Estimation of Rates of Drug Diffusion in Polymers

Chapter 4

Estimation of Rates of Drug Diffusion in Polymers C. G. Pitt, A. L. Andrady, Y. T. Bao, and Ν. K. P. Samuel

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Research Triangle Institute, P.O. Box 12194, Research Triangle Park, NC 27709

A method of estimating the solubility of drugs in rub bery polymers, based on the octanol-water partition coefficient of the drug, is described. This method, when combined knowledge of the drug diffusion coef ficient D, permits calculation of diffusion controlled release rates. Studies of the relationship between the solute structure and D are reviewed, to support the conclusion that D can be estimated from the solute molecular size or molecular weight; alternatively, D may be treated as a constant for a given polymer pro vided the molecular weight of the drug falls in the range of 250 - 350 au. Earlier methods of calculating the drug solubility in a polymer using drug melting points and solubility parameters are described. The present method is based on the correlation: log P(polymer) = a log P(octanol) + b, which is shown to apply for poly(dimethylsiloxane), poly(c-caprolactone), poly(ethylene-co-vinyl acetate), and poly(e-caprolactam-co-e-caprolactone), using a series of nine basic and steroidal drugs. When combined with the known or estimated drug water solubility, the correlation provides a simple method of estimating drug-polymer solubility and diffusion rates. Examples of the method are provided. The m a j o r i t y o f c o n t r o l l e d d r u g d e l i v e r y systems now b e i n g marketed or under development a r e based on d i f f u s i o n o f t h e d r u g t h r o u g h a semipermeable membrane t o a c h i e v e the r e q u i s i t e release rate. D i f f u s i o n c o n t r o l i s p a r t i c u l a r l y important to transdermal d e l i v e r y , where biodégradation and d i s s o l u t i o n a r e n o t v i a b l e mechanisms o f c o n t r o l l i n g the r e l e a s e r a t e . Provided the process i s F i c k i a n , the r a t e o f d i f f u s i o n t h r o u g h t h e semipermeable polymer i s d e t e r m i n e d by

0097-6156/87/0348-0049$06.50/0 © 1987 American Chemical Society

In Controlled-Release Technology; Lee, P., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

50

CONTROLLED-RELEASE TECHNOLOGY

pro

F i c k ' s f i r s t law; t h a t i s , the rate of d i f f u s i o n i s d i r e c t l y p o r t i o n a l to the diffusion coefficient (D) and the c o n c e n t r a t i o n g r a d i e n t (dC/dx) o f the drug i n the polymer (Equation 1). For a t r a n s d e r m a l p a t c h o r a subdermal c y l i n d r i c a l c a p s u l e o f u n i t l e n g t h , the r a t e s o f d i f f u s i o n a l drug r e l e a s e a r e g i v e n by e q u a t i o n s 2 and 3, r e s p e c t i v e l y (_1 ). dM/dt = -D.dC/dx

(1)

M/M

(2)

= DC A t / h oo

ρ

M/M

= 2TT DC


oo

t/ln(r ρ

/r.) Ο

(3)

1

(h = membrane t h i c k n e s s ; A = membrane a r e a ; C l i t y i n polymer, r . , r = inner,outer d r u g mass d i f f u s e d a t times t and i n f i n i t y )

= drug

cylindeP r a d i i ,

solubi M, M =

That i s , w i t h i n the geometric constraints of the d e l i v e r y system, the f e a s i b i l i t y o f u s i n g d i f f u s i o n to achieve a p r a c t i c a l rate of d e l i v e r y with a p a r t i c u l a r drug/polymer c o m b i n a t i o n depends on the v a l u e s o f D and C . This article will b r i e f l y r e v i e w methods o f e s t i m a t i n g these p r o p e r t i e s ( 2 ) , and i n t r o d u c e the i d e a o f u s i n g p a r t i t i o n c o e f f i c i e n t s as a s o u r c e o f C v a l u e s . Ρ Methods o f E s t i m a t i n g D i f f u s i o n

Coefficients

Methods o f estimating diffusion coefficients originate w i t h the e a r l i e r s t u d i e s o f gas t r a n s p o r t i n semipermeable membranes. D i f f u s i o n can be t r e a t e d as a t h e r m a l l y a c t i v a t e d p r o c e s s , the temperature dependence of which i s g i v e n by an A r r h e n i u s type o f e q u a t i o n (Equa t i o n 4 ) . The a c t i v a t i o n energy {K^) i s a c o n s t a n t f o r a p o l y m e r / d i f ~ fusant combination, D = D exp( E /RT) Q

(4)

d

and i s the energy r e q u i r e d to separate two polymer c h a i n s s u f f i c i e n t l y t o p e r m i t the d i f f u s a n t t o pass t h r o u g h . U s i n g p o l y e t h y l e n e , M i c h a e l s and B i x l e r (3) showed t h a t the d i f f u s i o n c o n s t a n t s o f e l e v en gases v a r y i n g i n s i z e from oxygen t o s u l f u r h e x a f l u o r i d e e x h i b i t the temperature dependence e x p r e s s e d by equation 4. In t h i s m i l e s t o n e paper, the a u t h o r s demonstrated t h a t t h e r e i s a s e m i - l o g a r i t h mic c o r r e l a t i o n o f D with the reduced m o l e c u l a r d i a m e t e r o f the d i f f u s a n t ( E q u a t i o n 5 ) . Here, d i s the d i a m e t e r o f the gas m o l e c u l e , and 0.5 φ / i s approximately equal t o the mean u n o c c u p i e d d i s t ance between two c h a i n segments. The experimental c o r r e l a t i o n i s shown i n F i g u r e 1. 1

