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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5/08 KlProj/AocaProlelRCIDatoDue.indd

Moisture Distribution in Blister Packages
By

Satish Muthu

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Chemical Engineering

2009

ABSTRACT
MOISTURE DISTRIBUTION IN BLISTER PACKAGES
By

Satish Muthu

Some blister packages are designed to protect pharmaceutical drugs from
environmental factors such as moisture. Moisture uptake by anhydrous solids stored in
these packages is primarily controlled by adsorption/absorption phenomena and can be
predicted if the thermodynamic and transport properties of the package constituents (esp,
the drug and the polymer barrier phases) are known. In this thesis, a quantitative metric
for the shelf life of a blister package is developed in terms of parameters associated with
individual components of the package and the environmental conditions associated with
storage. The adsorption/absorption model developed herein assumes that the moisture
distribution rapidly attains a pseudo-steady state profile within the polymer barrier phase
and within the air gap phase. This pseudo-steady state approximation (PSSA) is justified
based on an exact unsteady-state analysis of a conjugate absorption problem with linear
isotherms. A commercial finite element code (COMSOL MULTIPHYSICS 3.3) was used
to analyze the non-linear boundary value problem resulting from the use of a GAB-
adsorption isotherm at the solid-product/air-gap interface. The resulting mathematical
model is consistent with an earlier comprehensive experimental study of moisture uptake
by blister packages containing 20 mg Deltasone® tablets (see Allen, 1994). The PSSA
model provides a practical tool for estimating the shelf life of blister packages and for

evaluating testing protocols.

Dedicated to My Parents, Mr. and Mrs. Muthu, who have inspired me

to be what I am today.

iii

ACKNOWLEDGEMENT

The gratification and euphoria that accompany the successful completion of a project

would be incomplete without mentioning those who made it possible.

I would like to express my deep sense of gratitude and indebtedness to Professor Charles
Petty for his incomparable counsel and indefatigable efforts, which were invaluable for
the completion of this work. In addition, I would like to sincerely thank Professor Maria
Rubino, Professor Rafael Auras, and Professor Andre’ Bénard for providing their
expertise and valuable input regarding this research as well as Karuna Koppula for her
help in formulating the boundary value problem and the use of COMSOL Multiphysics

software.

I would like to thank the Department of Chemical Engineering and Materials Science, the
Department of Mechanical Engineering, the School of Packaging, Pfizer and the National
Science Foundation Industry/University Cooperative Research Center for Multiphase
Transport Phenomena (NSF/ECC 0331977) for all their financial and technical support

during the research

Finally, I would like to thank my parents, ma belle and friends for their unconditional

love and affection that kept me going through hard times.

iv

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... vi
LIST OF FIGURES ........................................................................................................ vii
LIST OF SYMBOLS ........................................................................................................ ix
CHAPTER 1 INTRODUCTION ..................................................................................... 1
1.1 MOTIVATION ........................................................................................................... 2
1.2 BACKGROUND ......................................................................................................... 3
1.3 OBJECTIVES AND OUTLINE .................................................................................... 4
CHAPTER 2 LITERATURE REVIEW ......................................................................... 7
2.1 WATER ACTIVITY ................................................................................................... 8
2.2 MOISTURE SORPTION ISOTHERMS ......................................................................... 8
2.3 GAB ISOTHERMS .................................................................................................. 11
2.4 WATER ABSORPTION PHENOMENA IN THE PRODUCT PHASE ............................. 14
2.5 WATER ABSORPTION PHENOMENA IN THE POLYMER BARRIER PHASE ............ 17
2.6 BLISTER PACKAGES .............................................................................................. 19
2.7 SHELF LIFE METRIC ............................................................................................ 22
CHAPTER 3 MASS TRANSFER MODEL .................................................................. 25
3.1 INTRODUCTION: ONE-DIMENSIONAL MASS TRANSFER MODEL ......................... 26
3.2 ASSUMPTIONS ....................................................................................................... 26
3.3 UNSTEADY AND PSEUDO-STEADY STATE MODELS ............................................. 28
3.3 PSEUDO STEADY STATE MODEL FOR THE PQI-TIME .......................................... 34
3.4 LINEAR ADSORPTION — ANALYTICAL SOLUTION ................................................ 35
3.6 LINEAR ADSORPTION — NUMERICAL SOLUTION ................................................. 37
3.7 UNSTEADY STATE MODEL FOR THE PQI-TIME ................................................... 37
3.8 GAB ADSORPTION - NUMERICAL SOLUTION ...................................................... 42
3.9 PSSA MODEL WITH A GAB ISOTHERM — EXPERIMENTAL VALIDATION ........... 42

CHAPTER 4 PARAMETRIC STUDIES OF THE PSEUDO-STEADY STATE

ABSORPTION (PSSA-) MODEL .................................................................................. 44
4.1 INTRODUCTION ..................................................................................................... 45
4.2 INPUT DATA FOR THE PSSA MODEL ................................................................... 45
4.3 PSSA MODEL --- LINEAR ISOTHERM ................................................................... 45
4.4 PSSA MODEL (GAB ISOTHERM) ......................................................................... 49

V

CHAPTER 5 CONCLUSIONS ...................................................................................... 60

CHAPTER 6 RECOMMENDATIONS ........................................................................ 64

APPENDICES ................................................................................................................. 66
APPENDIX A. ONE-DIMENSIONAL DIFFUSION FOR LINEAR ADSORPTION
ISOTHERMS ................................................................................................................... 67
APPENDIX B. PSEUDO-STEADY STATE ABSORPTION MODEL ................................. 71
APPENDIX C. UNSTEADY STATE CONJUGATE DIFFUSION ....................................... 77
APPENDIX D. PROBLEM SETUP FOR THE NON-LINEAR PSSA MODEL ................... 86
APPENDIX E. ABSORPTION: FLAT PLATE GEOMETRY AND LARGE BIOT NUMBERS
....................................................................................................................................... 90
APPENDIX F. JUSTIFICATION OF PSSA-MODEL FOR LINEAR ADSORPTION .......... 93

REFERENCES .............................................................................................................. 100

Vi

Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 8.1

Table 8.2

Table 8.3

Table F .1

LIST OF TABLES

Data used in the PSSA Model Calculations .................................... 46
Groups of the PSSA Model (Linear Isotherm) ................................ 47

Eigenvalues and Fourier Coefficients for the Pseudo-Steady
State Model ........................................................................ 48

Eigenvalues and Fourier Coefficients for Linear
Conjugate Absorption ............................................................ 50

Eigenvalues and Fourier Coefficients for the Linear PSSA-Model
for a = 0.1 .......................................................................... 74

Eigenvalues and Fourier Coefficients for the Linear PSSA—Model
for or = 0.2 ......................................................................... 75

Eigenvalues and Fourier Coefficients for the Linear PSSA-Model
for a = 0.3 ........................................................................... 76

Effect of the parameters NS , Np & NT on
the PQI-time ..................................................................... 99

Vii

Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5

Figure 3.1

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure F.1
Figure F2

Figure F3

LIST OF FIGURES

Classification ofAdsorption Isotherms .......................................... 9
GAB Moisture Sorption Isotherm ............................................... 15
Verification of the Product Diffusion Coefficient ............................ l8
Verification of the Polymer Diffusion Coefficient — PVC .................. 20
Verification of the Polymer Diffusion Coefficient — Aclar .................. 21

Schematic of a Blister Package and Coordinates for the
One-Dimensional Transport Model ............................................. 27

The Effect of on on the Relaxation of the POI-function

(Linear isotherm) .................................................................. 5 I
The Effect ofot on the PQI-time for O§ = 0.5 (Linear isotherm)...........52
Comparison of Experimental Results with the PSSA-model

at 80% Relative Humidity (GAB isotherm) ................................... 53
Comparison of Experimental Results with the PSSA-model

at 90% Relative Humidity (GAB isotherm) ................................... 54
Moisture Ingress as a Function of Time in the Individual Phases

and in the Whole Blister ......................................................... 56
Variation of Diffusive Flux at the Product - Air Gap Interface

with Time .......................................................................... 58
Effect of Relative Humidity on the Moisture Sorption of a Blister
Package ............................................................................ 59
Effect of NI on the PQI function ............................................. 96
Effect of NS on the PQI function ............................................. 97
Effect of Np on the PO] function ............................................. 98

viii

NOTATION
a1, a2 ,a3

An

Aw

Bi
Bim

CS,CG,CP,CE

DSaDGsDP

1331:}:

KSGaKPG

LIST OF SYMBOLS

GAB coefficients (temperature dependent).

Dimensionless Fourier coefficient defined in Table 4.1.

Water activity or the Relative Humidity (RH).

Biot number defined by Eq.(3.9).

Modified Biot number defined by Eq.(C.I 1)

Concentration of water in the S-phase, G-phase, P-phase, and the

surrounding (environmental) phase, respectively (units:
mass/volume).

Diffusivity of water in the S-phase, G-phase, and P-phase,
respectively (units: (lengthy/time).

Eigenfunctions in the S-phase and P-phase, respectively.

Convective mass transfer coefficient for water at the external PG-
interface (units: length/time).

Thermodynamic equilibrium distribution coefficients for water

across the SG-interface and the PG interface; units: (mass of water
per unit volume of S-phase)/(mass of water per unit volume of G-
phase). See Eq.(3.10).

ix

KGPaKPE

LSaLG’LP

MOI‘ Mt

PQI-

PVC

RH

Thermodynamic equilibrium distribution coefficients across the

GP-interface (between the air gap and the polymer barrier) and the
PE-interface (between the polymer and surroundings). In this

research KGP = KPG = KPE .

Half-thickness of the S-phase; effective gap of the G-phase, and

thickness of the P-phase, respectively (units: length). See Figure
3.].

Moisture content in the tablet in g / 100 g of solid tablet.

Equilibrium value of Mt-

...See Eq.(3.32).

 

PQI = Product Quality Index. This terminology is used to
designate a quantitative condition related to the moisture content of
the solid product phase. The PQI-function is a volume average of
the concentration difference function (see Eq.(3.15)); the PQI-time
is defined by Eq.(3.7) and gives the shelf life based on a criterion
for the moisture content in the product phase.

Polyvinyl chloride.

Relative humidity of the storage environment.

SGP

SSG

XG’XG

is, Xs

22p, X'p

89(81,82,83)

VS

VG

Mass transfer surface area between the air gap and the polymer
barrier.

Mass transfer surface area included for mass transfer between the
solid product phase and the air gap.

(disk: SSG = 21rR2,R 2 disk radius)

Dimensional and dimensionless times, respectively. Time is made

dimensionless by using the diffusion time scale of water in the S-
phase (see Eq.(3.3)).

Shelflife defined by Eqs.(3.7) and (3.8).

Dimensional and dimensionless spatial position variables

(see Figure 3.1).

Embedded coordinate in the G-phase.
Embedded coordinate in the S-phase.

Embedded coordinate in the P-phase.

Spatial position vector in a three dimensional space.
Volume of the solid product phase.
Volume of the G-phase.

Volume of the P-phase.

xi

Greek

<OS>(t)

PS

Dimensionless eigenvalue associated with the pseudo-steady state
model (see Eq.(3.16) and (3.17) and Table 4.2).

Dimensionless concentration difference functions associated with

the S-phase, the G-phase, and the P-phase, respectively (see
Eqs.(3.12)-(3.14).

Value of the PQI-function at the critical moisture concentration

(see Eq.(3.7). This dimensionless specification depends on three
factors: I) the critical moisture content of the S-phase; 2) the initial
moisture content of the S-phase; and, 3) the equilibrium moisture
content of the S-phase consistent with the storage conditions.

Volume average of the water concentration difference function
over the S-phase.

Density of the tablet

Operators and Functions

cos( AnX)

Laplacian operator

Function defined by Eq.(3.41). The zeros of this function

correspond to the eigenvalues that appear in Eq.(3.38).

Volume average operation associated with the S-phase (see Eq.(3.2)
and Eq.(3.15)).

Eigenfunction associated with the pseudo-steady state model

xii

3i
at’axz

F irst-order and second-order partial differential operators

Symbols, super/subscripts

S, G,P

Used to designate a critical condition associated with the
maximum allowable water concentration in the solid product phase
(S-phase). Also used to designate a convective mass transfer
coefficient (see kc above).

Used to designate an initial condition.

Index integer for the discrete eigenvalues (n = 1,2,3, ...)

Used to designate a property associated with the S-phase, the
internal or external G-phases, and the P-phase.

Used to designate a dimensional property, or variable.

Used to designate an equilibrium steady-state condition.

xiii

CHAPTER 1

INTRODUCTION

1.1 Motivation

Moisture protection is one of the most important functions of a blister package
containing a pharmaceutical drug inasmuch as the interaction between some drugs and
small amounts of water may cause physical and chemical modifications during storage
(Labuza, 1985; Ahlneckand Zografi ,1990; Bell and Labuza ,2000 ). Presently, extensive
long-time empirical testing is used to identify optimal package designs. This research is
partly motivated by the promise that a mathematical model for the shelf life of a blister
package can complement current testing protocols by reducing the design cycle time for
selecting package components and by providing additional assurances that FDA

standards will be met by the final package design ( PQRI, 2005).

The amount of moisture absorbed by drugs and excipients affects the flow,
compression characteristics, and hardness of granules and tablets. Most significantly, the
active form of the drug may interact with water to form undesirable products and loss of
drug benefit. In addition, moisture may affect the dissolution and transport of the drug
from the tablet. To avoid packaging failures, an overprotective package could be used,
but this is expensive. Instead, blister packages are often tested to identify low cost options
with sufficient barrier protection. However, the estimation of shelf life by experimental
methods alone is laborious and is constrained by both time and cost. A suitable
alternative is to use an appropriate mathematical model to predict the long-time
absorption rates of a blister package based on mass transfer principles. Absorption
experiments under extreme conditions, which would be relatively fast and cost effective,

can be used to determine the thermodynamic and transport properties of specific blister

constituents. This experimental/mathematical approach should reduce the cost and time
involved in shelf life estimation by eliminating repetitive and unnecessary tests. A
mechanistic mathematical model could also be used to optimize the size and shape of
blister packages as well as the selection and design of polymeric moisture barriers for

specific applications.

