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www.elsevier.com/locate/jconrel
Journal of Controlled Releas
Mathematical modeling and in vitro study of controlled drug
release via a highly swellable and dissoluble polymer matrix:
polyethylene oxide with high molecular weights
Ning Wua,1, Li-Shan Wanga, Darren Cherng-Wen Tana,
Shabbir M. Moochhalab, Yi-Yan Yanga,*
aInstitute of Bioengineering and Nanotechnology, 31 Biopolis Way, The Nanos, #04-01, Singapore 138669, SingaporebDSO National Laboratory, 27 Medical Drive, #12-00, Singapore 117510, Singapore
Received 1 June 2004; accepted 1 November 2004
Available online 8 December 2004
Abstract
A mathematical model is developed to describe the transport phenomena of a water-soluble small molecular drug
(caffeine) from highly swellable and dissoluble polyethylene oxide (PEO) cylindrical tablets. Several important aspects in
drug release kinetics were taken into account simultaneously in this theoretical model: swelling of the hydrophilic matrix and
water penetration, three-dimensional and concentration-dependent diffusion of drug and water, and polymer dissolution. The
moving boundary conditions are explicitly derived, and the resulting coupled partial differential equations are solved
numerically. In vitro study of swelling, dissolution behavior of PEOs with different molecular weights and drug release are
also carried out. When compared with experimental results, this theoretical model agrees with the water uptake, dimensional
change and polymer dissolution profiles very well for pure PEO tablets with two different molecular weights. Drug release
profiles using this model are predicted with a very good agreement with experimental data at different initial loadings. The
overall drug release process is found to be highly dependent on the matrix swelling, drug and water diffusion, polymer
dissolution and initial dimensions of the tablets. Their influences on drug release kinetics from PEO with two different
molecular weights are also investigated.
D 2004 Elsevier B.V. All rights reserved.
Keywords: Swelling; Diffusion; Dissolution; Controlled release; Polyethylene oxide (PEO) matrix; Caffeine
0168-3659/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jconrel.2004.11.002
* Corresponding author. Tel.: +65 68247106; fax: +65
64789084.
E-mail address: [email protected] (Y.-Y. Yang).1 Current address: The Department of Chemical Engineering,
Princeton University, USA.
1. Introduction
Hydrophilic polymeric matrices have attracted
many researchers due to their wide applications in
controlled drug delivery. When the release medium
(i.e. water) is thermodynamically compatible with a
e 102 (2005) 569–581
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581570
polymer, the polymer may undergo a relaxation
process so that the polymer chains become more
flexible and the matrix swells. This could allow the
encapsulated drug to diffuse more rapidly out of the
matrix. On the other hand, it would take more time for
drug to diffuse out of the matrix since the diffusion
path is lengthened by matrix swelling. Moreover, it
has been widely known that swelling and diffusion are
not the only factors that determine the rate of drug
release [1]. For dissolvable polymer matrix, polymer
dissolution is another important mechanism that can
modulate the drug delivery rate. While either swelling
or dissolution can be the predominant factor for a
specific type of polymer [2], in most cases drug
release kinetics is a result of a combination of these
two mechanisms [3,4].
Among a variety of hydrophilic polymers, poly-
ethylene oxide (PEO) is one of the most important
materials used in the pharmaceutical industries mainly
because of its non-toxicity, high water-solubility and
swellability, insensitivity to the pH of the biological
medium and ease of production. Recently, the
swelling and dissolution behavior of PEO tablets [5–
8] and hydrogels [9] as well as their influences on
drug release characteristics have been studied. It is
found that, compared with low molecular weight
PEO, the high molecular weight PEO tablet swells to
a greater extent and the swelling of the polymer rather
than the dissolution of the polymer is the governing
factor for drug release. The compression force applied
during the manufacturing process, pH of the release
medium and the stirring rate do not affect the drug
release behavior significantly. However, most of the
above studies of PEO matrices are mainly exper-
imental and there is no comprehensive mathematical
modeling of controlled drug release via PEO matrices
with different molecular weights. The effect of initial
dimensions of tablets on drug release kinetics is also
not studied.
