Advances in Chemical Engi neering and Science , 2011, 1, 256-270
doi:10.4236/aces.2011.14037 Published Online October 2011 (http://www.SciRP.org/journal/aces)
Understanding the Glycoproteins Release from
Alginate-Barium Capsules in Physiologic Enviroments
Edgar Perez Herrero, Eva M. Martin del Valle*, Miguel A. Galán
Department of Chemical Engineering, University of Salamanca, Salamanca, Spain
Received July 5, 2011; revised July 27, 2011; accepted August 12, 2011
Abstract
The authors carried out a steady and unsteady mass transfer studies to simulate both the release of proteins in
physiologic environments and proteins transport through a tissue or organ from polymeric capsules by using
a substance, the rhodamine B isothiocyanate dextran (RBID) that mimics the behaviour of glycoproteins
such as vascular endothelial growth factor (VEFG). These studies highlighted the importance of electrostatic
interactions between alginate and proteins in the release processes. Thereby, this fact has opened new per-
spectives in order to use these kinds of capsules in protein recognition processes. The electrostatic interac-
tions between alginate and RBID allow pH-dependent controlled release systems that simulate the behaviour
of glycoproteins.
Keywords: Controlled Release Systems, Proteins, Glycoproteins, Growth Factors, Mass Transfer, Unsteady,
Steady
1. Introduction
The most common methods of drug administration, tab-
lets or pills and injections, satisfy the requirements on
efficacy for traditional drugs, not being suitable for drugs
developed in recent years, such as drugs based on pro-
teins. These difficulties lead to the need of developing
new systems of drug delivery, such as encapsulation,
which maintain therapeutic levels of the drug without
producing the unwanted side effects. In general, the main
advantages of these new systems are: providing protec-
tion to the encapsulated material and to enable constant
release profiles over time, without repeated administra-
tion [1].
The key point in the administration of traditional drugs
is to maintain the blood level of the active agent between
a maximum value, which represents the toxic level, and a
minimum level below which the drug is not effective. In
the controlled drug delivery systems, like microcapsules,
the drug level in blood follows a constant profile be-
tween the minimum and maximum levels allowed, over a
long period of time [2].
In order to achieve the understanding of the real be-
haviour of the capsules in the human body for a later use
in any of the multiple applications in the field of the
medicine is needed to carry out studies of mass transfer.
To do that it has been simulated the release of proteins
from polymeric spheres into physiologic environments
(external medium of finite volume) such as eyes [3-5],
bones [6,7], and periodontal pockets [8-10] by means of
a steady mass transfer study. This sort of protein release
process may occur when the protein concentration in the
medium surrounding the drug delivery device builds up
to appreciable levels. But also, it has been developed an
unsteady mass transfer study to simulate the protein
transport through a tissue or organ after a therapeutic
leakage from storage cavities.
Macroscopically, both the physiological environments
and tissues or organs, are ideally assumed as an isotropic
porous medium as shown in Figure 1 [11], since it is
studied the release of proteins from capsules to a packed
bed, simulated by the other capsules.
Figure 1. Drug transport in the human body, assumed as an
isotropic porous medium.

