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Mathematics-aided quantitative analysis of diffusion characteristics of pHEMA sponge hydrogels.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2007; 2: 609–617
Published online 10 August 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI:10.1002/apj.062
Research Article
Mathematics-aided quantitative analysis of diffusion
characteristics of pHEMA sponge hydrogels
X. Lou,1 * S. Wang2 and S. Y. Tan1
1
2
Chemical Engineering Department & Nanochemistry Research Institute, Curtin University of Technology, Bentley WA 6102, Australia
School of Mathematics and Statistics, University of Western Australia, Crawley WA 6009, Australia
Received 4 April 2007; Revised 6 June 2007; Accepted 24 June 2007
ABSTRACT: This study reports the current progress in quantitative analysis of the release characteristics of pHEMA
spongy hydrogels using prednisolone 21-hemisuccinate sodium salt as a model drug. Extraction of effective diffusion
coefficients of the drug from various pHEMA matrices was made using a novel mathematical model that handles
both boundary layer and initial burst effects. Drug loading level and entrapment efficiency were also determined.
The computed diffusion coefficients and the drug loading capacity in relation to the device porous structure and drug
concentration of the loading solution, as well as the size of device are discussed. Mathematical modelling proves to be
a powerful tool not only for establishing and interpreting structure and performance relationships but also for handling
experimental ambiguity.  2007 Curtin University of Technology and John Wiley & Sons, Ltd.
KEYWORDS: pHEMA; diffusion coefficient; mathematical modelling; drug delivery system
INTRODUCTION
Quantitative analysis of drug release characteristics and
their relation to the properties of a drug carrier are
critical to the design and development of an effective
controlled drug delivery system which aims to optimise
a drug delivery process compatible with the absorption
rate of the human body and requires a device to
supply and release therapeutic agents to the desired
location with a precise therapeutic dose for a prolonged
period. The ultimate effect and of great significance
to a controlled delivery system include the optimal
drug response, minimum side effects, prolonged drug
efficacy, and therefore less drug wastage and follow-up
care, lower risk and better patient’s compliance.
Controlled delivery devices are generally composed
of polymers as matrices, which are capable of releasing the drugs continuously over time periods ranging
from hours to days or months. In a diffusion-controlled
polymeric system, the release of drugs is controlled by
the drug diffusing through a complex porous path in
the polymer matrix, which is driven by the concentration gradient between the carrier and the drug release
medium or body fluid. The effective diffusion coefficient of a drug in a delivery system, critical to its
functionality and performance, is largely determined by
*Correspondence to: X. Lou, Chemical Engineering Department &
Nanochemistry Research Institute, Curtin University of Technology,
Bentley WA 6102, Australia. E-mail: X.Lou@curtin.edu.au
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
the structural parameters of the system and the interactions between the drugs and the transport matrix. When
these properties are unknown or difficult to quantify,
extraction of the effective diffusion coefficient of the
system becomes a major concern. Various techniques
have been developed for the identification of effective
diffusivities of matrices with limited pore sizes. Little
has been done on such materials as phase-separation
spongy pHEMA, which possess a range of pore sizes
from several to a hundred micrometers.
Phase-separation spongy pHEMA hydrogels are often
produced in the presence of a large quantity of water
(reportedly above 40–45%) (Peppas and Lustig, 1986a;
Chirila et al ., 1993; Lou et al ., 1999; 2000). In comparison with their homogenous and transparent analogue, which has been extensively investigated and
well known for their applications as in contact lens
and intraocular lens, the spongy pHEMA shows an
opaque and softer form because of the pores that are
large in both size and number. These materials encourage cell and tissue integration and have demonstrated
excellent performance in ophthalmic implants (Chirila et al ., 1998; Crawford et al ., 2002; Hicks et al .,
2006). They have also shown potential in such applications as drug delivery systems and tissue scaffolds (Lou
et al ., 2004). In some degree, the drug diffusion mechanisms in the pHEMA sponge matrix become more
complicated owing to the contributions from both solute
diffusion and convection (Peppas and Lustig, 1986b;
610
X. LOU, S. WANG AND S. Y. TAN
Asia-Pacific Journal of Chemical Engineering
Lou et al ., 2004). Accurate determination and establishment of drug transportation parameters and their
correlations with the porous structure of such hydrogels become important for the optimal design of an
applicable device to deliver target drugs in a controlled
fashion. Investigation on the transportation phenomena and diffusion coefficient of various molecules in
porous pHEMA matrices would provide valuable information for their extended applications in the biomedical
field.
