Bulletin of Mathematical Biology

, Volume 81, Issue 1, pp 105–130

# A Nonlinear Mathematical Model of Drug Delivery from Polymeric Matrix

• Koyel Chakravarty
• D. C. Dalal
Original Article

## Abstract

The objective of the present study is to mathematically model the integrated kinetics of drug release in a polymeric matrix and its ensuing drug transport to the encompassing biological tissue. The model embodies drug diffusion, dissolution, solubilization, polymer degradation and dissociation/recrystallization phenomena in the polymeric matrix accompanied by diffusion, advection, reaction, internalization and specific/nonspecific binding in the biological tissue. The model is formulated through a system of nonlinear partial differential equations which are solved numerically in association with pertinent set of initial, interface and boundary conditions using suitable finite difference scheme. After spatial discretization, the system of nonlinear partial differential equations is reduced to a system of nonlinear ordinary differential equations which is subsequently solved by the fourth-order Runge–Kutta method. The model simulations deal with the comparison between a drug delivery from a biodegradable polymeric matrix and that from a biodurable polymeric matrix. Furthermore, simulated results are compared with corresponding existing experimental data to manifest the efficaciousness of the advocated model. A quantitative analysis is performed through numerical computation relied on model parameter values. The numerical results obtained reveal an estimate of the effects of biodegradable and biodurable polymeric matrices on drug release rates. Furthermore, through graphical representations, the sensitized impact of the model parameters on the drug kinetics is illustrated so as to assess the model parameters of significance.

## Keywords

Local drug delivery Polymer degradation Biodurable Specific/nonspecific drug binding Internalization

## Notes

### Acknowledgements

Authors are extremely grateful to all the anonymous reviewers for their fruitful comments and suggestions towards improvement of the present article.

