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Application of Probability
of Passing Multiple Stage Tests
in Benchmarking and
Validation of Processes
B Y P R A M O T E C H O L AY U D T H
❖
INTRODUCTION
The three basic principles of quality assurance (QA) in
the Food and Drug Administration1 (FDA) and World Health
Organization2 (WHO) validation guidelines may be reproduced as follows:
1. Quality, safety, and efficacy must be designed and
built into the product.
2. Quality cannot be inspected or tested into the
product.
3. Each step of the manufacturing process must be controlled to maximize the probability that the finished
product meets all quality and design specifications.
This article will discuss the scientific aspect of the last QA
principle mentioned above and describe how to demonstrate
the probability of meeting the product specifications, for multiple stage tests, for future quality control (QC) samples using
process optimization or validation test results. Demonstrating
the probability will have the following benefits:
• Providing a scientifically predictive tool for future
samples without re-sampling or re-testing
• Evaluating the product quality level, and subsequently, the benchmarking of the manufacturing
processes
• Providing the principle for establishing validation
acceptance criteria
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Journal of Validation Technology
There is a direct correlation between product quality level
and the probability of meeting a certain specification. For a
high quality level, the probability level for a QC sample to
meet the specification limit will be high or, in other words,
will have a high percentage of QC samples (e.g., 90 to 95%),
when taken in a large number, passing that particular test
limit. Specification limits are normally based on those of the
compendial monographs that include multiple stage tests.
The tests have been established based on normality assumption resulting in a typical sampling plan with multiple individual dosage units – e.g., Content Uniformity.3 The
assumption is the key statistical fundamental for computation of the probability of passing these multi-stage tests.
Compendial Multiple Stage Tests
The United States Pharmacopeia (USP) multiple stage
tests to be discussed in this article are noted as follows:
• <905> Uniformity of Dosage Units (Content Uniformity) – two-stage test
• <711> Dissolution – three-stage test
• <701> Disintegration – two-stage test
• <755> Minimum Fill – two-stage test
• <698> Deliverable Volume – two-stage test
The test selection and acceptance criteria may be summarized as seen in Figures 1-7:
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Figure 1
Uniformity of Dosage Units: Test Selection Criteria
rd
USP 20 (3 sup.) – 30
Dosage
Form
Type
Uncoated
Tablet
Coated
Hard
Capsule
Soft
USP 27 – 30
Subtype
≥ 50 mg &
≥ 50 %
–
WV
WV
CU
WV
CU
CU
CU
CU
WV
WV
CU
WV
CU
CU
CU
CU
CU
CU
CU
CU
WV
WV
WV
WV
Film
Others
–
Sus,
Emul
Solution
< 50 mg OR ≥ 25 mg OR < 25 mg OR
≥ 25 %
< 50 %
< 25 %
WV: Weight Variation, CU: Content Uniformity
Figure 2
Uniformity of Dosage Units: Test Acceptance Criteria
USP 20 (3rd sup.) – 30
USP 27 – 30 (New Criteria)
USP Criteria Stage 1: Assay 10 units.
Pass if the following criteria are met:
1) RSD is not more than 6.0% (n = 10)
2) Not more than c unit(s) is outside 85
– 115% LC and no unit is outside 75
– 125% LC
Where:
c = 0 for tablet and 1 for capsule
1) M - X + 2.4s ≤ 15
2) X min ≥ 0.75M and Xmax ≤ 1.25M
Where:
M = reference value
s = standard deviation
X = content uniformity data mean
Xmin = minimum
Xmax = maximum of 10 units
LC: Label Claim
USP Criteria Stage 2: Assay 20 additional units.
Pass if, for all 30 units, the following criteria are met:
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X = content uniformity data mean
Page 288
Xmin = minimum
Xmax = maximum of 10 units
Pramote Cholayudth
Figure 2 (Continued)
Uniformity of Dosage Units: Test Acceptance Criteria
USP Criteria Stage 2: Assay 20 additional units.
Pass if, for all 30 units, the following criteria are met:
Acceptance Value (AV) M - X + ks ≤ L1
X min ≥ (1- L2* 0.01) M, Xmax ≤ (1+ L2* 0.01) M
k = 2.4 (n = 10) & 2.0 (n = 30), L1 = 15, L2 = 25
The new criterion:
1) will provide at least 95% confidence (95% prediction interval) that a future test result
will fall within 0.85M - 1.15M% LC range while the criterion
2) will identify the presence of individual units outside 0.75M - 1.25M% LC range.
Figure 3
Uniformity of Dosage Units: Reference Value Criteria (USP 27 - 30)
Case
Target (T) ≤ 101.5 % LC
Target (T) > 101.5 % LC
Subcase
M
98.5 ≤ X ≤ 101.5
X
ks
X < 98.5
98.5
98.5 - X +ks
X > 101.5
101.5
X - 101.5+ks
98.5 ≤ X ≤ T
X < 98.5
X >T
288
AV =
Journal of Validation Technology
X
98.5
T
ks
98.5 - X +ks
X - T+ks
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Figure 4
Dissolution: Test Acceptance Criteria
Stage 1 (S1): Test 6 units. Pass if each unit is not less than Q+5%.
Stage 21 (S
(S1):
): Test 6 units. Pass if each unit is not less than Q+5%.
Stage
2 Test 6 additional units. Pass if mean of 12 units (S1+S2) is not less than
no unit isunits.
less than
Stage 2 (S ): Q%
Testand
6 additional
PassQ-15%.
if mean of 12 units (S1+S2) is not less than
Stage 1 (S12): Test 6 units. Pass if each unit is not less than Q+5%.
Q% and
no unit is units.
less than
12 additional
PassQ-15%.
if mean of 24 units (S1+S2+S3) is not less
Stage 3 (S ): Test
Stage 2 (S23): Test 6 additional units. Pass if mean of 12 units (S1+S
than
2) is not less
than
Q%
and
not
more
than
two
units are
less
than
of 24
units
(SQ-15%
notunit
lessis less
Stage 3 (S3): Test 12 additional units. Pass if mean
1+S2+Sand
3) is no
Q% and no unit is less than Q-15%.
than
than Q-25%.
Q% and not more than two units are less than Q-15% and no unit is less
12
additional units. Pass if mean of 24 units (S1+S2+S3) is not less
Stage 3 (S3): Test
than Q-25%.
Figure 5
than Q% and not more than two units are less than Q-15% and no unit is less
than Q-25%.
Disintegration: Test Acceptance
Criteria
Stage 1 (S1): Test 6 units. Pass if all of the units have disintegrated completely within the
time iflimit.
not units
morehave
than disintegrated
2 units fail, continue
with within
stage the
2.
Stage 1 (S1): disintegration
Test 6 units. Pass
all ofIf the
completely
disintegration
time
limit.
If
not
more
than
2
units
fail,
continue
with
stage
2.
Stage 2 (S ): Test 12 additional units. Pass if not less than 16 of the total of 18 units tested
Stage 1 (S12): Test 6 units. Pass if all of the units have disintegrated completely within the
completely
disintegration
limit.
Test 12 additional
units.within
Pass the
if not
less than 16 time
of the
total of 18 units tested
Stage 2 (S2): disintegrate
disintegration time limit. If not more than 2 units fail, continue with stage 2.
disintegrate completely within the disintegration time limit.