2

2

ln(D/d )

= K(d - 0.5

1

2

φ / )

(5)

K r e v e l e n (4) has summarized much of the p u b l i s h e d d a t a on gas d i f f u s i o n , i n c l u d i n g the graphic relationship between the a c t i v a t i o n energy o f d i f f u s i o n , Ε , the relative s i z e o f the d i f f u s i n g mole c u l e , ( d N / d X ) , and the g l a s s t r a n s i t i o n temperature (Tg) o f 2

2


PITT ET AL.

Estimation of Rates of Drug Diffusion in Polymers

10

I Δ - Doto of vanAfiwrongtn(2) Δ Ht

-JO"


τ»

\

\ Ν

(/)
SF,

ιο

U) 2J0 3J0 *0 1 &0 RE0UCE0 MOLECULAR DIAMETER/ d - * ) (Â) #

7.0

β

Figure 1. Correlation of diffusion coefficients (D*) of small gaseous molecules in amorphous polyethylene (natural rubber) with their reduced molecular diameters (d-0.5 φ" ). (Reproduced with permission from Ref. 3. Copyright 1961 John Wiley & Sons.) 2


52


the polymer (Figure 2). Here, the term dN /dX refers to the d i a meters of nitrogen gas and the molecule X. This permits the estimat i o n of the d i f f u s i o n c o e f f i c i e n t of the molecule X provided the Tg of the polymer i s known. Similar correlations between D and permeate size have been established f o r larger organic solutes, using the molecular volume or molecular weight i n place of the molecular d i a meter (5,6). For example, the values of E. i n polystyrene f o r eight a l k y l and a r y l derivatives were proportional to their molecular volumes (Figure 3). As the size of the diffusant increases i t i s possible to use the molecular weight as an approximation of molecular volume. Thus, Baker and Lonsdale (Jj noted that there i s an approximate log-log relationship between the d i f f u s i o n c o e f f i c i e n t and molecular weights f o r halogenated paraffins i n polystyrene and azonapthalene dyes i n natural rubber (7-9). A log-log r e l a t i o n ship (Figure 4) i s also observed f o r the d i f f u s i o n of low molecular weight siloxanes i n polydimethylsiloxane f l u i d (10) while, for four anticancer drugs, Chien (2) has noted the d i f f u s i o n c o e f f i c i e n t i n a methacrylate hydrogel i s proportional to M W - ° . . It has been suggested that, since many drugs f a l l into a similar size range, i t i s possible to treat D as a constant for a given drug class i n a s p e c i f i c polymer. As an example, most steroids have in common a t e t r a c y c l i c skeleton, and d i f f e r primarily i n their substitution pattern. Some l i t e r a t u r e values of d i f f u s i o n c o e f f i c i ents i n steroids i n semipermeable membranes are l i s t e d i n Table I and provide support f o r the v a l i d i t y of using an average d i f f u s i o n c o e f f i c i e n t . Table I l i s t s the d i f f u s i o n c o e f f i c i e n t s of several narcotics studied i n our laboratory; here also the values of D f a l l in the same r e l a t i v e l y narrow range. The importance of the Tg of the polymer i n determining permeab i l i t y i s evident from the relationship i n Figure 2. In most cases, the Tg i s available from l i t e r a t u r e compilations (11). Methods of estimating Tg from substituent group contributions have been described (12,13). Some q u a l i t a t i v e guidelines for predicting the change i n Tg with polymer structure are: 1. Chains based on Si-0, P-N, C-C, and C-0 links are f l e x i b l e and have low Tg's. Ring structures e.g. p-phenylene groups, i n the chain increase the Tg. Substituents, p a r t i c u l a r l y r i g i d , polar, or branched structures, increase Tg by impeding intramolecular motion or increasing intermolecular interaction by van der Waals forces, dipolar interaction, or hydrogen bonding. Long chain a l k y l substituents can reduce Tg by s e l f - p l a s t i c i z a tion. 2. The structural features which influence the Tg of a polymer are similar to those that determine i t s c r y s t a l l i n i t y . In fact the relationship (Eq. 6) i s often observed.


2

33

3.

Tg = 2/3T (°K) m

unsymmetrical polymers

(6a)

Tg = 1/2T (°K) m

symmetrical polymers

(6b)

If information on Tg i s not available, the density can be used as an estimate of the free volume of the polymer. The lower the density, the greater the permeability.


In Controlled-Release Technology; Lee, P., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987. 21 G U T T A P E R C H A

Do

7

ETHYLENE)

2

Figure 2. Relationship between activation energy of diffusion (E_), the relative size of the diffusing molecule, (dN /dX) and the glass transition temperature ( T ) of the polymer. (Reproduced with permission from Ref. 4. Copyright 1976 Elsevier.)