E. P. HERRERO ET AL.
257
To do that, in this paper, a novel and simple method
was used in order to develop the steady and unsteady
mass transfer studies. A simple perturbation method
based on the use of a colored substance, rhodamine B
isothiocyanate dextran (RBID), was used to mimic the
release of glycoproteins from polymeric capsules.
RBID is an amphoteric dye that is normally listed as
basic due to its overall positive charge. The structure of
the rhodamine B is shown in Figure 2. The main feature
of the structure of this dye is the quaternary nitrogen, but
it is also necessary to take into account the existence of a
carboxyl group, which would give the molecule an am-
photeric character at high pH values [12].
In addition, RBID is a biomacromolecule available in
three molecular weights, 10, 40, and 70 KDa, due to the
dextran molecules. Dextran is a complex, hydrophilic,
branched polysaccharide made of many chains of glu-
cose, characterized by its high molecular weight (MW),
good water solubility, low toxicity, and relative inactivity
[13].
Rhodamine B molecules can be functionalized with an
isothiocyanate group (-N=C=S), replacing a hydrogen
atom on the upper ring of the structure (see Figure 3).
The dextran molecule binds to the rhodamine B molecule
by the isothiocyanate group, leaving free the carboxyl
group (Figure 3). And then, RBID maintain the same
effective charge that rhodamine B, because of RBID re-
tains the carboxyl group that gives the amphoteric char-
acter to the molecule at high pH values (Figure 3).
2. Materials and Methods
2.1. Materials
Sodium alginate from macrocystis pyrifera (medium vis-
Figure 2. Rhodamine B structure.
(a) (b)
Figure 3. Structure of the Rhodamine B isothiocyanate (a)
and RBID (b).
cosity) was purchased from Sigma Chemicals, barium
chloride dihydrate, reagent grade was purchased from
Scharlau, RBID was purchased from Sigma Chemicals.
2.2. Production of Capsules
RBID-alginate solutions with different concentrations of
alginate (1.2 wt% and 3 wt%) were prepared dissolving
the proper amount of alginate in 1 mg/mL of RBID solu-
tion. Alginate-RBID solutions were extruded drop by
drop through a 25G needle using a sterile syringe into a 2
wt% barium chloride solution under stirring speed of 30
rpm. After two minutes of gelation, capsules were fil-
trated.
2.3. Experimental Device to Steady Mass
Transfer Study
In order to simulate the steady diffusional phenomenon,
one gram of the capsules generated were placed in a 25
mL spherical batch reactor with 5 mL of deionised water.
The reactor was placed over an orbital sake rotator with a
constant speed of 110 RPM. This stirring speed attempts
to simulate the movement of the human body in the pro-
cess of releasing the protein in a physiological medium.
2.4. Experimental Device to Unsteady Mass
Transfer Study
In order to simulate the phenomenon at the organ or tis-
sue, one gram of capsules were placed in a 25 mL con-
tinuous flow reactor with 5 mL of deionised water. The
reactor was placed over an orbital rotator with in con-
stant speed of 110 rpm. This stirring speed attempts to
simulate the movement of the human body in the process
of releasing the protein through a tissue or organ.
Two peristaltic pumps keep constant the volume inside
the reactor in addition to providing the advection effect
to the process. The amount of RBID released was deter-
mined by means of a spectrophotometer at 555 nm
(maximum peak absorbance for RBID) connect to the
system providing continuous data every minute. The
scheme of the equipment used to the simulation is shown
in Figure 4.
3. Experimental Mass Transfer Study
The cumulative release data of three different molecular
weight of RBID (10,000, 40,000, and 70,000 Da) were
obtained and plotted in Figures 5 and 6 to the steady and
unsteady processes, respectively. These data were ob-
tained for two different alginate concentrations (1.2 wt%
and 3.0 wt%).

258 E. P. HERRERO ET AL.
Figure 4. Equipment used to simulate the unsteady mass
transfer phenomena. 1 and 3 peristaltic pumps; 2 reactor
over an orbital shake rotator; 4 spectrophotometer; 5 ther-
mostatic bath.
(a)
(b)
Figure 5. Cumulative steady release of RBID at different
molecular weight with different alginate concentration. (a)
1.2 wt%; (b) 3.0 wt%.
(a)
(b)
Figure 6. (a) Cumulative unsteady release of RBID at dif-
ferent molecular weight (Alginate concentrat.: 1.2 wt%); (b)
Cumulative unsteady release of RBID at different mo-
lecular weight (Alginate concent.: 3.0 wt%).

E. P. HERRERO ET AL.
259
In these figures, the cumulative release of RBID has
been adimensionalized by dividing by the amount initially
loaded in the capsules. Therefore, it is possible to com-
pare the data from the three RBID molecular weights.
From Figures 5 and 6 it can be seen that the smallest
RBID exhibits a much lower cumulative release in both
alginate concentrations despite its smaller molecular
weight. Also, it is showed that high alginate concentra-
tion (3.0 wt%) leads to higher cumulative release. Both
behaviors can be observed quantitatively in Tables 1 and 2.
3.1. Effect of Molecular Weight of RBID in the
Cumulative Release of RBID
From the data shown it can be seen a concurrent release
of the molecules encapsulated since the smallest RBID
exhibits a much lower cumulative release in both algi-
nate concentrations despite its smaller molecular weight.
This behavior can be assigned to the presence of at-
tractive electrostatic interactions (Figure 7) between the
protonated amine group of RBID (positive charged) and
carboxyl groups of alginate (negative charged), that are
directly proportional to the charges, and indirectly pro-
portional to the square of the distance between charges,
i.e.,