In this work, we use a newly developed mathematical
model for the extraction of effective diffusion coefficients of a model drug, prednisolone 21-hemissucinate
sodium salt, in a series of porous pHEMA spongy
matrices. Boundary-layer effect and the initial burst
phenomenon in a diffusion process were treated mathematically. Devices made of pHEMA sponges containing various pore sizes were fabricated into cylindrical
geometry. Drug release profiles of each device were
monitored and their effective diffusion coefficients computed. The effect of the porous structure on the estimated effective diffusion coefficient, the drug loading
capacity and the drug entrapment efficiency are discussed.
MATERIALS AND METHODS
Materials
HEMA (Bimax, ophthalmic grade) was used as received. The cross-linking agent 1,5-hexadiene-3,4diol (DVG) with a purity of 97% was supplied
by Sigma-Aldrich. An aqueous solution of 10 wt%
ammonium persulphate (APS) (Ajax Chemicals) was
used together with N , N , N , N -tetramethylethylene
diamine (TEMED) (Aldrich Chemical Co.) as initiators. Prednisolone 21-hemissucinate sodium salt powder
(Sigma Chemical Co., Belgium) was used as a model
drug. Deionised water was used for all experiments in
the study.
Preparation of porous pHEMA discs
Nine sets of pHEMA discs were prepared according to
the formulas and sizes given in Table 1. The amount of
each chemical was adjusted proportionally depending
on the total volume of monomer mixture required.
To cast the polymer discs, HEMA and water were
well mixed in a beaker followed by addition of the
cross-linking agent (DVG) and the initiators (APS
and TEMED). The solution was then distributed in
a 24-well cell culture plate. Volume of the monomer
mixture in each well was determined by the height of
discs required. Polymerisation was carried out at room
temperature for 3 h and then at 50 ◦ C for 24 h.
Following the polymerisation, the samples were
removed from their moulds and immersed in deionised
water for 4 weeks with daily water exchange to remove
residual monomers and oligomers. The purified discs
were kept in deionised water for further work.
Measurements on pHEMA discs
Several measurements were carried out on each set
of pHEMA discs to determine their equilibrium water
content (EWC), water and polymer volume fractions
(1 and 2 , respectively) and polymer density of
fully hydrated pHEMA (ρp,w ), as well as that of dried
pHEMA (ρp,d ). These include
Ww,a : the weight of a fully hydrated disc in air,
Wd,a : the weight of the same disc in air after dehydration,
Ww,w : the weight of a fully hydrated disc in water, and
Wd,w : the weight of the same disc in water after
dehydration.
EWC, as a weight percentage, is given by Eqn (1):
EWC =
Ww,a − Wd,a
× 100%
Ww,a
(1)
Table 1. Formulation for pHEMA discs.
Sample
ID
A
B
B1
B2
B3
C
D
D1
E
Diametera
(mm), height (mm)
14,
14,
14,
14,
7,
14,
14,
14,
14,
7
7
14
3
7
7
7
3
7
HEMA
(g)
Water
(g)
DVG
(µl)
APS (10%)
(µl)
TEMED
(µl)
5
4
4
4
4
3
2
2
1.5
5
6
6
6
6
7
8
8
8.5
50
40
40
40
40
30
20
20
15
100
80
80
80
80
60
40
40
30
100
80
80
80
80
60
40
40
30
a
Data were taken on the basis of the moulds used for device preparation and are only indicative of the size
variation.
