JOURNAL OF BIOPHARMACEUTICAL STATISTICS Vol. 14, No. 3, pp. 757–779, 2004
Three-Stage Sequential Statistical Dissolution Testing Rules# Yi Tsong,1,* Meiyu Shen,1 and Vinod P. Shah2 1
Office of Biostatistics and 2Office of Pharmaceutical Sciences, Center for Drug Evaluation and Research, U.S. Food and Drug Administration, Rockville, Maryland, USA
ABSTRACT The U.S. Pharmacopoeia (USP) general monograph provides a standard for dissolution compliance with the requirements as stated in the individual USP monograph for a tablet or capsule dosage form. The USP monograph sets performance limit on dissolution in terms of a specific percentage Q that the drug product is required to be dissolved at a specified time. Japan Pharmacopoeia provides acceptance rules different from USP. However the objective of the acceptance rules was not defined in terms of the inference of the whole lot by either USP, European Pharmacopoeia (EP) or Japan Pharmacopoeia (JP). The operating characteristics’ curves of these rules are all shown to be sensitive to the true mean dissolution and do not reject a lot which has a large percentage of tablets that dissolve with less than the specified limit Q. This is especially true when the mean dissolution is close to the specification value. We proposed that the goal of the dissolution test sampling plan is to accept a lot at least 90% of the tablets dissolved more than a pre-specified amount Q at the specific time. The group
#
The views expressed in this paper are the authors’ professional opinions. They do not represent the official positions of the U.S. Food and Drug Administration. *Correspondence: Yi Tsong, HFD-705, 12720 Twinbrook Parkway, Rockville, MD 20857, USA; Fax: 301-827-4576; E-mail:
[email protected]. 757 DOI: 10.1081/BIP-200025689 Copyright # 2004 by Marcel Dekker, Inc.
1054-3406 (Print); 1520-5711 (Online) www.dekker.com
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Tsong, Shen, and Shah sequential procedure derived accordingly is shown to outperform both USP and JP in controlling the type I error rate under normality assumption. Key Words: Dissolution test; Sampling inspection; Hypothesis of coverage probability; Operating characteristics; 3-Stage test; Group sequential test; Tolerance interval.
I. INTRODUCTION The US Food and Drug Administration (US FDA) sets performance limit on dissolution in terms of a specific percentage Q that the drug product is required to be dissolved at a specified time. Based on inferences from correlated bioavailability studies, these dissolution specifications were intended to assure that at least 80% of the labeled quantity of drug was available. When carried out appropriately, dissolution analysis of pharmaceutical drugs has been identified as an important test to assure the quality of a drug product (Banakar, 1991). To ensure that the manufacturing procedure yields a product equivalent to what formed the basis of NDA (New Drug Application) approval, dissolution testing is required by US FDA to confirm that a product’s dissolution is consistent with the declared dosage (Bisaillon and Tawashi, 1971; Cox et al., 1978; Wood, 1991). The USP dissolution test provides as a standard to determine compliance with the dissolution specification(s) stated in the individual monograph for a tablet or capsule dosage form to be marketed in the United States. In the United States, the dissolution apparatus is typically constructed so that the dissolution testing may be performed on six tablets or capsules simultaneously, and so that samples may be drawn at various pre-determined time intervals to generate a dissolution profile. Each lot of tablets or capsules must meet the USP requirements as set forth in the compendium in order to be released as a satisfactory drug product in United States (Bisaillon and Tawashi, 1971; Cox et al., 1978; Wood, 1991). The general objective of the acceptance sampling procedure is to reject lots with large percentage of tablets or capsules dissolved less than Q, and accept lots with a small percentage of the tablets or capsules dissolved less than Q. The acceptance rules proposed by U.S. Pharmacopoeia (USP XIX, 1975; USP XX, 1980; USP XXII, 1990) have an important function in manufacturing processes. Although the rules were proposed for testing one sample (consisting of 6 to 24 tablets depending on the number of steps required for acceptance decision on the sample), typically, the manufacturer will accept or reject the lot based on the result of the tested sample and take more testing samples if any technical problem is detected. However, the objective of USP dissolution testing sampling plan was not clearly specified in terms of the population parameter. Without a clear objective referring to the lot, The USP dissolution test sampling plan was modified no fewer than five times since its first publication in USP XIX (1975). The two-stage acceptance rule proposed in the USP XIX is a sampling sequential procedure by attributes. In the first stage, randomly selected 6 tablets
