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**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.

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**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

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**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

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**Complex Kinetics—Example ken.waterman@freethinktech.com 2014**

Drug → primary degradant → secondary degradant

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**ken.waterman@freethinktech.com 2014**

Accelerated Aging Complex Kinetics 70°C 60°C 50°C Fixed time accelerated stability

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**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

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**Complex Kinetics: Real Example ken.waterman@freethinktech.com 2014**

Accelerated Aging Complex Kinetics: Real Example 80C 70C 50C 60C 30C Real time data CP-456,773/60%RH

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**Accelerated Aging—Isoconversion Approach**

0.2% specification limit Isoconversion: %degradant fixed at specification limit, time adjusted

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**Accelerated Aging—Isoconversion Approach Complex Kinetics**

Using 0.2% isoconversion 70°C 60°C 50°C 30°C

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**Accelerated Aging—Isoconversion Approach Complex Kinetics—Real Example**

ASAPprime Shelf Life yrs Experimental Shelf Life yrs 70C 80C 50C 60C 30C Real time data CP-456,773/60%RH

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**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)

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**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

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**ken.waterman@freethinktech.com 2014**

Note R2 for line = 0.998

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**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

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**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

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**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)

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**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

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

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**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%

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**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)

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**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)

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**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)

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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)

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Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 13 Nonlinear and Multiple Regression.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 13 Nonlinear and Multiple Regression.

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