## 3 Results and Discussions
### Human Skin with Malignancies  
Skin cancer, which includes melanoma, basal cell carcinoma and squamous cell carcinoma, frequently starts as precancerous lesions. It grows through several intermediate phases and can present as new growths or precancerous lesions, whereafter it establishes itself as a tumor on the outer surface of the skin. Firstly, it is seen in the basal layer (with a thickness of around 10 \(\upmu\)m), then it similarly develops on the other parts of neighboring cells of the epidermis layer. The neighboring medium contains the living epidermis (with a thickness of around 70 \(\upmu\)m) and the papillary dermis (with a thickness of around 80 \(\upmu\)m), as studied by [PERSON] et al. [21]. In the present work, consideration is given to a tool for the early detection of skin cancer, and the two intermediate phases of the development of cancer in human skin are analyzed. At first, the predictions are delivered for cases where the cancer appears only in the basal layer, and normal other layers of the human skin are conspired (Figure 6). Then, predictions are discussed for cases where the cancer develops similarly on both sides of the basal layer (i.e., the rest of the living epidermis, with a thickness of 70 \(\upmu\)m) and the papillary dermis (with thickness: 80 \(\upmu\)m), producing a 160-\(\upmu\)m total thickness of the affected skin tissue (Figure 7). Figure 6 presents temporal variations in the reflectance at x = 0 for the first case where the cancer is developed only in the basal layer (thickness: 10 \(\upmu\)m) in consideration of the three main types of malignancies (i.e., cutaneous melanoma (CM), nodular basal cell carcinoma (NBCC) and squamous cell carcinoma (SCC)). These skin malignancies differ in their optical properties, as tabulated in Table 3. Figure 6 shows that, for normal skin, the peak magnitude of the reflectance is obtained at time t = 3.85 ps. However, the peak of magnitudes in the existence of CM, NBCC and SCC malignancies in the basal layer manifest a little ahead of the normal tissue (at 0.06, 0.15 and 0.17 ps, respectively). In addition, these peak magnitudes of reflectance are less than that of the normal skin. By comparison to the last peak magnitude cited for the normal skin, the differences are 0.00154, 0.0016 and 0.002 for CM, NBCC and SCC, respectively (Figure 6). \begin{table}
\begin{tabular}{c c} \hline \hline
**Angular Discretization Scheme** & **Average Relative Error (\%)** \\ \hline FT4 & 7.28 \\ FT6 & 4.2 \\ FT8 & 1.8 \\ FT10 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Angular discretization effects on the average relative error (using \(400\times 21\times 21\) on spatial mesh). \begin{table}
\begin{tabular}{c c} \hline \hline
**Spatial Discretization Scheme** & **Average Relative Error (\%)** \\ \hline \((200\times 11\times 11)\) & 6.4 \\ \((200\times 21\times 21)\) & 5.2 \\ \((400\times 21\times 21)\) & 1.8 \\ \((400\times 41\times 41)\) & 1.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Spatial discretization effects on the average relative error (using the FT8 on angular mesh). Figure 6: Temporal signals of normalized reflectance from the normal skin and skin with malignancy in the basal layer (thickness: 10 \(\upmu\)m). In the second case, where the cancer has spread similarly on both sides of the basal layer (consisting of the living epidermis and papillary dermis) up to a total thickness of 160 \(\upmu\)m, Figure 7 shows the temporal signals of the reflectance for human skin for CM, NBCC and SCC. Variances in the peak magnitudes between the normal skin and the skin with CM, NBCC and SCC malignancies are clearer than they are in the previous case (i.e., where these three types of malignancies are only located in the basal layer (Figure 6)). For the CM, NBCC and SCC malignancies, these variances are 0.026, 0.0305 and 0.0351, respectively. Thus, the difference is approximately 16 times more than the same difference for the first case where the malignancy exists only in the basal layer (Figure 6).