From the Garrett Two-Stage Model to the Bayesian View: A Historical Survey of Stability Modeling 1954 to 2017

Transcription

1 From the Garrett Two-Stage Model to the Bayesian View: A Historical Survey of Stability Modeling 1954 to 2017 William R. Porter, PhD Principal Scientist, PPPP-LLC@comcast.net Stan Altan, PhD Manufacturing, Toxicology and Applied Statistics, Janssen R&D SALTAN@its.jnj.com

2 Outline Early period : Accelerated stability assessment using Arrhenius models to estimate shelf life at room temperature. First Transitional period : Moving from accelerated studies to RT studies Real time room temperature period : 1987 FDA guidance and 2004 Q1E. Second transitional period 2010 current: Movement from real time studies back to accelerated studies with focus on degradants. Example real time and accelerated studies 2

3 Arrhenius, 1889 Reaction rate depends on temperature! Z physik Chemie 4: (1889) That is, k T k T T T 0 exp C kt 0 T0T1 1 exp C T T 1 3

4 Prehistory: Bigelow, 1921 Early attempt to relate time-tofail to storage temperature J Infec Dis 29: (1921) 4

5 Prehistory: Higuchi et al., 1950 Graphical analysis of data no mention of any statistical modeling. Note that t½ was used instead of rate k. 5

6 McBride & Villars, 1954 Anal Chem 26(5): (1954) First 2-stage regression model for Arrhenius kinetics Rates were calculated in Stage One Logarithms of rates were calculated for Arrhenius plot Weighted regression used for Stage Two 6

7 Garrett, 1954 (1) J Amer Pharm Assoc 43(9): (1954) The Ur paper for statistical analysis of accelerated stability assessment studies. McBride & Villars were ignored for years. 7

8 Garrett, 1954 (2) Details of the statistical modeling did not appear in Garrett s first paper. 8

9 Huyberechts et al., 1955 Don t publish in French if you want to be cited. Uses weighted 2 nd stage (McBride & Villars) 9

10 Garrett, 1956A J Amer Pharm Assoc 45(3): (1956) First paper by Garrett with details of how to fit Arrhenius model to data using two-stage unweighted regression. Tedious explanation. 10

11 Garrett, 1956B J Amer Pharm Assoc 47(7): (1956) The predictions work 1 st real time stability data plot with confidence band. 11

12 McLeod et al., 1958 H. A. McLeod, O. Pelletier and J. A. Campbell., "The Prediction of Expiration Dates for Multivitamin Preparations by Accelerated Storage Tests." Canadian Pharmaceutical Journal Scientific Section 91: (1958). The authors provide complete worked out numerical details with raw data and all statistical calculations using a 2-stage Garrett pseudo-zero-order kinetic model. Individual pseudo-zero-order rates and initial values (time zero potency) were obtained for each elevated storage temperature. The rates were converted to base 10 logarithms. The base 10 logarithms were fitted to an Arrhenius model of the form log 10 (rate) = intercept - slope reciprocal absolute temperature. NO WEIGHTING was used, and the authors cite Edward Garrett's paper. The fitted Arrhenius model was used to generate an estimate of the base 10 logarithm of the rate at some other temperature and the standard error of the base 10 logarithm of the rate was also calculated at that temperature. The rate at the new temperature was obtained by taking antilog of the base 10 logarithm of the projected rate at the new temperature. The standard error of the rate at the new temperature was calculated by taking the antilogs of the difference between the base 10 logarithm of the rate plus its standard error (in log units) and the base 10 logarithm of the rate minus its standard error (in log units) and then dividing the result by 2. The shelf life was calculated by taking the initial potency and the potency at the end-of-shelf-life as fixed constants, finding their difference, and dividing this difference by the estimate rate plus or minus its estimated standard error. The standard error of the shelf life was calculated by calculating the shelf life at rate plus standard error and at rate minus standard error, then finding the difference and dividing by two. Student's t distribution was used to generate a confidence interval estimate using T 2 d.f., where T is the number of different rates used in the second stage Arrhenius model. 12

