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An Analog of the Galerkin Method in Problems of Drug Delivery in Biological Tissues

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Abstract

The authors propose an analog of the Galerkin method for the initial–boundary-value problem that describes drug delivery in the artery wall using a drug-coated stent. The method of numerical solution of the initial–boundary-value problem is constructed and the theorems on its convergence to the solution are proved.

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References

  1. N. Bulman-Fleming, “Numerical simulations of stent-based local drug delivery: 2D geometric investigations and the evaluation of 3D designs on the basis of local delivery effectiveness,” PhD Thesis, McGill University, Montreal (2003).

  2. G. Pontrelli and F. De Monte, “A multi-layer porous wall model for coronary drug-eluting stents,” Intern. J. of Heat and Mass Transfer, Vol. 53, No. 19–20, 3629–3637 (2010). https://doi.org/10.1016/j.ijheatmasstransfer.2010.03.031.

    Article  MATH  Google Scholar 

  3. M. Vahedi, V. Mohammad, and F. De Monte, “An advection-diffusion multi-layer porous model for stent drug delivery in coronary arteries,” J. Comp. and Applied Research in Mech. Engineering, Vol. 9, No. 1, 1–18 (2019). https://doi.org/10.22061/jcarme.2018.2741.1280.

  4. S. McGinty and G. Pontrelli, “Drug delivery in biological tissues: A two-layer reaction-diffusion-convection model,” in: Progress in Industrial Mathematics at ECMI 2014, ECMI 2014, G. Russo, V. Capasso, G. Nicosia, and V. Romano (eds.), Mathematics in Industry, Vol. 22, Springer, Cham (2016), pp. 355–363. https://doi.org/10.1007/978-3-319-23413-7_47.

  5. J. Weiler, E. Sparrow, and R. Ramazani, “Mass transfer by advection and diffusion from a drug-eluting stent,” Intern. J. Heat and Mass Transfer, Vol. 55, Iss. 1–3, 1–7 (2012). https://doi.org/10.1016/j.ijheatmasstransfer.2011.07.020.

  6. S. McGinty, S. McKee, R. M. Wadsworth, and C. McCormick, “Modeling arterial wall drug concentrations following the insertion of a drug-eluting stent,” SIAM J. Applied Math., Vol. 73, No. 6, 2004–2028 (2013). https://doi.org/10.1137/12089065X.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Pontrelli and F. de Monte, “Modeling of mass dynamics in arterial drug-eluting stents,” J. Porous Media, Vol. 12, No. 1, 19–28 (2009). https://doi.org/10.1615/JPorMedia.v12.i1.20.

    Article  Google Scholar 

  8. M. Grassi, G. Pontrelli, L. Teresi, G. Grassi, L. Comel, A. Ferluga, and L. Galasso, “Novel design of drug delivery in stented arteries: A numerical comparative study,” Mathematical Biosciences and Engineering, Vol. 6, No. 3, 493–508 (2009). https://doi.org/10.3934/mbe.2009.6.493.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. A. Klyushin, S. I. Lyashko, N. I. Lyashko, O. S. Bondar, and A. A. Tymoshenko, “Generalized optimization of processes of drug transport in tumors,” Cybern. Syst. Analysis, Vol. 56, No. 5, 758–765 (2020). https://doi.org/10.1007/s10559-020-00296-9.

    Article  MATH  Google Scholar 

  10. L. D. Karabin and D. A. Klyushin, “Two-phase Stefan problem for optimal control of targeted drug delivery to malignant tumors,” J. Coupled Systems and Multiscale Dynamics, Vol. 2, No. 2, 45–51 (2014). https://doi.org/10.1166/jcsmd.2014.1044.

    Article  Google Scholar 

  11. G. V. Sandrakov, S. I. Lyashko, E. S. Bondar, and N. I. Lyashko, “Modeling and optimization of microneedle systems,” J. Autom. Inform. Sci., Vol. 51, No. 6, 1–11 (2019). https://doi.org/10.1615/JAutomatInfScien.v51.i6.10.

    Article  Google Scholar 

  12. D. A. Klyushin, N. I. Lyashko, and Yu. N. Onopchuk, “Mathematical modeling and optimization of intratumor drug transport,” Cybern. Syst. Analysis, Vol. 43, No. 6, 886–892 (2007). https://doi.org/10.1007/s10559-007-0113-z.

    Article  MathSciNet  MATH  Google Scholar 

  13. S. I. Lyashko, D. A. Klyushin, A. A. Tymoshenko, N. I. Lyashko, and E. S. Bondar, “Optimal control of intensity of water point sources in unsaturated porous medium,” J. Autom. Inform. Sci., Vol. 51, No. 7, 24–33 (2019). https://doi.org/10.1615/JAutomatInfScien.v51.i7.20.

    Article  Google Scholar 

  14. A. Tymoshenko, D. Klyushin, and S. Lyashko, “Optimal control of point sources in Richards–Klute equation,” in: Advances in Computer Science for Engineering and Education, ICCSEEA 2018, Z. Hu, S. Petoukhov, I. Dychka, and M. He (eds.), Advances in Intelligent Systems and Computing, Vol. 754, Springer, Cham (2019), pp. 194–203. https://doi.org/10.1007/978-3-319-91008-6_20.

  15. I. V. Sergienko, S. I. Lyashko, and S. A. Voitsekhovskii, “An approximate solution for a class of second-order elliptic variational inequalities in arbitrary-form domains,” Cybern. Syst. Analysis, Vol. 40, No. 4, 486–490 (2004). https://doi.org/10.1023/B:CASA.0000047870.13325.c2.

