Numerical and Lyapunov-Based Investigation of the Effect of Stenosis on Blood Transport Stability Using a Control-Theoretic PDE Model of Cardiovascular Flow

TL;DR

This paper investigates how arterial stenosis affects blood flow stability using a PDE model, numerical simulations, and Lyapunov stability analysis, revealing that increased stenosis severity reduces flow stability and decay rates.

Contribution

It introduces a control-theoretic PDE model with Lyapunov analysis to study blood flow stability under stenosis, combining numerical and theoretical methods.

Findings

Stenosis severity decreases blood flow stability.

Lyapunov functional confirms asymptotic stability for various conditions.

Decay rate of stability diminishes with increased stenosis severity.

Abstract

We perform various numerical tests to study the effect of (boundary) stenosis on blood flow stability, employing a detailed and accurate, second-order finite-volume scheme for numerically implementing a partial differential equation (PDE) model, using clinically realistic values for the artery's parameters and the blood inflow. The model consists of a baseline $2\times 2$ hetero-directional, nonlinear hyperbolic PDE system, in which, the stenosis' effect is described by a pressure drop at the outlet of an arterial segment considered. We then study the stability properties (observed in our numerical tests) of a reference trajectory, corresponding to a given time-varying inflow (e.g., a periodic trajectory with period equal to the time interval between two consecutive heartbeats) and stenosis severity, deriving the respective linearized system and constructing a Lyapunov functional. Due to the fact that the linearized system is time varying, with time-varying parameters depending on the reference trajectories themselves (that, in turn, depend in an implicit manner on the stenosis degree), which cannot be derived analytically, we verify the Lyapunov-based stability conditions obtained, numerically. Both the numerical tests and the Lyapunov-based stability analysis show that a reference trajectory is asymptotically stable with a decay rate that decreases as the stenosis severity deteriorates.