What Happens When You Swallow a Hydrogel Pill?©
Keywords: hydrogel, drug delivery, modeling, transport, finite element method
BEE 4530: Computer-Aided Engineering: Applications to Biomedical Processes
Katherine Carroll, Rachel Connolly, Kaitlin Hearn, Imran Yasin
May 2017
Contents
1.0
EXECUTIVE SUMMARY ............................................................................................... 3
2.0
INTRODUCTION.............................................................................................................. 4
3.0
PROBLEM STATEMENT ............................................................................................... 4
3.1
ASSUMPTIONS ................................................................................................................... 5
4.0
DESIGN OBJECTIVES .................................................................................................... 5
5.0
SCHEMATIC ..................................................................................................................... 6
6.0
GOVERNING EQUATIONS ........................................................................................... 7
6.1
6.2
6.2
7.0
7.1
7.2
DEFORMATION GEOMETRY ............................................................................................... 7
DIFFUSION COEFFICIENT EQUATION ................................................................................. 8
VELOCITY EQUATIONS ...................................................................................................... 8
BOUNDARY AND INITIAL CONDITIONS ................................................................. 8
BOUNDARY CONDITIONS ................................................................................................... 8
INITIAL CONDITIONS ......................................................................................................... 9
8.0
PARAMETER VALUES .................................................................................................. 9
9.0
RESULTS ......................................................................................................................... 10
9.1
9.2
9.3
9.4
MESH CONVERGENCE ..................................................................................................... 10
COMPLETE SOLUTION ..................................................................................................... 11
ACCURACY CHECK.......................................................................................................... 12
SENSITIVITY ANALYSIS ................................................................................................... 14
10.0 CONCLUSION ................................................................................................................ 15
11.0 DISCUSSION ................................................................................................................... 16
12.0 REFERENCES ................................................................................................................. 17
13.0 APPENDICES .................................................................................................................. 18
13.1 APPENDIX A: INPUT PARAMETERS .................................................................................. 18
13.2 APPENDIX B: COMPUTATIONAL METHOD ....................................................................... 19
2
1.0
Executive Summary
The scientific world has limited knowledge on tablet properties, so scientists want a reliable
method to investigate these properties and their effects on the drug release profile. We developed
a model that is based on experimental data and used for various matrix combinations. In COMSOL,
we modeled how the three phases of a hydrogel – swelling, eroding and diffusing – facilitated the
overall delivery of a given drug. We used the principles of mass transport and solid mechanics,
respectively, to model the delivery of the drug in the body as well as the erosion and swelling of
the hydrogel.
In COMSOL, a university licensed software program, we modeled a semi-overall shaped
hydrogel through a 2D axisymmetric model. We initially modeled the geometry at its initial, nondeformed shape with the mass fractions of the water, drug, and polymer are at their initial values.
The geometry deformed with time as swelling and erosion affect the polymer matrix. Using this
geometry, we modeled and analyzed percent drug release, the mass of water inside the matrix, and
mass of drug and polymer release over time. We then used past research data to evaluate and to
understand texture analysis and to model phenomena of multiple diffusing species from a matrix.
To validate our model, we compared our modeling outcomes with experimental results to
understand how successfully our model depicts the experimental data. Finally, we conducted a
sensitivity analysis to determine how various parameters independently affected our results.
Based on our design objectives, our model aligned with our initial goals. We successfully
modeled the swelling and erosion of a hydrogel matrix over time. We also effectively modeled the
release of the drug from the polymer-drug matrix. We combined these two processes in COMSOL.
With the results, we were able to post-process our solution by refining our mesh and conducting a
sensitivity analysis.
The results of our study are limited to a set of assumptions made about the geometry. These
assumptions include the use of a 2D model for a 3D delivery process and negligible velocity in the
environment. Therefore, while the model provides a realistic analysis of hydrogel drug delivery,
some environmental factors may influence the overall outcome for each user.
Using mass transport and solid mechanics to model the three phases of the hydrogel, we
were able to create a realistic model of this drug delivery process. Our model allows scientists and
engineers investigate various matrix properties and their effects on the tablet's drug release profile.
Without invasive procedures, they can use our model to design the optimal tablet for their specific
pharmaceutical need. Therefore, our model is a necessary tool for the scientific world to develop
cost and time efficient hydrogel drug-matrices.
3
2.0
Introduction
One of the most important technological growth sectors is biomedical engineering.
Through new inventions, engineers can treat more ailments with increasing efficacy. There is now
a huge demand for novel, powerful technology. Hydrogels are three-dimensional, cross-linked
networks of swelling polymers (Hoare, 2008). Hydrogels are applicable in numerous areas,
including drug and protein delivery, tissue engineering, and nanotechnology due to their unique
material properties (Langer, 2003). The creation of new polymer materials and the invention of
new applications for hydrogels is revitalizing modern medicine.