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(or capsules) are compared with the label specification Q. The unit fails if three or more tablets have dissolution values less than Q. The unit is accepted if all six tablets have dissolution values of no less than Q. Otherwise, additional 6 tablets are subjected to testing. The unit (with a sample of 12 tablets) is accepted if at least 10 of the 12 tablets have dissolution values of no less than Q, rejected otherwise. It is revised in USP XX (1980) to a three-stage sequential procedure. In the first stage, 6 randomly selected tablets (or capsules) are compared with the label specification Q. The unit is accepted if all 6 tablets have dissolution values of more than Q þ 5%, rejected if at least three tablets have less than Q 15%. Otherwise, additional 6 tablets are tested. The unit is accepted if all 12 tablets have dissolution values of more than Q 15% and with the average dissolution being greater than Q. The unit is rejected if three or more tablets have dissolution values of less than Q 15%. Otherwise, additional 12 tablets will be added to the 12 tablets already tested. At the third stage, the unit is accepted if no more than 2 tablets (out of 24 tested) have dissolution values less than Q 15% with the average dissolution being greater than Q. The unit is rejected otherwise. Pheatt (1980) criticized that the USP XX procedure offered no improvement over the USP XIX procedure in the ability to discern acceptable and unacceptable materials and required a larger sample size to make a decision. Givand (1980) made the same criticism of the USP XX procedure and proposed a modification that replaces Q 15% by Q 5% in stages 2 and 3. Both authors drew the conclusions based on the operating characteristic curves generated by computer simulation. USP XXII (1990) published the 3-stage procedure that was the modified version of US XX procedure by including one condition that all 24 tablets have dissolution values of no less than Q 25% for acceptance at stage 3. U.S. Pharmacopoeia revised the sampling plan again in USP XXIII (2000). The USP XXIII rule accepts the unit at the first stage if all 6 tablets dissolve more than Q þ 5%. Otherwise, it accepts the unit at the 2nd stage, if all 12 tablets dissolve more than Q 15% and their sample mean is greater than Q. Otherwise, at the 3rd stage, it accepts the unit, if at least 22 tablets dissolve more than Q 5% with the sample mean of 24 tablets greater than Q, and the minimum value is greater than Q 15% (Fig. 1). European Pharmacopoeia (HARMEURODA, 2001) uses the same rule as USP XXIII. Japan Pharmacopoeia XIII (JP XIII, 1996; Katori et al., 1998) proposed a sampling plan based on sampling mean that it accepts the unit when X 0.8226s > Q and min(X) Q 10% in the first stage with 6 tablets, where X and s are the sample mean and sample standard deviation, respectively. If the unit is not accepted at the first stage, sample another 6 tablets, and it accepts the unit (of 12 tablets) at the 2nd stage, if X 0:5184s > Q and min(X) Q 10%. It rejects the unit otherwise (Fig. 2). Tsong et al. (1995) considered the statistical inference aspect of the dissolution test sampling plan and criticized procedures stated in all documents from USP XIX to USP XXII for the lack of the discriminating capability. To be more specific, the procedures often fail to reject batches or lots with large percentage of below specification tablets. Based on computer simulation, they showed also that the discriminating capability of USP rules varied with the let mean of the lot. They
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Figure 1. USP XXIII three stage acceptance rule.
proposed rules that would accept a lot only if there are more than 90% of the tablets dissolved more that Q at the specified time. In addition, they proposed a 3-stage maximum likelihood estimation procedure (Fig. 3) that outperformed the USP plans for the discriminating capability. In this paper, we consider the inference of a lot through hypothesis testing setup by testing against the coverage probability of no more than P of tablets dissolving more than Q. By rejecting the null hypothesis, the lot or batch is accepted. A three stage sequential procedure is proposed to control the type I error rate that a failed lot is falsely accepted as an acceptable lot. A comparison between the proposed 3-stage group sequential plan with the USP XXIII plan and JP plan is also performed through computer simulations.
II. HYPOTHESIS TESTING FOR QUALITY CONTROL BASED ON DISSOLUTION Consider the objective of the dissolution test sampling acceptance plan is to accept a lot if more than P percent tablets of the lot dissolve at least Q percent of the label claim at the specific time. The hypotheses involved in the
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Figure 2. Japan rule (sampling by measure-CI rule).
testing are H0 : PrðX QÞ P
vs:
Ha : PrðX QÞ > P
ð1Þ
where X is the amount dissolved of any randomly selected tablet. Hence a lot is accepted when the null hypothesis is rejected. An a-level test can be derived by controlling the type error rate for testing H0 : PrðX QÞ ¼ P
vs:
Ha : PrðX QÞ > P
Assuming X follows N (m, s2) distribution, then H0 : PrðX QÞ ¼ P implying that Pr½ðX mÞ=s ¼ ðQ mÞ=s P Hence, (1) can be rewritten as H0 : Pr½ðX mÞ=s ðQ mÞ=s ¼ P Ha : Pr½ðX mÞ=s ðQ mÞ=s > P Since (X m)=s follows N (0, 1), which means that under H0, ðQ mÞ=s ¼ z1P
ð10 Þ
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Figure 3. Proposed 3-stage group sequential testing method.
