Journal of Biopharmaceutical Statistics, 5(2), 171-183 (1995)

DISSOLUTION TEST ACCEPTANCESAMPLING PLANS*

Yi Tsong, J Thomas Hammerstrom, J Karl Un, i and Ting Eng Onl

IDivision of Biometrics, HFD-7102Division of Biopharmaceutics, HFD-423

Center for Drug Evaluation and ResearchU.S. Food and Drug Administration5600 Fishers Lane, Rockvile, Maryland 20857

Key words. Dissolution test; Sampling inspection; Operating char-acteristics; Three-stage test; Robustness

Abstract

The U.S. Pharmacopeia (USP) general monograph provides a stan-dard for dissolution compliance with the requirements as stated inthe individual USP monograph for a tablet or capsule dosage form.The acceptance rules recommended by USP have important roles inthe quality control process. The USP rules and their modificationsare often used as an industrial lot release sampling plan, where a lot isaccepted when the tablets or capsules sampled are accepted as proofof compliance with the requirement. In this paper, the operatingcharacteristics of the USP acceptance rules are reviewed and comparedto a selected modification. The operating characteristics curves showthat the USP acceptance rules are sensitive to the true mean dis-solution and do not reject a lot or batch that has a large percentageof tablets that dissolve with less than the dissolution specification.

*The views expressed in this paper are the authors' professional opinions. They do not represent theoffcial positions of the U.S. Food and Drug Administration.

171

Copyright ~ 1995 by Marcel Dekker, Inc.

..~~

172 Tsong et al.

This is especially true when the mean dissolution is close to thespecification. Although the modified USP rule, as proposed by Gi-vand, improves the performance substantially, it is stil sensitive tothe mean dissolution. In order to have the robustness (to the meandissolution), we propose an acceptance rule that is based on a sampling-by-measurements approach. It is preferable to the others in both lotdiscrimination and robustness when the assumption of normality holds.

1. Introduction

Dissolution testing was introduced during World War II as a military pro-curement tool to minimize the number of incidents of the passage of intacttablets through the full gastrointestinal system and their elimination in thefeces. A dissolution test was first introduced by U.S. Pharmacopeia (USP)XV in 1955. Due to a few cases of therapeutic failure with readily disinte-grated tablets recognized in the middle 1960s, and while dissolution testingappeared to be predictive for in vivo dissolution, the importance of the de-velopment of a dissolution procedure was recognized. In 1970, an officialdissolution procedure was first adopted in USP XVII (1,2).

By 1973, bioavailabilty differences between brands of a number of drugswere recognized and demonstrated through intensive screening of dissolutionprofiles. Hence, the Food and Drug Administration (FDA) set performancelimits on dissolution in terms ofa specific percentage that was required to

be dissolved at a specified time. Based on inferences from correlated bio-availabilty studies, these dissolution specifications were intended to assurethat at least 80% of the labeled quantity of drug was available.

When carried out appropriately, dissolution analysis of pharmaceuticaldrugs has been identified as an important test to assure the quality of a drugproduct (3). To ensure that the manufacturing procedure is yielding product

equivalent to that which formed the basis of new drug application approval,dissolution testing is required by the FDA to confirm that the dissolutionprofie is consistent with the declared dosage (4).

The USP dissolution test is provided as a standard to determine com-pliance with the dissolution specification(s) stated in the individual mono-graph for a tablet or capsule dosage form. The dissolution apparatus is typ-ically constructed so that the dissolution testing may be performed on sixtablets or capsules simultaneously, and so that samples may be drawn at var-ious predetermined time intervals to generate a dissolution profie. Each lotof tablets or capsules must meet the USP requirements as set fòrth in thecompendium in order to be released as a satisfactory drug product (1,2,4).

