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

IF 2.4 Q2 AUTOMATION & CONTROL SYSTEMS
Shantanu Singh;Nikolaos Bekiaris-Liberis
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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.
利用心血管流动的控制论 PDE 模型,基于数值和 Lyapunov 对狭窄对血液运输稳定性影响的研究
我们进行了各种数值测试来研究(边界)狭窄对血流稳定性的影响,采用了详细而精确的二阶有限体积方案来数值化一个偏微分方程(PDE)模型,使用临床上实际的动脉参数值和血液流入量。该模型由一个基线为 2 元/次 2 元的异方向非线性双曲偏微分方程系统组成,其中,狭窄的影响由动脉段出口处的压力降来描述。然后,我们研究了参考轨迹的稳定性(在数值测试中观察到),该轨迹对应于给定的时变流入量(例如,周期轨迹,其周期等于两次连续心跳之间的时间间隔)和狭窄严重程度,推导出各自的线性化系统并构建了 Lyapunov 函数。由于线性化系统是时变的,其时变参数取决于参考轨迹本身(而参考轨迹又以隐含的方式取决于狭窄程度),无法通过分析得出,因此我们通过数值方法验证了所获得的基于 Lyapunov 的稳定性条件。数值测试和基于 Lyapunov 的稳定性分析表明,参考轨迹是渐近稳定的,其衰减率随着狭窄严重程度的恶化而降低。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
IEEE Control Systems Letters
IEEE Control Systems Letters Mathematics-Control and Optimization
CiteScore
4.40
自引率
13.30%
发文量
471
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