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Simplex search in optimization of capsule formulation
Simplex Search in Optimization of Capsule Formulation EFRAIM SHEKs, MAHMOOD GHANI, and RICHARD E. JONES Received May 15,1978. from the Institute ofPhariaceutical Sciences, Syntex Research, Stanford Industrial Park, Palo Alto, CA Accepted for publication April 16.1980.
-
%tract The simplex method of optimization was applied to a capsule formulation using the dissolution rate and compaction rate as the desired responses to be optimized. The formulation parameters investigated included the levels of drug, disintegrant, lubricant, and fill weight. Following the successful optimization of the capsule formulation, the acm u l a t e d data were fitted to a polynomial regression model to plot response surface maps around the optimum. Keyphrases 0 Simplex method-optimization of capsule formulation 0 Dosage forms-capsule formulations, optimization by simplex method
In addition to the active material, pharmaceutical products contain several ingredients specially selected and present in definite concentrations. These ingredients or excipients, which Schroeter (1) named Ingredient X, are selected to make the most effective product from the drug. The substance and the excipients compose the dosage form. Development of dosage forms is essentially an optimization process. Certain goals ordinarily are defined for a dosage form, i.e., physical and chemical stability, physiological availability, ease of manufacture, and cost. Once a drug is chosen, the formulator must produce the best dosage form according to the criteria selected. Among the factors involved is the mixture of excipientsto be used, The desired properties can be termed the dependent variables, and the excipients are the independent variables. The optimization process is not always straightforward; one objective often must be sacrificed to improve another, or a balance must be achieved between conflicting aims. For example, adding additional lubricant to a capsule or tablet formulation may improve its ease of manufacturing but may adversely affect dissolution or bioavailability. Another problem is possible complex interactions among the excipients that affect desired properties. BACKGROUND Solution of such formulation optimization problems can be done by trial and error techniques. but this approach suffers from several draw backs. This approach is not only time consuming, unreliable, costly, and often u n s u d u l , but it also may provide only a provisionaly acceptable mlution rather than an optimum mlution. Reports of formulation optimization by mathematical optimization methodologies have been published recently. Bohidar et al. (2) used principal component analysis to select pharmaceutical formulations. Optimization studies by mathematical models were reported for enteric film coating of tablets (3), encapsulation of powders in hard gelatin capsules (4.61, and tablet formulation (5). Schwartz et al. (6,7) utilized factorial design and computer technique8 to optimize formulation for a tablet. Evolutionary operation (81, which is a statistical method for evaluating the process on a gradual basis, was used to optimize tablet manufacture (9) and particulate matter detection equipment (10). The simplex method is another mathematical optimization method that is an empirical approach. The sequential simplex technique was developed by Spendley et aL (11). Thi ingenious method for optimizing operating conditions the results of previous experiments in a mathematically rigorous fashion to define the parameters (i.e., values of the independent variables) for the next experiment in the search for optimum response. The simplex method is a stepwise optimization pro-
0022-3549/8L1/ 1000-1135$01.00/0 @ 1980,American PlvYmaceutlcalA.%W&tb
Scheme Z
XI
cedure that merely uses the response surface. Nelder and Mead (12) subsequently modified the original algorithm to improve the convergence speed of the system. The method has been used in chemistry for diverse applications. Long (13) applied simplex strategy to analytical chemistry problems. S i p l e x has been used in developing analytical methods for the determination of formaldehyde (14, 15) and cholesterol (16, 17). King et al. (18) discussed the difficulties in the application of simplex to analytical chemistry. Simplex optimization also has been employed for pattern recognition (19,20), for improving reaction yields (21). and for drug design (22, 23).
Two cases of utilization of simplex in pharmaceutical systems were reported recently. Zubkova et al. (24) reported optimiization of proroxan tablets with respect to active drug, lactose, talc, calcium stearate, starch, and compression pressure. Beyer (25) concluded that a simplex method cannot be used in all cases of tablet formulation because tablet properties do not always change with manufacturing conditionsin a predictable way. However, Beyer (25)did not present enough details to enable evaluation of thii conclusion. The objective of the present study was to evaluate the potential of the flexible simplex method, as modified by Nelder and Mead (12), for optimizing a capsule formulation. The capsule formulation was required at very early etages of drug development. At such times, enough raw material for evaluation of a formulation under production conditions usually is unavailable. However, the rate of compaction1 was reported (26) to be related to powder flowability. which is potentially indicative of manufacturing feasibilityon high-speed encapsulating machinery. The amount of material to measure this property is quite small. Thus, in the present study, optimization of the capsule formulation of a drug with respect to its dissolution and rate of compaction was desired. THEORY The simplex method referred to here was developed by Nelder and Mead (12) from the original approach of Spendley et a!. (11). The simplex procedure derives its name from the geometric figure that is moved along the response surface in search of the optimum. This optimization method approaches the optimum in stepwise fashion by moving away from low values of the response function rather than by moving in a line toward the maximum. The logical and mathematical details of the simplex have been discussed extensively (11-13, 16-18.27-29). Thus, only a brief general outline of simplex procedures is given here. The conceptual basis to the simplex approach is most easily comprehended in the two-dimensional (Le., two independent variables) case. In Scheme I, the points refer to several experiments run with independent variables X I and x2. The three experiments, W,B, and S, define the simplex, which in this case is a triangle. To obtain x 1and x g for the next experiment in the optimization search, one move8 away from the point that gave the least favorable response. For example, if W gives the worst response, the next point ( R ) is found by reflecting across the B-S exis, as shown in Scheme 11. The experiment at R is performed and the result is compared to the previous results from B and S.A decision now is made whether to retain the new experiment, R,as a replacement for W.ER is not accepted. then
* In measuring the rate of cornpaction (or of packing down),the change in volume of a quantity of powder caused by constant tapping with time is observed (26).
Journal of f'harmaceutical Sciences I 1135 Vol. 69.No. 10,October 1980
other levels of motion, such a~ expansion and contraction, are tried. These rules for simplex manipulation were outlined by Nelder and Mead (12). With a new simplex now defined, the procedure is repeated to obtain the next step on the optimization path. The process continues until a desired optimum is reached. The simplex procedure thus can be likened to a search for a mountain summit on which one is blindfolded, feeling one's way a step at a time. On the other hand, the typical factorial design plus response surface analysis could be likened to developing a contour map of the mountain first and thereby determining where the summit lies. More generally, the simplex is n dimensional, corresponding to n independent variables. The simplex manipulations are performed by rather simple vector algebra and are easily carried out manually or with a small calculator. As indicated, a given simplex is constructed of n 1vertexes represented as n-dimensional vectors W,S, . B. These vertexes are ranked in order of their responses: Rw (worst), Rs, . . . Re (best). The centroid is (Eq. 1) while W is the eliminated vertex. Now the response RR at R (Eq. 2) is evaluated: 1 B = -(S... B) (Eq. 1) n
..
I " Scheme I I
capsule was dropped in the vessel using a stainless steel coil around it. The stirring speed initially was 50 rpm and was increased after 30 min to 200 rpm for 10min, at which time the fmal concentration was recorded. The amounts of drug dissolved at 8 and 30 min were calculated as a fraction of the total drug dissolved. Normalization of Data-The factors in this study had different units and regions of interest. In such cases (171, it is useful to normalize the factors and to extend their domains from 0 to 1009'0. This process was performed using:
+
+ R = P + (P- W)
*-
x 100 (Eq. 6) H-M where N is the normalized factor, X is the original unnormalized value, and M and H are the lowest and highest values, respectively, in the region of interest for the specific factor. Both the independent and dependent (responses) variables were nonnalized using Eq.6 according to the boundaries shown in Table L For the rate of packing down, larger numbers of taps or slower packing down resulted in low values of response. This effect required modification of Eq. 6 to:
N=-
(reflection) (Es.2) If Rs I RR I Re, then R is retained. If RR > RB,RE at E (%.3) is evaluated. If RE > RR. E is retained. However, if RE 5 RR, then R is retained:
-
E = P + 2 8 W) (expansion) (Eq. 3) If Rw IRR < Rs,Rc, at Cr (&.4) is evaluated and retained. Now, if RR < Rw, then Rc, is evaluated and retained at C, (Eq. 5):
-
(Eq.7)
Cr = P + W) (contraction) (Es.4 ) C , = P - 1/@ W) (contraction) (Eq. 5) If Rc, or Rc. < Rs,then the following is being set: Rw = Rs, W = 5, and Rs = Rc, or Rc, and S = C. or C , . Additional rulesand procedureswhich
where R1and Y1 are the normalized and observed responses, respectively. Boundary values were chosen arbitrarily as reasonable limits on formulation variables and desired responses, To get a single value for the overall response for a given formulation to allow comparison with other formulations in the simplex, an arbitrary procedure was used. The total desired response was obtained by a weighted linear combination of the three observed responses according to:
were applied in this study are explained under Results.
EXPERIMENTAL
Formulations-Cornstarch USP2, stearic acid3, lactose USP4, and the drug substance 5-(2-thenoyl)-1,2-dihydro-3H-pyrrole-[l,2-a]-pyrRt 0.5Ri O A R 2 t OARS (Eq.8 ) role-1-carboxylicacid6 (I) were mixed thoroughly for 30 min. The rate where Rt is the symbol for the total desired response. of packing down of the formulationsand their dwlution rate from a hard Equation 8 shows that the rate of packing down was considered to be
+
gelatin capsule*were determined. Rate of Packing Down Measurement-A 10-mlgraduated cylinder containing -10 ml of loose-packed mixture was mechanically tapped. The mechanicaldevice consisted of a platform supported by a cam, which was rotated by a small motor. During each rotation, the platform was raised gradually and then allowed to drop 0.5 cm. The frequency of the tapping was 50 dropdmin. The initial volume was observed, and volume readings were taken at timed intervals until no further changes were observed. The rate of packing down was calculated by multiplying the ratio of the initialvolume to the f i i volume by the number of tap it took to reach the final volume. Dissolution Measurement-The dissolution vessel was a 1-liter round-bottom flask'. The stirring shaft was equipped with a polyethylene propellel.8, which was positioned 2.5 cm above the lowest inner surface ,of the vessel The shaft was stirred by a multiple-spindlestirring systemg. A peristaltic pumplo transported the solution continuously from the vessel through a variable-wavelength detector" adjusted at an appropriate wavelength. The dissolution process was registered by a chart recorder. Five hundred milliliters of 0.05 M pH 5.0 acetate buffer ( p = 0.5 M NaCl), which was used as the dissolution medium, was kept at 37O.The
I
of 50%importance as a measure of formulation success, and dissolution at 30 min was considered to be of 41% importance. The third response, dissolution at 8 min, was much less important than the other two. If any response fell outside of the limits in Table I, the total desired response was set at zero. Simplex Optimization-Initial experiments ( i e . , the initial simplex) were defined by the procedure of Spendley et af. (11) and Himmelblau (29).The first vertex was the experimental origin and corresponded to the lowest factor values in the region of interest. The remaining four vertexes in the initial simplex were obtained by multiplying the appropriate fraction of each factor according to the procedure mentioned earlier (11,29)and by the initial step size. The initial step size (28), which is the amount of change in a factor that takes place when a new vertex for the initial simplex is calculated, determines the size of this simplex. Yarbro and Deming (28) indicated that a large initial simplex size is useful in furnishing information about a wide range of factor levels. This suggestion was employed in the present work Table I-Factor
Boundaries Used to Normalize Their Levels Value(M) Low
Value(H) High Symbol (N)
Factor (X) 2 A. E. Staley ManufacturingCo., Decatur, IL 62525. 8 Emery Industries, Cincinnati, OH 45232.
