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Mathematical and numerical analysis for PDE systems modeling intravascular drug release from arterial stents and transport in arterial tissue

Xiaobing Feng, Tingao Jiang

TL;DR

This work analyzes a modified one-dimensional PDE-ODE system modeling intravascular drug release from arterial stents, establishing well-posedness via a Galerkin plus compactness argument and developing both semi-discrete and fully discrete finite-element schemes. It proves optimal-order error estimates in the energy norm for the semi-discrete method and the fully discrete Euler scheme, and demonstrates numerical performance using decoupling strategies and multi-rate time stepping. A detailed numerical study validates convergence, accuracy, and the practical impact of different discretization choices, while a two-dimensional generalization is briefly discussed. The results provide a rigorous, computationally robust framework for simulating trans-wall drug transport and can inform design and pharmacokinetic assessment of drug-eluting stents.

Abstract

This paper is concerned with the PDE and numerical analysis of a modified one-dimensional intravascular stent model originally proposed in [4]. It is proved that the modified model has a unique weak solution using the Galerkin method combined with a compactness argument. A semi-discrete finite element method and a fully discrete scheme using the Euler time-stepping are formulated for the PDE model. Optimal order error estimates in the energy norm are proved for both schemes. Numerical results are presented along with comparisons between different decoupling strategies and time-stepping schemes. Lastly, extensions of the model and its PDE and numerical analysis results to the two-dimensional case are also briefly discussed.

Mathematical and numerical analysis for PDE systems modeling intravascular drug release from arterial stents and transport in arterial tissue

TL;DR

This work analyzes a modified one-dimensional PDE-ODE system modeling intravascular drug release from arterial stents, establishing well-posedness via a Galerkin plus compactness argument and developing both semi-discrete and fully discrete finite-element schemes. It proves optimal-order error estimates in the energy norm for the semi-discrete method and the fully discrete Euler scheme, and demonstrates numerical performance using decoupling strategies and multi-rate time stepping. A detailed numerical study validates convergence, accuracy, and the practical impact of different discretization choices, while a two-dimensional generalization is briefly discussed. The results provide a rigorous, computationally robust framework for simulating trans-wall drug transport and can inform design and pharmacokinetic assessment of drug-eluting stents.

Abstract

This paper is concerned with the PDE and numerical analysis of a modified one-dimensional intravascular stent model originally proposed in [4]. It is proved that the modified model has a unique weak solution using the Galerkin method combined with a compactness argument. A semi-discrete finite element method and a fully discrete scheme using the Euler time-stepping are formulated for the PDE model. Optimal order error estimates in the energy norm are proved for both schemes. Numerical results are presented along with comparisons between different decoupling strategies and time-stepping schemes. Lastly, extensions of the model and its PDE and numerical analysis results to the two-dimensional case are also briefly discussed.
Paper Structure (17 sections, 15 theorems, 72 equations, 8 figures, 1 table)

This paper contains 17 sections, 15 theorems, 72 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical model and its weak formulation
  3. PDE analysis
  4. A priori estimates
  5. Well-posedness
  6. Error estimates and formulation of fully discrete scheme
  7. Error estimates for semi-discrete finite element method
  8. Formulation of fully discrete schemes
  9. Numerical simulations
  10. Comparison between two decoupling strategies
  11. Comparison between two time-stepping strategies
  12. Simulation results
  13. Two-dimensional generalization
  14. Summary and concluding remarks
  15. Parameters and cited facts
  16. ...and 2 more sections

Key Result

Theorem 3.1

\newlabelApriori Let $(c,c_1,c_2)$ be a weak solution to the system given by PDE--ICii in the sense of Definition WeakSoln. Then, the following holds: where $C>0$ is a constant that is independent of $(c_1,c_2,c)$.

Figures (8)

  • Figure 1.1: Sketch of arterial wall structure from blausen.
  • Figure 2.1: 1-d schematic diagram.
  • Figure 5.1: Algorithm I based on decoupling strategy #1.
  • Figure 5.2: Algorithm II based on decoupling strategy #2.
  • Figure 5.3: Stent concentrations.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 3.2: Uniqueness
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5: Existence
  • proof
  • ...and 16 more