## 6 Results
### The Effect of Tumor Thermal Conductivity  
Table 6 shows the relationship between tumor thermal conductivity and skin surface temperature at different depths within the breast tissue and different breast sizes. At  
T1 depth, the temperature variations range from 0.82 \({}^{\circ}\)C (2.46%) for a tumor thermal conductivity of 5 milli (W/mK) and a small breast size of 7.5 cm to 1.12 \({}^{\circ}\)C (3.36%) for a conductivity of 1000 (W/mK) and a large breast size of 17 cm. Similarly, at T3 depth, the temperature variations range from 0.74 \({}^{\circ}\)C (2.22%) to 1.1 \({}^{\circ}\)C (3.3%). However, at T9 depth, they range from 0 \({}^{\circ}\)C (0%) variation to 0.09 \({}^{\circ}\)C (0.27%) variation. Temperature variations, however, are not detectable at this depth for the small breast size. These results indicate that as the tumor thermal conductivity increases, the temperature variations from the healthy breast also increase. Higher thermal conductivity allows for better heat transfer, leading to a more significant detectable temperature difference. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**(a)**} \\ \hline  & \multicolumn{2}{c}{**Skin surface temperature \({}^{\circ}\)C**} & \multicolumn{2}{c}{**Temperature variations from healthy breast \({}^{\circ}\)C**} \\ \hline  & \multicolumn{2}{c}{**Breast size**} & \multicolumn{4}{c}{**Breast size**} \\ \hline K milli (W/mK) & 7.5 cm & 10 cm & 17 cm & 7.5 cm & 10 cm & 17 cm \\ \hline Healthy & 33.35 & 33.35 & 33.35 & & & \\ \hline
5 & 34.2 & 34.18 & 34.17 & 0.85 & 0.83 & 0.82 \\  & & & & (2.55\%) & (2.49\%) & (2.46\%) \\ \hline
10 & 34.25 & 34.2 & 34.19 & 0.9 & 0.85 & 0.84 \\  & & & & (2.7\%) & (2.55\%) & (2.52\%) \\ \hline
62 & 34.32 & 34.29 & 34.26 & 0.97 & 0.94 & 0.91 \\  & & & & (2.91\%) & (2.82\%) & (2.73\%) \\ \hline
1000 & 34.47 & 34.45 & 34.44 & 1.12 & 1.1 & 1.09 \\  & & & & (3.36\%) & (3.3\%) & (3.27\%) \\ \hline  & \multicolumn{4}{c}{**(b)**} \\ \hline  & \multicolumn{2}{c}{**Skin surface temperature \({}^{\circ}\)C**} & \multicolumn{2}{c}{**Temperature variations from healthy breast \({}^{\circ}\)C**} \\ \hline  & \multicolumn{2}{c}{**Breast size**} & \multicolumn{4}{c}{**Breast size**} \\ \hline K milli (W/mK) & 7.5 cm & 10 cm & 17 cm & 7.5 cm & 10 cm & 17 cm \\ \hline Healthy & 33.35 & 33.35 & 33.35 & & & \\ \hline
5 & 34.11 & 34.1 & 34.09 & 0.76 & 0.75 & 0.74 \\  & & & (2.28\%) & (2.25\%) & (2.22\%) \\ \hline
10 & 34.21 & 34.18 & 34.16 & 0.86 & 0.83 & 0.81 \\  & & & (2.58\%) & (2.49\%) & (2.43\%) \\ \hline
62 & 34.31 & 34.27 & 34.25 & 0.96 & 0.92 & 0.9 \\  & & & (2.88\%) & (2.76\%) & (2.7\%) \\ \hline
1000 & 34.45 & 34.42 & 34.4 & 1.1 & 1.07 & 1.05 \\  & & & & (3.3\%) & (3.21\%) & (3.15\%) \\ \hline  & \multicolumn{4}{c}{**(c)**} \\ \hline  & \multicolumn{2}{c}{**Skin surface temperature \({}^{\circ}\)C**} & \multicolumn{2}{c}{**Temperature variations from healthy breast \({}^{\circ}\)C**} \\ \hline  & \multicolumn{2}{c}{**Breast size**} & \multicolumn{4}{c}{**Breast size**} \\ \hline K milli (W/mK) & 7.5 cm & 10 cm & 17 cm & 7.5 cm & 10 cm & 17 cm \\ \hline Healthy & 33.35 & 33.35 & 33.35 & & & \\ \hline
5 & NA & 33.35 & 33.35 & NA & 0 & 0 \\ \hline
10 & NA & 33.38 & 33.37 & NA & 0.03 & 0.02 \\  & & & & (0.09\%) & (0.06\%) \\ \hline
62 & NA & 33.4 & 33.41 & NA & 0.05 & 0.06 \\  & & & & (0.15\%) & (0.18\%) \\ \hline
1000 & NA & 33.44 & 33.42 & NA & 0.09 & 0.07 \\  & & & & (0.27\%) & (0.21\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: (a) Tumor thermal conductivity (T1 depth), (b) tumor thermal conductivity (T3 depth), and (c) tumor thermal conductivity (T9 depth). Because the thermal conductivity of biological materials describes how well the material conducts heat, these results show that the tumor thermal conductivity has a similar behavioral trend as the tumor heat generation rate discussed above, and hence a similar argument applies. As the thermal conductivity increases, the detectable temperature variations from the healthy breast also increase.