13 Box & Lucas, 1959 Biometrika 46:77 90 (1959) Optimal design for Arrhenius model: Impractical Requires initial observation at 1/T 13

14 Toothill, 1961 Uses common intercept, isoconversion design to estimate rates in first stage. Adaptation of bioassay methods (Finney). Point & interval estimates for shelf life. 14

15 FDA cgmps, August,1969 Federal Register, 34(161): , August 22, 1969 No mention of statistical methods for evaluating shelf life assignment 15

16 Conference on Dating of Pharmaceuticals, October,1969 Comer JA. Processing of stability data for FDA submission. The Dating of Pharmaceuticals. Univ. Wisconsin pp , 1969 The computer system in use when this paper was written was the IBM The paper tape is read on the paper tape reader into the processing unit and onto magnetic tape. A cooperative effort of academic, governmental, and industrial organizations is needed to improve these systems until they become a common means of assisting the pharmaceutical scientist in the planning, interpretation, and submission of stability data. NOTE CONFIDENCE LIMITS USED TO DEFINE SHELF LIFE! 16

17 Carstensen & Su, 1971 Bull Parent Drug Assoc 25(6): ()1971) 1 st nonlinear regression fitting of Arrhenius model. Use pseudo-zero-order model if degradation is less than 10%. Use pseudo-first-order model if degradation exceeds 10%. 17

18 Carstensen & Nelson, Carstensen JT, Nelson E. Terminology regarding labeled and contained amounts in dosage forms. J Pharm Sci 65(2): (1976) Incorporated into 1987 FDA Stability Guidance 18

19 Davies & Budgett, 1980 (1) Distribution of Arrhenius parameter estimates: Prior distributions, posterior distributions. Lognormal distribution for Arrhenius frequency factor. Lognormal distribution for shelf life. 19

20 Davies & Budgett, 1980 (2) Prior distributions! 20

21 Davies & Budgett, 1980 (3) Prior distributions (cont.) Monte Carlo confirmation of posterior distributions 21

22 Davies & Budgett, 1980 (4) Posterior distributions 22

23 Statistics In the Pharmaceutical Industry, 1981 Editors: Charles Ralph Buncher, Jia-Yeong Tsay First edition covers ONLY accelerated stability studies. NO MENTION of real time studies. 23

24 King, Kung & Fung, 1984 J Pharm Sci 73(5): (1984) Direct point and interval estimation of shelf life at normal storage temperature from accelerated stability assessment. 24

25 FDA Stability Guideline, 1987 First guidance with detailed description of appropriate statistical methods for analysis and interpretation of real time stability data. Shelf life (expiration dating period) set using Carstensen & Nelson (1976). Fit pseudo-zero-order kinetic model to loss of potency as a function of time using linear regression. Construct one-sided lower confidence bound for the mean potency as a function of time. Shelf life is time when LCB reaches specification limit. Other approaches can be used but must be justified. 25

26 Statistics In the Pharmaceutical Industry, 2 nd Edition, 1991 Editors: Charles Ralph Buncher, Jia-Yeong Tsay Second edition covers BOTH accelerated stability studies AND real time studies. Only Frequentist methods described. 26

27 Shao & Chow, 1991 Release limits based on stability considerations; focus on potency loss. 27

28 Chow & Shao 1991 Confidence limits evaluated using constant normal prior by Monte Carlo simulation. 28

29 Shao & Chow, 1994 Confidence limits evaluated using constant normal prior by Monte Carlo simulation. 29

30 Su et al., 1994 Studied effect of varying kurtosis on estimates of ln(k) at various T conditions. ln(a) E a DID NOT look at prior distribution of raw data! 30

31 Shao & Chow, 2001 Three methods for estimating confidence limit approach to shelf life estimation using constant normal prior distribution evaluated using Monte Carlo simulation. 31

