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Drug Release from a Suspension with a Finite Dissolution Rate: Theory and Its Application to a Betamethasone 17-Valerate Patch
Drug Release from a Suspension with a Finite Dissolution Rate: Theory and Its Application to a Betamethasone 17-Valerate Patch KIYOSHIKUBOTA’§, E. H. TWIZELL~, AND HOWARD 1. MAIBACH” Received February 4, 1994, from the ’Department of Dermatology, University of Californiij-San Francisco, School of Medicine, Box 0989, Surge 110, San Francisco, CA 94 743-0989, and #Department of Mathematics and Statistics, Brunel University, Uxbridge, Middlesex, 5 Present address: Drug Safety Research Unit, Bursledon Hall, Accepted for publication July 21, 1994@. England UB8 3PH. Southampton, England SO31 1AA. Abstract 0 A random walk method for a diffusion equation is applied to the model for a suspension with a finite dissolution rate developed by Ayres and Lindstrom in 1977. In the method, the diffusion of dissolved drug and dissolution of crystal are calculated separately using a simple BASlC program. The random walk method strictly meets the principle of the conservation of mass as the drug amount in each sublayer rather than the concentration at each subinterval is concerned in the ointment. The model is used to analyze the release of betamethasone 17-valerate from a pressure-sensitive silicone adhesive into a sink. The drug release from the 1.50 mg/mL patch shows no substantial discrepancy from that predicted by the classic suspension model assuming an infinite dissolution rate. However, the classic model overestimates the release from the 3.08 and 5.88 mg/mL patches. The disagreement is lessened when the dissolution rate is assumed to be finite. However, the model does not give a perfect explanation because the drug release from the 3.08 and 5.88 mglmL patches in the early phase is faster than the model predicts.
Introduction In the classic model for drug release from a suspension, the dissolution of the suspended phase (crystal phase) is considered to be rapid in comparison to drug diffusion from an ointment matrix.l However, this is often not the case. In 1977, Ayres and Lindstrom developed a model for drug release from a suspension where the dissolution of the crystal is not necessarily f a ~ t . ~ , ~ In this paper, the model by Ayres and Lindstrom (finite dissolution rate model) is interpreted by the “random walk m e t h ~ d ” .The ~ , ~data for the release of a topical corticosteroid, betamethasone 17-valerate, from a pressure-sensitive silicone adhesive directly into aqueous receptor fluid are analyzed by the model.
Theoretical Section Random Walk Method Applied to the Drug Release from a Suspension with a Finite Dissolution Rate-As in the “random walk method” applied to the case of a simple soluti0n,4>~ the ointment, with thickness h, is divided into N sublayers. Each sublayer has a thickness Ax defined as Ax = h/N
(1)
When Ai(t)is defined as the total drug amount per unit area in the ith sublayer at time t, Ai(t) can be given as
A,(t)= A,,(t> + A c i @ ) (i = 1,2, ...,N )
(2)
where A ~ i ( tis) the amount of the dissolved drug per unit area @
Abstract publishedin Advance ACS Abstracts, September 1,1994.
0 1994, American Chemical Society and American Pharmaceutical Association
in the ith layer a t time t , and A&) is that of the crystal form. The amount per unit area of the crystal phase, A&), will decrease as time elapses because some fraction in this phase is released into the matrix as dissolved molecules. When E is defined as the volume fraction measure of crystal (given in mIJmL) and esis the crystal or powder density (given in pgl mL or mg/mL),2s3A&) can be rewritten as
It is known that adat is expressed as2l3
where K is the dissolution rate from the crystal phase, to is the initial volume fraction measure of crystal (60 = Aci(O)/ (esAx),from eq 3), Cs is the drug solubility in the matrix, and Ci(t) is the dissolved drug concentration in the zth layer. The dissolution rate K is given as K = K K , , E where ~ ~ ~ K is the surface area to volume constant and K,, is the crystallization rate c o n ~ t a n t . When ~ , ~ SOis defined as the initial surface area of the crystal phase per unit volume, the surface area to volume constant K is defined as K = S&02/3. The relation between Ci(t) and A ~ i ( tis) given as
if the initial volume fraction of the matrix (solvent), 1 - €0, is assumed unchanged throughout the study. The processes in each layer a t each discrete time step t = kAt (k = 0, 1, ...) are c o n ~ i d e r e d . The ~ , ~ time step, At, is given as
W A t = Ax2 or At = 1/(2k6\r2)
(6)
where D is the diffusion coefficient of the dissolved drug in the matrix and kd is the diffusion parameter defined as k d = D/h2. After the kth time step (when t = kAt) during the interval [t,t At], two consecutive processes, diffusion and then dissolution, are considered to occur. In each time step, half of the amount of dissolved drug in the ith layer, AD&), will “jump”forward t o the ti 1)th layer while the other half will “jump”back to the (i - 1)th layer. This relationship leads to Fick’s first and second laws. First, the net amount per unit area transported across the boundary between the (i - 1)th and ith layers in each time step is given as &-l(t)/2 - A ~ i ( t ) /=2 -AC,(t)Ax(l - E O ) / ~when ACi(t)is defined as ACiW = C,(t) - Ci-&). Therefore, the flux per unit area at this boundary during the interval [t,t At], Ji(t), is given as Ji(t) = -ACi(t)Ax(l - co)/2/At where At is defined in eq 6. This leads to the relationship Ji(t) = -D(1 co)ACi(t)/Ax,which approaches J = -D(1 - c0) aC/& (Fick‘s first law) when Ax 0. Second, the net change in the amount of dissolved drug per unit area during the interval [t, t At] in the ith layer or AA&) = (Ci(t A t ) - Ci(t))Ax(l - € 0 )
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Journal of Pharmaceutical Sciences / 1593 Vol. 83, No. 11, November 1994
results from (a) the difference of the flux a t two boundaries between the (i - 11th and ith layers and between the ith and (i 1)th layers or J,(t) - J,+l(t)and (b) the drug dissolution from the crystal phase in the ith layer during [t,t At]. The relation is given as AADM = At(J,(t) - J,+l(t))- AA&) or (C,(t At1 - C,(t))&(l - 60) = AtD(1 - ~o)(AC,+l(t) - AC,(t))/ Ax - AE@,& leading to aC/at = D a2C/ax2 - ~ $ (-l Eo)adat (Fick‘s second law) when Ax 0 and At 0. The above differential equation is solved by the random walk method, and a simple associated BASIC program is shown in Appendix 1 of the present paper. Being different from the method of Ayres and Lindstrom213and the method shown in Appendix 2, the two processes of diffusion and dissolution are calculated separately by the random walk method. Hereafter a line number in the BASIC program in Appendix 1which corresponds to the equation is also given. First, in the random walk method, the diffusion of the drug molecules which have been dissolved already occurs immediately aRer the previous, kth, time step. Conversely, after the “jump”,the amount per unit area in the ith layer is given as
+
+
+
+ 0 ) =A~,-,(t)/2+AD,+,(t)/2
= 2,
(2
...,N
- 1) (line 110) (7)
In the first layer, instead of eq 7, the following equation is used4p5
ADi(t
+ 0 ) = A~,(t)/2+ A~,(t)/2 (line 100)
(8)
In the Nth layer adjacent to the receptor phase, the following equation is used if the receptor phase works as a perfect sink and no “back diffusion” from the receptor to the membrane OCCU~S:~~~
A,dt
+ 0) = AD,-,(t)/2
k
-
-
A,(t
The cumulative drug amount per unit area released into the receptor phase till time t , Q(t),is given as
(line 100)
( 9)
Second, a part of the crystal phase is dissolved in each sublayer. From eqs 3-6, the change in A d t ) during the time interval [t,t At], AAa(t),can be given as in eq 10 below.
+
Q(t)= cADdO.- l)At)/2 (line 120)
In the calculation, the following two points are considered. First, the dissolved drug concentration cannot exceed the drug At) I C d l - to)&. In the solubility in the matrix or A,,(t initial several steps, the calculated value O f &,(it 4-At) in eq 12 may exceed the limit, cs(l - EO)&, where K >> k d unless N is sufficiently large. In such a case, eq 10 should not be used but the value of AA,,(t) in that layer is calculated from eq 15 below.
+
+ 0 ) - Cs(1 - c0)AX
AA,,(t) =A,,(t
(line 180) (15)
The second point is important immediately before the crystal phase is depleted. Obviously the drug amount released from the crystal phase cannot exceed the amount of this phase which exists. Therefore, the following relation should AA&) 2 0, where AAc,(t)is given in eq be satisfied: A&) AAc,(kAt) < 0, where 10 for any t. Thus, if Ac,(lzAt) AAc,(KAt) is calculated by eq 10 while Ac,(jAt) AAc,(jAt) L 0 w h e n j = 1, 2, ..., k - 1, the value for AAc,(kAt)should be adjusted so that
+
+
AA,,(KAt) = -A,,(kAt)
+
(line 160)
AD,(0)= Cs(1 - c O ) k ( i = 1, 2,
..., N)
(line 60)
A,,(O) = cOesAx (i = 1 , 2 , ...,N) (line 60)
(17)
(18)
When the initial total drug amount and area are defined as MDTand S, respectively, the total amount per unit area, MDT/ S , is given as
c N
= -K(E/EO)~’~(CSAX -ADJt
+ 0)/(1 - ~0))/[2Fz$’) o)/
= - { ~ ~ d [ A ~ ~ ( t ) / ( E o @ ~ k ) -A,,(t ] ~ ’ ~ [ cfs A X
MDdS =
(A,,(())
+ Ac,(O))
2=1
=N(Cd1 - c O ) k
+ c o Q , h ) = h ( C d 1 - c0) + toes)
(1- E ~ ) I } / ( ~ Nti~ )= 1,2, ...,A? (line 140) (10)
+
In eq 10, the use Of A ~ i ( t 0) but not A ~ i ( tin) eqs 7-9 (after the “jump”)is justified by the following consideration: if K >> k d and Ac;(O) > AD~(O), in the first several steps, the dissolved drug concentration may be maintained to be Cs in every layer, including the Nth layer, by the rapid supply from the crystal phase. However, if&(t) but not A,(t 0 )is used, AA&) in eq 10 is calculated to be zero, indicating that the supply from the crystal phase is zero. In the next, ( k l)th, time step when t = (k 1)At, the amounts of crystal and dissolved drug per unit area in the zth layer are given as
+
+
+
+ At) = A,,(t) + &I& (i =) 1, ...,A?
