The Posterior Probability of
Dissolution Equivalence
David J LeBlond 1 , John J Peterson 2 and Stan Altan 3)
1
Exploratory Statistics, Abbott, david.leblond@abbott.com
Research Statistics Unit, GlaxoSmithKline Pharmaceuticals
3 Pharmaceutical R&D, Johnson & Johnson
2
Midwest Biopharmaceutical Statistical Workshop
Muncie, Indiana
May 25, 2011
► Objective
Outline
► Background
 Why dissolution?
 Equivalence defined
 Current practice
► Why
a Bayesian approach?
► Posterior probability defined
► MCMC
► Examples
 Equivalence of 2 lots
 Equivalence of 2 processes (multiple lots)
 Model dependent comparisons
► Summary
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2
Objective
► Make
this tool available to you so
you can use it if you want to.




Statistical Modeling
Software (R, WinBUGS)
Example Data & Code
david.leblond@abbott.com
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Importance of in-vitro dissolution
►
Surrogate measure of in-vivo dissolution
►
In-vivo dissolution rate affects drug bio-availability
►
Bio-availability may affect PK (blood levels)
►
Blood levels may affect safety and efficacy
►
Compendial requirement for most solid oral dosage forms
►
Need to show “equivalence” for process/ method change
or transfer to obtain a bio-waiver.
►
Need to show “non-equivalence” to prove in-vitro method
can detect formulation / process differences.
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Measurement of in-vitro dissolution
►1
tablet/ stirred vessel
► 1 run usually = 6 tablets
► solution sampled at fixed
intervals
► samples
assayed
► cumulative concentration
► expressed as % of dosage
form Label Content (%LC)
► Are
and “equivalent”?
% Dissolved
100
0
Time
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5
Equivalence defined
►
Identify parameter space based on
 Difference in Dissolution at multiple time points
 Difference in profile model parameters
 Condensed univariate distance measure
►
Establish similarity region
 Constraints on parameter space
 Based on “customer requirements”
►
Obtain distance estimate from data
 Conforms to parameter space
►
Equivalence: distance estimate is
“sufficiently contained within” the
similarity region.
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?
6
Example: f2 similarity metric
(see reference 9)
►
parameter space: Dissolution differences, Di, at p time points.
►
similarity region:
D2
f2  50 or equvalently, D  99 p  10 p
10 2
where


p
100
D   D 2i and f2  50  log10 

2
i 1
D
 1

p

►
distance estimate =
►
Equivalence:
fˆ2
fˆ2  50
0







0
D1
(point estimate)
(no measure of uncertainty)
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The confidence set approach
TOST (one dimensional)
5%
5%
8%
Yes
2%
No
80
70
60
50
40
10
20
30
40
50
%LC at 30 Minutes
Yes
%LC at 45 Minutes
%LC at 45 Minutes
%LC at 45 Minutes
“MOST” (multi-dimensional)
80
70
60
50
40
10
20
30
40
50
%LC at 30 Minutes
No
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80
70
60
50
40
10
20
30
40
50
%LC at 30 Minutes
Maybe
8
Confidence set approach considerations
►
Must choose similarity region shape.
►
Must choose confidence region shape.
►
The number of shapes increases with number of
dimensions.
►
Lack of conformance between similarity and confidence
region shapes  conservative test
►
Conclusion depends on shape choices.
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Confidence set approach considerations
►
The confidence level is not the probability of equivalence.
►
It is the probability of covering the “true” difference in
repeated trials.
►
What if you really want to know the probability of
equivalence?
 risk based decision making (ICH Q9)
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Proposed Bayesian Approach
distance estimate: Joint Posterior of
Distance measures
Measure of Equivalence
= Integrated density
= Posterior Probability of Equivalence
Obtained by counting from MCMC
pdf
dimension 2
dimension 1
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Similarity region
(“customer requirement”)
11
Bayesian equivalency in a nutshell
Prior Information
(non-informative)
Probability Model
(Likelihood)
Dissolution Data
(Test and Reference)
MCMC
Draws from the joint posterior
distribution of distance parameters
(10-100 thousand)
Count the fraction of draws
within the similarity region
Conclude equivalency if fraction
exceeds some limit (e.g. 95%)
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Example 1: Is the Reference lot equivalent to
the Test lot?
6 tablets per lot
20
Reference 2
40
60
80
Test 1
100
Dissolution
80
60
40
20
20
40
60
80
Minutes
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Example 1: Multivariate Dissolution Model
% Dissolution vector, Y, for the ith tablet from the kth lot …
  1    12