11 POLYETHYLENE TEREPHTHALATE>f)

10 POLYVINYL ACETATE) it)

25 P0LY\TETRA-FLUOR0 E T H Y L E N E ) 26 POLY (2,6-0



Figure 3. Relationship between the activation energy for diffusion in polystyrene and the molar volume of 10 organic substrates; measured at temperatures greater than Τ . (Reproduced with permission from Ref. 4. Copyright 1976 Elsevier)

Me Si(OSiMe ) . OSiMe 3

2

M

2

3

LnD

Ln D =

4.4

4.6

4.8

11.01

5.0

5.2

- 1.59

Ln MW

5.4 5.6 5.8 Ln(M.WL)

6.0

6.2

64

6.6

Figure 4. Plot of Ln D versus Ln (molecular weight) for a series of low-molecular-weight linear dimethylsiloxanes in polydimethylsiloxane fluid. Numbers at each data point refer to N, defined by chemical formula. Data from Ref. 10.


PITT E T A L .

4.

57


Crosslinking w i l l increase the Tg, reducing the d i f f u s i o n coef ficient. 5. P l a s t i c i z e r w i l l often reduce the Tg; both water and drug may serve this function. Because d i f f u s i o n i s limited to the amorphous phase of semic r y s t a l l i n e polymers, and the c r y s t a l l i n e phase can a d d i t i o n a l l y r e s t r i c t chain motion i n the amorphous phase, the value of D i s dependent on the degree of c r y s t a l l i n i t y of the polymer. To a f i r s t approximation, t h i s e f f e c t may be expressed by equation 7, where χ is the c r y s t a l l i n e volume f r a c t i o n and D i s the d i f f u s i o n c o e f f i cient of the t o t a l l y amorphous polymer. For example, d i f f u s i o n c o e f f i c i e n t s for high density polyethylene are lower than for low density polyethylene (3).


4.

D = D (1-x) a

(7)

F i l l e r s such as s i l i c a in s i l i c o n e rubber have the same e f f e c t as c r y s t a l l i n i t y , reducing polymer motion by physical crosslinking and increasing the tortuosity of the d i f f u s i o n path (14,15). Estimation of the Drug S o l u b i l i t y (C^) The estimation of the s o l u b i l i t y of a drug in a polymer has general ly been approached using Hildebrand's theory of micro-solutes. Qua l i t a t i v e l y , comparison of the s o l u b i l i t y parameters (δ) of the ste r o i d and drug i s a useful means of assessing the l i k e l y m i s c i b i l i t y of a polymer-drug combination. The values of δ may be calculated from the part structures of the polymer and the drug using published tables of group contributions (4,16). The more s i m i l a r the values of δ, the greater the compatability of the drug and polymer. It i s possible to determine C quantitatively using Hilde brand's theory of microsolutes. An example of the accuracy that can be achieved i s provided by the calculation of the s o l u b i l i t i e s of a series of p-aminobenzoate esters in hexane (17,18). Michaels, et a l . (19) used t h i s approach to estimate the s o l u b i l i t y of steroids i n various polymers. The s o l u b i l i t i e s of seven steroids in s i x poly mers were calculated from the steroid melting points, heats of fus ion, and s o l u b i l i t y parameters. Equation 8 was derived, where Jjim i s the maximum steady state flux, h i s the membrane thickness, χ i s the product of V, the molar volume of the l i q u i d drug, and the square of the difference in the s o l u b i l i t y parameters of the drug and polymer, ρ i s the steroid density, Τ i s melting point ( Κ), Τ is the temperature of the environment, R i s the gas constant, and ΔΗ^ and AS are the enthalpy and entropy of fusion, respectively. β

f

ln [J.. .h.exp(l+x)] = AH.(1/T-1/T)/R + l n PD l im ι m Also, since In

ΔΗ- * Τ &S„, ι m i

[J .h.exp(H-xH llm

(8)

= AS (T^/T-D/R + l n oD f

(9)

Both the heat of fusion, AH , and the entropy of fusion, AS^ , vary with the steroid structure, although AS- i s more nearly constant. f


58


plot

Consequently, from equation 9, a of ln[J .h.exp(l+x)] versus (T^/T-l) i s expected to be approximately linear. This was found to be the case (Figure 5). Using equation 9, and assuming average v a l ues of AS and D, i t was possible to calculate the permeability of any steroid/polymer combination i n the series to within a factor of two. This approach to estimating s o l u b i l i t i e s and d i f f u s i o n rates has not been applied to other classes of solutes, even though the s o l u b i l i t y parameters can be e a s i l y estimated by group contribution methods and AHf and Τ can be determined by d i f f e r e n t i a l scanning calorimetry. The p o s s i b i l i t y of simplifying the method further arises i f χ contributes l i t t l e to the relationship and can be treated as a con stant. With t h i s assumption, and because = C D, Equation 9 may be rewritten as equations 10-12. liro


J

n

l

i

n

i

n

P

ln [ J . h.exp(l+x)/p.D] = AS.(T/T-1)/R ι lm t m

(10)

ln C +(1+χ)-1ηρ = -AS.(T /T-l)/R ρ r m

(11)

ln C = aT + b ρ m

(12)