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2007; 2: 609–617
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
DIFFUSION CHARACTERISTICS OF pHEMA SPONGE HYDROGELS
Water and polymer volume fraction (1 and 2 ) are
given by Eqn (2), where Vtotal represents the volume of
a fully hydrated disc and Vp the volume of polymer,
i.e. the same disc after dehydration. Both Vtotal and Vp
are calculated on the basis of Archimedes’ buoyancy
principle where the density of water, ρw , was taken as
1.0 mg/cm3 (Eqns (3) and (4)):
Vp
, and φ1 = 1 − φ2
Vtotal
Wd ,a − Wd ,w
Vp =
ρw
Ww ,a − Ww ,w
Vtotal =
ρw
φ2 =
(2)
30mL glass vial
Teflon Disc
15mL
Drug Loaded Device
(b)
r2
r1
(3)
(4)
Polymer density of a fully hydrated specimen, ρp,w ,
and that of the same sample when it is dehydrated, i.e.
the dry polymer ρp,d , are calculated using Eqns (5) and
(6).
Wd,a
Vtotal
Wd ,a
=
Vp
(a)
ρp,w =
(5)
ρp,d
(6)
Average values of five measurements for each sample
were used. Details of the measurements can be found
in our previous work (Lou et al ., 1997; 2000).
Drug loading
Three drug solutions containing 0.5, 1.0 and 1.5 wt%
of model drugs were prepared. After the total removal
of water in a freeze drier (Dynavac), pHEMA discs
were placed in 10 ml stock solution for up to 10 days
to facilitate drug loading. The dimensions and weight
of the device were recorded after loading of the drugs.
Diffusion experiment
The diffusion experiment was carried out in a 30-ml
glass vial at room temperature. The device pre-loaded
with the drug was placed in the glass vial after being
gently blotted with a tissue to remove the excessive drug
solution, and its height was measured. Deionised water
(15 ml) was then added carefully and the vial gently
agitated by an orbital shaker (Chiltern Scientific) at a
speed of 45 rpm. Five hundred microlitres of solution
was withdrawn from the vial at 15, 30, 45, 60 and
90 min, 2, 3, 4, 6 and 9 h, 1, 2 and 3 days to monitor
the concentration change. Teflon disks were used on
the top of the device to serve as an impermeable barrier
(Fig. 1(a)). The absorbance at 247 nm of the sample
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 1. (a) Diffusion experiment set-up. (b) Top view
of a cylindrical device of radius r1 in a container of
radius r2 . This figure is available in colour online at
www.apjChemEng.com.
collected from each vial was determined with a UV
spectrometer (GBC Scientific Equipment, Australia).
The concentration was calculated on the basis of a
calibration curve that was obtained from a series of
drug solutions with known concentrations. The amount
of drug diffused into the solution at various time points
was computed to generate the experimental time-release
series data (release profiles).
Determination of drug loading level and
entrapment efficiency
Devices that completed the diffusion experiments were
taken out from the diffusion glass vial and further
extracted with fresh water. The extracted drugs were
quantified using the UV/vis spectroscopy and summed
with the quantity of drugs already released into the
deionised water during the diffusion experiment. This
provided the total amount of drug loaded in a specific
device.
The drug loading level and entrapment efficiency of
each device were then determined by the following
equations.