2
f
Kqq d

 (1)
where K is the Coulomb constant.
As it was mentioned before, the variation of the mo-
lecular weight of the RBID is due to the dextran mole-
cules size. In this work, it was used three different mo-
lecular weight of RBID (10000, 40000, and 70000 Da).
RBID is a core, the rhodamine B, bond to the dextran
chains by means of the isothiocyanate group. The larger
size of dextran molecules, the smaller the attractive elec-
trostatic interactions due to sterically hinder of the dex-
tran molecules because of its conformation that separates
with a greater distance, d, the positive and negative
charges (Figure 8). The dextran molecules have a ran-
Table 1. Cumulative release data (steady process).
Molecular weight of RBID (KDa)
10 40 70
Alginate
concentration
(wt%) Cumulative release, C/Co = M/M∞ (%)
1.2 28 49 52
3.0 31 61 57
Table 2. Cumulative release data (unsteady process).
Molecular weight of RBID (KDa)
10 40 70
Alginate concentration
(wt%)
Cumulative release, M/M∞ (%)
1.2 44 67 68
3.0 54 90 92
Figure 7. Electrostatic interactions between RBID and algi-
nate.
dom-coiled conformation that covers the RBID core to a
greater or lesser extend, depending on the size of the
dextran molecules, hindering the electrostatic interac-
tions (Figure 8).
Therefore, the larger RBID molecules will be less at-
tracted to the alginate because of, based on the Equation
(1), the distance between charges will be longer, and
therefore there will be a larger cumulative release of high
molecular weight RBID [14,15].
This fact proves that the number of free molecules of
RBID in the capsules increases as the electrostatic inter-
actions decrease. So, it is as if the effective concentration
for the same initial concentration of RBID was greater
with increasing molecular weight of RBID. The con-
tinuous experiments should be taken into account for the
quantification of the electrostatic interactions between
RBID and alginate. Figures 6(a) and 6(b) showed that
the concentration of the supernatant is zero after finish-
ing the overall process. There are not free molecules ca-
pable of being released from the capsules at the end of
the process, and therefore the RBID molecules that do
not release from capsules will be linked by electrostatic
bonds to the alginate molecules. So, from a mass balance
to the capsules it is possible to quantify the electrostatic
bonds that are listed in the Table 3, where it is also
shown a summary of the influence of electrostatic inter-
actions in the cumulative release of RBID.
Table 3 shows that by increasing the molecular weight
of RBID, for the two concentrations of alginate, the elec-
trostatic interactions between alginate and RBID de-
crease, that is, when the molecular weight of RBID is
increased, the free molecules of RBID is also increased
into the capsules, increasing the cumulative release of
RBID.

260 E. P. HERRERO ET AL.
(a)
(b)
(c)
Figure 8. Simulation of the conformation of RBID mole-
cules in the capsules. Molecular weight of RBID: (a) 10
KDa, (b) 40 KDa, (c) 70 KDa.
The behavior found for RBID molecules opens new
ways of application for the capsules, taking into account
that the pKas of the RBID molecules, unknown at the
present time, will in turn determine the charge and the
acid-base behavior of the isomers as a function of pH. At
higher pH values, the attraction of RBID to the alginate
may be diminished as a result of unfavorable electrostat-
ics interactions when the carboxyl group gives the
molecule the amphoteric character. Knowing this, it is
possible to achieve a pH-dependent controlled release of
Table 3. Effect of the electrostatic interactions in the cumu-
lative release of RBID.
Molecular weight of
RBID (KDa)
10 40 70
Electrostatic interactions >>> >>>
Cumulative release + +++++
1.2 56 33 32
Alginate
concentration
(wt%) 3.0
% Electrostatic
interactions 46 108
proteins, resulting in protein imprinting processes.
This behaviour of RBID makes this macromolecule
very similar to proteins. But, in particular, RBID mimics
the behaviour of a special type of protein, the glycopro-
teins. These substances are proteins (core) that contain
oligosaccharide chains (glycans) covalently attached to
their polypeptide side-chains. Then, the structure of the
glycoproteins is very similar to the RBID, that is, RBID
is a core, the rhodamine B, bond to the dextran chains by
means of the isothiocyanate group (see Figure 9).
(a)
(b)
Figure 9. Analogy between RBID and Glycoprotein mole-
cules. (a) RBID; (b) Glycoprotein.