(19) The relationship between EO and MDTgiven below2s3is derived from eq 19. 60
= [MDd(J(Sh)- cs]/(@,-
+ At) =AD,(t + 0) - AA,i(t)
(i = 1, ..., N) (line 190) (12)
The drug flux during the interval [t, t given as 1594 / Journal of Pharmaceutical Sciences Vol. 83, No. 11, November 1994
+ At], J(t + At/2), is
cs)
(20)
Consistency with the Classic Model for the Drug Release from Suspension or Solution-When the crystal phase in suspension does not dissolve at all or K = 0, the drug is released from the dissolved phase only and the ointment behaves as a solution. Thus, the cumulative drug amount per unit area released from the ointment should be identical to the value predicted by the following equation for a ~ o l u t i o n ~ , ~
(line 190) (11)
AJt
(16)
The initial condition is given as
AA,i(t) = A&,AX)
A,,(t
(14)
J=1
Q = hCs(1 - c0){ 1 -
-c 8 ”
1
2m=0(2m
+ 1)’
O n the o t h e r hand, w h e n K >> kd, the process should b e r~ described b y the classic model for ~ u s p e n s i o n ol ~
Q = [D(2A- Cs*)Cs*t]'"
(t 5 to)
(22)
and
Q = ( 2 A - Cs*)h/2
+ hCs*(l exp[-l(t - t0)])/2 (t > t o )(23)
+
where A = C s ( 1 - E ~ ) toes, Cs* = C s ( 1 - to), t o = ( 2 A Cs*)h2/(4DCsh), and il= W/h2. In eqs 22 and 23, the quantity Cs* but not Cs should be used because in the classic model the solubility i s expressed as the drug amount per total volume (volume of matrix plus the initial volume of the crystal phase).lr7
M acetate buffer (pH 4.5) at a flow rate of 10 mL/h for the first 3 h, then the flow rate was decreased to 1-3 m u so that the receptor concentration was maintained t o be less than 5% of the aqueous saturation concentration of betamethasone 17-valerate ( ~ 5 - 6yg/mL, measured in room temperature). The receptor fluid was collected at 1, 2, 3, 4, 6, 10, 24, 34, 48, 72, 96, and 144 h and the amount of betamethasone in the receptor fluid was measured as previously.11J2 Estimation of Parameters-The saturation concentration of betamethasone 17-valerate in BIOPSA was estimated from the plots of the aqueous concentration versus concentration in BIOPSA at equilibrium. The diffusion coefficient of the dissolved drug, D, was estimated from the plots of the mean ( n = 5) Q versus t for the 1.50 mg/mL patch using eqs 22 and 23 because they did not show any substantial discrepancy from the prediction by the classic m ~ d e l l ? ~ (eqs 22 and 23). Using the value of I) estimated and N = 40, the values of the dissolution rate X in eq 10 for the 3.08 and 5.88 mg/mL patches were estimated from the plots of the mean ( n = 5 or 6 ) Q versus t by a nonlinear least squares methodI3 assuming e8= 1g/mL = 106 yg/mL.
Experimental Section
Results
Partition of Betamethasone 17-Valerate between Silicone Adhesive and Water-Betamethasone 17-valerate was a generous gift from Schering Plough (Bloomfield, NJ). A pressure-sensitive silicone adhesive, BIOPSA, in trichlorotrifluoroethane as solvent carrier was purchased from Dow Corning (Midland, MI). Betamethasone 17-valerate was dissolved in a mixture [50:50 (w/w)l of dichloromethane-BIOPSA in solvent carrier to obtain 10 different concentrations (0.01-10 mg/mL). On the tips of 7 x 60 mm strips of Scotchpak (Type 1009 backing membrane, 3M, St. Paul, MN) was mounted 30 yL of 10 various concentrations of betamethasone 17valerate in the dichloromethane-BIOPSA mixture (n = 4 for each concentration). After dichloromethane and solvent carrier were evaporated, the tip was dipped into 1.5-2.5 mL of 0.1 M acetate buffer (pH 4.5) for 10 days, and then the strip was separated from the acetate buffer, and amounts of betamethasone 17-valerate in buffer and BIOPSA were measured. Acetate buffer (pH 4.5) was used because, unless the aqueous pH is maintained between 4 and 5, betamethasone 17-valerate is rapidly converted to betamethasone 21-valerate and then betamethasone.s-10 To know the amount of betamethasone 17valerate per unit volume of BIOPSA attained after the solvent was evaporated, 10 solutions were also developed manually into an approximately 6 x 10 cm area on the backing membrane by a casting device (2.5 in. in width and 0.010 in. in thickness, designed and made by Stanford Research International (Menlo Park, CA)), and the thickness of this film was measured by light wave micrometer (Van Keuren Co., Watertown, MA). The drug amount per unit volume of BIOPSA was, on average, 3.1 times higher than that in the mixture of dichloromethane-BIOPSA in solvent carrier [50:50 (w/w)l. Betamethasone 17-valerate in aqueous solution was measured by a normal phase high-performance liquid chromatographic method as previously described, where the detection limit approximated 5 ng per sample.11J2 To measure betamethasone 17-valerate in BIOPSA, BIOPSA on the backing membrane was sonicated for 30 s in dichloromethane and the solution directly injected into the chromatograph. The mean (&SD,n = 7) recovery of betamethasone 17-valerate from BIOPSA was 96.0 f 4.3% when the peak height of a known sample (200 yg) was compared to that of the standard solution prepared in 2% methanol in dichloromethane. Release of Betamethasone 17-Valerate from BIOPSASolutions of three concentrations (1.5, 0.75, and 0.375 mg/mL) of betamethasone 17-valerate in the mixture [64:36 (w/w)l of dichloromethane-BIOPSA in solvent carrier were prepared and developed into an approximately 6 x 20 cm area on the backing membrane as earlier. The low adhesion polyester film, Scotchpak (Type 1022 release liner, 3M, St. Paul, MN), was placed on BIOPSA after the solvent was evaporated. The patch thickness was ~ 4 ym 0 and the final drug amounts per unit volume of BIOPSA were estimated as 5.88, 3.08, and 1.50 mg/mL, respectively. After confirming that the thickness of each area used in the experiment was 40 2 ym by light wave micrometer, a round piece (1cm2)was excised from the patch. The round piece was then placed on the flow-through diffusion cell (LG-1084-LPC, Laboratory Glass Apparatus Inc., Berkeley, CAI and the direct release of betamethasone 17-valerate from BIOPSA to the receptor fluid was measured. Into the receptor cell was pumped 0.1
Factors Influencing the Predicted Release Rates (Figure 1)-The Q-& profiles predicted by the model using arbitrary p a r a m e t e r values are shown in Figure la-c. The effects of the change in N ( t h e n u m b e r of sublayers), K (the dissolution rate), a n d 60 ( t h e initial volume fraction measure of the crystal) o n the prediction are illustrated in Figure lac, respectively. In Figure 2a, the Q-& profiles predicted w h e n K = 0 h-l (A and A, corresponding t o e q 21) a n d K = 1000 h-l (0 and 0, corresponding t o e q s 22 and 23) are depicted. F o r either value of K , t o examine the effect of the change in N (number of sublayers) o n the prediction, two calculation results w h e r e N = 10 (A a n d 0 ) a n d N = 40 (A a n d 0) are shown. O t h e r parameters employed are arbitrarily chosen as follows: C s = 1000 pg/mL, to = 0.01, h = 50 pm, eS cm2/h. Thus, kd = D/h2 = = 1 g/mL or lo6 pg/mL, D = 0.04 h-l a n d A in eqs 22 and 2 3 is given as A = Cs(1 - € 0 ) toes = 10.99 mg/mL. The initial a m o u n t per unit area t o be released in e q 23, Ah, is given as Ah = 54.95 pg/cm2. O n the o t h e r hand, the initial a m o u n t per unit area t o be released when K = 0 h-' (A and A, eq 21) is Cs(1- c0)h = 4.95 pg/cm2 (the initial amount of crystal phase, toesh = 50 pg/cm2, is not released at all because K = 0 h-l). When K = 1000 h-l, the values of Q predicted b y the r a n d o m walk method w h e n N = 10 (0)underestimate those yielded b y e q s 22 a n d 2 3 while the difference between the Q-& profiles predicted by eqs 22 a n d 23 a n d those b y the random w a l k method w h e n N = 40 (0) is less prominent. O n the o t h e r h a n d , w h e n K = 0 h-l, the Q-& profiles predicted b y the random walk method w h e n N = 10 (A)and 40 (A) are almost identical t o those predicted b y eq 21. Figure l b shows the effect of the change in the dissolution rate, K , o n the Q-& profiles, w h e r e N = 40 and the s a m e values of Cs, €0, h, es, a n d D as in Figure l a are used for all of the predictions. Drug release is slowed particularly in the early phase with a decrease in the dissolution rate from K = 1000 h-l(O, eqs 22 and 23) t o K = 1h-l (A)a n d 0.1 h-l (W). When K = 0.01 h-l (O), which is 4 times smaller than the diffusion parameter, k d (0.04 h-l), the Q-& profiles are almost identical t o those predicted by the classic model for the solution case6z7 (eq 21) or the case where K = 0 h-l ( A ) until almost all of the dissolved d r u g molecules are released t o the receptor at t x 16 h t& x 4 h). However, thereafter, Q continues t o increase slowly due to the slow dissolution of the crystal phase. Thus, w h e n K < kd, the Q-& profiles h a v e two phases; the first p h a s e where the dissolved d r u g is released rapidly, followed b y the second phase where the crystal phase is slowly dissolved and released.