The mathematical modeling of drug release is of
great importance in pharmaceutical science and
engineering because the idealized but key transport
mechanisms can be studied in the mathematical
model and the model itself can be used to predict
the effects of the composition and geometry on drug
release profiles, which is very helpful to the design
of new drug delivery system. The mechanism study
of drug release via a swellable and dissoluble
hydrophilic polymer matrix is not as extensive as
for purely diffusion, swelling or polymer dissolution-
controlled drug release systems since all these
processes are coupled, thus making the models more
intricate and difficult to solve. Both empirical [10,11]
and theoretical [12,13] mathematical models have
been reported for drug release via either cross-linked
[14] or uncross-linked [15] polymer matrix in
literature. However, most of the models consider
only certain physical characteristics and neglected
the others. For example, diffusion is not considered
[11]; a pseudosteady state assumption is made [16];
transport of active agent is considered only in one-
dimensional [17–19]. Recently, a two-dimensional
approach is reported [20] for HPMC tablets. Fick’s
second law of diffusion with concentration-depend-
ent diffusivity is used to describe water and drug
transport. A surface-normalized rate constant is used
to model polymer dissolution and swelling is
considered using moving boundary conditions. How-
ever, the moving boundary conditions are not
explicitly expressed.
The model developed here can be viewed as a
two-dimensional extension of the model developed
by Vrentas et al. [21] for the impurity removal in
polymer films by the addition of a second solvent, in
which the diffusion coefficients of two solvents in a
polymer–solvent–solvent system are related to the
concentration and properties of all the components
(two solvents and polymer) in the polymer film by
using a comprehensive free-volume theory of trans-
port. In our work, we extend that model into two-
dimensional form to account for the geometry of
common cylindrical tablet. Polymer swelling and
dissolution are included since PEO is a highly
swellable and dissoluble polymer and for simplicity
we use a simpler form of diffusion coefficient which
is expressed as a function of water concentration
only.
2. Mathematical model
A schematic for the model considered here is
plotted in Fig. 1. The matrix is cylindrical with initial
radius of r0 and initial height of 2z0. Only the upper
half part of the cylindrical matrix is considered in
mathematical modeling since it is symmetrical to the
t = 0 t > 0
r0
z0
r
z
z
r
n
m
Fig. 1. Schematic of the tablet considered in the model.
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581 571
lower half part. Swelling is assumed to be in radial
and axial directions and the diffusion of water and
drug are assumed Fickian diffusions with water
concentration dependent mechanism. Effect of
stresses arising in the swelling process on diffusion
and the influence of the diffusion-induced convection
associated with volume changes in mixing are not
considered here since for most polymer–solvent
systems these effects can be neglected [18,22]. The
swelling is assumed to be homogeneous and at any
time, the total volume of matrix is the sum of volume
of water, drug and polymer. Polymer dissolution is
assumed on the surface of the tablet. Thus, the
positions of swelling front are the same as the
dissolution front and they are denoted as m and n in
radial and axial directions.
The three-dimensional basic equation for diffusion
of water and drug through the matrix can be written
in cylindrical coordinate with neglecting concentra-
tion gradient of both water and drug in angular
direction
BCi
Bt¼ 1
r
B
BrrDi
BCi
Br
� �þ B
BzDi
BCi
Bz
� �ð1Þ
where subscript i=1 for water and i=2 for drug, Ci is
the mass concentration of component i within the
matrix.
The diffusivities of drug andwater are assumed to be
concentration-dependent and they can be represented
by a bFujita-likeQ free-volume model [23]
Di ¼ Di;eexp t� bi 1� C1=C1;e
�b
�ð2Þ
where Di,e is the diffusion coefficient of water or drug
in the fully swollen polymer matrix system, bi is a
constant which characterizes the concentration depend-
ence of water or drug diffusivity and C1,e is the
equilibrium water concentration in fully swollen
polymeric matrix. D1,e and b1 are to be determined
by fitting of experimental data in pure polymer
swelling and dissolution studies. Values of D2,e and
b2 can be obtained by fitting the drug release data.
Experimentally these parameters can also be obtained
by pulsed-field-gradient spin-echo (PFGSE) NMR
techniques [24] although not investigated in this work.
The Fujita model of diffusion is a simplified model that
describes the transport of small molecules in polymer
based on free-volume theory of Vrentas and Duda
[25,26], in which the diffusion coefficient for a
penetrant is related to the volumetric and free-volume
properties of itself and other components in the
polymeric matrix. It has been successfully applied in
a variety of models of controlled drug release at
different release mediums [20,27–29].