1.2 Background

Moisture is an important factor in determining the shelf life of some products
packaged in so-called "blisters" (see Labuza, 1985; Yoon, 2003; PQRI, 2005). In general,
the shelf life of a moisture sensitive drug (or food) is primarily determined by l) the
manufacturing and packaging processes; 2) the storage environment; 3) the water
sensitivity of the product; 4) the thermodynamic and transport properties of water in the
constituent phases; and, 5) the design of the package. When the volume average moisture
concentration of the product phase exceeds a pre-determined critical level, the blister

package is removed from the shelf.

This master thesis research relates to previous studies on the Shelf life of moisture
sensitive products conducted at Michigan State University by Kim (1992), Allen (1994),
Kim et al. (1998), and Yoon (2003) in the School of Packaging. Earlier experimental and
theoretical research by Zografi et a1. (1988), Howsmon and Peppas (1986), Anderson and
Scott (1991), Smith and Peppas (1991), Marsh et al.(1999), and, Badaway et al.(2001)

have also examined the impact of moisture on the shelf life of food and drug products.

1.3 Objectives and Outline

The objective of this research is to develop a quantitative measure of the shelf life
of a moisture sensitive product based on unsteady-state mass transfer principles and to
compare the model predictions with previously reported experimental data (Allen, 1994).
The focus of the research is to determine how the geometry and the physical properties of
a blister package influence the Shelf life of a moisture sensitive product. A mechanistic
understanding of the relationship between package design and product quality may
contribute to a reduction in the cost associated with testing the efficacy of different

package designs.

The scope of this study is limited to a simple package configuration that justifies
the use of a classical one-dimensional diffusion model (see Appendix A); however, the
pseudo-steady state absorption (PSSA-) model developed in Chapter 3 (and Appendix B)
can easily be extended to more complex geometries and to products with other rate
limiting phenomena including chemical reactions and moisture sensitive diffusion
coefficients (Philip, 1994). The need to consider explicit unsteady-state moisture
transport in the drug phase was clearly demonstrated in the earlier study by Kim (1992;
also see Kim et al., 1998). Many shelf life prediction models for drug and food products
have been proposed in the literature over the past thirty five years. A few noteworthy
references are Labuza et al., 1972, Khanna and Peppas, 1978; Peppas and Khanna, 1980;
Peppas and Sekhon,1980; Peppas and Kline, 1985; Khanna and Peppas, 1982; Peppas
and Kline, 1985; Howsmon and Peppas, 1986; Smith. and Peppas ,1991; Anderson and

Scott, 1991; and, Badawy et al., 2001.

In Chapter 3, a quantitative definition of the shelf life of a moisture sensitive
pharmaceutical tablet within a blister package is defined. A mathematical model for this
metric is developed based on the idea that a pseudo-steady state profile within the
polymer barrier phase occurs on a time scale which is small compared with the rate of
absorption by the product phase (Bischoff, 1963; Bischoff, 1965; Bowen, 1965; Hill,
1984). In the PSSA-model, the transport of water through the tablet — blister system is
assumed to be governed by unsteady-state Fickian diffusion with a constant diffusion
coefficient. A pseudo-steady-state diffusion model in the polymer and air gap phases
together with an unsteady-state diffusion model in the product phase provides a means to
determine the long-time absorption rates and, thereby, the shelf-life of a blister package.
In Appendix F, conditions that support the validity of the PSSA-model for linear
isotherms are identified by using an exact solution to an unsteady-state conjugate
boundary value problem for moisture transport through the polymer barrier and the drug
product (see Appendix C). The PSSA-model provides a conservative, albeit realistic,
estimate for the shelf life of a blister package (i.e., the shelf life predicted by the PSSA-
model will be less than the shelf life predicted by the unsteady state conjugate mass
transfer problem. Therefore, the proposed approach developed hereinafter provides a

practical means for screening design options.

The PSSA-model is comprehensive in the sense that it incorporates all the
physical and environmental parameters that influence the shelf life of a blister package.
Model predictions are used to interpret previously published experimental data for blister

packages by Allen (1994). Selected experimental data from Allen’s work were used to

confirm the thermodynamic and transport properties for individual components of the
blister package. This information defines a benchmark for a parametric study of the
PSSA-model in Chapter 4. The parametric results provide a basis to further develop a

strategy for improving the design of blister packages for moisture sensitive pills.

CHAPTER 2

LITERATURE REVIEW

2.1 Water activity
Water activity is a measure of the relative availability of water to hydrate a
material. The water activity is related to the chemical potential of the system by the

thermodynamic relation (Van den Berg and Bruin, 1981):
p=p0+RT In (f/fo) . (2.1)

In the above equation, u(J/mole) is the chemical potential of the system; “0 is the

chemical potential of the pure substance at temperatureT (OK); R is the gas constant; and,
f is the fugacity or the escaping tendency of a substance. The parameter fo is the escaping

tendency of the anhydrous material. The water activity of a substance is defined as f/fO
and is designated as AW (5 f/fO) . For all practical purposes involving the conditions

under which blister packages are found, the fugacity is approximately equal to the

relative vapor pressure, (i.e., Aw = f/fO E p/po ). Thus, the relative humidity and the

water activity are equal and can be calculated as the ratio of the vapor pressure of water
in air to the saturation vapor pressure. Clearly, the temperature and the relative humidity
(water activity) of the surrounding humid air will influence the quality of food and drug

products (Rockland and Stewart, 1981).

2.2 Moisture Sorption Isotherms
As illustrated by Figure 2.1, moisture sorption isotherms can be classified into six
major types (IUPAC, 1985). When the surface coverage is sufficiently low all the

isotherms reduce to a linear form referred to as the Henry’s law region. The Type I

 

 

 

I II III

 

 

 

 

 

 

 

 

 

 

Six different adsorption isotherms for a solid/air interface. M represents the mass
of water adsorbed per unit mass of dry solid and AW represents the relative
humidity, which increases from 0.0 (dry) to 1.0 (100%).

 

 

 

IV VI

 

 

 

 

 

 

 

 

 

Figure 2.1 Classification of Adsorption Isotherms (IUPAC, 1985).

isotherm is reversible and is concave to the Aw (i.e., p/pO) axis. The amount adsorbed

(i.e., number of molecules or moles per unit area) approaches a finite value as the activity
tends towards unity. Type I isotherms are sometimes referred to as Langmuir isotherms.
This type of behavior occurs for microporous solids that have relatively small external
surface areas (such as activated carbons and zeolites). Pores with diameters larger than 50
nm are called macropores; pores with diameters between 2 nm and 50 nm are called

mesopores; and, pores with diameters less than 2 nm are called micropores.

The Type II isotherm, which is commonly encountered, is also reversible. Most
adsorbents, which are non-porous (or macroporous), exhibit this kind of behavior. These
isotherms exhibit monolayer-multilayer adsorption phenomena. Point B on Figure 2.1
represents the onset of multilayer adsorption phenomena. Type III isotherms are convex
over the entire range of Aw and, thereby, do not show a distinct Point B. This type of
isotherm is not very common but few systems like nitrogen on polyethylene show this

type of behavior.

Type IV isotherms show adsorption/desorption hysteresis due to capillary

condensation occurring within the mesopores at the interface and the limited uptake over
a range of high Aw. The Type IV isotherm is similar to the Type II isotherm for small
values of Aw. Many meSOporous industrial adsorbents exhibit this type of behavior.

Type V isotherms are Similar to the Type III isotherms. It has weak adsorbent/adsorbate
interaction Similar to the Type II isotherm, but is distinguished by the fact that it has a

hysteresis loop. Type VI isotherms, which are characteristics of non-porous surfaces

10

Show step-wise multilayer adsorption phenomena. The height of each step represents the
monolayer capacity of each adsorbed layer. The sharpness of each step is dependent on
the system chosen and the ambient temperature. Type VI behavior is exemplified by

argon (or krypton) on graphitized carbon black at liquid nitrogen temperatures.

2.3 GAB Isotherms

In spite of its limitations, the classical Brunauer, Emmett and Teller (BET-)
multilayer sorption equation is still used to calculate monolayer values under different
physicochemical conditions (Perry and Green, 1997). From these data, specific area
values are obtained. It is mainly used because of the simplicity of its application and
because it has the approval of the International Union of Pure and Applied Chemistry
(IUPAC, 1985). A 1985 report by the Commission on Colloid and Surface Chemistry

recommends that the BET isotherm be used as a standard for monolayer adsorption for

Aw in the range: 0.05 < AW < 0.3. Multilayer sorption isotherms show a sigmoid, or S-

shaped, form and are well represented by the BET-isotherm defined by

v: va CB Aw . (2.2)
(l-Aw)(l+(CB-1)Aw)

In the above equation, v represents the amount of water (sorbate) adsorbed by a gram of

sorbant if the water activity of the humid air at the interface is Aw . The parameter va
represents the monolayer value in the same units as v and the constant CB is the

difference in the free enthalpy (standard chemical potential) of the sorbate molecules in
the pure liquid state and in the monolayer (first sorbed) state. To obtain the two

characteristic constants from experimental data, the BET equation is often rewritten as

11

 

 

A _.
W 1 + CB 1 AW. (2.3)

FBET E =
(1-Aw)V cB VmB cB VmB

If the BET-postulate is valid, then a plot of FBET vs. Aw is linear. This usually occurs
at low activities (0.05 < Aw < 0.3 ). For Aw > 0.3 , an upward curvature is observed for
FBET- This deviation shows that, at higher activities, less gas or vapor is sorbed than

that anticipated by the BET-equation based on constants deduced from low activity data

(i.e., the parameter v is weakly dependent on Aw for AW > 0.3).

The Guggenheim, Andersen and de Boer (GAB-) sorption equation has a similar
structure as the BET-isotherm (Anderson , 1946). This equation is Often employed to
account for deviations from the classical BET isotherm (Costantino et a1. 1997; Moreira
et al., 2002; Moreira et al., 2003). The main reason for its use is that the activity range
covered by this isotherm is much wider than that of the BET-equation

(0.05<AW <0.8—0.9 ). in other fields, the use of the GAB-isotherm is not well
established. The GAB-isotherm is defined by

v = VmG CG kAW . (2.4)
(l-kAw)(1+(cG-1)kAw)

Here va is the GAB-monolayer capacity and CG is the analogue of the BET energy
coefficient CB. The GAB-model assumes that sorption states of the water molecules in

the layers beyond the first are the same, but different from the pure liquid state. This idea
requires the introduction of an additional coefficient, k. This constant is a measure of the
difference of free enthalpy (standard chemical potential) of the sorbate molecules in these

two states (i.e., the pure liquid state and the second sorption stage above the monolayer).

l2

Likewise, the other GAB-energy constant measures the difference of the chemical
potentials of the sorbate molecule in the upper sorption layers and in the monolayer. The

GAB-isotherm can be rewritten as

A _
w 1+CG1A

= (2.5)
(l—kAw)V 001(va CG VmG

FGAB 5

With k =1, the GAB-isotherm reduces to the original BET equation

(VmB=VmG;CB= CG)-

Eq.(2.4) can also be rewritten as

M; V = 2 Aw , (2.6)
VmG a3 Aw+a2Aw+a1

 

 

which is used in this study (see Chapter 3) to account for the adsorption of water by the
solid drug phase. Also see Kim (1992), Allen (1994) and Kim et al. (1998). In Eq.(2.6),
the parameter M represents the mass of water adsorbed relative to the mass of solid. The

coefficients a] ,a2, and a3 are related to the GAB parameters by the following equations:

 

als

 

 

ch
32 5 CG — 2 (2.7a,b,c)
Cgk

__ _ (CG -1)k2

_ cgk
Note that

(co —1>(1—k2)

a1+a2 +a3 = . (2.8)

CG k

13

Generally, CG 21 and k S 1 . These parameters depend on the temperature and the
specific properties of the solid phase. It is noteworthy that for CG > land k ->1, the
GAB isotherm approaches the BET isotherm and M —->oo for AW —>1( see Type II
isotherm in Figure 2.1 above). For CG Zland k $1, M —>MC <00 for AW —+1( see

Type III isotherm in Figure 2.1 above).

Figure 2.2 illustrates the GAB-isotherm for 20 mg Deltasone® tablets at room
temperature. Deltasone® is a brand of prednisone, which is used for anti-inflammatory
purposes in the treatment of arthritis. The following GAB parameters,

a] = 0.064, a2 =l.922, and a3 = —1.79,
were measured by Allen (1994) and were used in the modeling study reported by Kim et

al. (1998). Figure 2.2 shows that the linear isotherm based on the GAB-parameter a1

significantly over predicts the concentration of moisture at the solid/air interface

for AW > 0.03.

2.4 Water Absorption Phenomena in the Product Phase

Water absorption by solid materials (esp., pharmaceutical drugs) surrounding by
humid air involves two steps: I) water adsorption onto the surface; and, 2) subsequent
diffusion of water through the solid phase. Moisture adsorption is characterized by a
sorption isotherm. The use of an isotherm at the solid/gas interface assumes that the solid
interface is in thermodynamic equilibrium with the contiguous gas phase. The isotherm is

an algebraic (usually non-linear) equation that relates the concentration of moisture at the

14

M

 

 

 

 

 

g H20 i i I

g dry product 3 i i
monolayer ; i capillary . i
adsorption 5 multilayer adsoprtion g condensation g

6 AW 5 i

M=Aw/a1 E @X=I,M(X,t)= i :

/ i a3Aw+a2Aw+a1 l i

I 2 s i

I 2 2 2

/ i E a

/ a e e

/ g : g

I z a

/ 5 3

/ g s

/ i :
/ = _ = = e
I a3 1.79, 32 1.922, a1 0.064 :

 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

A

Water Activity A ,RH

W

Figure 2.2 GAB Moisture Adsorption Isotherm.

