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
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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

RT
Where kT = Degradation Rate
A = Non-thermal Constant
Ea = Activation Energy
R = Universal Gas Constant (1.987 cal/mol)
T = Absolute Temperature
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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
𝐸𝑎
𝑙𝑛𝐾 = 𝑙𝑛𝐴 −
+𝐵×𝐻
𝑅×𝑇
Pre-exponential factor
gas constant (1.987cal/mol)
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Extended King-Kung-Fung Model
King-Kung-Fung (KKF) model is widely used for analyzing accelerated stability
data
𝑘 𝑇,𝐻 =
𝐸
− 𝑎 +𝐵×𝐻
𝐴𝑒 𝑅×𝑇
𝑨
Let T =298oK (25oC)
H = 60
kT ,H  k 298 ,60 e
𝐴=
𝐸𝑎
𝑘298,60 𝑒 298×𝑅−𝐵×60
Ea  1 1 
   B ( H  60 )

R  298 T 
𝐴𝑠𝑠𝑢𝑚𝑖𝑛𝑔 𝑧𝑒𝑟𝑜 𝑜𝑟𝑑𝑒𝑟 𝑘𝑖𝑛𝑒𝑡𝑖𝑐
𝐷𝑡 = 𝐷0 + 𝑘 𝑇,𝐻 × 𝑡
𝐸𝑎
𝐷0 − 𝑄
×
𝐷𝑡 = 𝐷0 −
×𝑡×𝑒𝑅
𝑡𝑆𝐿
1 1
298−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
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.
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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.
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Thank You!