It is convenient to introduce the following dimen-
sionless variables
C1 ¼ C1=C1;e C2 ¼ C2=C2;0 D1 ¼ D1=D1;e D2 ¼ D2=D1;e
r ¼ r=r0 z ¼ z=r0 s ¼ D1;et=r20
ð3Þ
so that we can write Eq. (1) in dimensionless form
BCi
Bs¼ 1
r
B
B rr Di
BCi
B r
� �þ B
B zDi
BCi
B z
� �ð4Þ
where C2,0 is the initial concentration of drug.
r
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581572
The initial conditions are
s ¼ 0 0V r VR 0V z VZC1 ¼ 0:0 C2 ¼ 1:0
ð5Þ
The boundary conditions are
sN0 0V r VR z ¼ Z C1 ¼ 1:0 C2 ¼ 0:00V z VZ r ¼ R C1 ¼ 1:0 C2 ¼ 0:0
ð6Þ
sN00V r VR z ¼ 0
BC1
B z¼ 0:0
BC2
B z¼ 0:0
0V z VZ r ¼ 0BC1
B r¼ 0:0
BC2
B r¼ 0:0
ð7Þ
where R=m/r0 and Z=n/z0.
Additional boundary conditions on swelling front
m (or R) in radial direction and n (or Z) in axial
direction are needed. They can be derived from a
volume balance.
2pm2n ¼ 21
q1
Z n
0
Z m
0
C1 r; z; tð Þ2prdrdz
þ 21
q2
Z n
0
Z m
0
C2 r; z; tð Þ2prdrdz
þ 1
qp
mp;0 �Z t
0
KpAsdt
� �ð8Þ
where qi is the density of water (i=1), drug (i=2) or
polymer (i=p).
Note that the third term on the right-hand side of
Eq. (8) represents the polymer dissolution process,
which is rather complex and involves polymer chain
disentanglement and reptation [30]. To simplify the
model analysis, we use a single parameter, dissolution
rate constant Kp, to characterize the surface (hetero-
geneous) dissolution of polymer. It is a mass rate
constant and defined as (1/As)(dmp/dt)=�Kp. As is the
total surface area of polymer matrix in contact with
the release medium and mp,0 is the initial mass of
polymer. Kp can also be viewed as a product of
dissolution/mass transfer coefficient and the equili-
brium polymer concentration at the dissolution
(swelling) front [18,31].
Differentiating Eq. (8) with respect to time and
using Leibniz’s rule yields
pm2 dn
dtþ 2pmn
dm
dt¼ C1;e
q1
pm2 dn
dtþ C1;e
q1
2pmndm
dt
þZ n
0
Z m
0
1
q1
BC1
Btþ 1
q2
BC2
Bt
!2prdrdz
� 1
qp
Kpðpm2 þ 2pmnÞ ð9Þ
Substituting Eq. (1) in Eq. (9) and simplifying it we
can get
1� C1;e
q1
� �pm2 dn
dtþ 1� C1;e
q1
� �2pmn
dm
dt
¼ 2pmZ n
0
1
q1
D1
BC1
Br
r¼m
þ 1
q2
D2
BC2
Br
r¼m
� �dz
þ 2pZ m
0
1
q1
D1
BC1
Bz
z¼n
þ 1
q2
D2
BC2
Bz
z¼n
� �rd
� 1
qp
Kpðpm2 þ 2pmnÞ ð10Þ
If we assume that the swelling in radial direction is
independent on the swelling in z direction (edge effect
is neglected), we can split Eq. (10) into two separate
equations, which describe the rate of swelling front
advancement in r and z direction, respectively. They
are
n 1� f1ð Þ dmdt
¼ 1
q1
Z n
0
D1
BC1
Br
r¼m
dz
þ 1
q2
Z n
0
D2
BC2
Br
r¼m
dz� nKp
qp
ð11Þ
m2 1� f1ð Þ dndt
¼ 1
q1
Z m
0
D1
BC1
Bz
z¼n
2rdr
þ 1
q2
Z m
0
D2
BC2
Bz
z¼n
2rdr � m2 Kp
qp
ð12Þ
where f1 is the equilibrium volume fraction of water in
the full-swollen matrix.
