A Model for Simultaneous Crystallisation and Biodegradation of

Transcription

1 A Modl for Simultanous Crystallisation and Biodgradation of Biodgradabl Polymrs iaoxiao Han and Jingzh Pan Dpartmnt of Enginring, Univrsity of Listr, Listr, LE 7RH, UK Corrsponding author. Tl.: ; fax: addrss: jp65@listr.a.uk

2 ABSTRACT: This papr omplts th modl of biodgradation for biodgradabl polymrs that was prviously dvlopd by Wang t al. (Wang Y, Pan J, Han, Sinka, Ding L. A phnomnologial modl for th dgradation of biodgradabl polymrs. Biomatrials 28;29: ). Crystallisation during biodgradation was not onsidrd in th prvious work whih is th topi of th urrnt papr. For many ommonly usd biodgradabl polymrs, thr is a strong intrplay btwn rystallisation and hydrolysis ration during biodgradation th hain lavag ausd by th hydrolysis ration provids an xtra mobility for th polymr hains to rystallis and th rsulting rystallin phas boms mor rsistant to furthr hydrolysis ration. This papr prsnts a omplt thory to dsrib this intrplay. Th fundamntal quations in th Avrami s thory for rystallisation ar modifid and oupld to th diffusion-ration quations that wr dvlopd in our prvious work. Th mathmatial quations ar thn applid to thr biodgradabl polymrs for whih long trm dgradation data ar availabl in th litratur. It is shown that th modl an aptur th bhaviour of th major biodgradabl polymrs vry wll. Kywords: Biodgradabl polymrs, biodgradation, rystallisation, modlling

3 Nomnlatur A non-dimnsionalisd form of A as dfind in Eqs. (9) and (2) C mol numbr of str bounds of amorphous polymr pr unit volum of smi-rystallin polymr C initial valu of C C m mol numbr of monomrs rmaind in th matrial pr unit volum of smi-rystallin polymr, D fftiv diffusion offiint of monomrs in dgrading polymr D diffusion offiint of monomrs in amorphous polymr D por diffusion offiint of monomrs in liquid-filld pors G N linar growth rat of a singl rystal mol onntration of nuli of rystallization N initial mol onntration of nuli of rystallisation R mol numbr of th amount of monomrs produd by hydrolysis ration pr unit volum of smi-rystallin polymrs. V sing volum of a singl rystal volum dgr of rystallinity xt xtndd volum dgr of rystallinity div divrgn of vtor in Cartsian oordinats x i x i grad gradint of sal variabl in Cartsian oordinats x i x i mol numbr of str bounds of amorphous polymr pr unit volum of th amorphous polymr

4 m mol numbr of monomrs rmaind in th matrial pr unit volum of amorphous polymr, r mol numbr of th amount of monomrs produd by hydrolysis ration pr unit volum of amorphous polymr, k ration rat of non-atalyzd hydrolysis k 2 ration rat of auto-atalyzd hydrolysis k Avrami onstant for rystallisation l n m haratristi lngth of dvi xponnt of dissoiation of aid nd groups Avrami xponnt n A Avogadro numbr ( = ) p r porosity of dgrading polymr du to laving monomrs siz of a singl rystal r max maximum siz of a singl rystal t dgradation tim t haratristi tim of auto-atalyzd hydrolysis ration ( n t / k2c ) x i (i=,2,3), Cartsian oordinats shap fator of a singl rystal (4 /3 for a sphr), impingmnt paramtr for rystallization, /( ) lif tim of a singl rystal probability of formation of growth nuli pr nulus pr unit tim 2

5 . Introdution Linar aliphati polystrs, poly(glyolid) (PGA), poly(latid) (PLA) and thir opolymrs in partiular, ar bing usd in ontrolld drug dlivry, orthopadi fixation, tissu nginring and many othr biomdial appliations to provid various tmporary funtions insid human body []. Th bakbons of ths polymrs ar highly hydrolysabl and th dgradation produts ar vntually mtabolizd into arbon dioxid and watr. Th dvis disappar aftr srving thir funtions to lt biology tak ovr. Baus of thir wll stablishd bioompatibility, PGA, PLA and thir opolymrs ar th most ommonly usd biodgradabl polymrs in mdial dvis. Suturs mad of ths polymrs hav bn safly usd sin 97. Biodgradabl srws, plats and films ar bing inrasingly usd in orthopadi fixations [2,3]. Drug dlivry dvis, tmporary barrirs for adhsion prvntion and tmporary vasular grafts ar som othr intnsivly invstigatd appliations [4]. Currntly th most pursud appliation is prhaps in tissu nginring to rrat or improv nativ tissu funtion using dgradabl saffolds [5]. PGA, PLA and thir opolymrs ar rtainly not matrials asy to work with. Among many othr things, two ompliations hav to b onsidrd: (a) th biodgradation of polymrs ontaining PLA is htrognous du to th autoatalyti natur of th hydrolysis ration [6], and (b) many of ths polymrs, PGA and poly(l-latid) (PLLA) for xampl, rystallis during biodgradation [7-9]. A ompliatd intrplay btwn th hydrolysis ration, diffusion of th ration produts and rystallisation maks th mhanial and funtional proprtis of th biodgradabl dvis diffiult to prdit. So far th dvlopmnt of mdial dvis mad of biodgradabl polymrs has bn 3

