УДК 517.9, 519.6

АНАЛОГ МЕТОДУ ГАЛЬОРКІНА В ЗАДАЧАХ ПЕРЕНОСУ ЛІКІВ
У БІОЛОГІЧНИХ ТКАНИНАХ

Клюшин Дмитро Анатолійович,
доктор фіз.-мат. наук, професор кафедри Київського національного університету імені Тараса Шевченка,  dokmed5@gmail.com

Ляшко Сергій Іванович,
чл.-кор. НАН Украины, доктор фіз.-мат. наук, професор, завідувач кафедри Київського національного університету імені Тараса Шевченка,  lyashko.serg@gmail.com

Ляшко Наталія Іванівна,
кандидат техн. наук, науковий співробітник Інституту кібернетики ім. В.М. Глушкова НАН України, Київ,  lyashko.natali@gmail.com

Бондар Олена Сергіївна,
інженер Київського національного університету імені Тараса Шевченка,  lyashko.serg@gmail.com

Тимошенко Андрій Анатолійович,
aсистент кафедри Київського національного університету імені Тараса Шевченка,
 inna-andry@ukr.net

СПИСОК ЛІТЕРАТУРИ

1. Bulman-Fleming N. 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. Pontrelli G., De Monte F. A multi-layer porous wall model for coronary drug-eluting stents. International Journal of Heat and Mass Transfer. 2010. Vol. 53, N 19–20. P. 3629–3637. https://doi.org/10.1016/j.ijheatmasstransfer.2010.03.031.


3. Vahedi M., Mohammad V., De Monte F. An advection-diffusion multi-layer porous model for stent drug delivery in coronary arteries. Journal of Computational and Applied Research in Mechanical Engineering. 2019. Vol. 9, N 1. P. 1–18. https://doi.org/10.22061/jcarme.2018.2741.1280.


4. McGinty S., Pontrelli G. (2016) Drug delivery in biological tissues: a two-layer reaction-diffusion-convection model. In: Progress in Industrial Mathematics at ECMI 2014. ECMI 2014. Russo G., Capasso V., Nicosia G., Romano V. (Eds). Mathematics in Industry, vol 22. Cham: Springer, 2016. P. 355-363. https://doi.org/10.1007/978-3-319-23413-7_47.


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


6. McGinty S., McKee S., Wadsworth R.M., McCormick C. Modeling arterial wall drug concentrations following the insertion of a drug-eluting stent. SIAM Journal of Applied Mathematics. 2013. Vol. 73, N 6. P. 2004–2028. https://doi.org/10.1137/12089065X.


7. Pontrelli G, de Monte F. Modeling of Mass Dynamics in Arterial Drug-Eluting Stents. Journal of Porous Media. 2009. Vol. 12, N 1. P. 19–28. https://doi.org/10.1615/JPorMedia.v12.i1.20.


8. Grassi M., Pontrelli G., Teresi L., Grassi G, Comel L., Ferluga A., Galasso L. Novel design of drug delivery in stented arteries: A numerical comparative study. Mathematical Biosciences and Engineering. 2009. Vol. 6, N 3. P. 493–508. https://doi.org/10.3934/mbe.2009.6.493.


9. Klyushin D.A., Lyashko S.I., Lyashko N.I., Bondar O.S., Tymoshenko A.A. Generalized optimization of processes of drug transport in tumors. Cybernetics and Systems Analysis. 2020. Vol. 56, N 5. P. 758–765. https://doi.org/10.1007/s10559-020-00296-9.


10. Karabin L.D., Klyushin D.A. Two-phase Stefan problem for optimal control of targeted drug delivery to malignant tumors. Journal of Coupled Systems and Multiscale Dynamics. 2014. Vol. 2, N 2. P. 45–51. https://doi.org/10.1166/jcsmd.2014.1044.


11. Sandrakov G.V., Lyashko S.I., Bondar E.S., Lyashko N.I. Modeling and optimization of microneedle systems. Journal of Automation and Information Sciences. 2019. Vol. 51, N 6. P. 1–11. https://doi.org/10.1615/JAutomatInfScien.v51.i6.10.


12. Klyushin D.A., Lyashko N.I., Onopchuk Yu.N. Mathematical modeling and optimization of intratumor drug transport. Cybernetics and Systems Analysis, Vol. 43, N 6, 886-892. https://doi.org/10.1007/s10559-007-0113-z.


13. Lyashko S.I., Klyushin D.A., Tymoshenko A.A., Lyashko N.I., Bondar E.S. Optimal control of intensity of water point sources in unsaturated porous medium. Journal of Automation and Information Sciences. 2019. Vol. 51, N 7, P. 24–33. https://doi.org/10.1615/JAutomatInfScien.v51.i7.20.


