Report

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 Janssen Research & Development 2 Arrhenius Equation Named for Svante Arrhenius (1903 Nobel Laureate in Chemistry) who established a relationship between temperature and the rates of chemical reaction: kT Ae Ea RT Where kT = Degradation Rate A = Non-thermal Constant Ea = Activation Energy R = Universal Gas Constant (1.987 cal/mol) T = Absolute Temperature Janssen Research & Development 3 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 = − +× × Pre-exponential factor gas constant (1.987cal/mol) Janssen Research & Development 4 Extended King-Kung-Fung Model King-Kung-Fung (KKF) model is widely used for analyzing accelerated stability data , = − +× × Let T =298oK (25oC) H = 60 kT ,H k 298 ,60 e = 298,60 298×−×60 Ea 1 1 B ( H 60 ) R 298 T = 0 + , × 0 − × = 0 − ×× 1 1 298− +× −60 + • 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 5 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: 1. Reliable estimates for temperature and relative humidity effects on degradation rates, 2. Accurate and precise shelf-life estimation, 3. 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. Janssen Research & Development 6 Illustration of ASAPprimeTM Approach Shelf Life (SL) Estimation using Zero-order, 2-Temperature Example Data Entry Condition Days Impurity 0 0 50C 14 0.14 0 0 60C 14 0.41 SD 0.020 0.015 0.020 0.124 SL (spec. = 0.2) Mean SD 20.6 2.6 6.9 1.3 Calculate SD of SL at accelerated conditions = Mean SL – Extrema SL 50C 0.20 ±SD: 0.015 % degradant 0.15 0.10 ±SD: 0.020 0.05 Mean SL Extrema SL 0.02 0.00 0 -0.02 0 5 10 15 18 20 20.6 25 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 • 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 Time (days) 7 Error propagation through a MC simulation Condition SL Mean (SD) MC simulated SL lnK (K = spec / SL) 50C 20.6 (2.6) days 24, 22, 21, 19, 18, 17, 23 -4.8, -4.7, -4.7, -4.6, -4.5, -4.4, -4.7 60C 6.9 (1.3) days 6, 8, 5, 7, 7, 9, 6.5 -3.4, -3.7, -3.2, -3.6, -3.6, -3.8, -3.5 -3.2 -3.4 Pairs of lnK at 50C and 60C form 49 regression lines across 1/T with slopes (=Ea/R) and intercepts (lnA) -3.6 ln K -3.8 -4.0 -4.2 50C Shelf Life: 20.6±2.6 days 60C Shelf Life: 6.9±1.3 days -4.4 -4.6 -4.8 0.00300 0.00302 0.00304 0.00306 1/T (kelvin) 0.00308 0.00310 Simulation lnA Ea (kcal/mol) lnK at 25C 1 2 ... 48 49 Mean SD 18.9 12.6 … 6.4 16.9 14.1 4.7 15.2 11.1 … 7.0 13.9 12.0 3.1 -6.78 -6.16 … -5.36 -6.56 -6.16 0.52 Calculate lnK at 25C for each simulation: lnK = lnA – Ea/(RT) • 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. ASAP Probability statement about lnK and SL at 25C Normal Distribution of lnK 0.8 0.7 Mean = -6.16 SD = 0.52 +1SD -1SD 0.5 0.4 Distribution of Shelf Life at 25C 0.8 0.3 0.2 0.7 +2SD -2SD 0.6 0.1 0.0 -7.5 -7.0 -6.5 -6.0 lnk -5.5 -5.0 probability probability 0.6 Mean 50% 0.5 0.4 84% 0.3 0.2 0.1 16% 98% 2.3% 0.0 50 100 150 200 Shelf Life (days) 250 300 350 • 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. 