The objective of the acceptance sampling procedure is to reject lots withlarge numbers of tablets or capsules with dissolution values less than Q, and

Dissolution Test Acceptance Sampling Plans 173

to accept lots with small numbers of the tablets or capsules dissolved less

than Q. The acceptance rules proposed by the USP (5-7) have an importantfunction in the process. Although the rules were proposed for one sample(consisting of six to 24 tablets depending on the number of steps required foracceptance decision on the sample), typically the manufacturer wil accept orreject the lot based on the result of the tested sample and take more testingsamples if any technical problem is detected. Modifications of USP proce-dures have also been proposed and used by both manufacturers and FDAreviewers (8,9). These rules are based on acceptance sampling by attributes.In this paper, the operating characteristics of sampling-by-attributes rules arereviewed and compared to the proposed sampling-by-measurements rules, whichare based on maximum-likelihood estimation (10).

2. The USP Multiple-Stage Sampling Rules

In the last two decades, the USP proposed three acceptance rules. The two-stage acceptance rule proposed in the USP XIX (5) is a sampling by attributessequential procedure. In the first stage, an initial randomly selected six tablets(or capsules) are compared with the label specification Q. The lot or batchfails if three or more tablets have dissolution values less than Q. The lot orbatch is accepted if all six tablets have dissolution values of no less than Q.Otherwise, an additional six tablets are subjected to testing. The lot or batch(with a sample of 12 tablets) is accepted if at least 10 of the 12 tablets havedissolution values of no less than Q and is rejected otherwise (see Fig. 1).

i step 1, 6 tablets I

-------- --- ------+---- ------------i I II I IAccept i Reject_____________________ i ______________________

IIIII

V

I al 1 tablets ~ Q J tablets c: Q

i step 2, additional 6 tablets I

-- ------ ---------+-- -- ------------I II IAccept Reject10 tablets ~ Q otherwise

Figure 1. The USP xix two-stage acceptance sampling rule.

of

174 Tsong et al.

The three-stage acceptance rule proposed in USP XX (6) is also a se-quential procedure. In the first stage, an initial six randomly selected tablets(or capsules) are compared with the label specification Q. The lot or batchis accepted if all six tablets have dissolution values of more than Q + 5%and rejected if at least three tablets have less than Q - 15%. Otherwise, anadditional six tablets are tested. The lot or batch is accepted if all 12 tabletshave dissolution values of more than Q - 15% and with the average disso-lution being greater than Q. The lot or batch is rejected if three or more tabletshave dissolution values of less than Q - 15%. Otherwise, an additional 12tablets wil be added to the 12 tablets already tested. At the third stage, thelot or batch is accepted if no more than two tablets (out of 24 tested) havedissolution values less than Q - 15% with the average dissolution being greaterthan Q. The unit is rejected otherwise (see Fig. 2).

In USP XXII (7), the USP XX three-stage procedure was modified byincluding a condition that all 24 tablets .have dissolution values of no lessthan Q - 25% for acceptance at stage 3 (see Fig. 3).

: step 1, 6 tablets :

: all tablets ~ Q + 5% :

---- -- - ------ ----+----- --- ---- - ---I I II I IAccept : RejectiIIIII

V

I J tablets C Q - 15%

---- -- ---------- ----------.----------------- ----- ---- ---- - - - - --------- ---

------------- ---- - - - --- - - ----- ---i step 2, additional 6 tablets :-- ----------------------- -- -- - ---

: 1. Min (X) ~ Q - 15% :i II _ iI 2. X ~ Q :

------------- -----+- - - --- --- - -- ------I I II I IAccept : Reject----------------------- : ------------------------

iIIIIIII

V

: 3 tablets C Q - 15%-- -- -- - - - - -- - - - - -- - -- ---

- - - ----- --------- - - ----

- -- ----- -. - - - - --- ---. - - - -- - - ------I step 3, additional 12 tablets :- - ---- - -- - - -. - ----- -- - -- - - - -- -- ----- --- -- -- - -- - -- -+---- ----- -~- ----I II IAccept Rej ect

- -. - -- -- - - - - - - -- - - - -. - - - - - - --1. no more than 2 tablets

c Q - 15%otherwise

2. X ~ Q- -------- -------- -- --- - ------

Figure 2. The USP XX three-stage acceptance rule.

