BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 Hydrogels in drug delivery Control of drug release kinetics by hydrogel structure 6,7 o Release from stable hydrogels is controlled by diffusion of solute through the network o Diffusion is described by Fick’s second law: ?C ? 2 C Eqn 1 ?t = D gel ?x 2 o Recall the solution to Fick’s second law for a semi-infinite slab contacting a perfect sink: Eqn 2 c 0 ? c(x) = 1 ? erf ? ? ? 2 tD x ? ? ? c 0 o Diffusion of drugs through a network is controlled by the mesh size (ζ) c(x) c 0 Increasing time erf(z) solution x Free surface Lecture 10 – Bioengineering Applications of Hydrogels 1 of 4 BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 o The mesh size is related to the network swelling Q and the end-to-end distance between cross-links: (<r 0 2 >) 1/2 =N c 1/2 a statistical segment length Number of segments between cross-links Eqn 3 r 0 2 1/ 2 ? 2M c ? 1/ 2 C 1/ 2 l() = ? ? ? M 0 ? ? ? n o …assuming a polymer chain that has 2 carbon-carbon bonds per repeat unit o derived from random walk chain statistics null Where l is the bond length in the polymer backbone null M c is the molecular weight between cross-links null M 0 is the molecular weight per repeat unit null Where C n is the characteristic ratio for the polymer chain () 2 1/2 Eqn 4 ξ = r 0 1/3 = Q 1/3 () r 0 2 1/2 = C n 1/2 Q 1/3 N 1/2 l φ 2,s null Q is the degree of swelling = V swollen polymer /V dry polymer null N is the degree of polymerization between cross-links null The mesh size is related to the diffusion constant of a solute in the network null Eyring theory of diffusion: ? ?G * ? ?H * ?S * Eqn 5 D = Tνe kT = Tνe kT e k o Where ?G* is the activation energy, ?H* is activation enthalpy, and ?S* is activation entropy o N = translational oscillating frequency of solute molecule (jump rate!) o T = temperature o k = Boltzman constant null The ratio of diffusion constant in the gel to that in solution is: * ?S gel k ? Eqn 6 D = D gel = e ?S 0 * D 0 e k o Where ?S* gel is the activation entropy for diffusion in the gel and ?S* 0 is the activation entropy in for diffusion in the solvent o This assumes the activation enthalpy and oscillation frequencies for diffusion are approximately the same in the gel and pure solvent (reasonable for dilute and chemically inert systems) null The activation entropies are: Eqn 7 ?S* gel = k ln P* - k ln P 0 Lecture 10 – Bioengineering Applications of Hydrogels 2 of 4 ? ? BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 Eqn 8 ?S* 0 = k ln P* 0 – k ln P 0 * * * Eqn 9 D = P gel = P gel,opening P gel,volume ? * * P 0 P 0,volume o Where P*volume is the probability that a solute-sized volume of free space exists to jump into o P*opening is the probability that the network has a solute-sized gap to jump through P* gel,opening drug r drug P* gel,volume * ξ? r Eqn 10 P gel,opening = ξ = 1? r ξ o Where r is the size of the solute (drug) and ξ is the network mesh size null The probability of a volume to jump into is an exponential of the ratio of the solute size to the available free volume per mole: v* ? * Eqn 11 P gel,volume ~ e v free,gel v* ? * Eqn 12 P 0,volume ~ e v free,1 o Where vfree is the specific free volume and v* is the volume of the solute (drug) o Refs for free volume theory applied here: null Yasuda et al. Makromol. Chem. 26, 177 (1969) null Peppas and Reinhart, J. Membrane Sci. 15, 275 (1983) null Now: * ? v* v* ? Eqn 13 P gel,volume = e ? ? ? v free,gel ? v free,1 ? ? * P 0,volume null The free volume in a swollen gel is approximately vfree,1 since the free volume contribution from polymer is extremely low (2.5% even in solid polymers at 25°C) Eqn 14 v free,gel = φ 1vfree,1 + φ 2 v free,2 null Therefore: Eqn 15 v free , gel ~ φ 1 v free,1 = (1-φ 2 )v free,1 = (1-1/Q)v free,1 o Where Q is the swelling degree = V swollen gel /V dry gel = 1/φ 2 null Therefore: Lecture 10 – Bioengineering Applications of Hydrogels 3 of 4 BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 ? ? ? ? ? v* ? v* ? ? * ? Q )v free,1 v free,1 ? ? ? v* ? ? 1 ? ? 1 ? Eqn 16 P gel,volume = e ? ? (1? 1 ? = e v free,1 ? Q?1? ? ≈ e ? ? ? Q?1? ? * P 0,volume o v*/v free,1 ~ 1 for most polymers, experimentally null Therefore: ? ? ? ? ?1 ? ? ? Eqn 17 D ? ? ? 1? r ? ? e ?(Q?1)? ξ null And thus finally: ? ? ? ? ?1 ? ? Eqn 18 D gel ? D 0 ? ? 1? r ? ? e ?(Q?1)? ξ o Insulin: MW – 5900 g/mole; hydrodynamic radius = 16 ? References 1. Byrne, M. E., Oral, E., Hilt, J. Z. & Peppas, N. A. Networks for recognition of biomolecules: Molecular imprinting and micropatterning poly(ethylene glycol)-containing films. Polymers for Advanced Technologies 13, 798-816 (2002). 2. Hart, B. R. & Shea, K. J. Molecular imprinting for the recognition of N-terminal histidine peptides in aqueous solution. Macromolecules 35, 6192-6201 (2002). 3. Tan, Y. Y. & Vanekenstein, G. O. R. A. A Generalized Kinetic-Model for Radical-Initiated Template Polymerizations in Dilute Template Systems. Macromolecules 24, 1641-1647 (1991). 4. Shi, H. Q., Tsai, W. B., Garrison, M. D., Ferrari, S. & Ratner, B. D. Template-imprinted nanostructured surfaces for protein recognition. Nature 398, 593-597 (1999). 5. Shi, H. Q. & Ratner, B. D. Template recognition of protein-imprinted polymer surfaces. Journal of Biomedical Materials Research 49, 1-11 (2000). 6. Lustig, S. R. & Peppas, N. A. Solute Diffusion in Swollen Membranes .9. Scaling Laws for Solute Diffusion in Gels. Journal of Applied Polymer Science 36, 735-747 (1988). 7. Canal, T. & Peppas, N. A. Correlation between Mesh Size and Equilibrium Degree of Swelling of Polymeric Networks. Journal of Biomedical Materials Research 23, 1183-1193 (1989). 8. Podual, K., Doyle, F. J. & Peppas, N. A. Dynamic behavior of glucose oxidase-containing microparticles of poly(ethylene glycol)-grafted cationic hydrogels in an environment of changing pH. Biomaterials 21, 1439-1450 (2000). 9. Podual, K., Doyle, F. J. & Peppas, N. A. Preparation and dynamic response of cationic copolymer hydrogels containing glucose oxidase. Polymer 41, 3975-3983 (2000). 10. Podual, K., Doyle, F. J. & Peppas, N. A. Glucose-sensitivity of glucose oxidase-containing cationic copolymer hydrogels having poly(ethylene glycol) grafts. Journal of Controlled Release 67, 9-17 (2000). Lecture 10 – Bioengineering Applications of Hydrogels 4 of 4