*
+
Journal of Pharmaceutical Sciences / 1595 Vo/. 83, No. 11, November 1994
360
4
10
5
0 0
2
6
4
a
60
40
20
0
0
5
10
15
0
5
10
\
15
20
25
30
Time, min
Figure 2-The effect of the change in N (number of sublayers) and the method of approximation to (&,dax) (x = 0) (11 = C/Cs) on the ,u-4profiles. See the text for the parameter values.
where the slope of the Q-& profiles is increased with an increase in €0. Comparison with the Ayres and Lindstrom Model (Figure 2)-In order to make the comparison between the method by Lindstrom and Ayres3 with the random walk 0 2 4 6 8 method (Appendix 11, we developed an original algorithm for this purpose as shown in Appendix 2. Lindstrom and Ayres3 1 Time (h) used a second-order approximation to aqGx (7= C/Cs) at the Figure 1-@& profiles predicted by the models. The Q 4profiles by the boundary x = 0, leading to eqs A12 and A14 in Appendix 2. classic model (lines) and the finite dissolution rate model (symbols) are shown. Their results (their Tables I-V, VIII, and are almost The effect of the change in N (number of sublayers) is shown with the prediction identical to the results obtained using the alternative secondby the classic suspension model (upper line) and solution model (lower line) in order algorithm detailed in Appendix 2 (eqs A9-Al2 and A14) part a. The @A profiles with different K values are shown in part b. The with N = 10. This similarity was observed for all values of effect of the change in EO is shown with the prediction by the classic suspension the model parameters which were used. Using a first-order model for 60 = 0.01 (upper line) and €0 = 0.0025 (middle line) and that by the approximation to avlibc (x = 0 )gives the alternative algorithm solution model (lower line) in part c. See the text for the other parameter values. described by eqs A9-Al3 in Appendix 2. The p-& profiles $i is the relative cumulative drug mass released and defined Figure l c shows the effect of the change in EO on the a s p = QMhCs)13obtained using this algorithm with the same Q-& profiles where K and Kc,, are assumed to be constant model parameters and N were very close t o those obtained even when €0 is varied. Thus, K ( K = K K ~ ~changes E ~ ~ are / ~ ) using the random walk method (Appendix 1). Therefore, the proportional to eO2/3. The other parameter values are the same comparison between the random walk method and the method as those in Figure la. In Figure lc, triangles (A and A ) of Lindstrom and Ayres3 reduces to that between the firstdelineate the profiles when K > kd ( K = 0.4 and 1 h-l) and and second-order approximations to (x = 0 ) in the circles ( 0 and 0)denote those when K < k d (K = 0.004 and algorithm detailed in Appendix 2. Generally, the estimate of Q or ,u by the first-order ap0.01 h-l). In all four cases, the Q-& profiles until t x 1 h proximation to aq/& (x = 0 ) is smaller than that by the secondare similar to one another. Thereafter, the Q-& profiles order approximation. However, the difference of Q or p differ substantially between the case where €0 = 0.01 and K between the two approximations is generally small (less than = 1h-l (A)and the case where EO = 0.0025 and K = 1.(0.0025/ 2% of the value) particularly when K x k d or K < kd. When 0.01)2/3= 0.4 h-l (A). However, it is only after almost all of K >> K d , the discrepancy is remarkable between the two the dissolved drug molecules have been released at t x approximations during the initial period. This problem is 16 h (or & x 4 h) that the substantial difference between illuminated in Figure 2, where the effect of using a first- or the Q-./i profiles where €0 = 0.01 and K = 0.01 h-l ( 0 )and second-order approximation t o ay/& (x = 0 ) and the effect of those where €0 = 0.0025 and K = 0.01~(0.0025/0.01~2/3 = 0.004 increasing the number of sublayers are shown for such a case. h-l(O) becomes prominent. In such a case ( K < Ka), the clear Note that in Figure 2, Cs = 100 pg/mL and €0 = 0.4, as assumed by Lindstrom and Ayres (see their Figure 4),3so that difference is observed only in the second phase (& > 4 h), 1596 / Journal of Pharmaceutical Sciences Vol. 83, No. 11, November 1994
U
10
100
1000
10000
100000
C o n c e n t r a t i o n in P a t c h (pg/mL) Figure 3-Plots of aqueous betamethasone 17-valerate concentration vs concentration in the silicone adhesive (BIOPSA) at equilibrium.
the maximum value of ,u is 4000. Therefore, only less than 10% of the initial total amount is released during the time interval shown in Figure 2. In addition, K or Kz is assumed to be >>kd (Kz = 1000 min-l while kd = 0.15 min-l, where Kz is defined as Kz = K Kor~K = ~ K2c02/3 = 540 min-l, Appendix 2). Other parameter values are D = 6.10-* cm2/min,h = cm, Cs = 100 ,ug/cm3, es = 1.0 g/cm3, and At = 0.5 min, as assumed by Lindstrom and A y r e ~ . ~ In Figure 2, any method in Appendix 2 underestimates the p ~ - & profiles (a) ( p= ~a / h , / a , corresponding t o eq 22),3 though the discrepancy decreases as t increases. The p-& profiles obtained using eqs A9-Al2 and A14 (second-order approximation) with N , the number of sublayers, increased from 10 (0)to 40 (A), become very close to the p ~ - & profiles already a t t x 2.5 h (i.e., & x 12 or 13 min). Those obtained using the first-order algorithm (eqs A9-Al3) are closer to the p ~ - & profiles when N is increased from 10 (0)to 40 (A),though not as close as the second-order algorithm (eqs A9-Al2 and A14) with N = 40 (A). Solubility of Betamethasone 17-Valeratein BIOPSA (Figure S)-Figure 3 shows the plots of the aqueous concentration versus patch concentration measured after the patches containing various amounts of betamethasone 17-valerate were dipped in 0.1 M acetate buffer (pH 4.5) for 10 days. The aqueous concentration was essentially unchanged when the drug amount per unit volume of BIOPSA was more than 1000 pg/rnL and the average aqueous concentration was 5.4 ,ug/mL (horizontal line). When the patch concentration was less than 1000,ug/mL, the partition coefficient between patch and water was weakly dependent on the concentration. The patcldwater partition coefficientwas e l 0 0 when the aqueous concentration was around 0.1 ,ug/mL while it was ~ 2 0 when 0 the aqueous concentration was around 5 ,ug/mL. Using a partition coefficient of ~ 2 0 0 the , solubility of betamethasone 17-valerate was estimated as around 1000pg/mL or 1mg/mL. This value of the solubility was compatible with the observation that the patch was translucent when the final concentration in BIOPSA was 5600 ,ug/mL but clouded when 21500 pg/mL. Release of Betamethasone 17-Valeratefrom BIOPSA (Figure 4)-Figure 4 shows the mean Q-& profiles of drug release from the patches with three final concentrations, 1.50 (a),3.08 (A), and 5.88 mg/mL (0). The mean (fSD, n = 5 or 6 ) percentages of the initial drug amount released until 144 h (&= 12 h) from the 1.50,3.08, and 5.88 mg/mL patches to the receptor fluid were 97.5 f2.7,93.5 f 1.1, and 57.8 f 9.7%, respectively, when the initial amount is calculated as that excreted to the receptor fluid until 144 h plus that remaining in the patch a t 144 h. As shown in Figure 4a, the Q-&
dTime (h) Figure 4-Gz/t profiles of betamethasone 17-valerate released from the silicone adhesive (BIOPSA). Lines I, 11, and 111 in part a are the predictions by n = 5), 3.08 (A, n = 6), and 1.50 the classic suspension model for the 5.88 (0, (0,n = 5) mg/rnL patches, respectively. Lines I and II in part b are the predictions n = 5) and 3.08 (A, n = 6) by the finite dissolution rate model for the 5.88 (0, mg/mL patches, respectively.
profiles from the 1.50 mg/mL patch (where cOes = 0.58 mg/ mL) predicted by eqs 22 and 23 were not substantially different from those observed (U),and the value of D was estimated as D = 2.79.10-7 cm2h when eqs 22 and 23 were fitted to the average Q-& profiles from the 1.50 mg/mL patch. However, the model (eqs 22 and 23) overestimated the observed Q-& profiles when this D value was assumed (Figure 4a). The magnitude of the discrepancy between the model prediction and observation was lessened when the model supposing the finite dissolution rate was used (Figure 4b). When the model was fitted to the observed average Q-& profiles assuming D = 2.79-1Op7cm2h, K was estimated as K = 0.145 h-l for the 3.08 mg/mL patch (n = 6)and K = 0.084 h-l for the 5.88 mg/mL patch (n = 5).