Dissolution Test Acceptance Sampling Plans

I step 1, 6 tablets :

- - -- - ---- -- - -- --+--- - - -- - ---- - ---I I II I IAccept : Rej ectiIIIII

V

L all tablets ). Q + 5% : : 3 tablets ~ Q - 15% I

: step 2, additional 6 tablets I

------ -- --------+-- - - ---- ------ --I I II I IAccept : Reject----------------------- I ----------------------

IIIIII

V

I 1. Min(x) ~ Q - 15% II 2. X ~ Q : : 3 tablets ~ Q - 15%1

: step 3, additional 12 tablets :

- ------- - ---- - - -+- - ---- - - - - - -----I lI IAccept Rej ect: 1. no more than 2 tablets: ~ Q - 15%: 2. Min(X) ~ Q -25%: 3. X ~ Q

otherwise

Figure 3. The USP XXII three-stage acceptance rule.

3. Modifed USP Procedures

175

I

Pheatt (8) criticized that the USP XX procedure offered no improvement overthe USP xix procedure in the abilty to discern acceptable and unacceptablematerials and required a larger sample size make a decision. Givand (9) madethe same criticism of. the USP XX procedure and proposed a modificationthat replaces Q - 15% by Q - 5% in stages 2 and 3. Both authors comparedthe USP procedures using operating characteristic curves generated by com-puter simulation. The modification can easily applied to the USP xxii pro-cedure and is given in Figure 4.

4. Sampling-by-Measurement Rules

A sampling procedure based on a studentized statistic can be derived basedon the assumption of normality of dissolution values at the prespecified test-ing time point. We describe a fixed-sample test rule and a three-stage rule

in the following.

176 Tsong et al.

-- - - - - -- - - - - - - ~ - - - - - --: step 1, 6 tablets :

-- - - --- - - - - - - - - -+- - - - - - - - - - - - - ---I I II I IAccept : Rej ect--------_______________ l ______________________

IIIII

V

i all tablets ~ Q + 5% i i 3 tablets c: Q - 5% :- - - - - - - - --- - - - - -- - - - - --

-- -- ---- - - - - -- - --- - -- - - - - -- - - - --: step 2, additional 6 tablets :- - - ---- -- -- -- - - - -- - - - - - - - - - - - - --

: 1. Min(X) ~ Q - 5%I 2. X ~ Q

- ---- -- - - - -- - - - -+-- ---- - - - - - - - - --I I II I IAccept : RejectiIIIIIII

V

: 3 tablets c: Q - 5%

- -- -- - ---- - -- --- - - -- ---

---------------- -------- ---- ---- - - ------- -- ---

--- -- - --- --- -- ----- ---- ----- - - - ---i step 3, additional 12 tablets :--- - - ---- - -- -- - -------- - --- -- -- ------ - - -------- ---+-- ----- --- - - ----I II IAccept Rej ect----- ----- ---- ---------- - - ---

: 1. no more than 2 tablets: c: Q - 5%: 2. Min(X) ~ Q - 25%I 3. X ~ Q

otherwise

--- - --------- - ----- ----- - ----

Figure 4. Modified USP XXII three-stage acceptance rule.

4.1 Fixed-Sample Procedure

Assuming a sample of 24 tablets (or capsules) are taken randomly from a lotwith dissolution values distributed as N(¡. a). The decision rule is: acceptthe lot or batch if the probabilty of any randomly selected tablet dissolves

in more than Q at greater than a prespecified significance level A; reject thelot or batch otherwise.

Let X be the amount a (future) randomly selected tablet dissolves:

P(X? Q) = P((X - QJ/a? 0)= P((X - ¡.J/a? -(¡. - QJ/a)

= Ø((¡. - QJ/a)

where ø is the cumulative distribution function, cdf, of the standard normalvariable.