E. P. HERRERO ET AL.
261
Several hormones, antibodies, various enzymes, re-
ceptor proteins, cell adhesion proteins, growth factors,
etc, are glycoproteins. This fact is very important be-
cause it is possible to study the controlled release of sev-
eral growth factors by means of RBID that is much
cheaper than growth factors but also mimics the behav-
iour of these proteins. Herrero et al. [16] achieved the
immobilization of mesenchymal stem cells and mono-
cytes in biocompatible microcapsules to cell therapy.
Microencapsulated cells survive at least two weeks after
preparation in vitro. Monocytes produce growth factors,
such as vascular endothelial growth factor (VEGF). VEGF
is a homodimeric glycoprotein of relative molecular
mass of 45,000 Da [17]. Therefore, to achieve study the
release of the growth factor, such us VEFG, secreted
from the cells immobilized inside the microcapsules, is
sufficient to study the release of RBID.
3.2. Effect of Alginate Concentration in the
Cumulative Release of RBID
Also, Figures 5 and 6 shows that high alginate concen-
tration (3.0 wt%) leads to a higher cumulative release.
This behaviour is due to the mechanism of formation of
microcapsules. Once the alginate-RBID drop takes con-
tact with the cationic solution, instantaneously starting
membrane formation.
An increase of the alginate concentration produces an
increase of the number of molecules of the biopolymer
per volume unit available to react with the divalent
cations of barium chloride, producing quickly a very
dense network of gel [18]. As a result, there will be
fewer available alginate effective charges to interact with
the RBID molecules, leaving more free RBID molecules
that are able to release to the supernatant, giving higher
values of cumulative release.
3.3. Effect of Forced Convection in the
Cumulative Release of RBID
In this section it is compared the experimental data from
the steady and unsteady processes to observe the effect
of the forced convection in the mass transfer process
(Table 4). As shown in the table, the values of cumula-
tive release of RBID in the unsteady process, for all the
variables studied, are higher that those of the steady
process.
In the unsteady mass transfer process, besides the
mass transfer by diffusion, the fluid supplied by the peri-
staltic pumps produces advection processes removing
more quantity of RBID from capsules because it has a
double effect. The fluid is continuously making the
RBID coming out of the capsules, but also renews the
Tabla 4. Cumulative release data.
Steady mass
transfer study
Unsteady mass
transfer study
Alginate concentration (wt%)
1.2 3.0 1.2 3.0
Molecular weight of
RBID (KDa)
Cumulative release, M/M∞ (%)
10 28 31 44 54
40 49 61 67 90
70 52 57 68 92
supernatant constantly, so that, the concentration of RBID
in the supernatant is maintained at zero, producing a
constant increase of the concentration gradient between
the capsule and supernatant.
4. Theoretical Mass Transfer Study
Due to the difficulties encountered in the validation of
the mass transfer in patients (in vivo), modelling plays an
important role to find a suitable release profile to opti-
mize the therapeutic efficacy of a substance immobilized
[11].
For these reasons, in this paper, based on the experi-
mental work, a theoretical model was developed to
simulate the mass transfer processes of the release of a
protein from capsules to different conditions.
4.1. Theoretical Steady Mass Transfer Study
Based on the work of Lewinska et al. [19], a mathemati-
cal model was developed in order to simulate the steady
mass transfer of the drug from capsules into a finite ex-
ternal volume, that is, a steady mass transfer process.
Several assumptions were taking into account:
1) Buoyancy/gravity effects in the fluid are neglected.
2) The initial concentration of the tracer in the sur-
rounding solution is zero.
3) There is not a concentration gradient inside the
sphere.
The last assumption implies that it is considered each
sphere as an ideal well-mixed batch reactor with an iden-
tical tracer concentration CS everywhere inside a capsule
at a given moment of time.
A schematic of the process is described in Figure 10,
where before the beginning of the mass transfer process,
all the RBID molecules are inside of the capsules, that is,
the concentration of RBID in the capsules, CS, is the
same that the initial concentration, C0. Without any loss
of generality, it is considered that the initial concentra-
tion of RBID in the supernatant is zero (initially
“RBID-free” medium, C = 0). At the inception of the
process (t = 0), RBID starts diffusing out from the cap-
sule, that is, the steady mass transfer process starts from

262 E. P. HERRERO ET AL.
Figure 10. Initial conditions and time system evolution.
the capsule to the supernatant of the reactor, where RBID
will be accumulating with time until the equilibrium state
is achieved (t = teq) with a equilibrium concentration, Ceq.
In this moment the concentration of RBID in the capsule
and in the supernatant will be the same and the mass
transfer process will stop.
The mass transfer process from the capsules to the su-
pernatant can be generally described [20] by Equation
(2), where the variation of the concentration in the su-
pernatant is due to the mass transfer from the capsules to
the supernatant:

0
d
dS
C
VhAC
t 

C (2)
where C is the concentration of the tracer in solution
outside capsules, and CS the concentration of the tracer
inside capsules. The symbol A designates the total exter-
nal surface area of capsules, with h representing the mass
transfer coefficient, V0 the volume of solution outside
capsules, and VC the volume of capsules. The mass
transfer coefficient, h, describes the resistance to mass
transfer, 1/h, in the fluid boundary layer at the surface of
the sphere. The driving force for mass transfer is the
concentration gradient of the tracer in the batch reactor.
Taking into account the following boundary condition:
in the equilibrium state there is a constant equilibrium
concentration, Ceq, inside and outside the capsules,
eq
, the Equation (2) can be
complemented by a mass balance Equation (3) in the
form:

S
Ct Ct C  


00CSC eq
VCV CVVC (3)
Then, the direct relationship between concentrations C
and CS is represented by Equation (4):
0
00
CC
eq S
VV V
CC
VV




 C
as a function of time, given by Equation (5):
(4)
The substitution of C (Equation 4) into Equation (2)
and solution of this resultant equation with the initial
condition CS (t = 0) = C0, yields a general expression
describing the RBID concentration CS inside the capsules
2
00
1C
VhA
CC
C
 
 


00 0
1e
t
VV
eq eq
S
CC C








 (5)
The value of V0 is known for a given experiment as
well as the total capsule volume VC. Knowing the ex-
perimental average capsule diameter (D) in a sample, by
an optical microscope (Leica DM 1000), the total exter-
nal surface area of capsules (A) can be approximated
from their geometrical surface. For the capsules with
small size dispersion (less than 10%), samples can be
represented by N spheres of diameter (dp) and their total
surface area (A) can be expressed as:
2
d


