Author(s): G. Bertaglia; A. Valiani; V. Caleffi

Linked Author(s): Alessandro Valiani

Keywords: Blood flow; Fluid-structure interaction; Compliant vessels; DOT scheme; 1D modelling

Abstract: The augmented system of the 1D fluid-structure-interaction (FSI) problem for blood flow is presented and solved with the first-order Dumbser-Osher-Toro (DOT) scheme using the simple linear path. Nowadays mathematical models are widely used in the field of haemodynamics, representing a valuable resource for different medical applications. The theory behind the phenomenon is closely related to the study of incompressible flow through compliant thin-walled tubes, even collapsible under certain circumstances in the case of veins. Recent works also showed the benefits of modelling the mechanical behaviour of the vessel wall using a viscoelastic law, considering in this manner the viscous damping of the pulse-waves. In this context, to have an efficient, robust and easily extensible model is one of the main relevant purposes. The basic equations of the blood flow in medium to large-size vessels are obtained from the principles of conservation of mass and momentum. To close the governing partial differential equation (PDE) system, a tube law representative of the interaction between vessel wall and blood, relating the internal pressure to the wall displacement via the cross-sectional area, is required. The elastic tube law here adopted is the standard one, with the corresponding distinct values of the parameters for arteries and veins. Differentiating with respect to time this equation, the following PDE is derived. Thus, adding this equation to the system, the sought 1D augmented fluid-structure-interaction (FSI) system is obtained. Furthermore, to accommodate discontinuous material properties along the vessel, such as equilibrium cross sectional area, instantaneous Young modulus E0 and external pressure pext, it has also to be considered that their time derivatives are zero. For the resolution of the system, the explicit path-conservative finite volume method with the first-order DOT solver has been used. To test the resolution of the augmented FSI system with the DOT solver, the three Riemann problems proposed, designed to schematically represent specific cardiovascular applications, have been chosen. Results confirmed the well-balancing property of the numerical scheme for the PDE system under consideration and its suitability to solve also unsteady problems in arteries. The same does not apply in the case which represents an idealised Valsalva maneuver effect on an internal jugular vein with incompetent valve, where the augmented system shows the critical points inherent in its own form and its inability to correctly describe shock waves when dealing with strongly non-linear configurations, as with collapsible veins.

DOI: https://doi.org/10.3850/978-981-11-2731-1_074-cd

Year: 2018