15

surface in the solid product phase to the concentration of moisture at the surface in the
adjacent gas phase. During the absorption process, the concentration at the interface
changes with time until an equilibrium is established between the bulk solid and bulk
gasphases. Local equilibrium at the interface occurs instantaneously, but equilibrium
between the bulk phases may require years, depending on the governing transport

phenomena.

Various mathematical equations have been reported in the literature to account for
moisture sorption phenomena in food and in pharmaceutical products (see, esp., Labuza
et al., 1972, Khanna and Peppas, 1978; Peppas and Khanna, 1980; Peppas and
Sekhon,1980; Peppas and Kline, 1985; Khanna and Peppas, 1982; Peppas and Kline,
1985; Howsmon and Peppas, 1986; Smith. and Peppas ,1991; Anderson and Scott, 1991;
Masaro and Zhu, 1999;and, Badawy et al., 2001). Diffusion through the solid product
phase is by gradient diffusion with a diffusion coefficient that may depend on the local
thermodynamic state (i.e., temperature and moisture concentration). Earlier sorption
models (for porous solids) assumed that the moisture content of the solid phase quickly
adjusted to the moisture content of the surroundings and that Shelf-life was primarily
controlled by external resistance to mass transfer. Over the past twenty years, a
significant amount of work has emphasized physicochemical phenomena associated with

the solid product phase (see, esp., Allen, 1994, Kim et al., 1998).

Allen measured the diffusion coefficient and the adsorption coefficients for

Deltasone® tablets. The one dimensional boundary value problem for the tablet (i.e., flat

I6

plate geometry) surrounded by humid air at constant concentration is described in
Appendix E. The boundary value problem was solved analytically by using standard
separation-of-variable techniques (see, Rice and Do, 1995; Bird et al., 2002) and
computationally by using a commercial PDF solver based on a finite element method
(COMSOL MULTIPHYSICS®). Allen used the results from Crank (1975) to estimate
the diffusion coefficients from the experimental data. For the analogous heat transfer
problem, see the classical paper by Gurney and Lurie (1923). Figure 2.3 compares the
theoretical prediction of the volume average moisture concentration with the

experimental data reported by Allen (1994). The characteristic diffusion time for the solid
phase, LZS/DS, is about 11 hours. The thermodynamic and transport properties of

Deltasone® tablets measured by Allen will be used as a reference case for the parametric

study summarized in Chapter 4.

2.5 Water Absorption Phenomena in the Polymer Barrier Phase

Capillary transport and activated diffusion are the two main modes of mass
transport through polymers (Brandrup et al., 1999). Capillary transport involves the
passage of molecules through pinholes and/or very porous media such as cellulose and
glass. Activated diffusion essentially involves three steps: the first step involves the
adsorption of the diffusing species onto the non-porous polymer film; the second step
involves diffusion through the film due to a concentration gradient; and, the final step
involves desorption from the film surface. Thus, mass transport across the polymer
barrier is controlled by adsorption, diffusion, and desorption. Diffusion can be Viewed as

a series of activated jumps from one cavity to another in a water matrix. The rate of

17

0.9 4

0.8 4

 

 

E]

   
 

------- 1 eigenvalue
—-— 1O eigenvalues

El COMSOL

Lp = 1.96 mm (half thickness)

A5, = 0.92; M,o = 3.514 g HZO/IOO g dry product

D3 = 0.996x 10 ’6 cmz/s
Eigenvalues and Fourier coefficients are listed in Table E.1

 

"VI “_r—*— T‘T—Ti T "_l'—'f ‘—T—T—TT _T_—'T—V flvi

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Figure 2.3 Absorption of Water by Deltasone® Tablets at Room Temperature.

18

diffusion is proportional to the number of cavities in a polymeric film. The presence of a

plasticizer increases the diffusivity; crystalline polymer structures tend to decrease the

diffusivity.

The polymeric film used by Allen (1994) was a composite of two polymers: PVC
and ACLAR®. The diffusion coefficients for both polymers were measured in the same
fashion as the pharmaceutical product by measuring the moisture uptake in the presence
of humid air. The rate data were used to estimate the diffusion coefficient. The diffusion
coefficients for both polymers were reported by Allen (1994) and verified in this study by
using the analysis briefly described in Appendix E. Figures 2.4 and 2.5 illustrate the

comparison between the experimental observations and the theoretical results.

2.6 Blister Packages

The stability, physical condition and potency of a drug tablet may be threatened
by exposure to moisture, heat, oxygen and light. Many drugs, such as penicillin, are
completely inactivated by exposure to moisture. Clearly, a well designed package is an
essential step in the distribution and use of these products. Typical barrier strategies
commonly employed in packaging include impenetrable moisture barriers (aluminum),
coated product pills, and permeable plastic films. Barrier polymer films or plastics can be
used to produce clear pre-formed plastic packages, which are shaped like a blister. These
blister packs store the drug pills and are sealed at the open end with metal foils such as
aluminum. Blister packs are also known as push through packs in some parts of the world,

because of the technique used to remove the tablet from the blister pack. Blister packages

19

1 a

09*

089

O]

06~

05-

04~

034

02*

01*

 

l

 

LP = 0.095 mm (half thickness)

AF, = 0.77; M00 = 0.133 g H20/100 g dry product

D1) = 5.04><10'9 cm2/s;
Eigenvalues and Fourier coefficients are listed in Table E.1

 

 

0 ‘F-‘T——T———T__T—-—T——T_ f I I T I I I __T_—T_‘T—fl— _T—’fi

000102030405060708091011121314151617181920

Figure 2.4 Absorption of Water by PVC Film at Room Temperature.

20

0.9

0.8 -

0.7

0.6

0.5

0.4 -

0.3

0.2 -<

0.1

 

    

— 10 eigenvalues

A Experimental (Allen, 1994)

LS = 0.02 mm (half thickness)

A5] = 0.58; M00 = 0.121 g H O/100 g dry product

2

DS = 7.03x10"° cm2/s

Eigenvalues and Fourier coefficients are listed in Table E.l

 

 

I I T T I I | I j I l I I I I I I I I I

0 0.10.20.30.40.50.60.70.80.9 1 1.11.21.31.41.51.61.71.81.9 2

Figure 2.5 Absorption of Water by AC LAR® Film at Room Temperature

21

have several advantages inasmuch as they prevent cross contamination of the drugs at
various stages and they reduce abrasion of the tablets during distribution. Blister
packages also reduce the chance of overdose for over-the-counter drugs by printing the

days of the week above the dose. They also save storage Space compared to bottles.

Blister packs are hard to be tampered with, and the transparency of the polymer
films ensures easy identification of the drug tablet. Figure 2.6 illustrates the idea of a

blister package for a drug tablet.

2.7 Shelf Life Metric

Water is a main component of many pharmaceutical products and hence it plays
an important role in determining their physical and chemical properties. Water controls
mass transfer rates, microorganism activity, and rates of various chemical reactions.
Hence, the quality of a pharmaceutical product may be influenced significantly by either
a gain or a loss of moisture. Drug quality is determined by the chemical reactivity of the
product as a function of time and the environmental conditions. Moisture can act as a
solvent and cause a dilution effect on the substrates, which in turn can lead to increased

reactivity.

The shelf life of a blister package containing a moisture sensitive drug tablet (or
pill) is defined as the time needed for the product phase to acquire an unacceptable
volume average moisture concentration. The shelf life will depend on a number of factors

including: 1) The initial moisture concentration of the product phase, the air-gap phase,

22

and the polymer barrier phase; 2) The temperature and the relative humidity of the
surrounding (environmental) phase; 3) The thermodynamic and transport properties of
the constituent phases of the blister package; and, 4) The surface area and the volume of
each constituent phase (i.e., product, air-gap, and polymer barrier). The manufacturing
and processing conditions will determine the initial conditions of the constituent phases
and may influence the transport properties of the polymer barrier phase during thermal
forming. Because of this possibility, it is important to use extreme environmental testing
(i.e., short time experiments as mentioned in Section 1.1 above) to confirm the
thermodynamic and transport properties of the polymer barrier. The shelf life for a
blister package can be estimated empirically by simply storing packages under given
conditions until the volume average moisture concentration of the drug tablet acquires an
unacceptable value. Accurate estimates for the shelf life can also be made by using
“accelerated testing” in combination with mathematical simulations. “Storage testing” is
obviously expensive and time consuming, but this approach is presently mandated by the
FDA as an integral part of a New Drug Application (see USP XXII “Stability
Considerations in Dispensing Practice”, Current Good Manufacturing Practice for
Finished Pharmaceuticals” 21 CFR). This paradigm has also been employed for testing

food packages for more than forty years (see Karel , 1967).

Allen (1992) explored the possibility of using an “accelerated testing” protocol
for blister packages under extreme storage conditions of relative humidity and
temperature. This high-rate absorption data can be used to evaluate the thermodynamic

and transport properties of constituent phases within a blister package. A mathematical

23

model can then be used to predict the time required for an unacceptable condition to
occur in the product phase. The prediction of shelf life at less extreme conditions based
on a mathematical model clearly depends on the assumptions used to develop the model
(see Section 3.2 below). Allen (1994) partially tested this strategy with an absorption
model previously developed by Kim (1992). Although predictions based on the non-
linear absorption model developed by Kim were not completely satisfactory (see pages
66-73 in Allen’s thesis), the non-linear PSSA-model developed in Chapter 3 below

provides an explanation of the blister results reported by Allen.

24

CHAPTER 3

MASS TRANSFER MODEL

25

3.1 Introduction: One-dimensional Mass Transfer Model
The moisture concentration in a drug tablet within a blister package is developed in
this section based on a one-dimensional diffusion model. The model requires the

specification of a single length scale for each phase. As illustrated by Figure 3.1 L3 is
defined as the half-thickness of the tablet (S-phase); Lp is the thickness of the polymer

barrier (P-phase); and, LG is the thickness of the air gap (G-phase). For this geometry,

the Shelf life will depend on the following two geometric ratios,
N1 ELG/LS and N2 ELp/Ls. (3.1a,b)

For blister packages, N1 <1 and N2 [1 1.

3.2 Assumptions
The main assumptions underlying the moisture absorption models are:

a. The product tablet can be treated as a flat plate (neglect end effects and
curvature);

b. Moisture transfer through the product phase and the polymer barrier is by
Fickian diffusion. The diffusion coefficients are assumed to be constant
(weak function of temperature and moisture concentration);

c. The moisture sorption isotherms for the drug tablet and the polymer barrier
are known;

d. The temperature and the relative humidity of the surrounding environment
are constant;

e. Absence of moisture sorption/desorption hysteresis;

f. Moisture concentration in the air-gap (G-phase) is spatially uniform, but time
dependent; and,

g. The initial conditions in the product phase, in the air-gap phase, and in the
polymer barrier phase are in thermodynamic equilibrium with the humid air
in the air-gap phase and the surrounding environment

26

Ar+HzO

 

     

 

 

 

 

polymer barrier
Wage headspm\ /
1 x
215 SG—Interface
*—'—-* solid product phase
= i
l impermeable hacking
i polymer phase
backing l
l > PGirterfam
solid product Ip
pm- 5 _ X
Xs=0 i szi
Xs =1 1

Xp =0

Figure 3.1 Schematic of a Blister Package for One-Dimensional Absorption.

27

3.3 Unsteady and Pseudo-Steady State Models
Unsteady-state absorption occurs by changing the relative humidity of the
environmental phase. In this study, a Product Quality Index (PQI-) function, which

incorporates the major factors that determine the shelf life of a moisture sensitive product,

is defined as

 

s S ~
C —-<C >(t)
<OS>(t).=_ 6“ . (3.2)
cS CS
eq- 0

In the above definition, ng is the steady-state, equilibrium moisture concentration in

the solid product phase (S-phase); CE is the initial moisture concentration; and,

< C S> (t) is the instantaneous volume average moisture concentration:

<CS>(E)E 7/1; (”Cso‘cbdvs. (3.3)

The instantaneous water concentration within the S-phase, C8023) , depends on the

position vector X and the timet ; VS denotes the volume of the S-phase. The

dimensionless POI-function depends on a dimensionless time defined by

DS is a constant diffusion coefficient for water in the S-phase and LS is a characteristic

length scale (half-thickness of the S-phase).

28

The quality of a moisture sensitive product degrades as it absorbs moisture. For t = 0,

the product phase has a moisture content determined by the manufacturing and packaging

processes. The PQI-function is defined so that<OS> (0)51. If the blister package is

exposed to a humid environment not in equilibrium with its initial state, then the PQI-

function relaxes to zero as time increases, <OS> (t) —> 0 as t—) 00. Consequently, the

POI-function decreases monotonically from unity to zero,

OS<OS>(t)Sl , OStSoo. (3.5)

For a one-dimensional model, the PQI-time, which provides an objective metric to
compare different package designs, depends on different dimensionless groups related to
the shape of the package and the physical properties that control the unsteady-state

absorption of water. The critical PQI-time is defined as follows
S _ S
<O >(tc) — <(~) >C ,OStC<oo. (3.6a)

As noted above, the parameter <E‘)S>C depends on Cesq, <CS>C, and CE. The“

moisture concentration of the solid product phase in equilibrium with the surrounding
humid air is designated as Cesq. The critical (i.e., unacceptable) volume average moisture
concentration of the solid product phase is designated as<CS >c- The initial moisture
concentration of the solid phase, which is in equilibrium with the air-gap phase, is

designated as Cg .

