Coupling between lumped and distributed models for blood flow problems

In this paper we propose a method for coupling distributed and lumped models for the blood circulation. Lumped parameter models, based on an analogy between the circulatory system and an electric or a hydraulic network are widely employed in the literature to investigate different systemic responses in physiologic and pathologic situations (see e.g. [13, 24, 30, 15, 4, 27, 11, 14]). From the mathematical viewpoint these models are represented by ordinary differential equations. On the other hand, for the accurate description of local phenomena, the Navier–Stokes equations for incompressible fluids are considered. In the multiscale perspective, lumped models have been adopted (see e.g. [16]) as a numerical preprocessor to provide a quantitative estimate of the boundary conditions at the interfaces. However, the two solvers (i.e. the lumped and the distributed one) have been used separately. In the present work, we introduce a genuinely heterogeneous multiscale approach where the local model and the systemic one are coupled at a mathematical and numerical level and solved together. In this perspective, we have no longer boundary conditions on the artificial sections, but interface conditions matching the two submodels. The mathematical model and its numerical approximation are carefully addressed and several test cases are considered.