kenneth c. waterman, ph.d. jon swanson, ph.d. freethink technologies, inc

[email protected] 2014 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


•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

Outline


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


0 7 14 21 280

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (days)

%De

grad

ant

Complex Kinetics—Example Drug → primary degradant → secondary degradant


0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time (days)

%De

grad

ant

50°C

60°C

Accelerated Aging Complex Kinetics

70°C

Fixed time accelerated stability


0.0029 0.00295 0.003 0.00305 0.0031 0.00315 0.0032 0.00325 0.0033-4

-3.8

-3.6

-3.4

-3.2

-3

-2.8

1/T

ln k 50°C

60°C

30°C?

70°C More unstable

Accelerated Aging Complex Kinetics

• Appears very non-Arrhenius• Impossible to predict shelf-life

from high T results

[email protected] 2014

0.00280 0.00290 0.00300 0.00310 0.00320 0.00330-7

-6

-5

-4

-3

-2

-1

0

1/T

ln k

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

7

30C

70C60C 50C

80C

CP-456,773/60%RH

Real time data

Accelerated Aging Complex Kinetics: Real Example


0 1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

0.3

Time (days)

%De

grad

ant

50°C60°C

70°C

Accelerated Aging—Isoconversion Approach

0.2% specification limit

Isoconversion: %degradant fixed at specification limit, time adjusted


0.0029 0.00295 0.003 0.00305 0.0031 0.00315 0.0032 0.00325 0.0033

-7

-6

-5

-4

-3

-2

-1

1/T

ln k

Using 0.5% isoconversion

50°C60°C

30°C

70°CUsing 0.2% isoconversion

Accelerated Aging—Isoconversion Approach Complex Kinetics


0.00280 0.00290 0.00300 0.00310 0.00320 0.00330 0.00340-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

1/T (K)

ln(k

)

30C

70C

60C50C

80C

CP-456,773/60%RH

Real time data

ASAPprime Shelf Life 1.2 yrsExperimental Shelf Life 1.2 yrs

Accelerated Aging—Isoconversion Approach Complex Kinetics—Real Example


More Detailed Example

• 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)

A B Ck1 k2


Method Shelf-life (yrs) @25°C

Spec. 0.2%

Spec. 0.5%

Exact 1.43 4.454 linear rate constants @ each T 0.62 1.561 linear rate constant through 4 points @ each T 0.29 0.71Single point at isoconversion @ each T 1.43 4.45Linear fitting of 4 points @ each T to determine intersection with specification 12.35 1.40Determining intersection with specification using 2 points closest to specification @ each T (or extrapolating from last 2 points, when necessary)

1.36 3.19

Primary Degradant (“B”) Formation


0 10 20 30 40 50 60 70

-0.1

2.77555756156289E-17

0.1

0.2

0.3

0.4

0.5

0.6

Time (days)

%De

grad

ant B

Example @40°C

Note R2 for line = 0.998


Method Shelf-life (yrs) @25°C

Spec. 0.2%

Spec. 0.5%

Exact 2.02 4.014 linear rate constants @ each T 16.64 41.611 linear rate constant through 4 points @ each T 3.29 8.21Single point at isoconversion @ each T 2.02 4.01Linear fitting of 4 points @ each T to determine intersection with specification 2.75 7.56Determining intersection with specification using 2 points closest to specification @ each T (or extrapolating from last 2 points, when necessary)

2.06 4.78

Secondary Degradant (“C”) Formation


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


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)


Isoconversion Uncertainty Methods• Confidence Interval:• Regression Interval:• 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


0 5 10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (days)

%De

grad

ant

Test Calculations: Model System


Calculation Method 5 Days (Interpolation)

40-Days (Extrapolation)

Regression Interval 0.023% 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%

Calculations Where Formulae Exist


• 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)

Isoconversion Uncertainty


Calculation Method

5 Days (Interpolation)

40-Days (Extrapolation)

Stochastic 0.016% 0.166%Non-Stochastic 0.016% 0.166%

Extrema 0.027% 0.223%

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

Calculations Where Formulae Do Not Exist


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)


Monte Carlo IsoconversionMonte Carlo Arrhenius

25°C Projected Rate Distributions60, 70, 80°C measurements @10 days; RSD=10%, LOD=0.02%; 25 kcal/mol

Extrema IsoconversionMonte Carlo Arrhenius

84.1%1.42 X 10-4%/d

50%2.34 X 10-4%/d

50%2.38 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

Rate from CI (RSD/LOD, 0.2%)

rate

frequ

ency

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

020

040

060

080

0

Rate from Deg Extrema (RSD/LOD, 0.2%)

ratefre

quen

cy

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

020

040

060

080

0



• 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



• 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


Extrapolated Shelf-life (years) at 25°C84.1% 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 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

Comparison of Arrhenius Fitting Methods



• 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


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


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


•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


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)

kenneth c. waterman, ph.d. jon swanson, ph.d. freethink technologies, inc

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