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


15. Sergienko I.V., Lyashko S.I., Voitsekhovskii S.A. An approximate solution for a class of second-order elliptic variational inequalities in arbitrary-form domains. Cybernetics and Systems Analysis. 2004. Vol. 40, N 4. 486–490. https://doi.org/10.1023/B:CASA.0000047870.13325.c2.


16. Lyashko S.I., Nomirovskii D.A. The generalized solvability and optimization of parabolic systems in domains with thin low-permeable inclusions. Cybernetics and Systems Analysis. 2003. Vol. 39, N 5. P. 737–745. https://doi.org/10.1023/B:CASA.0000012094.62199.de.


17. Lyashko S.I., Semenov V.V. Controllability of linear distributed systems in classes of generalized actions. Cybernetics and Systems Analysis. 2001. Vol. 37, N 1, P. 13–32. https://doi.org/10.1023/A:1016607831284.


18. Lyashko S.I., Nomirovskii D.A., Sergienko T.I. Trajectory and final controllability in hyperbolic and pseudohyperbolic systems with generalized actions. Cybernetics and Systems Analysis. 2001. Vol. 37, N 5. P. 756–763. https://doi.org/10.1023/A:1013871026026.


19. Lyashko S.I., Klyushin D.A., Palienko L.I. Simulation and generalized optimization in pseudohyperbolical systems. Journal of Automation and Information Sciences. 2000. Vol. 32, N 5. P. 64–72. https://doi.org/10.1615/JAutomatInfScien.v32.i5.80.


20. Lyashko S.I., Man’kovskii A.A. Simultaneous optimization of impulse and intensities in control problems for parabolic equations. Cybernetics and Systems Analysis. 1983. Vol. 19, N 5. P. 687–690. https://doi.org/10.1007/BF01068766.


21. Nakonechnyi A.G., Lyashko S.I. Minmax estimation theory for solutions of abstract parabolic equations. Cybernetics and Systems Analysis. 1995. Vol. 31, N 4. 626–630. https://doi.org/10.1007/BF02366418.


22. Lyashko S.I., Semenov V.V. Controllability of linear distributed systems in classes of generalized actions. Cybernetics and Systems Analysis. 2001. Vol. 37, N 1. P. 13–32. https://doi.org/10.1023/A:1016607831284.


23. Lyashko S.I., Nomirovskii D.A., Sergienko T.I. Trajectory and terminal controllability in hyperbolic and pseudohyperbolic systems with generalized actions. Cybernetics and Systems Analysis. 2001. Vol. 37, N 5. P. 756–763. https://doi.org/10.1023/A:1013871026026.


24. Lyashko S.I. Numerical solution of pseudoparabolic equations. Cybernetics and Systems Analysis. 1995. Vol. 31, N 5. P. 718–722. https://doi.org/10.1007/BF02366321.


25. Lyashko S.I. Approximate solution of equations of pseudoparabolic type. USSR Computational Mathematics and Mathematical Physics. 1991. Vol. 31, N 12. P 107–111.


26. Vlasenko L.A., Rutkas A.G., Semenets V.V., Chikriy A.A. On the optimal impulse control in descriptor systems. Journal of Automation and Information Sciences. 2019. V. 51, No. 5. P. 1–15. https://doi.org/10.1615/JAutomatInfScien.v51.i5.10.


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


28. Aralova N.I., Shakhlina L.Y.-G., Futornyi S.M. Mathematical model of high-skilled athlete’s immune system. Journal of Automation and Information Sciences. 2019. Vol. 51, N 3. P. 56–67. https:/doi.org/10.1615/JAutomatInfScien.v51.i3.60.


29. Aralova N.I., Shakhlina L.Y.-G., Futornyi S.M., Kalytka S.V. Information technologies for substantiation of the optimal course of interval hypoxic training in practice of sports training of highly qualified sports women. Journal of Automation and Information Sciences. 2020. Vol. 52, N 1. P. 41–55. https://doi.org/10.1615/JAutomatInfScien.v52.i1.50.


30. Bomba A.Ya. Asymptotic method for approximately solving a mass transport problem for flow in a porous medium. Ukrainian Mathematical Journal. 1982. Vol. 34, Iss. 4. P. 400–403.


31. Bomba A.Ya., Fursachuk O.A. Inverse singularly perturbed problems of the convection-diffusion type in quadrangular curvilinear domains. Journal of Mathematical Sciences. 2010. Vol. 171, N. 4. P. 490–498.


32. Ляшко И.И., Диденко В.П., Цитрицкий О.Е. Фильтрация шумов. Киев: Наукова думка, 1979. 232 с.