© 1991 Elsevier Science Publishers B.V. 0378-5173/91/$03.50 ADONIS 0378517391001990
IJP 02407
Measurement of drug diffusivity in stratum corneum membranes and a polyacrylate matrix *
A c h i m G r p f e r i c h a n d G e o f f r e y L e e
Institute for Pharmaceutical Technology and Biopharmaceutics, Heidelberg University, Heidelberg (Germany) (Received 5 July 1990)
(Modified version received 4 September 1990) (Accepted 25 September 1990)
Key words: S t r a t u m c o r n e u m ; P o l y m e r m a t r i x ; D i f f u s i o n ; M e a s u r e m e n t
Summa~
Diffusivities have been determined for the drug clenbuterol for three non-sink, diffusional problems: diffusion out of a polymer matrix, diffusion through excised, human stratum corneum, and the combined case of diffusion out of a polymer matrix through contiguous, excised stratum corneum. Numerical solution of the diffusion equation for these problems yields theoretical values of drug mass in the acceptor phase, ma(t)th, VS time. Diffusion was then measured experimentally and the best value for diffusivity obtained by comparing the values of drug mass in the acceptor phase, ma(t)exp, with the corresponding m a ( t ) t h data. In this fashion it was possible to obtain diffusivities for diffusion out of the polymer matrix for various values of drug loading and matrix thickness.
For the problem of diffusion through stratum corneum a value for the partition coefficient of clenbuterol as well as for its diffusivity could be determined. The two diffusivities obtained for the combined model (i.e., D m within the matrix and D s within the stratum corneum) agree quite well with those obtained for the two separate problems, but are more scattered. Partition coefficients could not be accurately determined for diffusion out of the matrix or the combined problem.
Introduction
T h i s s t u d y o r i g i n a t e d f r o m m e a s u r e m e n t s o f d r u g d i f f u s i o n o u t o f a p o l y m e r m a t r i x t h r o u g h a c o n t i g u o u s m e m b r a n e o f excised s t r a t u m c o r n e u m i n t o a well-stirred a c c e p t o r m e d i u m . W e wished to d e t e r m i n e the effects of the p r o p e r t i e s o f the m a - trix o n the d i f f u s i o n a l b e h a v i o u r of the d r u g w i t h i n
* Dedicated to Professor Helmut Stamm on the occasion of his 65th birthday.
Correspondence." Geoffrey Lee, Pharmazeutische Technologie und Biopharmazie, Im Neuenheimer Feld 366, D-6900 Heidel- berg, Germany.
e a c h l a y e r o f this b i l a y e r system. T o a v o i d intrin- sic e r r o r we c o n s i d e r e d realistic e x p e r i m e n t a l a n d b o u n d a r y c o n d i t i o n s , t h e r e b y recognising, for ex- ample, the r e m o v a l o f s a m p l e s f r o m the s y s t e m for analysis a n d the e x i s t e n c e o f a n o n - s i n k . This d i f f u s i o n a l p r o b l e m is r e l a t e d to a n o n - s t e a d y state m o d e l a n a l y s e d b y G u y a n d H a d g r a f t (1980) for p e r c u t a n e o u s a b s o r p t i o n w i t h release i n t o a sink in vivo. T h e analysis o f such m u l t i - l a y e r p r o b l e m s is, h o w e v e r , a s s o c i a t e d w i t h c o m p l i c a - tions arising f r o m the i n v e r s i o n p r o b l e m for the L a p l a c e t r a n s f o r m s u s e d as a s o l u t i o n m e t h o d . T h e s e c o m p l i c a t i o n s u l t i m a t e l y led G u y a n d H a d g r a f t to a d o p t a c o m p a r t m e n t a l m o d e l to rep- resent p e r c u t a n e o u s a b s o r p t i o n in vivo, w i t h diffu-
sivities being replaced by rate constants (Guy et al., 1982). Similar difficulties arise with the ana- lytical solution to the non-sink, non-steady state diffusional problem being considered here: it is not readily expressed in a form convenient for computation (see Appendix). The compartmental approach would not be a suitable alternative, as such a model would not directly relate the specific properties of a matrix-type delivery device (drug loading, matrix thickness and drug diffusivity within the matrix) to the rate of appearance of drug on the inner side of the stratum corneum.