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581 573
Physically, the initial and boundary conditions
mean
(i) At t=0, the matrix is dry and a homogeneous
mixture of drug and polymer;
(ii) After the matrix is immersed into release
medium, the concentration of drug is always
zero at the surface of the cylindrical matrix, i.e.
a perfect sink condition is assumed. While the
water concentration at the surface is a constant,
which is equal to the equilibrium concentration
at a full-swollen state with polymer;
(iii) At the two symmetric planes located in the center
of the cylinder, symmetry conditions are applied;
(iv) The rate of advancement of swelling front is
determined by the amount of water diffusing in,
drug diffusing out of and polymer dissolving
from the matrix.
Since the mathematical model developed here is a
moving boundary problem, special technique is
required. We use the front-fixing method, which was
proposed by Landau [32] and first applied to a finite-
difference scheme by Crank [33] to fix the swelling
boundary.
Let n=r/R and g=z/Z. By using the relationships
BCi
B r¼ 1
R
BCi
Bn
BCi
B z¼ 1
Z
BCi
Bg
BCi
Bs
!r ;z
¼ BCi
Bs
!n;g
� 1
2
BCi
BnnR2
B R2ð ÞBs
� 1
2
BCi
BggZ2
B Z2ð ÞBs
ð13Þ
to transform from C–i(r,z,s) to C
–i(n,g,s), the diffusion
Eq. (4) can be written
BCi
Bs¼ 1
nB
BnnDi
BCi
Bn
� �1
R2þ B
BgDi
BCi
Bg
� �1
Z2
þ 1
2
BCi
BnnR2
B R2ð ÞBs
þ 1
2
BCi
BggZ2
B Z2ð ÞBs
ð14Þ
The initial and boundary conditions can be rewritten as
s ¼ 0 0V nV 1 0V gV1C1 ¼ 0:0 C2 ¼ 1:0
ð15Þ
sN0 0V nV 1 g ¼ 1 C1 ¼ 1:0 C2 ¼ 0:00V gV 1 n ¼ 1 C1 ¼ 1:0 C2 ¼ 0:0
ð16Þ
sN01V nV 1 g ¼ 0
BC1
Bg¼ 0:0
BC2
Bg¼ 0:0
0V gV 1 n ¼ 0BC1
Bn¼ 0:0
BC2
Bn¼ 0:0
ð17ÞsN0 n ¼ 1 0V gV 1
1
21� f1ð Þ dR
2
ds¼ f1
Z 1
0
D1
BC1
Bn
n¼1
dg
þ f2;0
Z 1
0
D2
BC2
Bn
n¼1
dg � RKp
qp
ð18ÞsN0 g ¼ 1 0V nV 1
1
21� f1ð Þ dZ
2
dt¼ f1
Z 1
0
D1
BC1
Bg
g¼1
2ndn
þ f2;0
Z 1
0
D2
BC2
Bg
g¼1
2ndn � ZKp
qp
ð19Þwhere K
–p=Kpr0/D1,e and f2,0 is the volume fraction of
drug at s=0.Eq. (14) can be solved with the initial and
boundary conditions of Eqs. (15–19) numerically for
the spatial and time variations of C–1 and C
–2. An
explicit finite-difference scheme is applied for solving
the coupled equations at each time step.
To compare the fitness of the theoretical model and
experimental results, the coefficient of determination,
R2, is calculated [34]. It is defined as
R2 ¼ regression sum of squares
total sum of squares
¼ 1�
Xn
i¼1
yi � yyið Þ2
Xn
i¼1
yi � yi� �2 ð20Þ
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581574
where yi and yi are the experimental and theoretical y
co-ordinates at time point i, y is the arithmetic mean
of the experimental y co-ordinates of the whole
series. Generally, we find the fit acceptable if R2 is
close to 1.
0
20
40
60
80
100
120
0 6 12 18 24 30 36time (hr)
wat
er u
pta
ke (
%)
0
20
40
60
80
100
120
0 4 8 12 16 20 24time (hr)
wat
er u
pta
ke (
%)
(a)
(b)
Model
Experiment
Model
Experiment
Fig. 2. Water uptake of pure PEO tablets with two different
molecular weights. Squares denote experimental results and solid
line is the fitting curve using the model developed in this study. The
total weight of polymer tablet is 360 mg. The initial radius and
height are 0.5 and 0.381 cm, respectively. (a) PEO with Mw=8�106
(R2=0.990); (b) PEO with Mw=4�106 (R2=0.980).
3. Materials and methods
Poly(ethylene oxide) (PEO) with Mw of 8�106
and 4�106 (Aldrich Chem. Milwaukee, USA) were
used as the dissoluble matrix material, in which
caffeine powder (Sigma, St. Louis, USA) is mixed
thoroughly by manually grinding in a stone mortar.
