Modeling blood flow in viscoelastic vessels: the 1D augmented fluid–structure interaction system

Abstract

Nowadays mathematical models and numerical simulations are widely used in the field of hemodynamics, representing a valuable resource to better understand physiological and pathological processes in different medical sectors. The theory behind blood flow modeling is closely related to the study of incompressible flow through compliant thin-walled tubes, starting from the incompressible Navier-Stokes equations. Furthermore, the mechanical interaction between blood flow and vessels wall must be properly described by the model. Recent works showed the benefits of characterizing the rheology of the vessel wall through a viscoelastic law. Taking into account the viscous contribution of the wall material and not simply the elastic one leads to a more realistic representation of the vessel behavior, which manifests not only an instantaneous elastic strain but also a viscous damping effect on pulse pressure waves, coupled to energy losses. In this context, the aim of this work is to propose an easily extensible one-dimensional mathematical model able to accurately capture fluid-structure interactions. The originality of the model lies in the introduction of a viscoelastic tube law in PDE form, valid for both arterial and venous networks, leading to an augmented fluid-structure interaction system. In contrast to well established mathematical models, the proposed one is natively hyperbolic. The model is solved with an efficient and robust second-order numerical scheme; the time integration is based on an Implicit-Explicit Runge-Kutta scheme conceived for applications to hyperbolic systems with stiff relaxation terms. The validation of the proposed model is performed on several different test cases. Results obtained in Riemann problems, adopting a simple elastic tube law for the characterization of the vessel wall, are compared with available exact solutions. To validate the contribution given by the viscoelastic term, the Method of Manufactured Solutions has been applied. Specific tests have been designed to verify the well-balancing with respect to fluid-at-rest condition and the accuracy-preserving property of the scheme. Finally, a specific test case with an inlet pulse pressure wave has been designed to assess the effects of viscoelasticity with respect to a simple elastic behavior of the vessel wall. The complete code, written in MATLAB (MathWorks Inc.) language, with the implemented test cases, is made available in Mendeley Data repository.

Assessment of cardiac function by pulse wave analysis. A computational study

“…The 1-D model vascular network was modelled using the augmented fluid-structure interaction (a-FSI) system 3,5,2 , a hyperbolic set of equations. This includes equations for the conservation of mass and momentum, and a closure equation relating vessel cross-sectional area and internal pressure, the so-called tube law 4,3 . Key assumptions for the haemodynamic model are: laminar flow, incompressible and Newtonian fluid (blood density, ρ = 1060 kg/m 3 ; blood viscosity, µ = 2.5 mPa s), and no energy losses at bifurcations.…”

“…The aim of this study is to investigate the effect of cardiac properties on pulse wave morphology using a 1-D model of the arterial vasculature 33 coupled to a 0-D model of cardiac contraction. The 1-D model considers the viscoelastic behaviour of vessel walls 3,5 .…”

Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods

“…To obtain a fully discrete scheme, we consider a finite volume method for the spatial discretization, and uniform grid with mesh spacing ∆ = +1/2 − −1/2 . For each internal step of the IMEX scheme, numerical fluxes are evaluated following the Dumbser-Osher-Toro (DOT) solver, which coincides with the Godunov flux based on the exact Riemann solver for linear hyperbolic systems with constant Jacobian matrix [9,10,17]. Boundary-extrapolated values on the two sides of the interface within cell are computed by piecewise linear reconstruction, recurring to the minmod slope limiter to obtain a TVD scheme [40] and achieve second order of accuracy for smooth solutions also in space.…”

Computationally efficient finite element formulation for blood flow analysis in multi‐layered aorta modeled as viscoelastic material

“…However, most of the 1D wave propagation models of the blood flow consider elastic constitutive relation 4,6,7,21,34‐40 to model the arterial wall. There are a only few literature available on the one‐dimensional blood flow study considering viscoelastic arterial model 2,23,41‐43 . Viscoelasticity is one of the important features of blood vessels regulating their physical damping and significantly affecting the clinically important information of the blood flow such as pressure variation and flow rate.…”