42
p
AN







(6)
The total capsule volume VC can be estimated as:
3
d


4
32
p
C
VN






(7)
The experimental data corresponding to the cumula-
tiv
that the values of
bo
efficients given in Table 5
re
n coefficients obtained
Table 5. Values of mass transfer coefficients and effective
e release profiles were correlated using Mathematica
TM computer program, according to Equation (5), using
the mass transfer coefficient, h, in the model as a fitting
parameter. The mass transfer coefficient values, h, were
obtained for the different conditions, and were collected
in Table 5. Also, the values of the effective diffusivity,
calculated from the values of the mass transfer coeffi-
cient, h, and the Sherwood number (Explanation Table
1) [21,22] were listed in Table 5 to observe the effect of
the diffusion in the overall process.
From Table 5 it can be observed
th the external mass transfer coefficient and the effec-
tive diffusivity decrease when the molecular weight of
RBID and the alginate concentration increase, that is, the
mass transfer rate decrease.
The effective diffusion co
fer to the effective diffusion of the RBID in the cap-
sules structure that is different to that obtained in solu-
tion. In the case of this paper, RBID is not free, but in-
side the crosslinked gel network.
In comparison, data of diffusio
diffusivity at different conditions (steady process).
Alginate concentration (wt%)
1.2 3.0 3.0 1.2
Molecular
Mafer coent, Efdiffusiv e,
10000 3.10 × 1 × 10 2.12 × 1 × 10
weight of
RBID (Da)ss transfficie
h (cm/s)
0–4 9.62 –5
fective ity, D
(cm2/s)
0–8 4.69–9
40000 1.66 × 10–4 3.76 × 10–5 8.33 × 10–9 1.15 × 10–9
70000 1.43 × 10–4 3.20 × 10–5 6.66 × 10–9 9.00 × 10–1

E. P. HERRERO ET AL.
263
by pof
tr
ci
ree
m
e, must decrease
w
work of Kout-
w
Effect of alginate concentration in the mass transfer
coefficient, h, and the effective diffusivity, De:
Koutsoulos et al. [23] to the controlled release o
different proteins from croslinked gel networks and solu-
tion are attached in Table 6, including an estimation of
the Brownian diffusion by the Stokes-Einstein equation.
Effect of molecular weight of RBID in the mass
The values of both the external mass transfer coeffi-
cient and the effective diffusivity decreases when the
alginate concentration increase, that is, the mass transfer
rate decrease with a higher alginate concentration, be-
cause there is more resistance to the mass transfer. This
behavior is due to the mechanism of formation of micro-
capsules. Once the alginate-RBID drop takes contact
with the cationic solution, the formation of the mem-
brane starts instantaneously. The membrane grows from
outside to inside. An increase of the alginate concentra-
tion produces an increase of the number of molecules of
the biopolymer per volume unit available to react with
the divalent cations of barium chloride, producing
quickly a very dense network of gel [18]. This dense
membrane offers higher resistance to the mass transfer of
the RBID molecules, that is, the capsules will have less
porosity, decreasing the rate of mass transfer.
ansfer coefficient, h, and the effective diffusivity, De:
The values of both the external mass transfer coeffi-
ent and the effective diffusivity decrease when the mo-
lecular weight of RBID increases, that is, the mass trans-
fer rate decreases with a higher RIBD molecular weight,
because there is more resistance to the mass transfer.
In the case of high molecular weight of RBID, the f
olecules of RBID have a great difficulty to get out
through the pores of the capsules due to the steric hin-
drance of the large dextran molecules, hence the lower
transfer rate. When the molecular weight of RBID de-
crease, the steric hindrance decrease, and then the rate of
release of the free RBID molecules will be higher, since
there is less resistance to mass transfer.
The effective diffusion coefficient, DIn fact, these explanations were proved in a qualitative
way, since the capsules formed with a higher concentra-
ith increasing the size of molecules, since the equation
that describes Brownian diffusion—the Stokes Einstein
equation (Equation (8))—shows that the size of the mo-
lecules is indirectly proportional to the Brownian diffu-
sion coefficient [24], as can be observed in the values ob-
tained in this paper, shown in Table 5.
This effect was also reflected in the
Table 6. Diffusion coefficients estimated for the release of
different proteins at different conditions.
Diffusion coefficients (10–10 m2/s)
Protein Molecular
weight (KDa)Estimation (Ec.
Stokes-Einstein)
Effective
diffusion in
solution
Effective
diffusion
through gel
Lysozyme14.3 1.15 0.96 0.50
Trypsin
inhibitor 20.1 0.91 0.72 0.32
BSA 66.0 0.61 0.51 0.24
IgG 150.0 0.40 0.40 0.07
opulos et al. [23], as shown in Table 6.
3
b
kT
Dd

 (8)
where k is the Boltzmann constant, T is the temperature
in Kelvin scale,
 is the viscosity of the liquid, and d is
the diameter of the molecules. (Koutsopoulos et al., 2009).
Explanation Table 1. Determination of the effective diffusion coefficient.
The Sherwood number, Sh, is a dimensionless number that relates the resistance to mass transfer to the resistance to molecular diffusion, that is, it
measures the ratio of the mass transfer flux to the diffusive driving force.
p
M
hd
Sh
D
 (1)
where dp is the diameter of the capsules, DM is the molecular diffusion coefficient, and h is the mass transfer coefficient.
The Sherwood number can be expressed by means of two dimensionless numbers, the particle Reynolds number, Rep, and the Schmidt number,
Sc, by using the empirical correlation [27]:
1
0.63
21.1Re
p
Sh Sc (2) Rep
p
dv


