Transcript Bayesian Modeling in Accelerated Stability Studies

Modeling Approaches to
Multiple Isothermal Stability
Studies for Estimating Shelf Life
Oscar Go, Areti Manola,
Jyh-Ming Shoung and Stan Altan
Non-Clinical Statistics
Contents
 Overview of Statistical Aspect of Stability
Study
 Accelerated Stability Study
 Bayesian Methods
 Case Study
 Concluding Remarks
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Purpose of Stability Testing
 To provide evidence on how the quality of a
drug substance or drug product varies with
time under the influence of a variety of
environmental factors (such as temperature,
humidity, light, package)
 To establish a re-test period for the drug
substance or an expiration date (shelf life) for
the drug product
 To recommend storage conditions
3
Typical Design
 Randomly select containers/dosage units at
time of manufacture, minimum of 3 batches,
stored at specified conditions.
 At specified times 0, 1, 3, 6, 9, 12, 18, 24, 36,
48, 60 months, randomly select dosage units
and perform assay on composite samples
 Basic Factors : Batch, Strength, Storage
Condition, Time, Package
 Additional Factors: Position, Drug Substance
Lot, Supplier, Manufacturing Site
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Kinetic Models
 Orders 0, 1, 2 :
C ( 0) (t )  C0  k0  t
C (t )  C0  e
(1)
 k1 t
 1 

C (t )     k 2  t 
 C0 

1
( 2)
where C0 is the assay value at initial
 When k1 and k2 are small,
C (1) (t )  C0  C0  k1  t
C ( 2 ) (t )  C0  C02  k 2  t
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Estimation of Shelf Life
Data Plot with Regression Line and Lower Confidence Limit
Assay
(%Label)
100
95
Lower
Specification
(LS)
90
85
0
3
6
9
12
18
24
30
36
Time (months)
Intersection of specification limit with
lower 1-sided 95% confidence bound
6
Linear Mixed Model
yijk    i  B j Tijk   ijk
where
yijk = assay of ith batch at jth temperature and kth time point,
 = process mean at time 0 (intercept),
2
i = random effect due to ith batch at time 0:  i ~ N (0, )
Bj = fixed average rate of change,
Tijk = kth sampling time for batch i at jth temperature,
ijk = residual error:  ijk ~ N (0, 2 )
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Shelf Life

If Bi  0 , the expiration date ( TSL ) at condition i
is the solution to the quadratic equation
LSL  ˆ  Bˆi  TSL  q  Var(ˆ  Bˆi  TSL )  ˆ2
LSL = 90% = lower specification limit,
q = (1-)th quantile, (=0.05 and z-quantile
was used for the case study)
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Accelerated Stability Testing
 Product is subjected to stress conditions.
 Temperature and humidity are the most
common stress factors.
 Purpose is to predict long term stability and
shelf life.
 Arrhenius equation captures the kinetic
relationship between rates and temperature.
The usual fixed and mixed models ignore any
relationship between rate and temperature.
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Arrhenius Equation
Named for Svante Arrhenius (1903 Nobel
Laureate in Chemistry) who established a
relationship between temperature and the
rates of chemical reaction

kT  k (T )  Ae
Ea
RT
where kT = Degradation Rate
A = Non-thermal Constant
Ea = Activation Energy
R = Universal Gas Constant (1.987)
T = Absolute Temperature
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Assumptions Underlying
Arrhenius Approach
 The kinetic model is valid and applies to the
molecule under study
 Homogeneity in analytical error
NB: Humidity is not acknowledged in the
equation
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Nonlinear Parametrization
(King-Kung-Fung Model)

kT  Ae
Ea
RT
Let T =298oK (25oC)
kT  k298e
A  k298e
Ea
298R
Ea  1 1 
 

R  298 T 
CT (t )  C0  k298e
Ea  1 1 
 

R  298 T 
kT
t
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King-Kung-Fung
Nonlinear Mixed Model
*
Cijl  C0  ui  k 298  e
e Ea  1
1


R  298 T j




 tijl   ijl
Indices
 i = batch identifier
 j = temperature level
 l = time point
ui ~ N (0,  u2 )
 ijl ~ N (0,  2 )
*
2
2
C
,
k
,
E
(

ln
E
),

,

Parameters are : 0 298 a
a
u

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King-Kung-Fung ModelEstimation of Shelf Life
Shelf life at a given temperature Tj = T is the
solution tSL in the following equation
LSL  ˆ (Cijl (t SL ))  t0.95,df
where
 

E (Cijl (t ))  C0  k 298  e

Var(ˆ (Cijl (t SL )))

Ea*
e
 1 1
 

R  298 T 
t
t0.95,df is the Student’s 95th t-quantile with
df degrees
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Linearized Arrhenius Model
Take log on both sides of the Arrhenius
equation
kT  A  e

Ea
RT
Ea
 log kT  log A 
R T
Assuming a zero order kinetic model
CT t   C0  kT  t  log kT  logC0  CT t   logt
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Linearized Arrhenius Model
 Combining the two equations and solving for
log t
Ea
log t  log( C0  CT (t ))  log A 
R T
 Set t to t90 , time to achieve 90% potency for
each temperature level (CT ( t90 )=90 )
Ea 1
log t 90  log(C0  90)  log A 