Stage 2 (S2): Test 12 additional units. Pass if not less than 16 of the total of 18 units tested
disintegrate completely within the disintegration time limit.
Figure 6
to Acceptance
creams, gels,Criteria
lotions, ointments, etc.
MinimumApplied
Fill: Test
Applied1:toCheck
creams,
Stage
10 gels,
units.lotions,
Pass ifointments,
mean of alletc.
10 units is not less than 100% of label amount
no 10
unitunits.
is less
thanif P%
of of
label
amount.
Stage 1: and
Check
Pass
mean
all 10
units is not less than 100% of label amount
Applied to creams, gels, lotions, ointments, etc.
and no20
unit
is less than
P%Pass
of label
amount.
Stage 2: Check
additional
units.
if mean
of all 30 units is not less than 100%
Stage 1: Check 10 units. Pass if mean of all 10 units is not less than 100% of label amount
of
label
amount
and
not
more
than
1
unit
less
of label
amount.
Stage 2: Check 20 additional units. Pass if mean ofis all
30than
unitsP%
is not
less than
100%
and no unit is less than P% of label amount.
P = 90% of
not
more
thanis60
g or
mLP%
perofcontainer,
of label
label amount
amount for
andcontents
not more
than
1 unit
less
than
label amount.
Stage
2: Check
additional
units. Pass
mean
of all60
30g units
isper
not
less
100%
P
of
amount
for
between
more
than
60mL
and
150
g orthan
mL per
container.
P == 95%
90%
of label
label20
amount
for contents
contents
notifmore
than
or
container,
not morebetween
than 1 unit
is less
P% 150
of label
P = 95% of
oflabel
labelamount
amountand
for contents
more
thanthan
60 and
g or amount.
mL per container.
P = 90% of label amount for contents not more than 60 g or mL per container,
P = 95% of label amount for contents between more than 60 and 150 g or mL per container.
Figure 7
Deliverable Volume: Test Acceptance Criteria
Applied to oral liquids, e.g., solutions, syrups, suspensions, and emulsions of contents
not more than 250 mL per container.
Stage 1: Check 10 units. Pass if mean of all 10 units is not less than 100% of label
amount and no unit is less than 95% of label amount.
For single-unit containers: no unit is more than 110% of label amount.
Stage 2: Check 20 additional units. Pass if mean of all 30 units is not less than 100%
of label amount and not more than 1 unit is less than 95% of label amount.
For single-unit containers: not more than 1 unit is more than 110%, but not
more than 115% of label amount.
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Stratified Sampling Plans
Evaluation of the process to be optimized or validated
will require a process sampling. In either batch (e.g., mixing)
or continuous (e.g., tableting or capsulation) operation steps
of manufacturing processes, sampling is generally undertaken in a way that n unit(s) from each of L locations are
taken. This is called a stratified sampling plan where n = 1
and > 1 is for sampling plan 1 and 2 respectively (Figures 8
and 9). The analysts that test the samples must keep the
test results in the as-is sequence so that further statistical
evaluation can be made. Validation samples may also be
taken by multiple samplings – e.g., top, middle, and bottom in batch operations or beginning, middle, and end in
continuous operations for normal or less critical quality attributes (Figure 9).
Figure 8
Stratified Sampling Plans
Sampling Plan #
Individual Sample
Size (n)
Number of
Sampling
Location (L)
1*
1
L
L
>1
L
(> 1)L
2*
Total Sample
Size
(N = nL)
* SUGGESTED BY BERGUM.
4
Figure 9
Comparison of Quality Assurance and Validation Sampling Plans
Quality Attributes
Quality Attributes
Evaluation
QA Sampling
Dose Uniformity
Critical
Composite
Dissolution
Critical
Composite
Degradents
Critical
(depend on product)
Composite
Assay
Moisture
Microbial
Normal
Normal
Normal
Normal
Composite
Composite
Composite
FDA Guidance
Test result is evaluated for probability of meeting specification
(3)
Test result is evaluated statistically against specification
This figure is reproduced and modified from reference #13.
(1)
(2)
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Journal of Validation Technology
Validation
Sampling
Stratified samples
20 locations (L = 20)
n = 3 each location1, 2
Stratified samples
20 locations (L = 20)
n = 2 each location2
Stratified samples
20 locations (L = 20)
n = 1 each location3
Beginning, Middle, End
Beginning, Middle, End
Beginning, Middle, End
Beginning, Middle, End
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Probability Of Passing Compendial
Multiple Stage Tests
Probability of passing Content Uniformity test may be
statistically defined as below:
The multi-stage tests determined to be critical, – e.g.,
Content Uniformity and Dissolution, require to be performed during the validation exercise. Establishing the validation acceptance criteria is based on the tightened limits
that will generate the high probability level for passing the
tests. The passing level can be applied for use in benchmarking purpose of the processes – i.e., passing the specifications at consistent probabilities for either the two tests or
the remainders (e.g., Disintegration, Minimum Fill, and
Deliverable Volume) across a number of validation or other
production batches will allow us to determine the benchmark of the manufacturing processes.
➣ Probability of passing USP 20 (3rd sup.) - 30 criteria
➣ Content Uniformity (USP 20 - 30)
= Prob (first ten tablets meet stage 1 criteria
or all thirty tablets meet stage 2 criteria)
≥ MAX{Prob (first ten tablets meet stage 1
criteria), Prob (all thirty tablets meet
stage 2 criteria)}
The next steps will involve the estimation of the upper
and lower bounds for the sample mean and the upper bound
for sample SD. Detailed information can be searched from
reference # 10 from which some key part is reproduced on
the next page:
As discussed earlier, the normality assumption and the
testing plans comprising multiple individual dosage units
will provide the statistical advantage for further computation of the probability of passing the multi-stage tests.
James S. Bergum introduced how to compute the probability of passing the multiple stage tests for a QC sample in a
paper entitled Constructing Acceptance Limits for Multiple
Stage Tests. Bergum method is one of the most important
tools for Content Uniformity and Dissolution assessment. His
validated SASTM program (CuDAL) for computation of the
acceptance limits for sample means and relative standard deviations (RSDs) in response to the entry for sample size, confidence level, and probability level9 was provided. Using
Bergum method, an MS Excel® program for determination of
the probability of passing the USP tests was developed by the
author and published in 2004 and 2006.10, 11 The program is
directly applicable to sampling plan 1 – i.e., one dosage unit
(n = 1) from each of L locations (Figure 10). However, for the
data generated from sampling plan 2 (n > 1), one may use the
program after estimation for the upper and lower bounds for
sample mean and standard deviation (SD) where the use of
analysis of variance (ANOVA) approach (one-way nested
random effects model) is applied (Figure 11).