San Francisco, CA 94104. from The Institute of Organic Chemistry, Syntex Research, Palo Alto, CA 94304. e No. 1, Elaneo Products Co., Indianapolis,IN 46206. 7 Kimax 33710-S1. 8 Sar ent S-76680. e MAel72-R-115, Hansen Research, Northridge, CA 91328. 10 Model 1210, Harvard Apparatus Inc., Millis, M A 02054. 11 Model 770, Schoeffel Corp., Weatwood, NJ 07675. 4 Foremost. 6 Obtained
1136 I Journal of Pharmaceutical Sciences Vol. 69. No. 10. October 1980
Independent variables Drug (I), % w/w Disintegrant (cornstarch), 9i W/W Lubricant (steari? acid), % w/w Total capsule weight, mg De endent variables el, rate of packing down, taps Y2, dissolution a t 30 min, % Y3, dissolution a t 8 min, %
__ 1.0
41.0
0.0 0.2 100.0
50.0 2.2 400.0
300
500 100 75
50 25
A
D L W Rl
R2 R1
Table 11-Results of the Simplex Search. Vertex
SimDlex
Vertexes Retained
1 2 3 4 5 6 7 8 9 10
1'
1 1 1 1 1 2 2 3 3 4 4 5 5 6 6 6
5' 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 23'
11 12 13 14 15
34 35 36 37 38 39 40 41 42 43 44 45
Generated bv
Factor Level, Normalized, % A
D
L
W
1,39495 193,495 194,597 1,4,5,7 1,4,5,9 1,4,5,9 1,4,5, 11 1,4,5,11 1,5,11,13 1,5,11,13 1,5,11,13
0.00 92.56 21.85 21.85 21.85 -59.78 54.48 27.25 23.20 -21.02 35.60 16.46 21.52 17.63 20.80 0.00
0.00 21.85 92.56 21.85 21.85 46.29 27.96 -56.72 55.24 21.52 26.35 -20.22 36.38 20.45 21.50 0.00
0.00 21.85 21.85 92.56 21.85 46.29 27.96 49.33 28.72 43.60 31.87 44.42 32.65 -49.41 57.09 0.00
0.00 21.85 21.85 21.85 92.56 46.29 27.96 49.33 28.72 43.60 31.87 44.42 32.65 21.85 30.56 0.00
6
1,5,11,13
21.85
21.85
21.85
92.56
7 7 8
1,5,11,15 1,5,11,15 1,5,15,17 1,5,15,17 1,5,17,19 1,5, 17,19 1,5,19,21 1,5,21,22 1, 21,22,23 1,21,22,23 1,22,23,25 1,22,23,25 1,22,23,27 1,22,23,27 1,22, 23,29 1,22,23,29 1,23,29,31 1,23,29,31 1,23,29,31
17.60 20.54 -4.00 25.70 13.24 18.91 12.70 1.04 -5.53 15.00 -4.53 13.05 -1.60 10.85 -0.75 9.60 -1.95 9.03 1.04
-1.52 26.91 8.82 21.95 13.86 19.59 4.79 1.17 -9.07 14.12 -9.55 12.31 -4.98 9.35 -4.65 8.07 4.51 4.72 1.17
22.75 30.18 22.69 29.58 -16.29 38.75 14.92 8.18 9.07 18.66 -18.19 24.52 5.16 15.29 5.32 17.06 5.34 12.52 8.18
44.87 35.71 47.55 35.79 51.48 30.56 43.75 47.65 -31.58 61.53 30.56 34.40 1.37 46.49 34.40 34.50 20.57 37.95 47.65
1,23,29,33 1,23, 29,33 1,23,29,35 1,23,29,35 1,23,35,36 1,23,35,36 1,23,36,39 1, 23,36,39 1,23,36,41 1.23,36,41 1.36.41. ~,- ., .-,4.1 -1,36,41,43
0.85 7.40 0.61 -14.92 -6.33 6.56 -3.30 4.73 -3.36 4.08 3.68 1.70
-0.45 5.94 3.52 -10.73 -4.03 6.00 -0.60 4.31 -1.50 4.13 4.81 2.08
0.94 13.03 5.73 -10.36 -1.81 11.02 -0.57 9.63 0.76 8.46 3.73 7.08
31.54 33.76 26.00 -12.43 7.21 36.67 21.40 30.67 15.49 31.38 -3.65 34.83
8
9 9 10 11
12 12 13 13 14 14 15 15 16 16 16 17 17 18 18 19 19 20 20 21 ~~
21
Percent Desired Response, Normalized R1 R9 R.1 Rt 91.0
88.0 -231.0 -133.4 23.0 76.0
-29.0 27.0 33.0
12.6 13.8
3.8 -2.8
-
27.4 -
20.0 31.4 89.0 88.4 76.0 76.0
-
69.0
72.0 76.0 88.0 95.0
-
0.0
-
32.7 25.2 -
60.0 -8.0 -29.0 20.0 32.0
73.0 -5.9 7.0 -
33.7 90.7 91.0 1.5 17.2
22.0 30.0 29.0 45.0 3.0 18.0
4.2 26.7 35.7 66.5
14.0 16.0 18.0
3.0 -
6.0 5.0 -
-
24.0 7.0 57.0 6.0 -
31.0 67.7 67.0
18.0 43.0 31.0
-
40.0 78.4
25.5
100.0 100.0 93.0 94.0
22.4
-
-
100.0 100.0
41.2 82.2
37.0 52.0
86.0 -
-
-
99.0 100.0 100.0 100.0
80.0 44.7
47.7 68.5 83.5 75.0
-
34.0 59.0 76.0
-
45.0
86.4 O.Ob O.Ob
24.3 54.4 -1000.0' 15.1 -1000.0' O.Ob
-1000.0'pd 27.5 -1Ooo.oe 22.3 -1Ooo.O' 32.2 83.7* 85.1 38.gC 46.7 -1Ooo.O' 36.3 -1Ooo.O' 38.2 -1000.OC 50.1 59.9 75.9 -1000.0' 32.6 -1Ooo.oc.d
43.1 -1Ooo.OC 66.6 -1000.0' 61.8 -1Ooo.O' 64.2 77.94 76.9 -1000.oc 70.2 88.1 -1000.0' -1000.0' 67.0 -1Ooo.OC.d
83.3 -1Ooo.O' 91.0 -1Ooo.oc 84.5
a Key: R, reflection(Q.2); c,, contraction (Q.5); E , expansion (Jh. 3); R I ,normalized rate of packing down; Rs,normalized dissolution at 30 min; and Rs,normalized dissolutionat 8 min. * Boundary violation on one or more of the responses Boundary violation on one or more of the facto?. Second worst vertex is rejected. Response checked after vertex occurred in five successive simplices. The second line gives t h e average responseof the two observations.
by using 100 units of the normalized factor level as the initial step size. RESULTS Simplex Optimization-The formulations for the capsule were composed of four ingredients. The active ingredient (A) was the drug substance, I. Cornstarch and stearic acid were the disintegrant ( D )a d the lubricant (L),respectively, while the filler was lactose. Since the formulation expressed as percent by weight must add up to I@)%,only three of the four parameters are required to define the formulation and, therefore, are independent variables. In this study, the drug, cornstarch, and stearic acid were the three independent variables'2. Previous experimehts showed that the degree of fill of the capsule significantly affected the dissolution properties of the formulated capsules. Accordingly, the amount of the powder mixture fiied into the capsule ( W) was a fourth factor. Table I outlines the boundaries used for the various factors in this study. 12 Any three of the four parameters are equally appropriate as independent variables; the particular choice here waa for convenience.
The responses of the various experiments obtained by the simplex method on the four-factor response surface of the formulation are presented in Table 11. Initial experiments (i.e., the initial simplex) were defined by the procedure of Spendley et al. (11) and Himmelblau (29) and were discussed under Experimental. The remaining vertexes in Table I1 were generated by the simplex moves according to the appropriate rules. An example of a numerical step-by-step simplex calculation is outlined in the Appendix. During the optimum search, vertexes 1,5, and 23 occurred in five (n 1) successive simplices. According to the recommendation of Spendley et 01. (11). the responses at these threevertexes were checked (vertexes l', 5', and 23') and replaced by an average of the two results (Table 11). Vertexes 6,8,10,12,14,16.18,20,24,26,28,30,32,34,37,38,40,42, and 44 violated a boundary condition on one or more of the experimental factors. An arbitrarily very low response (-1Ooo) was assigned to these vertexes to constrain the factors inside the region of interest by means of a contraction (Cw). In the initial simplex, two vertexes (2 and 3) violated the boundaries with respect to various responses, and a total desired response of zero was arbitrarily assigned to them. However, to determine which of the two vertexes gave the worse response (and, therefore, which vertex to reject), total desired responses were calculated according to Q.8. Based on this
+
Journal of Pharmaceutical Sciences I 1137 Vol. 69, No. 10. October 1980
'
80 60 40
20
80t
A
L
\
s
2 80-
~
treatment, vertex 2 was the worst point in the initial simplex and waa rejected. A similar boundary violation of the dissolution response occurred in vertex 9. In two simplices (12and 19).the newly generated vertex was the worst one and was supposed to be the next one rejected. However, rejection of Table I I l - S t a n d a r d Deviation of Responses and Relative Simplex Size
SD,% of total desired response Relative Size R, R2 R3 Rt ofSimDiex
1 2 3 4
5
6 7 8 9 LO ii 12 13 14 15 16 17 18 19 20 21 22 Experimental error
I
140.5 88.3 36.3 33.8 32.4 31.1 27.3 21.2 7.4 7.6 8.4 22.1 22.0 5.9 5.8 6.4 6.4 6.4 5.6 5.6 5.2 5.1 6.3
44.6 42.8 35.0 34.0 27.7 29.2 33.4 36.3 36.3 33.3 30.5 28.6 33.2 31.3 25.5 26.5 24.0 22.1 21.4 16.5 10.3 8.6 10.0
'34.6 32.3 24.2 25.2 19.9 14.1 16.3 16.8 16.0 16.0 i2.2 12.4 14.1 21.1 21.2 20.1 14.3 10.2 8.2 11.4 16.5 13.2 15.0 ~
1138 t Journal of Pharmaceutical Sciences Vol. 69, No. 10, October lQ80
A
8olh
DRUO SUBSTANCE I0
60
SIMPLEX NUMBER
Figure 1-Response improvement during simplex search. The normalized response of t h most recent uerter is plotted against the U P propriate simplex number. Key: Rl,rate ofpackihg down reswFe; Rz, dissolution response at 30 min; Ra, dissolution response at 8 m n ;and Rt, total desired response.
