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Kenneth C. Waterman, Ph.D. Jon Swanson, Ph.D. FreeThink Technologies, Inc.

Kenneth C. Waterman, Ph.D. Jon Swanson, Ph.D. FreeThink Technologies, Inc. A Scientific and Statistical Analysis of Accelerated Aging for Pharmaceuticals: Accuracy and Precision of Fitting Methods. Accuracy in accelerated aging Point estimates Linear estimates Isoconversion

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Kenneth C. Waterman, Ph.D. Jon Swanson, Ph.D. FreeThink Technologies, Inc.

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  1. Kenneth C. Waterman, Ph.D. Jon Swanson, Ph.D. FreeThink Technologies, Inc. A Scientific and Statistical Analysis of Accelerated Aging for Pharmaceuticals: Accuracy and Precision of Fitting Methods ken.waterman@freethinktech.com 2014

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

  3. 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

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

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

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

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

  8. Accelerated Aging—IsoconversionApproach 60°C 70°C 50°C 0.2% specification limit Isoconversion: %degradant fixed at specification limit, time adjusted ken.waterman@freethinktech.com 2014

  9. Accelerated Aging—IsoconversionApproach Complex Kinetics Using 0.2% isoconversion 70°C 60°C 50°C 30°C ken.waterman@freethinktech.com 2014

  10. Accelerated Aging—IsoconversionApproach Complex Kinetics—Real Example ASAPprimeShelf Life 1.2 yrs Experimental Shelf Life 1.2 yrs 70C 80C 50C 60C 30C Real time data CP-456,773/60%RH ken.waterman@freethinktech.com 2014

  11. More Detailed Example k2 k1 A 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

  12. Primary Degradant (“B”) Formation ken.waterman@freethinktech.com 2014

  13. Example @40°C Note R2 for line = 0.998 ken.waterman@freethinktech.com 2014

  14. Secondary Degradant (“C”) Formation ken.waterman@freethinktech.com 2014

  15. 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

  16. Estimating Uncertainty • 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) ken.waterman@freethinktech.com 2014

  17. Isoconversion Uncertainty Methods • Confidence Interval: • Regression Interval: • Stochastic: Monte-Carlo distribution • Non-stochastic: 2npermutations of ±1σ • Extrema: 2n permutations of ±1σ; normalize using zero-error isoconversion- minimum time (maximum degradant) of distribution ken.waterman@freethinktech.com 2014

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

  19. Calculations Where Formulae Exist Fixed SD = 0.02% ken.waterman@freethinktech.com 2014

  20. 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

  21. Calculations Where Formulae Do Not Exist Fixed RSD = 10% with minimum error of 0.02% (LOD) ken.waterman@freethinktech.com 2014

  22. Arrhenius Fitting Uncertainty • 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) ken.waterman@freethinktech.com 2014

  23. 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 84.1% 1.42 X 10-4%/d 84.1% 1.43 X 10-4%/d 15.9% 4.05 X 10-4%/d 15.9% 3.83 X 10-4%/d Monte Carlo Isoconversion Monte Carlo Arrhenius ExtremaIsoconversion Monte Carlo Arrhenius ken.waterman@freethinktech.com 2014

  24. 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

  25. Arrhenius Fitting Uncertainty • 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

  26. Comparison of Arrhenius Fitting Methods • 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

  27. 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

  28. Low Degradantvs. 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

  29. 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

  30. 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 Conclusions ken.waterman@freethinktech.com 2014

  31. 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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