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Lower bound for probability of passing CU test for TABLETS =
MIN(Prob1,Prob2)
Probability using MS Excel
=MAX((FDIST(10/((0.06^2)*(1+10*(Mean/Sigma)^2)),(1+10*(Mean/Sigma)^2)^2/(1+2*10*(Mean/Sigma)
^2),10-1))+((NORMSDIST((Mean-85)/Sigma)+NORMSDIST((115-Mean)/Sigma)-1)^10)-1,(FDIST(30/
((0.078^2)*(1+30*(Mean/Sigma)^2)),(1+30*(Mean/Sigma)^2)^2/(1+2*30*(Mean/Sigma)^2),30-1))+
((NORMSDIST((Mean-85)/Sigma)+NORMSDIST((115-Mean)/Sigma)-1)^30+COMBIN(30,1)*((NORMS
DIST((Mean-85)/Sigma)+NORMSDIST((115-Mean)/Sigma)-1)^29)*(((NORMSDIST((Mean-75)/Sigma)NORMSDIST((Mean-85)/Sigma))+(NORMSDIST((125-Mean)/Sigma)-NORMSDIST((115-Mean)/
Sigma)))^1))-1,0) ……………… (I)
Where:
Prob1: Substitute 'Sigma' with upper bound for sample SD and substitute 'Mean'
with upper bound for sample mean, joint confidence level for both upper
bounds = 95% (i.e., confidence level for each bound = square root of 95%)
Prob2: Substitute 'Sigma' with upper bound for sample SD and substitute 'Mean'
with lower bound for sample mean, joint confidence level for upper and lower
bounds = 95% (i.e., confidence level for each bound = square root of 95%)
Upper bound (UB) for sample SD (SD) =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
Upper bound for sample mean (M) =M+UB*NORMSINV(1-(1-0.95^0.5)/2)/n^0.5
Lower bound for sample mean (M) =M-UB*NORMSINV(1-(1-0.95^0.5)/2)/n^0.5
Lower bound for probability of passing CU test for CAPSULES
= MIN(MAX(Part1/1,Part2/1,0),MAX(Part1/2,Part2/2,0))
Part1: Probability using MS Excel
=(FDIST(10/((0.06^2)*(1+10*(Mean/Sigma)^2)),(1+10*(Mean/Sigma)^2)^2/(1+2*10*(Mean/
Sigma)^2),10-1))+((NORMSDIST((Mean-85)/Sigma)+NORMSDIST((115-Mean)/Sigma)-1)^10
+COMBIN(10,1)*((NORMSDIST((Mean-85)/Sigma)+NORMSDIST((115-Mean)/Sigma)-1)^9)*
(((NORMSDIST((Mean-75)/Sigma)-NORMSDIST((Mean-85)/Sigma))+(NORMSDIST((125Mean)/Sigma)-NORMSDIST((115-Mean)/Sigma)))^1))-1 ……………… (II)
Part2: Probability using MS Excel
=(FDIST(30/((0.078^2)*(1+30*(Mean/Sigma)^2)),(1+30*(Mean/Sigma)^2)^2/(1+2*30*(Mean
Sigma)^2),30-1))+((NORMSDIST((Mean-85)/Sigma)+NORMSDIST((115-Mean)/Sigma)-1)^30
+COMBIN(30,1)*((NORMSDIST((Mean-85)/Sigma)+NORMSDIST((115-Mean)/Sigma)-1)^29)
*(((NORMSDIST((Mean-75)/Sigma)-NORMSDIST((Mean-85)/Sigma))+(NORMSDIST((125Mean)/Sigma)-NORMSDIST((115-Mean)/Sigma)))^1)+COMBIN(30,2)*((NORMSDIST((Mean85)/Sigma)+NORMSDIST((115-Mean)/Sigma)-1)^28)*(((NORMSDIST((Mean-75)/Sigma)NORMSDIST((Mean-85)/Sigma))+(NORMSDIST((125-Mean)/Sigma)-NORMSDIST((115-Mean)
/Sigma)))^2)+COMBIN(30,3)*((NORMSDIST((Mean-85)/Sigma)+NORMSDIST((115-Mean)/
Sigma)-1)^27)*(((NORMSDIST((Mean-75)/Sigma)-NORMSDIST((Mean-85)/Sigma))+(NORMS
DIST((125-Mean)/Sigma)-NORMSDIST((115-Mean)/Sigma)))^3))-1 ……………… (III)
Where:
Part1/1
& Journal
Part 2/1:
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of Validation Technology
Substitute, in Part1 & Part2, 'Sigma' with upper bound for sample SD and substitute 'Mean'
with upper bound for sample mean, joint confidence level for both upper bounds = 95%
Part1/2 & Part 2/2:
Substitute, in Part1 & Part2, 'Sigma' with upper bound for sample SD and substitute 'Mean'
with lower bound for sample mean, joint confidence level for upper and lower bounds = 95%
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Mean)/Sigma)-NORMSDIST((115-Mean)/Sigma)))^1)+COMBIN(30,2)*((NORMSDIST((Mean-
85)/Sigma)+NORMSDIST((115-Mean)/Sigma)-1)^28)*(((NORMSDIST((Mean-75)/Sigma)NORMSDIST((Mean-85)/Sigma))+(NORMSDIST((125-Mean)/Sigma)-NORMSDIST((115-Mean)
/Sigma)))^2)+COMBIN(30,3)*((NORMSDIST((Mean-85)/Sigma)+NORMSDIST((115-Mean)/
Pramote Cholayudth
Sigma)-1)^27)*(((NORMSDIST((Mean-75)/Sigma)-NORMSDIST((Mean-85)/Sigma))+(NORMS
DIST((125-Mean)/Sigma)-NORMSDIST((115-Mean)/Sigma)))^3))-1 ……………… (III)
Where:
Part1/1 & Part 2/1:
Substitute, in Part1 & Part2, 'Sigma' with upper bound for sample SD and substitute 'Mean'
with upper bound for sample mean, joint confidence level for both upper bounds = 95%
Part1/2 & Part 2/2:
Substitute, in Part1 & Part2, 'Sigma' with upper bound for sample SD and substitute 'Mean'
with lower bound for sample mean, joint confidence level for upper and lower bounds = 95%
Upper bound (UB) for sample SD (SD) =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
Upper bound for sample mean (M) =M+UB*NORMSINV(1-(1-0.95^0.5)/2)/n^0.5
Lower bound for sample mean (M) =M-UB*NORMSINV(1-(1-0.95^0.5)/2)/n^0.5
To demonstrate the application of Bergum method using the MS Excel program, the scenarios for sampling plans 1 and
2 are provided in Figures 10 and 11.
