\)** & **Dist. of (F) on \(x\)** & **H.** \\  & **x Axis** & & **Temp Diff** & **x Axis** & **to Center** & **Grad Diff** & **Axis** & **(G) to** \\ \hline \multirow{3}{*}{**Tumor Diam**} & **Left Bound (P\({}_{\text{6H}}\))** & **Right Bound (P\({}_{\text{7H}}\))** & **Center** & & & & & \\  & **(P\({}_{\text{6H}}\))** & **Right Bound (P\({}_{\text{7H}}\))** & **Center** & & & & & \\ \hline
5 & 4.50 & 9.5 & 7.00 & 78.88 & 3.15 & 3.85 & 67.62 & 6.40 & 0.60 \\ \hline
6 & 4.00 & 10.00 & 7.00 & 78.64 & 3.25 & 3.75 & 66.51 & 6.80 & 0.20 \\ \hline
8 & 3.00 & 11.00 & 7.00 & 78.03 & 3.35 & 3.65 & 64.67 & 7.50 & \(-\)0.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Projection of magnitude of gradient peaks related to tumor (mm). above (located on the right boundary of tumor, P\({}_{6H}\)--Table 9) was 0.341 \({}^{\circ}\)C/mm, almost an order of magnitude higher than the maximum temperature difference. This would be probably measurable due to future improvements in the temperature mapping precision of existent technologies. In superficial temperature measurements, IR thermography reached sensitivities up to tens of mK. Precision was not a concern in this matter because temperature increases regarding healthy cases were in the order of tenths of degrees Celsius and temperature differences between studied cases were in the order of hundredths. Using an IR camera with a sensitivity of 20 mK, the tumors of 5 and 6 mm in diameter considered in this work could be differentiated in terms of size; however, as discussed above, in terms of surface measurements, neither temperature data nor its associated gradients could be related to tumor malignancy since there seemed to be no correlation with the tumor \(Q_{m}\). Another significant observation was obtained from the external temperature and gradient data. In Table 6, the most relevant parameters obtained from the arc lengths traced onto breast surfaces that matched with the tumor's center are listed. For all cases with tumors, the arc length corresponding to the maximum temperature differences regarding a healthy case are listed in column (B). They had the same value as the arc length that corresponded to the intersection with a healthy case on the gradient graphs, shown in column (D).