Coupled system of ODEs and quasilinear hyperbolic PDEs arising in a multiscale blood flow model

We study a coupled system of ordinary differential equations and quasilinear hyperbolic partial differential equations that models a blood circulatory system in the human body. The mathematical system is a multiscale model in which a part of the system, where the flow can be regarded as Newtonian and homogeneous, and the vessels are long and large, is modeled by a set of hyperbolic PDEs in a one-spatial-dimensional network, and in the other part, where either vessels are too thin or the flow pattern is too complicated (such as in the heart), the flow is modeled as a lumped element by a set of ordinary differential equations as an analog of an electric circuit. The mathematical system consists of pairs of PDEs, one pair for each vessel, coupled at each junction through a system of ODEs. This model is a generalization of the widely studied models of arterial networks. We give a proof of the well-posedness of the initial-boundary value problem by showing that the classical solution exists, is unique, and depends continuously on initial, boundary and forcing functions and their derivatives. (c) 2008 Elsevier Inc. All rights reserved.