Drug loading level = Amount of drugs loaded (mg)/
Dry weight of the device (g)
Entrapment efficiency = Drugs loaded (mg)/
Drugs used for loading (mg)
Extraction of the effective diffusion coefficient
of each device
The basic model and its analytical solution
On the basis of the diffusion experimental set-up, the
top view of a device of cylindrical geometry placed in
a diffusion container of similar geometry is depicted
Asia-Pac. J. Chem. Eng. 2007; 2: 609–617
DOI: 10.1002/apj
611
612
X. LOU, S. WANG AND S. Y. TAN
Asia-Pacific Journal of Chemical Engineering
in Fig. 1(b) in which r1 is the device radius and r2
the container radius. As the drug release experiments
were carried out on an orbital shaker at a relatively
low resolution, it is reasonable to assume that release
environment is well mixed along the rotational direction
and the diffusion process is radial only (more details can
be found in Wang and Lou, 2007). That is to say, for
a device that is initially loaded with amount of M 0 of
drug, the concentration of drug in liquid in the container
is a function of r, and the diffusion process is governed
by the following diffusion equation in polar coordinates,
∂ 2 C (r, t) 1 ∂C (r, t)
∂C (r, t)
=D
,
+
∂t
r
∂r
∂r 2
× 0 < r < r2 , t > 0
(7)
∂C (r2 , t)
= 0, t > 0
∂r
0
M /Vd , 0 < r < r1
C (r, 0) =
0,
r1 < r < r2
(8)
(9)
where D, the effective diffusion coefficient, is independent of r and C (r, t) is the unknown concentration of
the drug at r at time t, and Vd denotes the volume of the
device. Using the technique of separation of variables,
the solution to Eqn (7) satisfying Eqn (8) is given by
the following series (Wang and Lou, 2007):
C (r, t) =
∞
An J0
n=0
αn r
r2
e −Dαn t/r2
2
2
(10)
where J0 denotes the zero-order Bessel function of the
first kind and An is a coefficient to be determined for
each n = 0, 1, 2, . . .. The constants α0 , α1 , . . . , αn , . . .
are the non-negative roots of the equation, J1 (αn ) = 0,
where J1 is the first-order Bessel’s function of the first
kind. The coefficient An is determined on the basis of
the initial condition in Eqn (9) and is given as follows:
A0 =
M 0σ 2
and
Vd
An =
2M 0 σ J1 (σ αn )
for n = 1, 2, . . . .
Vd αn J02 (αn )
(11)
where σ = r1 /r2 (details can be found in Wang and
Lou, 2007).
Since α0 = 0 and J0 (α0 ) = 1, Eqn (10) then becomes
C (r, t) =
∞
M 0 σ 2 2M 0 σ J1 (σ αn )
+
Vd
Vd n=1 αn J02 (αn )
αn r
2
2
e −Dαn t/r2 , 0 < r < r2 (12)
× J0
r2
Equation (12) defines an analytical solution to Eqns
(7)–(9).
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Finding Mt , total mass released at time t
For the above model, the total mass released, Mt , at
time t can be determined by integrating C (r, t)rdrdφ
over the solution region, 0 ≤ φ ≤ 2π and r1 ≤ r ≤ r2 ,
where C is given by Eqn (12)
Mt =
M 0σ 2
4M 0 σ 2 Vc
(Vs − Vd ) −
Vd
Vd
∞
2
J1 (σ αn ) −Dαn2 t/r 2
2
×
e
α 2 J 2 (αn )
n=1 n 0
(13)
where Vs is the volume of the solution in the container.
For t → ∞, the total mass release in infinite time is
given as follows:
M∞ =
M 0σ 2
(Vs − Vd )
Vd
Dividing Eqn (13) by M∞ yields
∞
Mt
4 J12 (σ αn ) −Dαn2 t/r 2
2
=1−
e
M∞
1 − σ 2 n=1 αn2 J02 (αn )
(14)
Equation (14) defines the ratio of the mass released
from the device into the liquid in the time interval [0,
t] and the total mass release from the device in infinite
time.