 (3)
M
Sc
D


 (4)
The empirical correlation (2) can be applied to spherical particles with a particle Reynolds number between 3 and 10000 [27]. The molecular
diffusion coefficient, DM, is determined from Equations (1-4).
The effective diffusion coefficient, De, is obtained from the molecular diffusion coefficient, DM, and the porosity, ε, taking into the account that
each capsule is immersed in a packed bed formed by the other capsules, resulting in an isotropic porous medium. According to the model of Wa-
kao and Smith [28], the effective diffusion coefficient may be calculated from equation (5):
2
eM
DD

 (5) 0.82
empytotal spheres
total total
VVV
VV


  (6)

E. P. HERRERO ET AL.
264
t
p
cients to different proteins determined both in solution
and through a gel, and through the Stokes-Einstein equa-
t
s
solution by 10% - 20%, but differed significantly from
those calculated through the gel.ss tht the
Brownian diffusion predicts the behaviour of networks of
gel
ousion
c
t
calculated for the diffusion of the protein in solution.
This reflects the importance of th
the gel network in the diffusion coefficient, that is, the
om the
m
earlier works it has been established that the pure Darcy
w not give satisfactory results when it is required
to take into account the no slip boundary condition [26].
millionth of the sphere’s radius. Inside the boundary
layeradt intangen-
tial direction, with the velocity increasing rapidly from
the very small relative thickness of the boundary layer, it
Darcy’s slip velocity in the tangential direction to the
surface [25].
A schematic of the process and of the coordinate sys-
te
y, it is considered that the initial
co
ion of alginate were more resistant from a mechanical
oint of view, due to the dense structure of the gel.
Table 6 [23] shows the values of the diffusion coeffi-
included—that takes into account the no slip condition
on the surface of the sphere, has been applied, because in
ion. The results show that the Stokes-Einstein equation However, after several considerati
lightly overestimates the diffusivities of the proteins in
ons it is possible to
conclude that the thickness of the boundary layer is a
Thi meanar, there is a very high velocity gien the
s of very low density that can be likened to the release
f a protein in solution. It is noticed that the diff
zero to the value calculated by Darcy’s law. Because of
oefficients calculated for the diffusion of proteins
hrough a polymer network are much lower than those
is acceptable to neglect it and to consider that there is a
step velocity change from zero (no slip condition) to
e grade of crosslink of
denser the network of the gel, the lower the diffusion
coefficient, as there will be greater resistance to diffusion.
This behaviour reflected in the data of Koutsopoulos et al.
[23] can be also observed in the results presented in this
paper in the Table 5, that is, the effective diffusion coef-
ficient decreases with increasing concentration of algi-
nate that increases the density of the gel network.
4.2. Theoretical Unsteady Mass Transfer Study
Macroscopically, the tissue is ideally assumed as an iso-
tropic porous medium, which is described by Darcy’s
law [11].
Based on the Feng and Michaelides [25] mass transfer
studies, in order to develop a mathematical model to
simulate the unsteady mass transfer of a protein from a
microcapsule in a porous medium, the following as-
sumptions were considered:
1) Buoyancy/gravity effects in the fluid are neglected.
2) It is considered a constant diameter of the micro-
capsules with time (no shrink).
3) There is not a concentration gradient inside the
sphere.
4) Velocity field inside the porous medium is gov-
rned by Darcy’s law and is unidirectional far fre
sphere.
5) Torsional flow.
6) The initial tracer concentration on the bulk fluid is
zero.
7) The velocity field around the spheres is a solution
to the potential flow.
The assumption 3) implies that each sphere is consid-
ered as an ideal well-mixed batch reactor where the con-
centration at any point in the sphere is the same.
The 4) assumption implies that there may be slip at the
surface of the sphere. However, here the Brinkman
odel—extended Darcy formulation with inertial term
la does
m is depicted in Figure 11.
The presence of the sphere creates a disturbance to this
velocity field, which is essentially confined to the vicin-
ity of the sphere. Far from the sphere, the velocity field is
unidirectional. At the inception of the process (t = 0),
RBID starts leaking from the sphere. The two fluids mix
freely and are transported in the porous medium. Without
any loss of generalit
ncentration of RBID in the porous medium is zero
(initially “RBID-free” medium). After the inception of
the leakage process (t > 0) it should be considerer a
boundary condition at the surface of the sphere: the con-
centration of RBID is constant, Cs0.
Then, the boundary conditions for the external fluid
are:
0at
eat
z
vnr a
vU r



 
(9)
From the velocity field in the porous medium and us-
ing the boundary conditions for the external fluid, which
initially saturates the porous medium, at the sur-
face/infinite, it is obtained the velocity field of the cap-
sules that is a solution to the potential flow problem.
P (x,y,z)
θ
y
x
u
z
Figure 11. The flow and the coordinate system.