R T
0
1
 Expressed as linear regression problem
1
log t 90   0  1   
T
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Linearized Arrhenius Mixed Model
To include batch-to-batch effect in the model,
we can add a random term to 0
1
logt 90ij   0  vi   1    ij
Tij
Indices
 i = batch identifier
 j = temperature level
vi ~ N (0,  v2 )
 ij ~ N (0,  2 )
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Linearized Arrhenius Mixed Model
To summarize, the (Garrett, 1955) algorithm:
1) Fit a zero-order kinetic model by batch and
and temperature level.
2) Estimate t90 and its standard error from each
zero-order kinetic model.
3) Fit a linear (mixed) model to log(t90) on the
reciprocal of Temperature(Kelvin scale).
4) Shelf life for a given temperature level is
estimated from the model in step 3.
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Comparison Between the Three
Approaches
 Linear Mixed Model
 Loses information contained in the Arrhenius
relationship when it is valid
 Linearized Arrhenius Model (Garrett)
 Simple and does not require specialized
software
 Not clear how to estimate shelf life in relation to
ICH guideline
 Ignores heteroscedasticity in the error terms
 Difficult to interpret the random effect
 Nonlinear Model (King-Kung-Fung)
 Computationally intensive
 Computing convergence issues
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King-Kung-Fung Model: Bayesian
Method
*
Cijl  C0  ui  k 298  e
e Ea  1
1


R  298 T j




 tijl   ijl
Indices
 i = batch identifier
 j = temperature level
 l = time point
ui ~ N (0,  u2 )
 ijl ~ N (0,  2 )
Parameters:
C0 , k298 , Ea* ( ln Ea ),  u2 ,  2
Additional Parameters:
t 90T  298 , kT 303 , t 90T 303
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Shelf Life
 Consider
kT  k298e
Ea  1 1 
 

R  298 T 
LSL  90  C0  kT  t90T  Z
Z is independent of data and symmetrical about 0,
eg, Z ~ N (0,  u2 ).
Z was added to take into account  u2 .
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King-Kung-Fung Model:
Method-Prior Distributions
Bayesian
 Provides a flexible framework for incorporating
scientific and expert judgment, incorporating past
experience with similar products and processes
 Expert opinions





Process mean at time 0 is between 99% and
101%
No information regarding degradation rate
No information regarding activation energy
Batch variability is between 0.1 and 0.5 with
99% probability
Analytical variability is between 0.1 to 1.0
with 99% probability
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Prior Distributions
C0 ~ N (100, 0.1)
k 298 ~ I (,)
E ~ I (,)
*
a
 u ~  (10, 2 )
2
1
  ~  (6, 2 )
2
1
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Case Study
24
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R/WinBUGS Simulation Parameters
 3 chains
 500,000 iterations/chain
 Discard 1st 100,000 simulated values in each
chain
 Retain every 100th simulation draw
 A total of 27,000 simulated values for each
parameter
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Model Parameter Estimates
Nonlinear Mixed Model
Parameters
C0
k298
Ea*
 u2
2
True
Value
Bayesian Nonlinear Mixed Model
Estimate
95%
Confidence
Interval
Mean (Median)
95% Credible
Interval
100.0
99.9
98.3 - 101.5
100.0 (100.0)
99.5 - 100.4
0.26
0.24
0.15 - 0.32
0.24 (0.24)
0.20 - 0.28
10.04
10.08
9.89 - 10.27
10.07 (10.07)
9.99 - 10.16
0.20
0.32
0.24 (0.22)
0.13 - 0.43
0.33
0.41
0.43 (0.42)
0.29 - 0.64
 Bayesian method provides the ability to
characterize the variability of parameter
estimates, even when data are limited.
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Shelf Life Estimates
Linear Mixed Model
Temperature Estimate
Shelf Life
25C
41.3
32.9
30C
21.8
18.3
40C
6.1
5.2
Nonlinear Mixed Model
Temperature
True
Value
90%
Shelf
Estimate Confidenc
Life
e Interval
Linearized Arrhenius
Model
Bayesian Nonlinear
Mixed Model
Estimate
90%
Confidence
Interval
Mean
(Median)
90%
Credible
Interval
25C
38.9
42.0
31.9 - 52.1 33.7
37.7
31.1 - 44.4
42.1 (41.8)
35.9 - 49.1
30C
20.5
21.6
17.8 - 25.4 18.3
20.4
17.2 - 23.6
21.7 (21.6)
19.1 - 24.5
40C
6.0
6.1
6.3
5.1 - 7.5
6.1 (6.1)
5.5 - 6.8
5.2 - 6.9
5.3
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29
Variance Component
30
Degradation Rate
31
Shelf Life
32
Summary
 King-Kung-Fung model is a practical way to
characterize multiple isothermal stability profiles
and has been shown to be extended easily to a
nonlinear mixed model context.
 Bayesian method permits integration of expert
scientific judgment in characterizing the stability
property of a pharmaceutical compound.
 The Bayesian credible interval can be interpreted
in a probabilistic way and provides a more natural
meaning to shelf life compared with the
frequentist repeated sampling definition.
 The problem of determining the appropriate
degrees of freedom in mixed modeling is
eliminated by Bayesian method.
 Bayesian method is flexible and can be easily
applied to a wide family of distributions.
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