Hence H0 of (1) can be rewritten again as H0 : ðQ mÞ=s ¼ z 1P
ð2Þ
And Ha can be written as Ha : ðQ mÞ=s < z 1P Hypotheses of (2) can also be rewritten as H0 : m ¼ Q sz 1P Ha : m > Q sz 1P
ð20 Þ
The above hypotheses in (2) can be tested with type I error rate a for a a-level significance test as follow. To reject H0 of (2), one needs to show that m þ sz 1P > Q
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When s is known, we may use the Wald’s statistic p p ZðN Þ ¼ ½ðX QÞ þ sz1P =ðs= N Þ ¼ ½ðX QÞ=s þ z1P N where N is the sample size, X is the sample mean. Under the null hypothesis, Z(N) has the asymptotic distribution N(0, 1). Z(N ) can be used to test against z1a (i.e., reject H0 if Z (N ) > z1a) for a a-level significance test. When s is unknown, we consider the statistic p TðN Þ ¼ fX Q þ sðN Þz1P g=½sðN Þ= N p ¼ f½X Q þ sz1P sz1P þ sðN Þz1P g=½sðN Þ= N
ð3Þ
where s(N ) is the unbiased estimate of s. Under H0: m ¼ Q s z1P, T(N ) can be re-written as p p p TðN Þ ¼ ½ðX mÞ N =s Nz1P =½sðN Þ=s þ Nz1P p p p Under H0, [(X mÞ N =s Nz1P =½s ðN Þ=s tðN 1; Nz1P Þ, an p N1 degrees of freedom noncentral t distribution with noncentrality value N z1P . p is to be compared with the ð1 aÞ 100th percentile, Hence TðN Þ Nz 1P p tðN 1;p Nz1P ; ð1 aÞÞ of p the noncentral t distribution. H0 is rejected if TðN Þ N z1P > tðN 1; Nz1P ; ð1 aÞÞ for an a-level significance test. K-Stage Group Sequential Procedure In the sequential setting with sample size N and K interim looks T1 ; T2 ; . . . ; TK and final look, under a pre-specified a spending function, the value a at the kth stage, ak can be determined under the constraint that the experimental type I error rate is a and nk is the cumulative sample size. Based on a predetermined a spending function such as O’Brien Fleming spending function (Lan and DeMets, 1983; O’Brien and by the perFleming, 1979) the critical value Cðak Þ at the kth-stage can be calculated p centiles of the noncentral t-distribution. Hence, Cðak Þ ¼ tðnk 1; nk z1P ; 1 ak Þ, the ð1 ak Þth quantile of t-distribution with degrees of freedom nk 1. We consider the traditional 3-stage procedure but use the O’Brien Fleming boundaries to determine ak at the kth stage in the rest of the paper. Modification based on different K and a spending function can be easily carried out. The statistic at each k-stage is defined by p p Tðnk Þ nk z1P ¼ ½ X Q þ sðnk Þz1P =½sðnk Þ= nk p We reject H0 and stop sampling at the kth look if Tðnk Þ nk ZlP > CðaK Þ for k ¼ 1, 2, . . . , K þ 1. We continue to the next stage otherwise. We reject the lot if H0 is not rejected at the final stage. In the remaining of this paper, we consider for immediate released drug product with Q ¼ 85%, P ¼ 0.90 and 0.80, a ¼ 0.05 and using O’Brien Fleming boundaries for
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a three-stage acceptance plan with n1 ¼ 6, n2 ¼ 12 and N ¼ 24. O’Brien and Fleming (1979) proposed the a-spending function with low a values at the early stages in order to give large power in comparison to test based on fixed sample size N . Accordingly, the three a values are a1 ¼ 0:0009,pa2 ¼ 0:00548, a3 ¼ 0:04439. Correspondingly boundary values for the tðnK Þ nk Zlp are Cða1 Þ ¼ 3:7498, C(a2) ¼ 2.5406, C(a3) ¼ 1.6626. Alternatively the three-stage dissolution testing sampling acceptance rule proposed can then be restated below (Fig. 3), At the first stage, sample 6 tablets, Accept the lot if X1 Q > A1 where X1 and s1 are the sample p mean and sample standard deviation of the 6 tablets, respectively, A1 ¼ Cða1 Þðs1 6Þ S1 Z0:1 : Otherwise, move to the 2nd stage and sample additional 6 tablets, Accept the lot if X2 Q > A2 where X2 and s2 are the sample mean and sample standard deviation of the 12 p tablets, respectively, A2 ¼ Cða2 Þðs2 12Þ S2 Z0:1 : Otherwise, move to the 3rd stage and sample additional 12 tablets, Accept the lot if X3 Q > A3 ; and sample standard deviation of the 24 where X3 and s3 are the sample mean p tablets, respectively, A3 ¼ Cða3 Þðs3 24Þ S3 Z0:1 . Otherwise reject the lot.
