Comparison of Shelf Life Estimates Generated by ASAPprimeTM with the King-Kung-Fung Approach Juan Chen1, Sabine Thielges2, William R. Porter3, Jyh-Ming Shoung1, Stan Altan1 1Nonclinical Statistics and Computing, Janssen R&D 2BE Analytical Sciences and COES, Janssen R&D 3Peak Process Performance Partners LLC

Content Arrhenius Equation and Extensions to include Humidity Effects Extended Arrhenius Equation Extended King-Kung-Fung (KKF) Model ASAP Approach Zero order, 2-temperature Example Case Study using Pseudo-data DoE of Pseudo-data Comparison of Outputs from ASAP and KKF Model Conclusion

Arrhenius Equation Named for Svante Arrhenius (1903 Nobel Laureate in Chemistry) who established a relationship between temperature and the rates of chemical reaction: Where kT = Degradation Rate A = Non-thermal Constant Ea = Activation Energy R = Universal Gas Constant (1.987 cal/mol) T = Absolute Temperature

Arrhenius Equation with Humidity Term A humidity term with coefficient B is introduced to account for the effect of relative humidity on rate parameter. activation energy humidity sensitivity factor degradation rate 𝑙𝑛𝐾=𝑙𝑛𝐴− 𝐸 𝑎 𝑅×𝑇 +𝐵×𝐻 Pre-exponential factor gas constant (1.987cal/mol)

Extended King-Kung-Fung Model King-Kung-Fung (KKF) model is widely used for analyzing accelerated stability data Let T =298oK (25oC) H = 60 𝑘 𝑇,𝐻 =𝐴 𝑒 − 𝐸 𝑎 𝑅×𝑇 +𝐵×𝐻 𝐴= 𝑘 298,60 𝑒 𝐸 𝑎 298×𝑅 −𝐵×60 𝑨 𝐴𝑠𝑠𝑢𝑚𝑖𝑛𝑔 𝑧𝑒𝑟𝑜 𝑜𝑟𝑑𝑒𝑟 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝐷 𝑡 = 𝐷 0 + 𝑘 𝑇,𝐻 ×𝑡 𝐷 𝑡 = 𝐷 0 − 𝐷 0 −𝑄 𝑡 𝑆𝐿 ×𝑡 ×𝑒 𝐸 𝑎 𝑅 × 1 298 − 1 𝑇 +𝐵× 𝐻−60 +𝜀 Directly estimate Shelf Life (SL) at 25C/60%RH and its uncertainty Parameter estimates are calculated based on the Arrhenius relationship conditional on an assumed zero order kinetic Can be further extended to a nonlinear mixed model context and Bayesian calculation

Introduction to ASAPprimeTM The ASAPprimeTM computerized system is a computer program that analyzes data from accelerated stability studies using a 2-week protocol accommodating both temperature and humidity effects through a extended Arrhenius model. The program makes a number of claims: Reliable estimates for temperature and relative humidity effects on degradation rates, Accurate and precise shelf-life estimation, Enable rational control strategies to assure product stability. These claims require careful statistical considerations of the modeling strategies proposed by the developers. Our objective is to evaluate the first two claims in relation to widely accepted statistical approaches and considerations.

Illustration of ASAPprimeTM Approach Shelf Life (SL) Estimation using Zero-order, 2-Temperature Example Data Entry Condition Days Impurity SD 50C 0.020 14 0.14 0.015 60C 0.41 0.124 SL (spec. = 0.2) Mean SD 20.6 2.6 6.9 1.3 Data SD Hierarchy: 1. Calculate from replicate data , if >LOD 2. User-defined SD (fixed) or RSD, if >LOD 3. Default 10%RSD, if >LOD 4. LOD Calculate SD of SL at accelerated conditions = Mean SL – Extrema SL Extrema SL Mean SL Method of SD calculation is not consistent with the standard definition of a SD No a priori variance structure proposed for analytical variability in terms of a statistical model  The uncertainty in the SL estimates cannot be understood in relation to statistical principles; empirical comparisons only

Simulation was drawn from an undocumented distribution. Error propagation through a MC simulation Condition SL Mean (SD) MC simulated SL 50C 20.6 (2.6) days 24, 22, 21, 19, 18, 17, 23 60C 6.9 (1.3) days 6, 8, 5, 7, 7, 9, 6.5 lnK (K = spec / SL) -4.8, -4.7, -4.7, -4.6, -4.5, -4.4, -4.7 -3.4, -3.7, -3.2, -3.6, -3.6, -3.8, -3.5 Pairs of lnK at 50C and 60C form 49 regression lines across 1/T with slopes (=Ea/R) and intercepts (lnA) Calculate lnK at 25C for each simulation: lnK = lnA – Ea/(RT) Simulation lnA Ea (kcal/mol) 1 18.9 15.2 2 12.6 11.1 ... … 48 6.4 7.0 49 16.9 13.9 Mean 14.1 12.0 SD 4.7 3.1 lnK at 25C -6.78 -6.16 … -5.36 -6.56 0.52 Simulation was drawn from an undocumented distribution. Arrhenius parameters (lnA and Ea) are determined from simulated degradation rates  Statistical properties of the estimated lnA and Ea are not unknown.

