ASAP: Theory and Fundamentals
Science of Stability Conference
Groton US, October 2015
Garry Scrivens
Modelling The Effects of Temperature and Humidity
ASAP Scope and Definition(Accelerated Stability Assessment Protocols)
Environmental conditions:
1. Temperature
2. Humidity
3. Light• Accepted rapid ICH accelerated conditions exist
• Packaging used for most solid drug products protect from light
4. Oxygen level
(etc.?)
Shelf-Life Limiting Attributes
1. Chemical degradation
– Degradation products
– Assay
– Sometimes appearance
2. Physical degradation, e.g.:
– Dissolution, disintegration
– Hardness
– Polymorph changes / hydrate formation
Not ASAP
ASAP
ASAP
Not ASAP
An ASAP study is a
stability assessment of
a solid-state
pharmaceutical in
which the effects of
temperature and
humidity on the extent
or rate of chemical
degradation are
modelled and
quantified. The
experimental protocol
comprises a range of
different humidity and
elevated temperature
conditions. The
purpose of an ASAP
study is typically to
estimate shelf-life.
For the purposes of
this presentation:
Temperature and Humidity…
It has long been known that temperature and humidity are the two most important environmental factors for pharmaceutical solid-state stability:
– Schumacher (1972) and Grimm (1986, 1998):
• Zone 1: “Temperate” 21°C/45%RH
• Zone 2: “Subtropical and Mediterranean” 25°C/60%RH
• Zone 3: “Hot and Dry” 30°C/35%RH
• Zone 4: “Hot and Humid” 30°C/70%RH
%Deg is dependent on Temperature, Humidity (i.e. moisture) and time
%Deg = f(T,H,t)
Or
Rate constant for degradation, k = f(T,H)
Temperature…
Arrhenius equation (ca. 1889):
k = Ae-Ea/(RT)
Collision frequency
“pre-exponential factor”
Activation energy
Gas constant
Temperature (in K)
Rate constant(e.g. %deg per day)
Log k = Log A – (Ea/R).(1/T)
Log k
(1/T)
x
xx
xx
Intercept of line = Ln A
Slope of line = -Ea/R
Accelerated (high
temperature) results
e.g. T = 25ºC
Predicted rate of
degradation at 25ºC
Obtaining ‘k’ from experimental data…
Simple if degradation is linear with time:
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 500 1000 1500 2000
% Degradation
Time (Days)
X
X
X
X
k = [Deg]t – [Deg]0t
“Zero Order”
Obtaining ‘k’ from non-linear experimental data
k = A . exp[(Ea/R).(1/T)]
Approach #1: convert [%Deg]t and ‘t’ into ‘k’ via a rate equation:
k =
k = [%Deg]infinity
% D
egra
dati
on
Time
Zero
First
Second
Power (^2)
Avrami-Erofeev (^2)
Contracting volume
1-D diffusion D1
3-D Diffusion D3
At low %Deg….
n=0.3
0.5
0.7
1
3
An introduction of a simple
degradation shape parameter, ‘n’
may improve many models
Obtaining ‘k’ from Non-Linear Experimental Data
Approach #2: ‘Isoconversion’ or ‘Time to Failure’
Time
%Deg
Prod
30ºC
60ºC
70ºC
Spec.
Limit
1 month
Constant Time
1 day 3 days
Isoconversion
‘k’ = Slope
of line
With an isoconversion approach, the shape of the degradation curve is unimportant, we’re just
interested in how long it takes to reach a certain level of degradation (e.g. spec level) under
different conditions; this enables us to simply calculate ‘k’: k = %deg / time
Caution: Isoconversion Approaches
Time
%Deg
Prod
Spec.
Limit
Actual
Degradation
Predicted
Degradation
Accurate prediction at
(e.g.) spec level
The case for isoconversion:Real-World API Micro-Environments in Solid-State
In Solid State:
– Molecules in different microenvironments:
• crystal lattice
• surface
• amorphous
• solid-solution
– Multiple k’s
– Heterogeneous kinetics – formation of product is a superposition of multiple rates
– Shape of degradation curve in solid state may not be well described by simple kinetics (e.g. 0th, 1st or 2nd order, Avrami, diffusion, power law etc.)
[Pt] = ikit (different k for each API state)
Degradation Curves at Different Conditions
An assumption for most current ASAP models is that tor a given system, the
shape of the curve (i.e. degradation kinetics) is usually very similar across
different stability conditions, just the timescale is different…
Real world example:
3.1x faster
3.1x faster
Etc.
