BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 Lecture 2: Molecular Design and Synthesis of Biomaterials I: Biodegradable Solid Polymeric Materials(continued) hemistry and physical chemistry of degrading polymeric solids for biomaterials Today Theory of polymer erosion Enzymatic degradation of synthetic biomaterials Designing degradable materials Reading A. Gopferich, "Mechanisms of polymer degradation and erosion, Biomaterials 17, 103 (1996) Ratner p. 243-259 Supplementary Reading R.J. Young and P A. Lovell, "Introduction to Polymers, ch. 4 Polymer Structure pp 241 309(crystallization of polymers, Tm, glass transition, etc. Surface vs. Bulk Hydrolysis: GOpferich's theory for polymer erosion4 Biodegradable solids may have differing modes of degradation Surface erosion -degradation from exterior only with little/no water penetration into bulk Bulk erosion- water penetrates entire structure and degrades entire device simultaneously surface erosion bulk-erosion degree of degradation Fig. I. Schematic illustration of the changes a polymer matrix undergoes during surface erosion and bulk erosion. Polymers hydrolyzing by mechanisms ll or Ill can be either surface or bulk eroding Assuming that a polymer is water insoluble(initially) and that hydrolysis is the only mechanism of breakdown, the factors listed above all vary two rates of importance rate of water diffusion into polymer rate of chain cleavage by water ions The balance of these rates determines whether a polymer erodes from the surface in or by simultaneous degradation throughout the materia Comparing velocities of water diffusion and chain cleavage Lecture 2- Biodegradable Solid Polymers1 of 12
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 Lecture 2: Molecular Design and Synthesis of Biomaterials I: Biodegradable Solid Polymeric Materials (continued) Last time: chemistry and physical chemistry of degrading polymeric solids for biomaterials Today: Theory of polymer erosion Enzymatic degradation of synthetic biomaterials Designing degradable materials Reading: A. Gopferich, “Mechanisms of polymer degradation and erosion,’ Biomaterials 17, 103 (1996) Ratner p. 243-259 Supplementary Reading: R.J. Young and P.A. Lovell, “Introduction to Polymers,” ch. 4 Polymer Structure pp. 241- 309 (crystallization of polymers, Tm, glass transition, etc.) Surface vs. Bulk Hydrolysis: Göpferich’s theory for polymer erosion1-4 Biodegradable solids may have differing modes of degradation: Surface erosion – degradation from exterior only with little/no water penetration into bulk Bulk erosion – water penetrates entire structure and degrades entire device simultaneously Polymers hydrolyzing by mechanisms II or III can be either surface or bulk eroding.5-7 Assuming that a polymer is water insoluble (initially) and that hydrolysis is the only mechanism of breakdown, the factors listed above all vary two rates of importance: rate of water diffusion into polymer rate of chain cleavage by water ions The balance of these rates determines whether a polymer erodes from the surface in or by simultaneous degradation throughout the material: Comparing velocities of water diffusion and chain cleavage: Lecture 2 – Biodegradable Solid Polymers1 of 12
BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 Accounting for rate of water diffusion Time required for water to diffuse a mean distance <x> into the solid polymer: tdm=<X>2/4D20 DH20=effective diffusivity of water in polymer See Atkins Phys. Chem p 770 for derivatio Random walk Fig. 10. 18 One possible path of a random walk in three dimensions In this general case, the step length is also a random variable Mean distance from origin traveled by water molecule after time t= <p>=(2D:20t)12 Mean distance traveled in x direction =<>= 2(DH2otarl/I) EXPLAIN Number of bonds in depth <x> ( 2)n=<x(bonds/cm)3= <x>(NAvp/Mo) NAv Avogadro's number polymer density Mo= molecular weight of polymer repeat unit Accounting for rate of chain cleavage( k): probability that a bonds breaks in the interval (o, t) (3)p(t)=ke where we have assumed that chain cleavage is a random event following Poisson kinetics k= rate constant for bond hydrolysis Therefore the mean lifetime of a single bond is given by t>=【pd=【edt=1(kt+1e=1 k Time to degrade n bonds is a zero-order waiting time distributed according to a zero-order Erlang distribution (5)<t(n)>=(1/k)∑[=1ton](1/≈(1/k)ln(n) (1/k)[In <x>+(1/3)In(NAvp/Mo) (substituting(2)) Lecture 2-Biodegradable Solid Polymers2 of 12
