Modeling Viscoelastic Behavior of Arterial Walls and Their Interaction with Pulsatile Blood Flow

Fluid-structure interaction describing wave propagation in arteries driven by the pulsatile blood flow is a complex problem. Whenever possible, simplified models are called for. One-dimensional models are typically used in arterial sections that can be approximated by the cylindrical geometry allowing axially symmetric flows. Although a good first approximation to the underlying problem, the one-dimensional model suffers from several drawbacks: the model is not closed (an ad hoc velocity profile needs to be prescribed to obtain a closed system) and the model equations are quasi-linear hyperbolic (oversimplifying the viscous fluid dissipation), typically producing shock wave solutions not observed in healthy humans. In this manuscript we derived a simple, closed reduced model that accounts for the viscous fluid dissipation to the leading order. The resulting fluid-structure interaction system is of hyperbolic-parabolic type. Arterial walls were modeled by a novel, linearly viscoelastic cylindrical Koiter shell model and the flow of blood by the incompressible, viscous Navier-Stokes equations. Kelvin-Voigt-type viscoelasticity was used to capture the hysteresis behavior observed in the measurements of the arterial stress-strain response. Using the a priori estimates obtained from an energy inequality, together with the asymptotic analysis and ideas from homogenization theory for porous media flows, we derived an effective model which is an 2-approximation to the three-dimensional axially symmetric problem, whereis the aspect ratio of the cylindrical arterial section. Our model shows two interesting features of the underlying problem: bending rigidity, often times neglected in the arterial wall models, plays a nonnegligible role in the 2-approximation of the original problem, and the viscous fluid dissipation imparts long-term viscoelastic memory effects on the motion of the arterial walls. This does not, to the leading order, influence the hysteresis behavior of arterial walls. The resulting model, although two-dimensional, is in the form that allows the use of one-dimensional finite element method techniques producing fast numerical solutions. We devised a version of the Douglas-Rachford time-splitting algorithm to solve the underlying hyperbolic-parabolic problem. The results of the numerical simulations were compared with the experimental flow measurements performed at the Texas Heart Institute, and with the data corresponding to the hysteresis of the human femoral artery and the canine abdominal aorta. Excellent agreement was observed.