  

2


2

Yik  MVN   2  ,  12
2
    



3
13
23
3
   

2 
  4  
 24  34  4  
k  14

 1 
  01 
 D1 
 


 


0
 2    2   J  D2 
 3 
  03  k  D 3 
 


 


0
 4 k  4 
 D 4 k

 0,0,0,0  for reference lot
Jk  

 1,1,1,1 for test lot(s)
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Example 1: Non-informative Prior Information




i
N 0,1002
Di
N 0,1002
for i  1,2,3,4
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Example 1: Non-informative Prior Information
  12
  1
  1


 
 

2
0

0



2
2
2
 12

Q

2
  13  23  3
  0 0 3
  0 0 3


 
 

2
0
0
0

0
0
0






4

4
24
34
4 
 14
 1
1
 1






1
0

0

2
2

  12


 0 0 3
 0 0 3
  13 23 1








1
0
0
0

0
0
0

24
34

4   14
4

i
uniform 0,Max 
Q ~ IW41 I which implies that
qii ~ SRIG  shape  1, rate  0.5 
ij
uniform  1, 1
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Example 1: Non-informative Prior Information
  12
  1
  1


 
 

2
0

0



2
2
2
 12

Q

2
  13  23  3
  0 0 3
  0 0 3


 
 

2
0
0
0

0
0
0






4

4
24
34
4 
 14
 1
1
 1






1
0

0

2
2

  12


 0 0 3
 0 0 3
  13 23 1








1
0
0
0

0
0
0

24
34

4   14
4

Since
i
 i  i qii
uniform 0,Max  and
qii ~ SRIG  shape  1, rate  0.5 
can be shown (see appendix) to have the distribution
 Max 2 
1
   Max   1
f  | Max  
 1  exp  
 2 

2 
Max 2 


2


 


where    is the standard normal cdf
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Example 1: Non-informative Prior Information
 Max 2 
1
   Max   1
f  | Max  
 1  exp  
 2 

2 
Max 2 
2

    


where    is the standard normal cdf
0.3
density
0.25
0.2
Max=5
Max=10
Max=20
0.15
0.1
0.05
0
0
5
10
15
20
25
30
sigma
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Example 1: Definition of Equivalence
20
Reference 2
40
60
80
Test 1
100
Dissolution
80
60
40
20
20
40
60
80
Minutes
Define a rectangular similarity region, S, as
4%LC  Di  4%LC, for i  1,2,3,4
and require that
Pr  Δ  S  0.95
to conclude equivalence.
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Example 1: Results
10
0
5
0
5 10
Delta4 0
-10 -5
0
-5
-10
10
0
500 of 10,000
draws plotted
5 10
5
0
Delta3 0
-10 -5
-5
0
-10
5
0
-5
5
0
0
Delta1
-5
0
0
5
Delta2 0
0
-5
5
0
-5
Scatter Plot Matrix
Pr  Δ  S  0.77  0.95  not equivalent
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Example 2: Equivalence of 2 processes
20
Test 1
40
60
80
20
Test 2
Test 3
40
60
80
Test 4
100
80
60
Dissolution
40
20
Reference 5
Reference 6
Reference 7
Reference 8
100
80
60
40
20
20
40
60
80
20
40
60
80
Minutes
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Example 2: Hierarchical Model
% Dissolution vector, y, for the ith tablet from the kth run …
μk
y ik

