1. Accuracy and Precision of Fitting Methods
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.

2. ken.waterman@freethinktech.com 2014
Outline
Accuracy in accelerated aging
Point estimates
Linear estimates
Isoconversion
Uncertainty in predictions
Isoconversion methods
Arrhenius
Distributions (MC vs. extrema isoconversion)
Linear vs. non-linear
Low degradant
Conclusions

3. Accuracy in Accelerated Aging ken.waterman@freethinktech.com 2014
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

4. Complex Kinetics—Example ken.waterman@freethinktech.com 2014
Drug → primary degradant → secondary degradant

5. ken.waterman@freethinktech.com 2014
Accelerated Aging
Complex Kinetics
70°C
60°C
50°C
Fixed time accelerated stability

6. ken.waterman@freethinktech.com 2014
Accelerated Aging
Complex Kinetics
More unstable
70°C
60°C
50°C
30°C?
Appears very non-Arrhenius
Impossible to predict shelf-life from high T results

7. Complex Kinetics: Real Example ken.waterman@freethinktech.com 2014
Accelerated Aging
Complex Kinetics: Real Example
80C
70C
50C
60C
30C
Real time data
CP-456,773/60%RH

8. Accelerated Aging—Isoconversion Approach
0.2% specification limit
Isoconversion: %degradant fixed at specification limit, time adjusted

9. Accelerated Aging—Isoconversion Approach Complex Kinetics
Using 0.2% isoconversion
70°C
60°C
50°C
30°C

10. Accelerated Aging—Isoconversion Approach Complex Kinetics—Real Example
ASAPprime Shelf Life yrs
Experimental Shelf Life yrs
70C
80C
50C
60C
30C
Real time data
CP-456,773/60%RH

11. ken.waterman@freethinktech.com 2014
More Detailed Example k1 k2
A B C
Time 0, 3, 7, 14 and 28 days
using 50, 60 and 70°C
k1 = %/d k2 = %/d
@50°C for “B” example (25 kcal/mol)
k1 = %/d k2 = 0.09%/d
@50°C for “C” example (25 kcal/mol)

12. Primary Degradant (“B”) Formation ken.waterman@freethinktech.com 2014
Method
Shelf-life
Spec. 0.2%
Spec. 0.5%
Exact
1.43
4.45
4 linear rate each T
0.62
1.56
1 linear rate constant through 4 each T
0.29
0.71
Single point at each T
Linear fitting of 4 each T to determine intersection with specification
12.35
1.40
Determining intersection with specification using 2 points closest to each T (or extrapolating from last 2 points, when necessary)
1.36
3.19

13. ken.waterman@freethinktech.com 2014
Note R2 for line = 0.998

14. Secondary Degradant (“C”) Formation
Method
Shelf-life
Spec. 0.2%
Spec. 0.5%
Exact
2.02
4.01
4 linear rate each T
16.64
41.61
1 linear rate constant through 4 each T
3.29
8.21
Single point at each T
Linear fitting of 4 each T to determine intersection with specification
2.75
7.56
Determining intersection with specification using 2 points closest to each T (or extrapolating from last 2 points, when necessary)
2.06
4.78

15. ken.waterman@freethinktech.com 2014
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

16. Estimating Uncertainty ken.waterman@freethinktech.com 2014
Need to use isoconversion for accuracy: defines a 2-step 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)

17. Isoconversion Uncertainty Methods ken.waterman@freethinktech.com 2014
Confidence Interval: 𝐶𝐼=𝜎 1 𝑛 + 𝑑 𝑜 − 𝑑 𝑑 𝑖 − 𝑑 2
Regression Interval: 𝑅𝐼=𝜎 𝑛 + 𝑑 𝑝 − 𝑑 𝑑 𝑖 − 𝑑 2
Stochastic: Monte-Carlo distribution
Non-stochastic: 2n permutations of ±1σ
Extrema: 2n permutations of ±1σ; normalize using zero-error isoconversion - minimum time (maximum degradant) of distribution

18. Test Calculations: Model System ken.waterman@freethinktech.com 2014

19. Calculations Where Formulae Exist ken.waterman@freethinktech.com 2014
Calculation Method
5 Days (Interpolation)
40-Days (Extrapolation)
Regression Interval
0.023%
0.102%
Confidence Interval
0.012%
0.100%
Stochastic
0.099%
Non-Stochastic
Extrema
0.020%
0.147%
Fixed SD = 0.02%

20. Isoconversion Uncertainty ken.waterman@freethinktech.com 2014
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)

21. Calculations Where Formulae ken.waterman@freethinktech.com 2014
Do Not Exist
Calculation Method
5 Days (Interpolation)
40-Days (Extrapolation)
Stochastic
0.016%
0.166%
Non-Stochastic
Extrema
0.027%
0.223%
Fixed RSD = 10% with minimum error of 0.02% (LOD)

22. Arrhenius Fitting Uncertainty ken.waterman@freethinktech.com 2014
Can use full isoconversion distribution from Monte-Carlo 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)

23. 25°C Projected Rate Distributions ken.waterman@freethinktech.com 2014
60, 70, 80°C days; RSD=10%, LOD=0.02%; 25 kcal/mol
50%
2.38 X 10-4%/d
50%
2.34 X 10-4%/d
84.1%
1.42 X 10-4%/d
84.1%
1.43 X 10-4%/d
15.9%
3.83 X 10-4%/d
15.9%
4.05 X 10-4%/d
Monte Carlo Isoconversion
Monte Carlo Arrhenius
Extrema Isoconversion
Monte Carlo Arrhenius

24. Arrhenius Fitting Uncertainty ken.waterman@freethinktech.com 2014
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

25. Arrhenius Fitting Uncertainty ken.waterman@freethinktech.com 2014
𝑘𝑖𝑠𝑜 𝑇 1 =𝐴 𝑒 − 𝐸 𝑎 𝑅 1 𝑇 2
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

26. Comparison of Arrhenius Fitting Methods
Extrapolated Shelf-life (years) at 25°C
84.1%
Median
15.9%
Mean
Linear
3.86
2.31
1.43
2.70
Non-linear
7.12
2.33
0.90
5.41
Arrhenius based on isoconversion 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

27. Arrhenius Fitting Uncertainty ken.waterman@freethinktech.com 2014
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

28. Low Degradant vs. Standard Deviation
For low degradation rate (with respect to the SD), isoconversion less symmetric
Becomes = 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

29. ken.waterman@freethinktech.com 2014
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

30. ken.waterman@freethinktech.com 2014
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

31. ken.waterman@freethinktech.com 2014
Notes on King, Kung, Fung “Statistical prediction of drug stability based on non-linear parameter estimation” J. Pharm. Sci. 1984;73:
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)