Abstract
The use of enhancers to increase drug release from medical devices and drug transport through tissues has been largely investigated. Researchers from different fields like polymer chemistry, materials science, pharmaceutics, bioengineering, and chemical engineering have addressed efforts to combine materials, stimuli and drugs to design effective drug delivery platforms. For instance heat has been used to increase transdermal drug delivery. Patches with iron batteries are today in the market where heat generated by the batteries increases the drug release from the patches and the permeability of the skin, increasing drug absorption. Heat has been also used to increase drug availability in the target tissue in other contexts like in chemotherapy. In this case, to avoid the side effects of the systemic chemotherapy administration, drugs are encapsulated in thermosensitive carriers that transport the drug to the target where the cargo release is enhanced by heat. The aim of the present work is to study a system of partial differential equations (PDEs), from a numerical point of view, that can been used to describe the drug transport through tissues enhanced by heat. The system is composed by nonlinear PDEs for the temperature and for the drug concentration where the drug diffusion coefficient depends on the temperature. A finite difference method is studied and the qualitative behaviour of the temperature and concentration is numerically illustrated.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Barbeiro, S., Ferreira, J.A., Grigorieff, R.D.: Supraconvergence of a finite difference scheme for solutions in \(H^s(0, l)\). IMA J. Numer. Anal. 25, 797–811 (2005)
Evanghelidis, A., Beregoi, M., Diculescu, V., Galatanu, A., Ganea, P., Enculescu, I.: Delivery patch systems based on thermoresponsive hydrogels and submicronic fiber heaters. Sci. Rep. 8, 17555 (2018)
Ferreira, J.A., Grigorieff, R.: On the supraconvergence of elliptic finite difference schemes. Appl. Numer. Math. 28, 275–292 (1998)
Ferreira, J.A., Grigorieff, R.: Supraconvergence and supercloseness of a scheme for elliptic equations on non-uniform grids. Numer. Funct. Anal. Optim. 27(5–6), 539–564 (2006)
Ferreira, J.A., de OLiveira, P., Silveira, E.: Drug release enhanced by temperature: an accurate discrete model for solutions in \(H^3\). Comput. Math. Appl. (to appear). https://doi.org/10.1016/j.camwa.2019.08.002
Ferreira, J.A., Pinto, L.: Supraconvergence and supercloseness in quasilinear coupled problems. J. Comput. Appl. Math. 252, 120–131 (2013)
Forsyth, P., Sammon, P.H.: Quadratic convergence for cell-centered grids. Appl. Numer. Math. 4, 377–394 (1988)
Mura, S., Couvreur, P.: Stimuli-responsive nanocarriers for drug delivery. Nat. Mater. 12, 991–1003 (2013)
Lax, P., Richtmyer, R.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9, 267–293 (1956)
Manteuffel, T., White Jr., A.: The numerical solution of second order boundary value problems on nonuniform meshes. Math. Comput. 47, 511–535 (1986)
Patra, J., Das, G., Fraceto, L., Campos, E., Rodriguez-Torres, M., Acosta-Torres, L., Diaz-Torres, L., Grillo, R., Swamy, M., Sharma, S., Habtemariam, S., Shin, H.: Nano based drug delivery systems: recent developments and future prospects. J. Nanobiotechnol. 16, 71 (2018)
Qiao, Y., Wan, J., Zhou, L., Ma, W., Yang, Y., Luo, W., Yu, Z., Wang, H.: Stimuli-responsive nanotherapeutics for precision drug delivery and cancer therapy. WIREs Nanomed. Nanobiotechnol. 11, e1527 (2019)
Senapati, S., Mahanta, A., Kumar, S., Maiti, P.: Controlled drug delivery vehicles for cancer treatment and their performance. Signal Transd. Targeted Ther. 3, 7 (2018)
Sahle, F., Gulfam, M., Lowe, T.: Design strategies for physical stimuli-responsive programmable nanotherapeutics. Drug Discov. Today 23, 992–1006 (2018)
Szunerits, S., Boukherroub, R.: Heat: a highly efficient skin enhancer for transdermal drug delivery. Fontier Bioeng. Biotechnol. 6, 15 (2018)
Wells, C., Harris, M., Choi, L., Murali, V., Guerra, F., Jennings, J.: Stimuli-responsive drug release from smart polymers. J. Funct. Biomater. 10, 34 (2019)
Acknowledgements
This work was supported by Centro de Matemática da Universidade de Coimbra UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. The first and the second authors were also supported by the project NEXT.parts - Next- generation of advanced hybrid parts, funded by EU’s Horizon 2020 science programme (Portugal 2020, COMPETE 2020).
Elisa Silveira was supported by the FCT PhD grant PD/BD/128058/2016 funded by the Portuguese Government through FCT/MCTES.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Ferreira, J.A., de Oliveira, P., Silveira, E. (2020). Coupling Temperature with Drug Diffusion: A Second Order Approximation. In: Pinelas, S., Graef, J.R., Hilger, S., Kloeden, P., Schinas, C. (eds) Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham. https://doi.org/10.1007/978-3-030-56323-3_33
Download citation
DOI: https://doi.org/10.1007/978-3-030-56323-3_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-56322-6
Online ISBN: 978-3-030-56323-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)