The initial burst
An initial burst often appears in a release process mainly
because of excessive drug load near the surface of
a device or/and some free drugs left on the device
surface during the loading process. In this case, the
initial release rate is substantially greater than that of
the rest of the process. It is also possible that the initial
release rate is smaller than the normal rate if a device
is overly pre-washed with the intension to remove the
free drugs on the device surface, which is a common
practice in the lab. In either case, the computed effective
diffusion coefficient would differ substantially from the
actual one should the process be treated as a single
phase. To overcome this problem, the diffusion process
might be divided into two phases: the initial burst and
a normal diffusion with the following assumptions:
D=
D0 ,
D1 ,
0 < t < tc ,
t > tc ,
where D0 and D1 are effective diffusion coefficients for
the burst phase and the normal phase respectively, and
tc is the threshold time. All of these parameters are yet
to be determined.
From the section ‘Basic Model and its Analytical
Solution’ one may see that when 0 ≤ t ≤ tc , the concentration C (r, t) is given by Eqn (12) with D = D0 ,
Asia-Pac. J. Chem. Eng. 2007; 2: 609–617
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
DIFFUSION CHARACTERISTICS OF pHEMA SPONGE HYDROGELS
while C(r, t) for t > tc can be derived as follows using
the same argument as that for Eqn (10):
An J0
n=0
αn r
r2
e −D1 αn t/r2 , for t > tc
2
2
(15)
Considering the continuity condition at tc that
C (r, tc− ) = C (r, tc+ ), the coefficient An is given below
A0 =
M 0σ 2
2M 0 σ J1 (σ αn )
and An =
Vd
Vd αn J02 (αn )
× e −(D0 −D1 )αn /r2 for n ≥ 1
2
2
By substituting these into Eqn (15), the expression of
C (r, t) at t > tc is given by the following equation:
∞
M 0 σ 2 2M 0 σ J1 (σ αn )
+
Vd
Vd n=1 αn J02 (αn )
αn r
2
2
e −(D0 −D1 )αn t/r2 , t > tc
× J0
r2
C (r, t) =
(16)
Using the same argument as that for Eqn (14), the
Mt for the entire process is then expressed as
ratio M
∞

∞
J12 (σ αn )

4

1 −
2

1 − σ n=1 αn2 J02 (αn )




2
2

Mt
×e −D0 αn t/r2 ,
=
∞
J12 (σ αn )
M∞ 
4

1
−

2


1 − σ n=1 αn2 J02 (αn )




2
2
×e −αn (D1 (t−tc )+D0 tc )/r2 ,
Drug loading level and drug entrapment
efficiency
All drug loading and release experiments were carried
out in duplicate. There were no significant variations
observed between the two sets of experimental results
(see Fig. 2 the comparison of the loading data). For
0 ≤ t ≤ tc ,
Comparison Between Duplicate Experiments
(In 0.5%, 1.0% and 1.5% drug
solutions repectively)
t > tc .
(17)
RESULTS AND DISCUSSION
Physical properties of porous pHEMA discs
Drug loading level,
mg/g
C (r, t) =
∞
are summarised in Table 2. In general, the higher the
initial water content, the greater the EWC and water
volume fraction. Both polymer volume fraction and
wet polymer density display a descending trend with
the increase of water content in the initial monomer
mixture. However, all dry spongy specimens show a dry
polymer density (ρp,d ) in the range of 1.27–1.30. While
the observation on dry polymer density demonstrates a
full collapse and densification of polymer mass during
the dehydrating process at a relative low temperature
(50 ◦ C), other results indicate clearly a greater porosity
of specimens made in the presence of larger quantity of
water which is not unexpected.
Quantitative analysis on the porosity of such materials
based on micrographic images has been proved difficult,
and the results are biased, in particular when the materials are fully hydrated. To examine the porosity effect
on the release characteristics of the sponge devices, wet
polymer density, which is inversely proportional to the
porosity, is used in this paper as a quantitative measure.
350
300
250
200
150
100
50
0
A B C D E
A B C D E
A B C D E
Devices
The spongy discs varying in the initial water content in
their monomer mixtures show significant differences in
their EWC, polymer volume fraction and water volume
fraction, as well as in the polymer density, which
Exp 1
Exp 2
Figure 2. Drug loading level in various devices. This figure
is available in colour online at www.apjChemEng.com.