6 almost ntirly basd on th trial and rror approah. Th lak of a mathmatial framwork for th biodgradation pross maks it diffiult to xtrapolat xprin and data obtaind in on dvi to anothr. Baus th biodgradation is dimnsion dpndant, it is vn diffiult to xtrapolat data btwn sam dvis of diffrnt dimnsions (srws of diffrnt diamtrs for xampl). On th othr hand, modrn omputr modlling thniqus hav bn routinly usd in nginring to optimis nginring dsigns. A los xampl is th dsign of nginring omponnts oprating at lvatd tmpraturs, turbin blads in jt ngins and hot stam pips in powr gnration plants for xampls. Alloys usd in ths appliations xprin mirostrutural dgradation ovr a priod of -3 yars, rduing th strngth of th omponnts and limiting thir srvi lif. Th mthodology, spially th numrial thniqus dvlopd in th nginring dsign, an b radily applid to th dsign of biodgradabl mdial dvis. Howvr th undrlying mhanism of polymr dgradation is vry diffrnt from that for th high tmpratur failur of nginring alloys. A mathmatial framwork is ndd for th biodgradation pross. In our prvious papr, a st of simplifid diffusion ration quations wr stablishd to modl th biodgradation []. Th modl was ompard with xprimntal data and a biodgradation map was prsntd showing th intrplay btwn th hydrolysis ration and th diffusion of th ration produts. Th htrognous natur of th biodgradation was fully onsidrd. Howvr th intrplay btwn rystallisation and biodgradation was not onsidrd in th prvious work. Th urrnt papr omplts th modl by inorporating 4

7 rystallisation into th diffusion-ration quations. During biodgradation th hain sissions provid th xtra mobility for th polymr hains to rystallis. Th rsulting rystallin phas boms mor rsistant to furthr hydrolysis ration. Th rystallisation thory du to Avrami [-3] has bn shown to b valid for a wid rang of matrials inluding polymrs. This thory prdits an xponntial dpndn of th dgr of rystallinity on tim. Howvr th simpl xponntial quation annot b dirtly applid to biodgradation baus it dos not onsidr th intration btwn hydrolysis, diffusion and rystallisation. In this papr, w r-xamin th fundamntals in th Avrami s thory and show that th thory an b modifid and oupld to th diffusion-ration quations to modl simultanous rystallisation and biodgradation. It is no longr possibl to obtain analytial solutions to th rsulting diffrntial quations. Instad, ths quations ar solvd numrially. Th modl is thn applid to thr diffrnt biodgradabl polymrs inluding poly(glyolid-o-l-latid), poly(l-latid), and blnds of poly(l-latid) and poly(vinyl alohol), for whih omplt biodgradation data ar availabl in th litratur [3-5]. Finally paramtri studis ar arrid out using th modl to dmonstrat th ffts of rystallisation rat on th dgradation rat and on th biodgradation map. 2. Govrning quations for simultanous rystallisation and biodgradation It has bn widly obsrvd that th dgr of rystallinity in ommonly usd biodgradabl polymrs inrass signifiantly during both short trm and long trm 5

8 dgradation [7-9]. Th dgradation-indud rystallisation an our in ithr initially amorphous or smi-rystallin polymrs. Important xampls of th biodgradabl polymrs ar PGA, PLLA and thir opolymrs. PGA is a highly rystallin polymr whil PLLA is smi-rystallin. Th rystallin phas provids ths polymrs with th nssary mhanial strngth for th mdial dvis. During biodgradation, th hydrolysis ration of th str bakbon in aquous nvironmnt lads to lavag of th polymr hains and produs short oligomrs. Th oligomrs thn diffus out of th matrial lading to a wight loss of th dvi. For a smi-rystallin polymr, th hain lavag ours prfrntially in th amorphous rgion. Thrfor vn if th total volum of th rystallin phas rmains onstant, th obsrvd dgr of rystallinity still inrass du to th loss of th amorphous phas. Mor importantly, lavag of th long and amorphous polymr hains provids highr mobility for th polymr hains, failitating th rystallisation of th amorphous polymr [7]. Th dtaild dgradation pathways hav bn suggstd [7] and long trm xprimntal data of simultanous dgradation and rystallisation ar availabl in th litratur [8,9]. Howvr a mathmatial modl for th dgradation-indud rystallisation dos not xist as far as th authors ar awar. Avrami s thory [-3] has bn shown to b gnrally valid for polymr rystallisation [4]. It prdits that th dgr of rystallinity, dpnds on tim, t, in an xponntial mannr: ) m xp ( kt () in whih m is a onstant oftn rfrrd to as th Avrami xponnt and k is a tmpratur dpndnt fator (oftn takn as an Arrhnius typ xprssion). This 6