For moisture sensitive products, the water concentration must not exceed some

critical limit, < CS >c- If the surrounding humid air leads to an equilibrium concentration

of water in the product phase for which C Eq 2 < CS >C , then the shelf life to must be

selected so that

S S

C s <CS>(i)S<CS>(ic)E<C >c—<-Ceq , OSiStcSoo. (3.6b)

S
o
Ineq. (3.6b) is equivalent to the following inequality for the PQI-function,

[0=<OS>(oo)] u [<OS>(tC)E<OS>C] s <OS>(t) _<_ [<oS>(0)=1]. (3.6c)

The critical concentration < C S>c is clearly product specific and has nothing to do with

either the initial moisture content of the S-phase or the humid air of the surroundings

(Ahlneck and Zografi, 1990). However, as implied by Eq. (3.2), the time needed for the

volume average water concentration in the S-phase to become equal to C§ (i.e.,

Therefore, if Cg , CS

<OS>(tC)E <OS>C) does depend on Cg and CS eq ,

eq '
and <CS >c are all specified, then the shelf life can be determined by solving the

following equation for the PQI-time (5 tc ):

 

 

S S ~ S S
C —<C > t C —<C >
<®S>(tc)5 eq 3 S(°) = eqs 8 C E<os>c. (3.7)

The dimensional shelf life is proportional to the PQI-time and can be expressed as

t t O C

30

Mass Transfer: F ickian Diffusion
In the approach adopted hereinafter, water is transported by ordinary diffusion with a

constant diffusion coefficient in each phase. DS ,DG , and Dp represent, respectively,

the diffusivity of water in the S-phase, G-phase, and P—phase of the blister package (see
Figure 3.1). The dimensionless PQI-time depends on the relative values of the diffusion

coefficients:

N3 5 DG /DS and N4 5 Dp /Ds. (3.9a,b)

The diffusivity ratio N3 is usually very large (N3 [1 104 ); however, N4 <<1 .

External Mass Transfer: Environmental Convection & Diffusion

The transport of water to the external surface of the blister package may be
influenced by forced convection, natural convection, and multi-component diffusion
phenomena (Bird et al., 2002). In this study, the simultaneous transport of oxygen and
nitrogen with water is neglected. The analysis also assumes that the temperature and the
total pressure of the internal and external gas phases are constant and that local heating
effects due to absorption (mixing) are unimportant. In this study, a convective mass

transfer coefficient kc accounts for the resistance to mass transfer near the external

surface of the blister package. Thus, the Biot number for interfacial mass transfer, defined

as

N5 5 Bi 5 kCLP , (3.9c)
DP

 

also influences the PQI-time. The Biot number depends on the environmental Schmidt

number, Sc 2 VB / DE , and an environmental Reynolds number, Re 5 [cue /VE, based

3]

on a characteristic length and velocity scale associated with the surroundings. Previous

studies usually assume thatN5 —> oo . Although this assumption yields the simplification

that the water at the polymer/ gas (PG-) interface is instantaneously at its final steady state
condition, the formal analysis developed hereinafter retains the possibility that the

external resistance to mass transfer may limit the rate of absorption.

Thermodynamic Equilibrium at Interfaces

The water in the S-phase near the SG-interface is assumed to be in thermodynamic
equilibrium with the water in the G-phase near the SG-interface (see Figure 3.1). The
local thermodynamic equilibrium assumption (see Bird et al., 2002; Rice and Do, 1995;
Smith and Peppas, 1991; Kim et al., 1998) is also imposed on the PG- and GP-interfaces.

In this research, the adsorption and desorption isotherms are equal (i.e. no hysteresis).

At steady state, thermodynamic equilibrium applies everywhere and the water
concentration is spatially uniform in each phase. The temperature-dependent,
thermodynamic distribution coefficients at the SG-interface and the PG-interface for

small values of AW (i.e., low relative humidity) are denoted as KSG and Kpg. These

two parameters, which influence the POI-time, are used in this research to account for the

discontinuity in water concentration between phases:

CS()A(SG,1): KSG Cg(xsg,t) (adsorption) (3.10a)
C p( )2 gp,i) = KGP C60? (313,1) (desorption) (3.10b)
CPO? p0,1) = KPG COO? p0,1) (adsorption). (3.100)

32

The thermodynamic distribution coefficients KSG sKGP and K136 are temperature

dependent and could vary significantly. In general, KpG / KSG U 1.

The adsorption isotherm for the drug tablet is given by the Guggenheim-
Anderson-deBoer (GAB-) sorption isotherm (see Eq.(2.6) and Figure 2.2). For small

values of Aw:

AW AM 1

M = 2 > AW ,KSG =L (3.10d)
a3 Aw +32 AW +a1 al 31
In the above equation, Aw is the activity of water in the air gap phase (0 S Aw S 1) and

the dimensionless coefficients a], a2 and a3 are the product specific GAB-parameters
(temperature dependent, but independent of Aw ). In summary, the dimensionless PQI-

time defined by Eq.(3.7) depends on the foregoing dimensionless parameters and the

critical PQI coefficient, < O S>C . Thus,

tc=F(N1,N2;N3,N4,N5;KSGaKPG;@§) (3.11)
where N] and N2 characterize the geometry of the blister package; N3 , N4 , and N5

relate to relative mass transfer influences; and, KSG and KPG are dimensionless

thermodynamic distribution coefficients.
Table 4.2 in Chapter 4 shows the range over which each of the above independent

dimensionless groups may vary for blister packages. Eq. (3.11), albeit complicated,

provides a means to objectively compare the performance of different blister package

33

designs. As demonstrated in the next section, this relationship simplifies considerably by

using a pseudo-steady state approximation for the G- and P-phases.

3.3 Pseudo Steady State Model for the PQI-time

Eq.(3.11) simplifies significantly if a pseudo-steady state assumption is used to
approximate the transport of water across the thin polymer barrier and the air gap (see
Appendix B). The mass transfer resistance between the surroundings and the outer
surface of the blister package is included in the analysis by using a convective mass
transfer coefficient. Appendix A defines the initial boundary-value problem in terms of
the following concentration differences between the final equilibrium water concentration

and the local instantaneous water concentration in each phase:

 

 

 

S S
C —C (X,t)
OS(X,t)_ 6“ , OSXSSl (3.12)
S S
Ceq Co
G G
C -—C (X,t)
OG(X,t)E 6“ , OSXGSI (3.13)
G G
P egg—epoch
9 (X4); , OSXP s1. (3.14)
P P

The dimensionless time is defined by Eq.(3.4) and the dimensionless spatial
position is X(EK/LS). Figure 3.1 illustrates the geometry for the one-dimensional

mass transfer problem. The POI-function, defined by Eq.(3.2), is related to Eq.(3.12) by

integrating over the solid product phase:

34

1
<OS>(t)E jos(x,t)dx. (3.15)
0

At t = 0, the concentration difference functions are spatially uniform and equal to unity

within their designated domain (i.e. phase). As t —> oo , these functions relax to zero.

3.4 Linear Adsorption - Analytical Solution
With the pseudo-steady state assumption (see Appendix B), the difference

function in the solid product phase can be represented in terms of the eigenfuctions
associated with the initial boundary-value problem defined by Eqs. (B.1)-(B.4):

CD
OS(X,t) = Z An cos(x,,X)cxp(—7t,2,t) , 0 3 x3 :1 , t> 0. (3.16)

n=1
The eigenvalues associated with the eigenfunctions (i.e.,cosOtnX) , n = l,2,3,...) are

the roots to the following transcendental equation:

Antan(7tn)=ot , n=l,2,3,.... (3.17)

The dimensionless group O. in Eq.(3.17) compares the mass transfer resistance of the
product phase with the barrier phase. This dimensionless group is a composite of the

seven independent groups discussed above and is defined by

LG Lp 1
+ + ——
{ DG KpGDp kc } _ {mass transfer resis tan cc of the barrier phase} (3.18)

 

 

l
or { LS } {mass transfer resis tan cc of the product phase}

35

Physically, l/ot compares the series resistance to mass transfer due to the air gap,
the polymer barrier, and the external surroundings to the resistance to mass transfer in the
solid product phase. A package design with or >> 1 corresponds to a situation where the
mass transfer resistance of the product phase is large compared with the mass transfer
resistance of the other phases. Under these conditions, the PQI-time is relatively small
and, consequently, the shelf life of the blister package is relatively short. Blister packages
with or <1 have desirable barrier characteristics. The ideal situation is to have or = 0(no
penetration of external water), but this is not practical. Therefore, the relationship
between the eigenvalues and the design parameters provides a quantitative means to
identify a practical design. Table 4.3 (see Chapter 4) tabulates the first few roots to

Eq.(3.17) for different values of 0t. The corresponding Fourier coefficients An in

Eq.(3.16) are also given in Table 4.3.

The POI-function can be calculated directly from Eq.(3.16) by integrating over the

S-phase:

oo 1 00 .
< GS > (t) = Z An[ [cos (AnX)d xleXP(-ltr21t) = Z An SInOtn)

n=1 0 n=1 n

cxp(—>.,2,t). (3.19)

 

For t >> 0, the above representation is accurate with only a few terms in the series.
However, for very short times, many eigenvalues must be included in the series

representation. For long times, the smallest eigenvalue controls the relaxation of Eq.(3.19)

 

with the result that
m < o3 > (t) ——> < OS > (tc) E < OS >C ; A1 3mg“) exp(—7l.12tc). (3.20)

t—->tC

36

Thus, for blister package designs that have a relatively long shelf life, the PQI-time can

be calculated by using the asymptotic equation given by Eq.(3.20). Therefore,

 

to = —171n[A1 SIn(AS.1)/A1

] (2 PQI-time). (3.21)
Eq.(3.21) shows that the PQI-time depends only on the smallest root of Eq.(3.17), which
provides the essential theoretical link between the POI-time (i.e., the shelf life) and the

design of the blister package.

3.6 Linear Adsorption — Numerical Solution

The unsteady-state boundary value problem defined in Section 3.5 above (also see
Appendix B) can be solved numerically. For this purpose, COMSOL MULTIPHYSICS®
was used. This commercial software supports the simulation of partial differential
equations and boundary conditions commonly found in heat, mass, and momentum
transport. It is based on a finite element method. “Mass Balance - Diffusion — Transient”
analysis was the model selected under the chemical engineering module. A one
dimensional flat plate of unit thickness (dimensionless) was developed. The partial
differential equation, the initial conditions, and the boundary conditions were provided as

user inputs. The results are summarized in Chapter 4.

3.7 Unsteady State Model for the PQI-time

Water is transported by ordinary diffusion with a constant diffusion coefficient in

each phase. DS ,DG , and Dp represent, respectively, the diffusivity of water in the S-

phase, G-phase, and P-phase of the blister package (see Figure 3.1). The dimensionless

37

PQI-time will also be influenced by the relative sizes of the three phases: Lp << LS,L(3.

Diffusion in the polymer phase is one-dimensional and the geometry is flat. The moisture

distribution is governed by the following differential equation,

P 2 P
a—Cr—sza—AC— , OSXpSLp , 1>0. (3-22)
at axfi

Diffusion in the solid product phase is also one-dimensional and the geometry is flat,

S 2 s
flaps—679— , OSXSSLS , {>0 . (3.23)
at axé

The geometry of the air gap is complicated. In general, the concentration of water

in the G-phase is governed by 3D unsteady-state diffusion:

G
6—61— = DGV2CG . (3.24)

Application of the divergence theorem and the condition that the water flux is continuous

across an interface gives (Kim, 1998)

d<C>G 6C1) G

dt GP Paxp

H)
V
O

 

-SSGDG ac: (3.25)

.. 6X3

P—

VG

 

 

is =1
The above equation assumes that the flux is spatially uniform over the 2D interface. SSG

represents the total interfacial area (both sides of the symmetrical tablet) available for

mass transfer between the S-phase and the G-phase (air gap); SGP represents the

interfacial area available for mass transfer between the G-phase (air gap) and the P-phase.

The parameter SBG represents the interfacial area between the backing and the G-phase.

38

This area does not appear in the above equation because the moisture flux across the
backing is zero. If the concentration of water on the gas side near the interface equals the
spatial average concentration of water in the G-phase (i.e., well-mixed assumption,

DG 1] Ds,Dp ), then
CS(LS,i)=KSG <cG >(t) (3.26)

CP(0,’t‘) = KpG < CG > (t) (3.27)
The continuity of flux across the two internal interfaces has already been incorporated
into the macroscopic equation for the G-phase given above (see Eq.(3.25)). The water in
the S—phase near the SG-interface is assumed to be in thermodynamic equilibrium with
the water in the G-phase near the SG-interface (see Figure 3.1). The local thermodynamic
equilibrium assumption (see Bird et al., 2002; Rice and Do, 1995; Smith and Peppas,
1991; Kim et al., 1998) is also imposed on the PG- and GP-interfaces. At steady state,
thermodynamic equilibrium applies everywhere and the water concentration is spatially
uniform in each phase. The temperature-dependent, thermodynamic distribution

coefficients at the SG-interface and the PG-interface are denoted as KSG and Kpg.

Thermodynamic equilibrium and continuity of flux across the polymer/environmental

interface implies that

 

CP(LP,i) = KpoCEOZE = 0,2) (3.28)
P
Dpég— = D 6C (3.29)
6X1) . E 6X13 . ..
XP=LP XE:O (XPZLP)

 

 

The flux in the surrounding gas (or environmental) phase (E-phase) is determined by

introducing a convective mass transfer coefficient:

39

‘DE _,__

E . E
2k C O,t—C] 3.30
6X5 c ( ) oo ( )

25:0 (xp=L")

 

The following condition for disk geometry assumes that the water concentration in the air

gap is spatially uniform.