The resultant powder mixture is then compressed with
a laboratory hydraulic press (Graseby Specac) under a
pressure of 38 MPa for 1 min using two tablet
punches in 10-mm diameter, which have convex
surfaces. The total mass of each 10-mm-diameter
tablet was maintained at 360 mg with different
loadings of caffeine. Apparatus 2 of the USP
dissolution test (VK 7000, VanKel Technology
Group, Weston Parkway, USA) is employed for
polymer swelling, dissolution and drug release
studies. Simulated intestinal fluid (SIF, without
pancreatin), which is prepared using monobasic
potassium phosphate and sodium hydroxide (Merck,
Darmstadt, Germany), is used as a release medium.
The dissolution tests are performed in 1000 mL of
receptor medium with an agitation rate of 110 rpm.
Samples of 1 mL, which are replaced by an equal
volume of fresh receptor medium, are periodically
taken. All samples are filtered with 0.2-Am syringe
filters (Whatman, Clifton, USA) before being ana-
lyzed with high performance liquid chromatography
(HPLC). The HPLC system consists of a 2690
separations module (Waters, USA), a 996 photodiode
array detector (Waters) and a dedicated personal
computer (Compaq, USA). The UV detector wave-
length is set at 270 nm for detection of caffeine. The
sample separation is achieved by using an InsertsilRC8 analytical column (4.6�150 mm, 5 Am, GL
Sciences, Tokyo, Japan), as well as an InsertsilR C8
guard column E (4.0�10 mm, 5 Am, GL Sciences),
at room temperature. The mobile phase consists of
methanol and ultrapure water in a ratio of 30:70 by
volume at a flow rate of 1.0 mL/min. An injection
volume of 10 AL is used.
For polymer swelling and dissolution study,
tablets of 360 mg pure PEO are made by the same
method as previously described and immersed in
1000 mL of water at 37 8C. The weight of wet and
dry PEO is accurately monitored as a function of
time. Dimensional change of tablet is measured
immediately after each sample is taken. Weight loss
is the difference between the original weight of tablet
and the weight of dry PEO and the amount of water
uptake is the weight difference between wet tablet
and dry PEO.
1
1.5
2
2.5
3
3.5
dim
ensi
on
al c
han
ge
X/X
0
n / z0
m / r0
(a)
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581 575
4. Results and discussion
4.1. Swelling and dissolution of pure PEO matrices
Before studying and modeling the drug release
kinetics of PEO matrices, both water uptake and
polymer dissolution studies of pure PEO tablets are
carried out and fitted with the theoretical results in
order to obtain water diffusion and polymer dissolu-
tion parameters. Fig. 2 shows the fit of the model to
the experimentally determined relative amount of
water uptake by pure PEO tablets of two different
molecular weights. As it can be seen, compared with
50
60
70
80
90
100
110
0 10 20 30 40time (hr)
rela
tive
dry
wei
gh
t of P
EO
(%)
0
20
40
60
80
100
120
0 6 12 18 24time (hr)
rela
tive
dry
wei
gh
t o
f P
EO
(%
)
Model
Experiment
Model
Experiment
(a)
(b)
Fig. 3. The profile of relative dry weight of pure PEO tablets as a
function of time after immersion in the medium (both theoretical
and experimental). (a) PEO with Mw=8�106 (R2=0.980); (b) PEO
with Mw=4�106 (R2=0.954).
0 5 10 15 20time (hr)
1
1.2
1.4
1.6
1.8
2
2.2
2.4
0 5 10 15time (hr)
dim
ensi
on
al c
han
ge
X/X
0
n / z0
m / r0
(b)
Fig. 4. Relative dimensional change of pure PEO tablets during
swelling and dissolution. Squares denote the experimental measure
ments. (a) PEO with Mw=8�106. (b) PEO with Mw=4�106.
-
PEO of lower molecular weight, PEO with
Mw=8�106 shows a larger tendency to be hydrated
and the hydration time is up to 22 h, then the
dissolution process prevails on the swelling. On the
other hand, PEO of Mw=4�106 is hydrated com-
pletely within first 8 h. This is in agreement with
previous report [8]. A good agreement between the
theory and experiment can also be seen in Fig. 2.