11.7x faster
Modelling the Effects of Humidity / Moisture
No universally accepted “Arrhenius Equation” for the effect of humidity / moisture on the rate of degradation
Many options for “humidity descriptors”
– Descriptors of the headspace (gas-phase):
• Vapour Pressure (VP), Absolute Humidity (AH), Relative Humidity (RH)
– Descriptors of the solid sample:
• Water Activity (aw), Water Content (WC), Non-Bound (‘available’) Water Content (AWC)
Different possibilities for the relationship between degradation rate and “H”
– Rate a exp(B.H)
– Rate a (H)B
Humidity DescriptorsThe relation between VP, AH and RH
SVP SAH
These curves represent 100%RH
at different temperatures
X ~66%RH
• VP = RH * SVP ; RH = VP / SVP
• AH = RH * SAH ; RH = AH / SAH
• VP and AH are equivalent for
the purposes of modelling
T
1kexp.kSVP 54
Humidity Descriptors
Water activity, aw
– When a solid sample is in equilibrium with its environment, aw = RH
– N.B. RH typically expressed as %RH
(Available) water content, AWC & WC
– Related to %RH / aw by the moisture sorption isotherm of the material:
%RH
aw
Aw
Avai
lable
Wat
er C
onte
nt
Aw
AWC WC
t1/2~hrs
-8
-7
-6
-5
-4
-3
-2
-1
2.9 3 3.1 3.2 3.3 3.4
ln k
1/T (X1000)
10% RH
75% RH
Modelling the Effects of Humidity
Degradation of Aspirin Tablets:
Waterman ca. 2004: observation:
Furthermore: solid-state
degradation rate seems to
increase exponentially with
%RH:
-9
-8.5
-8
-7.5
-7
-6.5
-60 20 40 60 80 100
ln k
%RH
(Constant Temperature)
Log k = Log A - Ea/(RT) + B(%RH)
Log k
1/T
%RH
Log k = Log A - Ea/R(1/T) + B(%RH)
40/75
30/75
30/65
25/60
70/75
80/40
70/ 5
50/75
60/40
Visualizing the “Classic” ASAP (Waterman) Model
Ln A
B
Ea/R
“Hand-waving” rationale forLog k = Log A – Ea/RT + B(RH)
1/T
Log k
0
Lower
Humidity
Log A
Arrhenius The intercept (Log A) is associated
with the probability of reaction when
every molecule has sufficient energy
to react, i.e. at “infinite temperature”
(e.g. probability that they
collide/move in the right way)
Log k = Log A –Ea/RT + B(RH) is an empirical model (i.e.
based on experiment / experience / observation; however…
“Hand-waving” rationale forLog k = Log A – Ea/RT + B(RH)
1/T
Log k
0
Higher
Humidity
Lower
Humidity
Log A + B(RH)
Log k = Log A –Ea/RT + B(RH)
Moisture increases the
probability that the molecules
collide/move in the right way for
reaction
Alternative Models
Classic ASAP ‘Waterman’ Model:
– Rate = k1.exp[k2(1/T)].exp(B.RH) [RH = aw]
Classic ASAP model with alternative humidity descriptors, E.g.:
– Use AH (or VP) instead of RH
– Use WC or AWC instead of RH
Other popular Models:
– Rate = k1.exp[k2(1/T)].(H)B (“Power” model)
– Where ‘H’ is RH, AH, or (A)WC
GLM:– Log k = b0 + b1(1/T) + b2(%RH) + b3(1/T).(%RH) + b4(1/T)2 + b5(%RH)2 + b6(etc.)
Humidity Descriptors
It is relatively easy to convert one humidity descriptor into another:
VP = AH RH = aw AWCT, SVP
Moisture
Sorption
Isotherm
(GAB
parameters)
)]RH.(K.C)RH.(K1)][RH.(K1[
)RH.(K.C.WAWC m
Moisture Sorption IsothermN.B. Measuring water
content directly (e.g. by
Karl-Fischer is prone to
high experimental variability
“Power” Model
“Power” Model:
– Rate = k1.exp[k2(1/T)].(H)B
=>Log (rate) = Log k1 + k2(1/T) + B. Log(H)
– cf. “Classic”:
– Log (rate) = Log k1 + k2(1/T) + B.(H)
Rationale for “Power” model:
– Water is treated as a reactant, and the ‘B’ term is the order of reaction with respect to water, i.e.:
– Rate a [H2O]B
“Power” model
same as
Waterman
model except
Log(rate) is
dependent on
Log(H) instead
of (H)
Using VP or RH with “Power” Model
With “Power” Model, it doesn’t matter whether VP or RH is used, the model is mathematically virtually the same (i.e. will always result in virtually the same predictions and RMSE):
– Log (rate) = Log k1 + k2(1/T) + B. Log(VP)
– VP = RH . SVP &
Log (rate) = Log (k1.k4B) + (k2+B.k5)(1/T) + B.Log(RH)
Log (rate) = Log k1 + k2 (1/T) + B.Log(VP)
T
1kexp.kSVP 54
Constant Constant Constant
The ‘Classic’ Waterman (RH) model often performs very similarly to the “Power” (AWC) model…
Why?