 

= (disk --- symmetry) (3.31)

 

 

2
L /Dp
% E g (3.32)
LS/Ds
v K
NS 2-3—SL3. (3.33)
VG
v K
Np a—P—LG— (3.34)
VG

N12: compares the characteristic diffusion time of the P-phase with the characteristic

diffusion time of the S-phase. The Biot number for interfacial mass transfer, defined as

 

Bi 2 , (335)

also influences the PQI-time. Previous studies usually assume that Bi —> oo . Although

this assumption yields the simplification that the water at the PG-interface is
instantaneously at its final steady state condition, the analysis developed hereinafter
retains the possibility that the external resistance to mass transfer may limit the rate of

absorption.

40

In summary, the dimensionless PQI-time introduced by Eq. (3.7) above depends on
the foregoing dimensionless groups and the critical PQI coefficient, OE. Thus,

tC =F(NS,Np,NT,Bi,®§) (3.36)

Figure 3.1 illustrates the geometry for the one-dimensional mass transfer problem.

The PQI-function, defined by Eq.(3.2), is related to Eq.(3.12) by integrating over the

solid product phase:
1

<o>3 (02 joS(xS,t)de. (3.37)
0

At t = 0, the concentration difference functions are spatially uniform and equal to unity
within their designated domain (i.e. phase). Ast —+ 00, these functions relax to zero. As
indicated in Appendix C, the difference function in the solid product phase can be
represented in terms of eigenfunctions associated with the initial boundary-value problem

defined by Eqs. (C.l)-(C.4):

S 00 S
O Fn 2
P = Z An P exp(—7tnt) , t > 0 (3.38)
S _ S . S
Fn (XS) — B1n smOtnXS) + an cosOchS) (3.39)
P _ P - P
Fn (Xp) — B1n srn(NT}tnXp)+ an cos(Nt?tnXp) (3.40)

The eigenvalues associated with the eigenfunctions are the roots to the following

transcendental equation:

3 (An) 2 7t.n+NStan(}tn)+NP (NtkntangNtkn)_Bl) = 0 (3.41)
NT (NTAn +BItan(NIAn))

 

41

3.8 GAB Adsorption - Numerical Solution

The use of a GAB-isotherm (see Eq.(2.6) above) at the product/air-gap interface
makes the boundary value problem non-linear. The presence of this non-linearity
warrants the use of a numerical method. A finite element solver supported by COMSOL
MULTIPHYSICS® was used to develop numerical solutions to the boundary value

problem. See Appendix D for the set up and Chapter 4 for the results.

3.9 PSSA Model with a GAB Isotherm -- Experimental Validation
The pseudo-steady state model was validated by comparing the predicted results

with the experimental results reported by Allen (1994). Allen conducted sorption

experiments on 20 mg Deltasone® tablets at room temperature (25°C) exposed to
humid air with a relative humidity varying from 0 to 92%. Fourteen sorption experiments
were conducted on the product tablets. The diffusion coefficients and the GAB-
coefficients were determined from this data. The diffusion coefficients reported in
Allen’s thesis were confirmed as part of this study (see Section 2.4 and Appendix E). The
diffusion coefficients are functions of the temperature and the local moisture
concentration. Allen used an average diffusion coefficient to analyze the shelf life of

blister packages based on a finite difference model developed by Kim (1992).

The polymer used for the studies was a laminate of PVC and ACLAR®. Allen de-

laminated the polymer composite to obtain separate films for direct testing. Both polymer

films were individually studied for moisture uptake. The data obtained were used to

42

determine the diffusion coefficients of the polymer films. Allen estimated an effective

diffusion coefficient of the polymer laminate barrier based on the following expression:

L L
PVC + ACLAR = LP. (342)
DPVC DACLAR DP

 

 

In the above equation, Dp represents the diffusion coefficient of the composite polymer
barrier; Lp is the thickness of the polymer barrier; vac and L ACL AR represent,

respectively, the thickness of the PVC and the ACLAR® films; and,
DPVC and D ACL AR represent the diffusion coefficients of the two individual polymer

films. The thermodynamic moisture distribution coefficients (i.e., Henry’s law constant
for linear adsorption isotherms) for the polymer/environment interface were also
determined from the moisture sorption data. A confirmation of diffusion coefficients
reported by Allen for the polymer phase was carried out in the same fashion as the

confirmation for the tablets (see Section 2.5 above).

The moistUre sorption data obtained from the pseudo-steady state model with the
GAB-isotherm was compared with the experimental data reported by Allen (1994). The
parameters used in the model to generate the moisture sorption results were also selected
from Allen’s work. The data used in the comparison were for samples tested at a relative
humidity of 80% and a relative humidity of 90%. Table 4.1 summarizes the
thermodynamic and transport properties reported by Allen(1994). These parameters were

used as the “base” case for the parametric study presented in Chapter 4.

'43

CHAPTER 4

PARAMETRIC STUDIES OF THE PSEUDO-STEADY STATE

ABSORPTION (PSSA-) MODEL

44

4.1 Introduction

The pseudo-steady state absorption (PSSA) model can be used with linear and
non-linear sorption isotherms. The effect of the various environmental and physical
parameters on the sorption of moisture through the blister packaged pharmaceutical

product at various storage conditions is discussed in the following sections.

4.2 Input Data for the PSSA Model

The moisture sorption isotherms at 25°C for the tablet and for the polymer barrier
were previously determined by Allen (1994) as summarized in Chapter 3 above. The
physical and environmental parameters used in the PSSA model are listed in Table 4.1

below.

4.3 PSSA Model --- Linear Isotherm

The pseudo-steady state absorption model with linear isotherms is defined in
Sections 3.3 and 3.4 above as well as in Appendix A and Appendix B. Table 4.2 shows
the range over which each of the dimensionless groups governing the linear PSSA model
was varied. Table 4.3 gives the first three eigenvalues and Fourier coefficients for
resistance ratios in the range: 0.1Sot S 0.5. The results Show that the eigenvalues are

well separated, which partially justifies the use of Eq.(3.21) for the shelf life estimate.

45

 

Table 4.1 Base Case Parameters for Component Phases (Allen, 1994)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Component Symbol Values
Phase
External T 25 C
E ' t 0 ° '
nVIronmen AW x 100 /o., ./orelat1ve 8 0% a 90%
humIdIty
DP (average) 2><10'9 cmZ/s
Polymer Lp (full thickness) 0.0023 cm
Composite 3
Barrier
(PVC and KPG = KGP (composite) 125 (g H20)/(cm p3olymer)
ACLAR®) (g H20)/(cm air)
Vp, total volume 30 mm3
Ds 0.996><10'6 cmz/s
LS (half thickness) 0.196 cm
GAB parameters (—1.79,+1.92, +0.064)
Deltasone® (a3, a2, all )
Tablet _1_ : 1 6 (g H20)/(cm3 product)
KSG 91 (g H20)/(cm3 air)
V5, volume 308 mm3
Air Gap v0, volume 492 mm3, LG = 0.02, mm

 

 

 

46

 

 

Table 4.2 Scope of Parametric Study for the Linear PSSA Model

L3 .2. 0.196cm(ha1fthickness) , DS 510—6,cm2 /s, L2

IDS 511 hours

 

 

 

 

 

 

 

 

 

 

 

 

 

(see Eq.(3.18))

 

 

 

S
PSSA-Parameter Base Sco e Reference
(Table 4.1) 1’
@§ 0.5 0 < 6% <1 Figure 4.1
N1 5 LG /LS 0.6 N1 <<l Figure 4.1
< <
N2 5 Lp /L3 0.1 0'00] ’ N2 ' 0'1 Figure 4.1
N3 5 DG /DS 00 N3 = 00 -
N4 a DP /DS 0.002 0.0001 3 N4 S1 Figure 4.1
BiENSEkch/Dp oo N5=oo -
N6 5 KSG 16 N6 =16 -
N7 EKPG 125 N7 =l25 -
N7 /N6 7.8 N7 /N6 = 7.8 -
L K D 1 N
or E-—S PG P = 7N Figure4.1
LP KSG DS N2 N6 0 15 0 001 < < 100 Figure 4‘2
' ° _ or — Table 4.3
Table 4.4

 

47

 

 

Table 4.3: Eigenvalues and Fourier Coefficients for the Linear PSSA-Model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
Icos(}ch)dX
An tanOtn) = or An = 10
IcoszOtnXMX
0
on An An
0.31106 + 1.0161
0.1 3.1731 —0.019661
6.2991 + 0.0050275
0.43285 + 1.0311
0.2 3.2039 — 0.038153
6.3148 + 0.009977
0.5218 + 1.045
0.3 3.2341 — 0.05554
6.3305 + 0.014844
0.59325 + 1.058
0.4 3.2636 — 0.071899
6.3461 + 0.019633
0.65328 + 1.0701
0.5 3.2923 — 0.087281
6.3616 + 0.024338

 

 

 

 

 

 

48

 

Figure 4.] illustrates the temporal relaxation of the PQI-function for different
values of the resistance ratio a. The graph is generated by using Eq.(??.21). For a blister

S

package withot = 0.1, Figure 4.2 shows that for <O > E 0.5 , the dimensionless time

ta 8. Therefore, if the PQI coefficient OE = 0.5 and if Lg / DS = 11 hours (see Table
1), then the shelf life predicted by the pseudo-steady state model is about 3.5 days. For a

fixed value of OE , the shelf life clearly increases as or decreases.

Figure 4.2 illustrates the variation of the PQI-time with on. AS expected, the PQI-
time increases as the resistance ratio or decreases for a fixed value of O§ (= 0.5). In the

“barrier” regime (ot < 1), small changes in blister design can have a significant influence
on the shelf life of the product. On the other hand, for ot > 1, “large” changes in the

blister design have little effect on the shelf life.

4.4 PSSA Model (GAB Isotherm)

Figure 4.3 and 4.4 compare the experimental packaged tablet moisture content
curves (Allen, 1994) with the curves generated by the PSSA-model (GAB isotherm)
using COMSOL MULTIPHYSICS® for different values of (1 at relative humidities of 80
& 90%. This result shows a marked difference in the sorption data between the pseudo
steady state model and the experiments. One variable which was not accounted for was

the diffusion coefficient of the laminated polymer in the formed blister, which may have

49

 

Table 4.4: Eigenvalues and Fourier Coefficients for Linear Conjugate Absorption

(see Appendix C)

 

4n +NS tan(kn)+

 

= 0
Nr (NT).n +Bimtan(NtAn))

 

An

1 1
NS jFdes +Np jFdep + F§(I)
0 0

l l
Ns j(Fr§)2dXS +Np [(#52de + F§(I)Fri)(0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0
Bim soo;NS=l2;Np :11
PS __ cosOchS)
cos A
NT =1.— 3’
S P Fp _ (Ntkn tan(N,xn)—Bim)
n _
a = Np _ 11 (NTAn +Bim tan(NT}tn))
NgNs 12N¥ x(sin(NtAnXp)+cos(NT}tnXp))
N1: (1 n An An
1 0.7474 +0.9191
1 0.917 2 1.5708 -4.68E-6
3 2.3274 +0.2750
1 , 0.2671 +1.0282
3 0.102 2 1.0472 -1.11E-5
3 1.0967 -0.0806
1 0.1344 +1.1007
6 0.025 2 0.5236 4.93135
3 0.5607 -0.1217

 

 

 

 

 

50

 

 

 

 

0.9 ~
<®S> (t)
0.8 ~
0.7 4
0.64 tC
0.5 ----- — -— -_------ (1:0.05 .----_---_____--_,________-________-______
: \
0.4 - : S
I @C
0.3 4 ,
E a=0.l
0.2 4 E
1 ot=015
0.1 ~ E
: (1:0.2
0 ....'...a=025.¥.m. .
o 5 15 20 25 30 35 40

Figure 4.1 The Effect of or on the Volume Average Moisture Concentration of the Drug
for the Linear PSSA-Model.

51

t =—1—1n Al sin(}tl)/7tl

 

C 2 S
5
)‘1 69¢

 

 

(1

Figure 4.2 The Effect of ot on the Shelf Life for the Linear PSSA-Model (OE = 0.5 ).

52

0.9 r

0.8 —

0.7 a

0.6 ~

0.5 7

0.4 ~

0.3 r

0.2 ~

0.1 i

 

 

6: 0.15292 //

a = 0.07646
/ /

or = 0.03823

o.=0.01911

0. = 0.009558

r ------------------------------
I

g E ° Experimental (Allen, 1994) g

G
= keffLSCeq

. (1
Dsps

L L

 

 

25 50 75 100 ke‘léfi H58 Ki)5GDfiOO kc 225 250

t:t_P§.
2
LS

Figure 4.3 Comparison of Experimental Results with the Nonlinear PSSA-Model (GAB

Isotherm) for AW = 0.80 (see Appendix D for the experimental parameters).

53

0.9 —

0.8 ~
0.7 « Mt
M
0.6 ~ 00 o
a = 0.03823
0.5 ~

01 = 0.01911
0.4 4

E - Experimental (Allen, 1994) E

0.3 .
0.2 4

0.1 ~

 

 

0 10 20 30 40 50 60 70 80 90 100110120130140150160170180190 200 210 220 230 240 250

R

t = ———DS
2
LS
Figure 4.4 Comparison of Experimental Results with the Nonlinear PSSA-Model (GAB

Isotherm) for Aw = 0.90 (see Appendix D for the experimental parameters).

 

 

54

undergone structural changes during the forming process itself on account of the heat and
stress involved. A more detailed comparison of the experimental and the calculated
results was undertaken with the diffusion coefficient of the package as a variable. This
result is also illustrated in the figures. As seen from the results, the experimental results

imply that for or = 0.019.