III. HYPOTHESIS TESTING AND TOLERANCE LIMITS Testing of hypothesis (1) can also be carried out by comparing the lower limit of a one-sided tolerance interval with Q. The 95% tolerance interval ( x Ks, 1) can be estimated by solving for K of the following equation (Faulkenberry and Daley, 1970), PrfPrðX > x Ks j ðX; SðN ÞÞ ¼ ðx; sÞÞ 1 Pg ¼ 0:95: K is often called the tolerance factor. Faulkenberry and Daley (1970) proposed to use p p K ¼ tðN 1; Nzð1PÞ ; 95%Þ= N p t distribution where t (N 1, Nz(1 P), 95%) is the 95th percentile of the noncentral p with degrees of freedom N 1 and the noncentral parameter N z(1 P). Decision to accept the lot can be made if X Ks > Q. This result is consistent with the result derived in Sec. II. IV. SIMULATED EXAMPLES To illustrate the application of USP XXIII, JP XIII and the proposed procedure, we consider the following examples.
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Examples Three selected samples were generated from batches of tablet distributed as normal distribution with mean 97% and standard deviation 12.5% and with mean 97% and standard deviation 6%. The three samples were selected with different estimate values of standard deviation. The dissolution values at 30 min of each sample of 24 tablets (6 in stage 1, 6 in stage 2 and 12 of stage 3) are given in Table 1 for mean ¼ 97%, standard deviation ¼ 12.5% and in Table 2 for mean ¼ 97%, standard deviation ¼ 6%. The cumulative sample mean and sample standard deviation at each of the three stages of the six samples are given in Table 3. The estimate of standard deviation has a large variation among the three samples generated from mean 97%, standard deviation 12.5%. With Q ¼ 85%, the first sample has a low minimum value of 69.71% at the first stage and fails both USP XXIII and JP XIII because of the minimum value restriction. The other two samples passing at the first stage have large minimum values and small sample standard deviations at the first stage. Meanwhile, the first two samples with large estimates of standard deviation fail the proposed procedure at both 80% and 90% levels. Sample #3, with small sample standard deviation passes the proposed
Table 1. Data generated for example 1 batch mean, m ¼ 97%, standard deviation, s ¼ 12.5%. Sample 1
Stage 1
2
3
Dissolution values 77.59 126.13 101.42 69.71 107.42 115.23 74.81 94.47 108.61 104.51 99.02 73.07 109.17 110.06 78.48 68.94 95.79 90.15 94.94 91.77 82.74 90.14 98.38 106.93
Sample
Stage
2
1
2
3
Dissolution values 93.61 96.92 101.47 101.11 112.56 102.40 103.39 106.97 71.16 71.77 94.34 88.36 85.01 91.38 94.79 93.44 86.88 79.72 119.82 76.77 88.06 97.53 121.78 76.33
Sample
Stage
3
1
2
3
Dissolution values 93.05 92.04 95.58 98.09 94.20 99.33 87.22 98.03 88.87 97.46 79.10 88.78 109.74 111.01 92.18 103.68 105.65 98.91 113.25 89.60 99.58 108.42 103.02 112.44
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Table 2. Data generated for example 1 batch mean, m ¼ 97%, standard deviation, s ¼ 6.0%. Sample
Stage
1
1
3
2
3
Dissolution value 94.00 97.61 85.70 94.77 83.97 81.14 91.85 98.00 92.05 111.90 94.64 100.40 98.68 93.17 97.80 95.09 88.75 94.29 92.97 90.35 103.58 98.85 92.64 99.76
Sample
Stage
2
1
2
3
Dissolution value 105.19 97.32 108.47 97.35 82.27 97.09 105.13 93.42 97.36 96.95 99.45 109.04 98.65 91.25 96.85 89.25 95.82 87.63 91.56 99.83 106.74 98.71 90.70 104.30
Sample
Stage 1
2
3
Dissolution value 98.01 96.87 92.11 82.37 98.02 101.14 93.96 93.80 98.12 99.83 92.75 94.33 90.80 86.73 105.11 96.86 96.57 97.59 99.55 90.01 99.79 105.52 95.00 96.00
procedure at 80% level but fails at the 90% level. Note that only values of the first two stages are sampled using JP XIII (See Table 3). For the samples generated from mean 97%, standard deviation 6%, all three samples fail USP XXIII at the first stage due to the minimum value. They all pass at the second stage. Sample #2 and #3 pass the JP XIII at the first stage. Sample #1 fails JP XIII because of the low sample mean of 89.53%. But it passes JP XIII at the second stage. All three samples pass the proposed procedure at 80% level at the third stage. When a 90% level is required, only sample #3 passes at the third stage (See Table 3). V. COMPARISONS OF THE OPERATING CHARACTERISTICS OF USP XXIII, JP PROCEDURES WITH PROPOSED PROCEDURE The operating characteristics of the proposed hypothesis testing procedure can be derived mathematically. However, because of complicated structure of the USP XXIII and JP XIII procedures, the curves of operating characteristics are generated with computer simulations. Let us assume that the dissolution value of any randomly selected tablet X follows a normal distribution N (m, s2). As stated