9 quation howvr dos not takn into aount of th intration btwn polymr hain lavag and rystallisation, and thrfor annot b dirtly applid to biodgradation. Howvr it is possibl to modify Avrami s diffrntial quations [-3] ld to Eq. () to modl th dgradation-indud rystallisation. Following our prvious work [], a biodgradabl polymr an b viwd as bing onsistd of four spis: I. amorphous polymr moluls, whih an hydrolys but ar too larg to diffus; part of th polymr moluls an also rystallis; II. monomrs, whih ar th produt of th hydrolysis ration and an diffus; III. polymr rystals, whih ar formd and grow but do not hydrolys. IV. watr moluls, whih ar assumd to b abundant anywhr in th dvi; Th stat of a biodgrading polymr an thrfor b ompltly dsribd using a) C - mol numbr of str bounds of amorphous polymr pr unit volum of smi-rystallin polymr, b) C m - mol numbr of monomrs rmaind in th matrial pr unit volum of smi-rystallin polymr, and ) - th volum dgr of rystallinity. It is assumd that th hydrolysis ration only ours in th amorphous rgion dspit that th str groups of polymr hains on th surfa of th rystallin rgion ar hydrolyzd. It is thn nssary to furthr dfin th following variabls: d) - mol numbr of str bounds of amorphous polymr pr unit volum of th amorphous polymr, 7

10 ) m - mol numbr of monomrs rmaind in th matrial pr unit volum of amorphous polymr, f) r - mol numbr of th amount of monomrs produd by hydrolysis ration pr unit volum of amorphous polymr. Th prodution rat of monomrs by th hydrolysis ration is givn by [5] dr dt k n k2m (2) in whih k and k 2 ar th ration onstants for th non-autoatalyti and autoatalyti hydrolysis rations. Th powr n in th sond trm aounts for th dissoiation of th aid nd groups. Th mol onntrations in th amorphous phas and thos in th smi-rystallin polymr ar onntd by C ; m Cm. (3) Using Eq. (3) in Eq. (2) givs dr dt k C C n m C k2. (4) n It turns out to b onvnint to dfining a nw variabl R using dr dt dr C C n m k C k2. (5) n dt R rprsnts th mols of monomrs produd pr unit volum of th smi-rystallin polymr. It also rflts th total numbr of hain lavags pr unit volum of th smi-rystallin polymr. Th rdution in th str bound onntration in th amorphous phas originats from two parts: (a) hydrolysis of th polymr hains and (b) rystallisation of th mobil polymr hains, whih an b xprssd as 8

11 dc dt dr dt C d dt. (6) Th sond trm on th lft hand sid in Eq. (6) rprsnts th lost of amorphous polymr phas du to rystallisation. Assuming Fik s law for monomr diffusion, w hav th following govrning quation for th monomr onntration: dc dt m dr dt div x i D grad C x i m (7) in whih D is th phnomnologial diffusion offiint. Th nomnlatur of vtor analysis is usd to shortn th xprssion of Eq. (6). Th fftiv diffusion offiint D of th dgrading polymr is a funtion of th porosity, p, and dgr of rystallinity,. In our prvious papr [] a linar rlation btwn D and p was assumd, whih is valid if th porosity is lss than 25%. To improv th linar approximation, finit lmnt alulations of th fftiv diffusion offiint wr arrid out using a thr dimnsional rprsntativ ubi matrial. A randomly distributd sond phas was gradually introdud into th unit and th fftiv diffusion offiint of th two phas matrial was alulatd numrially. Dtails of th analysis will b publishd lswhr [6]. Th onlusion of th numrial study is that th numrial rsults an b fittd into th following mpirial quation: D D slow.3v.3v D D (8) 2 fast 3 fast fast slow in whih D slow and D fast rprsnt th diffusion offiints of th fast and slow diffusion phass rsptivly and V fast rprsnts th volum fration of th fast diffusion phas. Eq. (8) is valid if D D. For a dgrading polymr fast / slow 9

