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.

[email protected] 2014 1

• • • Accuracy in accelerated aging

Outline

• • • 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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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

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1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0

Complex Kinetics—Example

Drug → primary degradant → secondary degradant

7 14

Time (days)

21 28

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Accelerated Aging Complex Kinetics

0,45 0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 0

70°C 60°C 50°C

1 2 3 4 5

Time (days)

6

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Fixed time accelerated stability

8 9 10

5

Accelerated Aging Complex Kinetics

-2,8 -3 70°C

60°C More unstable

-3,2 -3,4

50°C

-3,6 30°C?

-3,8 • • Appears very non-Arrhenius Impossible to predict shelf-life from high T results -4 0,0029 0,00295 0,003 0,00305 0,0031 0,00315 0,0032 0,00325 0,0033

1/T [email protected] 2014 6

Accelerated Aging Complex Kinetics: Real Example

0 -1 -2 -3 -4

80

-5 -6 -7 0,00280 

C Fixed-time Predicted Shelf Life Experimental Shelf Life 0.5 yrs 1.2 yrs 70



C 60



C 50



C Real time data 30



C

0,00290 0,00300 0,00310

1/T [email protected] 2014

0,00320 0,00330 CP-456,773/60%RH

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Accelerated Aging—Isoconversion Approach

0,3

60°C

0,25

70°C 50°C 0.2% specification limit

0,2 0,15 0,1 0,05 0 0 1 2 3 4

Time (days)

5 6 7

Isoconversion: %degradant fixed at specification limit, time adjusted [email protected] 2014 8

Accelerated Aging—Isoconversion Approach Complex Kinetics

-1,5

Using 0.2% isoconversion

-2,5

70°C

-3,5

60°C 50°C

-4,5 -5,5 -6,5

Using 0.5% isoconversion 30°C

-7,5 0,0029 0,00295 0,003 0,00305 0,0031 0,00315 0,0032 0,00325 0,0033

1/T [email protected] 2014 9

Accelerated Aging—Isoconversion Approach Complex Kinetics—Real Example

2 1 0 -1 -2

80

-3 -4 -5 -6 -7 -8 0,00280 

C 70



C 60



C ASAP

prime

Experimental Shelf Life 50



C Shelf Life Real time data 30



C 1.2 yrs 1.2 yrs

0,00290 0,00300 0,00310 0,00320

1/T (K) [email protected] 2014

0,00330 0,00340 CP-456,773/60%RH

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More Detailed Example

A

k 1

B

k 2

• • Time points @ 0, 3, 7, 14 and 28 days Shelf-life @25°C using 50, 60 and 70°C • • k 1 k 1 = 0.000113%/d k 2 = 0.01125%/d @50°C for “B” example (25 kcal/mol) = 0.000112%/d k 2 = 0.09%/d @50°C for “C” example (25 kcal/mol) C

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Primary Degradant (“B”) Formation

Method Shelf-life (yrs) @25°C

Spec. Spec. 0.2% 0.5%

Exact 4 linear rate constants @ each T 1.43

0.62

4.45

1.56

1 linear rate constant through 4 points @ each T

0.29

0.71

Single point at isoconversion @ each T

1.43

4.45

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)

12.35

1.36

1.40

3.19

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Example @40°C

0,6 0,5 0,4 0,3 0,2 0,1 0 0

Note R 2 for line = 0.998

10 20 30 40

Time (days)

50

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60 70

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Secondary Degradant (“C”) Formation

Method Exact 4 linear rate constants @ each T Shelf-life (yrs) @25°C

Spec. 0.2%

2.02

Spec. 0.5%

4.01

16.64 41.61

1 linear rate constant through 4 points @ each T

3.29

8.21

Single point at isoconversion @ each T

2.02

4.01

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)

2.75

2.06

7.56

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

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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)

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•

Isoconversion Uncertainty Methods

Confidence Interval: 𝐶𝐼 = 𝜎 1 𝑛 + 𝑑 𝑜 − 𝑑 𝑖 − 2 2 • Regression Interval: 𝑅𝐼 = 𝜎 1 + 1 𝑛 + 𝑑 𝑝 − 𝑑 𝑖 − 2 2 • • • Stochastic: Monte-Carlo distribution Non-stochastic: 2 n permutations of ±1σ Extrema: 2 n permutations of ±1σ; normalize using zero error isoconversion - minimum time (maximum degradant) of distribution

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1,8 1,6 1,4 1,2 1 0,8 0,6 0,4 0,2 0 0

Test Calculations: Model System

5 10 15 20 25

Time (days)

30 35 40 45

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Calculations Where Formulae Exist

Calculation Method Regression Interval Confidence Interval Stochastic Non-Stochastic Extrema 5 Days (Interpolation) 0.023%

0.012% 0.012% 0.012% 0.020%

40-Days (Extrapolation) 0.102%

0.100% 0.099% 0.100% 0.147%

Fixed SD = 0.02%

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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)

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Calculations Where Formulae Do Not Exist

Calculation Method Stochastic Non-Stochastic Extrema 5 Days (Interpolation)

0.016% 0.016% 0.027%

40-Days (Extrapolation)

0.166% 0.166% 0.223%

Fixed RSD = 10% with minimum error of 0.02% (LOD)

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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)

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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 Rate from CI (RSD/LOD, 0.2%) 84.1% 1.42 X 10 -4 %/d 15.9% 3.83 X 10 -4 %/d 84.1% 1.43 X 10 -4 %/d 50% 2.34 X 10 -4 %/d Rate from Deg Extrema (RSD/LOD, 0.2%) 15.9% 4.05 X 10 -4 %/d

0e+00 2e-04 4e-04 rate

Monte Carlo Isoconversion Monte Carlo Arrhenius

6e-04 8e-04 0e+00 2e-04 4e-04 rate

Extrema Isoconversion Monte Carlo Arrhenius

6e-04 8e-04

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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

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Arrhenius Fitting Uncertainty

𝑘𝑖𝑠𝑜 𝑇 1 = 𝐴𝑒 − 𝐸 𝑎 𝑅 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 [email protected] 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

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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

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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 [email protected] 2014 28

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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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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• • • •

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) [email protected] 2014 31