Figure 10
Example for Statistical Evaluation of Data from Sampling Plan 1
n
1
1, 2, 3
L
60
Sample Mean (% LC)
100.68
RSD (%)
3.42
SD (% LC)
3.44
Substitute in (I); probability (lower bound) of passing CU test: 99.96%
Note: In MS Excel formulae, the symbol ‘=’ must be attached to the formula itself
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Figure 11
Procedures for Statistical Evaluation of Data from Sampling Plan 2
Given Data: n = 3, L = 20, Required C onfidence Level (1- ): 95%
Degree of freedom or DF (between-location): L -1 = 19
Degree of freedom or DF (within -location): L (n-1) = 20* 2 = 40
(1 − 2p)(1 − q)(1 − q) = 1 − α = 0.95
Given
and q = 2p
(1-2p) = (1-q) = (0.95) 1/3 = 0.983048, 1-p =
0.991524
σ 2 = σ e2 + nσ 2L
N = nL = 60
1− 2p : confidence level for sample mean,
1− q : confidence level for within & between - location variability,
confidence level = 0.95
1 − α : joint
σ 2 : total variance, σ 2e : expect ed mean-squared error (within), σ 2L : expected
location mean square (between), n = 3
Ue =
L ( n − 1)se2
χ12− q, L ( n −1)
=
40 se2
χ 02.983048,40
= 1.708se2
(L − 1)s2L
19s2L
UL = 2
= 2
= 2.281s2L
χ1− q,( L −1) χ 0.983048,19
The upper lim it for
UL
UL
σ2
σ2
σ 2 ( UL
σ2
U e : upper limit
error,
for
σ 2e , s e2 : mean-squared
U L : upper lim it for σ 2L , s 2L : location
mean square ( see Excel formulae below and
how to estimate
se2
&
s2L
in Figures 17-19)
) is based on the folowing conditions
1
1
2
1
= (1 − ) U e + U L = U e + U L
n
n
3
3
= Ue
If
Ue < UL
If
Ue ≥ UL
Upper bound (UB) for sample SD (SD) = square root of
Confidence limits for sample mean ( x ) =
Upper bound for sample mean ( x ) =
UL
σ2
:
x ±(U L /N) 1/2Z 0.991524
x +0.308(U L )1/2
1/2
Lower bound for sample mean ( x ) = x -0.308(UL )
Z 0.991524 = 2.388 (see Excel
formula below), N = 60
Substitute the upper and lower bounds in (I) as detailed in Fi gures 17 through 19.
In Excel, DF/CHIINV(0.95^(1/3),DF) =40/CHIINV(0.95^(1/3),40) = 1.708
In Excel, DF/CHIINV(0.95^(1/3),DF) =19/CHIINV(0.95^(1/3),19) = 2.281
In Excel, NORMSINV(1 -(1-0.95^(1/3))/2) = 2.388
Note: More reading in Chapter 5 of reference #5 is recommended.
Using Bergum method and MS Excel program will help us imagine how each of RSD values
at various means will generate the probability distributions for passing the content uniformity
test. The distributions are illustrated in Figures 12 through 15.
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Journal of Validation Technology
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Figure 12
Probability of Passing Content Uniformity Test Distributions: Tablets
Figure 13
Probability of Passing Content Uniformity Test Distributions: Tablets
Figure 14
Probability of Passing Content Uniformity Test Distributions: Capsules
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Figure 15
Probability of Passing Content Uniformity Test Distributions: Capsules
For the as-is Content Uniformity (CU) data, the RSD
limit, as part of the validation acceptance criteria, may be
customized based on the designed probability levels, – e.g.,
50% or 90%, for passing the tests. The designed level (50%
in Figures 13 and 15, 90% in Figures 12 and 14) will cross
the distribution curve lines where one of the RSD values can
be determined as part of the validation acceptance criteria.
For example, from Figures 13 and 15, the RSD limit of not
more than 5.6% and 6% may be established for tablet and
capsule content uniformity test, respectively. In Figures 12
and 14, the 90% probability level may be determined as the
benchmark of the process.
The MS Excel program used for computation of the probability of passing the content uniformity test for tablets and
capsules may be downloaded from the website in reference #
14. As indicated earlier, one should realize that the program
is applied to sampling plan 1. Sampling plan 1 may be accomplished using properly designed techniques – e.g., taking
1x60 instead of 3x20 units using automatic sampling device.
In the PQRI Recommendation Report7 and FDA Draft
Guidance for Industry8, the 1st stage of validation sampling
plan (plan 2) requires 60 dosage units (n = 3, L = 20) with
weight-corrected RSD limit of not more than 6% as part of
the acceptance criteria (Figure 16).
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Journal of Validation Technology
To demonstrate how to evaluate sampling plan 2 data,
Figures 17 through 19 provide examples for evaluation of
the validation test results that are obtained after process validation exercises in the company the author works with. The
validation acceptance criteria are completely based on the
FDA guidance and PQRI Recommendation Report – i.e.:
n = 3, L = 20, RSD ≤ 6.0%, etc. All the validation test
results completely meet the recommended acceptance
criteria. For further evaluation, the as-is (non-weight corrected) test results require to be analyzed for between – and
within-location variability using analysis of variance
(ANOVA) approach and then estimated for the upper and
lower bounds for sample statistics prior to computing the
lower bound for probability of passing the content uniformity tests. For the weight-corrected results, further statistical evaluation is marginally justified.
(CONTINUED ON PAGE 300)
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Figure 16
FDA Draft Guidance Acceptance Criteria (Based on PQRIs)
As–is Data
Weight Corrected Data
Stage 1: Assay 60 units.
Pass if the following criteria are met:
1) All individual results are within
75 – 125% LC.
1) Mean of each location is within 90 –
110% target potency (TP).
2) RSD ≤ 6.0%.
Stage 2: Assay 80 additional units.
Pass if, for all 140 units, the following criteria are met:
1) All individual results are within 75 –
125% LC.
1) Mean of each location is within 90 –
110% target potency (TP).
2) RSD ≤ 6.0%.
Rationale for acceptance criteria is as follows:
A comparison of each individual to
75.0% and 125.0% of target is used to
identify the presence of super-potent or
sub-potent units. A value outside 25.0%
of the target potency may indicate
inadequate blend uniformity.
The RSD limit defines the uniformity
requirements when there is no betweenlocation variability.
A comparison of each location mean to
90.0% - 110.0% of target identifies betweenlocation variability.
Note: Weight-corrected data is defined as, for example, a tablet with potency of 19.4 mg and weight of 98
mg = 19.4/98 = 0.198 mg/mg. Label claim is 20 mg per 100 mg tablet (0.20 mg/mg), so the weightcorrected result is (0.198/0.20)*100 = 99% of target blend potency.
