# Blood-Flow Modelling Along and Trough a Braided Multi-Layer Metallic Stent

## TL;DR

This paper develops a homogenized mathematical model to analyze blood flow in stented arteries, providing explicit velocity profiles and pressure relations, validated by numerical simulations, with implications for understanding flow dynamics in medical devices.

## Contribution

It introduces a first-order homogenized approximation for blood flow in braided multi-layer metallic stents, including explicit velocity profiles and pressure relations, extending previous theoretical tools.

## Key findings

- Explicit velocity profile depending on pressure data
- Pressure in aneurysm sac equals averaged main artery pressure
- Stent presence reverses vortex rotation

## 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.