That i s , the logarithm of the drug s o l u b i l i t y i s d i r e c t l y propor t i o n a l to the drug melting point. This relationship was shown to hold approximately for the ste r o i d s o l u b i l i t i e s i n EVA and polyetherurethane l i s t e d i n Table I (20). A semilog plot of the steroid s o l u b i l i t y (C ) versus s t e r o i d melting point i s shown i n Figure 6. The s t a t i s t i c s of a squares c o r r e l a t i o n are:

least

ln(C ) = -0.0198T + 5.225 n = 9 r =Λ).79

EVA Series

ln(C ) =

Polyether-Urethane Series

P

P

n

- 9

-0.0198T r

-

+ 6.148 B

0.82

The low c o r r e l a t i o n c o e f f i c i e n t s r e f l e c t the approximations made i n Equation 12. In a d i f f e r e n t theoretical treatment, Chien (J21) used the Van't Hoff equation to derive the relationship (Eq. 13) between 1/T and C . Ρ m

log C = log[S /(S +X )] = - l o g Y - AHJ1-T/T )/2.303RT Ρ Ρ Ρ Ρ ρ f m

(13)

Here, C

i s the mole f r a c t i o n s o l u b i l i t y of the drug, S i s the mole drug, X i s the mole f r a c t i o n of the polymer, and Y i s the a c t i v i t y c o e f f i c i e n t of the drug i n the polymer. Thii relationship i s equivalent to equation 8, i n assuming that AH rather than AS i s constant. The c o r r e l a t i o n was tested using the

fractioR of the

f

f

s o l u b i l i t y o f s t e r o i d s i n s i l i c o n e r u b b e r ( F i g u r e 7 ) . The r e l a t i o n s h i p s i n L i s t I , f o r f a m i l i e s o f t e s t o s t e r o n e , p r o g e s t e r o n e , and e s t r a d i o l d e r i v a t i v e s , were o b s e r v e d .




PITT ET AL.

10~ I 0 10

ι

ι ι ι ι ι 20 40 60 (T /310 - 1) Χ 10

ι

ho

-5

80

2

M

Figure 5. Correlation of permeabilities (J &) of steroids in various polymers with their melting points (T ) and a solubility parameter term χ. (Reproduced with permission from Ref. 19. Copyright 1975 American Institute of Chemical Engineers.) POLY-ETHER B A S E D U R E T H A N E 100

Lee et a l , 1985

Cp 5

•

10

LnCp = - 0 . 0 2 T

I l

^^-^

+ 6.1 5

r=0.82

140

150

160

170

180

190

200

210

220

230

Melting Point ( ° C )

Figure 6. Semilogarithmic correlation of the solubility (C ) of a series of steroids in polyethylene-co-vinyl acetate, 40% vinyl acetate, and the steroid melting point. Data from Ref. 20.




2.4

-12 I 1.6

1 1.6

1 2.0

I

1

2.2

2.4

1 2.6

J 2B

1000/Tm

Figure 7. Semilogarithmic relationship between the mole fraction solubility (C ) of testosterone (o), progesterone (•), and estradiol (Δ) derivatives iS polydimethylsiloxane and the reciprocal of the melting point ( T ) . (Reproduced from Ref. 21. Copyright 1976 American -1

m

Chemical Society.)


4.

PITT ET AL.

List


I . R e l a t i o n s h i p s Between l o g C

p

and M e l t i n g P o i n t o f S t e r o i d s

Relationship

Steroid Family

Testosterone

61

Derivatives

log C = 2.855/T n=ll r=0.87 *

- 9.631

P

Progesterone

log C = 3.668/T - 11.469 n=9 r*0.79 "

Derivatives


P

Estradiol

log C - 3.644/T - 11.763 D m n=7 Γ=0.94

Derivatives

μ

A l l compounds

log C = 3.085/T - 10.249 n=27 r=0.85 " P

The c o r r e l a t i o n c o e f f i c i e n t s of low.

these

C -T relationships are also " p

The Use of P a r t i t i o n Coefficients If the s o l u b i l i t y (C ) of a drug i n a low molecular weight solvent is known, i t follows from equation 14 that the drug s o l u b i l i t y i n a polymer can be derived from the d i s t r i b u t i o n of the drug between the C /C - Ρ Ρ s

(14)

low molecular weight solvent and the polymer. Determination of the p a r t i t i o n c o e f f i c i e n t (P) eliminates the need to evaluate terms r e l a t i n g to the s o l i d to l i q u i d phase change of the drug, i . e . mp, enthalpy, entropy of fusion. Since the aqueous s o l u b i l i t y of most common drugs i s either known, e a s i l y determined or estimated, water i s the obvious choice f o r the low molecular weight solvent, and the determination of C i s reduced to determination or estimation of the polymer/water p a r t i t i o n c o e f f i c i e n t . An advantage of defining the problem i n t h i s manner i s that the p a r t i t i o n c o e f f i c i e n t has become a central property i n quantitative s t r u c t u r e - a c t i v i t y relationships (QSAR) and a large data base of Ρ values i s available i n the medicinal chemistry l i t e r a t u r e (22-24). In p a r t i c u l a r , i f a c o r r e l a t i o n (Equation 15) between the polymerwater and octanol-water p a r t i t i o n c o e f f i c i e n t s can be established for a series of solutes, i t becomes possible to u t i l i z e l o g Ρ (octanol/water) value as a reference point from which to calculate the polymer-water value. log Ρ (polymer) «= a l o g Ρ (octanol) + b