Simoiex r--
\
\
88.2 63.5 33.1 3 _1_.5 27.5 25.4 23.2 21.6 22.0 18.0 16.5 20.8 20.3 16.0 10.5 9.8 8.5 9.2 9.1 8.3 5.3 3.1 6.2
1.00 0.88 0.77 0.72
0.67 0.57 0.55 0.53 '0.48 0.48 0.48 0.39 0.33 0.29 0.26 0.25 0.24 0.24 0.21 0.19 0.18 0.15
I (
1 1
1 1 t i I I I 1 1 1 1 1 1 I
1
3
6
7
9
1 1 1
I
11 13 16 17 19 21
SIMPLEX NUMBER
Figure 2-Chapges in factor levekr during simplex search. The normalized feuel of the factor in the most recent vertex is plotted against the appropriate simplex number. this new vertex would stalemate the search by regeneration of the previous simplex after the prescribed reflection operation. Conceptually, this situation is the same as the two simplices straddling a ridge on the response surface. Instead, translation of the simplex by eliminating the second poorest vertex allowed continuous movement of the simpler. As was pointed out by Long (13),this kind of translation usually will even promote movement of the simplex up a ridge that is being straddled. Figure 1shows the normalized responsesfor dissdlution, rate of packing down, and totaldesired response 89 functions of the simplex number. The response of the most recent vertex is plotted against the appropriate simplex number. The branches represent vertexes of failed steps, while the continuous lines describe the movement of the simplex toward the optimum. The alterations of the responses for the first five vertexes were not generated by the progress of the simplex but were originated by the order in which the initial simplex was constructed. The standard deviations of the responses for the varioue completed simplices are outlined in Table 111. The relative simpiex size (defined as the sum of the standard deviations of the factors in the simplex as a fraction of that of the initial simplex) also is shown in Table 111. As the simpiex progressed and approached the optimum, there was a leveling off in each response value (Fig. 1). with a concomitant decrease In standard deviations approaching the experimental error. At the same time, a substantial decrease in the simplex size could be observed. The search WM halted after 45 vertexes and 22 completed simplices. The response finally achieved was considered to be satisfactory. Figure 2 shows the variations in the normalized levels of the four fac-
Table IV-Results
of the Last Simplex (Simplex 22)
1
Normalized Values, % Total Desired Response ~
Vertex 1 36 41 43 45
Mean* SD Unnormalized mean
SD
Units
8
L
W
0 3.52 4.31 4.13 2.08 2.81 1.80 1.41
0 5.73 9.63 8.46 7.08 6.18 3.75 0.324
0 26.0 30.67 31.38 34.83 24.58 14.09 173.7
0.9
0.075
42.2
%wlw
mg
A
D
0 0.61 4.73 4.08 1.70 2.22 2.09 1.89 0.84
%wlw %wlw
~~~~
85.1 88.1 83.3 91.0 84.5
-
-
-
Centroid of this simplex.
tors. Again, the continuous line describes the progress of the simplex, while the branches display the failed step. As the searchwaa approaching the last completed simplex, most factors were converging toward the optimum. The optimum levels of the four factors then were taken as the centroid of simplex 22 (the last completed one) and are outlined in Table
IV.
8
.-C
E
0
m
tz 2 I-
3 8 II n
Regression Analysis and Response Surface Mapping-After an optimum is achieved, it is desirable to map the response surface in the neighborhood of this optimum. If a wide enough range of factor levels haa
600
450 l
i
l
t
P, \
10 20 30 40 50
I
t
60 70
,
80
,
,
90 100
NORMALIZED FACTOR LEVEL, %
400
Figure 44omposite plot for dissolutionresponse as a function of each independent variable. Key: W, drug substance ( I ) ; 0 , cornstarch; 0. ~
n
stearic acid; and 0 , capsule fill. The three other Variables are kept conetant at the optimum level (centroid of simplex 22).
360
c
g
been employed in the simplex search, the response data generated can be used in a regression analysis to map the response surface. Yarbro and Deming (28) indicated that one advantage in using a large initial simplex is that it furnishes information on a wide range of factor levels. In the present investigation, the large initial simplex covered most of the factor domain; the information generated during the optimization process was analyzed further by regression techniques as a tool for constructing the response surface. An assumption that a mathematical relationship exists between the measured responses and levels of the independent variables is required before a regression analysis can be performed. In pharmaceutical product design, theoretical mathematical models are impossible to derive because of the complexity of these systems so an empirical model is assumed (5, 30). An often used model for approximating a response surface ia a second-order polynomial equation (4-6,17,31). In the study reported here, the data of 26 observations of the 22 simplices were fitted using a regression computerized programl3:
300
0 0
z
tf 250 2
6w 200 ca a
150
100
Yi = ab
+ a\A
50
a NORMALIZED FACTOR LEVEL, %
Figure 3-CompaPite plot for rate of packing down response as a function of each independent variable. Key: 0 ,drug substance ( I ) ; 0 , cornstarch; and 0 ,stearic acid. The two other variables are kept constant at the optimum level (centroid of simplex 22).
+
+
+
t abD + a $ aiW ujlA2 a&D2 ahW2 + &AD + a\aAL t a\,AW ajDL + a*&W + a'&W (Eq.9) The dependent variables here are the unnormalized values for the various responses, and the a' values are the various regression coefficients. The A, 0,L, and W values are the normalized levels of the formulation factors, aa described earlier. With the rate of packing down response, capsule fill ( W )was dropped as an independent variable since this factor obviously has no effect on the compaction rate as measured here. t ah#
+
+
lS.ProgramBMDOPR, Health Science Computing Facility, University of Cali-
fornia at Los Angela
Jouvnal of Pharmaceutical Sciences I 1139 Vol. 69. No. 10. October 1980
30
-
20
-
tt
10-
The equations generated by the regression analysis were used in two graphical procedures, The first procedure was a plot of a given response as a function of one independent variable; the other independent variables were kept constant at the optimum level (centroid of simplex 22). Figures 3,4,and 5 illustrate the dependence of Yi, Y2, and R t , respectively, on each factor. The ordinates are indicated by the physical units (unnormalized) and by the total desired response corresponding to the dependent variable plotted. The independent variables are represented by their normalized units. The second graphical method was the three-dimensional plot (Fig. 6). In these plots, which were drawn by computd4, two (or in the case of the rate of packing down, only one) factors are fixed at the optimum level, while the specific response is plotted as a function of the two remaining factors.
DISCUSSION The main objective of this study was to evaluate the effectiveness of a simplex search in optimizing a capsule formulation with respect to its dissolution and rate of packing down characteristics. The first step in a simplex optimization procedure is the definition of the quantities to be optimized. This decision depends on the use and final desired characteristics of the system. In the present study, a capsule formulation that would provide complete dissolution of the drug within 30 min and would have a rate of packing down of 300 taps was considered as a well-opti14 Program
OPS,Center for Information Processing,Stanford University.
1140 I Journal of Pharmaceutical Sciences Vot. 69, No. 70, October 1980
Figure 6-Two-factor response surface showing the total desired response (TDR)as a function of cornstarch and stearic acid. Drug eubstance ( I ) and capsule fill are kept constant at the optimum level (centroid of simplex 22). mized formulation. The dissolutionresponse at 8 min was introduced only to make sure that at least 25% of I would be released at that time. The next step in optimization is the selection of the factors. Generally, to simplify the optimization, it is preferable to choose only the most important factors. However, Yarbro and Deming (28) emphasized that only factors absolutely known to be insignificant should be excluded. They recommended that all factors that are considered to affect the response should be included. This inclusion would not greatly increase the total number of trials needed to locate the optimum. Thus, in the study presented here, all three components (I, cornstarch, and stearic acid) and the capsule fill were chosen as significant factors. For the next step in the process, locating the initial simplex, an initial step size must be selected. An initially small simplex would expand in the direction of an optimum (28). A large initial simplex would occupy most of the factor domain, with the optimum being reached by diminution of the simplex size. [It was suggested (28) that the latter technique generates information on a large domain of the factors. This approach is very useful when regression analysis is used. Since regression analysis and mapping of the response surface were intended to be used, a large step size was chosen in this work.] Most of the successful moves in the optimum search by simplex were extensive contractions (e.g., vertexes 7,9,11,13,15,17,19,21,25.27,29,31,33,35,39,41,43, and 45. Table 11). This and the many failed movements,which were violations of boundary conditions on some factors, are typical of a large initial simplex. As the simplex moved and approached the optimum, its size decreased (Table 111). A t the same time, the coordinates of its vertexes focused on the factor levels corresponding to the optimum. Simultaneously, the responses and their standard deviations were aligned (Fig. 1and Table 111). When simplex 22 was reached, the standard deviation of the total desired response in this simplex approached the experimental error (Table 111). This was a strong indication that the optimum had been reached. The optimum with respect to the rate of packing down was reached much earlier than the total optimum (Fig. 1and Table 111). The coordinates of the centroid of the last completed simplex were taken as estimates of the optimized factor levels. The responses of the vertexes in this simplex were satisfactory, giving a mean total desired response of 86.7%. The rate of packing down response reached ita desired optimum (300 taps) while 89.9%dissolved at 30 min, which was very close to the desired optimum dissolution. While the simplex procedure has been considered an efficient method for optimization, it alone does not provide information that describes the response surface leading to the optimum. (Indeed, a major advantage of the simplex approach, unlike response surface techniques, is that it makes no assumptions concerning the shape of the response surface.) Yet data
