[Paper Review] Blood-Flow Modelling Along and Trough a Braided Multi-Layer Metallic Stent

Vuk Milišić|arXiv (Cornell University)|Aug 18, 2009

Advanced Mathematical Modeling in Engineering15 references3 citations

TL;DR

This paper develops a first-order asymptotic model for blood flow through and along a braided multi-layer metallic stent using boundary layer theory and homogenization. It derives an explicit velocity profile dependent only on pressure drop and homogenized constants, showing that the aneurysmal sac pressure equals the averaged stent pressure and that the stent reverses vortex rotation, validated numerically with error estimates.

ABSTRACT

In this work we study the hemodynamics in a stented artery connected either to a collateral artery or to an aneurysmal sac. The blood flow is driven by the pressure drop. Our aim is to characterize the flow-rate and the pressure in the contiguous zone to the main artery: using boundary layer theory we construct a homogenized first order approximation with respect to epsilon, the size of the stent's wires. This provides an explicit expression of the velocity profile through and along the stent. The profile depends only on the input/output pressure data of the problem and some homogenized constant quantities: it is explicit. In the collateral artery this gives the flow-rate. In the case of the aneurysm, it shows that : (i) the zeroth order term of the pressure in the sac equals the averaged pressure along the stent in the main artery, (ii) the presence of the stent inverses the rotation of the vortex. Extending the tools set up in [Bonnetier et al, Adv. Math. Fluids, 2009, Milisic, Meth. Apl. Ann., 2009] we prove rigorously that our asymptotic approximation is first order accurate with respect to . We derive then new implicit interface conditions that our approximation formally satisfies, generalizing our analysis to other possible geometrical configurations. In the last part we provide numerical results that illustrate and validate the theoretical approach.

Motivation & Objective

• To mathematically model blood flow through and along a braided multi-layer metallic stent in hemodynamically relevant configurations.
• To characterize flow-rate and pressure in regions contiguous to the main artery, especially in collateral vessels and aneurysmal sacs.
• To rigorously derive a first-order asymptotic approximation of velocity and pressure with respect to the stent wire size ε.
• To establish new implicit interface conditions for fluid-structure interaction at the stent interface, generalizing to other geometries.
• To validate the theoretical model with numerical results and error estimates.

Proposed method

• Uses boundary layer theory and multi-scale asymptotic analysis to construct a first-order approximation of the Stokes flow in the presence of a stent with small wire size ε.
• Applies homogenization techniques to derive effective, macroscopic fluid equations by solving microscopic cell problems on the stent’s periodic structure.
• Introduces a fictitious interface representing the stent, where velocity and pressure are described by explicit formulas depending only on input pressure and homogenized constants.
• Derives new implicit interface conditions: normal velocity continuity and a discontinuous homotetic relationship for tangential velocities, differing from classical wall laws.
• Proves first-order accuracy of the approximation with respect to ε using rigorous error estimates and vertical boundary correctors.
• Extends the framework to 3D configurations and validates the model numerically, demonstrating consistency with theoretical predictions.

Experimental results

Research questions

• RQ1 How does the presence of a braided multi-layer stent affect the velocity profile and pressure distribution in a stented artery with a collateral vessel?
• RQ2 What is the asymptotic behavior of blood flow through a stent when the wire size ε tends to zero, and how can it be approximated with high accuracy?
• RQ3 What are the effective interface conditions that govern fluid flow across the stent, particularly for tangential and normal velocity components?
• RQ4 How does the stent influence hemodynamics in an aneurysmal sac, especially regarding pressure homogenization and vortex dynamics?
• RQ5 Can a first-order asymptotic model be rigorously derived and validated for pressure-driven flow through porous stents with variable roughness?

Key findings

• The velocity profile through the stent is explicitly determined by the input/output pressure drop and homogenized constants, independent of detailed stent geometry.
• In the aneurysmal sac, the zeroth-order pressure is constant and equal to the spatially averaged pressure along the stent in the main artery.
• The stent induces a reversal in the rotation direction of the vortex within the aneurysm, a key hemodynamic effect for promoting thrombosis.
• The asymptotic model is first-order accurate with respect to ε, the stent wire size, as proven by rigorous error estimates.
• New implicit interface conditions are derived: normal velocity is continuous, while tangential velocities satisfy a discontinuous homotetic relationship, generalizing existing wall-law models.
• Numerical results confirm the theoretical predictions, demonstrating convergence and consistency of the asymptotic approximation across different configurations.