Upscaling in Dynamical Heat Transfer Problems in Biological Tissues

The asymptotic behaviour of the solution of a nonlinear dynamical boundary-value problem describing the bio-heat transfer in microvascular tissues is analysed. Our domain Ω is an ε-periodic structure, consisting of two parts: a solid tissue part Ωε and small regions of blood Ω/Ω(bar)ε of a certain temperature, ε representing a small parameter related to the characteristic size of the blood regions. In such a domain, we shall consider a heat equation, with nonlinear sink and source terms and with a dynamical condition imposed on the boundaries of the blood zones. The limit equation, as ε→ 0, is a new heat equation, with extra-terms coming from the influence of the non-homogeneous dynamical boundary condition. (author)