Discussion Random Walk Method and the Ayres and Lindstrom Model-One of the advantages provided by the use of the random walk method is the simplicity of programming (Appendix 1). Being different from the method by Burnette14 where a random value is produced in each iteration, for each of the 100 000 molecules, the computing time of the random walk method shown in this paper is comparable to that of the conventional finite difference method in Appendix 2. Another advantage is that mass is strictly conserved in the random walk method, where the amount of each sublayer is considered rather than the concentration in each subinterval. The principle of mass conservation is not always satisfied in the method by Ayres and Lindstrom, particularly when N is not large and the second-order approximation to a?,d& (x = 0 ) is used (eq A14). For instance, In Tables I-V, VIII, and M in the paper by Lindstrom and Ayes: ,u approaches 1.03-1.11 as time elapses when K = 0 while this value should be exactly 1.00. Generally, the random walk method (Appendix 1)produces the prediction which approximates that by the method of Journal of Pharmaceutical Sciences / 1597 Vol. 83, No. 11, November 1994
Lindstrom and A y r e ~ .This ~ is particularly true when K < = 0, both methods give Q-& or p-& profiles sufficiently close to those produced by eq 21 even if N is small (e.g., N = 10, Figure la). However, when K >> kd, the discrepancy predominates between these two methods during the initial phase. It is also in this particular situation that the prediction by either method substantially deviates from that by the classic suspension model (eqs 22 and 23). In order to fully elucidate this complicated matter, we developed the original algorithm shown in Appendix 2 where the first-order approximation to &,hk(x = 0)( r = C/Cs, eq A14) yields the p-& profiles = &/(hCs))by the random walk method (Appendix 1A) while the second-order approximation (eq A15) yields those by the method of Lindstrom and Ayres.3 The results (Figures la and 2) indicate that when K >> kd (i.e., the dissolution of the crystal phase is much faster than the diffusion of the dissolved drug through the matrix), any method underestimates p or Q predicted by eq 22 during the initial phase unless N is sufficiently large. With larger N , the agreement with the classic suspension model (eqs 22 and 23) can be obtained earlier. When N is fixed, the secondorder approximation and the method of Lindstrom and Ayres3 give values closer to that predicted by the classic model (eqs 22 and 23) than the first-order approximation and random walk method (Figure 2). Factors Determining the Release of Betamethasone 17-Valeratefrom BIOPSA-Even if the drug release from the 1.50 mg/mL patch was explained by the classic model for suspension^',^ (eqs 22 and 23, Figure 4a), this model failed to explain the drug release from either of the 3.08 and 5.88 mg/ mL patches, which was slower than the prediction by eqs 22 and 23. The discrepancy between the model prediction and the observation was lessened when the finite dissolution rate model was used (Figure 4b), though the model did not give a perfect explanation because the drug release in the early period was faster than the prediction by the model in either of the 3.08 and 5.88 mg/mL patches (Figure 4b). One possible explanation for the discrepancy between the model prediction and the observation was that the crystal phase was not uniform and comprised more than two phases with different dissolution rates. In order to come to an unambiguous conclusion, however, more experiments are needed, including the examination of the distribution of the particle size in the patch by microscopic examination, as well as the characterization of the crystals with other physical techniques such as spectroscopy, X-ray diffraction, and differential scanning calorimetry.
& or K = kd. For instance, when K
References and Notes 1. Higuchi, T. J. Pharm. Sci. 1961, 50, 874-875. 2. Ayres, J. W.; Lindstrom, F.T. J. Pharm. Sci. 1977, 66, 654662. 3. Lindstrom, F.T.; Ayres, J. W. J. Pharm. Sci. 1977, 66, 662668. 4. Kubota, K.; Koyama, E.; Yasuda, K. J. Pharm. Sci. 1991, 80, 752-756. 5. Kubota. K.: Maibach. H. I. In In Vitro Percutaneous Absorotion: P&czples, Fundamentals and Applieations, Bronaugh, R. L.; Maibach, H. I. Eds.; CRC Press: Boca Raton, FL, 1991; pp 243-264. 6. Higuchi, W. I. J. Pharm. Sei. 1962,51, 802-804. 7. Kubota, K.; Yamada, T.; Ogura, A.; Ishizaki, T. J. Pharm. Sci. 1990, 79, 179-184. 8. Yip, Y. W.; Li, W. P. A. J. Pharm. Pharmacol. 1979, 31, 400402. 9. Cheung, Y. W.; Li, W. P. A,; Irwin, W. J. Znt. J . Pharm. 1985, 26, 175-189. 10. Smith, E. W.; Haigh, J. M.; Kanfer, I. Znt. J. Pharm. 1985,27, 185-192.
1598 / Journal of Pharmaceutical Sciences Vol. 83, No. 11, November 1994
11. Kubota, K.; Sznitowska, M.; Maibach, H. I. J. Pharm. Sci. 1993, 450, 450-456. 12. Kubota, K.; Sznitowska, M.; Maibach, H. I. Int. J.Pharm. 1993. 96, 105-110. 13. Yamaoka, K.; Tanigawara, Y.; Nakagawa, T.; Uno, T. J. Pharmacobio-Dyn. 1981,4, 879-885. 14. Burnette, R. R. Znt. J . Pharm. 1984,22,89-97. 15. Twizell, E. H. Numerical Methods, with Applications in the Biomedical Sciences; Ellis Horwood: Chichester, 1988; pp 228278.
Appendix 1-BASIC Program of the Random Walk Method for the Finite Dissolution Rate Model 10 REM RS, CS(MICROC/ML) H,DX(CM) D(CMWH) K,KD(H"- I ) DT(H) 20 REM AD,AC,BD, ADZ, ACZ, Q (MICROG/CMA2)J(MICROG/CMA2/H) 30 N=40K=1: D=lOA(-6):RS=10"6 CS=lOOo: EZ=O.l: H=0.005 40 KD=D/H/H: DT=l/uKDR\I/N: DX=H/N:ADZ=CS*( 1 -EZ)*DX:ACZ=RS*EZ*DX 50 DIM AD(N),AC(N),BD(N) 60 FOR I=1 TO N: AD(I)=ADZ AC(I)=ACZ: NEXT I 70 Q=O:T=O NEXT I 80 AD(O)=O: AC(O)=O:FORI=1 TON: AD(O)=AD(O)+AD(I):AC(O)=AC(O)+AC(I): YO PRINT "T=";T, "Q=";Q,"AD=";AD(O),"AC=";AC(O):PRINT"T=";T-DT/2, "J="; J 100 T=T+DTBD( I )=AD(1)/2+AD(2)/2:BD(N)=AD(N-l)/2 llOFOR 1=2TON-1: BD(I)=AD(I-l)/2+AD(I+1)/2: NEXT1 120 J=AD(N)/2/DT:Q=Q+AD(N)/2 130 FOR I=1 TO N 140 DAC = - WKD*(AC(I)/EZ/RS/I)X)"(2/3)*(CS*DX-BDfI)/(l-EZ))/2/N/N 150 IF DAC>O THEN DAC=O 160 IF DAC+AC(I)
Appendix 2-Comparison of the Random Walk Method (Appendix 1) with the Method of Lindstrom and Ayres3 Consider the nonlinear parabolic, integrodifferential,initial! boundary value problem for mass transport of a drug in an ointment-perfect sink system, consisting of the model equation2s3
in which 17 = v(x,t), for all (x,t) in the region R = {(-h 0) x ( t > O)}, the initial distribution
rk,O)= 1; -h
5
x
50
(A2)
and the boundary conditions
aq(-h ,t) -ax - 0 ; t ' O q(O,t) = 0; t > 0
(A3) (A41
In eq Al, q(x,t)= C(x,t)/Cswhere C(r,t) is the solution phase drug concentration distribution. The dissolution rate constant Kz employed in the paper by Lindstrom and Ayres3is defined ~ . that the constant K is defined as K = as Kz = K K ~(Note KK~,.+ in~the ~ ~text ~ and in another paper by Ayres and Lindstrom2 so that K = KZco2/3).Other parameters, €0, eS,D , and h are defined as in the text.
Numerically, the problem (eqs Al-A4) may be solved using finite-difference techniques by discretizing the interval -h I x 5 0 into N subintervals each of width Ax so that NAx = h (eq l ) , and by discretizing the time interval t 2 0 into steps each of length At. The open region R = {(x,t): -h < x < 0 , t > 01 and its boundary aR consisting of the lines x = -h, x = 0, and t = 0 are thus covered by a rectangular mesh, the mesh mAx ( m points having coordinates (xm,tn),where xm = -h = 0, 1, ..., N) and t , = nAt ( n = 0, 1, 2, ...). The solution a t the mesh point (x,,tn) of an approximating finite difference scheme for eqs Al-A4 is denoted by &. The numerical method to be used is developed by approximating the time derivative in eq A1 by the first-order forward difference approximant
Substituting eqs A5 and A6 with 0 = 1into eq A1 leads to a finite difference scheme approximating the solution to eqs Al-A4. This scheme is given by
+
+
av(x,t)/at= A t - l [ ~ ( ~ , A t t ) - v(x,t)l+ O(At1 (A51 and the space derivative by the weighted second-order approximant
+ A t ) - ~ v ( xt,+ A t ) + + h7t + ~ t )+) (1- e){v(x- hx,t) - 2q(x,t)+ V(X +
a2v(x,t)/&x2= A C 2 [ 8 { v ( x- hx, t
v(x
Ax, t)}I
where p = DAtlAx2. It is clear from eq A4 that
vk = 0 for n
= 1, 2,
....
Then,
It is also clear that eq A9 needs modification for use with m = 0, for then the mesh point ( - h - Ax, t n ) ,which is outside R U aR, is needed. This difficulty is circumvented by noting that, to second order, from eq A3, rfl = vy for n = 0, 1, .... This gives
+ 0 ( h 2 )(A6)
in which x = x, ( m = 0, 1, ..., N - 11, f = t, ( n = 0, 1,2, ...), and 8 ( 0 I 8 5 1)is a parameter. Hereafter, 0 is fixed as 0 = 1 (giving a fully implicit methodP5 as in the paper by Lindstrom and Other methods (e.g., 8 = 0 and l/2) and the features unique to each method are detailed elsewhere.15 The factor (1 - r ( x , t ) ) is evaluated at the point (xm,tn+l), giving 1 - &. Approximating the integral
S, = f"[l 0 - v(x,,t)]dz
(A71
The solution of $+'(i = 0,1, ..., N - 1)may be determined by solving eqs Ag-All, a linear algebraic system of order N (but not N - l).3 Finally, the relative cumulative mass release ,u ( = Q / ( ~ C S ) ) ~ , ~ at time pnfl, is computed for n = 0, 1, 2, ... from the formula
where, t o first order,
in eq A1 by the trapezoidal rule, it may be shown that or, t o second order,3
and it is convenient to define
Acknowledgments in which m = 0, 1, ..., N and n = 0, 1, 2, = ~ 0 for % ~m = 0, 1, ..., N . that
....
