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Experiences and
Challenges with
Isoconversional Kinetics
Stability Modeling of
Packaged Amorphous Solid
Dispersions
Russell Hertzler, Ph.D.
Principal Research Scientist, Analytical R&D
AbbVie
John C. Strong, Ph.D.
Associate Research Fellow, Formulation Sciences
AbbVie
Contents
• Accelerated stability studies
• Specifics of applying accelerated stability to amorphous solid
dispersions
• The Tg problem and designing the experimental protocol
• Isoconversion kinetics
• Packaging model
• Results and implications
• Conclusions & questions
Experiences & Challenges with ASDs | 2014 MBSW | May 21, 2014 | Copyright © 2014 AbbVie
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Accelerated Stability
Study Design and Solid
State Kinetics
Chemical Kinetics in Solids
Stability studies at accelerated condition are desired to determine the
shelf-life of pharmaceutical products without having to wait for the
entire real-time degradation to occur.
However, chemical kinetic models used to describe solid state systems
(heterogeneous samples) are complex.
Typical Examples are
o diffusion models
o phase boundary models
o nucleation and growth models
Determination of the exact model can be difficult in heterogeneous
systems due to the irreproducibility of rate data.
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Isoconversion (time-to-failure) approach
Then we can ignore “how” the
impurity got to the level of
failure for determination of
kinetic parameters to be used
in further modeling.
0.25%
Specification Limit
0.20%
% Degradant, w/w
The object is to design an
experiment such that at each
temperature & humidity
condition, we know how long it
takes to achieve some fractional
degradation (i.e. 0.2%), typically
the specification limit.
0.15%
0.10%
Detected,
Not Quantitated
0.05%
Not detected
0.00%
0
0.5
1
1.5
2
Time, years
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Moisture-Adjusted Arrhenius Equation*
If no physical changes in the dosage form occur, then the Arrhenius
model for chemical reaction rates should apply.
 In this case, we can establish a range of temperature (T) and
humidity (h) values within which the extended Arrhenius model
correctly predicts shelf-life:
𝑘 𝑇,𝐻 = 𝐴 𝑒𝑥𝑝
−𝐸
+ 𝐵ℎ
𝑅𝑇
where h is the equilibrium relative humidity
and B is a constant
If physical changes occur, even if they are reversible, reaction rates
will not follow Arrhenius model.
*Genton & Kesselring, Effect of temperature and relative humidity on nitrazepam stability in solid state. J Pharm Sci 66: 676–
680 (1977)
*Waterman KC, Carella AJ, Gumkowski MJ, Lukulay P, MacDonald BC, Roy MC, Shamblin SL. Improved protocol and data
analysis for accelerated shelf-life estimation. Pharm Res 24(4):780–790 (2007).
Experiences & Challenges with ASDs | 2014 MBSW | May 21, 2014 | Copyright © 2014 AbbVie
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Characteristics of Amorphous Solid Dispersion (ASD)
Formulations
• Dispersion of an API in an inert carrier in the solid state prepared
by solvent evaporation, melting or solvent-melting methods.
• Used to increase the bioavailability of poorly soluble drugs by
improving their rate and extent of dissolution.
• Displays a glass transition temperature (Tg), below which the
ASD has the appearance of a solid and are considered as a one
phase system in which all molecules of the API are intimately
mixed with the carrier molecules.
• The “Tg minus 50” rule-of-thumb states that the molecular
mobility of an ASD becomes negligible 50°C below Tg.
• As we approach the Tg, chemical instability can become
significant. Physical stability may also become a problem.
• The Tg is also a function of the water content of the system.