Figure 4.5 shows the experimental sorption of moisture in the tablet, the PVC, ACLAR®
and the blister package containing the tablet at 80% RH. On close evaluation of the
moisture sorption data for the tablet, the polymer and the blister packaged tablet; we can
see a clear anomaly in the time required by the moisture to reach its equilibrium
concentration. The difference is all the more magnified for the particular example chosen.
The time required for the moisture in the blister packaged tablet to reach its equilibrium
value is around 10,000 hours while the tablet, PVC and the ACLAR reach equilibrium in
10, 4 and 2 hours respectively. This result prompts a question as to why when the two

entities i.e. the tablet and the polymer which both have an equilibrium time of nearly a
day each are put together to form a package the equilibrium time for the moisture in the
blister packaged tablet increases to almost one year. When the two systems in question
are compared the only additional factor in the blister packaged tablet is the presence of an
air gap between the polymer and the tablet phase. Does the air gap phase cause this
difference? Examination of the diffusion coefficients in the three phases says otherwise.
The diffusion coefficient in the air gap phase is very negligible compared to the polymer
and the product phases which is a direct validation of our initial assumption that the

moisture concentration in the air gap phase is not variable and can be assumed as

55

~ 0.8

. 0.7

    
 

‘ [:1 A A Tablet
El PVC
A ACLAR
— Blister Package

~ 0.6

r 0.5

~ 0.4

r 0.3

0.2

L0.1

 

 

 

 

 

,____ .._- T ., . . fl 0

0.1 1 10 100 1000 10000 100000

t , hours

Figure 4.5 Moisture Absorption by Individual Constituents ofa Blister Package
(nonlinear PSSA-model AW = 0.80 ).

56

33

spatially uniform. This result still does not rule out the influence of the air gap phase.

Hence, a detailed study of the moisture distribution in the air gap phase was warranted.

Figure 4.6 explains this anomaly by studying the variation of the flux at the tablet
air gap interface with time in the blister packaged tablet. We can see that the flux
increases rapidly initially and then slowly approaches its equilibrium value i.e. zero flux,
which essentially means no more moisture diffusing into the package. Since the diffusive
flux decreases exponentially over a long duration of time, the moisture inside the blister
packaged tablet takes a long time to reach equilibrium. The results give us a better
understanding of the role of the air gap phase in protecting the product from moisture
ingress. It takes a long amount of time for the moisture concentration to build up in the
air gap phase even though the diffusion coefficient for the air gap phase is very high; the

air gap phase essentially acts as a time barrier for the moisture
Figure 4.7 illustrates the variation of moisture sorption in the blister packaged

tablet as a function of the relative humidity of the storage environment. As expected, the

moisture sorption increases with an increase in the environmental RH.

57

 

0.00E+O 1.81E+3 3.625'1-3 5.43E+3 7.24E+3 9.05E+3 1.09E+4
0.” I fl I f

 

-0.002 ~

-0.004 <

-0.006 4

-0.008 4

-0.010 .

~0.012 ~

-0.014 1

 

 

0X

X=l

-0.016

mm

-0.018

Figure 4.6 Transient Moisture Flux at the Product/Air Gap Interface in a Blister Package
for AW = 0.80 (nonlinear PSSA-Model).

58

2.5 —

 

 

 

 

 

 

 

2.0 T /
Mt Aw=09
H O
8 2
/_________._.
1.57 g dry product Aw = 0.8
AW = 0.7
1.0 ~ ——————————— - ---- ................................................
' l
I I I _
' : I AW — 0 5
I I
' l l
: : 1 Aw = 0.3
05 ‘ l r l
I I
l l l i
i i ' Aw = 0 1
' 1 l
I I I
I I I
0.0 IIIfiIIIIgr IEF :11IITIFIiIIIIrTTIIIITITjifrIIIT—T—TI-fl—I—‘Ifi—I
o 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240
1D
t= —S
2
L
S

Figure 4.7 Effect of Relative Humidity on Moisture Absorption of a Blister Package

Based on the Non-linear PSSA-Model (LZS/DS= 11 hours , a = 0.01911).

59

CHAPTER 5

CONCLUSIONS

60

Diffusion coefficient of the packaging polymer is a very important property and
has to be determined accurately especially for multi-laminates. Figure 4.3 Shows that the
PSSA (GAB isotherm) sorption curve agrees with experimental data (Allen, 1994)

for 01=0.01911 , which corresponds to a polymer diffusion coefficient of

Dp = 5X10'10cm2/s. But, Allen used a diffusion coefficient for the polymer laminate of

Dp =2X10'9cm2/s (01:0.03822 ). The diffusion coefficient of the laminate was

calculated based on the thickness and the individual diffusion coefficients of moisture in
PVC and ACLAR films (Crank, 1956) similar to heat transfer through a multi-layered
wall. The effect of the adhesive layer on the diffusion coefficient of the polymer laminate
was not taken into account. This is the reason, the experimental data did not agree with

the moisture sorption calculations from her model.

GAB moisture sorption isotherm parameters for the product which has to be
packaged also determines the ease with which the shelf life time can be assessed for the
system. Certain levels of the environment humidity allow the linearization of the
isotherm with negligible errors which in turn guarantees an analytical solution. An
analytical solution saves us time and resources involved in numerical computations.
Figure 2.2 depicts the GAB isotherm and the corresponding linear isotherm for GAB

parameters, a3 = -1.79, a2 = 1.922, a1 = 0.064 . The linear isotherm for the particular

case chosen always predicts a high value of M corresponding to the storage humidity

(Aw ). Hence, the linear isotherm will always predict a PQI-time which is way lower

than the actual PQI-time because of the presence of a higher amount of driving force (M).

61

Therefore, the linear isotherm is not a good approximation to the GAB isotherm for the

chosen example system.

The time taken for the moisture in a blister packaged product to attain equilibrium
when in storage is of the order of several months to years depending on the product, the
packaging system and the storage conditions. The time taken for the moisture in just the
packaging polymer or in the product alone as a single entity to attain equilibrium is just a
matter or hours or a day (Figure 4.6). This anomaly prompts a question as to whether the
introduction of such a small air gap can increase the equilibrium time to the extent as
shown. The variation of the diffusive flux at the product air gap interface was studied
(Figure 4.7). The diffusive flux is very high initially but reduces exponentially to zero as
time increases, thereby reducing the amount of moisture diffusing into the tablet. This

property of the diffusive flux is what prolongs the shelf life of the pharmaceutical product

Pseudo steady state assumption has been shown to be a valid approach to solve
the boundary value problem analytically. The rigorous analysis of the same is discussed

in section 3.6 (Appendix C & Appendix F).

Long time analysis of the pseudo steady state problem with linear moisture
sorption isotherms yields a Simple analytic expression for the shelf life time of the
product (Eq.(3.21)). This can be used as a simple tool for pre-testing a system before the

actual stability testing for a package is done.

62

PSSA-model comports very well with an earlier comprehensive experimental
study of moisture uptake by 20 mg Deltasone® tablets in blister packages (see Allen, MS
Thesis, The School of Packaging, Michigan State University, 1994). Figure 4.3 shows
that the moisture sorption curves generated by the PSSA model with GAB isotherms
agrees well with the experimental data at 80% RH for a value of 01 = 0.01911. For this
case, the parameter or depends explicitly on the physical and geometrical properties of the
PVC/ACLAR moisture barrier as well as the external mass transfer coefficient for a
stagnant humid air film. Thus, the PSSA-model provides a means to benchmark the shelf
life of a class of blister packages as well as a means to interpret the results of specific

testing protocols. The PSSA-model has been validated using the experimental data (Allen,

1994).

63

CHAPTER 6

RECOMMENDATIONS

64

The effect of pill geometry on the moisture distribution can be studied. A model
analogous to the one developed in the present study which employs the pseudo steady
state condition can be formulated for various geometries like cylinders, spheres etc. The
initial boundary-value problem defined by Appendix A can be formulated as an axi-
symmetric problem and as a spherically symmetric problem. A complementary model
that assumes a pseudo-state approximation for the S-phase and an explicit unsteady state
model for the P-phase can be developed. The problem can be designed to analyze the

impact of curvature on the PQI-time.

The PSSA-model can be used as the basis for developing shelf life design criteria
for blister packages. The design criteria could be used to design a minimum set of
experiments which have to be carried out to generate sufficient moisture distribution data
(short time analysis), which can be used to determine the different variables inithe blister

system i.e. the diffusion coefficients of the individual phases etc.
Shelf life models for products which can be influenced by the other constituents

of the air like nitrogen, oxygen etc. in addition to moisture can be determined by using

the principles of mass transport, those involving multi-component diffusion.

65

 

APPENDICES

66

APPENDIX A. One-Dimensional Diffusion for Linear Adsorption Isotherms

The objective of this appendix is to define the initial boundary-value problem for
one-dimensional absorption in a blister package. The moisture adsorption isotherms at the
phase interfaces are assumed to be linear. Appendix A supports the discussion in Chapter
3 as well as Appendices B and C below. The following dimensionless variables are used
to characterize the moisture concentration in each phase (see Figure 3.1):

S

S
C —C (X,t)
OS(X,t)E eq

S S
Ceq ‘ Co

 

(A.1)

G —CG(X,t)

eq
G G
Ceq " Co

C
OG (X,t) E

 

(A.2)

P
eq

C

—CP(X,t)

P P
eq ‘Co

C
OP(X,t) a

 

(A3)

The above dimensionless concentration difference functions satisfy the following initial
boundary value problem.

Diflerential Equations

 

 

S 2 S
69 =69 , 0SXSXSEI , t>0 (S—phase) (AA)
61 0X2
G 2 G
69 =N3Q—G— , OSXSXGEI , t>0 (G—phase) (A5)
at 5x2
P 2 P
£=N4§—®— , 0SXSXpEl , t>0 (P—phase) (A6)
at 5x2

67

Initial Conditions

The definitions of the dimensionless concentration differences imply that

OS(X,O)51 , osst (A.7)
(fhxbyn ,OSXS] (A&
opocm=1, OSXSI (A9)

Boundary Condition at the Center ofthe Product Phase
The assumption that the moisture distribution in the product phase is symmetric about the
plane of symmetry (see Figure 3.1) implies that

665

———- =0 , t>0. (A10)
ax

X=0

 

Equilibrium and Continuity ofFlux on Internal Interfaces
Thermodynamic equilibrium and continuity of water flux across the SG-interface at

XS = 1 and the GP-interface at XG = 1 require the following four conditions:

 

 

 

 

 

OS = OGI , t>0 (SG—interface); (A.11)
s G
N
2.: = ——3— 29— , t>0 (SG-interface); (A.12)
6X N6 6X
OGI = OPI , t>0 (GP—interface);and, (A.13)
xG=1 xG=1
G P
N N
£19_ 2 4 7 69 l , t>0 (GP—interface). (A.14)
ax N3 ax 1
X0 =1 XG:1

68

Boundary Condition at the PG-Interface
The idea of thermodynamic equilibrium and continuity of water flux across the PG-

interface at Xp =1 implies that

P
69 = _L®P , t>0 (PG—interface). (A.15)
6X X _1 N2N7 Xp=1
P_

 

 

In Eq.(A.15), a Biot number (i.e., N5) is introduced at the PG-interface to account for
convective mass transfer. If N5 N2N7, then OP [1 0 at Xp =1. Physically, this means

that the PG-interface is in thermodynamic equilibrium with the humid air far from the
PG-interface surrounding the blister package (i.e., no external resistance to mass transfer).
The characteristic length and time scales used in the above formulation are

defined as follows

A A

=_X_ 12$. (A.16)
Ls L2S

The geometric ratios are defined as N1 E LG /LS and N2 2 Lp /LS, where LSis the

half width of the symmetric drug tablet (see Figure 3.1). The diffusivity ratios are defined

as N3 2 DG /DS and N4 2 Dp /DS; and, the Biot number is

N5 EEC—LL . (A.17)
DP

The thermodynamic distribution coefficients at the SG-interface and the PG-interface

are N6 EKSG and N7 EKPG (linear adsorption isotherms). Desorption of moisture

from the polymer barrier at the GP-interface (see Figure 3.1) is assumed to follow the

same isotherm as the PG-interface (no hysteresis). For a composite polymer barrier, the

69

adsorption isotherm would be a property of the outer laminate and the desorption

isotherm would be a property of the inner laminate. For this situation

(PVC/ACLAR®), KGP ¢KpG . The parametric study in Chapter 4 assumes that

KGP = KPG-

70

APPENDIX B. Pseudo-Steady State Absorption Model

An approximate solution to the initial boundary-value problem defined in
Appendix A can be developed by assuming that the moisture concentration across the
thin polymer barrier and the air gap rapidly adjusts to the relatively slow unsteady
absorption process associated with diffusion into the solid product phase.
Thermodynamic equilibrium and continuity of water flux across each interface, together

with the idea that the moisture flux is quasi-steady across the polymer barrier and the air

 

gap, yields the following initial boundary value problem for the concentration difference

function within the solid product phase (Bird et al., 2002):

 

Differential Equation
S 2 S
66 =69 , OSXSI , t>0 (8.1)
at (3X2

Initial Condition

OS(X,0)EI , OSXSI (B.2)

Boundary Condition at the Center of the Product Phase

695

— =0 , t>0. B.3
ax ( )

X=0

 

Boundary Condition at the SG-Interface
The pseudo-steady state approximation applied to the air gap and the polymer barrier
yields the following boundary condition at the SG-interface:

aOS

—— : —(X®S , t>0 (8‘4)
6X

le X=1

 

71

 

LG + Lp +_1_
i DG KPGDP kc =N2N6 N1N4 + 1 + 1 1
or { LS } N4 N3 N7 N5

 

 

(B.5)

{mass transfer resistance of the blister package}

 

{mass transfer resistance of the product phase}

If or is a constant (i.e., not a function of the moisture concentration), then an analytical

solution to the linear boundary value problem, defined by Eqs.(B.l)-(B.4), can be

represented as a Fourier series,

OS(x,t) = E An cos(>.nX)exp(—xfit) , os x _<_1 , t> 0. (B6)
n=1

The eigenvalues are the roots to the following equation

411 tan(7tn)=ot , n=1,2,3,... ; and,

(B7)

the Fourier coefficients are given by

l

 

 

Icos(ltnX)dX
An=(1’Fn) s 10 ; Fn(X)=cos(an) (88)
(Faith) 2
Icos (an)dX
0

Base case:

The following physical property data of Allen (1994) define the “base” case for the

parametric study presented in Chapter 4:

D5 = 0.996><10'6 cm2/S ;Dp=2><10'9 cm2/s ,g—P-(a N4) = 0.002
s

72

Lp =9.1mils=0.023 cm ;L3 = 9—ii2— =0.196 cm; %(2 N2)=0.117
S

l
KSG (5N6): 21—1—516;KPG ( 2N7): 125.