produced during a simplex search have been suggested (28) and reported (31) for use in generating response surfaces. A secondary objective in the work described here was construction of a response surface using the data generated during the simplex search. Figure 6 shows that it is possible to obtain such response surface maps using simplex information. However, one must be careful in using these response surfaces. The data obtained by the simplex search are not equally dispersed over all of the response surface leading to the optimum. Thus, the surface might be reliable only in the proximity of the optimum where the information obtained by the search is weighted heavily. me main objective of this work was not to obtain basic pharmaceutical information. Yet, in addition to optimizing the system, the resulta revealed some pharmaceutical characteristics of the ingredients. Active Ingredient (1)-At the optimum, an increase in the level of I decreased the totaldesired response (Fig.5)and 1was converged toward low levels (Fig. 2). This pronounced negative effect on the total desired response was a reflection of effects of I on both the rate of packing down and h l u t i o n responses. An increase of the level of the drug substance in the formulation decreased its dissolution rate (Fig. 4). Apparently there ~ 8 no 9 optimum drug level other than no drug at all with respect to the diaolution over the domain studied. Thus, a decrease in the level of I would increase its rate of release from the capsule. This finding indicates that the drug substance is quite hydrophobic. An increase in the level of I also had a negative effect on the rate of packing down response (Fig. 3). Thus, high concentrations of I resulted in higher values for the rate of packing down. It has been shown (26) that the rate of packing down of powders correlates with their flowability The faster the rate of packing down and the smaller the difference between the loose and tight packing, the better is the flowability. In the present work, the particles of the drug substance were rod-shaped crystals, a shape that usually prevents free flow of powders. Thus, this shape might explain the prohibitive effect of I on the rate of packing down response. 4 Dihtegrant-A decrease in the level of cornstarch enhanced the total desired response (Fig. 5). Furthermore, the relationship between the total desired response and the disintegrant level might be defined as a hyperbolic function (Fig. 5). Thus, the function has no maximum. Figures 3 and 4 show that the inhibition of the total desired response by cornstarch was a result of its depressing effect on the rate of packing down and the dissolution. In this case. cornstarch did not improve the dissolution properties of the formulation. Thii general prohibitive effect of cornstarch in the formulation also is shown in Fig. 2: the optimum is being converged upon at a low level of cornstarch. Lubricant-As expected, a decrease in the lubricant level increased the dissolution rate of the capsule (Fig. 4). Stearic acid affected the formulation positively with respect to the rate of packing down. As demonstrated in Fig. 3, the lubricant decreased the rate of packing down dramatically. These two opposite effects of stearic acid on the rate of packing down and dissolution responses were reflected in the total desired response. The lubricant had both an enhancing and a prohibitive effect on the total desired response. A t low levels of stearic acid, the btal desired response was improved (Fig. 5). Thii improvement was attributed to a strong enhancement in the rate of packing down response with a mild negative effect on the dissolution response. At higher levels, the lubricant dramatically reduced the total desired response (Fig. 5). This result was caused by the prohibitive effects of the lubricant on dissolution, which could not be compensated for by the improvement in the rate of packing down response. Capsule Fill-An apparently linear relationship existed between the total desired response and the capsule fill. At the optimum, an increase in the fill level increased the totaldesired response (Fig. 5). This increase in the total desired response was a result of a similar increase in the dissolution rate. Athough the capsule fill dramatically affected the &solution response a t the optimum. a normalized level of 09b fill resulted in a satisfactow dissolution rate. This finding also is reflected in Fig. 2, which does not indicate real convergence of this fador at the optimum. The data generated during the optimization can also be used when specifying ranges for various excipient levels. Eventual large-scale manufacture often requires minor formulation changes to improve manufacturing characteristics depending on the machine used. Both the experimental data and the polynomial fit give considerable information concerning the shape of the mountaintop: it may be a very sharp are& or a broad, gently doping plateau. A measure of suggested ranges can be obtained from the standard deviations of the parameters calculated from the five vertexes of the last simples. These values are seen in Table IV
for the normalized parameters, along with actual values back-calculated for the unnormalized variables. These results could be useful indicators for further formulation development and scaleup work. CONCLUSION
The simplex method can be employed in optimization of pharmaceutical dosage forms. This method has the advantages of simplicity and efficiency. No sophisticated calculations are necessary; manual or small calculator computation suffices.A relatively small number of experiments (in this work, only 29) is required to look for and find a true optimum, assuming nothing about the response surface a priori other than that some optimum exists. Moreover, the technique does not preclude a posteriori response surface methodologies for a fuller understanding of that surface. Although the simplex technique is a powerful optimization method, it in no way makes formulation development a routine matter. As pointed out by Schwartz et al. (6), no optimization technique can replace the pharmaceutical scientist. With simplex, the researcher must predetermine the set of characteristics of an optimum dosage form and then select reasonable excipients and levels thereof as a starting point. Finally, he or she must interpret the results in a rational and objective manner. Therefore, the simplex technique affords the pharmaceutical researcher a significant and powerful tool for efficient and effective dosage form development. APPENDIX
The resulta for the initial simplex (simplex 1)are listed here. Obviously, vertex 2 gives the worst response. The hyperface centroid F and the difference P - W were calculated and also are listed. The P value was calculated by averaging the levels of each factor in vertexes I, 5,4, and 3. The difference F - W was calculated by vector subtraction of vertex 2 from fi;. factor level
vertex 1 5 4
; W (vertex2) P-W
A D L 0.00 0.00 0.00 21.85 21.85 21.85 16.39 92.56 -76.17
21.85 21.85
92.56 34.07 21.85 12.22
21.85 92.56 21.85 34.07 21.85 12.22
1
response
0.00
86.4 54.4 24.3 -82.4
92.56 21.85 21.85 34.07 21.85 12.22
-116.3 ,
The first choice in movement of a simplex is by a reflection, and Eq. 2 is used to calculate the new vertex that will replace vertex 2. The new
vertex then is the vector sum of P and F - W, giving: vertex 6
A
-59.78
D 46.29
L 46.29
W 46.29
The calculation indicates that a reflection of vertex 2 results in a boundary violation with respect to factor A. According to the rules,an arbitrarily very low response, -1000, is assigned to this vertex and it is not retained. Instead, C, (contraction) according to Eq.5 is performed half of the difference H W is subtracted from P. This procedure results in the factor levels for vertex 7:
vertex 7
A 54.48
D 27.96
L 27.96
W 27.96
Evaluation of this vertex results in a total desired response of 15.3% (Table 111, and vertex 7 is accepted to replace vertex 2 in simplex 2. Of the vertexes in the new simplex (1,3,4,5, and 7), vertex 3 now has the worst response, and the simplex research continues accordingly. REFERENCES (1) 1969. (2) 64,966 (3) (4)
L. C. Schroeter, “Ingredient X,” Pergamon, New York, N.Y.,
N. R.Bohidar, F. A. Restaino, and J. B. Schwartz, J. Pharrn. Sci., (1975).
S. Dincer and S.Ozdurmus, ibid., 66,1070 (1977). C. Reier, R. Cohen, S. Rock, and F. Wagenblast, ibid.. 57,660
(1968). (5) D. E. Former, Jr., J. T.Buck, and G. S. Banker, ibid., 59,1587 Journal of Pharmaceutical Sciences I 1141 Vol. 69, No. 10, October 1880
(1970). (6) J. B. Schwartz, J. R. Flamholz, and R. H. Press, ibid., 62,1165 (1973). (7) Ibid., 62,1581 (1973). (8) G.E.P. Box and N. R. Draper, “Evolutionary Operation,” Wiley, New York, N.Y., 1969. (9) M. H. Rubenstein, Drug Cosmet. Ind., 116,44(1975). (10) T.N.DiGaetano, Bull. Parenter. Drug Assoc., 29,183 (1975). (11) W.Spendley, G. R. Hext, and F. R. Himsworth, Technometrics, 4,441 (1962). (12) J. A. Nelder and R. Mead, Comput. J., 1,308 (1965). (13) D. E.Long, Anal. Chirn. Acta, 46,193 (1969). (14) F.P.Czech, J. Assoc. Off. Anal. Chem., 56,1489 (1973). (15) Ibid., 56,1496 (1973). (16) S.N.Deming and S. L. Morgan, Anal. Chem., 45,278A (1973). (17) S.L. Morgan and S. N. Deming, ibid., 46,1170 (1974). (18) P.G.King, S. N. Deming, and S. L. Morgan, Anal. Lett., 8,369 (1975). (19) G. L.Ritter, S. R. Lowry,C. L. Wilkins, and T. L. Isenhour, Anal. Chem., 47,1951 (1975). (20) T. Failam, C. L. Wilkins, T. R. Brunner, L. T. Soltzberg, and S. L.Kaberline, ibid., 48,1768 (1976). (21) W.K. Dean, K. Y. Heald, and S. N. Deming, Science, 189,805
(1975). (22) F.Darvas, J. Med. Chem., 17,799 (1974). (23) R. D.Gillian, W. P. Purcell, and T. R. Bosin, Eur. J.Med. Chem., 12.187 (1977). (24) N. K. Zubkova, S. A. Minina, and A. S.Bril. Khim. Farm. Zh.. 11, 107 (1977);through Chem. Abstr., 88,79039t (1978). (25) C. Beyer, Arch. Pharm., 311,128 (1978). (26) B.S.Newman, Adu. Pharm. Sci., 2,203 (1967). (27) D. M. Olsson and L. S. Nelson, Teehnometrics, 17,45 (1975). (28) L. A. Yarbro and S. N. Deming, Anal. Chim. Acta, 73, 391 (1974). (29) D. M. Himmelblau. “Applied Nonlinear Programming,” McCraw-Hill, New York, N.Y., 1972,p. 148. (30) V. Chew, “Experimental Design in Industry,” Wiley, New York, N.Y., 1958,pp. 108-137. (31) S. N. Deming, S. L. Morgan, and M. R. Willcott, Am. Lab., Oct. 1976,13. ACKNOWLEDGMENTS The authors thank Mr. W. Morgenstern for assistance in computer programming for the generation of three-dimensional plots.
IR and X-Ray Diffraction Study of Chlorpheniramine Maleate-Montmorillonite Interaction M. SANCHEZ CAMAZANO A. DOMINGUEZ-GIL +x
*, M. J. SANCHEZ *, M. T. VICENTE *, and
Received June 4,1979,from the V e n t r o de Edafologia y Biologia Aplicada del C.S.I.C. and the thactical Pharmacy Department, Faculty of Accepted for publication March 27,1980. Pharmacy. University of Salamanca, Salamanca, Spain.
Abstract 0 As a contribution to studies on the adsorption of drugs by clay minerals aimed at achieving sustained action in the oral administration of antihistamines. the adsorption of chlorpheniramine maleate by sodium montmorillonite was studied by X-ray diffraction and IR spectroscopy. The results indicated that the chlorphenirammonium ion penetrated into the interlayer space of montmorillonite, producing an increase in the basal spacing, dml, of the silicate. This increase was influenced, as was the amount adsorbed, by the pH and the concentration of the chlorpheniramine maleate solution. At pH 7.0, the amount adsorbed was close to the exchange ca acity. and the complex formed had a basal spacing of 17.14 A (A = 7.54 The only mechanism responsible for the interaction was cation exchange. The complex at 17.14 A must be a monolayer with the benzene rings positioned perpendicular to the surface of the oxygen atoms.
1).