The probabilty P(X ? Q) is estimated by Ø((X - QJ/a), when a is

known from previous testing, or from F123((X - QJ/s), when a is unknown

Dissolution Test Acceptance Sampling Plans 177

and must be estimated by s, the standard deviation of the current sample of24 tablets. Here, X is the sample mean of the 24 tablets and F'23((X - Ql/

s) is the cd! of student t with 23 degrees of freedom.The decision rule is then to accept the lot or batch when

(l((X - Ql/a) ? A, or X? Q + aZ(A)

where Z(A) is the 100A percentile of the standard normal distribution.If a is unknown, the decision rule is to accept the lot or batch when

Ft23((X - Ql/s) ? A, or X? Q + st23(A),

where t23(A) is the 100A percentile of the Student t~distribution with 23 de-grees of freedom.

If a is known, the probabilty of acceptance:

P(Accept) = P(X ? Q + aZ(A))

= P((X - ILl/a-? (Q - ILl/a + Z(A))

= P(Y2 (X - ILl/a? V2 (Q - ILl/a + V2 Z(A))= 1 - (l(V2 (Q - ILl/a + Y2 Z(A))

since \1 (X - ILl/a has a standard normal distribution.

If a is unknown, the probabilty of acceptance:

P(Accept) = P(X ? Q + sZ(A))= P((X - Ql/a? (s/a) Z(A))= P(Y2 (X - Ql/a? V2 (s/a) Z(A))~ P((Y2 (X - Ql/al/(s/a) ? Y2Z(A))= 1 - Ft23CV24(p.-QI/u) (Y2 Z(A))

where Ft,/(À)(x) is the cd! of a noncentral Student t-variate with d degrees offreedom and noncentrality parameter À. It follows from the fact that

V2 (X - Ql/ais distributed as N(\1 (IL - Ql/a, 1) and is independent to S2/a2, which

is X~3/23.Let F = Pr(X -: Q) be the percentage of below specification tablets;

then

F = Pr(X -: Q) = Pr((X - ILl/a -: (Q - ILl/a) = (l ((Q - ILl/a).

On the other hand,

IL = Q - CP-I(F)a.

This shows that, under normal assumption, P(Accept) is determined by F anda and is independent of IL.

178 Tsong et al.

4.2 Three-Stage Procedure

The fixed-sample procedure can be modified into a three-stage procedure asfollow (10,11):

Stage 1: Compare the sample mean of the six randomly selected tabletswith the given critical values A i and B i, The lot or batch is

Accepted if Y6(X1 - Q)/Si ? ts(A1)

Rejected if Y6(X, - Q)/Si oi ts(B1)

where Xl and Sl are the sample mean and standard deviation of the first sixtablets. Otherwise perform the stage 2 testing.

Stage 2: Randomly select six more tablets and compare the sample meanof the 12 tablets with prespecìfied critical values Ai and Bi. The lot or batchis

Accepted if Yi(Xi - Q)/si ? tli(Ai)

Rejected ifyí(Xi - Q)/si oi tll(Bi)

where Xi and Si are the sample mean and standard deviation of 12 tablets.Otherwise perform the stage J testing.

Stage 3: Randomly select 12 more tablets from the lot and compare thesample mean of 24 tablets with prespecified critical value A3. The lot or batchis

Accepted if \I(X3 - Q) / S3 ? ti3(A3)

and rejected otherwise, where X3 and S3 are the sample mean and standarddeviation of all 24 tablets.

This sampling procedure is ilustrated in Figure 5. The probabilty of

lot acceptance can be calculated using the joint distribution of the statisticsin the three stages.

5. Simulation of the USP XXII Procedure and Its ModifiedPlan

The acceptance probability of the acceptance rule varies with J. - Q, the

difference between the true dissolution mean and the labeled value of Q, andF, the true percentage of units dissolved less than Q. To evaluate the oper-ating characteristics of the USP XXII procedure and the modification basedon Givand's approach, we randomly generated 500 values from normal dis-tribution with mean J. - Q and a given percentage of below-specification

tablets, F (which determines the standard deviation). The three choices of

Dissolution Test Acceptance Sampling Plans 179

: step 1, 6 tablets :

- - - - - - - - - -- - -- - ~+- - --- --- - - - - - ---I I 1.1 I IAooept : Rejeot------~-------------- : ---=----~-------------