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Figure 17
Example for Statistical Evaluation of Data from Sampling Plan 2 (# 1)
Tablet Product / Batch # 1: Stratified Sampling Location #
Unit #
1
2
3
SQD
Unit #
1
2
3
SQD
1
98.01
96.77
98.27
1.29
11
99.44
101.55
99.77
2.58
2
3
98.91 96.37
99.11 97.93
98.96 99.83
0.02
6.01
12
13
100.20 100.86
97.95 98.35
97.25 99.33
4.75
3.20
Average: 99.03% LC
4
97.97
96.80
97.64
0.73
14
101.63
99.04
99.07
4.42
5
97.93
98.52
100.27
2.96
15
97.91
98.57
100.13
2.60
6
96.83
98.85
98.68
2.51
16
101.95
99.73
97.92
8.15
7
101.31
96.48
97.46
13.04
17
102.66
98.03
98.84
12.23
RSD: 1.61%
8
98.74
97.53
95.83
4.27
18
99.55
99.18
99.86
0.23
9
97.87
103.71
98.23
21.42
19
100.65
99.63
101.15
1.20
SQD: Square Deviation
Degree of freedom (between-location): L-1 = 19
Degree of freedom (within-location): L(n-1) = 20*2 = 40
Total square deviation: 149.07 (sum of squares of deviations between all individuals and mean)
Square deviation (within-location): 96.58
(Total sum of squares of deviations between individuals and mean within each of 20 locations)
2
Mean-squared error (within-location): S e = 96.58/40 = 2.414
Square deviation (between-location): 149.07-96.58 = 52.49
2
Location mean square (between-location): S L = 52.49/19 = 2.763
Upper limit for σ
2
L
(UL): 2.281 S 2L = (2.281)(2.763) = 6.302
Upper limit for σ
2
e
(Ue): 1.708 S e = (1.708)(2.414) = 4.123 ( Ue < UL, so UL
2
σ2
=
2
3
Therefore, upper bound for total variance ( UL 2 ): (2/3)(4.123)+(1/3)(6.302) = 4.849
σ
Upper bound for total SD (UB): (4.849)^1/2 = 2.202*
Upper bound for sample mean: 99.03+(0.308)(6.302)^1/2 = 99.80*
Lower bound for sample mean: 99.03-(0.308)(6.302)^1/2 = 98.26*
* Substitute in (I); probability (lower bound) of passing CU test: 100.00%
298
Journal of Validation Technology
1
Ue + UL )
3
10
99.44
99.18
98.14
0.95
20
98.35
100.61
100.97
4.03
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Figure 18
Example for Statistical Evaluation of Data from Sampling Plan 2 (# 2)
Tablet Product / Batch # 2: Stratified Sampling Location #
Unit #
1
2
3
SQD
Unit #
1
2
3
SQD
1
95.52
97.19
101.28
17.56
11
95.48
95.87
96.68
0.75
2
3
96.57 94.77
97.69 102.02
95.16 97.48
3.21 26.84
12
13
96.87 96.40
99.86 94.57
97.61 98.10
4.85
6.23
Average: 98.43% LC
4
99.82
101.27
97.29
8.11
14
99.44
98.19
98.74
0.79
5
96.73
97.52
99.86
5.30
15
98.41
99.07
98.89
0.23
6
98.83
99.12
98.46
0.22
16
102.75
96.72
99.79
18.18
7
98.44
96.72
97.43
1.49
17
108.56
102.21
103.98
21.48
RSD: 3.04%
8
99.43
96.67
97.33
4.15
18
94.86
93.25
96.34
4.78
9
95.96
95.67
97.94
3.05
19
94.82
102.98
94.88
44.07
10
98.20
99.85
97.04
3.99
20
102.52
106.68
104.13
8.80
SQD: Square Deviation
Degree of freedom (between-location): L-1 = 19
Degree of freedom (within-location): L(n-1) = 20*2 = 40
Total square deviation: 528.00 (sum of squares of deviations between all individuals and mean)
Square deviation (within-location): 184.09
(Total sum of squares of deviations between individuals and mean within each of 20 locations)
2
Mean-squared error (within-location): S e = 184.09/40 = 4.602
Square deviation (between-location): 528.00-184.09 = 343.91
2
Location mean square (between-location): S L = 343.91/19 = 18.100
Upper limit for σ
2
L
2
(UL): 2.281 S L = (2.281)(18.100) = 41.286
Upper limit for σ
2
e
(Ue): 1.708 S e = (1.708)(4.602) = 7.860 ( Ue < UL, soUL
2
σ2
=
2
3
1
Ue + UL )
3
Therefore, upper bound for total variance ( UL 2 ): (2/3)(7.860)+(1/3)(41.286) = 19.002
σ
Upper bound for total SD (UB): (19.002)^1/2 = 4.359*
Upper bound for sample mean: 98.43+(0.308)(41.286)^1/2 = 100.41*
Lower bound for sample mean: 98.43-(0.308)(41.286)^1/2 = 96.45*
* Substitute in (I); probability (lower bound) of passing CU test: 99.25%
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Figure 19
Example for Statistical Evaluation of Data from Sampling Plan 2 (# 3)
Tablet Product / Batch # 3: Stratified Sampling Location #
Unit #
1
2
3
SQD
Unit #
1
2
3
SQD
1
100.48
93.22
97.02
26.37
11
98.38
98.50
98.58
0.02
2
3
97.43 99.47
99.31 105.68
99.85 102.10
3.23 19.43
12
13
101.99 99.72
99.40 97.61
97.76 99.72
9.10
2.97
Average: 99.75% LC
4
99.64
102.91
99.72
6.96
14
101.42
101.53
98.15
7.38
5
101.93
104.99
98.23
22.92
15
102.46
98.17
98.80
10.73
6
101.30
100.19
98.07
5.39
16
97.66
98.27
98.07
0.19
7
101.22
100.35
100.89
0.39
17
103.72
99.46
99.98
10.80
RSD: 2.23%
8
99.66
99.59
99.22
0.11
18
98.71
99.15
104.18
18.47
9
98.31
97.82
100.91
5.52
19
99.81
98.38
99.36
1.07
10
98.24
102.67
102.13
11.68
20
101.31
97.27
94.89
21.07
SQD: Square Deviation
Degree of freedom (between-location): L-1 = 19
Degree of freedom (within-location): L(n-1) = 20*2 = 40
Total square deviation: 293.00 (sum of squares of deviations between all individuals and mean)
Square deviation (within-location): 183.79
(Total sum of squares of deviations between individuals and mean within each of 20 locations)
2
Mean-squared error (within-location): S e = 183.79/40 = 4.595
Square deviation (between-location): 293.00-183.79 = 109.21
2
Location mean square (between-location): S L = 109.21/19 = 5.748
Upper limit for σ
2
L
(UL): 2.281 S
Upper limit for σ
2
e
(Ue): 1.708 S
2
L
2
e
= (2.281)(5.748) = 13.111
= (1.708)(4.595) = 7.848 ( Ue < UL, so UL
σ2
=
2
3
1
Ue + UL )
3
Therefore, upper bound for total variance ( UL 2 ): (2/3)(7.848)+(1/3)(13.111) = 9.602
σ
Upper bound for total SD (UB): (9.602)^1/2 = 3.099*
Upper bound for sample mean: 99.75+(0.308)(13.111)^1/2 = 100.87*
Lower bound for sample mean: 99.75-(0.308)(13.111)^1/2 = 98.63*
* Substitute in (I); probability (lower bound) of passing CU test: 99.25%
(CONTINUED FROM PAGE 296)
Inconsistent probability levels amongst the validation
batches are required to initiate an investigation. Well-developed and under-control processes will reproducibly generate the consistent probability levels for passing the test.
300
Journal of Validation Technology
For capsule products using the similar sampling plans,
estimation for statistics from the test data is conducted in the
same way. Substitution can be made in (II) and (III).
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Content Uniformity (USP 27 - 30)
Whatever the new test acceptance criteria will look like,
they should conform to their conventional standards.
According to Torbeck’s comment,15
by Bergum. As evidenced in his chapter: Statistical Methods
for Uniformity and Dissolution Testing; Section V. Future
Developments, in reference # 6 (2003).
“… the United States Pharmacopeia should
not change its philosophical position on single testing. … If statistical procedures were
given in the USP, companies would have little
incentive to develop better procedures.”
“…as part of the international harmonization of test methods, a proposed change to
the USP <905> content uniformity test has
been made. … In anticipation of this happening, appropriate modifications to the
CuDAL approach have been determined to
evaluate the newly proposed test.”
And Bergum method is a good example for this statement.
The approach to approximate the probability of meeting
the new test acceptance criteria has been under development
His new article on the new test acceptance criteria, when
published, will be very interesting to all professionals in the
pharmaceutical industry.