E. P. HERRERO ET AL.
265
That is [27],
3
1cos
r
a
vU r










(10)
3
1
1sin
2
a
vU r





 





(11)
For convenience, it is introduced the following dimen-
sionless variables, which are denoted by an asterisk (*).
It is considered as the characteristic length of the process
the radius of the sphere, a, and as characteristic time the
diffusion time scale, τD = a2/De:
2
0
(,)
*,*(*, *),
*,
e
s
e
(*, *,*),,,*,
x
yz r
xyz r
aaa a




sio less number relating the rate of advection of a flow
to its rate of diffusion
Upon substitution into (10) and (11), the following
dimensionless velocity field is obtained in the spherical
coordinates:
tD cxt
tcxt
Ca
Ua v
vPe
UD




(12)
is noteworthy the Peclet number that is a dimen-It
n

3
1
*1 cos
*
r
v
r


  


(13)

3
1
*1 sin
2*
v
r



 


(14)
Uever,
ed to simulate real living systems (e.g heart
is in a continuous contraction-relaxing movement) it was
introduced stirring to the system by an o
for this reason it is assumed torsional flow
assumptions. Torsional flows are induced by rotating
solid boundaries in contact with liquids. Th
to the no-slip boundary condition, has to fo
tion of the boundary, and therefore a torsional flow is
ge
e velocity due to the
torsional flow can be calculated as the following [28]:
∞
as it is intend
is the fluid velocity far from the sphere. How
rbital agitator,
on the main
e liquid, due
llow the mo-
nerated. Then, U∞, the fluid velocity far from the
sphere will be calculated by adding the flow velocity
through the porous medium supplied by the pumps to the
velocity due to the torsional flow supplied by the veloc-
ity of the orbital agitator.
The macroscopic quantity of th
(15)
The governing equation of the tracer transport process
has to be expressed in dimensionless form [29]:
43
2
0
4R
VRz
g


2
**** **
*
cPev cc
t

 (16)
with the dimensionless initial and boundary conditions
defined as follows:
*(*, *)0at*1,*0
*(*,*)( *)at*1,*0
*(*, *)0
cxtx t
cxt Htxt
cxt as *.x





(17)


Equations (13)-(17) showed an unsteady convection-
diffusion problem.
For convenience, in the equations that follow, it will
be omitted the superscript * of the dime
ables. It must be remembered, however, t
ables used hitherto are dimensionless.
As the experimental results show the m
process is taken about 100 minutes and c
average time of this kind of processes is in the range of
ho
nsionless vari-
hat the vari-
ass transport
onsidering the
urs, therefore this process could be considerer as a
short-time process. The solution in this type of process is
constructed by satisfying the boundary conditions at the
surface of the sphere and at infinity.
The concentration of the tracer in the fluid may be
given by a regular expansion of the concentration func-
tion c(x, t) as follows [25]:
01
cc Pec

  (18)
A first order expansion of the concentration function is
sufficient for the development of the solution.
The final analytical expression for RBID concentra-
tion is obtained by solving the governing equation in the
time domain or the Laplace domain [25]:


23
1
2
31 1 1
e
42
rt r
erfc t
rrt
11
,, 2
r
cr terfc
rt

1
13 1
244 2
Pe er
rr t
P
1r
fc

 
 

 

 
 
 









rmation about the distribution of the
concentration of the substance immobilized around the
capsules throughout the diffusion process to
distance of the application point to be able t
therapies to treat different diseases.
sibl


(19)
For the practical applications of this mass transfer
problem, i.e. the repair of tissues (cell therapy), is im-
portant to know info
a certain
o design
From the model described by Equation (19) it is pos-
e to generate a prediction of the concentration distri-