: r-X - Ql/si ~ Z(Ai): I: (X - Ql/si 0( Z(:ii) :--------------------- : ----------------------V

-- - - - - - - - - - - - - - - - ---- -- --- - - - - - --: step 2, additional 6 tablets :- -- - - - - - - - - - - - - - ---- - - - - - - - - - - - --

: (X - Ql/s2 ~ Z(A2) :- - - - - - -- - - -- - - - - - ----

- -- --- ---- - - - - - -+- - - - -- - - - - - - ----1 1 II 1 IAccept : Rejecti _____________________I

: (X-Ql/s2 0( Z(B2)i _____________________I

V- ---- - --- - -- - -- - - -- - - - - - - - - - -- - ---: step 3, addi tional 12 tablets :- -------- - -- - ----- - - - - - - - - - - -- - - --- - - - - - - --- --- - --+- ---- --- -- - - ---I 11 IAccept Rej ect

- - - - - - - - ---- -- - -- - - - --: (X - Ql/s3 ~ Z(A3) : OTHERWISE 1

I.-- - - --~- -- - -- - -- - -----

Figure 5. Three-stage sampling by measurement rule.

tL - Q are 10% (for lots with mean dissolution much greater than Q), 2%(for lots with mean dissolution slightly greater than Q), and -1% (for lotswith mean dissolution slightly less than Q). Choices of Fare 5-45% by anincrement of 5% for tL - Q = 2% and 10%, 55-90% by an increment of5% for tL - Q = -1%.

6. Approximation of the Acceptance Probabilty of the By-Measurement Procedure

Tlie acceptance probabilty of the fixed-sample and the three-stage sampling-by-measurement procedures is approximated by using the sample standarddeviation as the true lot standard deviation in computing the joint distributionof the statistics from the three stages. They are calculated with the sameconfigurations of tL - Q and F as given in the simulation and various valuesof A (ranges from .60 to .95) for the one-stage procedure, combinations ofAi (Ai ranges from .80 to .95), and B1 = 82 = .05 with i= 1-3, for the

three-stage procedure. The results of A = .75 for the single-stage procedure

and A i = A3 = .95, A2 = .85 for the three-stage procedure are shown for

180 Tsong et al.

comparison to USP three-stage plans. The A's, and B's are the best amongour choices, but are not the optimal choices for discriminating lots with ahigh percentage of tablets with dissolution below specification.

7. Results

The results of the simulation are given in Figures 6-8. It is shown for therules that:

1. Sampling-by-measurement rules: The OC curves are independent öf

JL - Q, as shown in Figures 6 and 7. So, with appropriate choice

of the critical values, one can fixed the acceptance probabilty fora prespecified fraction of below-specification tablets. More flexi-bilty can be obtained using the three-stage procedure. For the se-

lected critical values of Ai = A3 = .95, A2 = .85 and B1 = B2 :;.05, the OC curve shows good discrimination between lots with alow percentage of below-specification tablets and lots with a highpercentage of below-specification tablets. The probabilty of accep-tance is less than .0001 when F, the true percentage of units dis-solved less than Q, is greater than 45%.

100

90-

BO-

70

Ii 60w

~ 50Ð-

~ 40-0.

30

20

10

05%

i

I;

10% 15% 20% 25% 30%PROPORTION .: Q

35% 40% 45%

-- NEW 3-STAGE __ NEW 1-ST AGE -a- MODIFIED USP XXII -- USPXXII

Figure 6. OC curves of acceptance rules for dissolution testing. Mean = Q + 10%.

Dissolution Test Acceptance Sampling Plans 181

100 .90

80

70

1i 60w8 50-if0a: 40Do

30

20

10

00.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

PROPORTION 0: Q

__ NEW 3-ST AGE --- NEW l-STAGE -1- MODIFIED USP XXII -G USPXXII

Figure 7. OC curves of acceptance rules for dissolution testing. Mean = Q + 2%.