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Table 3. Statistics and results of the dissolution sampling acceptance tests. Proposed procedure Sample and stage
Mean
Standard deviation
Minimum
USP XXIII
JP
80%
90%
21.87 18.30 15.37 6.42 12.84 13.63 2.86 5.88 8.85
69.71 69.71 68.94 93.61 — — 92.04 — —
Failed Failed Failed Passed NA NA Passed NA NA
Failed Failed NA Passed NA NA Passed NA NA
Failed Failed Failed Failed Failed Failed Failed Failed Passed
Failed Failed Failed Failed Failed Failed Failed Failed Failed
6.77 8.17 6.44 9.06 7.35 6.77 6.74 4.94 5.24
81.14 81.14 — 82.27 82.27 — 82.37 82.37 —
Failed Passed NA Failed Passed NA Failed Passed NA
Failed Passed NA Passed NA NA Passed NA NA
Failed Failed Passed Failed Failed Passed Failed Failed Passed
Failed Failed Failed Failed Failed Failed Failed Failed Passed
Three samples from N(97%, 12.5%) Sample 1 Stage 1 Stage 2 Stage 3 Sample 2, Stage 1 Stage 2 Stage 3 Sample 3, Stage 1 Stage 2 Stage 3
99.58 96.00 94.56 101.34 95.34 93.93 95.38 92.64 98.30
3 Samples from N(97, 6%) Sample 1 Stage 1 Stage 2 Stage 3 Sample 2, Stage 1 Stage 2 Stage 3 Sample 3, Stage 1 Stage 2 Stage 3
89.53 93.84 94.67 97.95 99.08 97.51 94.75 95.11 95.87
in Sec. II, the relationship among P(¼ Pr(X Q)), the proportion of the lot that dissolves at least Q at the specified time, and m and s can be stated below, ðQ mÞ=s ¼ z1P Alternatively, it can be written as s ¼ ðQ mÞ=z1P Note that m < Q if P < 0.5 from hypothesis (1). We will study the lot acceptance probabilities with small P when the lot mean is below Q (¼85%) and with large P value when the lot mean is greater than Q. The simulations will consist of the following configurations, m ¼ 80%; 84%; 87%; 90%; 95% and 100%: For m ¼ 80% and 84%; P ¼ 0:10 to 0:45 by 0:05; For m ¼ 87%; 90%; 95% and 100%; P ¼ 0:50 to 0:95 by 0:05: For each configuration, 5000 samples will be generated.
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Results of the Simulation Study The OC curves (acceptance probabilities) of USP XXIII, JP XIII and the proposed procedure for lot mean ¼ 95% and 100% are plotted against Relative Standard Deviation (RSD) in Figs. 4 and 5. It shows that all three procedures have higher acceptance probability at same RSD when the lot mean is greater. With mean ¼ 95% the lot is almost certain to be rejected by the proposed procedure when RSD is greater than 14%. On the other hand, the acceptance probabilities using the USP XXIII or JP XIII are near 80%. Even at RSD larger than 50%, the acceptance probability using USP XXIII is greater than 5% and it is greater than 10% if using JP XIII. Similar large acceptance probability was shown for lot mean ¼ 100% and RSD between 10% and 20% for USP XXIII and JP XIII criteria. We present the results of lot acceptance probability against the proportion of lot with tablets dissolved greater than Q in Figs. 6–11. Figure 6 shows that the lot acceptance probability using USP XXIII can be more than 2% when the lot mean is only 80%. It is interesting to note that it peaks at almost 2.5% when 66% (i.e., 1–33%) of lot dissolve less than Q. The JP XIII gives more than 0.5% acceptance probability under the similar condition. Figure 7 shows that the lot acceptance probability using USP XXIII can be more than 20% when the lot mean is 84% and about 60% (i.e., 1–40%) of the lot dissolve less than Q. It is near 0% probability that the lot can be accepted under the proposed procedure. The undesirable behaviors of USP XXIII were demonstrated earlier in Tsong et al. (1995). As for lot with mean larger than Q, it is shown in Figs. 6–9 that the proposed procedure has the identical OC curves under different lot mean indicating that the proposed procedure is lot mean independent. It has 5% acceptance probability when
Figure 4. OC curves of acceptance rules for dissolution testing. Mean ¼ Q þ 10%.