32 Statistics In the Pharmaceutical Industry, 3rd Edition, 2006 Editors: Charles Ralph Buncher, Jia-Yeong Tsay Third edition covers ONLY real time studies. No mention of accelerated studies. Only Frequentist methods described. 32

33 Chen, Zhong & Nie, 2008 Used uniform prior distributions for initial value and pseudo-zero-order degradation rates. 2 nd stage of 2-stage model. Did not consider estimation from raw data (1 st stage of 2-stage model). 33

34 Statistics in the Pharmaceutical Industry Edited by Charles Ralph Buncher, Jia-Yeong Tsay Second Edition 1991 RT and Accelerated First Edition 1981 Accelerated Studies Third Edition 2006 RT Studies 34

35 EXAMPLE Real Time Studies: Mixed Model Framework Mechanistic and empirical basis exists to forego pooling tests (constrained models by strength is recommended). Assume a fixed common package-temperaturespecific slope. Assume different batch-specific Intercepts. Main requirement is to estimate the parameters and account for incipient variation in such a way that control over the lot mean is assured. 35

36 Mixed Effects Model Model estimates: Process Mean at time of Manufacture Rate parameter Variance Structure Process (Batch-Batch) Analytical Variation Number of Parameters Number Form Fixed Effects Variance 1 n yijk ( 0 i ) Bj Tijk c +1 2 ijk n c +1 3/4 a 2 yijk ( 0 i ) B j i Tijk ijk Index: i=batch, j=condition, k=time. Model 4: Random Term in the Intercept, Model 5: Random Terms in Intercept and Slopes a if correlated Random Terms in Intercept and Slope Main objection small number of batches. 36

37 Case Study (Assay) 12 months stability study for the Assay of 3 Drug Product Batches at 2 storage conditions. Estimate Degradation Rate and Shelf Life for Specification limits, 90% to 110%. 37

38 Bayesian Approach Bayesian approach provides a flexible framework for incorporating scientific and expert judgment, exploiting past experience with similar products and processes and a more natural way to approach decision making. Posterior Distribution: X ~ f ( ), ~ prior( ) post( x) f f ( x f ( x ( x ) prior( ) ) prior( ) ) prior( ) d Posterior Distribution ~ Likelihood x Prior Distribution 38

39 Expiration Date Mixed Model 1 with random batch term on intercept, condition-specific slope: Y ijk ( ) X The expiration date is the solution to the equation: LSL ˆ ˆ T ED c Bayesian approach, consider: LSL T Z or T i SL j Z is independent of data and symmetric about 0: 2 Z ~ N(0, ). ijk ijk Var ˆ ( ˆ ˆ T SL ED ) ˆ 2 LSL Z 39

40 Prior Distributions and Simulation Parameters Prior Distributions: Expert opinion Process mean is likely between 99% and 101%: ~ N(100, 0.1) Lot to lot variance is likely between 0.1 and 0.5: 2 1 ~ (10, 2) Flat prior on the yearly degradation rates: 1, 2 ~ I(, ) Analytical variance is likely between 0.1 to 1.0: R/WinBUGS Simulation Parameters: 2 1 ~ (6, 3 chains, 500,000 iterations/chain, discard 1 st 100,000 simulated values in each chain. Retain every 100 th simulation draw, 27,000 simulated values for each parameter. 2) 40

41 Comparison of Parameter Estimates & Expiration Dates Frequentist Bayesian Parameters 95% Confidence Mean 95% Credible Estimate Interval (Median) Interval , , , , , , (0.20) 0.11, (0.26) 0.17, 0.42 Shelf Life Expiration Date Storage Condition 25C 30C 25C 30C Frequentist Bayesian 94 ( 89 ) 64 ( 62 ) Bayesian estimate of shelf life is based on the posterior mean (median) and the expiration date corresponds to the 5%-tile value. Specification limits, 90% to 110%. 41

