On 1-D PDE-Based Cardiovascular Flow Bottleneck Modeling and Analysis: A Vehicular Traffic Flow-Inspired Approach

Bekiaris-Liberis, Nikolaos · 2023 · Crossref

DOI: 10.1109/tac.2022.3195148

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Summary

This paper addresses the lack of control-theoretic frameworks for analyzing and managing congested blood flow caused by arterial stenosis, a leading cause of cardiovascular mortality. The authors propose a novel modeling approach that draws a direct analogy between one-dimensional (1-D) blood flow dynamics in arteries with stenosis and vehicular traffic flow dynamics in the presence of bottlenecks. By treating stenosis as a boundary bottleneck, the study aims to leverage established methods from traffic flow control for the monitoring, estimation, and feedback control of cardiovascular systems. The methodology involves formulating a control-oriented model based on a $2 \times 2$ hyperbolic partial differential equation (PDE) system that describes 1-D blood flow dynamics, capturing mass and momentum conservation. The stenosis is modeled as a boundary condition at the outlet of the arterial segment, characterized by either a static or dynamic description of the pressure drop. The static model assumes zero or negligible stenosis length, while the dynamic model accounts for larger stenosis lengths using a coupled nonlinear ordinary differential equation (ODE). The analysis transforms the system into Riemann variables to examine information propagation and derives a fundamental diagram relating blood speed to arterial cross-sectional area. This allows for a rigorous comparison with second-order traffic flow models, such as the Payne-Whitham and Aw-Rascle-Zhang models. Key findings reveal a strong structural correspondence between the two domains. The blood flow conservation equation mirrors the conservation of vehicles in traffic models, and the derived fundamental diagram for blood flow exhibits properties similar to those in traffic theory. The study highlights that physiological blood flow operates in a subcritical (congested) regime, analogous to congested traffic, whereas traffic models often focus on free-flow regimes. Furthermore, the static and dynamic stenosis models correspond to traffic bottleneck descriptions involving capacity drops and dynamic density adjustments, respectively. The paper establishes that the hyperbolic system is well-posed under realistic physiological parameters. The significance of this work lies in providing a theoretical foundation for applying PDE-based control techniques to cardiovascular health. The authors suggest future research directions, including the development of adaptive observers for real-time identification of stenosis location and characteristics using boundary measurements. Additionally, the framework supports the design of feedback control strategies, such as manipulating inflow via smart stents or using automated drug delivery for in-domain actuation, to mitigate the effects of stenosis. This cross-disciplinary approach opens new avenues for improving the detection and treatment of arterial diseases through advanced control theory.

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