Keyphrases Chlorpheniramine maleate-mechanism of interaction with montmorillonite 0 Montmorillonite-mechanism of interaction with chlorpheniraminemaleate o Adsorption-mechanism of interaction of chlorpheniramine with montmorillonite Clays-montmorillonite, mechanism of interaction with chlorpheniramine maleate
The presence of clays in the GI tract is of biopharmaceutical interest because of the resultant decrease in the bioavailability of orally administered drugs and the possibility of continued action following oral administration by means of a programmed adsorption speed, thus ensuring efficient plasma drug levels over long periods. After the first reports on the adsorption of organic ions (1,2)and nonionic organic molecules of a polar character by montmorillonite (3-5), it was demonstrated that numerous organic molecules can be adsorbed by clays, 1142 I Jownal of Pharmaceutical Sciences Vol. 69,No. 10, October 1980
especially by montmorillonite (6). Investigations also were carried out on the interaction mechanisms (7-10) and on the factors affectingthe interlayer expansion of clays with organic compounds (11-16). Among these factors is the influence of the phase (liquid or vapor) with which montmorillonite is equilibrated (17). BACKGROUND Numerous reporta have been concerned with the applications of clay minerals in pharmacy (18,191,and some presented the results of drug adsorption by montmorillonita (20).However, few reports have dealt with the mechanisms of adsorption responsible for the interaction of drugs with clays. One study concerned the nature of the bonding in adsorbates of montmorillonite with diazepam and with various benzodiazepine derivatives (21). Another study considered the adsorption of clindamycin and tetracycline by montmorillonite using adsorption isotherms, X-ray diffraction, and IR spectroscopy (22).One report discussed the adsorption of cationic, anionic, and nonionic drugs by montmorillonite using dissolution dialysis (23);chlorpheniramine maleate was among the drugs investigated. Knowledge of the mechanism of interaction of clays with drugs should allow prediction of whether a given compound will be adsorbed by a c l q based on the properties of the clay and the structure of the drug. The objective of this investigation was to study the interaction between chlorpheniramine maleate and sodium montmorillonite by X-ray diffraction and IR spectroscopy to obtain direct evidence regarding the penetration of the organic cation into the hterhyer space of the clay and the interaction mechanism. The study also considered the possible or. ientation adopted by the organic cations within the interlayer space of montmorillonite. X-ray diffraction and IR spectroscopy generally are considered as the
0
0022-3549/80/ 7000-1 142$01.00/0 1980, American Pharmaceutical Association
-
%tract The simplex method of optimization was applied to a capsule formulation using the dissolution rate and compaction rate as the desired responses to be optimized. The formulation parameters investigated included the levels of drug, disintegrant, lubricant, and fill weight. Following the successful optimization of the capsule formulation, the acm u l a t e d data were fitted to a polynomial regression model to plot response surface maps around the optimum. Keyphrases 0 Simplex method-optimization of capsule formulation 0 Dosage forms-capsule formulations, optimization by simplex method
In addition to the active material, pharmaceutical products contain several ingredients specially selected and present in definite concentrations. These ingredients or excipients, which Schroeter (1) named Ingredient X, are selected to make the most effective product from the drug. The substance and the excipients compose the dosage form. Development of dosage forms is essentially an optimization process. Certain goals ordinarily are defined for a dosage form, i.e., physical and chemical stability, physiological availability, ease of manufacture, and cost. Once a drug is chosen, the formulator must produce the best dosage form according to the criteria selected. Among the factors involved is the mixture of excipientsto be used, The desired properties can be termed the dependent variables, and the excipients are the independent variables. The optimization process is not always straightforward; one objective often must be sacrificed to improve another, or a balance must be achieved between conflicting aims. For example, adding additional lubricant to a capsule or tablet formulation may improve its ease of manufacturing but may adversely affect dissolution or bioavailability. Another problem is possible complex interactions among the excipients that affect desired properties. BACKGROUND Solution of such formulation optimization problems can be done by trial and error techniques. but this approach suffers from several draw backs. This approach is not only time consuming, unreliable, costly, and often u n s u d u l , but it also may provide only a provisionaly acceptable mlution rather than an optimum mlution. Reports of formulation optimization by mathematical optimization methodologies have been published recently. Bohidar et al. (2) used principal component analysis to select pharmaceutical formulations. Optimization studies by mathematical models were reported for enteric film coating of tablets (3), encapsulation of powders in hard gelatin capsules (4.61, and tablet formulation (5). Schwartz et al. (6,7) utilized factorial design and computer technique8 to optimize formulation for a tablet. Evolutionary operation (81, which is a statistical method for evaluating the process on a gradual basis, was used to optimize tablet manufacture (9) and particulate matter detection equipment (10). The simplex method is another mathematical optimization method that is an empirical approach. The sequential simplex technique was developed by Spendley et aL (11). Thi ingenious method for optimizing operating conditions the results of previous experiments in a mathematically rigorous fashion to define the parameters (i.e., values of the independent variables) for the next experiment in the search for optimum response. The simplex method is a stepwise optimization pro-
0022-3549/8L1/ 1000-1135$01.00/0 @ 1980,American PlvYmaceutlcalA.%W&tb
Scheme Z
XI
cedure that merely uses the response surface. Nelder and Mead (12) subsequently modified the original algorithm to improve the convergence speed of the system. The method has been used in chemistry for diverse applications. Long (13) applied simplex strategy to analytical chemistry problems. S i p l e x has been used in developing analytical methods for the determination of formaldehyde (14, 15) and cholesterol (16, 17). King et al. (18) discussed the difficulties in the application of simplex to analytical chemistry. Simplex optimization also has been employed for pattern recognition (19,20), for improving reaction yields (21). and for drug design (22, 23).
Two cases of utilization of simplex in pharmaceutical systems were reported recently. Zubkova et al. (24) reported optimiization of proroxan tablets with respect to active drug, lactose, talc, calcium stearate, starch, and compression pressure. Beyer (25) concluded that a simplex method cannot be used in all cases of tablet formulation because tablet properties do not always change with manufacturing conditionsin a predictable way. However, Beyer (25)did not present enough details to enable evaluation of thii conclusion. The objective of the present study was to evaluate the potential of the flexible simplex method, as modified by Nelder and Mead (12), for optimizing a capsule formulation. The capsule formulation was required at very early etages of drug development. At such times, enough raw material for evaluation of a formulation under production conditions usually is unavailable. However, the rate of compaction1 was reported (26) to be related to powder flowability. which is potentially indicative of manufacturing feasibilityon high-speed encapsulating machinery. The amount of material to measure this property is quite small. Thus, in the present study, optimization of the capsule formulation of a drug with respect to its dissolution and rate of compaction was desired. THEORY The simplex method referred to here was developed by Nelder and Mead (12) from the original approach of Spendley et a!. (11). The simplex procedure derives its name from the geometric figure that is moved along the response surface in search of the optimum. This optimization method approaches the optimum in stepwise fashion by moving away from low values of the response function rather than by moving in a line toward the maximum. The logical and mathematical details of the simplex have been discussed extensively (11-13, 16-18.27-29). Thus, only a brief general outline of simplex procedures is given here. The conceptual basis to the simplex approach is most easily comprehended in the two-dimensional (Le., two independent variables) case. In Scheme I, the points refer to several experiments run with independent variables X I and x2. The three experiments, W,B, and S, define the simplex, which in this case is a triangle. To obtain x 1and x g for the next experiment in the optimization search, one move8 away from the point that gave the least favorable response. For example, if W gives the worst response, the next point ( R ) is found by reflecting across the B-S exis, as shown in Scheme 11. The experiment at R is performed and the result is compared to the previous results from B and S.A decision now is made whether to retain the new experiment, R,as a replacement for W.ER is not accepted. then
* In measuring the rate of cornpaction (or of packing down),the change in volume of a quantity of powder caused by constant tapping with time is observed (26).
Journal of f'harmaceutical Sciences I 1135 Vol. 69.No. 10,October 1980
other levels of motion, such a~ expansion and contraction, are tried. These rules for simplex manipulation were outlined by Nelder and Mead (12). With a new simplex now defined, the procedure is repeated to obtain the next step on the optimization path. The process continues until a desired optimum is reached. The simplex procedure thus can be likened to a search for a mountain summit on which one is blindfolded, feeling one's way a step at a time. On the other hand, the typical factorial design plus response surface analysis could be likened to developing a contour map of the mountain first and thereby determining where the summit lies. More generally, the simplex is n dimensional, corresponding to n independent variables. The simplex manipulations are performed by rather simple vector algebra and are easily carried out manually or with a small calculator. As indicated, a given simplex is constructed of n 1vertexes represented as n-dimensional vectors W,S, . B. These vertexes are ranked in order of their responses: Rw (worst), Rs, . . . Re (best). The centroid is (Eq. 1) while W is the eliminated vertex. Now the response RR at R (Eq. 2) is evaluated: 1 B = -(S... B) (Eq. 1) n
..
I " Scheme I I
capsule was dropped in the vessel using a stainless steel coil around it. The stirring speed initially was 50 rpm and was increased after 30 min to 200 rpm for 10min, at which time the fmal concentration was recorded. The amounts of drug dissolved at 8 and 30 min were calculated as a fraction of the total drug dissolved. Normalization of Data-The factors in this study had different units and regions of interest. In such cases (171, it is useful to normalize the factors and to extend their domains from 0 to 1009'0. This process was performed using:
+
+ R = P + (P- W)
*-
x 100 (Eq. 6) H-M where N is the normalized factor, X is the original unnormalized value, and M and H are the lowest and highest values, respectively, in the region of interest for the specific factor. Both the independent and dependent (responses) variables were nonnalized using Eq.6 according to the boundaries shown in Table L For the rate of packing down, larger numbers of taps or slower packing down resulted in low values of response. This effect required modification of Eq. 6 to:
N=-
(reflection) (Es.2) If Rs I RR I Re, then R is retained. If RR > RB,RE at E (%.3) is evaluated. If RE > RR. E is retained. However, if RE 5 RR, then R is retained:
-
E = P + 2 8 W) (expansion) (Eq. 3) If Rw IRR < Rs,Rc, at Cr (&.4) is evaluated and retained. Now, if RR < Rw, then Rc, is evaluated and retained at C, (Eq. 5):
-
(Eq.7)
Cr = P + W) (contraction) (Es.4 ) C , = P - 1/@ W) (contraction) (Eq. 5) If Rc, or Rc. < Rs,then the following is being set: Rw = Rs, W = 5, and Rs = Rc, or Rc, and S = C. or C , . Additional rulesand procedureswhich
where R1and Y1 are the normalized and observed responses, respectively. Boundary values were chosen arbitrarily as reasonable limits on formulation variables and desired responses, To get a single value for the overall response for a given formulation to allow comparison with other formulations in the simplex, an arbitrary procedure was used. The total desired response was obtained by a weighted linear combination of the three observed responses according to:
were applied in this study are explained under Results.
EXPERIMENTAL
Formulations-Cornstarch USP2, stearic acid3, lactose USP4, and the drug substance 5-(2-thenoyl)-1,2-dihydro-3H-pyrrole-[l,2-a]-pyrRt 0.5Ri O A R 2 t OARS (Eq.8 ) role-1-carboxylicacid6 (I) were mixed thoroughly for 30 min. The rate where Rt is the symbol for the total desired response. of packing down of the formulationsand their dwlution rate from a hard Equation 8 shows that the rate of packing down was considered to be
+
gelatin capsule*were determined. Rate of Packing Down Measurement-A 10-mlgraduated cylinder containing -10 ml of loose-packed mixture was mechanically tapped. The mechanicaldevice consisted of a platform supported by a cam, which was rotated by a small motor. During each rotation, the platform was raised gradually and then allowed to drop 0.5 cm. The frequency of the tapping was 50 dropdmin. The initial volume was observed, and volume readings were taken at timed intervals until no further changes were observed. The rate of packing down was calculated by multiplying the ratio of the initialvolume to the f i i volume by the number of tap it took to reach the final volume. Dissolution Measurement-The dissolution vessel was a 1-liter round-bottom flask'. The stirring shaft was equipped with a polyethylene propellel.8, which was positioned 2.5 cm above the lowest inner surface ,of the vessel The shaft was stirred by a multiple-spindlestirring systemg. A peristaltic pumplo transported the solution continuously from the vessel through a variable-wavelength detector" adjusted at an appropriate wavelength. The dissolution process was registered by a chart recorder. Five hundred milliliters of 0.05 M pH 5.0 acetate buffer ( p = 0.5 M NaCl), which was used as the dissolution medium, was kept at 37O.The
I
of 50%importance as a measure of formulation success, and dissolution at 30 min was considered to be of 41% importance. The third response, dissolution at 8 min, was much less important than the other two. If any response fell outside of the limits in Table I, the total desired response was set at zero. Simplex Optimization-Initial experiments ( i e . , the initial simplex) were defined by the procedure of Spendley et af. (11) and Himmelblau (29).The first vertex was the experimental origin and corresponded to the lowest factor values in the region of interest. The remaining four vertexes in the initial simplex were obtained by multiplying the appropriate fraction of each factor according to the procedure mentioned earlier (11,29)and by the initial step size. The initial step size (28), which is the amount of change in a factor that takes place when a new vertex for the initial simplex is calculated, determines the size of this simplex. Yarbro and Deming (28) indicated that a large initial simplex size is useful in furnishing information about a wide range of factor levels. This suggestion was employed in the present work Table I-Factor
Boundaries Used to Normalize Their Levels Value(M) Low
Value(H) High Symbol (N)
Factor (X) 2 A. E. Staley ManufacturingCo., Decatur, IL 62525. 8 Emery Industries, Cincinnati, OH 45232.
