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)…

Thanks for Listening

Questions?

Approach #1b: working with [%Deg]t :

Where:

k = A . exp[(Ea/R).(1/T)]

Obtaining ‘k’ from non-linear experimental data

[%Deg]t =

[%Deg]t =

[%Deg]t =

+ [%Deg]0