Mathematical modeling and simulation of coronary stents Öffentlichkeit
Martinez, Irving (Summer 2023)
Published
Abstract
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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