42 Shelf Life Distribution Bayesian Approach 42

43 EXAMPLE Accelerated Stability Testing Product is subjected to stress conditions. Temperature and relative humidity are the most common stress factors. Purpose is to predict long term stability and shelf life. Arrhenius equation captures the kinetic relationship between rates and temperature. The usual fixed and mixed models used in real time studies ignore any relationship between rate and temperature. 43

44 Arrhenius Equation with Relative Humidity Term A humidity term with coefficient B is introduced to account for the effect of relative humidity on rate parameter. degradation rate ln(k T,H ) = ln A Pre-exponential factor activation energy humidity sensitivity factor E a R T + B h gas constant (8.314 x 10-3 kj mol -1 K -1 ) 44

45 Nonlinear Parameterization (Extended King-Kung-Fung Model) k T,H = Ae E a R T +B H k T, H A k Let T =298 o K (25 o C) H = ,60 e Ea 1 1 B( H 60) R 298 T A = k 298,60 e E a 298 R B 60 Assuming zero order kinetics, total degradation is: D t = D 0 + k T,H t Q D 0 k298,60 t SL D t = D 0 + Q D 0 t SL t e Ea R T +B H 60 +ε Estimate Shelf Life at 25C/60%RH and its uncertainty w.r.t spec = Q. Parameter estimates are calculated based on the Arrhenius relationship conditional on an assumed zero order kinetic model. 45

46 Linearized Expanded Arrhenius Model Two-stage approach (Garrett/McBride & Villars, 1954), modified to include relative humidity. Assume a zero order kinetic model. Stage 1 : fit a pseudo zero order kinetic model to the concentration measurements: D T t D 0 Stage 2 : Model the rate estimates according to Arrhenius relationship: ln kt, h ln A t Expressed as linear regression problem: ln, E R k a T T 1 T k T h 2 B h h 46

47 Case Study of Degradant under Accelerated Conditions Data Table day Temp RH Deg

48 Bayesian Considerations for Model Specification Priors: D 0 ~ Uniform(0, 4) lne A ~ Uniform(2, 7) B ~ Uniform(0, 0.20) lnt SL ~ Uniform(1, 8) σ ε 2 ~ Inverse-Gamma(1.41, 0.055) JAGS specifications: Burn-in of 100,000 iterations followed by 3,000,0000 iterations. Thinning rate of chains. Total of 30,000 simulated values retained for each parameter. Randomly generated initial values for each parameter from corresponding prior distribution. 48

49 JAGS model specification 49

50 Accelerated Degradant Study Results Parameter (se) King, Kung, Fung Garrett Two- Stage Bayesian Posterior Estimates Mean Median 5% D initial 0.27 (0.40) 0.36(0.50) E a (kj Mol -1 ) (3.9) (4.4) B (RH) 0.06(0.002) 0.06(0.002) t SL, 25C/60%RH (Exp 107) (Day) (Exp 72) (Exp 56) t SL, 30C/75%RH (Exp 23) (Day) (Exp 17) (Exp 14) MSE

51 Comparison of Approaches Linearized Arrhenius Model (Garrett): Simple and does not require specialized software. Requires GLS approach for appropriate weighting. Nonlinear Model (King-Kung-Fung): Computationally intensive. Computing convergence issues. 51

52 Summary Current regulatory guidelines for assessing stability and shelf life claims are being challenged. Poolability paradigm is a concern. A Mixed Effects model is a natural representation of a manufacturing process. Bayesian framework can incorporate process engineering and scientific judgment. Addresses poolability concerns. Accelerated study models are being reassessed and updated through linear and non-linear models. King, Kung and Fung model has been extended to include a humidity term. Technology is evolving rapidly which will require greater statistical collaborations. 52

53 Future 1. Degradation rates are parameters, not data. 2. Consider effect of informative prior distribution on: a. Measurement and within-batch sampling errors: i. Repeatability (within test day errors). ii. iii. Intermediate precision (over testing interval errors). Reproducibility (between lab errors). b. Batch variation (manufacturing errors). c. Environmental errors (temperature, relative humidity). 3. Most drug products fail due to excess degradation products, not loss of potency. 4. A drug product is not stable unless some other analyst working in some other lab testing some other sample from some other batch stored in some other environment says so. 53

