Analysis of a Geometrical Multiscale Model Based on the Coupling of ODE and PDE for Blood Flow Simulations

In hemodynamics, local phenomena, such as the perturbation of flow pattern in a specific vascular region, are strictly related to the global features of the whole circulation (see, e.g., L. Formaggia et al., Comput. Vis. Sci., 2 (1999), pp. 75--83). In A. Quarteroni, S. Ragni, and A. Veneziani, Comput. Vis. Sci., 4 (2001), pp. 111--124 we have proposed a heterogeneous model, where a local, accurate, three-dimensional description of blood flow by means of the Navier--Stokes equations in a specific artery is coupled with a systemic, zero-dimensional, lumped model of the remainder of circulation. This is a geometrical multiscale strategy, which couples an initial-boundary value problem to be used in a specific vascular region with an initial-value problem in the rest of the circulatory system. It has been successfully adopted to predict the outcome of a surgical operation (see K. Laganà et al., Biorheology, 39 (2002), pp. 359--364, G. Dubini et al., Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, 2000). However, its interest goes beyond the context of blood flow simulations. In this paper we provide a well-posedness analysis of this multiscale model by proving a local-in-time existence result based on a fixed-point technique. Moreover, we investigate the role of matching conditions between the two submodels for the numerical simulation.