DISSOLUTION
Probability of passing Dissolution test may be statistically defined as below:
Probability of passing USP criteria
≥ MAX {P(meeting stage 1), P(meeting stage 2), P(meeting stage 3)}
≥ MAX {P(meeting criteria of stage 1), P(meeting 1st criteria of stage 2) +
P(meeting 2nd criteria of stage 2) - 1, P(meeting 1st criteria of stage 3) +
P*(meeting 2nd criteria of stage 3) - 1}
The next steps will involve the estimation of the lower bound for the sample mean and
the upper bound for sample SD. Detailed information can be searched from reference
# 11 from which some key part is reproduced below:
Probability using MS Excel
=MAX((1-NORMSDIST(((Q+5)-Mean)/Sigma))^6,(1-NORMSDIST(((Q-15)Mean)/Sigma))^12+(1-NORMSDIST((12^0.5)*(Q-Mean)/Sigma))-1,(1NORMSDIST(((Q-15)-Mean)/Sigma))^24+COMBIN(24,1)*(1-NORMSDIST(((Q-15)Mean)/Sigma))^23*(NORMSDIST(((Q-15)-Mean)/Sigma)-NORMSDIST(((Q-25)-Mean)
/Sigma))+COMBIN(24,2)*(1-NORMSDIST(((Q-15)-Mean)/Sigma))^22*(NORMSDIST(((Q-15)Mean)/Sigma)-NORMSDIST(((Q-25)-Mean)/Sigma))^2+(1-NORMSDIST((24^0.5)*(Q-Mean)/
Sigma))-1) ……………… (IV)
Where:
‘Sigma’ is substituted with upper bound for sample SD
‘Mean’ is substituted with lower bound for sample mean
Joint confidence level for upper and lower bounds = 95%
(i.e., confidence level for each bound = square root of 95%)
Upper bound (UB) for sample SD (SD) =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
Lower bound for sample mean (M) =M-UB*NORMSINV(1-(1-0.95^0.5))/n^0.5
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Figure 20
Probability of Passing Dissolution Test: Sampling Plan 1
The dissolution test results obtained from validation
batches testing will also require to be estimated for some
statistics in the same way, depending on which sampling
plan (1 or 2) is employed, prior to computation of the lower
bound for the probability. Figure 21 is the example for the
dissolution test results from three validation batches in the
company where the author provides consulting services.
Figure 21
Probability of Passing Dissolution Test: Multiple Sampling
Validation Batches Data for Film-Coated Tablets/Multiple Sampling
Tablet
Batch # 1 (% LC)
Batch # 2 (% LC)
Batch # 3 (% LC)
#
Top
Middle Bottom
Top
Middle Bottom
Top
Middle
Bottom
1
102.34 100.10 100.73 100.12 104.60
98.20 100.27
98.96
103.78
2
99.23 100.70 98.36 105.09
99.50 103.68 100.87
101.37
100.46
3
102.94
98.92 103.55 103.63
99.50 102.79 101.33
100.61
97.45
4
101.01 100.55 103.70 104.36 105.62 102.64 101.93
103.17
100.61
5
100.71
98.62 99.55 102.61
98.19 103.97 101.63
100.77
98.96
6
98.79
98.77 101.48
99.39
98.34 103.08 100.87
103.17
102.87
Mean
100.56
101.96
101.06
LB Mean
99.39
100.19
99.95
SD
1.70
2.57
1.61
UB SD
2.54
3.85
2.41
Prob
100.00%
100.00%
100.00%
Since the coated tablets were freely distributed during the coating process, no between-location
variability is expected. The composite data can be directly used for statistical estimation.
N = nL = 18, Q = 80% LC
Example for
estimation of
batch # 1 data
302
UB for sample SD =1.70*((18-1)/(CHIINV(0.95^0.5,18-1)))^0.5 = 2.54
LB for sample mean =100.56-2.54*NORMSINV(1-(1-0.95^0.5))/18^0.5 = 99.39
Substitute UB (SD) & LB (mean) values in (IV)
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DISINTEGRATION
As suggested by Chow and Liu,5 the exact probability of passing the USP Disintegration test may be given as below:
Exact probability = p6+6p17(1-p)+87p16(1-p)2
Where:
p : Probability that all dosage units fall below the upper limit for disintegration time.
Probability using MS Excel
=(NORMSDIST((UL-Mean)/Sigma))^6+6*((NORMSDIST((ULMean)/Sigma))^17)*(1-(NORMSDIST((ULMean)/Sigma)))+87*((NORMSDIST((UL-Mean)/Sigma))^16)*(1(NORMSDIST((UL-Mean)/Sigma)))^2 ……………… (V)
Where:
‘Sigma’ is substituted with upper bound for sample SD
‘Mean’ is substituted with upper bound for sample mean
‘UL’ is substituted with upper limit for disintegration time
Joint confidence level for upper and lower bounds = 95%
(i.e., confidence level for each bound = square root of 95%)
Upper bound (UB) for sample SD (SD) =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
Upper bound for sample mean (M) =M+UB*NORMSINV(1-(1-0.95^0.5))/n^0.5
When appropriate amount of disintegration time data
(e.g., 3x of QC sample size = 18) is obtained, estimation for
the required statistics is carried out and the resulting statistical estimates are substituted in (V). Disintegration data obtained by sampling plan 2 may be approximated for
statistical estimates in the same way under assumption that
there is no between-location variability.
Figure 22 illustrates how various proportions (1-p) above
the upper limit for the disintegration time (p is the probability
below the upper limit) generate the probability distributions.
‘LB Probability’ in the figure means the lower bound for
probability of passing the test that was suggested by Bergum
(1990). According to Chow and Liu,5 it is rather conservative
when compared to the exact probability method.
Figure 22
Probability of Passing Disintegration Test
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MINIMUM FILL AND DELIVERABLE VOLUME
How to set-up the target fill weights or volumes for dosage forms requiring Minimum Fill or Deliverable Volume is
described in Establishing Target Fills for Semisolid and Liquid Dosage Forms,12 the general target fill formula is summarized
as follows:
T = L +1.27σ / n (For in-process control (IPC) of fill weight or fill volume average)
Where:
T = Target fill value in g or ml per container
L = Labeled amount (content) value in g or ml per container (= 100% labeled amount (LA))
n = In-process control (IPC) sample size that is equal to QC sample size e.g., 10
(1st stage test)
σ = Lot sigma for fill weight/volume data under normality assumption
Using the target fill formula will provide the batch after
filling with approximately 95% probability level for a QC
sample to pass the Minimum Fill or Deliverable Volume test.
However, after filling, both the batch average and variability
i.e., standard deviation (for fill weight or fill volume) may
shift from what is earlier expected – that is, the actual fill
data will generate the different mean and SD. The lower and
upper estimates for the two statistics are used to compute the
lower bound for probability of passing the QC test to determine the process benchmark. The benchmarking approach
for a filling process is defined in the following two sections.
