Mathematical modeling and simulation of coronary stents Open Access

Martinez, Irving (Summer 2023)

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Every year approximately 3 million people in the US suffer from atherosclerosis, which is the condition in which one or more arteries get clogged up from excessive cholesterol and other residue build up. In spite of being introduced into the market decades ago, coronary stents remain the most popular solution, given their low surgery risk. However, stents are prone to malfunction after some time, with each type having its own set of complications. The introduction of newer types of stents to resolve the problems of their predecessors comes at the expense of creating different drawbacks. To have better insight into the physiological consequences of stent development, we give a contribution to fully understand stents through rigorous mathematical theory and modeling. \\


  In this work, we emphasize the understanding and application of PDEs such as Navier-Stokes and advection-diffusion equations in the context of hemodynamics to explore the blood velocity and pressure, the concentration of solutes, and the dissipation of drug across the stent artery system. Given the presence of lumen, wall, and stent regions, it is necessary to develop domain decomposition techniques through adaptations of Gauss-Seidel and Jacobi solvers. We extend Steklov-Poincaré theory to multiple domains by taking into account the interplay of distinct meshed domains. And meshing reassignment methods are elaborated with the purpose of sculpting geometries or transforming meshes over time. Overall, the composition and combination of our methods provides a theoretical and numerical groundwork to model different types of stent. 

Table of Contents

Introduction and Mathematical Preliminaries 1

Introduction 1

Mathematical Preliminaries 4

Steklov-Poincaré analysis of the basic three-domain stent problem 8

Stent Geometry 8

Mathematical Analysis of simplified stent 10

Blood and concentration equations 10

Definitions 12

Weak formulation and analysis of the NS problem 12

Time discretization 17

Full discretization 18

Convergence analysis of Two-domain problems 19

Convection dominated case considerations 27

Iterative-by-subdomain solution of the problem 28

The Jacobi variant 30

Domain decomposition methodology through Steklov-Poincaré operators 31

Steklov-Poincaré operators 34

Steklov-Poincaré analysis of the stent problem 35

Auxiliary operators 36

Weak formulation of the SP system 39

Space-discretization of the SP system 40

The substructuring method and the SP system 42

Numerical Results 45

Mathematical modeling of drug dynamics in the stent 54

Drug in stent motivation 54

Geometrical description 55

Initial drug coating 56

Drug coating evolution 57

Mathematical modeling 59

Drug release 60

Drug dynamics 64

Drug dynamics in the artery wall 64

Drug dynamics in the stent 67

Drug dynamics in the lumen 67

General drug dynamics 68

Weak formulation of the problem 69

Numerical Approximation 75

Numerical results (before meshing reassignment) 80

Discussion 83

Remeshing-free sculpting algorithms via reassignment for multidomain geometry modification 84

Motivation 84

Background 85

Multidomain sculpting transformations on uniform Eulerian meshes 86

Tetrahedral element reassignment 86

Pathfinding 87

Algorithms for mesh transformation 88

Computational results 105

Example: Cube 105

Example: Creating a stent 107

Example: Volume-less stents 110

Discussion 111

Future Work 112

Elution and erosion via meshing reassignment 112

Mapping of time-evolving domains 113

Patient-specific modeling 113

Optimization of stents 113

Appendix A: Summary of the numerical schemes implemented 115

Advection-diffusion equations 115

Navier-Stokes equations 117

Bibliography 119

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