## 4 Conclusions  
This works presents a numerical radiative heat transfer analysis of an ultrafast laser interaction with heterogeneous human tissue, with the goal of developing an optical biopsy for the early detection of skin cancer. Human brain and skin tissues subjected to ultrafast lasers have been analyzed using a formulation based on the FTn finite volume method. A combination of the curved-line advection method scheme and the FTn for spatial and angular discretization of the transient radiative transfer equation have been examined for several benchmarks.  
Human skin was modeled based on the material specifics presented in other studies. Malignancies considered in the skins were cutaneous melanoma (CM), nodular basal cell carcinoma (NBCC) and squamous cell carcinoma (SCC). In the initial phase of the skin cancer, the above malignancies were situated in the basal layer, after which they were developed in the full living epidermis. Predictions showed that transmittance decreases with an increase of the extinction coefficient, and that it becomes very weak with a higher extinction coefficient. Reflectance signals are important due to strongly scattered human skin. Therefore, the effects of short-pulse laser widths were investigated.  
Malignancies in the basal layer and living epidermis were considered. Further, the effects of laser light wavelength, the volume of the cancerous region and the scattering coefficient on their temporal signals of reflectance and transmittance were studied and compared against those for normal human skin (i.e., without malignancies). The results demonstrated that in the maximum zone a high reflectance was obtained for human skin with CM, while a low reflectance was observed in the case of skin with SCC. When living epidermis was complete infected with any of these malignancies, the reflectance was more than its values predicted in the case of infection of the basal layer on its own. Further, the effects of laser light wavelength, the volume of the cancerous region and its scattering coefficient on reflectance and transmittance were studied.  
Such predictions reported here for nonhomogeneous human skin tissues with a small cancerous zone subjected to an ultrafast laser may be explored for the early detection of small cancerous tumors. In fact, the obtained results from the radiative signals of skins with cutaneous melanoma, nodular basal cell carcinoma or squamous cell carcinoma are clearly different from normal skin, allowing an early detection of skin cancer. The reflectance of human skin with these types of cancer also changes from stage to stage as a result of the carcinoma thickness. Lower and slightly different profile peaks were obtained with increases of this thickness. Also, for cancerous skin, the results demonstrated a small advancement of the peaks and a presence of a second local profile maximum in more advanced stages that could allow a good and early identification of skin cancer. Thus, the obtained modifications of the optical properties and conditions represent a potential for the early detection of skin cancer. Further complete theoretical, numerical and experimental research will be surveyed soon. Moreover, in future works, the effects of optical characteristics of the granules will be introduced in the proposed mathematical formulation for improving the numerical accuracy of the present tool.  
Formal analysis, K.G. and M.O.; Funding acquisition, S.S.A.; Methodology, K.G., M.O. and S.S.A.; Visualization, K.G., M.O. and S.S.A.; Writing--original draft, K.G. and M.O.; Writing--review & editing, K.G. All authors have read and agreed to the published version of the manuscript  
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No (DF-191-980-1441). The authors therefore gratefully acknowledge DSR technical and financial support.  
The authors gratefully acknowledge DSR, King Abdulaziz University, technical and financial support.  
The authors declare no conflict of interest.