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 Accounting for rate of water diffusion: Time required for water to diffuse a mean distance <x> into the solid polymer: (1) tdiff = <x>2 π/4DH2O DH2O = effective diffusivity of water in polymer See Atkins Phys. Chem p. 770 for derivation Random walk: Mean distance from origi (Atkins8 ) n traveled by water molecule after time t = <r> = (2DH2Ot)1/2 Mean distance traveled in x direction = <x> = 2(DH2Otdiff/π) 1/2 EXPLAIN Number of bonds in depth <x>: (2) n = <x>(bonds/cm3 ) 1/3 = <x>(NAvρ/M0) 1/3 NAv = Avogadro’s number ρ = polymer density M0 = molecular weight of polymer repeat unit Accounting for rate of chain cleavage (k): probability that a bonds breaks in the interval (0,t): (3) p(t) = ke-kt where we have assumed that chain cleavage is a random event following Poisson kinetics k = rate constant for bond hydrolysis Therefore the mean lifetime of a single bond is given by: � � � <tc> = �t p(t) dt = �t e-kt dt = -1 (kt + 1)e-kt = 1 (4) 0 0 k 0 k Time to degrade n bonds is a zero-order waiting time distributed according to a zero-order Erlang distribution: (5) <tc(n)> = (1/k)Σ[i=1 to n] (1/i) ≈ (1/k)ln (n) = (1/k)[ln <x> + (1/3)ln (NAvρ/M0)] (substituting (2)) Lecture 2 – Biodegradable Solid Polymers2 of 12
BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 Mechanism(surface vs bulk) is controlled by ratio of time for diffusion to time for hydrolysis, a dimensionless parameter analogous to a Deborah number Erosion number s (5)s=taifl<t(n)>=<x>keT/[4DH20(In <x>+(1/3)In(NAvp/Mo)11 note <x> in denominator In should have same units as p, i.e. cm if p is in g/cm If ex> is replaced by the total thickness of a degrading sample, we can predict the mechanism of erosion: bulk erosion change in erosion mechanism surface erosion surface erosion 10 bulk erosion 30 10 Fig 2. Dependence of the erosion number, s, on the diffusivity of water inside the polymer, D,, the dimensions of a polymer matrix, L, and the polymer bond reactivity, i, calculated from equation 7. The white plane represents the area of surface erosion, the gray one the area of bulk erosion Lecture 2- Biodegradable Solid Polymers3 of 12
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 Mechanism (surface vs. bulk) is controlled by ratio of time for diffusion to time for hydrolysis, a dimensionless parameter analogous to a Deborah number: Erosion number = ε (5) ε ≡ tdiff/<tc(n)> = <x>2 kcπ/[4DH2O{ln <x> + (1/3)ln (NAvρ/M0)}] • note <x> in denominator ln should have same units as ρ, i.e. cm if ρ is in g/cm3 If <x> is replaced by the total thickness of a degrading sample, we can predict the mechanism of erosion: ε > 1 bulk erosion ε = 1 change in erosion mechanism ε < 1 surface erosion Lecture 2 – Biodegradable Solid Polymers3 of 12
BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 mass loss is linear for surface-eroding Table I Estimated values of s and Lentical for selected degradable polymers Chemical structure i(s") Poly(anhydrides) 1.9×10-3Ref.30 75 um C-O-C 64×10-3Ref.0 04 mt Polyfortho esters 48×10Ref:3 0.6mm =0-R 27×108Rer,130l O-R Poly( caprolactone) 97×10Re.3 H Poly(z-hydroxy-esters) 66x·10Ref.3ol 40x10- Poly(amides) 2.6×1013Ref.t3l 1.5x10 13.4m For a lcm thick device. D=10-'cm's"(estimated from Ref [32) and InVM/NA(N =-16.5. D= estimated from Ref. 3 2D) and in/VM/NA(N-19[--165 surface bulk eroding polyacetals oolyketals polyesters polyan. poly(ortho- hydrides esters) polyure polyamide 102 10 10 Fig 3. Critical thickness, Leritical, that a polymer device has to exceed to undergo surface erosion (calculated from Eq (7). data shown in Lecture 2- Biodegradable Solid Polymers of 12
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 • mass loss is linear for surface-eroding devices only “surface eroding” “bulk eroding” Lecture 2 – Biodegradable Solid Polymers4 of 12
BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 Experimental demonstration of theory: Transition of pl ga erosion from bulk to surface mode degraded at basic pH (12)-increased kc, thus decreasing s < 1 Bulk(normal erosion at pH 7. 4) Surface(pH> 12) Erosion profiles of poly( a-hydroxy esters)at PH 7.4:(a) lh() and PLAs017(-),(b)PLA2sGAs沿h(◆).PAs (·) and PLA2GAs47h(■) 2p8P21 SEM shown previously(Fig. 13)confirms transition to surface mode Synthesizing biodegradable macromolecules to tailor properties Approaches to molecular design o Control polymer hydrophobicity -> degradation rate o Control concentration of reactive groups o Alter biocompatibility What are the degradation products? Acidity/basicity? Toxicity? Biological effects? Lecture 2-Biodegradable Solid Polymers5 of 12
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 Experimental demonstration of theory: Transition of PLGA erosion from bulk to surface mode: degraded at basic pH (>12)- increased kc, thus decreasing ε << 1 Bulk (normal erosion at pH 7.4): Surface (pH > 12): SEM shown previously (Fig. 13) confirms transition to surface mode Synthesizing biodegradable macromolecules to tailor properties Approaches to molecular design • Copolymerization o Control polymer hydrophobicity -> degradation rate o Control concentration of reactive groups o Alter biocompatibility What are the degradation products? Acidity/basicity? Toxicity? Biological effects? Lecture 2 – Biodegradable Solid Polymers5 of 12