12 ontaining pors whih ar gnratd by monomrs diffusing out, its fftiv diffusion offiint an b alulatd using Eq. (8) as D D matrix 2 3.3p.3p D D, (9) por matrix in whih D matrix and D por rprsnt th diffusion offiints of monomrs in th polymr matrix and pors rsptivly. Th porosity p an b stimatd as p Cm C, () in whih C C / C and C m Cm / C whr C C t ; is th initial dgr of rystallinity. Th polymr matrix is onsistd of an amorphous and a rystallin phas. It an b assumd that th diffusion offiint of th monomrs in th rystallin phas is zro ( D slow = ). Using D to rprsnt th diffusion offiint of monomrs in th amorphous polymr, th fftiv diffusion offiint of th polymr matrix an b obtaind from Eq. (8) as D 2 C C C C.3.3 D m m matrix () Cm C /( ) Cm C /( ) 3 Th fftiv diffusion offiint of th dgrading polymr an b dtrmind by ombining Eqs (9-). Th ntr pi of Avrami s thory [-3] is th rlation btwn th volum dgr of rystallinity,, and a so-alld xtndd volum fration of th rystallin phas, xt whih is givn by d d xt. (2) Th xtndd volum is a fititious volum of th rystals imagining that th rystal growth is unimpdd by impingmnt upon ah othr. xt is thrfor muh asir

13 to alulat than. Th Avrami s xprssion of Eq. () was dirtly drivd from Eq. (2). In this study, it was quikly ralizd that Avrami s thory basd on Eq. (2) is unabl to aptur th obsrvd rystallisation bhavior in biodgradabl polymrs. Th first problm is that Eq. (2) always prdits to a signifiant inubation priod for rystallisation whih is vry short during biodgradation [7-9]. Th sond problm is that Eq. (2) always prdits full rystallisation as tim approahs infinity whih is not tru for most biodgradabl polymrs. Th following modifiation to Eq. (2) is usd to ovrom ths problms: d d xt. (3) Hr, or its anothr form, /( ), is rfrrd to as th impingmnt paramtr in th litratur, whih was introdud by prvious rsarhrs to provid a bttr fit with xprimntal data [7]. Th nxt fundamntal lmnt in Avrami s thory is th govrning quation for th rystallisation nuli. In Avrami s thory, it is assumd that thr xist a fixd numbr of nuli at th bginning of th rystallisation and that thy ar gradually usd up as th rystallisation ontinus. Using N to rprsnt th mol onntration of th nuli at tim t, Avrami proposd th following quation [-3]: dn Ndt N d. (4) Th first trm on th lft hand sid rprsnts th numbr of nuli that bom ativ growth during tim intrval dt in onsqun of fr nrgy flutuation. N rprsnts th probability of this happning. Th sond trm rprsnts th numbr

14 of nuli that ar swallowd by th growing rystals. Eq. (4) is subjt to th initial ondition that N N at t. During lavag-indud rystalisation, th amorphous polymr hains only start to rystallis whr hain lavag ours, i.. a nulus (a forign inlusion for xampl) an only bom availabl if a hain lavag ours narby. In a small tim intrval of dt, th inras in hain lavag is quantifid by dr givn by Eq. (5). Kping in mind that that N and C rprsnt th initial onntrations of th nuli and th str bounds rsptivly, w propos to modify Eq. (4) as dn Ndt N d N C dr, (5) subjt to th initial ondition that N at t. Th nwly addd trm on th right hand sid rprsnts th nuli rlasd by hain lavag ovr tim intrval of dt. In Eq. (5), N is th numbr of nuli that ar mad availabl by polymr hain lavag. N inrass from zro to a maximum numbr and thn drass as rystallisation ontinus. Following Avrami [-3] th xtndd volum fration of th rystals, xt, an b alulatd as xt t V sing t N n A d (6) in whih 23 n is th Avogadro s numbr ( 6.2 ) and V sing t is th volum A of a singl rystal at tim t that is nulatd at tim. Avrami s thory assums linar growth for a singl rystal. For polymrs th rystal grows through hain folding whih is onstraind by th ntropi frustration of th partiipating polymr hains [8]. Consquntly th rystallisd lamlla an only rah a limitd siz, 2

15 whih is rfrrd to as r max. In this papr, th linar growth limitd by a maximum siz is approximatd by th following funtion: r G r t max rmax (7) in whih G is th linar growth rat. W thn hav 3 V sing t r and Eq. (6) boms xt t r 3 max G r t 3 max N n d (8) A in whih is a numrial onstant dpnding on th shap of th rystal. As dmonstratd in our prvious papr [], it is onstrutiv to us th non-dimnsional form of th govrning quations. Th following non-dimnsional variabls ar introdud: C C C C ; m m C C ; R R ; C N xi N ; xi C l ; t t t n ( / k C t 2 ) (9) togthr with th following non-dimnsionalisd paramtrs in th modl: D D k ; 2 n ; n k2l C k2c k N N C ; 3 rmax C nar max ; t G G ; rmax t. (2) Hr C is th mol onntration of th amorphous str bounds at th bginning of biodgradation and l is a haratristi lngth of dvi. Th govrning quations thn bom dr dt k C C C n m ; (2) dc dt m dr dt div x i D grad C x i m ; (22) dc dt dr dt C d dt ; (23) 3