If the phase resistances to moisture transport through the air gap and through the

surrounding humid air are negligible, then 0t for the base case is approximately 0.13

LQ/+ Lp +1
)9/6 KPGDP c _N2N6[N1N4+1 1

1
—- +
or { LS } N4 N3 N7 N5

inasmuch as

 

 

 

l

 

: N2N6 _ (0.117)(l6)
_ N4N7 (0.002)(125)
=7.488 2) OL = 0.133 D 0.13

 

The first ten eigenvalues and Fourier coefficients for or = 0.1, 0.2, and 0.3 are tabulated
in Tables A.1-A.3. The following MATLAB® program was written to calculate the

eigenvalues and the Fourier coefficients defined by Eq.(B.7) and Eq.(B.8), respectively.

for z=1:5

y=O;

for i=1:10

while y*tan(y)-(z/10)<=0
y=y+0.0000l;

end

1(Zrilzy;

y=y+2i

end

end

tum=0;

for z=1z5

for j=1:41

for k=1:10

constant(z,k)=4*sin(l(z,k))/(l*(2*l(z,k)+sin(2*l(
Zrklllli

73

end
end

bonstant(z,k)=4*sin(l (
2*l(z,k)+sin(2*l(2 kl)
sum=4*sin(l(z,k))* sin(
1)*1)/(l.hz,k)* (2*1_(z ,k
tum=tum+sum;

z,k))* sin(l(z,k))/(l(z,k)*(
H;

1(2 k-))*exp( (l(z,k)“2)*(j-
)+sin(2*l(z, k))));

end
ans(z,j)=tum
tum=0;
tau(j)=(j—1)*1

 

Table B. 1: Eigenvalues and Fourier Coefficients for the

Linear PSSA-Model for a = 0.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lIcos(ltnX)dX
Anumflm)=a An: P

[coszanxmx
0

n An An

1 0.31106 +1.0161

2 3.1731 -0.01966

3 6.2991 +0.00502

4 9.4354 -0.00224

5 12.574, 10.00126

6 15.714 -0.00081

7 18.855 +0.00056

8 21.996 -0.00041

9 25.137 +0.00032

10 28.278 -0.00025

 

 

 

74

 

 

Table B.2: Eigenvalues and Fourier Coefficients for the

Linear PSSA-Model for a = 0.2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IICOSOchNX
An tanOtn) = or An = 10

Icosz (AUX)dX
0

n An An

1 0.43285 +1.0311

2 3.2039 -0.03815

3 6.3148 +0.00998

4 9.4459 -0.00447

5 12.582 +0.00252

6 15.721 -0.00162

7 18.86 +0.00112

8 22 -0.00083

9 25.141 +0.00063

10 28.281 -0.0005

 

 

 

 

75

.I

 

Table B.3: Eigenvalues and Fourier Coefficients for the

Linear PSSA-Model for or = 0.3

 

 

 

 

 

 

 

 

 

 

 

 

 

1_}‘cos(}tnX)dX
An tanOtn) = or An = 10

[cos2 (xnxmx
0

n An An

1 0.5218 +1.045

2 3.2341 -0.05554

3 6.3305 +0.01484

4 9.4565 -0.00669

5 12.59 +0003 78

6 15.727 -0.00242

7 18.865 +0.00168

8 22.005 -0.00124

9 25.145 +0.00095

 

 

 

76

 

 

APPENDIX C. Unsteady State Conjugate Diffusion

In this appendix, the absorption model defined in Appendix A is solved for
unsteady state diffusion in both the polymer and the solid phases. The moisture
concentration in the air gap is spatially uniform, but unsteady. For this situation, the three
dimensionless concentration difference functions defined by Eqs.(A.1)-(A.3) satisfy the
following initial boundary value problem.

Differential Equations

A material balance on the solid product phase implies thast

 

S 2 S
69 =69 , OSXSSI , t>0. (C.l)
at axg

A material balance on the air gap phase implies that (cf. Kim, 1992),

G P
Nqd<O> =NP 6O

, t>0. (C.2)
dt 8X1)

 

 

A material balance on the polymer barrier phase implies that

2 6OP _ 6281)

NT 9
at 6X12)

 

OSXpSl , t>0. (C3)

The physical property parameters are related to the following dimensionless groups:

 

2
N2=LP/DP=(N2)2
,_

L‘é/DS N4
VK V
NSE SVSG :Vs N6

G G

 

77

v
NP 5 PKGP = VP N7

, K = K E N
VG VG GP PG 7

Initial Conditions

The initial conditions are

OS(Xs,0) 21 , 0 s xg s1 (C.4)
< GO > (0) :1 (C5)
®P(Xp,0)=l , OSXP s1. (C.6)
Boundary Conditions

Uniformity of the concentration of water in the air gap justifies the following symmetry

condition within the product tablet phase (no end effects):

dos

-—— =0 , t>0. (C.7)
6X5

X320

 

Equilibrium and Continuity of Flux on Internal Interfaces

Thermodynamic equilibrium and a well-mixed gas phase at the SG-interface (i.e., XS =1 )
and the GP-interface (i.e., X p = 0) require the following conditions:

OS =<O>G (t) , t>0 (C.8)

XS=1

 

(t) , t > 0 (C9)

 

Continuity of flux across the SG-interface and the GP-interface has been used in the

application of the divergence theorem to obtain Eq. (C.2) above.

78

Ill-J

Boundary Condition at the PE-lnterface
Thermodynamic equilibrium and continuity of water flux across the PE-interface at
Xp =1 require the following condition:

66"

= —Bim OP . t>0. (C.10)
0X1)

XI) =1

 

Xp =1
In the above equation,

k L '
Bims #— a 8‘ (C.11)

KPGDP KPG °

 

An eigenfunction representation that satisfies the above initial boundary-value problem

can be written as

P = 2 An ‘1‘) exp(—Afit) (C.12)

F§(xg) = B15n sin(AnXS) + B3“ cosOtnXS) (C.13)
P _ P . _ P .
Fn (Xp) — B1n SIn(NT}tnXp) + an COS(NIAnXP) (C.14)

Using (C. l 2) through (GM) with the boundary condition (C.7) through (C.10) and the air

gap material balance (Eq.(C.2)), it follows that
F§(1)= F§(0) (C15)

3 _

B1n _ 0 (C.16)
P

B

—2—“- =cos(7t,,) (C.17)

S
BZn

79

P .
B1,, _ (NT).n tan(N,xn)—Brm) Bim—roc . —1 (C 18)
B21)n (Ntkn + Bim tan(N,x,,)) t911(Nr4n)

 

From Eqs.(C.8) and (C.12), it follows that

 

G(t)=OS|1 — -ZlneA J)” ntBS ncosltn , t>0. (C.19)
S:
Therefore,
d<® >G(t) ~22» t S
dt 21%,). Ane nB2 nncoslt , t> 0 (C20)
Eigenvalues

Inserting Eqs.(C.12) through (C.19) into (C.2) yields the following equation for the

eigenvalues:

N _ -m.
An+Nstan(7tn)+NP ( Tin’anmif‘“) B‘ )ES(An)=0 (C21)

Nr (Ntltn + Bim tan(N,i.n))

 

Eq. (C.21) reduces to Eq. (B.7) when the pseudo steady state is applied to the polymer

phase. The development of the same is discussed in Appendix F.

Fourier Coefficients

The eigenfunctions satisfy the following boundary value problem:

   

     

S
dF s
n _ n 1:n

2 9
dXS

0 3 x3 31 (C22)

80

(113,1:

 

 

     

 

 

   

 

 

     

 

 

 

 

2 = - N93133: , 0 3 XP s1 (C23)
dXP
P S
N323 FE =NP 5%— an (C24)
XS=1 pr _ dXS _
Xp—O XS—l
P S
-N%x% n _ =Np Eli—n — NgN 3:" (C25)
XP_O P XP— S XS:1

Equation (C.22) for a value n = m gives
dFS 2 S
——2=—xm Fm , osxssl (C26)
dXS
Thus, it follows from Eqs. (C22) and (C26) that

S S l

an l:s dFm 2 2 s s
(Fm dXS ndXS) — - (kn-km) IFn Fdes ((3.27)
:1 XS=1 0

The polymer phase provides a similar result inasmuch as

P P l

dF de

(F5 534) -F( m m ——> = - (1% ->»%1)N% [1:31:31pr (C28)

P xp=0 P xp=o 0

 

 

Multiplying Eq.(C.24) by Fgfil) and Eq.(C.24) by Fr? (1) and subtracting the resulting

equations and using (C. 15) yields

 

 

 

 

 

P
_ N%(}\2_x2m)fil FP _ ‘NP [FP an
nXS=1 Xp=0 dXP
P
S
2 S an
-N N F
r s l m dXS

81

 

S

P
-FEdF_m

' n

l

dX
P XP=O

 

(C29)
S 011%

dXS ]

X32]

 

Combining Eqs.(C27), (C28), and (C29) shows that the eigenfuctions satisfy the

following condition:

1 1
Ns IF§<Xs>F§1<Xs>dXs + Np IF§<X1>>F£1<XP>pr + FE<1>F$<0> = 0 (C30)
0 O

The moisture distribution in the solid phase can be represented as
s °° s 32

o =ZAnFne' n‘ (C31)
n=l

Therefore, for t = 0, it follows that
°° s

1 = Z AnFn (Xs) (032)
n=1

For XS =1 , Eq.(C.32) implies that

1 = Z AnF§(1) . (C33)
n=1

The moisture distribution in the polymer barrier phase can be represented as
P °° P 32
o = Z AnFn e' n‘ (C34)
n=1
Therefore, for t = 0, it follows that
°° P
1 = Z AnFn (Xp). (C35)
n=1

For Xp = O , Eq.(C.35) implies that .

1 = z AnFrl’m). (C36)
n=1

82

Multiplying (C32) and (C35) by F3 and F11; , respectively, and integrating yields

1 1 1
113%,de = Am 11:31:3de + Z An ($13,?de (C37)
0 0 mm o
1 1 1
jrgdxp = Am (13,212};de + Z An (3311311,) pr (C38)
0 0 mm 0

Eqs. (C37) and (C38) can be combined with the result that

1 1 1 1
NS (13$,de + Np [ngxp = Am[NS (133$,de + Np [nggdxp]
o 0 o 0 (C39)

- Z AnFEmHEm
n¢m

Eqs. (C36) and (C.15) imply that

P = p S p S

Fm<0> AmFm(0)Fm(1)+ Z AnFn<0)Fm<1). (C40)
n¢m

Combining Eqs.(C.40) and (C39) gives the following equation for the Fourier

 

coefficients:
1 S 1 P S
NS lFm(xS)dXS +NP lFm(xP)de +Fm“)
Am = fl 0 1 0 (C41)
Ns [Fr§1(Xs)F§1(Xs)dXs +Np JF§1<XP)F£(XP>pr + F§1<I)F};<0)
0 0

The first ten eigenvalues and Fourier coefficients for ???? are tabulated in Tables Cl-
C.?. The following MATLAB® program was written to calculate the eigenvalues and the
Fourier coefficients for the conjugate mass transfer problem developed in this appendix

(i.e., see Eqs.(C21) and (C41) above).

83

for 2:1:6

y=0.02;

for i=1:10
Ns=l2;
Np=(z*4)-1;
Nt=3;
Bi=1000OO;

while(-1)“(i)*(y+Ns*tan(y)+Np/Nt*(Nt*y*tan(Nt*y)-
Bi)/(Nt*y+(Bi*tan(Nt*y))))>=O

y=y+0.0000l;
end
l(z,i)=y;
y=y+0.000l;
end
end
for 2:1:6
for j=2:101

ans=0;

an32=0;
for i=1:10
t=(j-l)*1;
Ns=12;
Np=(z*4)—l;
Nt=3;
Bi=IOOOOO;
a(z,i)=(Nt*l(z,i)*tan(Nt*l(z,i))-
Bi)/(Nt*l(z,i)+Bi*tan(l(2,1)*Nt));
num(z,i)=Ns*tan(l(z,i))/l(z,i)+Np/(Nt*l(z,i))*(a(z,i)*(1-
cos(Nt*l(z,i)))+sin(Nt*l(z,i)))+1;
denom(z,i)=NS/((cos(l(z,i)))A2)/(4*l(z,i))*(2*l(z,i)+sin(2*
1(z,i)))+Np*(((a(z,i)*a(z,i))+1)/2—
(a(z,i)*c08(2*Nt*l(z,i))/(2*Nt*l(z,i)))-(((a(z,i)*a(z,i))-
l)*sin(2*Nt*l(z,i))/(4*Nt*l(z,i)))+a(z,i)/(2*Nt*l(z,i)))+1;
An(z,i)=num(z,i)/denom(z,i);
sum(z,i)=An(z,i)*cos(l(z,i)*1.0)/(cos(l(z,i)))*exp(-
(1(ZIi))A2*t)
sum2(z,i)=An(z,i)*(a(z,i)*sin(Nt*l(2,1)*O)+Cos(Nt*l(z,i)*O)
)*exP(-(l(z,i))“2*t)
ans=ans+sum(z,i);
an82=an52+sum2(z,i);
end

84

ansl(z,j)=ans
an53(z,j)=an32
ansl(z,1)=l;
an53(z,1)=1;
taU(j)=(j-l)*l,
end

t(2)=log(An(z,1)*tan(l(z,l))*2/l(z,1))/(l(z,1))“2;

85

APPENDIX D. Problem Setup for the Non-Linear PSSA Model

The non-linear PSSA-model was solved numerically. A GAB-isotherm at the
product-gas interface makes the boundary value problem non-linear (see Sections 3.8 and

3.9). The setup of the computational problem using COMSOL MULTIPHYSICS® is

described in this appendix.

