Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Shantanu Singh, Nikolaos Bekiaris-Liberis

TL;DR

This work investigates how boundary stenosis affects blood transport stability in a 1-D $2\times2$ nonlinear hyperbolic PDE model of cardiovascular flow, using a second-order finite-volume scheme and clinically realistic arterial parameters. A boundary stenosis is represented by a pressure drop at the outlet, and stability is analyzed by linearizing around a reference trajectory and constructing a Lyapunov functional for the time-varying error system. Numerical experiments show that the reference trajectory remains asymptotically stable, but the decay rate declines with increasing stenosis severity, and stability guarantees based on fixed Lyapunov functions hold only up to moderate stenosis (approximately $<84\%$). The findings highlight the potential and limitations of control-theoretic stability analysis in cardiovascular flow and point to time-varying Lyapunov approaches for stronger stenosis regimes.

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.

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 work investigates how boundary stenosis affects blood transport stability in a 1-D nonlinear hyperbolic PDE model of cardiovascular flow, using a second-order finite-volume scheme and clinically realistic arterial parameters. A boundary stenosis is represented by a pressure drop at the outlet, and stability is analyzed by linearizing around a reference trajectory and constructing a Lyapunov functional for the time-varying error system. Numerical experiments show that the reference trajectory remains asymptotically stable, but the decay rate declines with increasing stenosis severity, and stability guarantees based on fixed Lyapunov functions hold only up to moderate stenosis (approximately ). The findings highlight the potential and limitations of control-theoretic stability analysis in cardiovascular flow and point to time-varying Lyapunov approaches for stronger stenosis regimes.

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 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.
Paper Structure (7 sections, 1 theorem, 45 equations, 8 figures)

This paper contains 7 sections, 1 theorem, 45 equations, 8 figures.

Table of Contents

  1. Introduction
  2. 1-D PDE model of blood flow with stenosis
  3. Numerical Analysis of the blood flow model
  4. Lyapunov stability of the linearized system
  5. Stability Analysis of Error Eystem
  6. Verification of Stability Conditions
  7. Conclusions

Key Result

Proposition IV.1

Consider system Linearized_e, with A, B, and boundary conditions given by BC1_linear and BC2_linear. Assume that $\frac{\partial}{\partial v}G(u^*,v^*)\neq 0$ and $[u^*\ v^*] \in C^1([0,L]\times[0,+\infty); \mathbb{R}^2)$ is such that $[A^*\ V^*] \in \Omega$ and $[u^*\ v^*]$, $[u^*_x\ v^*_x]$ are u where $\mathcal{P}(x)={\rm diag}\{p_1e^{-\mu x}, p_2e^{\mu x}\}$.

Figures (8)

  • Figure 1: Flow $Q_1(L,t)$ for $0\%$, $47.11\%$, $70.25\%$, $84\%$, $90\%$, and $92.56\%$ stenosis, computed according to $100\times\frac{A_0-A_{\rm s}}{A_0}\%$ and corresponding to $r_L^+$ values of $0.55~cm$, $0.4~cm$, $0.3~cm$, $0.22~cm$, $0.17~cm$, and $0.15~cm$, respectively.
  • Figure 2: Pressure drop $\Delta P(t)$ for $0\%$, $47.11\%$, $70.25\%$, $84\%$, $90\%$, and $92.56\%$ stenosis.
  • Figure 3: Flow $Q_1(x,t)$ at $x=\{\frac{L}{4}, \frac{L}{2}, \frac{3L}{4}, L\}$ when the stenosis is $92.56\%$, i.e., $r_L^+=0.15~cm$.
  • Figure 4: Area $A_1(x,t)$ at $x=\{0, \frac{L}{4}, \frac{L}{2}, \frac{3L}{4}, L\}$ when the stenosis is $92.56\%$, i.e., $r_L^+=0.15~cm$.
  • Figure 5: $L^\infty$-norm of $[e_u(\cdot,t) \ e_v(\cdot,t)]^\top$ for various levels of stenosis, where $e_u=u-u^*$ and $e_v=v-v^*$.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Proposition IV.1