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Figure 5. OC curves of acceptance rules for dissolution testing. Mean ¼ Q þ 15%.
Figure 6. OC curves of acceptance rules for dissolution testing. Mean ¼ Q 5%.
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Figure 7. OC curves of acceptance rules for dissolution testing. Mean ¼ Q 1%.
Figure 8. OC curves of acceptance rules for dissolution testing. Mean ¼ Q þ 2%.
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Figure 9. OC curves of acceptance rules for dissolution testing. Mean ¼ Q þ 5%.
10% of the lot dissolve no more than 85%. The power of accepting the lot is lower than 80% when there is more than 3% dissolved less than Q when using the small sample size of 6=12=24 at the three stages. On the other hand, the acceptance probability of USP XXIII and JP XIII change with the unknown lot mean. For example, When the lot mean is 87%, a lot has more than 80% probability of acceptance even when 38% of the lot dissolve less than Q using the USP XXIII procedure. However, when the lot mean becomes larger, the probability reduces. With the same percentage of lot dissolving less than Q, the acceptance probability becomes 20% when the lot mean is 90%. The probability is around 5% when the lot mean is 95% and less than 5% when the lot mean is 100%. The JP XIII gives a more stable probability of acceptance regarding to the changes of lot mean though the probability of acceptance reduces when the lot mean increases when the percentage of lot dissolve less than Q is the same. With the closed formulae, the procedure can be easily modified for different combinations of number of stages, sample sizes at each stage, coverage percentage requirement and even the sequential boundary. For example, we studied the OC curves of the procedure using larger sample sizes. A comparison of three different sample sizes (cumulative sample size 6=12=24, 12=24=48 and 18=36=72) are shown in Fig. 12. The power of the procedure is greatly enhanced with 18=36=72 sample, when 5% of the lot dissolve less than Q, the probability of acceptance increases from 30% with 6=12=24 size to 65% with 18=36=72 size.
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Figure 10. OC curves of acceptance rules for dissolution testing. Mean ¼ Q þ 10%.
Figure 11. OC curves of acceptance rules for dissolution testing. Mean ¼ Q þ 15%.
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Figure 12. The influence of sampling size on OC curves of acceptance rules.
VI. EFFECTS OF NON-NORMALITY AND ZERO TOLERANCE CRITERIA OF THE USP PROCEDURE The effect of non-normality on two-sided tolerance limits was studied by Rao et al. (1973). They computed the actual confidence level r of a tolerance interval with various P and confidence level 1 a ¼ 0.90. The non-normal data were simulated from a density function of Z ¼ (X m)=s such that fðZÞ ¼ G0 ðZÞ ðl3 =4ÞGð4Þ ðZÞ þ ðl4 =24ÞGð5Þ ðZÞ þ ðl32 =72ÞGð6Þ ðZÞ where E(X) ¼ m, V(X) ¼ s2, G(r)(Z) is the (r 1)th derivation of the standard normal density, l3 and l4 are, respectively, the standardized 3rd and 4th cumulants of X. The density f(Z) was considered to be moderately non-normal (Gayan, 1949). Rao et al. also used 0 < l4 < 2.4 and l32 0.2 in order to make the density function unimodal and positive definite (Barton and Dennis, 1952). The results of Rao et al. indicated that r is smaller than 0.90 (i.e., 1 a) when P is 0.90 or greater and sample size is larger than 6. The r value decreases when the sample size increases. These results imply that the proposed sequential approach may be slightly liberal (lower producer’s risk and higher consumer’s risk) when the dissolution data are not normally distributed. The general perception is that the minimum value criterion (i.e., min(X) Q 15) of USP XXIII provides a sense of protection against approving a lot with tablets
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of extremely low dissolution values. Data with such extreme value often represent a very skewed distribution. The proposed sequential procedure based on normal tolerance limit provides no criterion for such protection. In order to study the effect of such criterion in lot rejection, a small simulation study was conducted to find the probability of rejecting a lot by the minimum criterion of USP XXIII procedure while it was accepted with the proposed sequential procedure. Note that by accepting the lot with the proposed procedure means that no less than 90% of the tablets dissolved greater than Q (i.e., 85% of label claim). In order to emphasize the effect of the minimum criterion of USP XXIII, we consider the distribution of the dissolution data of a lot is uni-modal left skewed. A simple simulation is carried out on dissolution data that follows a Beta distribution. Beta distribution was used in data generation because the skewedness of the distribution can be easily determined by one of the two population parameters. The probability density function of beta distribution (Johnson et al., 1995) is of the form fðyÞ ¼ ½1=Bðb; aÞ½ðy LÞb1 ðU yÞa1 =½ðU LÞaþb1 ;
LyU
where y is the dissolution value; a > 0, b > 0 are the two beta parameters; B(a, b) is the beta function; L and U are the minimum and maximum dissolution value, respectively.