The problem is better tackled numerically, giving an easily-manipulated, algebraic solution to Fick's Second Law. We wished to determine if this numerical solution could be used to calculate dif- fusivity in the matrix and stratum corneum from in vitro experimental data obtained for various matrix properties. As this is not necessarily a straightforward task for a multilayer system we first separated the original bilayer model into its two components problems, namely, non-sink dif- fusion out of a rectangular matrix (analytical solu- tion: Crank, 1975) and non-sink diffusion through a plane membrane (analytical solution: Spacek and Kubin, 1967). Experimental data were de- termined for each of these problems, to which the theoretical diffusional behaviour predicted by the corresponding numerical solution were fitted. Val- ues for diffusivity and partition coefficient could thereby be obtained.
Materials and Methods
Measurement of diffusion
For each of the three problems the diffusion of the water-soluble drug clenbuterol (Mol. Wt. 277 and pK a 9.5) was measured at 35°C using an isotonic, pH 8 phosphate buffer solution as recep- t o r / d o n o r phase:
C1
C1
Polymethylacrylate-ester matrices (freeze-dried Eudragit NE30D, R/Shm Pharma, Weiterstadt, Germany) of various thickness and containing various concentrations of drug were prepared on a
la
O
O
2
\5
2
O
lb
6 I
O O
lc
O
2
O
Fig. 1. The glass diffusion cell. (a) Construction for measure- ment of diffusion out of polymer matrix: (1) acceptor chamber (V a = 4 ml); (2) water jacket (35°C); (3) sampling port; (4) polymer matrix (A = 1 cm2); (5) magnetic stirrer (600 rpm). (b) Construction for measurement of diffusion through excised stratum corneum: (1) donor chamber (V d = 4 ml); (2) acceptor chamber (V a = 4 ml); (3) sampling port; (4) stratum corneum (A =1 cm2); (5) magnetic stirrers (600 rpm); (6) water jacket (35 o C). (c) Construction for measurement of diffusion out of polymer matrix through excised stratum corneum: (1)-(3) and (5) as for panel a; (4) polymer matrix with attached stratum
corneum.
TABLE 1
Initial and boundary conditions for Eqn
1
(1) Diffusion out of a matrix
matrix acceptor
I c(x,t) Ca(t)
D
x = 0
x=L
K
x = 0[B* ]:
~c(O,t)
~x = 0 , t > 0 ; (2) Diffusion through stratum corneumdonor stratum
corneum
Cd(t) 0 c(x,t)D [
x= -a x~ x=L
K K
x = L + a I
x=O[F*]:a'dCd(t)K.dt + D~OC(O t) =0,
(3) Combined problem
matrix stratum
corneum
Cm(X't)Dm 0 Cs(X't)Ds
x= - L x= x=a
K] K 2
x = r [ F * ] : - -
t > O ;
aCm(-L,t )
x = - L [ B * ] : ~x - 0 , t > 0 ;
x=a[F.]:b.dc~(t) OCs(a t)
K2.d t I - D s ~ = O ,
t > O .c ( x , o ) = Co; c o ( O ) = o.
a - d c a ( t )
Oc(Lm,t )
K ' d t ~-D 0x = 0 , t"
acceptor Ca(t)
x = L + b
cd(O) = Co; c ( x , O ) = q ( o ) = o.
x=L[F.];b'dca(t) F DOC(Ls 't)
K'dt
~ = 0 , t > 0 .acceptor Ca(t)
x=a+b I
Cm(X,0 ) = CO, Cs(X,0 ) = Ca(O ) = O.
Ocm(O,t)L
aCs(0,t)x = O [ G * ] : D m
axK]
- D s ~ - 0 , t > O ;* Classification according to Carslaw and Jaeger (1959b).
siliconised paper base by solvent evaporation (Zierenberg, 1985). Stratum c o m e u m membranes were prepared from whole human skin excised from the inner thighs of cadavers within 48 h post-mortem (Kligmann and Christophers, 1963).
The thickness of each matrix or piece of stratum corneum was measured at 20 positions using an Elcometer (Elcometer Instruments, Manchester,
U . K . )
The diffusion of the drug out of the polymer matrix was measured by affixing the latter
i n aglass diffusion cell as illustrated in Fig. la. Sam- ples of the acceptor solution were removed at
regular intervals, divided into three aliquots and
each individually assayed by HPLC. Drug diffu-
sion through excised, human stratum corneum
membranes was determined with the cell construc-
tion shown in Fig. l b and using a method de-
scribed by Swarbrick et al. (1984). Drug diffusion
out of the matrix through excised, human stratum
corneum was measured with the cell arrangement
shown in Fig. lc. Care was taken to ensure as-
near-as-possible complete contact between the
outer matrix surface and the inner side of the
contiguous stratum corneum. Each experiment was
performed at least four times and the results rep-
resented as plots of the experimentally determined mass of drug in the acceptor solution, ma(t)exp, VS time, t. The relative error in the H P L C - d e t e r m i n a - tion of ma(t)exp was found to be < 1%, even for the smallest amounts determined in the receiving chamber at short times.