San Francisco, CA 94104. from The Institute of Organic Chemistry, Syntex Research, Palo Alto, CA 94304. e No. 1, Elaneo Products Co., Indianapolis,IN 46206. 7 Kimax 33710-S1. 8 Sar ent S-76680. e MAel72-R-115, Hansen Research, Northridge, CA 91328. 10 Model 1210, Harvard Apparatus Inc., Millis, M A 02054. 11 Model 770, Schoeffel Corp., Weatwood, NJ 07675. 4 Foremost. 6 Obtained
1136 I Journal of Pharmaceutical Sciences Vol. 69. No. 10. October 1980
Independent variables Drug (I), % w/w Disintegrant (cornstarch), 9i W/W Lubricant (steari? acid), % w/w Total capsule weight, mg De endent variables el, rate of packing down, taps Y2, dissolution a t 30 min, % Y3, dissolution a t 8 min, %
__ 1.0
41.0
0.0 0.2 100.0
50.0 2.2 400.0
300
500 100 75
50 25
A
D L W Rl
R2 R1
Table 11-Results of the Simplex Search. Vertex
SimDlex
Vertexes Retained
1 2 3 4 5 6 7 8 9 10
1'
1 1 1 1 1 2 2 3 3 4 4 5 5 6 6 6
5' 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 23'
11 12 13 14 15
34 35 36 37 38 39 40 41 42 43 44 45
Generated bv
Factor Level, Normalized, % A
D
L
W
1,39495 193,495 194,597 1,4,5,7 1,4,5,9 1,4,5,9 1,4,5, 11 1,4,5,11 1,5,11,13 1,5,11,13 1,5,11,13
0.00 92.56 21.85 21.85 21.85 -59.78 54.48 27.25 23.20 -21.02 35.60 16.46 21.52 17.63 20.80 0.00
0.00 21.85 92.56 21.85 21.85 46.29 27.96 -56.72 55.24 21.52 26.35 -20.22 36.38 20.45 21.50 0.00
0.00 21.85 21.85 92.56 21.85 46.29 27.96 49.33 28.72 43.60 31.87 44.42 32.65 -49.41 57.09 0.00
0.00 21.85 21.85 21.85 92.56 46.29 27.96 49.33 28.72 43.60 31.87 44.42 32.65 21.85 30.56 0.00
6
1,5,11,13
21.85
21.85
21.85
92.56
7 7 8
1,5,11,15 1,5,11,15 1,5,15,17 1,5,15,17 1,5,17,19 1,5, 17,19 1,5,19,21 1,5,21,22 1, 21,22,23 1,21,22,23 1,22,23,25 1,22,23,25 1,22,23,27 1,22,23,27 1,22, 23,29 1,22,23,29 1,23,29,31 1,23,29,31 1,23,29,31
17.60 20.54 -4.00 25.70 13.24 18.91 12.70 1.04 -5.53 15.00 -4.53 13.05 -1.60 10.85 -0.75 9.60 -1.95 9.03 1.04
-1.52 26.91 8.82 21.95 13.86 19.59 4.79 1.17 -9.07 14.12 -9.55 12.31 -4.98 9.35 -4.65 8.07 4.51 4.72 1.17
22.75 30.18 22.69 29.58 -16.29 38.75 14.92 8.18 9.07 18.66 -18.19 24.52 5.16 15.29 5.32 17.06 5.34 12.52 8.18
44.87 35.71 47.55 35.79 51.48 30.56 43.75 47.65 -31.58 61.53 30.56 34.40 1.37 46.49 34.40 34.50 20.57 37.95 47.65
1,23,29,33 1,23, 29,33 1,23,29,35 1,23,29,35 1,23,35,36 1,23,35,36 1,23,36,39 1, 23,36,39 1,23,36,41 1.23,36,41 1.36.41. ~,- ., .-,4.1 -1,36,41,43
0.85 7.40 0.61 -14.92 -6.33 6.56 -3.30 4.73 -3.36 4.08 3.68 1.70
-0.45 5.94 3.52 -10.73 -4.03 6.00 -0.60 4.31 -1.50 4.13 4.81 2.08
0.94 13.03 5.73 -10.36 -1.81 11.02 -0.57 9.63 0.76 8.46 3.73 7.08
31.54 33.76 26.00 -12.43 7.21 36.67 21.40 30.67 15.49 31.38 -3.65 34.83
8
9 9 10 11
12 12 13 13 14 14 15 15 16 16 16 17 17 18 18 19 19 20 20 21 ~~
21
Percent Desired Response, Normalized R1 R9 R.1 Rt 91.0
88.0 -231.0 -133.4 23.0 76.0
-29.0 27.0 33.0
12.6 13.8
3.8 -2.8
-
27.4 -
20.0 31.4 89.0 88.4 76.0 76.0
-
69.0
72.0 76.0 88.0 95.0
-
0.0
-
32.7 25.2 -
60.0 -8.0 -29.0 20.0 32.0
73.0 -5.9 7.0 -
33.7 90.7 91.0 1.5 17.2
22.0 30.0 29.0 45.0 3.0 18.0
4.2 26.7 35.7 66.5
14.0 16.0 18.0
3.0 -
6.0 5.0 -
-
24.0 7.0 57.0 6.0 -
31.0 67.7 67.0
18.0 43.0 31.0
-
40.0 78.4
25.5
100.0 100.0 93.0 94.0
22.4
-
-
100.0 100.0
41.2 82.2
37.0 52.0
86.0 -
-
-
99.0 100.0 100.0 100.0
80.0 44.7
47.7 68.5 83.5 75.0
-
34.0 59.0 76.0
-
45.0
86.4 O.Ob O.Ob
24.3 54.4 -1000.0' 15.1 -1000.0' O.Ob
-1000.0'pd 27.5 -1Ooo.oe 22.3 -1Ooo.O' 32.2 83.7* 85.1 38.gC 46.7 -1Ooo.O' 36.3 -1Ooo.O' 38.2 -1000.OC 50.1 59.9 75.9 -1000.0' 32.6 -1Ooo.oc.d
43.1 -1Ooo.OC 66.6 -1000.0' 61.8 -1Ooo.O' 64.2 77.94 76.9 -1000.oc 70.2 88.1 -1000.0' -1000.0' 67.0 -1Ooo.OC.d
83.3 -1Ooo.O' 91.0 -1Ooo.oc 84.5
a Key: R, reflection(Q.2); c,, contraction (Q.5); E , expansion (Jh. 3); R I ,normalized rate of packing down; Rs,normalized dissolution at 30 min; and Rs,normalized dissolutionat 8 min. * Boundary violation on one or more of the responses Boundary violation on one or more of the facto?. Second worst vertex is rejected. Response checked after vertex occurred in five successive simplices. The second line gives t h e average responseof the two observations.
by using 100 units of the normalized factor level as the initial step size. RESULTS Simplex Optimization-The formulations for the capsule were composed of four ingredients. The active ingredient (A) was the drug substance, I. Cornstarch and stearic acid were the disintegrant ( D )a d the lubricant (L),respectively, while the filler was lactose. Since the formulation expressed as percent by weight must add up to I@)%,only three of the four parameters are required to define the formulation and, therefore, are independent variables. In this study, the drug, cornstarch, and stearic acid were the three independent variables'2. Previous experimehts showed that the degree of fill of the capsule significantly affected the dissolution properties of the formulated capsules. Accordingly, the amount of the powder mixture fiied into the capsule ( W) was a fourth factor. Table I outlines the boundaries used for the various factors in this study. 12 Any three of the four parameters are equally appropriate as independent variables; the particular choice here waa for convenience.
The responses of the various experiments obtained by the simplex method on the four-factor response surface of the formulation are presented in Table 11. Initial experiments (i.e., the initial simplex) were defined by the procedure of Spendley et al. (11) and Himmelblau (29) and were discussed under Experimental. The remaining vertexes in Table I1 were generated by the simplex moves according to the appropriate rules. An example of a numerical step-by-step simplex calculation is outlined in the Appendix. During the optimum search, vertexes 1,5, and 23 occurred in five (n 1) successive simplices. According to the recommendation of Spendley et 01. (11). the responses at these threevertexes were checked (vertexes l', 5', and 23') and replaced by an average of the two results (Table 11). Vertexes 6,8,10,12,14,16.18,20,24,26,28,30,32,34,37,38,40,42, and 44 violated a boundary condition on one or more of the experimental factors. An arbitrarily very low response (-1Ooo) was assigned to these vertexes to constrain the factors inside the region of interest by means of a contraction (Cw). In the initial simplex, two vertexes (2 and 3) violated the boundaries with respect to various responses, and a total desired response of zero was arbitrarily assigned to them. However, to determine which of the two vertexes gave the worse response (and, therefore, which vertex to reject), total desired responses were calculated according to Q.8. Based on this
+
Journal of Pharmaceutical Sciences I 1137 Vol. 69, No. 10. October 1980
'
80 60 40
20
80t
A
L
\
s
2 80-
~
treatment, vertex 2 was the worst point in the initial simplex and waa rejected. A similar boundary violation of the dissolution response occurred in vertex 9. In two simplices (12and 19).the newly generated vertex was the worst one and was supposed to be the next one rejected. However, rejection of Table I I l - S t a n d a r d Deviation of Responses and Relative Simplex Size
SD,% of total desired response Relative Size R, R2 R3 Rt ofSimDiex
1 2 3 4
5
6 7 8 9 LO ii 12 13 14 15 16 17 18 19 20 21 22 Experimental error
I
140.5 88.3 36.3 33.8 32.4 31.1 27.3 21.2 7.4 7.6 8.4 22.1 22.0 5.9 5.8 6.4 6.4 6.4 5.6 5.6 5.2 5.1 6.3
44.6 42.8 35.0 34.0 27.7 29.2 33.4 36.3 36.3 33.3 30.5 28.6 33.2 31.3 25.5 26.5 24.0 22.1 21.4 16.5 10.3 8.6 10.0
'34.6 32.3 24.2 25.2 19.9 14.1 16.3 16.8 16.0 16.0 i2.2 12.4 14.1 21.1 21.2 20.1 14.3 10.2 8.2 11.4 16.5 13.2 15.0 ~
1138 t Journal of Pharmaceutical Sciences Vol. 69, No. 10, October lQ80
A
8olh
DRUO SUBSTANCE I0
60
SIMPLEX NUMBER
Figure 1-Response improvement during simplex search. The normalized response of t h most recent uerter is plotted against the U P propriate simplex number. Key: Rl,rate ofpackihg down reswFe; Rz, dissolution response at 30 min; Ra, dissolution response at 8 m n ;and Rt, total desired response.
