A Scientific and Statistical Analysis of
Accelerated Aging for Pharmaceuticals:
Accuracy and Precision of Fitting Methods
Kenneth C. Waterman, Ph.D.
Jon Swanson, Ph.D.
FreeThink Technologies, Inc.
ken.waterman@freethinktech.com
2014
1
• Accuracy in accelerated aging
• Point estimates
• Linear estimates
• Isoconversion
• Uncertainty in predictions
• Isoconversion methods
• Arrhenius
Outline
• Distributions (MC vs. extrema isoconversion)
• Linear vs. non-linear
• Low degradant
• Conclusions
ken.waterman@freethinktech.com
2014
2
Accuracy in Accelerated Aging
• Statistics must be based on accurate models
• Most shelf-life today determined by degradant
growth not potency loss
• >50% Drug products show complex kinetics: do
not show linear behavior
• Heterogeneous systems
• Secondary degradation
• Autocatalysis
• Inhibitors
• Diffusion controlled
ken.waterman@freethinktech.com
2014
3
Complex Kinetics—Example
1
Drug → primary degradant → secondary degradant
0.9
0.8
%Degradant
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
7
14
21
Time (days)
ken.waterman@freethinktech.com
2014
28
4
Accelerated Aging
Complex Kinetics
0.45
0.4
70°C
%Degradant
0.35
0.3
60°C
0.25
0.2
0.15
50°C
Fixed time
accelerated
stability
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
Time (days)
ken.waterman@freethinktech.com
2014
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Accelerated Aging
Complex Kinetics
-2.8
-3
70°C
60°C
ln k
-3.2
50°C
-3.4
-3.6
-3.8
More
unstable
• Appears very non-Arrhenius
• Impossible to predict shelf-life
from high T results
-4
0.0029 0.00295
0.003
30°C?
0.00305 0.0031 0.00315 0.0032 0.00325 0.0033
1/T
ken.waterman@freethinktech.com
2014
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Accelerated Aging
Complex Kinetics: Real Example
0
Fixed-time Predicted Shelf Life
Experimental Shelf Life
-1
-2
80C
70C
50C
60C
-3
ln k
0.5 yrs
1.2 yrs
-4
-5
30C
-6
-7
0.00280
Real time data
0.00290
0.00300
0.00310
1/T
ken.waterman@freethinktech.com
0.00320
0.00330
CP-456,773/60%RH
2014
7
Accelerated Aging—Isoconversion Approach
0.3
60°C
70°C
%Degradant
0.25
50°C
0.2% specification limit
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
Time (days)
Isoconversion: %degradant fixed at specification limit, time adjusted
ken.waterman@freethinktech.com
2014
8
Accelerated Aging—Isoconversion Approach
Complex Kinetics
-1.5
-2.5
Using 0.2% isoconversion
70°C
60°C
-3.5
ln k
50°C
-4.5
-5.5
30°C
Using 0.5% isoconversion
-6.5
-7.5
0.0029 0.00295
0.003
0.00305 0.0031 0.00315 0.0032 0.00325 0.0033
1/T
ken.waterman@freethinktech.com
2014
9
Accelerated Aging—Isoconversion Approach
Complex Kinetics—Real Example
2
70C
1
0
80C
ASAPprime Shelf Life
Experimental Shelf Life
1.2 yrs
1.2 yrs
-1
50C
ln(k)
-2
60C
-3
-4
-5
30C
-6
Real time data
-7
-8
0.00280
0.00290
0.00300
0.00310
0.00320
1/T (K)
ken.waterman@freethinktech.com
0.00330
0.00340
CP-456,773/60%RH
2014
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More Detailed Example
A
k1
k2
B
C
• Time points @ 0, 3, 7, 14 and 28 days
• Shelf-life @25°C using 50, 60 and 70°C
• k1 = 0.000113%/d k2 = 0.01125%/d
@50°C for “B” example (25 kcal/mol)
• k1 = 0.000112%/d k2 = 0.09%/d
@50°C for “C” example (25 kcal/mol)
ken.waterman@freethinktech.com
2014
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Primary Degradant (“B”) Formation
Method
Exact
4 linear rate constants @ each T
1 linear rate constant through 4 points @ each T
Single point at isoconversion @ each T
Linear fitting of 4 points @ each T to determine
intersection with specification
Determining intersection with specification using 2
points closest to specification @ each T (or
extrapolating from last 2 points, when necessary)
ken.waterman@freethinktech.com
2014
Shelf-life (yrs)
@25°C
Spec. Spec.