Differential Equation (dimensional)

s zs
6C _DaC

——.-_ —— , osst , t>0 12.1
at 85x2 8 ( )
Initial Condition

S ‘ _ S ‘

C (x,0)=CO , osstS (D2)

Symmetry Boundary Condition
As indicated by Figure 3.1, the boundary condition on the symmetry plane in the product

tablet is

aCS

. =0 , {>0 (13.3)
ax

A

X=0

 

Boundary Condition at the SG-Interface
A pseudo-steady state approximation for the moisture flux across the air gap and the

polymer barrier yields the following condition at the SCI-interface:

s

93-6-9.— =+keff(CE-CG , ) , t>0. (D.4a)
6X . szQ
Xst ‘

 

The effective mass transfer coefficient keff in Eq.(D.4a) is defined as follows

86

l =LG + Lp
keff Do KPGDP

 

+ -1—. (D.4b)
kc
With

M(X,t) s Cs(x,E)/ps = M§q -(M§q —M§)oS(x,t)

AW(LS,t)ECG(LS,t)/CG —<CG>(i)/Cgiq , AaacE/CG

. eq ‘ eq , (D.4c)
X E X/LS

t E EDS / L28

Eq.(D.4a) can be re-written as

aM ,, E - keffLSng o p t

— =+a(Aw - Awlle) , a a —— , Ceq=18la—e—L. (D.4d)
6X X=1 DSPS RgT

The nonlinear PSSA-model is defined by the following boundary value problem:

2 .
ELEM. , 03x31 (D.S)
at 6X2
PM :0 (D6)
5X X=0
6M _ .. E
5(— X=1 —+(1(Aw - AWIX=1) . (D2)

The GAB-isotherm at the PE-interface relates the mass ratio to the relative humidity AW

as follows (see Chapter 2):

AWlX=1

 

M|x=l = (D.8)

a3 (AWlX:1 )2+32 (AWIX=1 )+al

Eqs.(D.5) through (D8) were solved using a finite element code supported by COMSOL

MULITIPHYSICS®. The results are reported in Chapters 3 and 4.

87

In COMSOL MULTIPHYSICS® 3.3, under the model library for 1D space
dimension. The user option, “Mass Balance - Diffusion - Transient Analysis”, was
selected under the chemical engineering module. Using the “draw” menu, a one
dimensional flat plate of unit thickness (dimensionless units) was drawn. The faces of the
plate were marked to identify the boundaries. The model constants, initial values, and
properties of each boundary were set in the “Physics” menu and the FEM grid was
generated with the default values (15 mesh elements) and a time step of 0.1 was selected.
The boundary value problem was solved for the time scale desired.

Eq. (D.8) was rearranged to obtain an explicit equation for AW in terms of M. For a
given value of M, Eq.(D.8) is a quadratic equation for Aw:
a3 MAE, + (azM-1)Aw + a1 M = o. (12.9)

The first and second GAB-coefficients are positive, and the third coefficient is negative:

31 > 0 , a2 > O , and a3 < 0. Therefore, the positive root of Eq.(D.9) is

 

A : +(a2lVll-l)+‘/(a2M-l)2+4|a3|a]M2 forX=l (D10)
W 2|a3|M ’ ' °

 

Eq.(D.lO) was substituted into (D2) to give a non-linear boundary condition at the
solid/air gap interface consistent with thermodynamic equilibrium and continuity of
moisture flux at the interface.

Base Case:

The physical property data of Allen (1994) are summarized in Table 4.1 above. The
following values are taken as the “base” case for the parametric study presented in

Chapter 4:

88

123 = 0.996><10'6 cm2/s , L5 = 99222 = 0.196 cm

12p=2><10'9 cm2/s , Lp = 9.1 mils = 0.023 cm
p8 = 1.39 g/(cm3) , KpG =125

CG

9 t ._
eq 18 waer—18

?
RgT ??

= ? 0.028 g/(cm3)
1

k ff -=—
e LQ/+ Lp +)le
)56 KPGDP c

keffLSng = N4 ng = (0.002)(0.028)

Dsps N2 PS (0.ll7) (1.39)

 

 

E 0.038

 

 

d:

89

APPENDIX E. Absorption: Flat Plate Geometry and Large Biot Numbers

The absorption of moisture in a flat plate of thickness 2L3 surrounded by humid

air at a relative humidity of A15” is described by the following boundary value problem,

which is just a special case of the problem described in Appendix A. The purpose of this
appendix is to define the boundary value problem that governs the absorption
experiments conducted by Allen(1994) on “flat plate drug tablets and “flat” plate polymer
barriers. These experiments are designed to measure the thermodynamic adsorption
isotherms and the diffusion coefficients of the three materials in the blister package. A
microbalance is used to directly measure the mass of water absorbed per mass of dry
solids as a function of time. Allen’s experiments were designed so that l) the convective

mass transfer coefficient kC was large compared with a characteristic velocity for

diffusion through the solid phase (i.e., large Biot number); and, 2) the surrounding
moisture concentration in the humid air was the same over the entire surface of the solid
and did not change with time.

Solid Phase --- Boundary Value Problem fdrug tablets or polymer barriers)

With M 2 C( X,i)/ p equal to the mass of water absorbed per unit mass of dry solid, it

follows directly from the discussion in Appendix B (and elsewhere in this thesis) that

2 E122 *
6—M22—M— , OSXSI , t>0 , tE— , X51. (13.1)
at 6X2 L2 L
M (X,0) a C(X,O)/p = M0 (constant) , 0 S X 51 (E2)

90

 

—— = O, t > 0 (E3)
l
KSG Aw = constant , linear isotherm
M(1, t) = 3(AW(1, t) =< (B4)
2 AW = constant , GAB — isotherm
(a3Aw +a2Aw +a1

 

A Fourier series representation of a solution to Eqs.(E.l)-(E.4), which is equivalent to the

special case on = 00 (see Appendix B above), is given by (cf. Eq.(B.6):

0° 2

M(X,t)=M1—(M1—MO)Z Ancosomxx'ln‘, osxs1 , t>0. (13.5)
n=0
+

cos(xn)=0 , kn=(2n% , n=0,1,2,3,.... (E6)

The Fourier coefficients are defined by Eq.(B.8) for a z 00:

l

 

 

 

jcosotnxxix
An E 10 : 451n()tn) (E2)
2 27m + sm(2 kn)
J‘cos (an)dX
0
The volume average of Eq.(E.5) gives (see p. 45, Crank, 1956):
1 oo 2
_ _ -)t t
Mt = IM(X,t)dX —M]1 —(M1—M0) Z An[£cos(}tn X)dX]e "
0 “:0 (E8)
8 °° 1 - 2
= M1—(M1—Mo)—2 2 e ln‘

7t n=0 (2n+l)2
Eq.(E.8) was used by Allen (1994) to estimate the thermodynamic parameters in Eq.(E.4)

by measuring M] for t—) 00 for different values of Aw. The diffusion coefficient was

91

also estimated by fitting Eq.(E.8) to the unsteady state absorption data. The following
examples (and many more) were reported by Allen.
_Deltasone ® -— drug tablet

Ldrug = 0.196 cm (half thickness)
M0 = 0 (dry solid)
AW = 0.873 RH

g water
100 g dry solid

 

M1 = 1.976
deg = 0.996x10—6 cm2 /s

PVC— polymer barrier

LPVC = 0.0095 cm (half thickness)
Mo = 0 (dry solid)
AW = 0.77 RH

g water
100 g dry solid

 

M1=O.133
DpVC _—. 5.045(10‘9 cm2 /s

A CLAR ® -- polymer barrier

L ACL AR = 0.0020 cm (half thickness)
M0 = 0 (dry solid)

AW = 0.58 RH

g water

M1=0.121 .
100g dry sol1d

 

DACLAR = 7.03x10_10 cm2 /8

Figures (2.3)-(2.5) in Chapter 2 compare the experimental data reported by Allen
(1994) and the theoretical result given by Eq.(E.8) for the three diffusivities given above.

This type of information is used in Chapters 3 and 4 to simulate the absorption of

moisture in a blister package.

92

 

APPENDIX F. Justification of PSSA-Model for Linear Adsorption

PSSA-model
In this appendix, the PSSA-model developed in Appendix B is compared with the
exact solution of the conjugate mass transfer problem developed in Appendix C.

Application of the pseudo-steady state assumptions to Eq.(C2) and Eq.(C3) implies that

 

 

 

G P s
$d<® >= pfl— —N%NSfl .20, t>0 (F.1)
dt axp axs
P 2 P
N369 :69 g0 , OSXpSl , t>0 (F2)
5t 6X12,

Because (93(14): <®G>(t) = ®P(0,t) and (for Bi :00) (91’ (1,0 =0, Eq.(F2) implies

 

that

P P _ P
9.9— :9 (1") 9 (O’t)=—®P(0,t)=—®S(l,t) , t>0. (F3)
6Xp l—O

 

Xp=0
Therefore, the PSSA-boundary condition at the SG-interface follows by combining

Eqs.(F.1) and (F3):

as
6X8

NP
NgNs

 

— —£®S(1,t) , a =

— 2 , t > 0. (F4)
Xs=1 NTNS

 

Eq.(F .4) is a special case for which Bi 2 co and the resistance to mass transfer across the
air gap due to accumulation of moisture is neglected (see approximation (F.1) above).
Therefore, the absorption of moisture into the solid product phase is governed by Eq.(F.4)

and the following two conditions:

93

 

= , O.<_X $1 , t>0 F.5
at 5x2 5 ( )
S
S
6.: =0 (F.6)
6X3
XS=O

The solution to Eq.(F.5) subject to Eqs.(F.4) and (F6) has been developed in Appendix B.
In what follows, the exact solution of the linear conjugate absorption problem developed
in Appendix C is compared with the corresponding exact solution of the linear PSSA-
model. The following set of physical property parameters from Allen’s thesis (Allen,
1994) is used in the comparison.

Base Case

L3 = 0.196 cm , LG = 0.258 cm , L1) = 0.0023 cm

125 ——- 0.1179><10'5 cm2/s , Dp=2x10'9 cmz/s

 

 

 

 

 

1 1
K = —=——=16, K =125

80 a3 0.064 PG
VS LS 0196_0.76
VG LG 0258
y}: = Lp =0.0023 #0089
VG LG 0.258
Ell: 00023 = 0.0117
LS 0.196

—5

12 .

S ___ 01179x10 :590

DP 2x10_9

94

L
N17; e(—11)2 gs— =(0.0117)2 x 590 = 0.0808 :5 NT = J0.0808 5 0.2842

 

Ls
V L
NS s—S—KSG =——S—KSG = (0.76)(16) s 12
VG LG
Np EXP-K126 =E-KPG =(0.0089)(125) 21.1
VG LG
NP (1.1)
a: = E .
NgNS (0.0808)(12)
Figures F.1, F2 and F3 show the relaxation of the PQI-function for different
values of the dimensionless groups: NT, NS and Np . For

NI=028, NS=12, and Np =1.1, the PQl-time (defined as < GS > (to) = 0.5) predicted

by the linear conjugate absorption model is about 11 L2S / DS , which is about 100 hours

for the problem defined above. For the linear PSSA-model, the PQl-time is about

8 L28 /DS (see Figure 4.1 in Chapter 4).

95

 

 

.

.

.

.

_

_

.

_

_

.

_ v

_

- Y

- Y

. _2

. 2 .

.

. 22 .

. .2 _

2 2 ,

. 2 .

. 2 r

. .2 .
2

2 .2 w

. 2

. \ 2 e

- \ v

. 2 .

_ 2 v

. 2 .

.

— Y

— 1

_ f

. e

_

. fl

— 2.

. f

2

. 2

_ 22.. w

. .

. \

a s.

_ 2

_

 

09
08
0.7
0

0

O4
03

20 30 40 50 60 70 80 90 100

10

Figure F.1 Effect of NT on the PQI-function (Ns=12, NP=1 1).

96

 

 

0.3 -
0.2 - \ NS =24
20 \
0.1 « 16 ~\ \
2\2:x\--22~

 

0 ~222222222222 8 H 4 -22, . .T“fi ;;:?:1
0 10 20 "" 40 50 60 70 80 90 100
4
.138.
— 2
Ls
Figure F2 Effect of NS on the PQI-function (NP=11, NI=3 ).

97

 

 
   

 

 

 

 

0.9 l
0.8 «
0.7 ~
0.6 4
0.5 . \
0 4
<os>u)
l
0.3 J : N 3
I _
l P g
I
. .
0.2 « : \u \\
I \ \\
I \\
I \\\
I
I
: 7 \\\\
I \\, -\\~~‘_
o 10 20 1 9 1 4o 50 60 7o 80 90 100
23
tDS
‘ = ‘2—
L
S

Figure F3 Effect of N on the PQl-function (NT=3, NS=12 ).

P

98

 

 

Table F.1 Effect of NS , Np & Nt on the Shelf-Life for the Linear Model

 

 

 

 

 

 

 

 

 

 

 

 

<®S >(tc)E®S =05 ;Ec =tCLZS/DS; Lé/DS sllhours
tC a = NP t
NS NP NI Exact NgNS PSSA
12 11 1 1 1.1 1
12 11 3 11 0.12 8
12 11 6 44 0.03 51
12 3 3 31 0.04 46
12 11 3 11 0.12 8
12 23 3 7 0.24 3
4 11 3 5 0.44 5
12 11 3 11 0.12 8
24 11 3 20 0.06 25

 

 

 

 

 

 

99

 

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100

 

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