Simoiex r--
\
\
88.2 63.5 33.1 3 _1_.5 27.5 25.4 23.2 21.6 22.0 18.0 16.5 20.8 20.3 16.0 10.5 9.8 8.5 9.2 9.1 8.3 5.3 3.1 6.2
1.00 0.88 0.77 0.72
0.67 0.57 0.55 0.53 '0.48 0.48 0.48 0.39 0.33 0.29 0.26 0.25 0.24 0.24 0.21 0.19 0.18 0.15
I (
1 1
1 1 t i I I I 1 1 1 1 1 1 I
1
3
6
7
9
1 1 1
I
11 13 16 17 19 21
SIMPLEX NUMBER
Figure 2-Chapges in factor levekr during simplex search. The normalized feuel of the factor in the most recent vertex is plotted against the appropriate simplex number. this new vertex would stalemate the search by regeneration of the previous simplex after the prescribed reflection operation. Conceptually, this situation is the same as the two simplices straddling a ridge on the response surface. Instead, translation of the simplex by eliminating the second poorest vertex allowed continuous movement of the simpler. As was pointed out by Long (13),this kind of translation usually will even promote movement of the simplex up a ridge that is being straddled. Figure 1shows the normalized responsesfor dissdlution, rate of packing down, and totaldesired response 89 functions of the simplex number. The response of the most recent vertex is plotted against the appropriate simplex number. The branches represent vertexes of failed steps, while the continuous lines describe the movement of the simplex toward the optimum. The alterations of the responses for the first five vertexes were not generated by the progress of the simplex but were originated by the order in which the initial simplex was constructed. The standard deviations of the responses for the varioue completed simplices are outlined in Table 111. The relative simpiex size (defined as the sum of the standard deviations of the factors in the simplex as a fraction of that of the initial simplex) also is shown in Table 111. As the simpiex progressed and approached the optimum, there was a leveling off in each response value (Fig. 1). with a concomitant decrease In standard deviations approaching the experimental error. At the same time, a substantial decrease in the simplex size could be observed. The search WM halted after 45 vertexes and 22 completed simplices. The response finally achieved was considered to be satisfactory. Figure 2 shows the variations in the normalized levels of the four fac-
Table IV-Results
of the Last Simplex (Simplex 22)
1
Normalized Values, % Total Desired Response ~
Vertex 1 36 41 43 45
Mean* SD Unnormalized mean
SD
Units
8
L
W
0 3.52 4.31 4.13 2.08 2.81 1.80 1.41
0 5.73 9.63 8.46 7.08 6.18 3.75 0.324
0 26.0 30.67 31.38 34.83 24.58 14.09 173.7
0.9
0.075
42.2
%wlw
mg
A
D
0 0.61 4.73 4.08 1.70 2.22 2.09 1.89 0.84
%wlw %wlw
~~~~
85.1 88.1 83.3 91.0 84.5
-
-
-
Centroid of this simplex.
tors. Again, the continuous line describes the progress of the simplex, while the branches display the failed step. As the searchwaa approaching the last completed simplex, most factors were converging toward the optimum. The optimum levels of the four factors then were taken as the centroid of simplex 22 (the last completed one) and are outlined in Table
IV.
8
.-C
E
0
m
tz 2 I-
3 8 II n
Regression Analysis and Response Surface Mapping-After an optimum is achieved, it is desirable to map the response surface in the neighborhood of this optimum. If a wide enough range of factor levels haa
600
450 l
i
l
t
P, \
10 20 30 40 50
I
t
60 70
,
80
,
,
90 100
NORMALIZED FACTOR LEVEL, %
400
Figure 44omposite plot for dissolutionresponse as a function of each independent variable. Key: W, drug substance ( I ) ; 0 , cornstarch; 0. ~
n
stearic acid; and 0 , capsule fill. The three other Variables are kept conetant at the optimum level (centroid of simplex 22).
360
c
g
been employed in the simplex search, the response data generated can be used in a regression analysis to map the response surface. Yarbro and Deming (28) indicated that one advantage in using a large initial simplex is that it furnishes information on a wide range of factor levels. In the present investigation, the large initial simplex covered most of the factor domain; the information generated during the optimization process was analyzed further by regression techniques as a tool for constructing the response surface. An assumption that a mathematical relationship exists between the measured responses and levels of the independent variables is required before a regression analysis can be performed. In pharmaceutical product design, theoretical mathematical models are impossible to derive because of the complexity of these systems so an empirical model is assumed (5, 30). An often used model for approximating a response surface ia a second-order polynomial equation (4-6,17,31). In the study reported here, the data of 26 observations of the 22 simplices were fitted using a regression computerized programl3:
300
0 0
z
tf 250 2
6w 200 ca a
150
100
Yi = ab
+ a\A
50
a NORMALIZED FACTOR LEVEL, %
Figure 3-CompaPite plot for rate of packing down response as a function of each independent variable. Key: 0 ,drug substance ( I ) ; 0 , cornstarch; and 0 ,stearic acid. The two other variables are kept constant at the optimum level (centroid of simplex 22).
+
+
+
t abD + a $ aiW ujlA2 a&D2 ahW2 + &AD + a\aAL t a\,AW ajDL + a*&W + a'&W (Eq.9) The dependent variables here are the unnormalized values for the various responses, and the a' values are the various regression coefficients. The A, 0,L, and W values are the normalized levels of the formulation factors, aa described earlier. With the rate of packing down response, capsule fill ( W )was dropped as an independent variable since this factor obviously has no effect on the compaction rate as measured here. t ah#
+
+
lS.ProgramBMDOPR, Health Science Computing Facility, University of Cali-
fornia at Los Angela
Jouvnal of Pharmaceutical Sciences I 1139 Vol. 69. No. 10. October 1980
30
-
20
-
tt
10-
The equations generated by the regression analysis were used in two graphical procedures, The first procedure was a plot of a given response as a function of one independent variable; the other independent variables were kept constant at the optimum level (centroid of simplex 22). Figures 3,4,and 5 illustrate the dependence of Yi, Y2, and R t , respectively, on each factor. The ordinates are indicated by the physical units (unnormalized) and by the total desired response corresponding to the dependent variable plotted. The independent variables are represented by their normalized units. The second graphical method was the three-dimensional plot (Fig. 6). In these plots, which were drawn by computd4, two (or in the case of the rate of packing down, only one) factors are fixed at the optimum level, while the specific response is plotted as a function of the two remaining factors.
DISCUSSION The main objective of this study was to evaluate the effectiveness of a simplex search in optimizing a capsule formulation with respect to its dissolution and rate of packing down characteristics. The first step in a simplex optimization procedure is the definition of the quantities to be optimized. This decision depends on the use and final desired characteristics of the system. In the present study, a capsule formulation that would provide complete dissolution of the drug within 30 min and would have a rate of packing down of 300 taps was considered as a well-opti14 Program
OPS,Center for Information Processing,Stanford University.
1140 I Journal of Pharmaceutical Sciences Vot. 69, No. 70, October 1980
Figure 6-Two-factor response surface showing the total desired response (TDR)as a function of cornstarch and stearic acid. Drug eubstance ( I ) and capsule fill are kept constant at the optimum level (centroid of simplex 22). mized formulation. The dissolutionresponse at 8 min was introduced only to make sure that at least 25% of I would be released at that time. The next step in optimization is the selection of the factors. Generally, to simplify the optimization, it is preferable to choose only the most important factors. However, Yarbro and Deming (28) emphasized that only factors absolutely known to be insignificant should be excluded. They recommended that all factors that are considered to affect the response should be included. This inclusion would not greatly increase the total number of trials needed to locate the optimum. Thus, in the study presented here, all three components (I, cornstarch, and stearic acid) and the capsule fill were chosen as significant factors. For the next step in the process, locating the initial simplex, an initial step size must be selected. An initially small simplex would expand in the direction of an optimum (28). A large initial simplex would occupy most of the factor domain, with the optimum being reached by diminution of the simplex size. [It was suggested (28) that the latter technique generates information on a large domain of the factors. This approach is very useful when regression analysis is used. Since regression analysis and mapping of the response surface were intended to be used, a large step size was chosen in this work.] Most of the successful moves in the optimum search by simplex were extensive contractions (e.g., vertexes 7,9,11,13,15,17,19,21,25.27,29,31,33,35,39,41,43, and 45. Table 11). This and the many failed movements,which were violations of boundary conditions on some factors, are typical of a large initial simplex. As the simplex moved and approached the optimum, its size decreased (Table 111). A t the same time, the coordinates of its vertexes focused on the factor levels corresponding to the optimum. Simultaneously, the responses and their standard deviations were aligned (Fig. 1and Table 111). When simplex 22 was reached, the standard deviation of the total desired response in this simplex approached the experimental error (Table 111). This was a strong indication that the optimum had been reached. The optimum with respect to the rate of packing down was reached much earlier than the total optimum (Fig. 1and Table 111). The coordinates of the centroid of the last completed simplex were taken as estimates of the optimized factor levels. The responses of the vertexes in this simplex were satisfactory, giving a mean total desired response of 86.7%. The rate of packing down response reached ita desired optimum (300 taps) while 89.9%dissolved at 30 min, which was very close to the desired optimum dissolution. While the simplex procedure has been considered an efficient method for optimization, it alone does not provide information that describes the response surface leading to the optimum. (Indeed, a major advantage of the simplex approach, unlike response surface techniques, is that it makes no assumptions concerning the shape of the response surface.) Yet data
produced during a simplex search have been suggested (28) and reported (31) for use in generating response surfaces. A secondary objective in the work described here was construction of a response surface using the data generated during the simplex search. Figure 6 shows that it is possible to obtain such response surface maps using simplex information. However, one must be careful in using these response surfaces. The data obtained by the simplex search are not equally dispersed over all of the response surface leading to the optimum. Thus, the surface might be reliable only in the proximity of the optimum where the information obtained by the search is weighted heavily. me main objective of this work was not to obtain basic pharmaceutical information. Yet, in addition to optimizing the system, the resulta revealed some pharmaceutical characteristics of the ingredients. Active Ingredient (1)-At the optimum, an increase in the level of I decreased the totaldesired response (Fig.5)and 1was converged toward low levels (Fig. 2). This pronounced negative effect on the total desired response was a reflection of effects of I on both the rate of packing down and h l u t i o n responses. An increase of the level of the drug substance in the formulation decreased its dissolution rate (Fig. 4). Apparently there ~ 8 no 9 optimum drug level other than no drug at all with respect to the diaolution over the domain studied. Thus, a decrease in the level of I would increase its rate of release from the capsule. This finding indicates that the drug substance is quite hydrophobic. An increase in the level of I also had a negative effect on the rate of packing down response (Fig. 3). Thus, high concentrations of I resulted in higher values for the rate of packing down. It has been shown (26) that the rate of packing down of powders correlates with their flowability The faster the rate of packing down and the smaller the difference between the loose and tight packing, the better is the flowability. In the present work, the particles of the drug substance were rod-shaped crystals, a shape that usually prevents free flow of powders. Thus, this shape might explain the prohibitive effect of I on the rate of packing down response. 