Homogenization for a non-local coupling model

Abstract

In [P. Deuflhard and R. Hochmuth, On the thermoregulation in the human microvascular system, Proc. Appl. Math. Mech. 3 (2003), pp. 378–379; P. Deuflhard and R. Hochmuth, Multiscale analysis of thermoregulation in the human microsvascular system, Math. Meth. Appl. Sci. 27 (2004), pp. 971–989; R. Hochmuth and P. Deuflhard, Multiscale analysis for the bio-heat transfer equation–the nonisolated case, Math. Models Methods Appl. Sci. 14(11) (2004), pp. 1621–1634], homogenization techniques are applied to derive an anisotropic variant of the bio-heat transfer equation as asymptotic result of boundary value problems providing a microscopic description for microvascular tissue. In view of a future application on treatment planning in hyperthermia, we investigate here the homogenization limit for a coupling model, which takes additionally into account the influence of convective heat transfer in medium-size blood vessels. This leads to second-order elliptic boundary value problems with non-local boundary conditions on parts of the boundary. Moreover, we present asymptotic estimates for first-order correctors.

Keywords:

• homogenization
• non-local boundary conditions
• Robin boundary conditions
• correctors
• heat transfer
• bio-heat equation
• hyperthermia

Mathematics Subject Classifications: :

• 35B27
• 80A20
• 92C50
• 35J25

Acknowledgement

The author has been supported by a Konrad-Zuse-Fellowship. In particular, he is grateful to Peter Deuflhard for stimulating and valuable comments during the preparation of this article.