1. 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

2. 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

3. 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

4. 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)

5. 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 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

6. 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.

7. 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

8. 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.

9. 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.

10. 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

11. 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 %

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

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

14. 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.

15. 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.

16. Thank You!