0.2%
0.5%
1.43 4.45
0.62 1.56
0.29 0.71
1.43 4.45
12.35
1.40
1.36
3.19
12
Example @40°C
0.6
%Degradant B
0.5
0.4
0.3
0.2
0.1
Note R2 for line = 0.998
0
0
10
20
30
40
50
60
70
Time (days)
ken.waterman@freethinktech.com
2014
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Secondary Degradant (“C”) Formation
Method
Exact
4 linear rate constants @ each T
1 linear rate constant through 4 points @ each T
Single point at isoconversion @ each T
Linear fitting of 4 points @ each T to determine
intersection with specification
Determining intersection with specification using 2
points closest to specification @ each T (or
extrapolating from last 2 points, when necessary)
ken.waterman@freethinktech.com
2014
Shelf-life
(yrs) @25°C
Spec. Spec.
0.2% 0.5%
2.02 4.01
16.64 41.61
3.29 8.21
2.02 4.01
2.75
7.56
2.06
4.78
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Accuracy
• Both isoconversion and rate constant
methods accurate when behavior is simple
• Only isoconversion is accurate when
degradant formation is complex
• Carrying out degradation to bracket
specification limit at each condition will
increase reliability of modeling
ken.waterman@freethinktech.com
2014
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Estimating Uncertainty
• Need to use isoconversion for accuracy: defines a 2step process
• Estimating uncertainty in isoconversion from
degradant vs. time data
• Propagating to ambient using Arrhenius equation
• Error bars for degradant formation are not uniform
• Constant relative standard deviation (RSD)
• Minimum error of limit of detection (LOD)
ken.waterman@freethinktech.com
2014
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Isoconversion Uncertainty Methods
• Confidence Interval:
• Regression Interval:
𝐶𝐼 = 𝜎
1
𝑛
+
𝑑 𝑜 −𝑑 2
𝑑 𝑖 −𝑑 2
1
𝑛
𝑅𝐼 = 𝜎 1 + +
𝑑 𝑝 −𝑑
2
𝑑 𝑖 −𝑑 2
• Stochastic: Monte-Carlo distribution
• Non-stochastic: 2n permutations of ±1σ
• Extrema: 2n permutations of ±1σ; normalize using zeroerror isoconversion - minimum time (maximum
degradant) of distribution
ken.waterman@freethinktech.com
2014
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Test Calculations:
Model System
1.8
1.6
%Degradant
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
Time (days)
ken.waterman@freethinktech.com
2014
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45
Calculations Where Formulae Exist
Calculation Method
Regression Interval
5 Days
(Interpolation)
0.023%
40-Days
(Extrapolation)
0.102%
Confidence Interval
0.012%
0.100%
Stochastic
0.012%
0.099%
Non-Stochastic
0.012%
0.100%
Extrema
0.020%
0.147%
Fixed SD = 0.02%
ken.waterman@freethinktech.com
2014
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Isoconversion Uncertainty
• CI too narrow in interpolation regions (<
experimental σ); also does not take into
account error of fit
• RI better represents error for predictions
• RI and CI converge with extrapolation
• Extrema mimics RI in interpolation; more
conservative in extrapolation
• Note: scientifically less confident in
isoconversion extrapolations (model fit)
ken.waterman@freethinktech.com
2014
20
Calculations Where Formulae
Do Not Exist
Calculation
Method
Stochastic
Non-Stochastic
Extrema
5 Days
(Interpolation)
0.016%
0.016%
40-Days
(Extrapolation)
0.166%
0.166%
0.027%
0.223%
Fixed RSD = 10% with
minimum error of 0.02% (LOD)
ken.waterman@freethinktech.com
2014
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Arrhenius Fitting Uncertainty
• Can use full isoconversion distribution from MonteCarlo calculation
• Can use extrema calculation
• Normalized about time (x-axis, degradant set by
specification limit)
• Normalized about degradant (y-axis, time set by
zero-error intercept with specification limit)
ken.waterman@freethinktech.com
2014
22
25°C Projected Rate Distributions
60, 70, 80°C measurements @10 days; RSD=10%, LOD=0.02%; 25 kcal/mol
50%
2.38 X 10-4%/d
50%
2.34 X 10-4%/d
Rate from CI (RSD/LOD, 0.2%)
800
Rate from Deg Extrema (RSD/LOD, 0.2%)
84.1%
1.43 X 10-4%/d
400
15.9%
4.05 X 10-4%/d
0
200
200
frequency
400
15.9%
3.83 X 10-4%/d
0
frequency
600
600
800
84.1%
1.42 X 10-4%/d
0e+00
0e+00
2e-04
4e-04
6e-04
8e-04
2e-04
4e-04
6e-04
rate
rate
Monte Carlo Isoconversion
Monte Carlo Arrhenius
ken.waterman@freethinktech.com
Extrema Isoconversion
Monte Carlo Arrhenius
2014
