Skip to main content
SpringerLink
Log in

The Mathematics of Drug Delivery

  • Conference paper
  • First Online:
Book cover Mathematical Models of Tumor-Immune System Dynamics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 107))

Abstract

There are a variety of devices for the delivery of pharmaceutical substances, tablets of course being the most prominent. Pharmaceutical scientists and physicians have formulated goals, such as release of a drug in a controlled fashion over an extended period of time or the targeted delivery of a drug to a specific site in a patient’s body. Since experiments with these delivery devices can be costly and sometimes only partially conclusive, mathematical modeling can play a role in understanding the mechanisms behind experimental release profiles and in developing delivery systems. Here we review mathematical models for drug delivery by matrix tablets and liposomes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Altin, J.G., Parish, C.R.: Liposomal vaccines–targeting the delivery of antigen. Methods 40, 39–52 (2006)

    Article  Google Scholar 

  2. Azarmi, S., Farid, J., Nokhodchi, A., Bahari-Saravi, S.M., Valizadeh, H.: Thermal treating as a tool for sustained release of indomethacin from Eudragit RS and RL matrices. Int. J. Pharm. 246, 171–177 (2002)

    Article  Google Scholar 

  3. Baeumer, B., Chatterjee, L., Hinow, P., Rades, T., Radunskaya, A., Tucker, I.: Predicting the drug release kinetics of matrix tablets. Discr. Contin. Dyn. Sys. B 12, 261–277 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Buchla, E., Nájera, A., Hinow, P., Radunskaya, A.: Swallowing a cellular automaton pill: predicting drug release from a matrix tablet. Simulation 90, 227–237 (2014). Available at http://arxiv.org/abs/1208.3447

    Article  Google Scholar 

  5. Buchla, E.: Celldiff. Available at https://github.com/catfact/celldiff (2012)

  6. Caraballo, I., Millan, M., Rabasco, A.M.: Relationship between drug percolation threshold and particle size in matrix tablets. Pharm. Res. 13, 387–390 (1996)

    Article  Google Scholar 

  7. Cecchelli, R., Berezowski, V., Lundquist, S., Culot, M., Renftel, M., Dehouck, M.-P., Fenart, L.: Modelling of the blood-brain barrier in drug discovery and development. Nat. Rev. Drug Discov. 6, 650–661 (2007)

    Article  Google Scholar 

  8. Chatterjee, L., Baier, A.-K., Rades, T., Tucker, I.G.: Effect of particle size of mannitol on polymethacrylate matrices prepared by direct compression. 35th Controlled Release Society Annual Meeting, New York (2008)

    Google Scholar 

  9. Donev, A., Torquato, S., Stillinger, F.H., Conelly, R.: Jamming in hard sphere and disk packings. J. Appl. Phys. 95, 989–999 (2004)

    Article  Google Scholar 

  10. DuBois, C., Farnham, J., Aaron, E., Radunskaya, A.: A multiple time-scale computational model of a tumor and its micro environment. Math. Biosci. Eng. 10, 121–150 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Enden, G., Schroeder, A.: A mathematical model of drug release from liposomes by low frequency ultrasound. Ann. Biomed. Eng. 37, 2640–2645 (2009)

    Article  Google Scholar 

  12. Goltsev, A.V., de Abreu, F.V., Dorogovtsev, S.N., Mendes, J.F.F.: Stochastic cellular automata model of neural networks. Phys. Rev. E 81, 061921 (2010)

    Article  MathSciNet  Google Scholar 

  13. Higuchi, T.: Mechanism of sustained medication: theoretical analysis of rate release of solid drugs dispersed in solid matrices. J. Pharm. Sci. 52, 1145–1149 (1963)

    Article  Google Scholar 

  14. Hinow, P.: A nonsmooth program for jamming hard spheres. Optim. Lett. 8, 13–33 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hinow, P., Radunskaya, A., Tucker, I., Yang, L.: Kinetics of bile salt binding to liposomes revealed by carboxyfluorescein release and mathematical modeling. J. Liposome Res. 22, 237–244 (2012)

    Article  Google Scholar 

  16. Kavousanakis, M.E., Liu, P., Boudouvis, A.G., Lowengrub, J., Kevrekidis, I.G.: Efficient coarse simulation of a growing avascular tumor. Phys. Rev. E 85, 031912 (2012)

    Article  Google Scholar 

  17. Knott, G.M., Jackson, T.L., Buckmaster, J.: Random packings of heterogeneous propellants. AIAA J. 39, 678–686 (2001)

    Article  Google Scholar 

  18. Kosmidis, K., Argyrakis, P., Macheras, P.: Fractal kinetics in drug release from finite fractal matrices. J. Chem. Phys. 119, 6373–6377 (2003)

    Article  Google Scholar 

  19. Lemaire, V., Bélair, J., Hildgen, P.: Structural modeling of drug release from biodegradable porous matrices based on a combined diffusion/erosion process. Int. J. Pharm. 258, 95–107 (2003)