Note, J", = 0 so
We thank Drs. David Friend and Harold W. Nolen, 111, of Stanford Research International (SRI) Menlo Park, CA, for their help in developing the patch containing betamethasone 17-valerate.
Journal of Pharmaceutical Sciences / 1599 Vol. 83, No. 1 1, November 1994
Introduction In the classic model for drug release from a suspension, the dissolution of the suspended phase (crystal phase) is considered to be rapid in comparison to drug diffusion from an ointment matrix.l However, this is often not the case. In 1977, Ayres and Lindstrom developed a model for drug release from a suspension where the dissolution of the crystal is not necessarily f a ~ t . ~ , ~ In this paper, the model by Ayres and Lindstrom (finite dissolution rate model) is interpreted by the “random walk m e t h ~ d ” .The ~ , ~data for the release of a topical corticosteroid, betamethasone 17-valerate, from a pressure-sensitive silicone adhesive directly into aqueous receptor fluid are analyzed by the model.
Theoretical Section Random Walk Method Applied to the Drug Release from a Suspension with a Finite Dissolution Rate-As in the “random walk method” applied to the case of a simple soluti0n,4>~ the ointment, with thickness h, is divided into N sublayers. Each sublayer has a thickness Ax defined as Ax = h/N
(1)
When Ai(t)is defined as the total drug amount per unit area in the ith sublayer at time t, Ai(t) can be given as
A,(t)= A,,(t> + A c i @ ) (i = 1,2, ...,N )
(2)
where A ~ i ( tis) the amount of the dissolved drug per unit area @
Abstract publishedin Advance ACS Abstracts, September 1,1994.
0 1994, American Chemical Society and American Pharmaceutical Association
in the ith layer a t time t , and A&) is that of the crystal form. The amount per unit area of the crystal phase, A&), will decrease as time elapses because some fraction in this phase is released into the matrix as dissolved molecules. When E is defined as the volume fraction measure of crystal (given in mIJmL) and esis the crystal or powder density (given in pgl mL or mg/mL),2s3A&) can be rewritten as
It is known that adat is expressed as2l3
where K is the dissolution rate from the crystal phase, to is the initial volume fraction measure of crystal (60 = Aci(O)/ (esAx),from eq 3), Cs is the drug solubility in the matrix, and Ci(t) is the dissolved drug concentration in the zth layer. The dissolution rate K is given as K = K K , , E where ~ ~ ~ K is the surface area to volume constant and K,, is the crystallization rate c o n ~ t a n t . When ~ , ~ SOis defined as the initial surface area of the crystal phase per unit volume, the surface area to volume constant K is defined as K = S&02/3. The relation between Ci(t) and A ~ i ( tis) given as
if the initial volume fraction of the matrix (solvent), 1 - €0, is assumed unchanged throughout the study. The processes in each layer a t each discrete time step t = kAt (k = 0, 1, ...) are c o n ~ i d e r e d . The ~ , ~ time step, At, is given as
W A t = Ax2 or At = 1/(2k6\r2)
(6)
where D is the diffusion coefficient of the dissolved drug in the matrix and kd is the diffusion parameter defined as k d = D/h2. After the kth time step (when t = kAt) during the interval [t,t At], two consecutive processes, diffusion and then dissolution, are considered to occur. In each time step, half of the amount of dissolved drug in the ith layer, AD&), will “jump”forward t o the ti 1)th layer while the other half will “jump”back to the (i - 1)th layer. This relationship leads to Fick’s first and second laws. First, the net amount per unit area transported across the boundary between the (i - 1)th and ith layers in each time step is given as &-l(t)/2 - A ~ i ( t ) /=2 -AC,(t)Ax(l - E O ) / ~when ACi(t)is defined as ACiW = C,(t) - Ci-&). Therefore, the flux per unit area at this boundary during the interval [t,t At], Ji(t), is given as Ji(t) = -ACi(t)Ax(l - co)/2/At where At is defined in eq 6. This leads to the relationship Ji(t) = -D(1 co)ACi(t)/Ax,which approaches J = -D(1 - c0) aC/& (Fick‘s first law) when Ax 0. Second, the net change in the amount of dissolved drug per unit area during the interval [t, t At] in the ith layer or AA&) = (Ci(t A t ) - Ci(t))Ax(l - € 0 )
0022-3549/94/1200-1593$04.50/0
+
+
+
-
+
+
Journal of Pharmaceutical Sciences / 1593 Vol. 83, No. 11, November 1994
results from (a) the difference of the flux a t two boundaries between the (i - 11th and ith layers and between the ith and (i 1)th layers or J,(t) - J,+l(t)and (b) the drug dissolution from the crystal phase in the ith layer during [t,t At]. The relation is given as AADM = At(J,(t) - J,+l(t))- AA&) or (C,(t At1 - C,(t))&(l - 60) = AtD(1 - ~o)(AC,+l(t) - AC,(t))/ Ax - AE@,& leading to aC/at = D a2C/ax2 - ~ $ (-l Eo)adat (Fick‘s second law) when Ax 0 and At 0. The above differential equation is solved by the random walk method, and a simple associated BASIC program is shown in Appendix 1 of the present paper. Being different from the method of Ayres and Lindstrom213and the method shown in Appendix 2, the two processes of diffusion and dissolution are calculated separately by the random walk method. Hereafter a line number in the BASIC program in Appendix 1which corresponds to the equation is also given. First, in the random walk method, the diffusion of the drug molecules which have been dissolved already occurs immediately aRer the previous, kth, time step. Conversely, after the “jump”,the amount per unit area in the ith layer is given as
+
+
+
+ 0 ) =A~,-,(t)/2+AD,+,(t)/2
= 2,
(2
...,N
- 1) (line 110) (7)
In the first layer, instead of eq 7, the following equation is used4p5
ADi(t
+ 0 ) = A~,(t)/2+ A~,(t)/2 (line 100)
(8)
In the Nth layer adjacent to the receptor phase, the following equation is used if the receptor phase works as a perfect sink and no “back diffusion” from the receptor to the membrane OCCU~S:~~~
A,dt
+ 0) = AD,-,(t)/2
k
-
-
A,(t
The cumulative drug amount per unit area released into the receptor phase till time t , Q(t),is given as
(line 100)
( 9)
Second, a part of the crystal phase is dissolved in each sublayer. From eqs 3-6, the change in A d t ) during the time interval [t,t At], AAa(t),can be given as in eq 10 below.
+
Q(t)= cADdO.- l)At)/2 (line 120)
In the calculation, the following two points are considered. First, the dissolved drug concentration cannot exceed the drug At) I C d l - to)&. In the solubility in the matrix or A,,(t initial several steps, the calculated value O f &,(it 4-At) in eq 12 may exceed the limit, cs(l - EO)&, where K >> k d unless N is sufficiently large. In such a case, eq 10 should not be used but the value of AA,,(t) in that layer is calculated from eq 15 below.
+
+ 0 ) - Cs(1 - c0)AX
AA,,(t) =A,,(t
(line 180) (15)
The second point is important immediately before the crystal phase is depleted. Obviously the drug amount released from the crystal phase cannot exceed the amount of this phase which exists. Therefore, the following relation should AA&) 2 0, where AAc,(t)is given in eq be satisfied: A&) AAc,(kAt) < 0, where 10 for any t. Thus, if Ac,(lzAt) AAc,(KAt) is calculated by eq 10 while Ac,(jAt) AAc,(jAt) L 0 w h e n j = 1, 2, ..., k - 1, the value for AAc,(kAt)should be adjusted so that
+
+
AA,,(KAt) = -A,,(kAt)
+
(line 160)
AD,(0)= Cs(1 - c O ) k ( i = 1, 2,
..., N)
(line 60)
A,,(O) = cOesAx (i = 1 , 2 , ...,N) (line 60)
(17)
(18)
When the initial total drug amount and area are defined as MDTand S, respectively, the total amount per unit area, MDT/ S , is given as
c N
= -K(E/EO)~’~(CSAX -ADJt
+ 0)/(1 - ~0))/[2Fz$’) o)/
= - { ~ ~ d [ A ~ ~ ( t ) / ( E o @ ~ k ) -A,,(t ] ~ ’ ~ [ cfs A X
MDdS =
(A,,(())
+ Ac,(O))
2=1
=N(Cd1 - c O ) k
+ c o Q , h ) = h ( C d 1 - c0) + toes)
(1- E ~ ) I } / ( ~ Nti~ )= 1,2, ...,A? (line 140) (10)
+
In eq 10, the use Of A ~ i ( t 0) but not A ~ i ( tin) eqs 7-9 (after the “jump”)is justified by the following consideration: if K >> k d and Ac;(O) > AD~(O), in the first several steps, the dissolved drug concentration may be maintained to be Cs in every layer, including the Nth layer, by the rapid supply from the crystal phase. However, if&(t) but not A,(t 0 )is used, AA&) in eq 10 is calculated to be zero, indicating that the supply from the crystal phase is zero. In the next, ( k l)th, time step when t = (k 1)At, the amounts of crystal and dissolved drug per unit area in the zth layer are given as
+
+
+
+ At) = A,,(t) + &I& (i =) 1, ...,A?