MINIMUM FILL
Probability of passing Minimum Fill test may be statistically defined as below:
Probability of passing USP criteria
≥ MAX {P(meeting stage 1), P(meeting stage 2)}
Probability using MS Excel
=MAX((1-NORMSDIST((100-Mean)/(Sigma/10^0.5)))+((1-NORMSDIST((PMean)/Sigma)))^10-1,(1-NORMSDIST((100-Mean)/(Sigma/30^0.5)))+(1NORMSDIST((P-Mean)/Sigma))^30+COMBIN(30,1)*(NORMSDIST((PMean)/Sigma))^29-1,0) ……………… (VI)
Where:
‘Sigma’ is substituted with upper bound for sample SD
‘Mean’ is substituted with lower bound for sample mean
‘P’ is 90 for fill size ≤ 60 g or mL per container and 95 for fill size > 60 and 150 g or mL per container.
Joint confidence level for upper and lower bounds = 95%
(i.e., confidence level for each bound = square root of 95%)
Upper bound (UB) for sample SD (SD) =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
Lower bound for sample mean (M) =M-UB*NORMSINV(1-(1-0.95^0.5))/n^0.5
304
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When appropriate amount of filling data is obtained, estimation for the required statistics is carried out and the resulting statistical estimates are substituted in (VI).
Figures 23 and 24 illustrate the probability distribution
curves obtained from composite samples – e.g., each sam-
ple comprising 10 units from each of 3 segments (n = 30) at
the beginning, middle, and end of a filling cycle. Fill data
obtained by sampling plan 2 may be approximated for statistical estimates in the same way.
Figure 23
Probability of Passing Minimum Fill Test (≤ 60 g or mL Size)
Figure 24
Probability of Passing Minimum Fill Test (> 60 - 150 g or mL Size)
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DELIVERABLE VOLUME
Probability of passing Deliverable Volume test may be statistically defined as below:
Probability of passing USP criteria
≥ MAX {P(meeting stage 1), P(meeting stage 2)}
Probability using MS Excel (multi-dose products)
=MAX((1-NORMSDIST((100-Mean)/(Sigma/10^0.5)))+((1NORMSDIST((95-Mean)/Sigma)))^10-1,(1-NORMSDIST((100Mean)/(Sigma/30^0.5)))+(1-NORMSDIST((95Mean)/Sigma))^30+COMBIN(30,1)*(NORMSDIST((95Mean)/Sigma))^29-1,0) ……………… (VII)
Probability using MS Excel (single-dose products)
=MAX((1-NORMSDIST((100-Mean)/(Sigma/10^0.5)))+(NORMSDIST((110Mean)/Sigma)-NORMSDIST((95-Mean)/Sigma))^10-1,(1-NORMSDIST((100Mean)/(Sigma/30^0.5)))+(NORMSDIST((110-Mean)/Sigma)-NORMSDIST((95Mean)/Sigma))^30+COMBIN(30,1)*((NORMSDIST((110-Mean)/Sigma)NORMSDIST((95-Mean)/Sigma))^29)*(((NORMSDIST((95-Mean)/Sigma)NORMSDIST((90-Mean)/Sigma)))^1)-1,0) ……………… (VIII)
Where:
‘Sigma’ is substituted with upper bound for sample SD
‘Mean’ is substituted with lower bound for sample mean
Joint confidence level for upper and lower bounds = 95%
(i.e., confidence level for each bound = square root of 95%)
Upper bound (UB) for sample SD (SD) =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
Lower bound for sample mean (M) =M-UB*NORMSINV(1-(1-0.95^0.5))/n^0.5
Figure 25
Probability of Passing Deliverable Volume Test (≤ 250 mL Size)
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Figure 26
Probability of Passing Deliverable Volume Test (Single Unit)
When appropriate amount of filling data is obtained as
described above, estimation for the required statistics is carried out and the resulting statistical estimates are substituted
in (VII) or (VIII).
Figures 25 and 26 illustrate the probability distribution
curves obtained from composite samples – e.g., each sample comprising 10 units from each of 3 segments (n = 30) at
the beginning, middle, and end of a filling cycle. Fill data
obtained by sampling plan 2 may be approximated for statistical estimates in the same way.
From Figure 26, it implies that a too high fill volume average, say 105% LA or greater, will have a tendency to fail
the test. Filling process should be strictly controlled about
the predetermined filling target.12
higher potency), three sample mean locations may be determined at 98, 100, and 102% LC with a constant sample
RSD predetermined at 5.0%. Upon estimation for the required statistics and substitution into formula (I), the probability results are 61.09, 78.53, and 55.34%, respectively.
The RSD value greater than 5.0% generates the probability
results less than 50%. Therefore, the RSD limit of 5.0% can
be established for an acceptance criteria limit with the sample mean (n = 45) within 98-102% LC. When the validation
test results are available e.g., sample mean = 100.17% LC
and RSD = 3.5%, the lower bound for probability of passing the CU test is computed (formula I) and the result is
99.92%. The trial and error approach may be applied in case
of the other tests and products.
APPLICATION FOR ESTABLISHING
ACCEPTANCE CRITERIA LIMITS
SUMMARY
As discussed earlier, the RSD limit of 5.6% and 6% in
Figures 13 and 15, giving probability values slightly above
50%, may be established as part of the validation acceptance criteria for tablet and capsule content uniformity tests,
respectively. For sample sizes other than those specified in
the figures above, a trial and error approach may be used.
For example, if N = nxL = 3x15 = 45 for content uniformity
test in process validation (of less critical tablet products e.g.,
A step-by-step procedure to compute the probability is
provided in Figure 27. Prior to substitution into those probability formulae, sample statistics are estimated for their statistical bounds where their MS Excel formulae are
summarized in Figure 28.
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Figure 27
Step-by-Step Procedure to Compute the Probability
Procedure
References
1. Collect data – e.g. actual QC data
available or preferably data from
3x QC sample size)
—
2. Compute sample mean and SD
Paste Function in Excel
3. Estimate Upper and/or lower
bounds for mean
Upper bound for SD
Figure 28, Paste
Function in Excel
4. Substitute in the formula in Excel
Figure 29, Paste
Function in Excel
Tools
Ready-for-use MS Excel
files for computing the
probability of passing
those USP tests are
obtainable upon request.*
*Interested readers may contact the author, [email protected], asking for the MS
Excel files.
Note: Constructing a formula may be made by carefully copying the desired formula in this paper bracket
by bracket into the MS Excel sheet.