16 d d xt ; (24) dn Ndt N d N dr ; (25) xt t G t rmax N d. (26) 3 Eqs. (2-26) form th govrning quations for simultanous biodgradation and rystallisation. Eqs. (2-23) govrn th biodgradation whil Eqs. (24-26) govrn th rystallisation. Th last trm in Eq. (23) onnts biodgradation with rystallisation in th simpl sns that rystallisation rdus th onntration of th amorphous polymrs. Th last trm in Eq. (25) onnts rystallisation with biodgradation in th simpl sns that an xisting nulus an only bom availabl for rystallisation if a hain lavag ours narby. It is usful to point out th following issus whn omparing th modl prditions with xprimntal data: a) As biodgradation approahs its nd, most of th amorphous polymr hains ar xhaustd and th validity of Eqs. (5) and (7) bom qustionabl. W ar howvr lss intrstd in th last part of th dgradation as a dvi would hav brokn apart by thn. b) For simpliity th molular wight distribution of th amorphous polymr hains has bn simplifid into a bimodal distribution, whih is haratrisd by C and C m. Assuming th monomrs ar too small to dtt using standard xprimntal thniqus, th masurd avrag molular wight M an b rlatd to C suh that / M C / C M in whih M and C ar th initial valus of th avrag molular wight and str bound 4

17 onntration rsptivly. Th modl dos not distinguish btwn numbr avragd and wight avragd molular wights, whih is a shortoming of th simplifiation. On nds to hoos on of th avragd molular wights. Th paramtrs in th modl, k and k 2, ar thn dfind aordingly. ) Th volum dgr of rystallinity,, dos not inlud th fft of wight-loss whih has to b onsidrd whn omparing with xprimntal data. Th obsrvd dgr of rystallinity is givn by, /( W ), in whih W rprsnts th wight-loss in prntag whih an b alulatd from knowing th monomrs diffusing out of th matrial. obs 3. Modl validation A omputr programm is dvlopd to solv th quations (2-26) numrially for infinitivly larg plat of thiknss 2l. At th surfa of th plat, it is assumd that any monomr arriving at th surfa is immdiatly takn away by th aquous mdium. Th numrial dtails ar not disussd hr in ordr to fous on th physis of th modl. Howvr to dmonstrat th robustnss of th numrial produr and to hk th omputr od, th numrial modl is rdud to Avrami s thory by swithing off biodgradation and diffusion. A initial valu, N, was st for th numbr of nuli and a vry larg valu was usd for r max. Analytial solutions to Avrami s thory ar givn by [-3] 3 3 G t N, (27) for larg, and by 5

18 3 4 N G t / 4, (28) for small. Fig. shows th omparison btwn th numrial and analytial solutions. It an b sn from th figur that our modl an b sussfully rdud to th Avrami s thory. Th dgradation and diffusion parts of th numrial modl hav bn validatd in our prvious work against finit lmnt solution obtaind using a ommrial pakag []. Tsuji and his o-workrs arrid out a sris of long trm biodgradation xprimnts and publishd omplt sts of data of avrag molular wights, wight loss, dgr of rystallinity and mhanial proprtis as funtions of tim for a rang of PLA and its opolymrs. Th first as that w xamind hr is thir xprimnt on pur PLLA and two blnds of PLLA and poly(vinyl alohol) (PVA) [8]. Th dgradation data wr olltd ovr a priod of 2 months. PVA was introdud as a hydrophili watr-insolubl polymr to alrat th biodgradation. In th blnd films, PLLA and PVA wr phas-sparatd and both th PLLA and PVA phass wr ontinuous and disprsd. Th two blnds hav wight prntags of PLLA of 8% and 6% rsptivly. In all th films th PLLA phas was initially amorphous whil th PVA phas was smi-rystallin. Th introdution of PVA ompliats th dgradation bhavior, hn provids a tst for th flxibility of th mathmatial framwork proposd in this papr. To tak aount of th PVA phas in th modl, Eqs. (2)-(23) ar modifid to nsur that hydrolysis ration only ours in th amorphous rgion of th PLLA. This is ahivd by adding th volum fration of PVA to in Eqs. (2) and (23). Th rystallin phas of th PVA was xludd 6