(19) The relationship between EO and MDTgiven below2s3is derived from eq 19. 60
= [MDd(J(Sh)- cs]/(@,-
+ At) =AD,(t + 0) - AA,i(t)
(i = 1, ..., N) (line 190) (12)
The drug flux during the interval [t, t given as 1594 / Journal of Pharmaceutical Sciences Vol. 83, No. 11, November 1994
+ At], J(t + At/2), is
cs)
(20)
Consistency with the Classic Model for the Drug Release from Suspension or Solution-When the crystal phase in suspension does not dissolve at all or K = 0, the drug is released from the dissolved phase only and the ointment behaves as a solution. Thus, the cumulative drug amount per unit area released from the ointment should be identical to the value predicted by the following equation for a ~ o l u t i o n ~ , ~
(line 190) (11)
AJt
(16)
The initial condition is given as
AA,i(t) = A&,AX)
A,,(t
(14)
J=1
Q = hCs(1 - c0){ 1 -
-c 8 ”
1
2m=0(2m
+ 1)’
O n the o t h e r hand, w h e n K >> kd, the process should b e r~ described b y the classic model for ~ u s p e n s i o n ol ~
Q = [D(2A- Cs*)Cs*t]'"
(t 5 to)
(22)
and
Q = ( 2 A - Cs*)h/2
+ hCs*(l exp[-l(t - t0)])/2 (t > t o )(23)
+
where A = C s ( 1 - E ~ ) toes, Cs* = C s ( 1 - to), t o = ( 2 A Cs*)h2/(4DCsh), and il= W/h2. In eqs 22 and 23, the quantity Cs* but not Cs should be used because in the classic model the solubility i s expressed as the drug amount per total volume (volume of matrix plus the initial volume of the crystal phase).lr7
M acetate buffer (pH 4.5) at a flow rate of 10 mL/h for the first 3 h, then the flow rate was decreased to 1-3 m u so that the receptor concentration was maintained t o be less than 5% of the aqueous saturation concentration of betamethasone 17-valerate ( ~ 5 - 6yg/mL, measured in room temperature). The receptor fluid was collected at 1, 2, 3, 4, 6, 10, 24, 34, 48, 72, 96, and 144 h and the amount of betamethasone in the receptor fluid was measured as previously.11J2 Estimation of Parameters-The saturation concentration of betamethasone 17-valerate in BIOPSA was estimated from the plots of the aqueous concentration versus concentration in BIOPSA at equilibrium. The diffusion coefficient of the dissolved drug, D, was estimated from the plots of the mean ( n = 5) Q versus t for the 1.50 mg/mL patch using eqs 22 and 23 because they did not show any substantial discrepancy from the prediction by the classic m ~ d e l l ? ~ (eqs 22 and 23). Using the value of I) estimated and N = 40, the values of the dissolution rate X in eq 10 for the 3.08 and 5.88 mg/mL patches were estimated from the plots of the mean ( n = 5 or 6 ) Q versus t by a nonlinear least squares methodI3 assuming e8= 1g/mL = 106 yg/mL.
Experimental Section
Results
Partition of Betamethasone 17-Valerate between Silicone Adhesive and Water-Betamethasone 17-valerate was a generous gift from Schering Plough (Bloomfield, NJ). A pressure-sensitive silicone adhesive, BIOPSA, in trichlorotrifluoroethane as solvent carrier was purchased from Dow Corning (Midland, MI). Betamethasone 17-valerate was dissolved in a mixture [50:50 (w/w)l of dichloromethane-BIOPSA in solvent carrier to obtain 10 different concentrations (0.01-10 mg/mL). On the tips of 7 x 60 mm strips of Scotchpak (Type 1009 backing membrane, 3M, St. Paul, MN) was mounted 30 yL of 10 various concentrations of betamethasone 17valerate in the dichloromethane-BIOPSA mixture (n = 4 for each concentration). After dichloromethane and solvent carrier were evaporated, the tip was dipped into 1.5-2.5 mL of 0.1 M acetate buffer (pH 4.5) for 10 days, and then the strip was separated from the acetate buffer, and amounts of betamethasone 17-valerate in buffer and BIOPSA were measured. Acetate buffer (pH 4.5) was used because, unless the aqueous pH is maintained between 4 and 5, betamethasone 17-valerate is rapidly converted to betamethasone 21-valerate and then betamethasone.s-10 To know the amount of betamethasone 17valerate per unit volume of BIOPSA attained after the solvent was evaporated, 10 solutions were also developed manually into an approximately 6 x 10 cm area on the backing membrane by a casting device (2.5 in. in width and 0.010 in. in thickness, designed and made by Stanford Research International (Menlo Park, CA)), and the thickness of this film was measured by light wave micrometer (Van Keuren Co., Watertown, MA). The drug amount per unit volume of BIOPSA was, on average, 3.1 times higher than that in the mixture of dichloromethane-BIOPSA in solvent carrier [50:50 (w/w)l. Betamethasone 17-valerate in aqueous solution was measured by a normal phase high-performance liquid chromatographic method as previously described, where the detection limit approximated 5 ng per sample.11J2 To measure betamethasone 17-valerate in BIOPSA, BIOPSA on the backing membrane was sonicated for 30 s in dichloromethane and the solution directly injected into the chromatograph. The mean (&SD,n = 7) recovery of betamethasone 17-valerate from BIOPSA was 96.0 f 4.3% when the peak height of a known sample (200 yg) was compared to that of the standard solution prepared in 2% methanol in dichloromethane. Release of Betamethasone 17-Valerate from BIOPSASolutions of three concentrations (1.5, 0.75, and 0.375 mg/mL) of betamethasone 17-valerate in the mixture [64:36 (w/w)l of dichloromethane-BIOPSA in solvent carrier were prepared and developed into an approximately 6 x 20 cm area on the backing membrane as earlier. The low adhesion polyester film, Scotchpak (Type 1022 release liner, 3M, St. Paul, MN), was placed on BIOPSA after the solvent was evaporated. The patch thickness was ~ 4 ym 0 and the final drug amounts per unit volume of BIOPSA were estimated as 5.88, 3.08, and 1.50 mg/mL, respectively. After confirming that the thickness of each area used in the experiment was 40 2 ym by light wave micrometer, a round piece (1cm2)was excised from the patch. The round piece was then placed on the flow-through diffusion cell (LG-1084-LPC, Laboratory Glass Apparatus Inc., Berkeley, CAI and the direct release of betamethasone 17-valerate from BIOPSA to the receptor fluid was measured. Into the receptor cell was pumped 0.1
Factors Influencing the Predicted Release Rates (Figure 1)-The Q-& profiles predicted by the model using arbitrary p a r a m e t e r values are shown in Figure la-c. The effects of the change in N ( t h e n u m b e r of sublayers), K (the dissolution rate), a n d 60 ( t h e initial volume fraction measure of the crystal) o n the prediction are illustrated in Figure lac, respectively. In Figure 2a, the Q-& profiles predicted w h e n K = 0 h-l (A and A, corresponding t o e q 21) a n d K = 1000 h-l (0 and 0, corresponding t o e q s 22 and 23) are depicted. F o r either value of K , t o examine the effect of the change in N (number of sublayers) o n the prediction, two calculation results w h e r e N = 10 (A a n d 0 ) a n d N = 40 (A a n d 0) are shown. O t h e r parameters employed are arbitrarily chosen as follows: C s = 1000 pg/mL, to = 0.01, h = 50 pm, eS cm2/h. Thus, kd = D/h2 = = 1 g/mL or lo6 pg/mL, D = 0.04 h-l a n d A in eqs 22 and 2 3 is given as A = Cs(1 - € 0 ) toes = 10.99 mg/mL. The initial a m o u n t per unit area t o be released in e q 23, Ah, is given as Ah = 54.95 pg/cm2. O n the o t h e r hand, the initial a m o u n t per unit area t o be released when K = 0 h-' (A and A, eq 21) is Cs(1- c0)h = 4.95 pg/cm2 (the initial amount of crystal phase, toesh = 50 pg/cm2, is not released at all because K = 0 h-l). When K = 1000 h-l, the values of Q predicted b y the r a n d o m walk method w h e n N = 10 (0)underestimate those yielded b y e q s 22 a n d 2 3 while the difference between the Q-& profiles predicted by eqs 22 a n d 23 a n d those b y the random w a l k method w h e n N = 40 (0) is less prominent. O n the o t h e r h a n d , w h e n K = 0 h-l, the Q-& profiles predicted b y the random walk method w h e n N = 10 (A)and 40 (A) are almost identical t o those predicted b y eq 21. Figure l b shows the effect of the change in the dissolution rate, K , o n the Q-& profiles, w h e r e N = 40 and the s a m e values of Cs, €0, h, es, a n d D as in Figure l a are used for all of the predictions. Drug release is slowed particularly in the early phase with a decrease in the dissolution rate from K = 1000 h-l(O, eqs 22 and 23) t o K = 1h-l (A)a n d 0.1 h-l (W). When K = 0.01 h-l (O), which is 4 times smaller than the diffusion parameter, k d (0.04 h-l), the Q-& profiles are almost identical t o those predicted by the classic model for the solution case6z7 (eq 21) or the case where K = 0 h-l ( A ) until almost all of the dissolved d r u g molecules are released t o the receptor at t x 16 h t& x 4 h). However, thereafter, Q continues t o increase slowly due to the slow dissolution of the crystal phase. Thus, w h e n K < kd, the Q-& profiles h a v e two phases; the first p h a s e where the dissolved d r u g is released rapidly, followed b y the second phase where the crystal phase is slowly dissolved and released.