Thus, the Fujita model of concentration-dependent
diffusivity of water describes the swelling process of a
hydrophilic polymer matrix (PEO) very well. It is
found that the diffusion coefficient of water within the
fully swollen PEO tablet of lower molecular weight
(2.9�10�6 cm2/s) is slightly higher than that in higher
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581576
molecular weight PEO (2.6�10�6 cm2/s), although
the concentration dependence constant b1 is 2.0 in
both polymers. It is reasonable since the weaker chain
entanglement within lower molecular weight PEO
matrix can make water molecules more easily diffuse
through. The commonly used glassy–rubbery inter-
face is not included in our model because small water
molecules are able to diffuse across this interface
[18,35], as indicated by the low concentration depend-
ence constant of water. Mathematically, at a glassy–
rubbery interface, there is a discontinuity in the slope
of concentration profile. In our model, the slope of the
concentration profile within the matrix is always
continuous. Nonetheless, if the concentration depend-
ence constant is high enough (i.e. solvent molecules
0
20
40
60
80
100
0 6 12 18 24time (hr)
rela
tive
dru
g r
elea
se (
%)
0
20
40
60
80
100
0 6 12 18 24time (hr)
rela
tive
dru
g r
elea
se (
%)
(a)
(c)
Fig. 5. Fit of the model to the experimentally determined relative amount
Squares denote experimental data. (a) 8.33% w/w (R2=0.998); (b) 16.6
(R2=0.995).
with low compatibility with polymer), the diffusion
front (or concentration profile) can be rather steep
[28], thus the glassy–rubbery interface still can be
simulated approximately in our model.
In this model, a dissolution rate constant, Kp, is
considered characterizing the polymer mass loss
velocity normalized to the actual surface of the
system: (1/As)(dmp/dt)=�Kp. This constant can be
obtained from the fitting of modeling dry polymer
weight to experimental results. From Fig. 3, it can be
seen that dissolution of higher molecular weight PEO
is rather slow. It takes more than 40 h for the polymer
to reduce its weight to half, while PEO of lower
molecular weight dissolves totally within 24 h. The
dissolution constant is found to be 1.1�10�4 mg/cm2-s
0
20
40
60
80
100
0 6 12 18 24time (hr)
rela
tive
dru
g r
elea
se (
%)
0
20
40
60
80
100
0 6 12 18 24time (hr)
rela
tive
dru
g r
elea
se (
%)
(d)
(b)
of caffeine released at different loadings from PEO of Mw=8�106.
7% w/w (R2=0.997); (c) 33.33% w/w (R2=0.998); (d) 50% w/w
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581 577
for PEO (Mw=8�106) and 8.0�10�4 mg/cm2 s for
PEO (Mw=4�106). This agrees with the finding that the
presence of PEO of higher molecular weight causes a
formation of stronger matrix, which is less liable to
dissolve, than PEO of lower molecular weight [8]. It is
also possible to develop scaling laws for predicting the
dissolution constant of different molecular weight
PEOs as what Ju et al. [36] did.
With the knowledge of water diffusion and
polymer dissolution parameters, we are able to predict
the dimensional changes of the PEO tablet during
swelling and dissolution process from the numerical
calculation. Prediction of the relative dimensional
change in radial and axial directions during swelling
0
20
40
60
80
100
0 5 10 15 20time (hr)
rela
tive
dru
g r
elea
se (
%)
rela
tive
dru
g r
elea
se (
%)
0
20
40
60
80
100
0 5 10 15 20time (hr)
rela
tive
dru
g r
elea
se (
%)
rela
tive
dru
g r
elea
se (
%)
(a)
(c)
Fig. 6. Fit of the model to the experimentally determined relative amount
Squares denote experimental data. (a) 8.33% w/w (R2=0.994); (b) 16.6
(R2=0.969).
and dissolution is shown in Fig. 4. The experimental
results of radial change as function of time are also
compared with the model predictions. We do not
measure the dimensional change of the tablets in axial
direction due to difficulty in handling the tablet. The
profiles of dimensional change are consistent with the
water uptake profiles (Fig. 2). PEO of Mw=4�106
swells fast initially and then decreases significantly in
dimensions. Therefore, swelling is dominant for PEO
ofMw=8�106 over 20 h while for PEO of Mw=4�106
polymer dissolution dominates over swelling after
initial 8 h. It also shows that the relative increase in
height is much faster (approximately two folds) than
that in radius during swelling for both PEOs. Since the
0
20
40
60
80
100
0 5 10 15 20time (hr)
0
20
40
60
80
100
0 5 10 15 20time (hr)
(d)
(b)
of caffeine released at different loadings from PEO of Mw=4�106.