    Article  Google Scholar 

  20. Löwen, H.: Fun with hard spheres. In: Statistical Physics and Spatial Statistics. Lecture Notes in Physics, vol. 554, pp. 295–331. Springer, New York/Berlin/Heidelberg (2000)

    Google Scholar 

  21. Lubachevsky, B.D., Stillinger, F.H.: Geometric properties of random disk packings. J. Stat. Phys. 60, 561–583 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nowak, S., Schadschneider, A.: Quantitative analysis of pedestrian counterflow in a cellular automaton model. Phys. Rev. E 85, 066128 (2012)

    Article  Google Scholar 

  23. Oude Blenke, E., Mastrobattista, E., Schiffelers, R.M.: Strategies for triggered drug release from tumor targeted liposomes. Expert Opin. Drug Deliv. 10, 1399–1410 (2013)

    Article  Google Scholar 

  24. Richardson, G., Kaouri, K., Byrne, H.M.: Particle trapping by an external body force in the limit of large Peclet number: applications to magnetic targeting in the blood flow. Eur. J. Appl. Math. 21, 77–107 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Siepmann, J., Peppas, N.A.: Hydrophilic matrices for controlled drug delivery: an improved mathematical model to predict the resulting drug release kinetics (the “Sequential Layer” Model). Pharm. Res. 17, 1290–1298 (2000)

    Article  Google Scholar 

  26. Torchilin, V.P.: Recent advances with liposomes as pharmaceutical carriers. Nat. Rev. Drug Discov. 4, 145–160 (2005)

    Article  Google Scholar 

  27. Torquato, S., Truskett, T.M., Debenedetti, P.G.: Is random close packing of spheres well defined? Phys. Rev. Lett. 84, 2064–2067 (2000)

    Article  Google Scholar 

  28. Tucker, I., Yang, L., Mujoo, H.: Delivery of drugs to the brain via the blood brain barrier using colloidal carriers. J. Microencapsul. 29, 475–486 (2012)

    Article  Google Scholar 

  29. Villalobos, R., Ganem, A., Cordero, S., Vidales, A.M., Domínguez, A.: Effect of the drug-excipient ratio in matrix-type-controlled release systems: Computer simulation study. Drug Dev. Ind. Pharm. 31, 535–543 (2005)

    Article  Google Scholar 

  30. Villalobos, R., Cordero, S., Vidales, A.M., Domínguez, A.: In silico study on the effects of matrix structure in controlled drug release. Physica A 367, 305–318 (2006)

    Article  Google Scholar 

  31. Yang, L., Zhang, H., Fawcett, J.P., Mikov, M., Tucker, I.G.: Effect of bile salts on the transport of morphine-6-glucuronide in rat brain endothelial cells. J. Pharm. Sci. 100, 1516–1524 (2011)

    Article  Google Scholar 

Download references

Acknowledgements

The authors have been supported by the grant “Collaborative Research: Predicting the Release Kinetics of Matrix Tablets” (DMS 1016214 to Peter Hinow and DMS 1016136 to Ami Radunskaya) of the National Science Foundation of the United States of America. We thank Boris Bäumer (Department of Mathematics and Statistics), Lipika Chatterjee, Lin Yang, and Ian Tucker (School of Pharmacy) at the University of Otago in Dunedin, New Zealand, Aisha Nájera (Claremont Graduate University) and Ezra Buchla for inspiring and cheerful collaboration. We thank Max Strater (Pomona College) for generating the images in Fig. 4. We thank the School of Pharmacy at the University of Otago in Dunedin, New Zealand, for hospitality during several collaboration visits. The original Workshop on the Application of Mathematics to Problems in Biomedicine (December 17–19, 2007) at the University of Otago in Dunedin, New Zealand was supported by NSF grant DMS-0737537. Peter Hinow thanks Amina Eladdadi and Peter Kim for organizing the Workshop on Mathematical Modeling of Tumor–Immune System Dynamics at the University of Sydney, Australia, in January 2013. Much of the material presented here appears in [3, 4] and [15].

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Wisconsin, 413, Milwaukee, WI, 53201-0413, USA

    Peter Hinow

  2. Department of Mathematics, Pomona College, 610 N. College Ave., Claremont, CA, 91711, USA

    Ami E. Radunskaya

Authors
  1. Peter Hinow
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ami E. Radunskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Hinow .

Editor information

Editors and Affiliations

  1. Mathematics Department, The College of Saint Rose, Albany, New York, USA

    Amina Eladdadi

  2. School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales, Australia

    Peter Kim

  3. Mathematical Sciences School, Queensland University of Technology, Brisbane, Queensland, Australia

    Dann Mallet

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this paper

Cite this paper

Hinow, P., Radunskaya, A.E. (2014). The Mathematics of Drug Delivery. In: Eladdadi, A., Kim, P., Mallet, D. (eds) Mathematical Models of Tumor-Immune System Dynamics. Springer Proceedings in Mathematics & Statistics, vol 107. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1793-8_5

Download citation

Publish with us

Policies and ethics

Search

Navigation