 0,0,0,0  for reference runs
MVN μ0  Jk Δ, Vrun  , Jk  
1,1,1,1 for test runs


MVN μk ,Vtablet 
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Example 2: Non-informative prior
information
 0 N  0,100 
D N  0,100 
elements of V
2
i
2
tablet
i
  tablet ij uniform  1, 1
Q tablet IW41  I

tablet i uniform(0, Max )   tablet i f  | Max 
  run ij uniform  1, 1
Q run IW4 1  I

 run i uniform(0, Max )   run i f  | Max 
elements of Vrun
Max = 40
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20
Test 1
40
60
80
20
Test 2
Test 3
40
60
80
Test 4
100
80
60
Dissolution
40
20
Reference 5
Reference 6
Reference 7
Reference 8
100
80
60
40
20
20
40
60
80
20
Minutes
40
60
80
Example 2: Definition of Equivalence
(same as Example 1)
Define a rectangular similarity region, S, as
4%LC  Di  4%LC, for i  1,2,3,4
and require that
Pr  Δ  S  0.95
to conclude equivalence.
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24
Example 2: Results
0
4
2
4
2
0
Delta4
0
-2
-4 -2 0
4
0
2
4
Delta3
0
-4
1000 of ~2,000
draws plotted
2
0
-2
-4 -2 0
4
0
2
-4
4
2
0
Delta2
0
-2
-4 -2 0
4
0
2
-4
4
2
0
Delta1
0
-2
-4 -2 0
-4
Scatter Plot Matrix
Pr  Δ  S  0.94  0.95  not equivalent
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Example 3: A model dependent comparison
• Data from reference 12
• 3 lots: 1 reference and 2 post-change lots
Pre-change
100
80
60
Dissolution (%LC)
40
20
Major Change
• A minor change and a major change lot
• 12 tablets per Lot
• Pre-change and Test Lots have different
time points
Minor Change
• Comparison requires a
parametric dissolution
profile model
100
80
60
• Similarity region
defined on the model
parameter space
40
20
0
50
100
150
200
250
Minutes
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26
Dissolution profile models
Probit:
Y       log  t   ,   standard normal cdf

Logistic:
Y  M  exp     log  t   1  exp     log  t  
Weibull:

  t   
Y  M   1  exp      
 T  







Exponential: Y  M  1  exp    t  ( 1st order kinetics )
Quadratic: Y    1  t  t   2  t  t
 ,t
2
 average of all time points
…and some others (Higuchi, Gompertz, Hixson-Crowell,…)
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27
Weibull parameters
M=80,100,120 T=100 beta=1


Y  M  1  exp  t
 T



% Dissolution
M measures content
100
80
60
40
20
0
0
50
100
150
200
250
Time in Minutes
M=100 T=100 beta= 0.5, 1.0, 2.0
M=100 T=50,100,150 beta=1
T is time to 63.2% Dissol.
beta measures delay
100
% Dissolution
% Dissolution
100
80
60
40
20
80
60
0.5
40
2.0
20
0
0
0
50
100
150
200
0
250
50
100
150
200
250
Time in Minutes
Time in Minutes
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28
Weibull parameterization in
WinBUGS
► The
following seemed to reduce colinearity and
improve convergence.
 Replace T with ln = - lnT
 Replace  with ln
 transform time (t) from minutes to hours


Y  M  1 exp  exp ln  t
MBSW May 25, 2011
expln  

29
Nonlinear mixed model in WinBUGS
% Dissolution, Y, for the ith tablet from the kth lot at the jth time (t) point…


Yijk  Mik  1 exp  exp lnik  t
expln ik 
  
ijk
 M 
 DM    M 
 M 

 0,0,0  for reference lot



 



Jk  
 ln    ln    Jk  Dln      ln  
1,1,1 for test lot(s)