Experiences & Challenges with ASDs | 2014 MBSW | May 21, 2014 | Copyright © 2014 AbbVie
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Example of a Glass Transition Temperature (Tg) diagram
Tg vs. %RH profile
80
25
20
15
40
10
% Water
Temp (C)
60
20
5
Moisture Uptake
Glassy State
0
0
0
20
40
60
80
% RH
Experiences & Challenges with ASDs | 2014 MBSW | May 21, 2014 | Copyright © 2014 AbbVie
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Stability Points in Relation to Glass Transition Temperature (Tg)
Stability Study Designs for ASD formulations:
o Stability study protocol needs to be constructed based on Tg–RH relationship
o ASDs typically will require more time to reach isoconversion levels of degradant than
“conventional formulations”
Graphical Representation of
Stability Study Design
Example Stability Protocol,
based on Tg–RH relationship
80
% Relative
Humidity
Days
st
(1 sample)
Days
nd
(2 sample)
40
60
14
28
40
43
21
42
50
38
14
21
50
60
14
28
60
20
28
56
60
46
7
14
70
10
14
28
Fast, non predictive
Degradation Rates
60
Temp (C)
Temperature
40
20
Slow Degradation
Rates
0
0
20
40
60
80
% RH
Experiences & Challenges with ASDs | 2014 MBSW | May 21, 2014 | Copyright © 2014 AbbVie
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ASD Protocols vs. Conventional ASAP Stability Protocol Design
Space
Conventional ASAP study design space is too broad for some ASD formulations.
80
Non representative
Degradation
Temp (C)
60
40
Slow
Degradation
Rates
20
0
0
20
40
60
80
% RH
Conventional ASAP study design condition
Formulation Specific study design condition
Formulation Specific Tg
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Packaging Model
Applying Chemical Degradation to Packaging
• Input Data
A. Degradant kinetics
1) isoconversion experiments
2) regression of degradation rate constants k from experimental data
3) regression of kinetic parameters from rate constants k
B. Measurement of moisture isotherms, packaging permeability, initial
moisture content and mass of the drug product & desiccant
• Packaging Model
A. Numerical integration of an ODE for Fickian moisture transfer through
package wall
B. Solve for moisture content and internal RH at any time during the shelf
life
C. Degradation growth can be determined by solving another ODE at the
same time with known environmental conditions inside the package.
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The Packaging Model
Basic Idea
• Packaging represents a moisture
transfer resistance but not a
perfect barrier – water diffuses
through the wall but slowly.
• Headspace holds a negligible
amount of moisture, however it
functions as a “middle man”
between external water vapor
and sorbed water equilibrium.
• Internal mass transfer is fast
relative to transfer across
packaging material and is not
rate limiting.
Internal
water
vapor
slow
External water
vapor
fast
Drug product
exchanges
sorbed water
with headspace
water vapor
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The Packaging Model
Simplifying Assumptions
• The bottle wall represents a
mass transfer resistance but no
capacity to adsorb moisture 
assume Fick’s 1st law applies
• No internal gradients in water
vapor concentration in
headspace
• The bottle contents represent
capacity to adsorb moisture but
no mass transfer resistance 
thermodynamic equilibrium
Internal
water
vapor
slow
External water
vapor
fast
Drug product
exchanges
sorbed water
with headspace
water vapor
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The Packaging Model
Simplifying Assumptions – “Package in Package”
• The bottle wall represents a
mass transfer resistance but no
capacity to adsorb moisture 
assume Fick’s 1st law applies
• No internal gradients in water
vapor concentration in
headspace
• The bottle contents represent
capacity to adsorb moisture but
no mass transfer resistance 
thermodynamic equilibrium
DES
Secondary
packaging, i.e.,
“package in
package”
fast
slow
Internal
water
vapor
External water
vapor
slow
fast
Drug product
exchanges
sorbed water
with headspace
water vapor
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The Packaging Model
Mass transfer details
Internal
water
slow
External
water vapor
vapor
Drug
product
fast
• Fickian diffusion driving force is the
difference in water vapor partial
pressure between external
environment and internal
headspace
• Headspace vapor
concentration is
determined by the
respective isotherms of
packaging contents
Desiccant
Interior air
E
Q
Capsules
Fickian
diffusion
mass transfer
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The Packaging Model
Fickian mass transfer
Fick:
Permeability
coefficient
Mass flux
j  CPpV
Gradient of partial
pressure
Drug
product
Dessicant
Assumed water
vapor partial
pressure profile
Capsules
WALL
OUTSIDE
pv
BOTTLE
INTERIOR
Equivalently:
Surface area
dmW
pV
 A CP
dt
l
Wall
thickness
Experiences & Challenges with ASDs | 2014 MBSW | May 21, 2014 | Copyright © 2014 AbbVie
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The Packaging Model
At any time t in the solution, we know the
amount of water inside the packaging
(based on initial water contents and
calculated mass transfer rates)
Equilibrium internal RH (symbol h) is
determined such that the adsorbed
moisture amounts in of each material
inside the package add up to the total
known moisture inside the packaging at
that time point. It will in general require a
numerical root-finding algorithm (e.g.,
Newton’s method, secant method, etc.