2. USP XXII rule: The property that the acceptance probabilties varywith t. - Q makes the procedure less applicable when t. is un-

known. The procedures can be very liberal (high acceptance prob-abilty) when the mean dissolution is only slightly higher than Q.For example; it accepts a lot with less than 15% below-specificationtablets with over 95% probabilty when the mean lot dissolution isonly 2% above Q. The acceptance probabilty is reduced to 80%when the lot mean dissolution is 10% higher than Q.

3. Modified USP XXII procedure: It is more discriminating than theregular USP procedure.

In Figures 6 and 7, it is shown that the USP XXII procedure and itsmodification were unable to reject a lot with a high fraction of units dissolvingless than Q when the mean dissolution is low. In Figure 8, a lot with meandissolution being 1 % less than Q and the percentage of units dissolving lessthan Q being 55%, almost 40 in 100 times the USP XXII procedure wilaccept the lot; the modified USP XXII procedure reduced the probabilties toless than 1 %. However, the acceptance probabilty of the modified USP XXIImay not decrease monotonically when the proportion of below-specificationtablets increases. The sampling-by-measurement rules wil almost certainlyreject the lot when F is greater than 50% regardless of t. - Q.

182

40

35

30

25

~UJ

820~0æ 15

10

5-

055%

Tsong et al.

60% 0.65

-l USP XXII

0.7 0.75PROPORTION .: Q

-l- MODIFIED USPXXII

0.8 0.85 0.9

Figure 8. OC curves of acceptance rules for dissolution testing. Mean == Q - 1%.

8. Conclusion

The USP acceptance rule is a function of both mean and variance of disso-lution for a given percentage of dosage forms. Hence, its operating charac-teristics vary with the mean dissolution, as shown. This property makes theprocedure less applicable when taken as a standard to discriminate amonglots with a large percentage of below-specification tablets from those of lowpercentages. Although the modified USP procedure as proposed by Givandimproved the performance substantially, it is also sensitive to mean disso-lution. Both of these procedures are robust to the uhderlyingdistribution ofdissolution and are poor in lot discrimination.

The proposed sampling-by-measurement rule is robust to mean disso-lution and hence is easy to apply as an acceptance rule with a unique oper-

ating characteristics curve once the critical values are specified. It also showsfavorable properties for. discriminating lots with a high percentage of below-specification tablets from those with a low percentage below specification.The precision of the. approximation and robustness to the skewness in un-derlying distribution are. yet to be evaluated for the OC curves.

Improvement of sampling plans to control for the maximum allowablepercentage of. below-specification tablets can be achieved by modifying the

Dissolution Test Acceptance Sampling Plans 183

critical points of the USP sampling plans or by use of the multiple-stagesampling-by-measurement plans.

Acknowledgments

The authors thank Dr. Wiliam Fairweather for his helpful discussions andsupport in the progress of this research project. They are also in debt to thereferees for their comments and recommendations.

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UV, Ed). Marcel Dekker, New York, 1991, pp. 413-421.3. Bisailon S, Tawashi R: Influence of fluid motion in dissolution-rate determinations. J

Pharm Sci 60:1874-1877, 1971.4. Cox DC, Douglas CC, Furman WB, Kirkchhoefer RD, Myrick JW, Wells CE: Guide-

lines for dissolution testing. Pharm Tech 2(4):40-53, 1978.

5. The United States Pharmacopeia xix. Mack Printing Company, Easton, PA, 1975.

6. The United States Pharmacopeia xx. Mack Printing Company, Easton, PA, 1980.

7. The United States Pharmacopeia xxii. Mack Printing Company, Easton, PA, 1990.

8. Pheatt CB: Evaluation of U.S. Pharmacopeia sampling plans for dissolution. J QualTech 1l(3):158-l64," 1980.

9. Givand TE: An evaluation of the dissolution test acceptance sampling plan of USP XX.Pharmacopeial Forum, Mar-Apr.: 186-190, 1980.

10. Grant EL, Leavenworth RS: Statistical Quality Control, 6th ed. McGraw-Hil, NewYork, 1988.

11. Bowker AH, Lieberman GJ: Engineering Statistics, 2nd ed. Prentice-Hall, EnglewoodCliffs, NJ, 1972.