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

Xiaobing Feng; Tingao Jiang; Xiaobing Feng; Tingao Jiang

doi:10.3934/mbe.2024248

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 4: 5634-5657. doi: 10.3934/mbe.2024248

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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

Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA

Received: 18 January 2024 Revised: 15 February 2024 Accepted: 27 February 2024 Published: 18 April 2024

This paper is concerned with the PDE (partial differential equation) and numerical analysis of a modified one-dimensional intravascular stent model. It is proved that the modified model has a unique weak solution by 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 have been 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.
- intravascular stent,
- drug release,
- diffusion-advection-reaction equation,
- well-posedness,
- Galerkin finite-element method,
- error estimates,
- pharmacokinetics
Citation: Xiaobing Feng, Tingao Jiang. Mathematical and numerical analysis for PDE systems modeling intravascular drug release from arterial stents and transport in arterial tissue[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5634-5657. doi: 10.3934/mbe.2024248

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Abstract

This paper is concerned with the PDE (partial differential equation) and numerical analysis of a modified one-dimensional intravascular stent model. It is proved that the modified model has a unique weak solution by 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 have been 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.

References

[1]	Blausen.com staff (2014), Medical gallery of Blausen Medical 2014, WikiJ. Med., 1 (2014). https://doi.org/10.15347/wjm/2014.010
[2]	G. Pontrelli, F. Monte, Mass diffusion through two-layer porous media: An application to the drug-eluting stent, J. Heat Mass Transfer, 50 (2007), 3658–3669. https://doi.org/10.1016/j.ijheatmasstransfer.2006.11.003 doi: 10.1016/j.ijheatmasstransfer.2006.11.003
[3]	S. McGinty, S. McKee, R. M. Wadsworth, C. McCormick, Modeling drug-eluting stents Math. Med. Biol., 28 (2011), 1–29. https://doi.org/10.1093/imammb/dqq003
[4]	S. McGinty, S. McKee, R. M. Wadsworth, C. McCormick, Modeling arterial wall drug concentrations following the insertion of a drug-eluting stent, SIAM J. Appl. Math., 73 (2013), 2004–2028. https://doi.org/10.1137/12089065X doi: 10.1137/12089065X
[5]	B. Balakrishnan, J. F. Dooley, G. Kopia, E. R. Edelman, Intravascular drug release kinetics dictate arterial drug deposition, retention, and distribution, J. Controll. Release, 123 (2007), 100–108. https://doi.org/10.1016/j.jconrel.2007.06.025 doi: 10.1016/j.jconrel.2007.06.025
[6]	A. Borghi, E. Foa, R. Balossino, F. Migliavacca, G. Dubini, Modelling drug elution from stents: Effects of reversible binding in the vascular wall and degradable polymeric matrix, Computer Methods Biomechan. Biomed. Eng., 11 (2008), 367–377. https://doi.org/10.1080/10255840801887555 doi: 10.1080/10255840801887555
[7]	J. Escuer, M. Cebollero, E. Peña, S. McGinty, M. A. Martínez, How does stent expansion alter drug transport properties of the arterial wall, J. Mechan. Behav. Biomed. Mater., 104 (2020), Article 103610. https://doi.org/10.1016/j.jmbbm.2019.103610 doi: 10.1016/j.jmbbm.2019.103610
[8]	P. Feenstra, C. Taylor, Drug transport in artery walls: A sequential porohyperelastic-transport approach, Computer Methods Biomath. Biomed. Eng., 12 (2009), 263–276. https://doi.org/10.1080/10255840802459396 doi: 10.1080/10255840802459396
[9]	M. Grass, G. Pontrelli, L. Teresi, G. Grassi, L. Comel, A. Ferluga, et al., Novel design of drug delivery in stented arteries: A numerical comparative study, Math. Biosci. Eng., 6 (2009), 493–508. https://doi.org/10.3934/mbe.2009.6.493 doi: 10.3934/mbe.2009.6.493
[10]	M. Horner, S. Joshi, V. Dhruva, S. Sett, S. F. C. Stewart, A two-species drug delivery model is required to predict deposition from drug-eluting stents, Cardiovascular Eng. Technol., 1 (2010), 225–234. https://doi.org/10.1007/s13239-010-0016-4 doi: 10.1007/s13239-010-0016-4
[11]	D. R. Hose, A. J. Narracott, D. Griffiths, S. Mahmood, J. Gunn, D. Sweeney, et al., A thermal analogy for modelling drug elution from cardiovascular stents, Computer Methods Biomechan. Biomedical Eng., 7 (2004), 257–264. https://doi.org/10.1080/10255840412331303140 doi: 10.1080/10255840412331303140
[12]	J. M. Weiler, E. M. Sparrow, R. Ramazani, Mass transfer by advection and diffusion from a drug-eluting stent, Int. J. Heat Mass Transfer, 55 (2012), 1–7. https://doi.org/10.1016/j.ijheatmasstransfer.2011.07.020 doi: 10.1016/j.ijheatmasstransfer.2011.07.020
[13]	P. Zunino, Multidimensional pharmacokinetics models applied to the design of drug-eluting stents, Cardiovasc. Eng. Int. J., 4 (2004), 181–191. https://doi.org/10.1023/B:CARE.0000031547.39178.cb doi: 10.1023/B:CARE.0000031547.39178.cb
[14]	S. McGinty, A decade of modelling drug release from arterial stents, Math. Biosci., 257 (2014), 80–90. https://doi.org/10.1016/j.mbs.2014.06.016 doi: 10.1016/j.mbs.2014.06.016
[15]	S. C. Brenner, L. R. Scott The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, NY (2002). https://doi.org/10.1007/978-0-387-75934-0
[16]	J. Escuer, A. F. Schmidt, E. Peña, M. A. Martìnez, S. McGinty, Mathematical modelling of endovascular drug delivery: Balloons versus stents, Int. J. Pharmaceut., 620 (2022), Article 121742. https://doi.org/10.1016/j.ijpharm.2022.121742 doi: 10.1016/j.ijpharm.2022.121742
[17]	A. McQueen, J. Escuer, A. F. Schmidt, A. Aggarwal, S. Kennedy, C. McCormick, et al., An intricate interplay between stent drug dose and release rate dictates arterial restenosis, J. Control. Release, 349 (2022), 992–1008. https://doi.org/10.1016/j.jconrel.2022.07.037 doi: 10.1016/j.jconrel.2022.07.037
[18]	L. C. Evans, Partial Differential Equations, AMS, Providence, RI (2010). https://doi.org/10.1090/gsm/019
[19]	P. Hartman, Ordinary Differential Equations, SIAM, Philadelphia (1964).
[20]	A. Friedman, Foundations of Modern Analysis, Dover, New York (2010).

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