On rugged shape of skin tumor (basal cell carcinoma)

Basal cell carcinoma (BCC) is the most common cancer of the skin that is made of transformed basal cells of the human epidermis, and spreads along the epidermis--dermis junction. It often forms tumor cell mass that protrudes towards the dermal connective tissue with many branches. Histopathological examination of the cancer demonstrates the cell mass of a rugged shape appearing as many regular spaced islands in a two-dimensional section. We study the process of cell proliferation and spatial pattern formation of the skin tumor by a pair of partial differential equations of tumor cells and nutrients. The assumptions are: (1) proliferation rate of tumor cells depends on the availability of nutrients, which simply diffuse out of capillaries through connective tissue; (2) nutrients are consumed by active tumor cells; (3) cell diffusion coefficient expressing tumor cell movements increase with the cell density and the nutrient availability. Starting from the initial condition with a single layer of tumor cells. the model develops a smooth colony if n'0 is large, but a characteristic rugged spatial pattern of tumor cell mass if n'0 is small, in which n'0 is nutrient concentration multiplied by square root of growth efficiency divided by diffusion coefficient of nutrients. The proportion of the area occupied by tumor cells increases with n'0. The coefficient of variation in the width of "islands" of tumor cell mass is rather small (0.2 reverse similar0.6), implying the regularity of the spatial pattern. We also analyses the photographs of a two-dimensional section of tumor cell mass and compare the spatial patterns generated by the model.