Figure 13. Simulated beta distributions.
B
2
a
15 15 15 15 15 15 15 17 17 17 17 17 17 17 19 19
110 115 120 125 130 135 140 110 115 120 125 130 135 140 110 115
Max value
Parameters of beta distribution
97.06 101.47 105.88 110.29 114.71 119.12 123.53 98.42 102.89 107.37 111.84 116.32 120.79 125.26 99.52 104.05
Mean 171 102 62 33 13 14 8 69 39 19 12 5 2 2 28 18
# failed by USP min (x) < Q 15% criterion 7 11 13 16 7 13 8 3 8 14 9 5 2 2 2 8
Accepted by TL of Col. 5 4.09 10.78 20.97 48.48 53.85 92.86 100.00 4.35 20.51 73.68 75.00 100.00 100.00 100.00 7.14 44.44
% of Col. 5
Based on 90% tolerance limit criterion
33 64 55 33 13 14 8 22 32 19 12 5 2 2 17 17
Accepted by TL of Col. 5
(continued)
19.30 62.75 88.71 100.00 100.00 100.00 100.00 31.88 82.05 100.00 100.00 100.00 100.00 100.00 60.71 94.44
% of Col. 5
Based on 80% tolerance limit criterion
Table 4. Number of 1000 simulated lot failed USP XXIII by minimum criterion and the number of lots failed by USP XXIII minimum criterion but accepted by the proposed procedure.
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B 120 125 130 135 140 110 115 120 125 130 135 140 110 115 120 125
Max value 108.57 113.10 117.62 121.39 126.67 100.43 105.00 109.57 114.13 118.70 123.26 128.23 101.20 105.80 110.40 115.00
Mean 15 5 2 0 1 12 8 5 2 1 1 0 9 4 0 1
# failed by USP min (x) < Q 15% criterion 7 5 2 0 1 5 6 5 1 1 1 0 5 4 0 1
Accepted by TL of Col. 5 46.67 100.00 100.00 0 100.00 41.67 75.00 100.00 50.00 100.00 100.00 0 55.56 100 0 100.00
% of Col. 5
Based on 90% tolerance limit criterion
15 5 2 0 1 11 8 5 2 1 1 0 9 4 0 1
Accepted by TL of Col. 5
100.00 100.00 100.00 0 100.00 91.67 100.00 100.00 100.00 100.00 100.00 0 100.00 100.00 0 100.00
% of Col. 5
Based on 80% tolerance limit criterion
776
19 19 19 19 19 21 21 21 21 21 21 21 23 23 23 23
a
Parameters of beta distribution
Table 4. Continued.
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To make simulation simple we use only L ¼ 0, U ¼ 110 to 140 (percentage of label claim) by 5, a ¼ 12 to 23 by 1, b ¼ 2. A selected few of the density functions are shown in Fig. 13. Thousand samples of 24 tablets were randomly generated from each of the 84 beta distributions. Each sample is then tested through the three-stages of USP XXIII in order to determine if it is rejected by the minimum criterion. The rejected sample is then tested using the proposed procedure to identify the sample that is accepted by the proposed procedure. Results of a ¼ 15 to 23 by 2 are presented in Table 4. First, there is in general a small percentage of the simulated sample failed USP XXIII minimum criterion but accepted by the proposed three-stage group sequential procedure using P ¼ 0.90. Actually, as shown in column 6 of Table 4, the percentages are mostly less than 1. Second, samples simulated from highly skewed distribution (e.g., high a value for fixed maximum value U, or high U value for a fixed a value) may be more likely to be rejected by the USP XXIII minimum criterion but accepted by the proposed procedure. The results are less consistent with the proposed approach using P ¼ 0.80.