*
+
Journal of Pharmaceutical Sciences / 1595 Vo/. 83, No. 11, November 1994
360
4
10
5
0 0
2
6
4
a
60
40
20
0
0
5
10
15
0
5
10
\
15
20
25
30
Time, min
Figure 2-The effect of the change in N (number of sublayers) and the method of approximation to (&,dax) (x = 0) (11 = C/Cs) on the ,u-4profiles. See the text for the parameter values.
where the slope of the Q-& profiles is increased with an increase in €0. Comparison with the Ayres and Lindstrom Model (Figure 2)-In order to make the comparison between the method by Lindstrom and Ayres3 with the random walk 0 2 4 6 8 method (Appendix 11, we developed an original algorithm for this purpose as shown in Appendix 2. Lindstrom and Ayres3 1 Time (h) used a second-order approximation to aqGx (7= C/Cs) at the Figure 1-@& profiles predicted by the models. The Q 4profiles by the boundary x = 0, leading to eqs A12 and A14 in Appendix 2. classic model (lines) and the finite dissolution rate model (symbols) are shown. Their results (their Tables I-V, VIII, and are almost The effect of the change in N (number of sublayers) is shown with the prediction identical to the results obtained using the alternative secondby the classic suspension model (upper line) and solution model (lower line) in order algorithm detailed in Appendix 2 (eqs A9-Al2 and A14) part a. The @A profiles with different K values are shown in part b. The with N = 10. This similarity was observed for all values of effect of the change in EO is shown with the prediction by the classic suspension the model parameters which were used. Using a first-order model for 60 = 0.01 (upper line) and €0 = 0.0025 (middle line) and that by the approximation to avlibc (x = 0 )gives the alternative algorithm solution model (lower line) in part c. See the text for the other parameter values. described by eqs A9-Al3 in Appendix 2. The p-& profiles $i is the relative cumulative drug mass released and defined Figure l c shows the effect of the change in EO on the a s p = QMhCs)13obtained using this algorithm with the same Q-& profiles where K and Kc,, are assumed to be constant model parameters and N were very close t o those obtained even when €0 is varied. Thus, K ( K = K K ~ ~changes E ~ ~ are / ~ ) using the random walk method (Appendix 1). Therefore, the proportional to eO2/3. The other parameter values are the same comparison between the random walk method and the method as those in Figure la. In Figure lc, triangles (A and A ) of Lindstrom and Ayres3 reduces to that between the firstdelineate the profiles when K > kd ( K = 0.4 and 1 h-l) and and second-order approximations to (x = 0 ) in the circles ( 0 and 0)denote those when K < k d (K = 0.004 and algorithm detailed in Appendix 2. Generally, the estimate of Q or ,u by the first-order ap0.01 h-l). In all four cases, the Q-& profiles until t x 1 h proximation to aq/& (x = 0 ) is smaller than that by the secondare similar to one another. Thereafter, the Q-& profiles order approximation. However, the difference of Q or p differ substantially between the case where €0 = 0.01 and K between the two approximations is generally small (less than = 1h-l (A)and the case where EO = 0.0025 and K = 1.(0.0025/ 2% of the value) particularly when K x k d or K < kd. When 0.01)2/3= 0.4 h-l (A). However, it is only after almost all of K >> K d , the discrepancy is remarkable between the two the dissolved drug molecules have been released at t x approximations during the initial period. This problem is 16 h (or & x 4 h) that the substantial difference between illuminated in Figure 2, where the effect of using a first- or the Q-./i profiles where €0 = 0.01 and K = 0.01 h-l ( 0 )and second-order approximation t o ay/& (x = 0 ) and the effect of those where €0 = 0.0025 and K = 0.01~(0.0025/0.01~2/3 = 0.004 increasing the number of sublayers are shown for such a case. h-l(O) becomes prominent. In such a case ( K < Ka), the clear Note that in Figure 2, Cs = 100 pg/mL and €0 = 0.4, as assumed by Lindstrom and Ayres (see their Figure 4),3so that difference is observed only in the second phase (& > 4 h), 1596 / Journal of Pharmaceutical Sciences Vol. 83, No. 11, November 1994
U
10
100
1000
10000
100000
C o n c e n t r a t i o n in P a t c h (pg/mL) Figure 3-Plots of aqueous betamethasone 17-valerate concentration vs concentration in the silicone adhesive (BIOPSA) at equilibrium.
the maximum value of ,u is 4000. Therefore, only less than 10% of the initial total amount is released during the time interval shown in Figure 2. In addition, K or Kz is assumed to be >>kd (Kz = 1000 min-l while kd = 0.15 min-l, where Kz is defined as Kz = K Kor~K = ~ K2c02/3 = 540 min-l, Appendix 2). Other parameter values are D = 6.10-* cm2/min,h = cm, Cs = 100 ,ug/cm3, es = 1.0 g/cm3, and At = 0.5 min, as assumed by Lindstrom and A y r e ~ . ~ In Figure 2, any method in Appendix 2 underestimates the p ~ - & profiles (a) ( p= ~a / h , / a , corresponding t o eq 22),3 though the discrepancy decreases as t increases. The p-& profiles obtained using eqs A9-Al2 and A14 (second-order approximation) with N , the number of sublayers, increased from 10 (0)to 40 (A), become very close to the p ~ - & profiles already a t t x 2.5 h (i.e., & x 12 or 13 min). Those obtained using the first-order algorithm (eqs A9-Al3) are closer to the p ~ - & profiles when N is increased from 10 (0)to 40 (A),though not as close as the second-order algorithm (eqs A9-Al2 and A14) with N = 40 (A). Solubility of Betamethasone 17-Valeratein BIOPSA (Figure S)-Figure 3 shows the plots of the aqueous concentration versus patch concentration measured after the patches containing various amounts of betamethasone 17-valerate were dipped in 0.1 M acetate buffer (pH 4.5) for 10 days. The aqueous concentration was essentially unchanged when the drug amount per unit volume of BIOPSA was more than 1000 pg/rnL and the average aqueous concentration was 5.4 ,ug/mL (horizontal line). When the patch concentration was less than 1000,ug/mL, the partition coefficient between patch and water was weakly dependent on the concentration. The patcldwater partition coefficientwas e l 0 0 when the aqueous concentration was around 0.1 ,ug/mL while it was ~ 2 0 when 0 the aqueous concentration was around 5 ,ug/mL. Using a partition coefficient of ~ 2 0 0 the , solubility of betamethasone 17-valerate was estimated as around 1000pg/mL or 1mg/mL. This value of the solubility was compatible with the observation that the patch was translucent when the final concentration in BIOPSA was 5600 ,ug/mL but clouded when 21500 pg/mL. Release of Betamethasone 17-Valeratefrom BIOPSA (Figure 4)-Figure 4 shows the mean Q-& profiles of drug release from the patches with three final concentrations, 1.50 (a),3.08 (A), and 5.88 mg/mL (0). The mean (fSD, n = 5 or 6 ) percentages of the initial drug amount released until 144 h (&= 12 h) from the 1.50,3.08, and 5.88 mg/mL patches to the receptor fluid were 97.5 f2.7,93.5 f 1.1, and 57.8 f 9.7%, respectively, when the initial amount is calculated as that excreted to the receptor fluid until 144 h plus that remaining in the patch a t 144 h. As shown in Figure 4a, the Q-&
dTime (h) Figure 4-Gz/t profiles of betamethasone 17-valerate released from the silicone adhesive (BIOPSA). Lines I, 11, and 111 in part a are the predictions by n = 5), 3.08 (A, n = 6), and 1.50 the classic suspension model for the 5.88 (0, (0,n = 5) mg/rnL patches, respectively. Lines I and II in part b are the predictions n = 5) and 3.08 (A, n = 6) by the finite dissolution rate model for the 5.88 (0, mg/mL patches, respectively.
profiles from the 1.50 mg/mL patch (where cOes = 0.58 mg/ mL) predicted by eqs 22 and 23 were not substantially different from those observed (U),and the value of D was estimated as D = 2.79.10-7 cm2h when eqs 22 and 23 were fitted to the average Q-& profiles from the 1.50 mg/mL patch. However, the model (eqs 22 and 23) overestimated the observed Q-& profiles when this D value was assumed (Figure 4a). The magnitude of the discrepancy between the model prediction and observation was lessened when the model supposing the finite dissolution rate was used (Figure 4b). When the model was fitted to the observed average Q-& profiles assuming D = 2.79-1Op7cm2h, K was estimated as K = 0.145 h-l for the 3.08 mg/mL patch (n = 6)and K = 0.084 h-l for the 5.88 mg/mL patch (n = 5).