19 as wll as th rystallin phas in th PLLA whn alulating th fftiv diffusion offiint. Fig. 2 shows th fitting btwn th modl and th xprimntal data for wight prntag of PLLA of % (Fig. 2(a)), 8% (Fig. 2(b)) and 6% (Fig. 2()) rsptivly. Tabl on shows th paramtrs usd in th modl to fit th xprimntal data whih will b disussd togthr with othr ass at th nd of this stion. Th sond as that w studid is thir xprimnt on pur PLLA films with diffrnt dgrs of initial rystallinity [9], whih wr ahivd by annaling th PLLA films at diffrnt tmpraturs. Th xprimntal data wr olltd ovr a priod of 36 months. Th atual morphology of th smi-rystallin PLLA is ompliatd. Thr ar diffrnt dimnsional and spaing paramtrs of th rystallin lamlla as wll as th siz of th rystallin sphrulits. Thr ar also amorphous phas insid and outsid th rystallin sphrulits. All ths strutural dtails ar ignord in our modl and th dgr of rystallinity is th only paramtr usd to dsrib th rystallin phas. This as, thrfor, provids a tst for th simplifiation in th modl. Fig. 3 shows th fitting btwn th modl and th xprimntal data for PLLA with initial dgr of rystllinity of 4% (Fig. 3(a)), 47% (Fig. 3(b)), and 54% (Fig. 3()) rsptivly. Tabl on shows th paramtrs usd in th modl in ordr to fit th xprimntal data. Th third as that w xamind is th poly(glyolid-o-l-latid) studid by Zong t al. [7]. Fig. 4 shows th fitting btwn th modl and th xprimntal data. Again th paramtrs usd in th modl ar givn in Tabl on. This is a fast dgradation as whih took just two wks to omplt. Similar to th sond as, th rystallin morphology xprind a 7

20 sophistiatd volution as larly xplaind by Zong t al. [5]. It an b obsrvd from Figs. 2-4 that th modl fits with all th xprimntal data vry wll xpt for th last data points whr th modl is invalid. During th biodgradation, th rystallisation, hydrolysis ration and diffusion of th hydrolysis ar highly intronntd. Th rystallisation rdus th rgion whr th hydrolysis ration oprats, th hydrolysis ration nourags furthr rystallisation and th diffusion pross rtards th auto-atalyti hydrolysis ration and lads to wight-loss. Figs 2-4 show that th modl dvlopd in this papr an aptur this sophistiatd intrplay for thr vary diffrnt biodgradabl polymrs. Th matrial paramtrs obtaind for th thr biodgradabl polymrs ar prsntd togthr in Tabl on so that thy an b xamind togthr. Th paramtrs ar prsntd in th non-dimnsional form for an asy omparison btwn th diffrnt matrials. As will b sn in th following stion of this papr, four of th rystallisation paramtrs, inluding N, r max, G and, afft th rystallisation rat as a group. W thrfor fixd N, r max, and for all th matrials and only varid G. N mans that w hav assumd ah hain lavag givs a nw nulation sit; Using C 73 3 mol/ m for PLLA and 23 n A 6.2, r orrsponds 5 max 4 /3 to a maximum siz of th polymr rystals of 2nm; orrsponds to th uppr limit of in th original Avrami s thory. G orrsponds to a growth rat at whih a rystal rahs its maximum siz of r max at t t. Th valus of G in Tabl on orrspond to rathr slow growth but this is baus th larg valus of N and usd in th modl. Th impingmnt fator,, is a fitting paramtr in 8

21 th modl whih has a profound impat on th rystallisation bhavior. It will b furthr disussd in th following stion of this papr. Th intrinsi diffusion offiint, D, was mainly dtrmind by th wight-loss. k and t ar mainly dtrmind by th rdution rat of th avrag molular wight. D, k and t an thrfor b rgardd as bing masurd through th modl and th xprimnts. 4. Th intrplay btwn dgradation and rystallisation It is usful to study what th modl prdits in trms of th fft of rystallisation rat on th apparnt hydrolysis rat of th matrial. Th apparnt hydrolysis rat an b haratrisd using th tim rquird for C to rah a fix valu, say.5. Th rystallisation rat an thn b haratrisd using th orrsponding valu of at C. 5. Howvr thr ar fiv paramtrs in th rystallisation modl, inluding N, r max, G,, and, whih all afft th rystallisation rat. Fig. 5 shows th alulatd dgrs of rystallinity as a funtion of tim using a wid rang of valus of th fiv paramtrs whih all giv. 2 at C.5. It an b larly sn from th figur that if th impingmnt fator is fixd, thn at a fix valu of C almost ompltly dtrmins th rystallisation bhavior. In othr words, N, r max, G, and, at as a group to ontrol th rystallisation rat. Th impingmnt fator, howvr, has an indpndnt fft on th rystallisation bhavior whih annot b aommodatd into th group. Fig. 6 shows th fft of rystallisation rat on th hydrolysis rat. Th modl prdits that fast rystallisation lads to fast apparnt hydrolysis in th amorphous phas. It is oftn said 9