Calculation of diffusivity
Diffusivities for the drug were calculated for each of the three problems by fitting the theoreti- cal values of m a ( t ) t h to the corresponding experi- mentally determined values of ma(t)exp. The former were obtained using a finite-difference method (Crank and Nicolson, 1947; Smith, 1987) to approximate numerically the diffusion equation for the linear m o v e m e n t of a drug with constant diffusivity, D, through a finite plane sheet:
D . c ( x , t ) x x - c ( x , t ) , = O . (1) c is the drug concentration, x the space coordi- nate, and t the time, with subscripts denoting partial derivatives. We thus assume isotropic phases and spontaneous partitioning at bounda- ries. For each of the three problems a three- diagonal matrix comprising Eqn 1 in finite dif- ference form together with the corresponding ini- tial and non-sink b o u n d a r y conditions (sum- marised in Table 1) was solved by a p r o g r a m m e in Pascal on an Epson PC AX3 personal computer with an 80 386/80 387 processor/co-processor.
Solution by Gauss's elimination method yielded the theoretical concentration profile of the drug within the sheet, c(x,t)th, from which the theoret- ical mass in the acceptor solution, ma(t)th, was obtained as a function of time. All experimental conditions capable of influencing the diffusion rate were taken into account in the calculation, for example, the removal of samples from the accep- tor chamber of the diffusion cell. The sum of the accumulated formula and rounding-off errors was assessed from ma(48 h)t h and found to be < 1.5%.
The m a ( t ) t h values were then fitted to the ma(t)exp values using the improved simplex method of Nelder and Mead (1967) to yield the best values for the diffusivity, D, and the partition coeffi- cient, K. A smooth curve was then drawn through
the coordinates of m a ( t ) t h using polynomial splines of third degree.
Results and Discussion
Diffusion out of polymer matrix
Fig. 2a and b shows results for the effect of drug loading ( m ~ ) and matrix thickness ( L ) on the coordinates of ma(t)exp plotted vs t for diffu- sion out of the polymer matrix. The curves passing through the coordinates are the best fits of m a ( t ) t n to ma(t)exp. The increased release rates seen with either greater drug loading or greater matrix thick- ness are in agreement with curves generated from the analytical solution to this non-sink p r o b l e m (Crank, 1975). Since the matrix used to determine m a ( t ) t h was non-singular it was possible to de- termine D unequivocally from the fitting of m a ( t ) t h tO ma(t)ex. Table 2 shows these diffusivi- ties as m e a n values with standard deviations. For a constant matrix thickness of 45.2 /~m mean diffusivity increases with greater drug loading, as already observed by Zierenberg (1985) for the same polymer. As to be expected, there is no clear influence of matrix thickness on diffusivity, and the scatter observed must be due to variation in the polymer system itself or error in the method.
The accuracy of the method depends on the HPLC-analysis error ( < 1%), the accumulated for- mula and rounding-of errors ( < 1.5%), and the errors in the m e a s u r e m e n t of L (approx. 2%) and m ~ ( < 2%, when determined from the exact com- position of the p o l y m e r / d r u g / s o l v e n t solution used to prepare a matrix of known weight). The effects of these errors alone are not of sufficient magnitude to be responsible for the observed scatter in diffusivity. The use of Eqn 1, however, assumes constant diffusivity within an isotropic medium, b o t h of which m a y be questionable con- ditions for this system.
It was not possible to determine an unequivocal
partition coefficient, the values obtained showing
either great variation or being unrealistic (Table
2). This results f r o m the strongly non-linear in-
fluence of K on the diffusion rate, d m a ( t ) / d t , as
illustrated theoretically by the dimensionless plots
of ma(t)th/m ~ vs D t / L 2 for this problem in Fig.