    Article  MATH  Google Scholar 

  16. S. I. Lyashko and D. A. Nomirovskii, “The generalized solvability and optimization of parabolic systems in domains with thin low-permeable inclusions,” Cybern. Syst. Analysis, Vol. 39, No. 5, 737–745 (2003). https://doi.org/10.1023/B:CASA.0000012094.62199.de.

    Article  MATH  Google Scholar 

  17. S. I. Lyashko and V. V. Semenov, “Controllability of linear distributed systems in classes of generalized actions,” Cybern. Syst. Analysis, Vol. 37, No. 1, 13–32 (2001). https://doi.org/10.1023/A:1016607831284.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. I. Lyashko, D. A. Nomirovskii, and T. I. Sergienko, “Trajectory and final controllability in hyperbolic and pseudohyperbolic systems with generalized actions,” Cybern. Syst. Analysis, Vol. 37, No. 5, 756–763 (2001). https://doi.org/10.1023/A:1013871026026.

    Article  MATH  Google Scholar 

  19. S. I. Lyashko, D. A. Klyushin, and L. I. Palienko, “Simulation and generalized optimization in pseudohyperbolical systems,” J. Autom. Inform. Sci., Vol. 32, No. 5, 64–72 (2000). https://doi.org/10.1615/JAutomatInfScien.v32.i5.80.

    Article  Google Scholar 

  20. S. I. Lyashko and A. A. Man’kovskii, “Simultaneous optimization of impulse and intensities in control problems for parabolic equations,” Cybern. Syst. Analysis, Vol. 19, No. 5, 687–690 (1983). https://doi.org/10.1007/BF01068766.

    Article  MathSciNet  Google Scholar 

  21. A. G. Nakonechnyi and S. I. Lyashko, “Minmax estimation theory for solutions of abstract parabolic equations,” Cybern. Syst. Analysis, Vol. 31, No. 4, 626–630 (1995). https://doi.org/10.1007/BF02366418.

    Article  Google Scholar 

  22. S. I. Lyashko and V. V. Semenov, “Controllability of linear distributed systems in classes of generalized actions,” Cybern. Syst. Analysis, Vol. 37, No. 1, 13–32 (2001). https://doi.org/10.1023/A:1016607831284.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. I. Lyashko, D. A. Nomirovskii, and T. I. Sergienko, “Trajectory and terminal controllability in hyperbolic and pseudohyperbolic systems with generalized actions,” Cybern. Syst. Analysis, Vol. 37, No. 5, 756–763 (2001). https://doi.org/10.1023/A:1013871026026.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. I. Lyashko, “Numerical solution of pseudoparabolic equations,” Cybern. Syst. Analysis, Vol. 31, No. 5, 718–722 (1995). https://doi.org/10.1007/BF02366321.

    Article  MATH  Google Scholar 

  25. S. I. Lyashko, “Approximate solution of equations of pseudoparabolic type,” USSR Comp. Math. and Math. Physics, Vol. 31, No. 12, 107–111 (1991).

  26. L. A. Vlasenko, A. G. Rutkas, V. V. Semenets, and A. A. Chikriy, “On the optimal impulse control in descriptor systems,” J. Autom. Inform. Sci., Vol. 51, No. 5, 1–15 (2019). https://doi.org/10.1615/JAutomatInfScien.v51.i5.10.

    Article  Google Scholar 

  27. M. R. Petryk, A. Khimich, M. M. Petryk, and J. Fraissard, “Experimental and computer simulation studies of dehydration on microporous adsorbent of natural gas used as motor fuel,” Fuel, Vol. 239, 1324–1330 (2019). https://doi.org/10.1016/j.fuel.2018.10.134.

  28. N. I. Aralova, L. Y.-G. Shakhlina, and S. M. Futornyi, “Mathematical model of high-skilled athlete’s immune system,” J. Autom. Inform. Sci., Vol. 51, No. 3, 56–67 (2019). https://doi.org/10.1615/JAutomatInfScien.v51.i3.60.

    Article  Google Scholar 

  29. N. I. Aralova, L. Y.-G. Shakhlina, S. M. Futornyi, and S. V. Kalytka, “Information technologies for substantiation of the optimal course of interval hypoxic training in practice of sports training of highly qualified sports women,” J. Autom. Inform. Sci., Vol. 52, No. 1, 41–55 (2020). https://doi.org/10.1615/JAutomatInfScien.v52.i1.50.

    Article  Google Scholar 

  30. A. Ya. Bomba, “Asymptotic method for approximately solving a mass transport problem for flow in a porous medium,” Ukr. Math. J., Vol. 34, Iss. 4, 400–403 (1982).

  31. A. Ya. Bomba and O. A. Fursachuk, “Inverse singularly perturbed problems of the convection-diffusion type in quadrangular curvilinear domains,” J. Math. Sci., Vol. 171, No. 4, 490–498 (2010).

  32. I. I. Lyashko, V. P. Didenko, and O. E. Tsitritskii, Noise Filtering [in Russian], Naukova Dumka, Kyiv (1979).

    Google Scholar 

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Authors and Affiliations

  1. Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    D. A. Klyushin, S. I. Lyashko, O. S. Bondar & A. A. Tymoshenko

  2. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    N. I. Lyashko

Authors
  1. D. A. Klyushin
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  2. S. I. Lyashko
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  3. N. I. Lyashko
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  4. O. S. Bondar
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  5. A. A. Tymoshenko
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Correspondence to D. A. Klyushin.

Additional information

Translated from Kibernetyka ta Systemnyi Analiz, No. 3, May–June, 2021, pp. 21–29

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Klyushin, D.A., Lyashko, S.I., Lyashko, N.I. et al. An Analog of the Galerkin Method in Problems of Drug Delivery in Biological Tissues. Cybern Syst Anal 57, 354–362 (2021). https://doi.org/10.1007/s10559-021-00360-y

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