54 References (1) Regulatory Guidance 1. International Conference on Harmonization ICH Harmonised Tripartite Guideline: ICH Q1A(R2) Stability Testing of New Drug Substances and Products. Revision 2 2. International Conference on Harmonization ICH Harmonised Tripartite Guideline: ICH Q1D Bracketing and Matrixing Designs for Stability Testing of New Drug Substances and Products. 3. International Conference on Harmonization ICH Harmonised Tripartite Guideline: ICH Q1E Evaluation of Stability Data if New Drug Substances and Products. Revision 2 4. International Conference on Harmonization ICH Harmonised Tripartite Guideline: ICH Q6B Specifications: Test Procedures and acceptance Criteria for Biotechnological /Biological Products. 5. International Conference on Harmonization ICH Harmonised Tripartite Guideline: ICH Q8(R2) Pharmaceutical Development. 6. International Conference on Harmonization ICH Harmonised Tripartite Guideline: ICH Q9 Quality Risk Management. Approaches to Pooling 1. Chen WJ, Tsong Y. Significance levels for stability pooling test: a simulation study. Journal of Biopharmaceutical Statistics 2003; 13: Chow SC, Shao J. Test for batch-to-batch variation in stability analysis. Statistics in Medicine 1989; 8: Liu W, Jamshidian M, Zhang Y, Bretz F, Han XL. Pooling batches in drug stability study by using constant-width simultaneous confidence bands. Statistics in Medicine 2007; 26: Ruberg SJ, Stegeman JW. Pooling data for stability studies: testing the equality of batch degradation slopes. Biometrics 1991; 47: Ruberg SJ, Hsu JC. Multiple comparison procedures for pooling batches in stability studies. Technometrics 1992; 34: Tsong Y, Chen WJ, Lin TY, Chen CW. Shelf life determination based on equivalence assessment. Journal of Biopharmaceutical Statistics 2003; 13:

55 References (2) Mixed and random Models 1. Chow SC, Shao J. Estimating drug shelf-life with random batches. Biometrics 1991; 47: Chen JJ, Hwang JS, Tsong Y. Estimation of the shelf-life of drug with mixed effects models. Journal of Biopharmaceutical Statistics 1995; 5: Shao J, Chow SC. Statistical inference in stability analysis. Biometrics 1994; 50: Michelle Quinlan, W. Stroup, J.D. Christopher, J. Schwenke. 2013a. On the Distribution of Batch Shelf Lives. Journal of Biopharmaceutical Statistics, 23:4, Michelle Quinlan, W. Stroup, J. Schwenke, J.D. Christopher. 2013b. Evaluating the Performance of the ICH Guidelines for Shelf Life Estimation, Journal of Biopharmaceutical Statistics, 23:4, Robert Capen, J.D. Christopher, P. Forenzo, C. Ireland, O. Liu, S. Lyapustina, J. O Neill, N. Patterson, M. Quinlan, D. Sandell, J. Schwenke, W. Stroup, T. Tougas On the Shelf Life of Pharmaceutical Products. AAPS PharmSciTech, 13: Norwood, T.E.: Statistical analysis of pharmaceutical stability data, Drug Development and Industrial Pharmacy, 12(4), (1986) 8. Kiermeier, A.; Jarett, R.G., Verbyla, A.P.: A new approach to estimating shelf-life, Pharmaceutical Statistics, 3, 3-11 (2004) Bayesian Modeling 1. Chen, J, Zhong,J, Nie,L. Bayesian hierarchical modeling of drug stability data. Statist. Med. 2008; 27:

56 Acknowledgements Thanks to (Janssen): Madelyn Drevets Areti Manola Jyh-Ming Shoung 56

57 Questions? 57