00¢
700
B00
500 m a ( t l / A
4oo
(lag cm -21 3oo
zo0
100
0 2a
m . : 137, w/~
- - rn.: B O'/.~,v
m. : I. 8 't. w/w
0 t z 28 30 40 50 BO 70 80
Time (h]
m, ltl/A
400 300[gg c m -2 ] 2oo
2b
o
o L= I 0 ~
16¢ ,~m
{" I I I I I t I I r
~l 10 20 30 4B SO BO 7 9 8 0 g o
Time [ h l
1 , 0
2C cool
o, g ~0
o, B
Ya
0,7 2S
0, 0
m,(Pth . _ - ~2s
m . o, 5
g, 4 0 , 3 0, 2
0 , 1
g, g 0, 2 0, 4 o, 6 0, s 1, 0
D.t/L ~
Fig. 2. Diffusion of clenbuterol out of polymer matrix. (a) Effect of drug loading (moo) on least-squares fits of theoretical
m a ( l ) th v a l u e s (continuous lines) to experimentally d e t e r m i n e d
coordinates of ma(t)exp; (b) effect of matrix thickness ( L ) on least-squares fits of theoretical m a(t) th values (continuous lines) to experimentally determined coordinates of ma(t)exp; (C) di- mensionless plots of m a ( t ) t h / m ~ vs D t / L 2 as a function of
the dimensionless variable V a / K L A .
TABLE 2
Diffusion of clenbuterol out of a polymer matrix
Matrix Drug Diffusivity Partition
thickness loading ( x 1011 ) (cm 2 s - 1) coefficient (/*m) (% w / w )
45.2 6 2.14 (n = 2) 84.6
45.2 8 2.57+0.405 (n = 8) 17.0+ 48.1 45.2 11 3.89+2.29 (n = 4) 83.8+ 76.2 45.2 13 6.75 + 5 . 0 9 (n = 6) 100 + 1 6 9 23.1 8 5.92+0.41 (n = 4 ) 9 . 9 8 × 1 0 - 8
+ 9 . 5 x 1 0 -8 160 8 4 . 0 9 + 0 . 7 2 4 ( n = 4 ) 0.924+1.85
2c. It can be shown from this figure that if K has a value below approx. 10 it exerts practically no influence on d m , ( t ) / d t . The receding concentra- tion profile of drug within the polymer matrix with time rapidly leads to low concentrations at the outer edge of the matrix (shown for sink-con- ditions in: Carlsaw and Jaeger, 1959a). Partition coefficients close to unity thus have little effect on the magnitude of d m , ( t ) / d t . This is exactly the case for clenbuterol, whose measured partition coefficient between hydrated Eudragit N E 3 0 D and water is 0.4. The three-dimensional space used for the Nelder-Mead calculations will, therefore, be flat in the K-direction. Large changes in the fitted values of K will have little effect on the residual sum of squares, making K fluctuate greatly.
Diffusion through excised, human stratum corneum Plots of ma(t ) vs t for diffusion through a plane membrane under non-sink conditions are sigmoidal in shape, as can be observed even with a single non-sink condition at x = L (Jenkins et al., 1970). The typical curves of best fit shown in Fig.
3a for diffusion through stratum corneum with
non-sink conditions at both boundaries are, how-
ever, not obviously sigmoidal. This occurs because
the points of inflexion of the three curves (as
determined by setting the second derivative of the
splined polynomial for each curve to zero) occur
at 71, 70 and 57 h, respective to increasing drug
concentration. U p to these times, only some 5% of
the total amount of drug had diffused through the
membrane. Table 3 shows the diffusivities calcu-
l o 0
off
m, ltllA
BO
[IJg crrr~
40
20
t . o ] b o, 0
0, B
B, 7
0 , 6 l~.c~u o s
0 , 4
÷
c, : 526 Rg cm~
t,=23~mcm 2
10 2 o a o 40 5 0
Time [h]
fl, 3
r , 2
0 , 1 - - 0 I~:@~ 0005
_ _ I
0,0 ~,2 0,4 0, o o,B 1,1~
x/L
Fig. 3. Diffusion of clenbuterol through excised s t r a t u m corneum. (a) Effect of drug loading (Co) on least-squares fits of theoretical m a ( t ) t h values (continuous lines) to experimentally determined coordinates of ma(t)exp; (b) concentration profile within stratum c o r n e u m , C(X,t)th//Kcd(O) VS t, as a function of
D t / L 2.
