Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and hyperbolic PDEs

For the numerical simulation of the circulatory system, geometrical multiscale models based on the coupling of systems of differential equations with different spatial dimensions are becoming common practice [L. Formaggia et al., Comput. Vis. Sci., 2 (1999), pp. 75-83, A. Quarteroni and A. Veneziani, Multiscale Model. Simul., 1 (2003), pp. 173-195, L. Formaggia et al., Comput. Methods Appl. Mech. Engrg., 191 (2001), pp. 561-582]. In this paper we address the mathematical analysis of a coupled multiscale system involving a zero-dimensional (0D) model, describing the global characteristics of the circulatory system, and a one-dimensional (1D) model giving the pressure propagation along a straight vessel. We provide a local-in-time existence and uniqueness of classical solutions for this coupled problem. To this purpose we reformulate the original problem in a general abstract framework by splitting it into subproblems (the 0D system of ODEs and the 1D hyperbolic system of PDEs); then we use fixed-point techniques. The abstract result is then applied to the original blood flow case under very realistic hypotheses on the data. This work represents the 1D-0D counterpart of the 3D-0D mathematical analysis reported in [A. Quarteroni and A. Veneziani, Multiscale Model. Simul., 1 (2003), pp. 173-195].