(15)

62


Correlations of log Ρ f o r a number of different low molecular weight solvent pairs, for example octanol/water vs ether/water and o l e y l alcohol/water, have been established (22-24) and provide a precedent for the application to high molecular weight solvents. The log Ρ values can be deduced for an even greater variety of structures by use of the method of substituent group contributions (25,26). As with the calculation of s o l u b i l i t y parameters using group contributions, the method i s based on use of c h a r a c t e r i s t i c TT values which represent the additive contributions of substituent groups (X) to the log Ρ value of the parent compound (Equation 16).


log P(RX) = log P(RH) + Ζ Tt(X)

(16)

The calculation of log Ρ (octanol) i s f a c i l i t a t e d by the a v a i l a b i l i t y of computer programs that use the group contribution method (27,28). The derivation of log Ρ (octanol) values from r e l a t i v e HPLC retention times (29-31), and from atomic charge densities calculated using semi-empirical molecular o r b i t a l methods has also been des cribed (32). Several laboratories have measured s o l u b i l i t i e s and/or p a r t i tion c o e f f i c i e n t s of solutes i n higher molecular weight media, and their data provides a test of this approach to estimating polymer s o l u b i l i t i e s . Flynn and Yalkowsky (17,18) studied the transport and s o l u b i l i t y properties of a series of p-aminobenzoate esters, pHjNCgH^COOR, R = methyl to hexyl, i n poly(dimethylsiloxane) f l u i d (PDMS). We find that their values of log P(PDMS) correlate well with reported (33) values of log Ρ values of the same series of solutes i n o l e y l alcohol/water, as i l l u s t r a t e d by the plot i n Figure 8 and the correlation s t a t i s t i c s (Equation 17). log P(PDMS) = 1.04 log Ρ (oleyl alcohol)-1.83 η = 8 r = 0.999

(17)

By combining the data on s o l u b i l i t i e s from different laboratories (34,35), and assuming Ρ i s the r a t i o of the s o l u b i l i t i e s , i t i s possible to test the c o r r e l a t i o n of log Ρ f o r octanol/water versus PDMS/water (Figure 9). Although the data are r e s t r i c t e d to s i x ste roids, the correlation i s excellent (Equation 18). log P(PDMS) = 1.79 log Ρ (octanol) - 5.14 η = 6 r = 0.981

(18)

The s o l u b i l i t y data i n Table I may be used to test the log Ρ corre lations i n poly(ethylene-co-vinyl acetate) and polyether-urethanes. The correlations i n Equations 19 and 20 are derived by combining this data with the reported (20) water s o l u b i l i t i e s and octanolwater p a r t i t i o n c o e f f i c i e n t s of the steroids (22-24). log Ρ (EVA) = 0.936 log Ρ (octanol) η = 4 r = 0.97

0.535

log P (EU) = 0.809 l o g Ρ (octanol) + 0.124 η = 4 r = 0.99


(19)

(20)



PITT ET AL.

ΟΊ

-

1

1

0

1

1

1

1

1

2

3

4

Log P(Silicone Oil)

Figure 8. C o r r e l a t i o n of log Ρ of the solvent pairs silicone o i l - w a t e r and oleyl a l c o h o l - w a t e r for the series of aminobenzoate esters, p - N H - C H C O O ( C H ) H . Numbers at each data point refer to n, the ester chain length. Data from Refs. 17 and 18. 2

4

n

Figure 9. C o r r e l a t i o n of water and polydimethylsiloxane solubilities of six steroids and their log P(octanol) values. Data from Refs. 34 and 35.