## References

1. Anderson JM, Shive MS (1997) Biodegradation and biocompatibility of PLA and PLGA microspheres. Adv Drug Deliv Rev 28:5–24
2. Argemí A, Ellis JL, Saurina J, Tomasko DL (2011) Development of a polymeric patch impregnated with naproxen as a model of transdermal sustained release system. J Pharm Sci 100:992–1000
3. Avtar R, Tandon D (2008) Modeling the drug transport in the anterior segment of the eye. Eur J Pharm Sci 35:175–182
4. Batycky RP, Hanes J, Langer R, Edwards DA (1997) A theoretical model of erosion and macromolecular drug release from biodegrading microspheres. J Pharm Sci 86(12):1464–1477
5. Border NI, Buchwald PE (2005) Ophthalmic drug design based on the metabolic activity of the eye: soft drugs and chemical delivery systems. AAPS J 7:820–833
6. Bozsak F, Chomaz JM, Barakat AI (2014) Modeling the transport of drugs eluted from stents: physical phenomena driving drug distribution in the arterial wall. Biomech Model Mechanobiol 13:327–347
7. Castellot JJ, Wong K, Herman B, Hoover RL, Albertini DF, Wright TC, Caleb BL, Karnovsky MJ (1985) Binding and internalization of heparin by vascular smooth muscle cells. J Cell Physiol 124(1):13–20
8. Chakravarty K, Dalal DC (2016) A two-phase model of drug release from microparticle with combined effects of solubilisation and recrystallisation. Math Biosci 272:24–33
9. Chakravarty K, Dalal DC (2018a) An analytical study of drug release kinetics from a degradable polymeric matrix. Int J Biomath 11(1):1850011-1–1850011-23Google Scholar
10. Chakravarty K, Dalal DC (2018b) An analytical study of drug release to biological tissues through endocytosis. Int J Dyn Control 6:167–178
11. Charlier A, Leclerc B, Couarraze G (2000) Release of mifepristone from biodegradable matrices: experimental and theoretical evaluations. Int J Pharm 200(1):115–120
12. Connell BMO, Walsh MT (2010) Demonstrating the influence of compression on artery wall mass transport. Ann Biomed Eng 38(4):1354–1366
13. Costa MA, Simon DI (2005) Molecular basis of restenosis and drug-eluting stents. Circulation 111:2257–2273
14. Deux JF, Meddahi-Pelle A, Blanche AFL, Feldman LJ, ColliecJouault S, Bree F, Boudghéne F, Michel J-B, Letourneur D (2002) Low molecular weight fucoidan prevents neointimal hyperplasia in rabbit iliac artery in-stent restenosis mode. Arterioscler Thromb Vasc Biol 22(10):1604–1609
15. Faisant N, Siepmann J, Benoit JP (2002) PLGA-based microparticles: elucidation of mechanisms and a new, simple mathematical model quantifying drug release. Eur J Pharm Sci 15(4):355–366
16. Ferron GM, Conway WD, Jusko WJ (1997) Lipophilic benzamide and anilide derivatives as high-performance liquid chromatography internal standards: application to sirolimus (rapamycin) determination. J Chromatogr B Biomed Sci Appl 703(1–2):243–251
17. Formaggia L, Minisini S, Zunino P (2010) Modeling polymeric controlled drug release and transport phenomena in the arterial tissue. Math Models Methods Appl Sci 20(10):1759–1786
18. Frenning G (2003) Theoretical investigation of drug release from planar matrix systems: effects of a finite dissolution rate. J Control Release 92:331–339
19. Grassi M, Grassi G, Lapasin R, Colombo I (2008) Understanding drug release and absorption mechanisms: a physical and mathematical approach. CRC Press, Boca RatonGoogle Scholar
20. Hara H, Nakamura M, Palmaz JC, Schwartz RS (2006) Role of stent design and coatings on restenosis and thrombosis. Adv Drug Deliv Rev 58:377–386
21. Hickson RI, Barry SI, Mercera GN, Sidhu HS (2011) Finite differences schemes for multi-layer diffusion. Math Comput Model 54:210–220
22. Hixson AW, Crowell JH (1931) Dependence of reaction velocity upon surface and agitation. Ind Eng Chem 23:923–931
23. Lauffenburger DA, Linderman JJ (1993) Receptors: models for binding, trafficking, and signalling. Oxford Univ. Press, New YorkGoogle Scholar
24. Liu X, Kruger P, Maibach H, Colditz P, Roberts M (2014) Using skin for drug delivery and diagnosis in the critically ill. Adv Drug Deliv Rev 77:40–49
25. Lu X, Yang J, Zhao JB, Gregersen H, Kessab GS (2003) Shear modulus of porcine coronary artery: contributions of media and adventitia. Am J Physiol Heart Circ Physiol 285:H1966–H1975
26. Maisel WH (2007) Unanswered questions—drug-eluting stents and the risk of late thrombosis. N Engl J Med 356:981–984
27. Manitz R, Lucht W, Strehmel K, Weiner R, Neubert R (1998) On mathematical modeling of dermal and transdermal drug delivery. J Pharm Sci 87:873–879
28. Noyes A, Whitney W (1897) The rate solution of solid substances in their own solutions. J Am Chem Soc 19:930–934
29. Prausnitz MR, Langer R (2008) Transdermal drug delivery. Nat Biotechnol 26:1261–1268
30. Raman C, Berkland C, Kim K, Pack DW (2005) Modeling small-molecule release from plg microspheres: effects of polymer degradation and nonuniform drug distribution. J Control Release 103(1):149–158
31. Rim JE, Pinsky PM, van Osdol WW (2009) Multiscale modeling framework of transdermal drug delivery. Ann Biomed Eng 37:1217–1229
32. Salahudeen MS, Nishtala PS (2016) An overview of pharmacodynamic modelling, ligand-binding approach and its application in clinical practice. Saudi Pharm J.
33. Saltzman WM (2001) Drug delivery: engineering principles for drug therapy. Oxford University Press, OxfordGoogle Scholar
34. Sousa JE, Serruys PW, Costa MA (2003a) New frontiers in cardiology, drug-eluting stents: part I. Circulation 107:2274–2279
35. Sousa JE, Serruys PW, Costa MA (2003b) New frontiers in cardiology, drug-eluting stents: part II. Circulation 107:2382–2389Google Scholar
36. Tzafriri AR, Levin AD, Edelman ER (2009) Diffusion-limited binding explains binary dose response for local arterial and tumour drug delivery. Cell Prolif 42(3):348–363
37. Tzafriri A, Groothuis A, Price GS, Edelman E (2012) Stent elution rate determines drug deposition and receptor-mediated effects. J Control Release 161:918–926
38. Weiler JM, Sparrow EM, Ramazani R (2012) Mass transfer by advection and diffusion from a drug-eluting stent. Int J Heat Mass Transf 55:1–7
39. Zhu X, Braatz RD (2015) A mechanistic model of drug release in plga biodegradable stent coatings coupled with polymer degradation and erosion. J Biomed Mater Res Part A 103A:2269–2279