# A Mathematical Model for the Release of Peptide-Binding Drugs from Affinity Hydrogels

## Abstract

A mathematical model for the release of peptide-binding drugs from affinity hydrogels is analyzed in detail. The model is not specific to any particular peptide/drug/gel system, and can describe drug release from a large class of affinity systems. In many cases, it is shown that the model can be reduced to a coupled pair of nonlinear partial differential equations for the total drug and peptide. Quantitative information relating the rate of drug release to the values of the model parameters is presented. Numerical solutions are displayed that illustrate the rich variety of release behaviors the system is capable of exhibiting. Theoretical release profiles generated by the model are compared with experimental release data from three different studies, and good agreement is found. The development of reliable mathematical models for affinity hydrogels will provide useful design tools for these systems.

## Keywords

Drug delivery Affinity-based delivery system Peptide-binding drug Mathematical model## Notes

### Acknowledgments

We gratefully acknowledge the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland (SFI) Investigator Award 12/IA/1683. Dr Meere thanks NUI Galway for the award of a travel grant. We thank the referees for their helpful suggestions to improve the paper.

### Conflict of interest

Tuoi T. N. Vo and Martin G. Meere declare that they have no conflicts of interest.

### Ethical Standards

No human or animal studies were carried out by the authors for this article.

## Supplementary material

## References

- 1.Alberts, B., A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter. Molecular Biology of the Cell, 5th edition. New York: Garland Science, 2008Google Scholar
- 2.Bradbury, E. J., L. D. F. Moon, R. J. Popat, V. R. King, G. S. Bennett, P. N. Patel, J. W. Fawcett, and S. B. McMaho. Chondroitinase ABC promotes functional recovery after spinal cord injury.
*Nature.*416:636–640, 2002.CrossRefGoogle Scholar - 3.Crank, J. The Mathematics of Diffusion, 2nd edition. Oxford: Oxford University Press, 1975.Google Scholar
- 4.Fu, A. S., T. R. Thatiparti, G. M. Saidel, and H. A. von Recum. Experimental studies and modeling of drug release from a tunable affinity-based drug delivery platform.
*Ann. Biomed. Eng.*39:2466–2475, 2011.CrossRefGoogle Scholar - 5.Hoffman, A. S. Hydrogels for biomedical applications.
*Adv. Drug Deliv. Rev.*64:18–23, 2012.CrossRefGoogle Scholar - 6.Hu, S. M. and S. Schmidt. Interactions in sequential diffusion processes in semiconductors.
*J. Appl. Phys.*39:4272, 1968.CrossRefGoogle Scholar - 7.Hubbell, J. A. Matrix-bound growth factors in tissue repair.
*Swiss. Med. Wkly.*136:387–391, 2006.Google Scholar - 8.King, J. R. and C. P. Please. Diffusion of dopant in crystalline silicon: an asymptotic analysis.
*IMA J. Appl. Math.*37:185–197, 1986.CrossRefzbMATHMathSciNetGoogle Scholar - 9.Lin, C. C. and K. S. Anseth. Controlling affinity binding with peptide-functionalized poly(ethylene glycol) hydrogels.
*Adv. Funct. Mater.*19:2325–2331, 2009.CrossRefGoogle Scholar - 10.Lin, C. C., P. D. Boyer, A. A Aimetti, and K. S. Anseth. Regulating mcp-1 diffusion in affinity hydrogels for enhancing immuno-isolation.
*J. Control. Release*142:384–391, 2010.CrossRefGoogle Scholar - 11.Lin, C. C., and A. T. Metters. Metal-chelating affinity hydrogels for sustained protein release.
*J. Biomed. Mater. Res. A*83A:954–964, 2007.CrossRefGoogle Scholar - 12.Lin, C. C., A. T. Metters, and K. S. Anseth. Functional peg-peptide hydrogels to modulate local inflammation induced by the pro-inflammatory cytokine tnf\(\alpha\).
*Biomaterials.*30:4907–4914, 2009.CrossRefGoogle Scholar - 13.Maxwell, D. J., B. C. Hicks, S. Parsons, and S. E. Sakiyama-Elbert, Development of rationally designed affinity-based drug delivery systems.