22 in th litratur that fast rystallisation rtards biodgradation. This is obvious baus th rystallin phas rsists furthr hydrolysis ration. Howvr Tsuji and Ikada [9] arfully distinguishd this from th fft of rystallisation on th hydrolysis rat in th amorphous rgion and obsrvd that th hydrolysis rat was highr in th amorphous rgion btwn th rystallin rgions than that of th fr amorphous rgion suh as in a ompltly amorphous spimn. Our modl prdition is thrfor onsistnt with th xprimntal obsrvation by Tsuji and Ikada [9]. It is intrsting to obsrv from Fig. 6 that th fft of rystallisation on th hydrolysis is almost indpndnt of k, th rlativ rat btwn th non-atalyzd and auto-atalyzd rations. An important onpt proposd in our prvious work [] is th biodgradation map whih shows th ontrolling mhanism for biodgradation in th landsap of D and k. Th map for infinitivly larg plat is prsntd in Fig 7 whr th dash lins shows th boundaris btwn th diffrnt zons for amorphous polymrs. W rall that zon B is th fast diffusion zon whr th polymr dgradation is ontrolld by th non-atalysd hydrolysis. Zon C is th slow diffusion zon whr th dgradation is ontrolld by th auto-atalysd hydrolysis. Zon D is th fast non-atalysd hydrolysis zon and zon A is whr hydrolysis and diffusion intrat to ontrol th dgradation rat. Undr th assumption of fast watr pntration into th dvi, biodgradation is spatially uniform xpt in zon A. Th shadd rgion on th map is th nwly alulatd zon A for C.7 to.8 at C =.3 to.4. Ths valus rflt a fast rystallisation rat in th biodgradation. Th rangs in ths 2

23 valus ar usd baus it is diffiult to onstrut th map and ontrol th valus of and C at th sam tim. Th impingmnt fator was st as. 5. Fig. 7 shows that rystallisation maks it mor likly for th biodgradation to b spatially uniform. Most of th biodgradation xprimnts wr prformd using thin sampls to avoid th aumulation of aid nd groups insid th film, i.. to oprat in zon B. Th map in Fig. 7 shows that on an us thikr sampls for smi-rystallizd polymrs than thos for amorphous polymrs, whih should mak th following mhanial tsts asir. 5. Conluding rmarks Th typial dgradation tim for ommonly usd biodgradabl polymrs in orthopadi surgris an b svral yars. Th trial and rror approah in dvi dvlopmnt is vry problmati. Th mathmatial modl dvlopd in th papr an b solvd for any sophistiatd dvi using th modrn finit lmnt mthod []. Many xprin and dgradation data hav bn olltd for various xisting dvis. It is thn possibl to us th finit lmnt analysis to bak-alulatd th paramtrs in th modl and to apply thm to nw dvi dsign using th sam polymrs. This is a powrful approah whih will alrat th dvlopmnt of various biodgradabl dvis. Aknowldgmnts: iaoxiao Han gratfully aknowldgs a partial PhD studntship from an EPSRC rsarh grant (S57996). 2

24 Rfrns [] Sodrgard A, Stolt M. Proprtis of lati aid basd polymrs and thir orrlation with omposition. Progrss in Polymr Sin 22;27: [2] Wood GD. Inion biodgradabl plats: th first ntury. British Journal of Oral and Maxillofaial Surgry 26;44:38-4. [3] Bll RB, Kindsfatr CS. Th Us of Biodgradabl Plats and Srws to Stabiliz Faial Fraturs. Journal of Oral and Maxillofaial Surgry 26;64:3-39. [4] Sipmann J, Sipmann F, Florn AT. Loal ontrolld drug dlivry to th brain: mathmatial modling of th undrlying mass transport mhanisms. Intrnational Journal of Pharmautis 26;34:-9. [5] Sngrs BG, Taylor M, Plas CP, Orffo ROC. Computational modling of ll sprading and tissu rgnration in porous saffolds. Biomatrials 27; 28: [6] Grizzi I, Garrau H, Li S and Vrt M. Hydrolyti dgradation of dvis basd on poly[dl-lati aid] siz dpndn. Biomatrials 995;6:35-3. [7] Zong H, Wang ZG, Hsiao BS, Chu B, Zhou JJ, Jamiolkowski DD, t al. Strutur and morphology hangs in absorbabl poly(glyolid) and poly(glyolid-o-latid) during in vitro dgradation. Maromoluls 999;32:87 4. [8] Tsuji H, Muramatsu H. Blnds of aliphati polystrs: V Non-nzymati and nzymati hydrolysis of blnds from hydrophobi poly(l-latid) and hydrophili poly(vinyl alohol). Polymr Dgradation and Stability 2;7: [9] Tsuji H, Ikada Y. Proprtis and morphology of poly(l-latid) 4: ffts of strutural paramtrs on long-trm hydrolysis of poly(l-latid) in phosphatbuffrd solution. Polymr Dgradation and Stability 2;67:79 89.