64


The c o r r e l a t i o n s , which a r e l i m i t e d t o t h o s e s t e r o i d s i n T a b l e I f o r which l o g Ρ ( o c t a n o l ) v a l u e s have been determined, a r e v e r y good c o n s i d e r i n g t h e f a c t t h a t t h e e x p e r i m e n t a l d a t a on t h e polymer s o l u b i l i t i e s were d e r i v e d indirectly from the a n a l y s i s of d i f f u s i o n kinetics. These p r e l i m i n a r y a n a l y s e s have encouraged an e v a l u a t i o n o f a wider range o f drugs and polymers. The f i v e polymers i n T a b l e II a r e b e i n g used t o determine t h e polymer-water p a r t i t i o n c o e f f i c i e n t s directly. Four o f these polymers a r e commonly used f o r d r u g d e l i very. Their solubility parameters range from 15.1 t o 22.9 J / c m - ' , which c o v e r s most o f t h e p o l a r i t y range o f common polymers. P r e l i m i n a r y r e s u l t s i n our l a b o r a t o r y u s i n g f o u r o f t h e s e polymers and a s e r i e s o f drugs which i n c l u d e d both s t e r o i d s and n i t r o g e n bases suggest t h e s e c o r r e l a t i o n s are quite general (Figure 10, T a b l e I I I ) . There was no e v i d e n c e t h a t t h e p a r t i t i o n c o e f f i c i e n t s o f the drugs s t u d i e d were dependent on t h e i r c o n c e n t r a t i o n s i n the two phases. The c o r r e l a t i o n f o r PDMS i n T a b l e I I I i s c o n s i d e r e d more a c c u r a t e than the c o r r e l a t i o n i n E q u a t i o n 18, the l a t t e r h a v i n g been d e r i v e d by combining solubility data from d i f f e r e n t l a b o r a t o r i e s w i t h the assumption t h a t Ρ i s the r a t i o o f t h e r e p o r t e d s o l u bilities . T h i s approach t o e s t i m a t i n g C does n o t r e q u i r e t h e water s o l u b i l i t y be known o r d e t e r m i n e d e x p e r i m e n t a l l y . S e v e r a l l a b o r a t o r i e s have s t u d i e d t h e r e l a t i o n s h i p between t h e water s o l u b i l i t y of a compound and i t s l o g Ρ ( o c t a n o l ) v a l u e . Hansch, e t al. (36) showed t h a t f o r 156 o r g a n i c l i q u i d s , t h e molar water s o l u b i l i t i e s (S ) were c o r r e l a t e d t o Ρ by e q u a t i o n 21a. The c o r r e l a t i o n c o e f f i c i e n t was i n c r e a s e d t o as h i g h as 0.99 by s e g r e g a t i n g compounds by c h e m i c a l


L

2

3

2

-log

S W

-log

r

S r

= 1.339 (±0.07)log Ρ - 0.978 = 0.935 η = 156

(±0.15)

= 1.07 l o g Ρ - 0.67 = 0.954

(21a)

(21b)

c l a s s , eg, a l c o h o l s , a l k a n e s , etc. U s i n g more r e c e n t e x p e r i m e n t a l s o l u b i l i t y and p a r t i t i o n d a t a , Yalkowsky and Morozowich (37) r e p o r t ed t h a t E q u a t i o n 21b i s a more a c c u r a t e c o r r e l a t i o n . This correla t i o n o f l i q u i d s i s not c o m p l i c a t e d by d i f f e r e n c e s i n the c o n t r i b u t i o n s of heats o f f u s i o n of c r y s t a l l i n e s o l i d s . Despite the v a r i a b i l i t y o f AH , Yalkowsky, e t aJL (38) have shown t h a t even w i t h c r y s t a l l i n e s o l i d s i t i s p o s s i b l e t o d e r i v e a good c o r r e l a t i o n be tween mp, C and P. Equation 22 (where S = molar aqueous s o l u b i l i t y ) was to apply t o a s e t o f 36 n o n - e l e c t r o l y t e s and weak electrolytes lacking a long flexible polymethylene chain. This c o r r e l a t i o n i s shown i n F i g u r e 11. f

sKown

log

S r

= -0.01 mp (°C) - l o g Ρ + 1.05 = 0.955 η = 36

(22)

The use o f t h e s e correlations can be i l l u s t r a t e d by comparing the e x p e r i m e n t a l polymer s o l u b i l i t i e s o f p r o g e s t e r o n e and n a l t r e x o n e with the values d e r i v e d u s i n g equation 22 t o c a l c u l a t e t h e water


4.

Table II.

Polymers and their S o l u b i l i t y Parameters for which Log Ρ Correlations are being Studied

Solubility Parameter (J cm- )

Polymer

1 /2


65


PITT ET AL.

poly(dimethylsiloxane) poly(ethylene), low density poly(ethylene-co-vinyl acetate) poly(e-caprolactone) poly(e-caprolactam-co-e-caprclactone)

Table I I I .

3/2

15.1 17.5 20.0 20.9 22.9

Preliminary Correlations of Log Ρ (Octanol) Versus Log Ρ (Polymer) for Poly(dimethylsiloxane), P o l y e t h y lene -co-vinyl acetate), 40* VA, Poly(ε-caprolactone), and Poly(e-caprolactam-co-e-caprolactone)

Log Ρ (PDMS) = 1.41 η = 8

Log Ρ (Octanol) - 2.95 r = 0.98

Log Ρ (EVA) = 1.14 η = 10

Log Ρ (Octanol) r = 0.98

-1.16

Log Ρ (PCL) = 0.91 η = 10

Log Ρ (Octanol) - 0.50 r = 0.97

Log Ρ (PAE) = 0.59 ii = 8

Log Ρ (Octanol) +0.78 r = 0.93

Solutes: codeine, cortisone, corticosterone, naltrexone, amobarbital, meperidine, androst-4-ene-3,17-dione, testo sterone, progesterone, methadone.



Correlation of Log P(Esteramide) and Log P(PDMS)


Versus Log P(Octanol)

Log P(Octanol) Correlation of Log P(EVA) and Log P(PDMS) Versus Log P(Octanol)

Ί

'

1

1

2

'

1

3

Log P(Octanol)

'

1

4

'

1

5

Figure 10. Correlation of log Ρ of the solvent pairs, poly(ecaprolactone-co-€-caprolactam)-water, poly(ethylene-co-vinyl ace tate), polydimethylsiloxane-water, and octanol-water. Unpublished results.