lated from six individual experiments on stratum corneum excised from the upper thigh of a single cadaver. The mean value of 3.97 x 10 -12 _ 2.33 X
this structure and size (Swarbrick et al., 1984). The standard deviation compares very favourably with literature data. To take a recent example, a diffu- sivity of 2.1 × 10 - s + 2.9 x 10 -5 cm 2 h -1 (n = 6) was found for benzoic acid by fitting experimental data to an analytical solution for this problem (Parry et al., 1990). Error in the determination of m o~ is smaller and of much less importance here than with the polymer matrix, since the drug con- centration in the donor solution can be directly measured with an error of < 1%. The scatter seen in diffusivity within stratum corneum is due to the highly non-isotropic nature of this tissue. Much less variation is found when diffusivities are meas- ured with the same method in an isotropic silicone membrane (GSpferich and Lee, 1990), with coeffi- cients of variation of < 5% being obtained. By way of comparison Table 3 also includes for each curve the lag time (~-) as measured by extrapola- tion of the tangent to the point of in flexion of the numerical curve of best fit. Diffusivity was calcu- lated from , using the appropriate lag-time rela- tionship for sink conditons: ~- =
L2/6D
(Table 3).The mean valueof 3.34 × 1 0 - 1 2 -I- 1.79 × 1 0 - 1 2 c m 2 s -1 so obtained does not, however, substantially differ from that obtained from the curve fitting.
The partition coefficients shown in Table 3 for this problem are reasonable in value and shown less scatter than those obtained for the polymer matrix. The mean value of all the results is 212 + 165. The better fitting of these values is due to the stronger influence of K on the diffusion rate for this problem. The lipophilic nature of the stratum corneum gives rise to high drug concentrations in T A B L E 3
Diffusion of clenbuterol through excised human stratum corneum (individual experiments)
M e m b r a n e D o n o r D i f f u s i v i t y Partition Lag Diffusivity a
thickness concentration ( x 1012) coefficient time ( x 1012)
( L ) (/Lm) (/xg cm - 3 ) (cm 2 s - 1 ) ( K ) (~') (h) ( c m 2 S - 1 )
9.00 234 1.55 306 22.9 1.64
10.5 526 2.17 291 20.6 2.48
11.5 656 5.83 109 10.1 6.06
12.0 671 6.08 70.8 13.0 5.13
10.7 702 6.34 36.5 21.7 2.44
9.50 708 1.87 458 18.1 2.31
a D ' = L2/6T.
its outer edges and, large partition coefficients at x = 0 and x = L. Fig. 3b illustrates theoretically for release into a non-sink h o w the drug con- centration profile within the stratum corneum changes with time. These profiles are quite differ- ent to those found for sink conditions (Barrer, 1951). The concentration directly adjacent to the donor solution, c(0,t) is initially zero, but jumps at t > 0 to a m a x i m u m value due to partitioning into this layer. Curved concentration profiles are seen which becomes more linear with time as c(O,t) progressively decreases and c(L,t) in- creases. High drug concentrations within the stra- tum corneum thus render d m a ( t ) / d t sensitive to the magnitude of K. In contrast to the polymer matrix model, the multi-dimensional space used for the curve fitting procedure will, in this case, be highly contoured in the K-direction. The residual sum of squares is n o w sensitive to K, enabling a good fit of this parameter. Indeed, when isotropic, synthetic membranes are examined with this method much less variation is found than seen here with stratum corneum. For the diffusion of clenbuterol through the above-mentioned silicone membrane, for example, a coefficient of variation of 12% was obtained (GSpferich and Lee, 1990).
These two contrasting examples of diffusion out of a polymer matrix and diffusion through stra- tum corneum thus illustrate well the contrasting restrictions imposed on the determination of K from diffusional data.
Diffusion out of polymer matrix through excised, human stratum corneum
The effect of drug loading o n the diffusion rate for the combined problem of diffusion out of a matrix through stratum corneum is s h o w n in Fig.
4a. The curves are all sigmoid having an initial lag time due to diffusion through the stratum corneum.
Furthermore, there is a simultaneous, continual decrease in diffusion rate of the drug out of the contiguous matrix with time, as illustrated by the receding theoretical concentration profile o f the drug within the matrix for this bilayer system (Fig.
