BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 Hydrogels in drug delivery Control of drug release kinetics by hydrogel structure6, o Release from stable hydrogels is controlled by diffusion of solute through the network o Diffusion is described by Fick,s second law. n 1 o Recall the solution to Fick's second law for a semi-infinite slab contacting a perfect sink co-c(x) cqn 2 =1-e viD o Diffusion of drugs through a network is controlled by the mesh size(o) c(x) Increasing time erf(z) solution X Lecture 10-Bioengineering Applications of Hydrogels 1 of 4
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 Hydrogels in drug delivery Control of drug release kinetics by hydrogel structure6,7 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 = Dgel ∂x 2 o Recall the solution to Fick’s second law for a semi-infinite slab contacting a perfect sink: Eqn 2 c0 − c(x) =1 − erf 2 tD x c0 o Diffusion of drugs through a network is controlled by the mesh size (ζ) c(x) c0 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 (≤r2>)12=N212a between cross-links qn o. assuming a polymer chain that has 2 carbon-carbon bonds per repeat unit o derived from random walk chain statistics Where /is the bond length in the polymer backbone Mc is the molecular weight between cross-links Mo is the molecular weight per repeat unit Where Cn is the characteristic ratio for the polymer chain 70) Ean 4 (o)=Cm0 Q is the degree of swelling = swollen polymer/var ry polyme N is the degree of polymerization between cross-links The mesh size is related to the diffusion constant of a solute in the network Eyring theory of diffusion Egn 5 Where△G′ is the activation energy,△H* is activation enthalpy,and△ S* is activation entropy o N=translational oscillating frequency of solute molecule (ump rate!) o T= temperature o k= Boltzman constant The ratio of diffusion constant in the gel to that in solution is Eqn 6 o Where As gel is the activation entropy for diffusion in the gel and As o 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 The activation ent Ean 7 △sga=klnP*- kIn Po Lecture 10-Bioengineering Applications of Hydrogels 20f4
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: (<r0 2>)1/2=Nc 1/2a statistical segment length Number of segments between cross-links Eqn 3 r0 2 1/ 2 2M c 1/ 2 C1/ 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 Where l is the bond length in the polymer backbone Mc is the molecular weight between cross-links M0 is the molecular weight per repeat unit Where Cn is the characteristic ratio for the polymer chain ( )2 1/ 2 Eqn 4 ξ = r0 1/ 3 = Q1/ 3 ( ) r0 2 1/ 2 = Cn 1/ 2 Q1/ 3 N1/ 2 l φ2,s Q is the degree of swelling = Vswollen polymer/Vdry polymer N is the degree of polymerization between cross-links The mesh size is related to the diffusion constant of a solute in the network 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 The ratio of diffusion constant in the gel to that in solution is: * ∆Sgel k ˆ Eqn 6 D = Dgel = e ∆S0 * D0 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) The activation entropies are: Eqn 7 ∆S*gel = k ln P* - k ln P0 Lecture 10 – Bioengineering Applications of Hydrogels 2 of 4
BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 Egn 8 k In P Ean 9 d=_pel, opening Pe, rome. 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 gel, opening d qn o Where r is the size of the solute(drug) and s is the network mesh size 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 Eqn 11 ,volume e,ged Eqn 12 olme e bree,I o Where free is the specific free volume and v* is the volume of the solute(drug) o Refs for free volume theory applied here Yasuda et al. Makromol. Chem. 26, 177(1969) Peppas and Reinhart, J Membrane Sci. 15, 275(1983) Now Eqn 13 0. volume The free volume in a swollen gel is approximately free, 1 since the free volume contribution from polymer is extremely low(2.5% even in solid polymers at 25C) qn Free,gel=Q1vfree,1 92Vfree, 2 Therefore. n 15 Vfreegel-P1Vfree, 1=(1-92) Vree, 1=(1-1/Q) ree. o Where Q is the swelling degree =Swollen ge vary gel =1/02 Therefore. Lecture 10-Bioengineering Applications of Hydrogels 3 of 4
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 Eqn 8 ∆S*0 = k ln P*0 – k ln P0 * * * Eqn 9 D = Pgel = Pgel,openingPgel,volume ˆ * * P0 P0,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 Pgel,opening = ξ =1− r ξ o Where r is the size of the solute (drug) and ξ is the network mesh size 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 Pgel,volume ~ e v free,gel v* − * Eqn 12 P0,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: Yasuda et al. Makromol. Chem. 26, 177 (1969) Peppas and Reinhart, J. Membrane Sci. 15, 275 (1983) Now: * v* v* Eqn 13 Pgel,volume = e − v free,gel − v free,1 * P0,volume 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 vfree,gel = φ1vfree,1 + φ2vfree,2 Therefore: Eqn 15 vfree,gel ~ φ1vfree,1 = (1-φ2)vfree,1 = (1-1/Q)vfree,1 o Where Q is the swelling degree = Vswollen gel/Vdry gel = 1/φ2 Therefore: Lecture 10 – Bioengineering Applications of Hydrogels 3 of 4
BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 Egn 16 =e o V/free 1-1 for most polymers, experimentally Eqn 17 And thus finally Eqn 18 o Insulin: MW-5900 g/mole; hydrodynamic radius =16 A References 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 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) Tan,Y.Y.& Vanekenstein, G O.R. AA Generalized Kinetic-Model for Radical-Initiated Template Polymerizations in Dilute Template Systems. Macromolecules 24, 1641-1647(1991) 5. sb protein recognition. Nature 398, 593-597(1999 Ratner, B D. Template-imprinted nanostructured surfaces ather, 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 Biomedica/ 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 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
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 − v* − v* * Q )v free,1 v free,1 − v* 1 1 Eqn 16 Pgel,volume = e (1− 1 = e v free,1 Q−1 ≈ e − Q−1 * P0,volume o v*/vfree,1 ~ 1 for most polymers, experimentally Therefore: −1 ˆ Eqn 17 D ≅ 1− r e(Q−1) ξ And thus finally: −1 Eqn 18 Dgel ≅ D0 1− r e(Q−1) ξ o Insulin: MW – 5900 g/mole; hydrodynamic radius = 16 Å 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