PITT ET AL.

-5

-4 -3 -2 -1 PREDICTED LOG MOLAR SOLUBILITY

ο

Figure 11. Observed and predicted aqueous solubilities of nonelectrolytes (o) and weak electrolytes ( · ) . The solid line is the theoretical line described by Equation 22. The dashed line is the regression line of the experimental data. (Reproduced with permission from Ref. 38. Copyright 1983 American Pharmaceutical Association.)


68


s o l u b i l i t y and F i g u r e 10 t o c a l c u l a t e t h e polymer s o l u b i l i t y . This comparison i s shown i n T a b l e IV, where i t c a n be seen t h a t t h e r e i s good agreement between t h e c a l c u l a t e d and e x p e r i m e n t a l v a l u e s . Table

IV. C a l c u l a t e d and E x p e r i m e n t a l Solubilities (mg/ml) o f Progesterone and N a l t r e x o n e i n Water, Poly(ε-caprol a c t o n e ) and P o l y ( e t h y l e n e - c o - v i n y l a c e t a t e )


Calculated

(Experimental)

Progesterone

Water

29.3 (14.1) χ 1 θ "

PCL

20.8 (14.7)

EVA

39.6

Solubility

Naltrexone

3

0.94 (1.33) 13.5 (16.4) 8.71 (9.2)

I t i s n o t n e c e s s a r y t o know o r d e r i v e t h e water s o l u b i l i t y o f the d r u g i n o r d e r t o make use o f t h e p a r t i t i o n c o e f f i c i e n t d a t a . For example, i t f o l l o w s from equation 23 t h a t , f o r any drug, t h e v e r t i c a l d i s p l a c e m e n t o f t h e two c o r r e l a t i o n l i n e s i n F i g u r e 10 i s a measure o f t h e r a t i o o f t h e d r u g s o l u b i l i t y i n t h e two polymers. log

(C /C ») = log (C /C )(C /C p

p

p

w

w

·) = l o g Ρ ( p o l ) - l o g P ( p o l ' )

(23)

G i v e n i n f o r m a t i o n on t h e c h a r a c t e r i s t i c d i f f u s i o n c o e f f i c i e n t s of the two polymers, i t i s t h e n possible to estimate t h e i r r e l a t i v e p e r m e a b i l i t i e s . The s l o p e o f t h e c o r r e l a t i o n l i n e i s a measure o f the p o l a r i t y o f t h e polymer; t h e lower the slope, the g r e a t e r the s o l u b i l i t y of a h y d r o p h i l i c drug. The a n t i c i p a t e d c o r r e l a t i o n be tween t h e s l o p e and t h e s o l u b i l i t y parameter o f t h e polymer i s ap proximately observed ( c f Tables I I , I I I ) . It i s important t o r e c o g n i z e t h a t these c o r r e l a t i o n s only apply to a s p e c i f i c polymer and, as d i s c u s s e d above, w i l l be s e n s i t i v e t o changes i n t h e polymer crystallinity, t h e i n c l u s i o n o f f i l l e r , and the e x a c t c h e m i c a l c o m p o s i t i o n . The s e n s i t i v i t y o f s o l u b i l i t y i n p o l y d i m e t h y l s i l o x a n e t o the f i l l e r content has been noted (14,15) and t h e c o r r e l a t i o n i n T a b l e I I I f o r PDMS a p p l i e s ony t o t h e u n f i l led f l u i d . The c r y s t a l l i n i t y o f many polymers depends on t h e i r m o l e c u l a r weight, and may change i f t h e polymer i s s u b j e c t t o biodégradation. The s o l u b i l i t y parameter, i . e . t h e p o l a r i t y , o f p o l y u r e thanes, i s s e n s i t i v e t o t h e n a t u r e and r a t i o o f t h e e t h e r ( o r e s t e r ) and u r e t h a n e segments.


4.

PITT ET AL.


69

Conclusions The v a r i o u s approaches to estimating diffusion c o e f f i c i e n t s and s o l u b i l i t i e s o f drugs i n polymers have been r e v i e w e d . The polymers t y p i c a l l y used f o r d r u g delivery have d i f f u s i o n c o e f f i c i e n t s t h a t a r e c h a r a c t e r i s t i c o f t h e polymer and r e l a t i v e l y c o n s t a n t f o r drugs of a s i m i l a r m o l e c u l a r s i z e . Drug s o l u b i l i t i e s i n a polymer c a n be e s t i m a t e d from t h e s o l u b i l i t y parameters and m e l t i n g p o i n t s ( s t e r o i d s ) , from t h e m e l t i n g point alone, o r from t h e c o r r e l a t i o n o f partition coefficients.