4b). This occurs at practically by the same rate as that seen with the matrix alone (cf. Carslaw and Jaeger, 1959a), indicating that the presence of the stratum corneum does not substantially hinder
480
350
500
251~ - - m,(t)/A
2 8 0 - - I N cm-q
158
18o
5~
2 . 0
1 , 8 - -
1 , 6 - -
1 . 4
CmlxO) ~,~
0 , 8 ~
e, 4 - -
0,2
m=: lO'/=w~
m.= 8%w~
18 20 30 40 5 0 6 8 70 88 glo
Time [ h ]
4b
x ( L + a )
0, 5 @,8 1,0
0 , 8
4c
0, /
KI =100 8 , 6
~ 5 m,lflt~
m,.
0,4
I( 1:10 0,3
0,2
0,1
iq:l 8,1~8 0, 2 5 8, 5 0 8, 7 5 l , 0 0 1, 2 5 1, 5 8 l , 7 5
U
Om't/L 2
Fig. 4. Diffusion of clenbuterol out of a polymer matrix through excised stratum corneum. (a) Effect of drug loading ( m ~ ) on least-squares fits of theoretical m a ( t ) t h values (con- tinuous lines) to experimentally determined coordinates of ma(t)exp; (b) concentration profile within matrix and stratum corneum, C(X,t)th/Cm(X,O ) VS x / ( L + a), as a function of Dmt/L 2 for K a = 300; (c) dimensionless plots of ma(t)th/moo
vs Dmt/L 2 as a function of K 1.
TABLE 4
Diffusion of clenbuterol out of polymer matrix through excised human stratum corneum
Matrix Drug loading Diffusivity Partition coefficient
thickness (/t m) (% w/w) (cm 2 s - 1 ) ( x 1011 )
50 8 D m = 5.39 + 4.08 (n = 6)
D s = 0.178 + 0.034
50 10 D m = 5.62 + 5.01 (n = 7)
Ds = 0.246 + 0.145
50 13 D m = 3.91 + 3.07 (n = 3)
D s = 0.536 + 0.115
K 1 = 7.89 x 1 0 4 _ 9.18 x 104 K 2 = 6.64 x 103 4- 3.94 x 1 0 3
K 1 = 8.42 X 105 + 0.21 x 105 K 2 = 1.10 X 1 0 3 + 2.92 x 1 0 3
K 1 = 6.95 x 104+ 8.79 x 104 K 2 = 1.19 x 103 -J- 1.68 x 103
d i f f u s i o n o u t of the m a t r i x , even t h o u g h D s is a l m o s t a n o r d e r of m a g n i t u d e l o w e r t h a n D m. T h i s is e v i d e n t l y d u e to the large p a r t i t i o n c o e f f i c i e n t existing at the b o u n d a r y b e t w e e n the p o l y m e r a n d l i p o p h i l i c s t r a t u m c o r n e u m , w h i c h results in large a m o u n t s o f d r u g b e i n g t a k e n u p i n t o the s t r a t u m c o r n e u m at e a r l y times. Fig. 4b shows h o w the s o l u t e ' c o n c e n t r a t i o n in the i n n e r m o s t r e g i o n o f the s t r a t u m c o r n e u m d i r e c t l y a d j a c e n t to the m a - trix,
cs(O,t ),
j u m p s at t = 0 d u e to the i m m e d i a t e n a t u r e o f the p a r t i t i o n i n g process. T h e r e a f t e r Cs(0,t ) declines w i t h t i m e a n d the c o n c e n t r a t i o n p r o f i l e s w i t h i n the s t r a t u m c o r n e u m b e c o m e m o r e l i n e a r as the d r u g diffuses slowly t h r o u g h this m e m b r a n e .T h e values o b t a i n e d for D m a n d D s ( T a b l e 4) a r e c o m p a r a b l e to t h o s e f o u n d for t h e m a t r i x o r s t r a t u m c o r n e u m alone. T h e r e is o n l y o n e i n c o n - sistency: the D m v a l u e s d o n o t s h o w the d e p e n - d e n c e on d r u g l o a d i n g seen b e f o r e , w h e r e a s D~
n o w a p p e a r to b e c o n c e n t r a t i o n d e p e n d e n t . T h e s t a n d a r d d e v i a t i o n s are, however, large e n o u g h to m a k e this o b s e r v a t i o n q u e s t i o n a b l e . D e s p i t e this s c a t t e r it is e v i d e n t t h a t the two diffusivities, w h i c h a r e an o r d e r o f m a g n i t u d e d i f f e r e n t in value, c a n b e c a l c u l a t e d f r o m a single p r o f i l e of m a ( t ) t h vs t t h a t is the b e s t fit to e x p e r i m e n t a l d a t a .