7% w/w (R2=0.997); (c) 33.33% w/w (R2=0.991); (d) 50% w/w
0 6 12 18 24
0.7
0.8
0.9
1
0.6
time (hr)
rela
tive
dry
wei
ght o
f PE
O
8.33%w/w
16.67%w/w
33.33%w/w
50.00%w/w
0 5 10 15 20
0.4
0.6
0.8
0.2
0
time (hr)
rela
tive
dry
wei
ght o
f PE
O8.33%w/w
16.67%w/w
33.33%w/w
50.00%w/w
(a)
(b)
Fig. 7. Model prediction of relative dry weight of PEO polymers
during drug release process at different initial drug loadings (weigh
percentage). (a) PEO of Mw=8�106; (b) PEO of Mw=4�106.
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581578
surface area in the axial direction (1.57 cm2) is higher
than that in radial direction (1.20 cm2), the majority of
water is diffusing in the matrix from the axial
direction.
4.2. Drug release from PEO matrices
The results of caffeine release from PEO matrices
of two different molecular weights at different
caffeine loadings into simulated intestinal fluid are
shown in Figs. 5 and 6. The model is fitted to the
experimentally determined relative amount of drug
released vs. time. It can be seen that drug release
profiles using this model are predicted with a very
good agreement with experimental data at different
initial loadings. Drug release from PEO matrices
involves two mechanisms, diffusion through swel-
ling polymer and release via polymer dissolution.
Thus the swelling and dissolution behaviors of
tablets made of pure polymer play important roles
in the overall drug release process. As shown in
Figs. 5 and 6, both lower and higher molecular
weight PEO tablets do not provide zero-order release
kinetics. This is in agreement with the polymer
swelling and dissolution experiments we have done
before, in which no synchronization of swelling and
dissolution is shown. It can also be seen that the
caffeine release is faster in lower molecular weight
PEO matrix compared to the release rate in PEO of
higher molecular weight. This is, probably, due to
the faster swelling and dissolution rate of lower
molecular weight polymer. The model developed
here can simultaneously calculate the amount of
water uptake and polymer dissolution as a function
of time in the drug release process, which is difficult
to evaluate in experiment. Therefore more informa-
tion on the release kinetics may be revealed by the
mathematical modeling. In Fig. 7, the predicted
relative amount of remaining polymer during caf-
feine release at different initial drug loadings is
plotted vs. time. It is found that as initial drug
loading increases, polymer dissolution becomes
more and more obvious since relative amount of
polymer dissolved in the release medium also
increases. Within the range of loadings investigated
in this paper, swelling and diffusion govern the
release kinetics for higher molecular weight PEO
since most of the polymer does not dissolve even
t
when almost all drugs are depleted. On the other
hand, polymer dissolution plays at least an equal
role with swelling during the caffeine release via a
lower molecular weight PEO matrix.
It is worthwhile to find out expressions which
describe relationship between the equilibrium drug
diffusion coefficient D2,e, the concentration depend-
ence factor b2 and the initial drug loadings in order to
predict drug release profiles at other initial loadings
and reduce the amount of experimental work. It is
found that power functions can be used to correlate
the equilibrium diffusion coefficient and the concen-
tration dependence factor with initial drug loadings
approximately, as shown in Fig. 8.
0 5 10 15 200
0.2
0.4
0.6
0.8
1
time (hr)
rela
tive
drug
rel
ease
2z0 / r0 = 0.5
2z0 / r0 = 1.0
2z0 / r0 = 2.0
0 5 10 15 200
0.2
0.4
0.6
0.8
1
time (hr)
rela
tive
drug
rel
ease
r0 / 2z0 = 0.5
r0 / 2z0 = 1.0
r0 / 2z0 = 2.0
(a)
(b)
Fig. 9. The effect of tablet (PEO ofMw=8�106) size on drug release
profile. The initial drug loading is 25.0% w/w. (a) Constant radius
(0.5 cm) with different heights (0.25, 0.5 and 1.0 cm); (b) constan
heights (0.5 cm) with different radii (0.25, 0.5 and 1.0 cm).
