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

Abstract

Abstract. 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 NavierStokes 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 initialboundary 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) 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.Key words. geometrical multiscale models, blood flow simulations, fixed-point techniques AMS subject classifications. 35M20, 35Q30, 76D03, 65L05, 76Z05PII. S1540345902408482 1. Introduction. In computational fluid dynamics there might be the need for accurately simulating the flow in a subregion of a complex system that can be reasonably represented by a hydraulic network. One may think, for instance, of a system for water supply for which there is the interest of ascertaining the spatial variations of velocity and pressure only in a specific pipeline or in a reservoir. Another instance, which has motivated our investigation, is the blood circulatory system. Here the underlying (low cost) model is based on an analogy with electric circuits and can predict the time evolution of average physical quantities (flow rate and pressure) in the different compartments of the overall system (see, e.g., [15,27]). Yet, there is the interest of carrying out three-dimensional (3D) simulations on a sensible region (e.g., a coronary bypass, a carotid bifurcation, a stented artery, etc.). In these cases, the local investigation requires the specification of boundary conditions on the interface between the region of interest and the remainder of the network. In fact, the significance of numerical results is strictly related to the capability of the numerical device of properly accounting for the exchange of information with the global network. Actually, it is well known that, in the circulatory system, local behavior of blood flow in a specific region is strictly related to the systemic features of the circulation. For

Geometric multiscale modeling of the cardiovascular system, between theory and practice

“…The portion of the lumped parameter model receiving this input reflects the nature of the data. This leads to the concept of bridging region [167].…”

“…We observe that for the hydraulic network topology, Robin conditions correspond to introducing a lumped resistance on the interface branch of the bridging region. This case has been considered in [167]. Instead, for the 3D problem, the Robin/Robin scheme would lead to a defective resistance condition (see (27) and the related paragraph).…”

“…In this perspective, existence (at least locally in time) follows from the application of fixed point theorems. In particular, in Proposition 5.1 of [167] the Schauder fixed point principle was used to analyze the case of 3D problem with rigid boundaries, after showing that (i) for suitable assumptions on the initial data and under bridging region compatibility, each subproblem is well posed; at the numerical level, this implies that the the block operators L and F are invertible; (ii) the operator T = L 0D L 3D is locally-in-time compact. In particular, the proposition was proved for a problem where all the data used as input to the 0D model where flow rates and the boundary conditions for the 3D one were of (defective) pressure types completed with the "do-nothing" approach.…”

3D Printing Cardiovascular Anatomy: A Single-Centre Experience

“…An MCL was built simulating first-stage palliation of HLHS with a modified Blalock-Taussig shunt [18] from the innominate artery to the pulmonary bed of the circuit and then modified to simulate first-stage palliation of HLHS with a Sano shunt [19] from the ventricle itself (in the experimental setup, from the de-airing valve of the Berlin Heart®) to the pulmonary bed of the circuit. The coupling of a detailed 3D model with a circuit summarising the remainder of the circulation can be considered a multi-scale [20] in vitro modelling approach for CHD. With regard to HLHS palliative surgery and the valuable insight that can be gathered experimentally, MCLs can be modified to incorporate the effect of respiration [21] which becomes relevant when these patients receive their second and third surgeries, i.e.…”

Computational Reduction for Parametrized PDEs: Strategies and Applications

“…A natural framework where suitable geometric multiscaling techniques have been developed is the numerical simulation of blood flows in the circulatory system [39,47]. Due to different physiological and morphological aspects, three-dimensional models for blood flows in (local) portions of large vessels have been coupled with onedimensional (global) models for the network of arteries and veins, or even to zero-dimensional (ODE) models for the capillary network.…”