3
Controlled
Release from E r o d i b l e Slabs, C y l i n d e r s ,
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a n d Spheres H. B. HOPFENBERG Department of Chemical Engineering, North Carolina State University, Raleigh, N.C. 27607
An extremely attractive c l a s s o f c o n t r o l l e d d e l i v e r y devices is based upon the concept o f an eroding host polymer which liber ates an agent which is, potentially, a c t i v e biologically. The ag ent may be d i s s o l v e d , suspended, or dispersed w i t h i n or c h e m i c a l l y bonded to the host polymer. The device concept is based upon the premise t h a t some zero-order process, e i t h e r c h e m i c a l , physical, o r p h y s i c o - c h e m i c a l , will completely c o n t r o l the r e l e a s e kinetics. It is implicit, i n the o v e r a l l concept, t h a t the r a t e d e t e r mining process will occur a t a boundary between essentially un affected host polymer and p r e v i o u s l y s w o l l e n , degraded, or per meated m a t e r i a l . The device functions ideally in the absence o f boundary l a y e r s external to the device or w i t h i n the p r e v i o u s l y degraded or swollen polymer. Specifically, it is assumed t h a t d i f f u s i o n o f h y d r o l y z i n g or s w e l l i n g water to the r e a c t i o n site and e l i m i n a t i o n o f a c t i v e agent from t h i s zone o f r e a c t i o n is ex tremely r a p i d compared with the r a t e determining process or pro cesses o c c u r r i n g a t t h i s well defined boundary r e g i o n . Chemical r e l a x a t i o n s controlling a d d i t i v e r e l e a s e from an implant device may be d i v i d e d i n t o two explicit c a t e g o r i e s . Neogi and A l l a n (1) have reviewed the concept o f a d d i t i v e s which are l i n k e d c h e m i c a l l y to a host polymer whereby, consequent to p l a c e ment, water penetrates the device h y d r o l y z i n g the chemical bond linking the would-be a d d i t i v e to the host polymer. I t is assumed t h a t m i g r a t i o n o f the l i b e r a t e d a d d i t i v e proceeds r a p i d l y com pared to the r a t e - d e t e r m i n i n g chemical " r e l a x a t i o n " or " r e a c t i o n " ( 2 , 3 ) . A l t e r n a t i v e l y , c h e m i c a l l y r e l a x i n g devices f u n c t i o n as a consequence o f chemical degradation o f the host polymer, liberat ing biologically a c t i v e compounds from the matrix o f hydrolytically unstable host polymer ( 4 - 7 ) . This concept r e q u i r e s , o f c o u r s e , t h a t the degradation products o f the host polymer be toxicologically acceptable to the organism or environment being t r e a t e d by the d e l i v e r y system. The most simple p h y s i c a l r e l a x a t i o n might be dissolution o f the host polymer thus liberating d i s s o l v e d or suspended biologi cally a c t i v e m a t e r i a l s . I f the dissolution proceeds at a constant
26 In Controlled Release Polymeric Formulations; Paul, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1976.
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3.
HOPFENBERG
Erodible
Shbs, Cylinders,
and
Spheres
27
r a t e , a zero-order process would, t h e r e f o r e , ensue and c o n t r o l l e d and constant d e l i v e r y might be a n t i c i p a t e d . In p r a c t i c e , e r o d i b l e devices may a c t u a l l y be c o n t r o l l e d by more than one l i m i t i n g mechanism. For i n s t a n c e , c h e m i c a l l y e r o d ing devices may be k i n e t i c a l l y c o n t r o l l e d by chemical r e a c t i o n as well as d i f f u s i o n o f e i t h e r water to or by-product from the r e action s i t e . Quite c o n c e i v a b l y , the o v e r a l l r e a c t i o n constant d e s c r i b i n g the r a t e - d e t e r m i n i n g process may be an "apparent" con s t a n t which d e s c r i b e s phenomenologically, a somewhat complex o v e r a l l mechanism. In any event, there are a wide v a r i e t y o f i n dependent mechanisms t h a t c o u l d r e s u l t i n c o n t r o l l e d r e l e a s e o f a d d i t i v e s from formed, e r o d i b l e polymers. Although very s t r i n g e n t physico-chemical requirements are imposed to o b t a i n t r u e " z e r o orderness;" a f u r t h e r , even more s t r i n g e n t geometrical requirement i s i m p l i c i t i n t h i s p a r t i c u l a r device d e s i g n . For an e r o s i o n process which i s , i n f a c t , t r u l y z e r o - o r d e r , the eroding device w i l l o n l y d e l i v e r b i o l o g i c a l l y a c t i v e compo nents a t a c o n t r o l l e d and constant r a t e i f the device i s a p e r f e c t s l a b . Cooney has analyzed the somewhat r e l a t e d case o f a s u r f a c e c o n t r o l l e d d i s s o l u t i o n o f pharmaceutical t a b l e t s and has suggested v a r i o u s t a b l e t geometries which would tend to s t a b i l i z e the d i s s o l u t i o n k i n e t i c s (8). The u n i f i e d a n a l y s i s presented here deals w i t h i d e a l i z e d d e l i v e r y k i n e t i c s which would be expected from eroding polymeric s l a b s , c y l i n d e r s , and spheres which c o n t a i n d i s s o l v e d , d i s p e r s e d , o r c h e m i c a l l y bonded a d d i t i v e . C l e a r l y , a capsule-shaped device would be best described as a composite c y l i n d e r and sphere and, t h e r e f o r e , the k i n e t i c s described here should form the g u i d e l i n e s f o r r a t i o n a l device design and a n a l y s i s . Analysis A n a l y t i c a l r e a l t i o n s h i p s d e s c r i b i n g the k i n e t i c s o f a d d i t i v e r e l e a s e from e r o d i b l e polymer formulations may be d e r i v e d simply i f one assumes t h a t there i s a s i n g l e zero-order process, charac t e r i z e d by a s i n g l e r a t e c o n s t a n t , c o n t r o l l i n g the o v e r a l l r e l e a s e process. In g e n e r a l , t h i s k i n e t i c process might be c h a r a c t e r i z e d by a r a t e c o n s t a n t , k , although i t i s not necessary to s p e c i f y the exact mechanism c o n t r o l l i n g the e r o s i o n process. Clearly this e r o s i o n process could i n v o l v e d i s s o l u t i o n , s w e l l i n g , chemical r e a c t i o n o f host polymer, or chemical r e a c t i o n o f l i g a n d s , b i n d i n g a d d i t i v e to a host polymer. Moreover, t h i s phenomenological con s t a n t , k , c o u l d d e s c r i b e a process i n v o l v i n g combinations o f the i n d i v i d u a l r e l a x a t i o n s o r , i n s p e c i a l c a s e s , c o u l d d e s c r i b e pro cesses i n v o l v i n g d i f f u s i o n o f reactant-water to or b i o l o g i c a l l y agent from the r e a c t i o n zone. The models which are presented here are useful f o r d e s c r i b i n g the k i n e t i c s o f any e r o s i o n process r e gardless o f the fundamental nature o f the mechanism; however, the models assume t h a t the i d e a l i z e d r e l e a s e k i n e t i c s are not con founded by time dependent d i f f u s i o n a l r e s i s t a n c e s i n t e r n a l to or external from the eroding d e v i c e . 0
0
In Controlled Release Polymeric Formulations; Paul, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1976.
CONTROLLED RELEASE POLYMERIC
28
FORMULATIONS
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Release K i n e t i c s from E r o d i b l e Spheres. Consider the c r o s s s e c t i o n o f a sphere undergoing e r o s i o n as presented i n Figure 1. In the region R