Table 2. Equilibrium water content, volume fraction and density of hydrated pHEMA devices.
Sample
ID
A
B
C
D
E
HEMA : water
ratio
EWC
(%)
ρp,w , Wet polymer
(g/cm3 )
ρp,d , Dry polymer
(g/cm3 )
Polymer volume
fraction (2 )
Water Volume
Fraction (1 − 2 )
50 : 50
40 : 60
30 : 70
20 : 80
15 : 85
51.5 ± 2.1
62.9 ± 0.8
72.4 ± 0.5
76.3 ± 0.5
80.3 ± 2.5
0.547 ± 0.027
0.406 ± 0.009
0.295 ± 0.060
0.250 ± 0.006
0.208 ± 0.027
1.28 ± 0.02
1.27 ± 0.00
1.29 ± 0.00
1.29 ± 0.01
1.32 ± 0.05
0.428 ± 0.021
0.320 ± 0.008
0.228 ± 0.004
0.195 ± 0.003
0.159 ± 0.026
0.572
0.680
0.772
0.805
0.841
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2007; 2: 609–617
DOI: 10.1002/apj
613
Asia-Pacific Journal of Chemical Engineering
simplicity and convenience, the discussions in this paper
are based on one set of experiments only. Conclusions
are applicable to all results unless specifically indicated.
Figure 3 displays the influence of polymer density
and the drug concentration of the loading solutions on
the amount of drug loaded into a device. A drastic and
near-linear ascending dependence on polymer density
is shown in samples C, D and E, while little difference
in loading level is seen between A and B. The drug
concentration in loading solution has also a more
significant effect on samples C, D, E than A and B.
It is worthy of mention that sample A and B were
respectively prepared in the presence of 40 and 50
wt% of water, which are within (for B) and above
(for A) the critical water concentration at which a
transparent, and therefore more likely a non-porous
pHEMA hydrogel would have been produced. Drug
loading and diffusion would be impeded significantly
in such a matrix by the non-porous nature, which has
been a major limitation for the conventional pHEMA
hydrogel being used as a drug carrier for the delivery
of large molecules (Oxley et al ., 1993). Since C,
D and E were produced in more than 50 wt% of
water, they contain large pores ranging from several
to hundreds of micrometers (Chirila et al ., 1998; Lou
et al ., 2000, 2004). At this water concentration range,
the pore size increases further with an increase in initial
water concentration, which also leads to a decrease in
hydrated polymer density. As the pore voids increase,
the drug loading level rises as well, while for A and B,
only free volume and very limited pores are available to
accommodate the drug molecules (Yasuda et al ., 1969;
Yasuda and Lamaze, 1971; Lou et al ., 2004). The slight
change of drug loading level between A and B caused
by drug concentration might be due to some free drugs
left on the device surface. More details on surface free
drugs are discussed in a later section.
Drug entrapment efficiency, i.e. the ratio of loaded
drug and drug used for loading, is important for the
application of any useful device. The drug loading onto
each device was achieved by a simple diffusion process,
which allows the highest loading level and repeated use
of the drug solution possible, though the latter is not
the intention of this study. Comparison of drug entrapment efficiency in all devices is shown in Fig. 4. The
400
320
240
160
80
0
0.00
0.5 wt%
1.0 wt%
1.5 wt%
Sample E
Sample D
Sample C
0.10
0.20
0.30
Sample B
0.40
Polymwer density,
0.50
Sample A
0.60
g/cm3
Figure 3. Dependence of drug loading level on hydrated
polymer density. This figure is available in colour online
at www.apjChemEng.com.
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Entrapment efficiency, wt%
X. LOU, S. WANG AND S. Y. TAN
Drug loading
level, mg/g
614
60
50
40
30
20
10
0
A
B
C
Sample ID
0.5
1
D
E
1.5
Figure 4.