Figure 28
Summary of Upper and Lower Bounds for Sample Statistics
USP Tests
Content Uniformity
Sample Statistics
Sample SD (SD)
Sample Mean (M)
95% Joint Confidence Interval/Upper & Lower Bounds
UB =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
UB(Mean) =M+UB*NORMSINV(1-(1-0.95^0.5)/2)/n^0.5
LB(Mean) =M-UB*NORMSINV(1-(1-0.95^0.5)/2)/n^0.5
Dissolution
Sample SD (SD)
UB =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
Sample Mean (M)
LB(Mean) =M-UB*NORMSINV(1-(1-0.95^0.5))/n^0.5
Disintegration
Sample SD (SD)
UB =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
Sample Mean (M)
UB(Mean) =M+UB*NORMSINV(1-(1-0.95^0.5))/n^0.5
Minimum Fill
Sample SD (SD)
UB =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
Sample Mean (M)
LB(Mean) =M-UB*NORMSINV(1-(1-0.95^0.5))/n^0.5
Deliverable
Sample SD (SD)
UB =SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
Volume
Sample Mean (M)
LB(Mean) =M-UB*NORMSINV(1-(1-0.95^0.5))/n^0.5
UB: Upper Bound (Upper Limit), LB: Lower Bound (Lower Limit)
How to estimate the upper bound for SD is based on sampling plan type – 1 or 2
Note: Remember that the symbol ‘=’ must be attached to the formula itself
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Figure 29
Quick Reference for Related Figures and Formulae
Description Items
CU
DISS
DIST
MF
DV
Probability Figure #
12-15
20, 21
22
23-24
25-26
Estimation Figure # (Plan 1)
10, 27
10, 27
10, 27
10, 27
10, 27
Estimation Figure # (Plan 2)
11, 27
11, 27
11, 27
11, 27
11, 27
MS Excel Formula #
(I), (II), (III)
(IV)
(V)
(VI)
(VII), (VIII)
CU: Content Uniformity, DISS: Dissolution, DIST: Disintegration,
MF: Minimum Fill, DV: Deliverable Volume
CONCLUSION
ABOUT THE AUTHOR
Probability of passing the compendial (USP) multi-stage
tests requires estimation for statistical lower and/or upper
bounds for the mean and standard deviation of the samples
of appropriate size. The estimation results will generate the
probability to pass the multi-stage tests. The sample size
used is the multiple (e.g., at least triple) of the compendial
QC test sample size so that the sample quality will comprehensively represent that of the batch. The validation acceptance criterion with respect to the RSD limit, when
customized, is based on the critical RSD value that provides
at least 50% probability, or higher as required, of meeting the
specification. Expressing the high probability of passing the
multi-stage tests (for critical quality attributes e.g., content
uniformity) will directly fulfill the validation guideline requirements i.e. “… maximize the probability that the finished
product meets all quality and design specifications.” As one
of the benefits, the benchmark of a certain process can be
quantitatively measured through these probability results. ❏
Pramote Cholayudth is Executive Director of Valitech
Co., Ltd., a well-established validation and compliance consultant services company for Pharmaceutical Industry in Thailand. He is a guest speaker on
Process Validation to industrial pharmaceutical scientists organized by local FDA. Prior to this he was a
full time lecturer in a School of Pharmacy in a private
university for four years (1998-2001). Before entering
the academic arena, he spent 23 years in the pharmaceutical industry with Bayer Laboratories (19741981) and OLIC (Thailand), Limited (1981-1997) – a
leading and largest pharmaceutical toll manufacturer
for multinational companies. Pramote is the author of
Concepts and Practices of Pharmaceutical Process
Validation. He is currently a JVT Editorial Advisory
Board Member and can be contacted by fax at 662740-9586, by e-mail at [email protected],
or at the following mailing address:
Pramote Cholayudth
6/756 Number One Complex, Bangkok-Ram 2 Road,
Pravate District, Bangkok, 10250
Thailand
REFERENCES
1. Division of Manufacturing and Product Quality, Office of
Compliance, Center for Drugs and Biologics, U.S. FDA,
Guideline on General Principles of Process Validation, May
1987, www.fda.gov/CDER/GUIDANCE/pv.htm.
A u g u s t 2 0 0 7 • Vo l u m e 1 3 , N u m b e r 4
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Pramote Cholayudth
2. WHO Working Document QAS/03.055/Rev.2: Supplementary Guidelines on Good Manufacturing Practices (GMP):
Validation, 2005, www.who.int/medicines/ services/expertcommittees/pharmprep/Validation_QAS_055_Rev2combined.pdf.
3. Presentation on “FDA Regulation of Drug Quality: New
Challenges,” Janet Woodcock, Director, Center for Drug
Evaluation and Research, Food and Drug Administration,
April 9, 2002. www.fda.gov/ohrms/dockets/ac/02/briefing/3869B1 _08_woodcock.ppt.
4. Bergum, J. S., “Constructing Acceptance Limits for Multiple
Stage Tests,” Drug Development and Industrial Pharmacy,
Marcel Dekker, Inc, 1990, Vol.16, No.14, pp. 2153-2166.
5. Chow, S. C. and J. P. Liu, “USP Tests and Specifications,”
in Statistical Design and Analysis in Pharmaceutical
Science: Validation, Process Controls, Practical and Clinical
Applications, 3rd edition New York: Marcel Dekker, Inc,
1995, pp. 158-165.
6. Bergum, J. S. and M. L. Utter, “Statistical Methods for Uniformity and Dissolution Testing,” Pharmaceutical Process
Validation: An International Third Edition, Revised and
Expanded, R. A. Nash and A. H. Wachter, New York,
Marcel Dekker, Inc, 2003, pp. 667-697.
7. Blend Uniformity Working Group (BUWG), Product Quality
Research Institute (PQRI), “The Use of Stratified Sampling
of Blend and Dosage Units to Demonstrate Adequacy of
Mix for Powder Blends,” Final Report on Blend Uniformity
Recommendations, December 30, 2002.
www.pqri.org/datamining/imagespdfs/123002sam.pdf.
8. Center for Drug Evaluation and Research (CDER), Food
and Drug Administration, “Powder Blends and Finished
Dosage Units - Stratified In-Process Dosage Unit Sampling
and Assessment,” Draft Guidance for Industry, October
2003. www.fda.gov/cder/guidance/5831dft.pdf and
www.fda.gov/cder/guidance/5831AttmtR.pdf.
9. J. S. Bergum’s SASÍ Programs – “Appendix E: Lower
Bound Calculations,” Content Uniformity and Dissolution
Acceptance Limits (CUDAL), version: 1.0, dated 7/26/03.
10. Cholayudth, P., “Use of the Bergum Method and MS Excel
to Determine the Probability of Passing the USP Content
Uniformity Test,” Pharmaceutical Technology, Volume 28,
Number 9, September 2004. www.pharmtech.com.
11. Cholayudth, P., “Using the Bergum Method and MS Excel
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to Determine the Probability of Passing the USP Dissolution
Test,” Pharmaceutical Technology, Volume 30, Number 1,
January 2006. www.pharmtech.com.
12. Cholayudth, P., “Establishing Target Fills for Semisolid and
Liquid Dosage Forms,” Pharmaceutical Technology, Volume
29, Number 4, April 2005. www.pharmtech.com.
13. Pluta, P., “Benchmarking” Journal of Validation Technology,
Volume 12, No. 1, November 2005, pp. 69-71.
14. MS Excel formula for computation of probability of passing
content uniformity test for tablets and capsules,
Pharmaceutical Technology website,
www.pharmtech.com, search ‘Pramote.’
15. Torbeck, L. D., “In Defense of USP Single Testing,”
Pharmaceutical Technology, Volume 29, Number 2,
February 2005. www.pharmtech.com.
Article Acronym Listing
CU
FDA
IPC
LA
LB
LC
MS
PQRI
QA
QC
RSD
SD
SqD
UB
UL
USP
WV
WHO
Content Uniformity
Food and Drug Administration
In-Process Control
Labeled Amount
Lower Bound
Label Claim
Microsoft
Product Quality Research Institute
Quality Assurance
Quality Control
Relative Standard Deviation
Sample Standard Deviation
Square Deviation
Upper Bound
Upper Limit
United States Pharmacopeia
Weight Variation
World Health Organization
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KEYNOTE ADDRESS
Notes From the Field:
A District Office’s
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