 ln  


 D   

ik  ln  
 ln  k  ln  i
 M 


  ln  
 
 ln  i
 ijk
MVN  0,V 
MBSW May 25, 2011

N 0, 2

30
Weibull Example: Judging similarity by
confidence set approach
“…At present, some issues are
unresolved such as
(i) how many standard
deviations (2 or 3) should be
used for a similarity criterion,
(ii) what to do if the ellipse is
only marginally out of the
similarity region …”
from Sathe, Tsong, Shah (1996)
Pharm Res 13(12) 1799-1803
MBSW May 25, 2011
31
Weibull Example: Posterior Probability of
Dissolution Equivalence
Major Change Lot
Minor Change Lot
Difference in LN(beta)
0.3
Prob = 0
0.2
2SD Similarity Region
0.1
0.0
-0.1
Prob = 0.949
-0.4
-0.2
0.0
0.2
0.4
Difference in LN(alpha)
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32
Pros and Cons of a Bayesian Approach
►
Pros
 Based on simple counting exercise (MCMC)
 Probability estimate for risk assessment
 Exact conformity between the similarity region and the estimate
(integrated posterior)
 Incorporation of prior information (or not) as appropriate
 True equivalence (not significance) test
 Rewards high data information content
►
Cons
 Requires (usually) MCMC
 Coverage properties require calibration studies.
 Regulatory acceptance?
MBSW May 25, 2011
33
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Schuirmann DJ (1981) On hypothesis testing to determine of the mean of a normal distribution is contained in a known interval, Biometrics 37:617
Berger RL (1982) Multiparameter hypothesis testing and acceptance sampling, Technometrics 24(4) 295-300
Schuirmann DJ (1987) Comparions of two one-sided procedures and power approach of rassessing the equivalence of average bioavailability, Journal
of Pharmokinetics and Biopharmaceutics 15:657-680.
Shah VP, Yamamoto LA Schirmann D, Elkins J and Skelly JP (1987) Analysis of in vitro dissolution of whole versus half controlled release
theophilline tablets, Pharm Res 4: 416-419
Food and Drug Administration. Guidance for Industry: Immediate Release Solid Oral Dosage Forms. Scale-Up and Postapproval Changes (SUPACIR): Chemistry, Manufacturing and Controls, In Vitro Dissolution Testing and In Vivo BE. 1995
Tsong Y, Sathe P, an dShah VP (1996) Compariong 2 dissolution data sets fro similarity ASA Proceedings of the Biopharmaceutical Section 129-134
Berger RL and Hsu JC (1996) Bioequivalence trials, intersection-union tests and equivalence confidence sets, Statistical Science 11(4) 283-319
J.W.Moore and H.H.Flanner, Mathematical Comparison of curves with an emphasis on in vitro dissolution profiles. Pharm. Tech. 20(6), : 64-74, 1996
Moore JW and Flanner HH (1996) Mathematical comparison of dissolution profiles, Pharmaceutical Technology 24:46-54
Tsong Y, Hammerstrom T, Sathe P, and Shah VP (1996) Statistical assessment of mean differences between two dissolution data sets, Drug
Information Journal 30: 1105-1112
Polli JE, Rekhi GS, and Shah V (1996) Methods to compare dissolution profiles, Drug Information Journal 30: 1113-1120.
Sathe PM, Tsong Y, Shah VP (1996) In vitro dissolution profile comparion: statistics and analysis, model dependent approach, Pharmaceutical research
13(12): 1799-1803.
Polli JE, Rekhi GS, an dShah VP (1996) Methods to compare dissoltuion profiles and a rationale for wide dissoltuion specifications for metroprolol
tartrate tablets j pharm Sci 86:690-700
FDA (1997) Guidance for industry: extended release oral dosage forms: development, evaluation, and application of in vitro/ in vivo correlations
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Appendix
Derivation of prior distribution of i shown on slide 17
Goal : pdf, h  |  , where    r , r  q
Since 
uniform  0, M  , f  | M  
1
[1]
M

 1, 0.5
2  e r
SRIG(  1,   0.5), g  r |   1,   0.5  
   r 2 1
Since r
Property of SRIG: Conditional on  ,  r

2
[2]

SRIG   1,     0.5 so g  |   
2
2
1

3
 e
2

2
2 2
[3]
By definition of a mixture distribution: h  | M    g  |    f  | M  d  [4]
M
0
Substituting [1] and [3] into [4] and simplifying, then using the following integration rule
1 
x  ax 2
erf
x
a

e
C
4 a3
2a
and some painful algebra, we obtain the pdf given on slide 17
2  ax
 x e dx 
2


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