Comparison of drug entrapment efficiency. This figure is available in colour online at
www.apjChemEng.com.
efficiency of the drug entrapment is generally below
50% for all cases. This is due to the use of excessive
volume of loading solutions in the loading experiment.
An increase in efficiency is also demonstrated when the
device becomes more porous (from A to E) for both
0.5 and 1.0 wt% drug solutions, which is typical for
a diffusion-driven loading process. For 1.5 wt% drug
solution, a descending trend is shown from device A
to E. Some of these devices have turned slightly transparent on their surface after the drug loading, which
indicates the occurrence of polymer-drug interaction at
such high drug concentration. The interaction first took
place in the near-surface region of the discs, which hindered further diffusion of drugs into the matrix. More
details will be discussed in a future paper.
Computation of effective diffusion coefficients
Diffusion processes appear in many areas, including
geophysics, engineering, material science and biomedical sciences. In many cases, diffusion coefficients are
unknown and need to be identified using experimental or exploratory data. While the diffusion coefficient
can be a function of space, time and even concentration of the substance, in practice we normally seek a
constant approximation to the diffusion coefficient, i.e.
the effective diffusion coefficient that yields a diffusion process matching the observed one optimally in
the least-squares sense at the observation time points.
Most existing techniques for the extraction of diffusion coefficients are based on either empirical or
semi-empirical models for drug delivery mechanisms
or on analytical solutions of the diffusion equation (Fu
et al ., 1976; Abdekhodaie and Cheng, 1997; Siepmann
et al ., 1998; Siepmann and Peppas, 2001; Siepmann
and Göpferich, 2001; Grassi and Grassi, 2005). Analytical solutions can be obtained for certain geometries of
devices under some assumptions. Some classic results
Asia-Pac. J. Chem. Eng. 2007; 2: 609–617
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
DIFFUSION CHARACTERISTICS OF pHEMA SPONGE HYDROGELS
on analytical solutions for devices of the disc or spherical geometry in ‘well-stirred’ liquids (Crank, 1975)
have been widely used in practice. Although these models are popular, they are based on the assumption that
the fluid around the device is ‘well stirred’, implying
that the concentration is uniform in the liquid. This can
never be true in real cases, as the width of a layer can
never be zero and only through this layer can the drug
diffuse into the external liquid. Another phenomenon
in the drug delivery process that has so far not been
handled by the existing models is the ‘initial burst’ that
can be caused by the experimental error due to the limitation of available techniques in controlling such error.
This initial burst phase may severely pollute the true
effective diffusion coefficient of a process.
Using the mathematical model and the novel technique described in Section on Extraction of the Effective
Diffusion Coefficient, the experimental diffusion process was divided into two stages separated by a critical
time tc . The effective diffusion coefficient was given as
D = D0 when 0 ≤ t ≤ tc and D = D1 when t > tc . D0
and D1 represent, respectively, the diffusion coefficients
of the initial burst and a corrected normal diffusion
process of the drug. Release profiles of a set of samples are displayed in Fig. 5. The computed diffusion
coefficients of each device, D0 and D1 , as well as the
least-squares fitting errors and the critical time tc are
tabulated in Table 3. In general, D0 is greater than D,
indicating a burst effect of excessive drugs on or near
the device surface. Four out of thirty cases in this study
show a smaller D0 , presumably due to the overly blotted
device surface when removal of surface drug solution
was intended (Section on Diffusion Experiment). The
critical time tc is in the range 900–3600 s. The high
end is seemingly larger than expected, which is probably due to the very frequent collection of samples in
the first few hours.
Figure 5. Time-release series curves based on the new
mathematical model and technique. This figure is available
in colour online at www.apjChemEng.com.