VII. CONCLUSIONS The proposed procedure is based on testing the hypothesis that acceptable lot has a proportion dissolving less than the pre-specified Q (percentage of label claim). With fixed sample size, the hypothesis testing procedure is equivalent to the decision based on one-sided tolerance limit that can be derived from Faulkenberry and Daley (1970). The 3-stage sequential procedure is based on O’Brien-Fleming boundary. As it is shown with simulation comparison, it has the following properties. (1) The procedure can be used to control type I error rate and avoid high false acceptance rate. (2) The proposed procedure has unique OC curve (curve of acceptance probability) that is invariant to the unknown lot mean. USP XXIII and JP XXIII are not capable to reject a lot with a large proportion dissolving less than the specification limits Q. As shown in Figs. 5 and 6, they have large acceptance probability for a lot even when the lot mean is lower than the specification limit Q. They have the largest acceptance probability when the lot mean is slightly greater than Q. With no closed form of the procedure in terms of the lot requirement, one is difficult to modify the procedure without intensive simulation study even when a manufacturer is willing to test for more tablets. It has been shown by Rao et al. (1973) that the effect of non-normality on tolerance limits is minor when the skewedness is moderate. It leads to somewhat liberal results. The zero tolerance feature with the minimum value criterion (i.e., min(x) > Q 15% in USP XXIII and min(x) > Q 10%) of the USP and JP procedures provide a guard against highly skewed dissolution distribution. A small simulation study based on highly skewed Beta distribution showed that the probability of a lot being rejected by the minimum value criterion but accepted
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by the proposed group sequential procedure may be less than 1% of all lots tested when using the proposed procedure with P ¼ 90%.
ACKNOWLEDGMENT The authors want to thank the two referees’ careful reading and comments that lead to the much improved version of the manuscript.
REFERENCES Banakar, U. V. (1991). Pharmaceutical Dissolution Testing. New York: Marcel Dekker, Inc. Barton, D. E., Dennis, K. E. (1952). The conditions under which Gram–Charlier and Edgeworth curves are positive definite and unimodal. Biometrika 39:425–427. Bisaillon, S., Tawashi, R. (1971). Influence of fluid motion in dissolution-rate determinations. J. Pharm. Sci. 60:1874–1877. Cox, D. C., Douglas, C. C., Furman, W. B., Kirkchhoefer, R. D., Myrick, J. W., Wells, C. E. (1978). Guidelines for dissolution testing. Phar. Tech. 2(4):40–53. Faulkenberry, G. D., Daley, J. C. (1970). Sample size for tolerance limits on a normal distribution. Technometrics 12(4):813–821. Gayan, A. K. (1949). The distribution of student’s t in random sample of any size drawn from non-normal universes. Biometrika 36:353–369. Givand, T. E. (1980). An evaluation of the dissolution test acceptance sampling plan of USP XX. Pharmacopoeia Forum Mar–Apr: 186–190. HARMEURODA. (2001). The European Pharmacopoeia Forum, 13.2 English version, pp. 315–326. Johnson, N. L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions. Vol. 2. 2nd ed. New York, NY: John Wiley & Sons. JP XIII. (1996). The Japanese Pharmacopoeia. 13th ed. The Society of Japanese Pharmacopoeia, pp. 33–36. Katori, N., Kaniwa, N., Aoyagi, N., Kojima, S. (1998). A new acceptance plan for the official dissolution test. JP Forum 7(4):166–173. Lan, K. K. G., DeMets, L. L. (1983). Discrete sequential boundaries for clinical trials. Biometrika 70:659–63. O’Brien, P. C., Fleming, T. R. (1979). A multiple testing procedure for clinical trials. Biometrics 35:549–556. Pheatt, C. B. (1980). Evaluation of U.S. Pharmacopoeia sampling plans for dissolution. J. Qual. Tech. 12(3):158–164. Rao, J. N. K., Subrahmaniam, K., Owen, D. B. (1973). Effect of non-normality on tolerance limits which control percentages in both tails of normal distribution. Technometrics 14(3):571–575. The United States Pharmacopoeia XIX. (1975). Mack Printing Company: Easton, Penn. The United States Pharmacopoeia XX. (1980). Mack Printing Company: Easton, Penn.
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The United States Pharmacopoeia XXII. (1990). Mack Printing Company: Easton, Penn. Tsong, Y., Hammerstrom, T., Lin, K., Ong, T. E. (1995). Dissolution test acceptance sampling plans. J. Biopharm. Statist. 5(2):171–183. U.S. Pharmacopoeia XXIII. (2000). Mack Printing Company: Easton Penn. Wald, A., Wolfowitz, J. (1946). Tolerance limits for a normal distribution. Ann. Math. Statist. 17:208–215. Wood, J. H. (1991). Dissolution rediscovered. In: Banakar, U. V., ed. Pharmaceutical Dissolution Testing. New York: Marcel Dekker Inc., pp. 413–421. Received September 2003 Accepted November 2003
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