The determination of the equilibrium RH
allows the solution to progress to the next
time step via solution of the ODE.
m (moisture content, g / g)
Adsorbed moisture equilibrium
Dessicant
m1
Gelatin capsule
m2
m3
Drug product
h (Relative Humidity)
Known at each time step
mtotal  m1 h   m2 h   m3 h 
h is solved for such that equality
𝑑𝑚𝑤 𝐴𝐶𝑃 ∆𝑝𝑣
∆ℎ
=
, ∆𝑝𝑣 = 𝑝𝑠𝑎𝑡 𝑇
𝑑𝑡
𝑙
100%
is satisfied
Experiences & Challenges with ASDs | 2014 MBSW | May 21, 2014 | Copyright © 2014 AbbVie
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The Packaging Model
Degradants
What about degradants?
If the degradant growth can be modeled using an ODE that can likewise
be integrated WRT time alongside the Fickian diffusion equation, then
it is straightforward to solve for degradant growth in the packaging
model.
Degradant concentration
growth rate (zero-order)
dCi
 ki T , h 
dt
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Isoconversion Solution
Methods
Data Analysis
Open-dish degradant concentration data
• Due to limited resources, open dish data typically consists of 2 points
for each condition:
— Concentration at time t = 0
— Concentration at some time t predicted to be close to the true ttf
• Zero-order rate constants k(T,h) are slopes calculated from the
measured concentration and time of measurement
— Since zero order is assumed (straight line), this rate constant is equivalent to one
derived from the ttf at Ciso
T1,h1
T2,h2
𝑘 𝑇,𝐻
𝐶𝑖𝑠𝑜
=
𝑡𝑇,𝐻
T3,h3
T4,h4
Ciso
C
ttf1
ttf2
t
ttf3
ttf4
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Data Analysis
Regression approach
1. Calculate rate constants k(T,h) for each set of T,h conditions
2. Assume a model of the form
𝑘 𝑇,𝐻
−𝐸
= 𝐴 𝑒𝑥𝑝
+ 𝐵ℎ
𝑅𝑇
Three parameters
• E – apparent activation energy
• A – frequency factor
• B – humidity factor
Two independent variables
• T – temperature
• h – relative humidity
3. Regress rate constants to obtain model parameters
• Weighted Levenberg-Marquardt method for nonlinear regression
– Sensitive to starting estimates, can use linearized model to generate
– Frequency factor A moved inside exp() term to improve numerical stability
• In general, 5-10 observations (T,h conditions) used to predict 3 parameters
– Enough to get a good estimate of standard error of regression?
– Some overfitting may be occurring
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Simulation Results &
Discussion
Analysis of Open-Dish Data
• Data for purpose of
example is not actual
measured data, but
representative of many
ASDs
• Good fit obtained to
open dish data
• One points could be an
outlier – but is it?
