Multiscale Modeling of Light Absorption in Tissues: Limitations of Classical Homogenization Approach

In biophotonics, the light absorption in a tissue is usually modeled by the Helmholtz equation with two constant parameters, the scattering coefficient and the absorption coefficient. This classic approximation of "haemoglobin diluted everywhere" (constant absorption coefficient) corresponds to the classical homogenization approach. The paper discusses the limitations of this approach. The scattering coefficient is supposed to be constant (equal to one) while the absorption coefficient is equal to zero everywhere except for a periodic set of thin parallel strips simulating the blood vessels, where it is a large parameter omega. The problem contains two other parameters which are small: epsilon, the ratio of the distance between the axes of vessels to the characteristic macroscopic size, and delta, the ratio of the thickness of thin vessels and the period. We construct asymptotic expansion in two cases: epsilon -> 0, omega -> infinity, delta -> 0, omega delta -> infinity, epsilon(2)omega delta -> 0, and epsilon -> 0, omega -> infinity, delta -> 0, epsilon(2)omega delta(2) -> infinity and prove that in the first case the classical homogenization (averaging) of the differential equation is true while in the second case it is wrong. This result may be applied in the biomedical optics, for instance, in the modeling of the skin and cosmetics.