Analytical Expressions of Concentrations inside the Cationic Glucose-Sensitive Membrane

Swaminathan Sevukaperumal, Shunmugham Loghambal, Lakshmanan Rajendran

Department of Mathematics, The Madura College, Madurai, India.

DOI: 10.4236/am.2012.34058                

Abstract

A mathematical model of Wu et al. [J. Membr. Sci 254 (2005) 119-127] of a cationic glucose-sensitive membrane is discussed. The model involves the system of non-linear steady-state reaction-diffusion equations. Analytical expres-sions pertaining to concentration of oxygen, glucose, and gluconic acid for all values of parameters are presented. We have employed Homotopy analysis method to evaluate the approximate analytical solutions of the non-linear boundary value problem. A comparison of the analytical approximation and numerical simulation is also presented. A good agreement between theoretical predictions and numerical results is observed.

Keywords

Homotopy Analysis Method; Cationic Glucose-Sensitive Membrane; Non-Linear Reaction/Diffusion Equations

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	K. Park, “Nanotechnology: What It Can Do for Drug Delivery,” Journal of Controlled Release, Vol. 120, No. 1-2, 2007, pp. 1-3. doi:10.1016/j.jconrel.2007.05.003
[2]	J. Kost and R. Langer, “Responsive Polymer Systems for Controlled Delivery of Therapeutics,” Trends in Biotechnology, Vol. 10, 1992, pp. 127-131. doi:10.1016/0167-7799(92)90194-Z
[3]	L. A. Klumb and T. A. Horbett, “Design of Insulin Delivery Devices Based on Glucose Sensitive Membranes,” Journal of Controlled Release, Vol. 18, No. 1, 1992, pp. 59-80. doi:10.1016/0168-3659(92)90212-A
[4]	J. Kost and R. Langer, “Responsive Polymeric Delivery Systems,” Advanced Drug Delivery Reviews, Vol. 46, No. 1, 2001, pp. 125-148.
[5]	S. W. Kim and H. A. Jacobs, “Self-Regulated Insulin Delivery-Artificial Pancreas,” Drug Development and Industrial Pharmacy, Vol. 20, No. 4, 1994, pp.575-580. doi:10.3109/03639049409038319
[6]	S. J. Lee and K. Park, “Glucose-Sensitive Phase-Reversible Hydrogels,” In: R. M. Ottenbrite, S. J. Huang and K. Park, Eds., Hydrogels and Biodegradable Polymers for Bioapplications, ACS, Washington DC, 1996, pp. 11-16.
[7]	J. Kost, T. A. Horbett, B. D. Ratner and M. Singh, “Glucose-Sensitive Membranes Containing Glucose Oxidase: Activity, Swelling, and Permeability Studies,” Journal of Biomedical Materials Research, Vol. 19, No. 9, 1985, pp. 1117-1133. doi:10.1002/jbm.820190920
[8]	G. Albin, T. A. Horbet and B. D. Ratner, “Glucose-Sensitive Membranes for Controlled Delivery of Insulin: Insulin Transport Studies,” Journal of Controlled Release, Vol. 2, No. 3, 1985, pp. 153-164. doi:10.1016/0168-3659(85)90041-0
[9]	G. Albin, T. A. Horbet, S. R. Miller and N. L. Ricker, “Theoretical and Experimental Studies of Glucose Sensitive Membranes,” Journal of Controlled Release, Vol. 6, No. 1, 1987, pp. 267-291. doi:10.1016/0168-3659(87)90081-2
[10]	G. Albin, T. A. Horbett and B. D. Ratner, “Gulcose-Sensitive Membranes for Controlled Delivery of Insulin,” In: J. Kost, Ed., Pulsed and Self-Regulated Drug Delivery, CRC Press, Boca Raton, 1990, pp. 159-185.
[11]	T. Traitel, Y. Cohen and J. Kost, “Characterization of a Glucose Sensitive Insulin Release System in Simulated in Vivo Conditions,” Biomaterials, Vol. 21, No. 16, 2000, pp. 1679-1687. doi:10.1016/S0142-9612(00)00050-8
[12]	M. Glodrich and J. Kost, “Glucose Sensitive Polymeric Matrices for Controlled Drug Delivery,” Clinical Materials, Vol. 13, No. 1-4, 1993, pp. 135-142. doi:10.1016/0267-6605(93)90100-L
[13]	K. Podual, F. J. Doyle III and N. A. Peppas, “Glucose Sensitivity of Glucose Oxidase Containing Cationic Copolymer Hydrogels Having Poly(Ethylene Glycol) Grafts,” Journal of Controlled Release, Vol. 67, No. 1, 2000, pp. 9-17. doi:10.1016/S0168-3659(00)00195-4
[14]	K. Podual, F. J. Doyle III and N. A. Peppas, “Dynamic Behavior of Glucose Oxidase-Containing Microparticles of Poly(Ethylene Glycol)-Grafted Cationic Hydrogels in an Environment of Changing pH,” Biomaterials, Vol. 21, No. 14, 2000, pp.1439-1450. doi:10.1016/S0142-9612(00)00020-X
[15]	J. K. Leypoldt and D. A. Gough, “Model of a Two-Substrate Enzyme Electrode for Glucose,” Analytical Chemistry, Vol. 56, No. 14, 1984, pp. 2896-2904. doi:10.1021/ac00278a063
[16]	D. A. Gough, J. Y. Lusisano and P. H. S. Tse, “Two-Dimensional Enzyme Electrode Sensor for Glucose,” Analytical Chemistry, Vol. 57, No. 12, 1985, pp. 2351-2357. doi:10.1021/ac00289a042
[17]	J. Y. Lusisano and D. A. Gough, “Transient Response of the Two Dimensional Glucose Sensor,” Analytical Chemistry, Vol. 60, No. 13, 1988, pp. 1272-1281. doi:10.1021/ac00164a007
[18]	M. J. Abdekhodaie and X. Y. Wu, “Modeling of a Cationic Glucose-Sensitive Membrane with Consideration of Oxygen Limitation,” Journal of Membrane Science, Vol. 254, No. 1-2, 2005, pp. 119-127.
[19]	S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.
[20]	S. J. Liao, “On the Homotopy Analysis Method for Nonlinear Problems,” Applied Mathematics and Computation, Vol. 147, No. 2, 2004, pp. 499-513. doi:10.1016/S0096-3003(02)00790-7
[21]	S. J. Laio, “Notes on the Homotopy Analysis Method: Some Definitions and Theorems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 4, 2009, pp. 983-997. doi:10.1016/j.cnsns.2008.04.013
[22]	S. J. Liao and Y. Tan, “A General Approach to Obtain Series Solutions of Nonlinear Differential Equations,” Studies in Applied Mathematics, Vol. 119, No. 4, 2007, pp. 297-355. doi:10.1111/j.1467-9590.2007.00387.x
[23]	S. J. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method,” Chapman and Hall, CRC Press, Boca Raton, 2003, p. 336.
[24]	S. Loghambal and L. Rajendran, “Analytical Expressions of concentration of Nitrate Pertaining to the Electrocatalytic Reduction of Nitrate Ion,” Journal of Electroanalytical Chemistry, Vol. 661, No. 1, 2011, pp. 137-143. doi:10.1016/j.jelechem.2011.07.027
[25]	G. Domairry and M. Fazeli, “Homotopy Analysis Method to Determine the Fin Efficiency of Convective Straight Fins with Temperature-Dependent Thermal Conductivity,” Communications in Nonlinear Science and Numerical Simulation, Vol.14, No. 2, 2009, pp.489-499. doi:10.1016/j.cnsns.2007.09.007