E. P. HERRERO ET AL.
266
nction of the radius, the angular co-
or
itial concentration
co
e effective diffu-
si
Figure 12 shows the concentration di
which results for a molecular weight of RBID of 100
an
that very
cl
However, there is an appreciable change in concentra-
tion profile far from the surface (r < 1.5). This is an in-
dication that an almost equilibrium state is quickly estab-
lished in the immediate vicinity of the sphere. Howeve
the RBID migrates at a faster rate towards the outer re
gion due to the influence of the peristaltic pumps. The
e 13 it is possible to observe that the
co
gradient from
to
dvection with respect to the
di
s increased the molecular weight of
RB
se. The small-
es
aspect of the Figure 12 (from top to bottom) underlines
the small significance of the advection with respect to the
diffusion process.
Figure 13 shows the concentration distribution field,
which results for a molecular weight of RBID of 70,000
and an alginate concentration of 1.2 wt%.
From Figur
bution fields as a fu
dinate, at different times. To make that, the model
equation was represented in parametric coordinates, and
the dimensionless radius was varied continuously from 0
to 3 in both sides of the capsule. The in
ndition is a unit step change at the surface of the
sphere [25]. This corresponds to the process of the leak-
age at t = 0. In order to generate the concentration distri-
butions were set different Peclet numbers (Pe = U∞ a/De)
that takes into account the relationship between the
process of convection and diffusion. Based on the veloc-
ity data of the peristaltic pumps, and th
ncentration gradient do not vary significantly from left
to right (y direction) during the process of mass transfer.
However, it is shown a large concentration
p to bottom (flow direction, z) since the inception of
the mass transfer process. RBID spread out in the flow
direction while not in the opposite direction. This shows
the importance of the a
vity data previously obtained in the steady process, it
was set the different Peclet numbers to each experimen-
tal condition. ffusion process at high molecular weight of RIBD. A
cumulative release increase has to be produced by a
reduction of the rate of diffusion to increase the
advection effect with regard to the diffusion effect.
From Figure 14 it is possible to observe the evolution
of the distribution profiles with increasing molecular
weight of RBID. It can be observed an increase in the
contribution of convection in the overall mass transfer
process when it i
stribution field,
00
d an alginate concentration of 1.2 wt%.
From Figure 12 it is possible to observe
ose to the surface also of the sphere (r < 1.2) the con-
centration profile does not vary significantly during the
process of mass transfer.
ID.
It was shown that there is an effect of the molecular
weigh of the protein on the cumulate relea r,
-t molecules exhibit a much lower cumulative release.
This behavior can be assigned to the presence of attrac-
(a) (b) (c)
Figure 12. Concentration distribution profiles (RBID MW = 10000; alginate concent. 1.2 wt%). (a) 2 minutes; (b) 22 minutes;
(c) 99 minutes.

E. P. HERRERO ET AL.
267
(a) (b) (c)
Figure 13. Concentration distribution profiles (RBID MW = 70000; alginate concent. 1.2 wt%). (a) 6 minutes; (b) 22 minutes;
(c) 99 minutes.
(a) (b) (c)
Figure 14. Concentration distribution profiles (Time = 88 minutes/alginate concentration = 1.2 wt%). RBID MW (a) 10000
Da; (b) 40000 Da; (c) 70000 Da.

E. P. HERRERO ET AL.
268
tive electrostatic interactions between protein and alginate.
Due to the amphoteric character, at higher pH values, the
attraction of protein to the alginate may be diminished as
a result of unfavorable electrostatics interactions. Know-
ing this, it is possible to achieve a pH-dependent con-
trolled release of proteins, resulting in protein imprinting
processes. In particular, it was mimicked the behaviour
of a special type of proteins, glycoproteins.
Based on the experimental work, and several theoreti-
cal mass transfer studies, it was predicted the coefficients
of the mass transfer process to the steady and unsteady
processes.
From the steady mass transfer study it can be con-
cluded that the rate of mass transfer decreases when the
molecular weight of RBID increases due to the size of
the RBID molecules that have more difficulties to be
released through the pores of the capsules. Also, the rate
of mass transfer decreases when the alginate concentra-
tion increases due to the mechanism of formation of the
capsules.
5. Conclusions
From the experim
om alginate-barium cap-
inguez from the Mathematics Department of
alamanca for the help with Mathe-
rch was supported by funds from the
the financial support.
7. References
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on the release of proteins fr
sules.
From the final analytical expression for the unsteady
mass transfer it is possible to predict the concentration
fields around the capsules as a function of the angular
coordinate, the radius and the time. From the con-
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The authors gratefully acknowledge Prof. Dr. D. Jose
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the University of S
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E. P. HERRERO ET AL.
270
g/ml
Concentration of the tracer in solution outside capsules mg/ml
eq Concentration in the equilibrium state mg/ml
S Concentration of the tracer inside capsules mg/ml
dp Particle diameter cm
De Effective diffusion coefficient cm2/s
DM Molecular diffusion coefficient cm2/s
De Effective diffusivity cm2/s
er, ez, eθ Unit vectors
erfc Complementary error function
g Gravity m/s
2
h Mass transfer coefficient cm/s
K Permeability m2
n Outward vector
N Number of capsules
p Pressure N/m
2
P Power input W
Pn Legendre polynomial
Pe Peclet number adimensional
r Radial coordinate cm
Re Reynolds Number (dp ρ v/μ) adimensional
s variable in Laplace domain to t
Sc Schmidt number (μ/ρ DM) adimensional
Sh Sherwood number (h dp/DM) adimensional
t Time variable s
U Fluid velocity far from the sphere cm/s
v Fluid velocity near the sphere, linear fluid velocity cm/s
V0 Volume of solution outside capsules cm3
x,y,z Coordinates cm
Greek letters
δ Stagnant boundary layer thickness μm
εp Porosity of the capsules
Ψ Stream function
ρ Fluid density kg/m
3
ζ cosθ radians
τD Diffusion time scale adimensional
θ Angular coordinate radians
μ Fluid viscosity kg/m s
Nomenclature
Symbol Description
a Radius of the sphere
A Total external surface area of capsules
c Concentration function m
Units
cm
cm2
C
C
C