Fig. 8. Correlation of b2 and D2,e with different initial caffeine
loadings (w) for two different molecular weight PEOs. Square:
D 2,e�10�6 cm2/s; triangle: b2. (a) PEO of Mw=8�106,
D2,e=1.5008�10�6(w)�0.2464 cm2/s, b2=2.6056�10�6(w)0.069; (b)
PEO of Mw=4�106, D 2,e=1.5353�10�6(w )�0.2143 cm2/s,
b2=3.945�10�6(w)0.086.
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581 579
4.3. Influence of tablet dimensions on the drug release
kinetics
The influence of initial tablet dimensions (radius
and height) on drug release profile is also studied
since they are the factors that can be easily changed in
both modeling and experiment. Fig. 9a shows the
modeling predictive effect of initial height on release
profile with constant radius, and the effect of changing
initial radius with constant height is shown in Fig. 9b.
In all cases, the initial drug loading in high molecular
weight PEO (Mw=8�106) tablets is kept in a constant
of 16.67% w/w. The predictions show that relative
drug release rate is faster when the initial height
decreases with constant radius or the initial radius
increases with constant height. This is because the
ratio of surface area to volume is lager with
decreasing radius if the height is kept constant. On
the other hand, if the radius is a constant the ratio of
surface area to volume is smaller at larger initial
height. Thus, the ratio of surface area to volume (i.e.
relative surface area) is an important parameter in
drug release kinetics.
t
Fig. 10. The effect of tablet (PEO of Mw=8�106) aspect ratio on
drug release profile. The initial volume is 0.25 cm3 and drug loading
is 25.0% w/w. From top to bottom, the release profiles correspond to
the tablet of (a) r0=0.2 cm, 2z0=1.989 cm (As=2.75 cm2); (b) r0=0.3
cm, 2z0=0.884 cm (As=2.23 cm2); (c) r0=0.8 cm, 2z0=0.124 cm
(As=4.64 cm2); and (d) r0=0.5 cm, 2z0=0.381 cm (As=2.57 cm2).
N. Wu et al. / Journal of Controlled Release 102 (2005) 569–581580
We are also interested in the effect of aspect ratio on
drug release profile. Four types of tablets from the
shape of a flat disc to a slim cylinder are modeled with
constant initial volume (0.25 cm3) and drug loading
(25% w/w). As one can see in Fig. 10, the tablets with
shapes of a flat disc or a slim cylinder release drug
much faster than the other two bregularQ shape tablets.The difference in relative surface area is one of the
reasons why different shape of tablet has so different
release profile. However, the large difference in
relative surface area between the flat disc and the
slim cylinder shape does not show any significant
influence in terms of releasing rate. This probably
indicates that the length of diffusion path is also
another important factor that can affect drug release
rate. In tablets with either vary large or very small
aspect ratio, one of the diffusion paths (either in axial
or radial direction) is extremely short, which can
expedite the diffusion of drug in release medium.
5. Conclusions
In this work, a mathematical model is developed
and in vitro study is carried out for the controlled
release of caffeine from polyethylene oxide (PEO)
cylindrical tablets with two different molecular
weights (Mw=8�106 and Mw=4�106). The swelling
of the hydrophilic matrix and water penetration, three-
dimensional and concentration-dependent diffusion of
drug and water, and polymer dissolution are taken into
account simultaneously in this model. The moving
boundary conditions for the swelling and dissolution
front are explicitly derived, and the resulting coupled
partial differential equations are solved numerically
by using a front-fixing method. The model agrees
with the water uptake, polymer dissolution and
dimensional change profiles very well for pure PEO
tablets. No synchronization of swelling and dissolu-
tion is found in experiments and modeling for both
PEOs. It is found that swelling is the dominant factor
in drug release kinetics for higher molecular weight of
PEO (Mw=8�106) while both swelling and dissolu-
tion are important to caffeine release for lower
molecular weight PEO (Mw=4�106). Drug release
profiles using this model are predicted with a very
good agreement with experimental data at different
initial loadings. It is also found that when initial drug
loading increases, polymer dissolution becomes more
and more important in the release process. Besides
swelling and dissolution properties of polymer, the
ratio of surface area to volume and the aspect ratio of
initial tablets are also found to be influential in the
overall release profile. For possible further improve-
ment of the model, the more complicated expression
of diffusion coefficient, which is related to the
microstructure of diffusion components and polymer
[21], may be incorporated into the analysis.
Acknowledgements
This research was funded by Agency for Science,
Technology and Research, as well as Defence Science
and Technology Agency, Singapore.
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