Mathematical modeling and in vitro study of controlled drug release via a highly swellable and dissoluble polymer matrix: polyethylene oxide with high molecular weights

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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


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.


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


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)


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


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


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.


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.


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).


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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Documents

Mathematical modeling and in vitro study of controlled drug release via a highly swellable and dissoluble polymer matrix: polyethylene oxide with high molecular weights