4 Dihtegrant-A decrease in the level of cornstarch enhanced the total desired response (Fig. 5). Furthermore, the relationship between the total desired response and the disintegrant level might be defined as a hyperbolic function (Fig. 5). Thus, the function has no maximum. Figures 3 and 4 show that the inhibition of the total desired response by cornstarch was a result of its depressing effect on the rate of packing down and the dissolution. In this case. cornstarch did not improve the dissolution properties of the formulation. Thii general prohibitive effect of cornstarch in the formulation also is shown in Fig. 2: the optimum is being converged upon at a low level of cornstarch. Lubricant-As expected, a decrease in the lubricant level increased the dissolution rate of the capsule (Fig. 4). Stearic acid affected the formulation positively with respect to the rate of packing down. As demonstrated in Fig. 3, the lubricant decreased the rate of packing down dramatically. These two opposite effects of stearic acid on the rate of packing down and dissolution responses were reflected in the total desired response. The lubricant had both an enhancing and a prohibitive effect on the total desired response. A t low levels of stearic acid, the btal desired response was improved (Fig. 5). Thii improvement was attributed to a strong enhancement in the rate of packing down response with a mild negative effect on the dissolution response. At higher levels, the lubricant dramatically reduced the total desired response (Fig. 5). This result was caused by the prohibitive effects of the lubricant on dissolution, which could not be compensated for by the improvement in the rate of packing down response. Capsule Fill-An apparently linear relationship existed between the total desired response and the capsule fill. At the optimum, an increase in the fill level increased the totaldesired response (Fig. 5). This increase in the total desired response was a result of a similar increase in the dissolution rate. Athough the capsule fill dramatically affected the &solution response a t the optimum. a normalized level of 09b fill resulted in a satisfactow dissolution rate. This finding also is reflected in Fig. 2, which does not indicate real convergence of this fador at the optimum. The data generated during the optimization can also be used when specifying ranges for various excipient levels. Eventual large-scale manufacture often requires minor formulation changes to improve manufacturing characteristics depending on the machine used. Both the experimental data and the polynomial fit give considerable information concerning the shape of the mountaintop: it may be a very sharp are& or a broad, gently doping plateau. A measure of suggested ranges can be obtained from the standard deviations of the parameters calculated from the five vertexes of the last simples. These values are seen in Table IV
for the normalized parameters, along with actual values back-calculated for the unnormalized variables. These results could be useful indicators for further formulation development and scaleup work. CONCLUSION
The simplex method can be employed in optimization of pharmaceutical dosage forms. This method has the advantages of simplicity and efficiency. No sophisticated calculations are necessary; manual or small calculator computation suffices.A relatively small number of experiments (in this work, only 29) is required to look for and find a true optimum, assuming nothing about the response surface a priori other than that some optimum exists. Moreover, the technique does not preclude a posteriori response surface methodologies for a fuller understanding of that surface. Although the simplex technique is a powerful optimization method, it in no way makes formulation development a routine matter. As pointed out by Schwartz et al. (6), no optimization technique can replace the pharmaceutical scientist. With simplex, the researcher must predetermine the set of characteristics of an optimum dosage form and then select reasonable excipients and levels thereof as a starting point. Finally, he or she must interpret the results in a rational and objective manner. Therefore, the simplex technique affords the pharmaceutical researcher a significant and powerful tool for efficient and effective dosage form development. APPENDIX
The resulta for the initial simplex (simplex 1)are listed here. Obviously, vertex 2 gives the worst response. The hyperface centroid F and the difference P - W were calculated and also are listed. The P value was calculated by averaging the levels of each factor in vertexes I, 5,4, and 3. The difference F - W was calculated by vector subtraction of vertex 2 from fi;. factor level
vertex 1 5 4
; W (vertex2) P-W
A D L 0.00 0.00 0.00 21.85 21.85 21.85 16.39 92.56 -76.17
21.85 21.85
92.56 34.07 21.85 12.22
21.85 92.56 21.85 34.07 21.85 12.22
1
response
0.00
86.4 54.4 24.3 -82.4
92.56 21.85 21.85 34.07 21.85 12.22
-116.3 ,
The first choice in movement of a simplex is by a reflection, and Eq. 2 is used to calculate the new vertex that will replace vertex 2. The new
vertex then is the vector sum of P and F - W, giving: vertex 6
A
-59.78
D 46.29
L 46.29
W 46.29
The calculation indicates that a reflection of vertex 2 results in a boundary violation with respect to factor A. According to the rules,an arbitrarily very low response, -1000, is assigned to this vertex and it is not retained. Instead, C, (contraction) according to Eq.5 is performed half of the difference H W is subtracted from P. This procedure results in the factor levels for vertex 7:
vertex 7
A 54.48
D 27.96
L 27.96
W 27.96
Evaluation of this vertex results in a total desired response of 15.3% (Table 111, and vertex 7 is accepted to replace vertex 2 in simplex 2. Of the vertexes in the new simplex (1,3,4,5, and 7), vertex 3 now has the worst response, and the simplex research continues accordingly. REFERENCES (1) 1969. (2) 64,966 (3) (4)
L. C. Schroeter, “Ingredient X,” Pergamon, New York, N.Y.,
N. R.Bohidar, F. A. Restaino, and J. B. Schwartz, J. Pharrn. Sci., (1975).
S. Dincer and S.Ozdurmus, ibid., 66,1070 (1977). C. Reier, R. Cohen, S. Rock, and F. Wagenblast, ibid.. 57,660
(1968). (5) D. E. Former, Jr., J. T.Buck, and G. S. Banker, ibid., 59,1587 Journal of Pharmaceutical Sciences I 1141 Vol. 69, No. 10, October 1880
(1970). (6) J. B. Schwartz, J. R. Flamholz, and R. H. Press, ibid., 62,1165 (1973). (7) Ibid., 62,1581 (1973). (8) G.E.P. Box and N. R. Draper, “Evolutionary Operation,” Wiley, New York, N.Y., 1969. (9) M. H. Rubenstein, Drug Cosmet. Ind., 116,44(1975). (10) T.N.DiGaetano, Bull. Parenter. Drug Assoc., 29,183 (1975). (11) W.Spendley, G. R. Hext, and F. R. Himsworth, Technometrics, 4,441 (1962). (12) J. A. Nelder and R. Mead, Comput. J., 1,308 (1965). (13) D. E.Long, Anal. Chirn. Acta, 46,193 (1969). (14) F.P.Czech, J. Assoc. Off. Anal. Chem., 56,1489 (1973). (15) Ibid., 56,1496 (1973). (16) S.N.Deming and S. L. Morgan, Anal. Chem., 45,278A (1973). (17) S.L. Morgan and S. N. Deming, ibid., 46,1170 (1974). (18) P.G.King, S. N. Deming, and S. L. Morgan, Anal. Lett., 8,369 (1975). (19) G. L.Ritter, S. R. Lowry,C. L. Wilkins, and T. L. Isenhour, Anal. Chem., 47,1951 (1975). (20) T. Failam, C. L. Wilkins, T. R. Brunner, L. T. Soltzberg, and S. L.Kaberline, ibid., 48,1768 (1976). (21) W.K. Dean, K. Y. Heald, and S. N. Deming, Science, 189,805
(1975). (22) F.Darvas, J. Med. Chem., 17,799 (1974). (23) R. D.Gillian, W. P. Purcell, and T. R. Bosin, Eur. J.Med. Chem., 12.187 (1977). (24) N. K. Zubkova, S. A. Minina, and A. S.Bril. Khim. Farm. Zh.. 11, 107 (1977);through Chem. Abstr., 88,79039t (1978). (25) C. Beyer, Arch. Pharm., 311,128 (1978). (26) B.S.Newman, Adu. Pharm. Sci., 2,203 (1967). (27) D. M. Olsson and L. S. Nelson, Teehnometrics, 17,45 (1975). (28) L. A. Yarbro and S. N. Deming, Anal. Chim. Acta, 73, 391 (1974). (29) D. M. Himmelblau. “Applied Nonlinear Programming,” McCraw-Hill, New York, N.Y., 1972,p. 148. (30) V. Chew, “Experimental Design in Industry,” Wiley, New York, N.Y., 1958,pp. 108-137. (31) S. N. Deming, S. L. Morgan, and M. R. Willcott, Am. Lab., Oct. 1976,13. ACKNOWLEDGMENTS The authors thank Mr. W. Morgenstern for assistance in computer programming for the generation of three-dimensional plots.
IR and X-Ray Diffraction Study of Chlorpheniramine Maleate-Montmorillonite Interaction M. SANCHEZ CAMAZANO A. DOMINGUEZ-GIL +x
*, M. J. SANCHEZ *, M. T. VICENTE *, and
Received June 4,1979,from the V e n t r o de Edafologia y Biologia Aplicada del C.S.I.C. and the thactical Pharmacy Department, Faculty of Accepted for publication March 27,1980. Pharmacy. University of Salamanca, Salamanca, Spain.
Abstract 0 As a contribution to studies on the adsorption of drugs by clay minerals aimed at achieving sustained action in the oral administration of antihistamines. the adsorption of chlorpheniramine maleate by sodium montmorillonite was studied by X-ray diffraction and IR spectroscopy. The results indicated that the chlorphenirammonium ion penetrated into the interlayer space of montmorillonite, producing an increase in the basal spacing, dml, of the silicate. This increase was influenced, as was the amount adsorbed, by the pH and the concentration of the chlorpheniramine maleate solution. At pH 7.0, the amount adsorbed was close to the exchange ca acity. and the complex formed had a basal spacing of 17.14 A (A = 7.54 The only mechanism responsible for the interaction was cation exchange. The complex at 17.14 A must be a monolayer with the benzene rings positioned perpendicular to the surface of the oxygen atoms.
1).
Keyphrases Chlorpheniramine maleate-mechanism of interaction with montmorillonite 0 Montmorillonite-mechanism of interaction with chlorpheniraminemaleate o Adsorption-mechanism of interaction of chlorpheniramine with montmorillonite Clays-montmorillonite, mechanism of interaction with chlorpheniramine maleate
The presence of clays in the GI tract is of biopharmaceutical interest because of the resultant decrease in the bioavailability of orally administered drugs and the possibility of continued action following oral administration by means of a programmed adsorption speed, thus ensuring efficient plasma drug levels over long periods. After the first reports on the adsorption of organic ions (1,2)and nonionic organic molecules of a polar character by montmorillonite (3-5), it was demonstrated that numerous organic molecules can be adsorbed by clays, 1142 I Jownal of Pharmaceutical Sciences Vol. 69,No. 10, October 1980
especially by montmorillonite (6). Investigations also were carried out on the interaction mechanisms (7-10) and on the factors affectingthe interlayer expansion of clays with organic compounds (11-16). Among these factors is the influence of the phase (liquid or vapor) with which montmorillonite is equilibrated (17). BACKGROUND Numerous reporta have been concerned with the applications of clay minerals in pharmacy (18,191,and some presented the results of drug adsorption by montmorillonita (20).However, few reports have dealt with the mechanisms of adsorption responsible for the interaction of drugs with clays. One study concerned the nature of the bonding in adsorbates of montmorillonite with diazepam and with various benzodiazepine derivatives (21). Another study considered the adsorption of clindamycin and tetracycline by montmorillonite using adsorption isotherms, X-ray diffraction, and IR spectroscopy (22).One report discussed the adsorption of cationic, anionic, and nonionic drugs by montmorillonite using dissolution dialysis (23);chlorpheniramine maleate was among the drugs investigated. Knowledge of the mechanism of interaction of clays with drugs should allow prediction of whether a given compound will be adsorbed by a c l q based on the properties of the clay and the structure of the drug. The objective of this investigation was to study the interaction between chlorpheniramine maleate and sodium montmorillonite by X-ray diffraction and IR spectroscopy to obtain direct evidence regarding the penetration of the organic cation into the hterhyer space of the clay and the interaction mechanism. The study also considered the possible or. ientation adopted by the organic cations within the interlayer space of montmorillonite. X-ray diffraction and IR spectroscopy generally are considered as the
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0022-3549/80/ 7000-1 142$01.00/0 1980, American Pharmaceutical Association