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8e-04
Arrhenius Fitting Uncertainty
• Distribution of ambient rates from Monte-Carlo
or extrema calculations very similar
• In both cases, rate is not normally distributed
• Probabilities need to use a cumulative
distribution function
ken.waterman@freethinktech.com
2014
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Arrhenius Fitting Uncertainty
𝑘𝑖𝑠𝑜𝑇1 = 𝐴𝑒 −𝐸𝑎
𝑅 1 𝑇2
• Can be solved in logarithmic (linear) or exponential (non-linear)
form
• With perfect data, point estimates of rate (shelf-life) will be
identical
• A distribution at each point will generate imperfect fits
• Least squares will minimize difference between actual and
calculated points
• Non-linear will weight high T more heavily
• Constant RSD means that higher rates will have greater
errors
ken.waterman@freethinktech.com
2014
25
Comparison of Arrhenius Fitting Methods
•
•
•
•
•
Linear
Extrapolated Shelf-life (years) at 25°C
84.1%
Median
15.9%
Mean
3.86
2.31
1.43
2.70
Non-linear
7.12
2.33
0.90
5.41
Arrhenius based on isoconversion values @60, 70, 80°C
Origin + point at 10 days; spec. limit (0.20%)
RSD=10%; LOD = 0.02%
Isoconversion distribution using extrema method
True shelf-life equals 2.31 years
ken.waterman@freethinktech.com
2014
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Arrhenius Fitting Uncertainty
• Non-linear least squares fitting gives larger, less
normal distributions of ambient rates
• Non-linear fitting’s greater weighting of higher
temperatures makes non-Arrhenius behavior
more likely to cause inaccuracies
• Since linear is also less computationally
challenging, recommend use of linear fitting
ken.waterman@freethinktech.com
2014
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Low Degradant vs. Standard Deviation
• For low degradation rate (with respect to the SD),
isoconversion less symmetric
• Becomes discontinuous @Δdeg = 0 (isoconversion = ∞)
for any sampled point
• Can resolve by clipping points with MC
• Distribution meaning when most points removed?
• Can use extrema
• Define behavior with no regression line isoconversion
• Can define mean from first extrema intercept (2 X value)
• No perfect answers—modeling better when data show
change
ken.waterman@freethinktech.com
2014
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Notes
• ICH guidelines allow ±2C and ±5%RH—
average drug product shows a factor of
2.7 shelf-life difference within this range
• ASAP modeling uses both T and RH, both
potentially changing with time—errors
will change accordingly
• Assume mathematics the same, but
need to focus on instantaneous rates
ken.waterman@freethinktech.com
2014
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Conclusions
• Modeling drug product shelf-life from accelerated
data more accurate using isoconversion
• Isoconversion more accurate using points bracketing
specification limit than using all points
• With isoconversion, regression interval (not
confidence interval) includes error of fit, but difficult
to calculate with varying SD
• Extrema method reasonably approximates RI for
interpolation; more conservative for extrapolation
• Linear fitting of Arrhenius equation preferred
ken.waterman@freethinktech.com
2014
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Notes on King, Kung, Fung “Statistical prediction of drug
stability based on non-linear parameter estimation” J.
Pharm. Sci. 1984;73:657-662
• Used rates based on each time point independently
• Changing rate constants not projected accurately for shelf-life
• Gives greater precision by treating each point as equivalent, even when
far from isoconversion (32 points at 4 T’s gives better error bars than
just 4 isoconversion values: more precise, but more likely to be wrong)
• Non-linear fitting to Arrhenius
• Weights higher T more heavily (and where they had most degradation)
• Made more sense with constant errors used for loss of potency
• Non-linear fitting in general bigger, less symmetric error bars, more
likely to be in error if mechanism shift with T
• Used mean and SD for linear fitting, even when not normally distributed
(i.e., not statistically valid method)
• Do not recommend general use of KKF method (fine for ideal behavior,
loss of potency)
ken.waterman@freethinktech.com
2014
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