Model fitting cannot be confirmed by standard statistical procedures. ASAP Probability statement about lnK and SL at 25C SL at 50C and 60C previously simulated from an unknown distribution  Cannot verify lognormal distribution at 25C Model fitting cannot be confirmed by standard statistical procedures.

Case Study using Pseudo-data 4 x 4 factorial design of temperature and humidity 9 sets of data simulated from combinations of D0, Ea, B values each at L, M, H, 73-day sampling design assuming a zero-order model lnA back-calculated to obtain SL at 2 years at 25C/60%RH Normally distributed random errors with mean 0 and SD 0.1 were added as analytical variability H M L D0 0.3 0.15 Ea (kcal/mol) 38.2 29.9 19.1 B 0.09 0.035 0.006 Temp (0C) RH (%) HHH HHL HLH HLL LHH LHL LLH LLL MMM 40 11.1 x 31.6 55.1 74.6 48 11.0   30.8 53.3 56 10.9 29.8 52.0 74.4 65 10.8 28.6 51.5 74.2

ASAP different user specified SD affect probability statement about SL KKF model carried out in SAS Proc nlin (Initial values: SL = 0.27 year, Ea = 10, B = 0.001, D0 = 0) KKF estimated residual errors range from 0.08 to 0.12, whereas the true value is 0.1 ASAP different user specified SD affect probability statement about SL Data Sets  Parm  True Value KKF ASAP True Value Est. se se (SD=0.1) se (SD= 10%RSD) HHH lnA 53.1 54.3   43.0 4.68 2.37 LHL 58.3 56.9 56.6 1.45 2.61 Ea 38.2 39.1 0.65 31.2 3.18 1.60 37.4 0.46 37.2 0.91 1.64 B 0.09 0.001 0.08 0.007 0.003 0.006 0.000 SL 2 2.2 0.13 1.1 57.2 % 63.3 % 1.86 0.075 1.83 100 % HHL 58.1 57.7 57.2 1.55 2.85 LLH 21.0 20.7 1.21 1.97 38.0 0.45 37.5 0.98 1.80 19.1 19.2 0.23 18.8 0.81 1.30 0.005 0.087 0.002 2.0 99.98 % 0.074 1.84 99.78 % HLH 20.8 21.4 8.0 3.23 LLL 26.0 25.8 26.2 1.24 0.92 19.6 9.3 2.09 1.01 19.0 0.27 0.83 0.60 0.0010 0.06 0.004 0.0003 0.5 0 % 2.03 HLL 24.1 1.22 1.07 MMM 42.3 42.6 39.9 1.37 1.76 0.25 17.9 0.82 0.70 29.9 30.0 0.28 28.1 1.19 0.035 0.032 1.6 99.99 % 2.05 1.68 LHH 53.2 52.5 50.5 3.71 1.57 Note: ASAP reports probability of SL greater than 1 year rather than standard error. 37.8 0.56 36.3 2.53 1.04 0.089 0.086 1.90 0.10 1.66 97.32 %

ASAP generally underestimated the shelf life. KKF estimated SL were generally closer to true values than ASAP. ASAP generally underestimated lnA.

ASAP generally underestimated Ea. Standard errors of Ea are affected by user specified SD, and are generally larger than KKF estimates.

ASAP underestimated B for data HHH and HLH, and overestimated B for LLL. Standard errors of B are affected by user specified SD, and are generally larger than KKF estimates.

Conclusion Uncertainty measure for ASAP estimated shelf life is derived from an “error propagation” calculation using either replicate error or a user defined quantity. Statistical rationale for uncertainty limits is not clear. Does not lead to a statistical confidence statement. ASAP simulation of SL to predict room temperature SL: Underlying distribution of SL at accelerated conditions is not documented. The precision of Arrhenius model parameter estimates is influenced by user specified SD and cannot be validated statistically. Overall model fitting is unclear and lacks documentation. Manufacturing variability cannot be accommodated.

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