Acknowledgments The s u p p o r t o f our work by g r a t e f u l l y acknowledged.

the

National Institute

on Drug Abuse i s

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Baker, R. W.; Lonsdale, H. K. In "Controlled Release of Bio logically Active Agents", Tanquary, A. C.; Lacey, R. E., Eds.; Plenum, New York, 1973, pp. 87-72. For previous discussions of this subject, see Chien, Y. W. "Novel Drug Delivery Systems: Fundamentals, Developmental Con cepts, Biomedical Assessments", Dekker, New York, NY, 1982. Michaels, A. S.; Bixler, H. J. J. Polym. Sci. 1961, 50, 393. Van Krevelen, D.W. "Properties of Polymers: Their Estimation and Correlation with Chemical Structure"; Elsevier, New York, N.Y., 1976. Zhurkov, S. N.; Ryskin, G. Y. J. Techn. Phys.(USSR), 1954, 24, 797. Duda, J. L.; Vrentas, J. S. J. Poly. Sci. 1968, A2(6), 675. Grun, F. Experientia, 1947, 3, 490. Park, G. S. Trans. Far. Soc. 1950, 46, 684; 1951, 47, 1007. Stannett, V.; In "Diffusion in Polymers", Crank, J.; Park., G. S.; Eds.; Academic Press, London, 1968. McCall, D. W.; Anderson, E. W.; Huggins, C. M. J. Chem. Phys. 1961, 34, 804. Brandrup, J . ; Immergut, Ε. Η.; Eds.; "Polymer Handbook", Wiley, New York, N.Y., 1975. Weyland, H. G.; Hoftyzer, P. J.; Van Krevelen, D. W. Polymer. 1970, 11, 79. Lee, W. A. J. Poly. Sci. 1970, A2(8), 555. Most, C. F. J. Appl. Poly. Sci. 1970, 14, 1019. Flynn, G. L.; Roseman, T. J. J. Pharm. Sci. 1971, 60, 1789 . Barton, A. F. M. "Handbook of Solubility Parameters and Other Cohesion Parameters", CRC Press, Boca Raton, Fla., 1983. Yalkowsky, S. H.; Flynn, G. L.; Slunick, T. G. J. Pharm. Sci. 1972, 61, 853. Flynn, G. L.; Yalkowsky, S. H. J. Pharm. Sci. 1972, 61, 838. Michaels, A. S.; Wong, P. L. S.; Prather, R.; Gale, R. M. Am. Inst. Chem. Eng. J. 1975, 21, 1073. Lee, Ε. K. L.; Lonsdale, Η. K.; Baker, R. W.; Driolli, E.; Bresnahan, P. A. J. Membrane Sci. 1985, 24, 125.


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26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

CONTROLLED-RELEASE TECHNOLOGY Chien, Y. W. In "Controlled Release Polymeric Formulations", Paul, D. R.; Harris, F. W., Eds.; ACS Symposium Series, Vol. 33, 1976, pp 53-71. Leo, Α.; Hansch, C .; Elkins, D. Chem. Rev. 1971, 71, 525. Hansch, C.; Leo, A. "Substituent Constants for Correlation Analysis in Chemistry and Biology"; Wiley-Interscience, New York, N.Y., 1979. Rekker, R. F. In "The Hydrophobic Fragmentation Constant"; Nauta, W. Th.; Rekker, R. F., Eds.; Elsevier, New York, 1977. Leo, Α.; Yow, P. Y. C.; Silipo, C.; Hansch, C. J. Med. Chem. 1975, 18, 865. Taft, R. W.; Abraham, M. H.; Famini, G. R.; Doherty, R. M.; Abboud, J-L. M.; Kamlet, M. J. J. Pharm. Sci. 1985, 74, 807 (and references therein). Cory, M.; Johnson, H.; Leibrand, K. "Log P", Prophet System, NIH. Chou, J. T.; Jurs, P. C. J. Chem. Inf. Computer Sci. 1979, 19, 172. Mirrless, M. S.; Moulton, S. J.; Murphy, C. T.; Taylor, P. J. J. Med. Chem. 1976, 19, 615. Unger, S. F.; Cook, J. R.; Hollenberg, J. S. J. Pharm. Sci. 1978, 67, 1364. Caron, J. C.; Schroot, B. J. Pharm. Sci. 1984, 73, 1703 . Klopman, G.; Iroff, L. D. J. Comput. Chem. 1979, 2, 172 . Buchi, J.; Perlia, X.; Strassle, A. Arzneim.-Forsch. 1966, 16, 1657. Chien, Y. W.; Jefferson, D. M.; Cooney, J. G.; Lambert, H. J. J. Pharm. Sci. 1979, 68, 689. Tomoda, H. T.; Yotsuyanagi; Ikeda, K. Chem. Pharm. Bull. 1978, 26, 2832. Hansch, C.; Quinlan, J. E.; Lawrence, G. L. J. Org. Chem. 1968, 33, 347. Yalkowsky, S. H. and Morozowich, W. A Physical Chemical Basis for Prodrugs In "Drug Design", Ariens, E. J . , Ed.; Vol. 9, Academic Press, New York, NY, 1980, pp. 122-185. Yalkowsky, S. H.; Valvani, C. C.; Roseman, T. J. J. Pharm. Sci. 1983, 72, 866 .

RECEIVED December 9, 1986


Estimation of Rates of Drug Diffusion in Polymers

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