Diffusion coefficient and drug loading level in
relation to the drug concentration and device
porosity
Displayed in Fig. 6(a) and (b) are the diffusion coefficient and drug loading level of each device in relation
to the drug concentration in the loading solution and
the device porosity. An indicative ascending trend of
drug loading level is observed with the increase in both
the drug concentration and device porosity (inversely
proportional to the polymer density). A similar trend is
also demonstrated in the diffusion coefficient, although
the dependence of diffusion coefficient on the drug concentration is less noticeable than that of drug loading
level. In terms of the dependence on the polymer density, a much smaller slope is apparent in the range
Table 3. Computed diffusion coefficient, least-squares fitting error and the critical time.
Drug conc. (wt %)
0.50
1.00
1.50
Sample ID
D0
(×10−6 cm2 /s)
D1
(×10−6 cm2 /s)
Squared error
(×10−2 )
tc (s)
A
B
C
D
E
A
B
C
D
E
A
B
C
D
E
12.4
10.4
6.97
25.7
24.8
32.0
38.4
18.6
33.5
68.9
51.5
19.5
35.2
101
187
2.18
3.25
1.99
1.28
1.52
4.14
3.43
3.73
7.90
16.1
3.80
3.01
3.80
9.26
49.3
4.71
0.84
3.82
1.85
1.69
1.52
0.93
1.00
0.21
0.88
3.15
1.36
0.39
0.80
0.58
3600
900
1800
900
900
3600
3600
1800
3600
3600
1800
2700
900
2700
1800
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2007; 2: 609–617
DOI: 10.1002/apj
615
616
X. LOU, S. WANG AND S. Y. TAN
Asia-Pacific Journal of Chemical Engineering
Table 4. Diffusion coefficient, D1 × 10−6 cm2 /s, for
devices of various sizes.
D1 (×10−6 cm2 /s)
Sample
ID
B
B1
B2
B3
D
D1
a
a
Diameter,
height (mm, mm)
0.5 wt%
drug
1.0 wt%
drug
14.0, 7.0
14.0, 14.0
14.0, 3.0
7.0, 7.0
14.0, 14.0
14.0, 3.0
2.26
2.76
2.96
–
10.4
12.1
4.07
2.62
2.87
1.61
20.1
18.2
Indicative data based on the moulds used for device preparation.
CONCLUSIONS
Figure 6. Drug loading level (a) and effective diffusion
coefficient (b) in relation to polymer density of a device.
0.35–0.55 for both the diffusion coefficient and drug
loading level. Again this is due to the non-porous nature
of homogenous pHEMA hydrogels. It is worthy of mention that the change in drug loading level in the porous
range, ρp,w = 0.1–0.3, is more significant than that of
diffusion coefficient. This is a very useful piece of information in the optimal design of drug delivery systems
for which a high loading capacity and a slow release
are often desirable.
Diffusion coefficient vs size of the device
A set of devices of various heights and diameters were
prepared and analysed on their diffusion coefficients.
Table 4 shows the values of D1 for these devices.
Change in device height has little influence on D1 for
both the less porous device B and more porous D. There
is a small reduction in D1 value shown in the device
that has a reduced diameter; however no sufficient data
is available to make it conclusive at this stage.
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Materials such as phase-separation spongy pHEMA
hydrogels contain much larger pores than their transparent counter materials. Mathematical modelling proves to
be a powerful tool not only for the extraction of drug
diffusion coefficients from these materials, but also for
reducing experimental errors that are not easy to handle in the common practice. The pHEMA hydrogels
reported in this study vary significantly from their conventional non-porous analogue in the porous structure,
and therefore lead to different performances in applications such as a drug delivery system. The results have
shown a great potential for them to be used to deliver
large molecules. The geometry parameter of the devices
has shown little influence on the diffusion characteristic of these matrices. However, the drug concentration
of the loading solution and device porosity affect drug
diffusion in various ways. The performance of these
matrices can be further improved by altering the porosity of the surface layer of each device, which is under
study by our group.
Acknowledgements
The authors wish to acknowledge the support of the
Australian Research Council via the ARC Discovery
Project Grant DP0557148 to this study.
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