25 [] Wang Y, Pan J, Han, Sinka IC, Ding L. A phnomnologial modl for th dgradation of biodgradabl polymrs. Biomatrials 28;29: [] Avrami M. Kintis of phas hang I: gnral thory. Journal of Chmial Physis 939;7:3 2. [2] Avrami M. Kintis of phas hang II: transformation-tim rlations for random distribution of Nuli. Journal of Chmial Physis 94;8: [3] Avrami M. Kintis of phas hang III: granulation, phas hang, and mirostrutur. Journal of Chmial Physis 94;9: [4] Long YU, Shanks RA, Stahurski ZH. Kintis of Polymr Crystallisation. Prog. Polym. Si. 995;2:65-7. [5] Siparsky GL, Voorhs KJ, Miao F. Hydrolysis of polylati aid (PLA) and polyaprolaton (PCL) in aquous atonitril solutions: autoatalysis. Journal of Environmntal Polymr Dgradation 998;6:3-4. [6] Jiang WG, Han, Pan J. Efftiv diffusion offiint of biodgradation polymrs and its rol in dgradation. Univrsity of Listr Rsarh Rport 28. [7] Starink MJ. On th maning of th impingmnt paramtr in kinti quations for nulation and ration rations. Journal of Matrials Sin. 2; 36: [8] Muthukumar M. Molular modlling of nulation in polymrs. Phil. Trans. R. So. Lond. A. 23;36:

26 Tabl on Paramtrs usd in th modl to fit th xprimntal data for th diffrnt ass. Paramtrs Matrials D k N r max G t (wk) PLLA/PVA 5 V PLLA =% / V PLLA =8% / V PLLA =6% / = / PLLA 5 = / = / PLA-o-PGA 2 4 /

27 Dgr of rystallinity = = Normalisd tim Fig.. Dgr of rystallinity as a funtion of tim for larg and small valus of - a omparison btwn th analytial solutions of th Avrami s thory (solid lins) and th numrial solutions (disrt symbols) of th simultanous dgradation and rystallisation modl whn it is rdud to th Avrami s thory. N ; G ; ; 8 r max 4 / 3. 2

28 Normalisd wight loss Dgr of rystallinity Normalisd molular wight Tim (months) (a) % PLLA Normalisd wight loss Dgr of rystallinity Normalisd molular wight Tim (months) (b) 8% PLLA 3

29 Normalisd wight loss Dgr of rystallinity Normalisd molular wight Tim (months) () 6% PLLA Fig. 2. Comparison btwn th modl and th xprimntal data obtaind by Tsuji and Muramatsu [8] for avrag molular wight of PLLA, volum dgr of rystallinity of PLLA and wight loss of th film as funtions of tim for volum prntag of PLLA = % (a), 8% (b) and 6% () rsptivly. Th ontinuous lins rprsnt th modl prdition whil th disrt symbols ar th xprimntal data. Th paramtrs usd in th modl ar givn in Tabl. 4

30 Normalisd molular wight Dgr of rystallinity Normalisd wight loss Tim (months) (a) 4% Normalisd molular wight Dgr of rystallinity Normalisd wight loss Tim (months) (b) 47% 5

31 Normalisd molular wight Dgr of rystallinity Normalisd wight loss Tim (months) () 53% Fig. 3. Comparison btwn th modl and th xprimntal data obtaind by Tsuji and Ikada [9] for PLLA films of diffrnt initial dgr of rystallinity, showing avrag molular wight, volum dgr of rystallinity, and wight loss as funtions of tim. Th ontinuous lins rprsnt th modl prdition whil th disrt symbols ar th xprimntal data by Tsuji and Ikada. Th paramtrs usd in th modl ar givn in Tabl. 6

32 Dgr of Crystallinity Normalisd molular wight Tim (wks) Fig. 4. Comparison btwn th modl and th xprimntal data obtaind by Zong t al. [7] for poly(glyolid-o-latid), showing avrag molular wight and volum dgr of rystallinity as funtions of tim. Th ontinuous lins rprsnt th modl prdition whil th disrt symbols ar th xprimntal data by Zong t al. [7]. Th paramtrs usd in th modl ar givn in Tabl. 7

33 ŋ= Dgr of r yst al l i ni t y ŋ=. ŋ= Nor mal i zd t i m Fig 5. Efft of th fiv paramtrs in th rystallisation modl on th dgr of rystallinity as a funtion of tim for a fixd valu of =.2 at C. 5. For all th urvs D. 2, k and 4 r max 4 /3. For th urv of, th following paramtrs wr usd: N ; G. 29;. For th urv of.2, th following paramtrs wr usd: N ; G 4. 7;. For th urv of., th following rang of paramtrs wr usd: N. to ; G.2 to ;. 3 to. 8

34 Dgr of rystallinity at C. 5 Fig. 6. Efft of rystallisation rat on th apparnt hydrolysis rat showing fast rystallisation lads to fast apparnt hydrolysis. k =,, for th thr urvs rsptivly; D. 2. 9

35 Fig. 7. Biodgradation map for infinitivly larg plat showing th ontrolling mhanism for biodgradation in th landsap of D and k. Th dash lins show th boundaris btwn th diffrnt zons for amorphous polymrs. Th shadd ara is th nwly alulatd zon A for smi-rystallin polymrs.