Modeling blood flow in networks of viscoelastic vessels with the 1-D augmented fluid–structure interaction system

Abstract

A noteworthy aspect in blood flow modeling is the definition of the mechanical interaction between the fluid flow and the biological structure that contains it, namely the vessel wall. Particularly, it has been demonstrated that the addition of a viscous contribution to the mechanical characterization of vessels brings positive results when compared to in-vivo measurements. In this context, the numerical implementation of boundary conditions able to keep memory of the viscoelastic contribution of vessel walls assumes an important role, especially when dealing with large circulatory systems. In this work, viscoelasticity is taken into account in entire networks via the Standard Linear Solid Model. The implementation of the viscoelastic contribution at boundaries (inlet, outlet and junctions), is carried out considering the hyperbolic nature of the mathematical model. Specifically, a non-linear system is established based on the definition of the Riemann Problem at junctions, characterized by rarefaction waves separated by contact discontinuities, among which the mass and the total energy are conserved. Basic junction tests are analyzed, such as a trivial 2–vessels junction, for both a generic artery and a generic vein, and a simple 3–vessels junction, considering an aortic bifurcation scenario. The chosen asymptotic-preserving IMEX Runge-Kutta Finite Volume scheme is demonstrated to be second-order accurate in the whole domain and well-balanced, even when including junctions. Two different benchmark models of the arterial network are then implemented, differing in number of vessels and in viscoelastic parameters. Comparison of the results obtained in the two networks underlines the high sensitivity of the model to the chosen viscoelastic parameters. In these numerical tests, the conservation of the contribution provided by the viscoelastic characterization of vessel walls is assessed in the whole network, including junctions and boundary conditions.