*Acta Biomater.*1:101–113, 2005.CrossRefGoogle Scholar - 14.Maynard, H. D. and J. A. Hubbell. Discovery of a sulfated tetrapeptide that binds to vascular endothelial growth factor.
*Acta Biomater.*1:451–459, 2005.CrossRefGoogle Scholar - 15.Mohtaram, N. K., A. Montgomery, and S. M. Willerth. Biomaterial-based drug delivery systems for the controlled release of neurotrophic factors.
*Biomed. Mater.*8(1–13):022001, 2013.Google Scholar - 16.Pakulska, M. M., K. Vulic, and M. S. Shoichet. Affinity-based release of chondroitinase ABC from a modified methylcellulose hydrogel.
*J. Control. Release*171:11–16, 2013.CrossRefGoogle Scholar - 17.Please, C. P. & J. R. King. One- and two-dimensional nonlinear dopant diffusion in crystalline silicon - some analytical results.
*Solid State Electron.*31:299–305, 1988.CrossRefGoogle Scholar - 18.Sakiyama-Elbert, S. E. and J. A. Hubbell, Development of fibrin derivatives for controlled release of heparin-binding growth factors.
*J. Control. Release*65:389–402, 2000.CrossRefGoogle Scholar - 19.Shepard, J. A., P. J. Wesson, C. E. Wang, A. C. Stevans, S. J. Holland, A. Shikanov, B. A. Grzybowski, and L. D. Shea. Gene therapy vectors with enhanced transfection based on hydrogels modified with affinity peptides.
*Biomaterials.*32:5092–5099, 2011.CrossRefGoogle Scholar - 20.Siepmann, J. and F. Siepmann. Mathematical modelling of drug delivery.
*Int. J. Pharm.*364:328–343, 2008.CrossRefGoogle Scholar - 21.Tzafriri, A. R., A. D. Levin, and E. R. Edelman. Diffusion-limited binding explains binary dose response for local arterial and tumour drug delivery.
*Cell Prolif.*42:348–363, 2009.CrossRefGoogle Scholar - 22.Varki, A., R. D. Cummings, J. D. Esko, H. H. Freeze, P. Stanley, C. R. Bertozzi, G. W. Hart, and M. E. Etzler. Essentials of Glycobiology, 2nd edition. New York: Cold Spring Harbor Laboratory Press, 2009.Google Scholar
- 23.Vo, T. T. N. Mathematical Analysis of Some Models for Drug Delivery, PhD thesis. Galway: National University of Ireland Galway, 2012.Google Scholar
- 24.Vo, T. T. N. and M. G. Meere. Minimizing the passive release of heparin-binding growth factors from an affinity-based delivery system.
*Math. Med. Biol.*29:1–26, 2012.CrossRefMathSciNetGoogle Scholar - 25.Vo, T. T. N. and M. G. Meere. The mathematical modelling of affinity-based drug delivery systems.
*J. Coupled Syst. Multiscale Dyn.*(submitted)Google Scholar - 26.Vo, T. T. N., R. Yang, Y. Rochev, and M. G. Meere. A mathematical model for drug delivery.
*Prog. Ind. Math. ECMI*2010:521–528, 2010.Google Scholar - 27.Vulic, K., M. M. Pakulska, R. Sonthalia, A. Ramachandran, and M. S. Shoichet. Mathematical model accurately predicts protein release from an affinity-based delivery system.
*J. Control. Release.*197:69–77, 2015.CrossRefGoogle Scholar - 28.Vulic, K. and M. S. Shoichet. Tunable growth factor delivery from injectable hydrogels for tissue engineering.
*J. Am. Chem. Soc.*134:882–885, 2012.CrossRefGoogle Scholar - 29.Vulic, K. and M. S. Shoichet. Affinity-based drug delivery systems for tissue repair and regeneration.
*Biomacromolecules.*15:3867–3880, 2014.CrossRefGoogle Scholar - 30.Wang, N. X. and H. A. von Recum. Affinity-based drug delivery.
*Macromol. Biosci.*11:321–332, 2011.CrossRefGoogle Scholar - 31.Willerth, S. M., P. J. Johnson, D. J. Maxwell, S. R. Parsons, M. E. Doukas, and S. E. Sakiyama-Elbert. Rationally designed peptides for controlled release of nerve growth factor from fibrin matrices.
*J. Biomed. Mater. Res. A*80A:13–23, 2007.CrossRefGoogle Scholar - 32.Wood, M. D., A. M. Moore, D. A. Hunter, S. Tuffaha, G. H. Borschel, S. E. Mackinnon, and S. E. Sakiyama-Elbert. Affinity-based release of glial-derived neurotrophic factor from fibrin matrices enhances sciatic nerve regeneration.
*Acta Biomater.*5:959–968, 2009.CrossRefGoogle Scholar - 33.Wood, M. D. and S. E. Sakiyama-Elbert. Release rate controls biological activity of nerve growth factor released from fibrin matrices containing affinity-based delivery systems.
*J. Biomed. Mater. Res. A*84A:300–312, 2008.CrossRefGoogle Scholar