9 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 Temp (0C) 40 48 56 65 RH (%) HHH HHL HLH HLL 11.1 31.6 55.1 74.6 11.0 30.8 53.3 74.6 10.9 29.8 52.0 74.4 10.8 28.6 51.5 74.2 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x H LHL x x x x x x x x x x x x x x x x L D0 0.3 0.15 0 Ea 38.2 29.9 19.1 (kcal/mol) B 0.09 0.035 0.006 LHH x x M LLH x x x x x x x x x x LLL MMM x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 10 • • • Data Sets HHH HHL HLH HLL LHH 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 True Parm Value KKF Est. ASAP se Est. se se (SD= (SD=0.1) 10%RSD) lnA Ea B SL lnA Ea B SL lnA Ea B SL 53.1 54.3 38.2 39.1 0.65 0.09 0.09 0.001 2 2.2 0.13 58.1 57.7 38.2 38.0 0.45 0.006 0.005 0.000 2 2.0 0.08 20.8 21.4 19.1 19.6 0.23 0.09 0.09 0.0010 2 2.2 0.08 43.0 31.2 0.08 1.1 57.2 37.5 0.005 2.0 8.0 9.3 0.06 0.5 4.68 3.18 0.007 2.37 1.60 0.003 57.2 % 63.3 % 1.55 0.98 0.001 2.85 1.80 0.001 100 % 99.98 % 3.23 2.09 0.004 1.60 1.01 0.002 0% 0% lnA 25.8 26.0 24.1 1.22 Ea B SL lnA Ea B SL 19.1 19.2 0.25 0.006 0.006 0.0003 2 2.0 0.08 53.2 52.5 38.2 37.8 0.56 0.09 0.089 0.001 2 1.90 0.10 17.9 0.006 1.6 50.5 36.3 0.086 1.66 Data True Sets Value KKF Est. ASAP se Est. se se (SD= (SD=0.1) 10%RSD) 58.3 56.9 56.6 38.2 37.4 0.46 37.2 0.006 0.006 0.000 0.006 2 1.86 0.075 1.83 21.0 21.0 20.7 19.1 19.2 0.23 18.8 0.09 0.09 0.001 0.087 2 2.0 0.074 1.84 26.0 25.8 26.2 19.1 19.0 0.27 19.2 0.006 0.006 0.0003 0.06 2 2.0 0.09 2.03 1.45 0.91 0.001 2.61 1.64 0.001 100 % 100 % 1.21 0.81 0.002 1.97 1.30 0.006 100 % 99.78 % 1.24 0.83 0.001 0.92 0.60 0.001 100 % 100 % 1.07 42.3 39.9 1.37 1.76 0.82 0.001 0.70 0.001 1.19 0.002 100 % 28.1 0.032 1.68 0.83 0.001 99.99 % MMM 29.9 30.0 0.035 0.035 2 2.05 100 % 100 % 3.71 2.53 0.005 1.57 1.04 0.002 97.32 % 100 % LHL LLH LLL 42.6 0.28 0 0.06 Note: ASAP reports probability of SL greater than 1 year rather than standard error. 11 2.25 2.00 • ASAP generally underestimated the shelf life. • KKF estimated SL were generally closer to true values than ASAP. 1.50 1.25 1.00 ASAP Estimate KKF Estimate True Value 0.75 0.50 HHH HHL HLH HLL LHH LHL Data Sets LLH LLL MMM 60 50 40 ASAP generally underestimated lnA. lnA Shelf Life (years) 1.75 30 20 ASAP Estimate KKF Estimate True Value 10 HHH HHL HLH HLL LHH LHL Data Sets LLH LLL MMM 40 35 ASAP generally underestimated Ea. 25 20 15 ASAP Estimate KKF Estimate True Value 10 HHH HHL HLH HLL LHH LHL Data Sets LLH LLL MMM 3.5 ASAP SD=0.1 ASAP SD=10%RSD KKF 3.0 Standard errors of Ea are affected by user specified SD, and are generally larger than KKF estimates. Standard Error of Ea Ea 30 2.5 2.0 1.5 1.0 0.5 0.0 HHH HHL HLH HLL LHH LHL Data Sets LLH LLL MMM13 0.09 ASAP Estimate KKF Estimate True Value 0.08 0.07 ASAP underestimated B for data HHH and HLH, and overestimated B for LLL. 0.05 0.04 0.03 0.02 0.01 0.00 HHH HHL HLH HLL LHH LHL Data Sets LLH LLL MMM 0.007 ASAP SD=0.1 ASAP SD=10%RSD KKF 0.006 Standard errors of B are affected by user specified SD, and are generally larger than KKF estimates. Standard Error of B B 0.06 0.005 0.004 0.003 0.002 0.001 0.000 HHH HHL HLH HLL LHH LHL Data Sets LLH LLL MMM 14 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. Janssen Research & Development 15 Thank You!