Discussion Random Walk Method and the Ayres and Lindstrom Model-One of the advantages provided by the use of the random walk method is the simplicity of programming (Appendix 1). Being different from the method by Burnette14 where a random value is produced in each iteration, for each of the 100 000 molecules, the computing time of the random walk method shown in this paper is comparable to that of the conventional finite difference method in Appendix 2. Another advantage is that mass is strictly conserved in the random walk method, where the amount of each sublayer is considered rather than the concentration in each subinterval. The principle of mass conservation is not always satisfied in the method by Ayres and Lindstrom, particularly when N is not large and the second-order approximation to a?,d& (x = 0 ) is used (eq A14). For instance, In Tables I-V, VIII, and M in the paper by Lindstrom and Ayes: ,u approaches 1.03-1.11 as time elapses when K = 0 while this value should be exactly 1.00. Generally, the random walk method (Appendix 1)produces the prediction which approximates that by the method of Journal of Pharmaceutical Sciences / 1597 Vol. 83, No. 11, November 1994
Lindstrom and A y r e ~ .This ~ is particularly true when K < = 0, both methods give Q-& or p-& profiles sufficiently close to those produced by eq 21 even if N is small (e.g., N = 10, Figure la). However, when K >> kd, the discrepancy predominates between these two methods during the initial phase. It is also in this particular situation that the prediction by either method substantially deviates from that by the classic suspension model (eqs 22 and 23). In order to fully elucidate this complicated matter, we developed the original algorithm shown in Appendix 2 where the first-order approximation to &,hk(x = 0)( r = C/Cs, eq A14) yields the p-& profiles = &/(hCs))by the random walk method (Appendix 1A) while the second-order approximation (eq A15) yields those by the method of Lindstrom and Ayres.3 The results (Figures la and 2) indicate that when K >> kd (i.e., the dissolution of the crystal phase is much faster than the diffusion of the dissolved drug through the matrix), any method underestimates p or Q predicted by eq 22 during the initial phase unless N is sufficiently large. With larger N , the agreement with the classic suspension model (eqs 22 and 23) can be obtained earlier. When N is fixed, the secondorder approximation and the method of Lindstrom and Ayres3 give values closer to that predicted by the classic model (eqs 22 and 23) than the first-order approximation and random walk method (Figure 2). Factors Determining the Release of Betamethasone 17-Valeratefrom BIOPSA-Even if the drug release from the 1.50 mg/mL patch was explained by the classic model for suspension^',^ (eqs 22 and 23, Figure 4a), this model failed to explain the drug release from either of the 3.08 and 5.88 mg/ mL patches, which was slower than the prediction by eqs 22 and 23. The discrepancy between the model prediction and the observation was lessened when the finite dissolution rate model was used (Figure 4b), though the model did not give a perfect explanation because the drug release in the early period was faster than the prediction by the model in either of the 3.08 and 5.88 mg/mL patches (Figure 4b). One possible explanation for the discrepancy between the model prediction and the observation was that the crystal phase was not uniform and comprised more than two phases with different dissolution rates. In order to come to an unambiguous conclusion, however, more experiments are needed, including the examination of the distribution of the particle size in the patch by microscopic examination, as well as the characterization of the crystals with other physical techniques such as spectroscopy, X-ray diffraction, and differential scanning calorimetry.
& or K = kd. For instance, when K
References and Notes 1. Higuchi, T. J. Pharm. Sci. 1961, 50, 874-875. 2. Ayres, J. W.; Lindstrom, F.T. J. Pharm. Sci. 1977, 66, 654662. 3. Lindstrom, F.T.; Ayres, J. W. J. Pharm. Sci. 1977, 66, 662668. 4. Kubota, K.; Koyama, E.; Yasuda, K. J. Pharm. Sci. 1991, 80, 752-756. 5. Kubota. K.: Maibach. H. I. In In Vitro Percutaneous Absorotion: P&czples, Fundamentals and Applieations, Bronaugh, R. L.; Maibach, H. I. Eds.; CRC Press: Boca Raton, FL, 1991; pp 243-264. 6. Higuchi, W. I. J. Pharm. Sei. 1962,51, 802-804. 7. Kubota, K.; Yamada, T.; Ogura, A.; Ishizaki, T. J. Pharm. Sci. 1990, 79, 179-184. 8. Yip, Y. W.; Li, W. P. A. J. Pharm. Pharmacol. 1979, 31, 400402. 9. Cheung, Y. W.; Li, W. P. A,; Irwin, W. J. Znt. J . Pharm. 1985, 26, 175-189. 10. Smith, E. W.; Haigh, J. M.; Kanfer, I. Znt. J. Pharm. 1985,27, 185-192.
1598 / Journal of Pharmaceutical Sciences Vol. 83, No. 11, November 1994
11. Kubota, K.; Sznitowska, M.; Maibach, H. I. J. Pharm. Sci. 1993, 450, 450-456. 12. Kubota, K.; Sznitowska, M.; Maibach, H. I. Int. J.Pharm. 1993. 96, 105-110. 13. Yamaoka, K.; Tanigawara, Y.; Nakagawa, T.; Uno, T. J. Pharmacobio-Dyn. 1981,4, 879-885. 14. Burnette, R. R. Znt. J . Pharm. 1984,22,89-97. 15. Twizell, E. H. Numerical Methods, with Applications in the Biomedical Sciences; Ellis Horwood: Chichester, 1988; pp 228278.
Appendix 1-BASIC Program of the Random Walk Method for the Finite Dissolution Rate Model 10 REM RS, CS(MICROC/ML) H,DX(CM) D(CMWH) K,KD(H"- I ) DT(H) 20 REM AD,AC,BD, ADZ, ACZ, Q (MICROG/CMA2)J(MICROG/CMA2/H) 30 N=40K=1: D=lOA(-6):RS=10"6 CS=lOOo: EZ=O.l: H=0.005 40 KD=D/H/H: DT=l/uKDR\I/N: DX=H/N:ADZ=CS*( 1 -EZ)*DX:ACZ=RS*EZ*DX 50 DIM AD(N),AC(N),BD(N) 60 FOR I=1 TO N: AD(I)=ADZ AC(I)=ACZ: NEXT I 70 Q=O:T=O NEXT I 80 AD(O)=O: AC(O)=O:FORI=1 TON: AD(O)=AD(O)+AD(I):AC(O)=AC(O)+AC(I): YO PRINT "T=";T, "Q=";Q,"AD=";AD(O),"AC=";AC(O):PRINT"T=";T-DT/2, "J="; J 100 T=T+DTBD( I )=AD(1)/2+AD(2)/2:BD(N)=AD(N-l)/2 llOFOR 1=2TON-1: BD(I)=AD(I-l)/2+AD(I+1)/2: NEXT1 120 J=AD(N)/2/DT:Q=Q+AD(N)/2 130 FOR I=1 TO N 140 DAC = - WKD*(AC(I)/EZ/RS/I)X)"(2/3)*(CS*DX-BDfI)/(l-EZ))/2/N/N 150 IF DAC>O THEN DAC=O 160 IF DAC+AC(I)
Appendix 2-Comparison of the Random Walk Method (Appendix 1) with the Method of Lindstrom and Ayres3 Consider the nonlinear parabolic, integrodifferential,initial! boundary value problem for mass transport of a drug in an ointment-perfect sink system, consisting of the model equation2s3
in which 17 = v(x,t), for all (x,t) in the region R = {(-h 0) x ( t > O)}, the initial distribution
rk,O)= 1; -h
5
x
50
(A2)
and the boundary conditions
aq(-h ,t) -ax - 0 ; t ' O q(O,t) = 0; t > 0
(A3) (A41
In eq Al, q(x,t)= C(x,t)/Cswhere C(r,t) is the solution phase drug concentration distribution. The dissolution rate constant Kz employed in the paper by Lindstrom and Ayres3is defined ~ . that the constant K is defined as K = as Kz = K K ~(Note KK~,.+ in~the ~ ~text ~ and in another paper by Ayres and Lindstrom2 so that K = KZco2/3).Other parameters, €0, eS,D , and h are defined as in the text.
Numerically, the problem (eqs Al-A4) may be solved using finite-difference techniques by discretizing the interval -h I x 5 0 into N subintervals each of width Ax so that NAx = h (eq l ) , and by discretizing the time interval t 2 0 into steps each of length At. The open region R = {(x,t): -h < x < 0 , t > 01 and its boundary aR consisting of the lines x = -h, x = 0, and t = 0 are thus covered by a rectangular mesh, the mesh mAx ( m points having coordinates (xm,tn),where xm = -h = 0, 1, ..., N) and t , = nAt ( n = 0, 1, 2, ...). The solution a t the mesh point (x,,tn) of an approximating finite difference scheme for eqs Al-A4 is denoted by &. The numerical method to be used is developed by approximating the time derivative in eq A1 by the first-order forward difference approximant
Substituting eqs A5 and A6 with 0 = 1into eq A1 leads to a finite difference scheme approximating the solution to eqs Al-A4. This scheme is given by
+
+
av(x,t)/at= A t - l [ ~ ( ~ , A t t ) - v(x,t)l+ O(At1 (A51 and the space derivative by the weighted second-order approximant
+ A t ) - ~ v ( xt,+ A t ) + + h7t + ~ t )+) (1- e){v(x- hx,t) - 2q(x,t)+ V(X +
a2v(x,t)/&x2= A C 2 [ 8 { v ( x- hx, t
v(x
Ax, t)}I
where p = DAtlAx2. It is clear from eq A4 that
vk = 0 for n
= 1, 2,
....
Then,
It is also clear that eq A9 needs modification for use with m = 0, for then the mesh point ( - h - Ax, t n ) ,which is outside R U aR, is needed. This difficulty is circumvented by noting that, to second order, from eq A3, rfl = vy for n = 0, 1, .... This gives
+ 0 ( h 2 )(A6)
in which x = x, ( m = 0, 1, ..., N - 11, f = t, ( n = 0, 1,2, ...), and 8 ( 0 I 8 5 1)is a parameter. Hereafter, 0 is fixed as 0 = 1 (giving a fully implicit methodP5 as in the paper by Lindstrom and Other methods (e.g., 8 = 0 and l/2) and the features unique to each method are detailed elsewhere.15 The factor (1 - r ( x , t ) ) is evaluated at the point (xm,tn+l), giving 1 - &. Approximating the integral
S, = f"[l 0 - v(x,,t)]dz
(A71
The solution of $+'(i = 0,1, ..., N - 1)may be determined by solving eqs Ag-All, a linear algebraic system of order N (but not N - l).3 Finally, the relative cumulative mass release ,u ( = Q / ( ~ C S ) ) ~ , ~ at time pnfl, is computed for n = 0, 1, 2, ... from the formula
where, t o first order,
in eq A1 by the trapezoidal rule, it may be shown that or, t o second order,3
and it is convenient to define
Acknowledgments in which m = 0, 1, ..., N and n = 0, 1, 2, = ~ 0 for % ~m = 0, 1, ..., N . that
....
Note, J", = 0 so
We thank Drs. David Friend and Harold W. Nolen, 111, of Stanford Research International (SRI) Menlo Park, CA, for their help in developing the patch containing betamethasone 17-valerate.
Journal of Pharmaceutical Sciences / 1599 Vol. 83, No. 1 1, November 1994