• With only 7 data points,
removal of one outlier
could significantly
change parameter
estimates
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Analysis of Open-Dish Data
80
Fast, non predictive
Degradation Rates
• Selection of outliers also based on
likelihood of physical instability
60
• In general, if data point has large
studentized residual and is likely to
be in physical instability region, it is
removed from the analysis
Temp (C)
• Can be subjective
40
20
Slow Degradation
Rates
0
0
20
40
60
80
% RH
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Analysis of Open-Dish Data
New prediction
Previous result
including outlier point
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Packaging Simulation – moisture and RH
Inputs:
• 25°C, 60% RH external
• 5 oz HDPE bottle, 1.13E-2
mg/day/%RH
• 10 g of drug product, modeled as
microcrystalline cellulose
• Starting moisture 2% w/w
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Packaging Simulation – degradant predictions
• Due to small data set, removal
of one outlier can significantly
impact the solution
• Selection and removal of
outliers must be done
cautiously, based not just on
statistical reasoning but
knowledge of physical stability
• If in doubt, most conservative
estimate may be the more
prudent choice.
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Temperature and Humidity Accuracy
• This places extra emphasis on precise
control of temperature and RH in open dish
isoconversion studies
• Although T and RH are treated as known
quantities in the regression, practically
speaking they do have an uncertainty
associated with them
60
Temp (C)
• Open dish conditions need to be within
narrow range of Tg (e.g., ± 5°C) in order to
achieve sufficiently fast rates yet not be at
risk of physical instability
80
40
20
0
0
20
40
60
80
% RH
o RH sensors typically ±2% uncertainty
o Temperature sensors can be as low as ±0.2°C
uncertainty, but temperature uniformity in a
chamber may be ± 2°C or greater.
• What is the impact of temperature and humidity
uncertainty?
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Sensitivity of rate constant to uncertainty in T and h
𝜕𝑘
2
𝜎𝑘 =
𝜕𝑇
2
𝜕𝑘
2
𝜎𝑇 +
𝜕ℎ
2
𝜎ℎ2 = 𝑘 2
𝐸
𝑅𝑇 2
2
𝜎𝑇2 + 𝐵ℎ 2 𝜎ℎ2
Suppose we use the parameter estimates for the example data, and R
is the gas constant 0.008314 kJ/mol K. At 25°C and 60% RH,
𝜎𝑘2 = 𝑘 2 3.3E−2 𝜎𝑇 + 9.0E−4 𝜎ℎ2
𝜎𝑘
≅ 0.2 𝜎𝑇
𝑘
For example, if sT is ~ 1°C (perhaps due to chamber temperature nonuniformity), the magnitude of error in rate estimate will be large
relative to the rate itself. It could be even larger if physical stability is
compromised.
Experiences & Challenges with ASDs | 2014 MBSW | May 21, 2014 | Copyright © 2014 AbbVie
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Sensitivity of predicted degradation to uncertainty in T
• Simulation was run
using estimates for
rate constant k plus
an error term due to
uncertainty in
temperature control
𝑘 = 𝑘 ± 𝑛𝜎
n = 1, 2, 3
Experiences & Challenges with ASDs | 2014 MBSW | May 21, 2014 | Copyright © 2014 AbbVie
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Conclusions
Conclusions & Questions
• A “screening design” can be proposed where Tg is not a factor, but
this is more difficult to implement with an ASD due to Tg
restrictions, and may not be practical for ASD formulations in
general.
• It does not seem straightforward how to obtain a good estimate of
uncertainty in rate constant estimates from small data sets, and the
uncertainty from imprecise temperature control may be just as large
(depending on quality of stability chamber)
• What can we do to improve parameter estimation with small data
sets?
• What can we do to minimize impact of temperature uncertainty in
open dish studies?
• Is there a better quantitative way of detecting outliers or minimizing
their impact?
Experiences & Challenges with ASDs | 2014 MBSW | May 21, 2014 | Copyright © 2014 AbbVie
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