A s w i t h the p o l y m e r m a t r i x m o d e l it was n o t p o s s i b l e to o b t a i n r e a s o n a b l e p a r t i t i o n c o e f f i c i e n t s for the c o m b i n e d p r o b l e m . T h e v a l u e s in T a b l e 4 a r e u n a c c e p t a b l y large a n d s h o w m u c h scatter.
T h i s is due, once again, to the low d r u g c o n c e n t r a - t i o n s existing at the o u t e r edge of the m a t r i x ( x = 0) even a f t e r s m a l l times (Fig. 4b), w h i c h p r o d u c e s a h i g h l y n o n - l i n e a r i n f l u e n c e o f K1 o n
the d i f f u s i o n rate. Fig. 4c i l l u s t r a t e s t h e o r e t i c a l l y t h a t
d m a ( t ) / d t
h a r d l y c h a n g e s o n c e K 1 exceeds 100 in m a g n i t u d e . T h e v a l u e o f K 1 for c l e n b u t e r o l is, however, a p p r o x . 300, a n d will, therefore, lie o n a v e r y flat s e c t i o n o f the f i t t i n g p l a n e m a k i n g its i n f l u e n c e o n the r e s i d u a l s u m o f s q u a r e s negligi- ble.It t h u s a p p e a r s p o s s i b l e to c a l c u l a t e diffusivi- ties for the b i l a y e r m o d e l , b u t n o t p a r t i t i o n coeffi- cients, b y f i t t i n g the n u m e r i c a l s o l u t i o n to F i c k ' s S e c o n d L a w to e x p e r i m e n t a l d r u g release d a t a . A g r e a t e r d e g r e e o f s c a t t e r is, however, o b s e r v e d c o m p a r e d to m e a s u r e m e n t s m a d e o n the two sep- a r a t e p r o b l e m s .
Appendix
F o r t h e c o m b i n e d p r o b l e m as s h o w n in T a b l e 1, a s o l u t i o n to the f o l l o w i n g s y s t e m o f e q u a t i o n s is r e q u i r e d :
D m . c m ( X , t ) x x - C m ( X , t ) t = O , - Z < x < 0 ,
/ > 0 ;D s . c s ( X , t ) x x - C s ( X , t ) t = O , O < x < a ,
/ > 0 ; g I • Cs(0,t ) - Cm(0,/) = 0, t > 0;g 2 • Ca(t ) --
Cs(a,t ) = O, t > O.
T h e m e t h o d o f L a p l a c e t r a n s f o r m s y i e l d s the fol- l o w i n g e x p r e s s i o n for Ao
( m a ( t ) } ( s ) :
* ~ { m a ( t ) } ( s )
= (c0 D ~ - m b s i n h
qm L )
X ( s { ~ [ b q s c o s h qs a + K 2 s i n h qsa]
x c o s h q m L
+ K l ~ m m [bq~ s i n h q~a + K 2 c o s h qsa]
× s i n h qm L }) 1
w h e r e q s : v ~ / ~ a n d q m = V / S / ( - ~ . D e - t e r m i n a t i o n o f t h e i n v e r s e t r a n s f o r m u s i n g t h e r e s i d u e t h e o r y is n o t s t r a i g h t f o r w a r d . I t c o u l d b e a p p r o x i m a t e d for s h o r t o r l o n g t i m e s ( G u y a n d H a d g r a f t , 1980) o r else i n v e r t e d n u m e r i c a l l y ( A d - d i c k s et al., 1989). N e i t h e r of t h e s e p r o c e d u r e s a p p e a r e d to us p a r t i c u l a r l y a d v a n t a g e o u s c o m - p a r e d to t h e d i r e c t n u m e r i c a l a n a l y s i s o f t h e c o m - p l e t e p r o b l e m s , as p r e s e n t e d i n this p a p e r .
Acknowledgements
W e w i s h to t h a n k H e i d e l b e r g U n i v e r s i t y for p r o v i d i n g f u n d s for t h e p u r c h a s e of a p e r s o n a l c o m p u t e r a n d also B o e h r i n g e r I n g e l h e i m for t h e i r g e n e r o u s f i n a n c i a l s u p p o r t of this p r o j e c t . T h a n k s a r e also d u e to P r o f e s s o r H. Stricker, w h o k i n d l y m a d e s o m e d e p a r t m e n t a l f u n d s a v a i l a b l e .
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