Often AWC ~ Exp(RH)
(i.e. Log(AWC) ~ RH)
“Power”: Log (rate) = Log k1 + k2(1/T) + B.Log(AWC)
~ “Classic”: Log (rate) = Log k1 + k2(1/T) + B(RH)
How Do the Different Models Perform?
RH AH/VP AWC
Waterman 1 3 4
“Power” 2 2 1
“Classic” Waterman(RH) ~ “Power”(AWC)
“Power”(RH) ~ “Power”(AH)
Type 1 > Type 2 > Type 3 > Type 4
• When Type 1 is good, it is very good
• When Type 1 is poor, none of these models are
particularly good
So far: These models have been compared using a
small number of real products…
Classic ASAP
Ln k = Ln A - Ea/R(1/T) + B(%RH)
vs
GLM:Ln k = b0 + b1(1/T) + b2(%RH) + b3(1/T).(%RH) + b4(1/T)2 + b5(%RH)2 + b6(etc.)
“Classic ASAP” vs General Linear Models
Classic ASAP GLM
If there is a discontinuous (non-smooth) effect of temperature or humidity (e.g. a
phase change / physical change at a particular threshold), then applying a GLM
model (i.e. fitting a smooth curve to the data) would lead to error
GLMs inevitably have better “fits”, but not necessarily a better prediction; better to
attempt to understand why “Classic” ASAP isn’t working rather than introduce
new terms
Classic ASAP vs General Linear Models
ln k
1/T
%RHln k
1/T
%RH
B
Ea/R
x
x80/40 x
x
xo
o
oo
40/75
30/75
30/65
25/60
70/75
50/75
60/40
70/5
Possible Sources of inaccuracy and variability in ASAP
Lack of representativeness of an accelerated condition to long-term storage conditions.
ln k
1/T
%RHln k
1/T
%RH
B
Ea/R
x
x80/40 x
x
xo
o
oo
40/75
30/75
30/65
25/60
70/75
50/75
60/40
70/5
E.g.:
• Polymorph changes
• Hydrate formation
• Melting point
• Glass Transition
• Deliquescence
For all the models discussed so far, it is assumed that the shape of the
curve (i.e. degradation kinetics) remains similar at the different conditions,
and that only the timescale changes …
…sometimes this is not the case
3.1x faster
3.1x faster
Etc.
11.7x faster
Possible Sources of inaccuracy and variability in ASAP
…But from this same experiment…the curve for 60°C / 75% RH could not be made to map on to the 80°C / 40% RH curve
• The reason for this is that it is likely that >1 process is occurring in this sample and the rates of
the different processes relative to each other are different at 60°C/75% vs 80°C/40%
• This leads to inaccuracies with all the models discussed so far
Apply the same factor
Multiple competing / consecutive processes
• Recipe instructions: Bake at 160°C for 30 mins
• Impatient ASAP Scientist: Bake at 240°C for 1
minute
Multiple simultaneous / competing / consecutive processes:• Many different chemical reactions
• Multiple physical processes (e.g. melting, evaporation, diffusion, thermal
conductivity etc.)
Each process has different profiles of progression and different dependencies
on temperature and humidity etc.
Result: crème-brûlée
instead of cake
Conclusions
There are many ways to model the effect of T and “H” on Deg Rate
“Classic” ASAP performed the best out of the models assessed here (caveat: so far only a small number of products evaluated)
– N.B. the “Power”(AWC) model performs equally well, maybe even better. The “Power”(AWC) model is essence very similar to “Classic ASAP”
Log (rate) = Log k1 + k2(1/T) + B.Log(AWC)
Room for Improvement: there a number of products where none of these current models perform satisfactorily
– Phase changes and multiple competing processes across the different accelerated stability conditions present the biggest challenge
An additional degradation “shape” parameter (and maybe a degradation shape –humidity interaction term) may improve many models. However, must avoid the “GLM” pitfalls of ‘over-fitting’), and this will not solve all poor-fits:
Log (%Deggrowth) = Log k1 + k2(1/T) + B.(RH) + n1.Log(t) + n2.Log(t).(RH)
Kinetic Simulations based on an improved molecular-level understanding of solid-state degradation processes…(empirical approaches can only take us so far)…