Innovative areas of hydrogel research are: enhancing the mechanical properties, super
porous hydrogels with heightened response times, self-assembling hydrogels from hybrid graft
copolymers with protein-controlling domains, and genetically engineered copolymers (Kopecek,
2007). Hydrogel's success is attributed to its simplicity of production, low development costs, and
high adaptability (Caccavo, 2015, MP). Nonetheless, their drug release mechanisms are complex
in nature and depending on the geometry and the material, require advanced modeling.
One of the most important hydrophilic hydrogel materials is hydroxypropyl
methylcellulose (HPMC); it is valued for its physical, chemical, and biological properties, which
make it an excellent drug delivery receptacle (Caccavo, 2015, IJP). It is used in capsule form for
controlled drug release. Its release is a complex process that has been studied extensively in the
past decade (Caccavo, 2015, IJP). This study is examining the release mechanism, which can be
divided into two primary processes: drug diffusion and gel deformation (swelling). Once a solvent
contacts the hydrogel's matrix, the polymer "undergoes relaxation" caused by the unfolding of the
polymer chains (Caccavo, 2015, IJP). This reaction prompts swelling, as a gel-like layer grows
from the surface polymer chains. The drug can then diffuse more readily from this outer surface
due to both the increased surface area and the relaxed polymeric structure (Caccavo, 2015, MP).
Thus three “fronts” are formed: the inner “swelling front” made up of the relaxed matrix; the outer
“diffusion front” made up of the highly concentrated, non-matrix-bound drug in solution before
dispersion; and the dividing “erosion front” where the hydrophilic polymeric matrix is hydrated
and begins to dissolve. This process has been examined using several techniques including, but
not limited to: magnetic resonance imaging (MRI), atomic force microscopy (AFM), texture
analyzer, and ultrasound techniques (Caccavo, 2015, MP). However, our study will use computer
simulation on the program COMSOL to analyze this process.
3.0
Problem Statement
We created an oral drug tablet in solution in COMSOL using a 2D axisymmetric model.
Our goal was to develop a model that can be utilized for various matrix combinations. To conduct
this investigation, we considered the following: the swelling of the hydrogel matrix; the diffusion
4
of the drug through the hydrated matrix; and the erosion of the polymers on the outside of the
matrix.
3.1
Assumptions
To simplify the model, we made the following assumptions about the Transport of
Concentrated Species equation. We based our assumptions on Caccavo’s research (2015, MP).






The volume does not change due to mixing water, drug, and polymer, which follows the
ideal thermodynamic behavior.
The drug dissolution is much faster than diffusion within the matrix.
The drug that has been released from the matrix is immediately taken away. The
environment in which the tablet is dissolving remains unchanged by the dissolution.
The convective term is equal to zero.
The swelling of the polymer is independent of the erosion of the tablet.
The erosion velocity is constant over the course of the simulation.
We based these assumptions on our understanding of the mass transport and deforming
geometry terms. We simplified our problem statement to its lowest complexity without losing the
integrity of the formulation. Therefore, the assumptions maintained the physics of the process.
These assumptions are critical in enabling the modeling of this complex process.
4.0
Design Objectives
Our design objectives, or goals, of this project were split into the following four statements:
1.
2.
3.
4.
Model the swelling and erosion of a hydrogel matrix over time;
Model the release of a drug out of the hydrogel drug-matrix;
Solve the above modeling processes in COMSOL;
Model multiple diffusion species to investigate a more realistic delivery.
We used these four design objectives as the criteria to measure the successfulness of our
COMSOL model. If we meet all of our goals, our hydrogel tablet model will be a reliable tool for
scientists to model various matrix combinations.
5
5.0
Schematic
We created two schematics to illustrate the initial tablet geometry as well as the three
phases that is will undergo. We modeled our hydrogel as a 2-D, axisymmetric geometry that will
swell, erode and diffuse drug over time.
The initial geometry is a rectangle that is 1 mm in axial height and 6.5 mm in radial height.
The geometry is axisymmetric across boundary one and is mirrored across boundary two. These
boundaries will not be affected by swelling and erosion. Additionally, boundaries three and four
are the erosion and swelling fronts. These boundaries will be affected during the three phases
described in Figure 2a, 2b, and 2c.
Figure 1. Schematic of Hydrogel Matrix with Dimensions. The dashed lines
represent the radial and axial lines of symmetry, and each of the numbers
corresponds to one of the boundaries.
The geometry described in Figure 1 will undergo three phases. Initially, the tablet is dry
and only contains polymer and drug. Once the hydrogel is consumed, the tablet will begin
absorbing water. The absorption of water causes the hydrogel to swell and allows the drug to
diffuse out of the tablet. Finally, the tablet boundary will begin to significantly erode when the
swelling is no longer significant. The drug will continue to dissolve into the body.
6
Figure 2a, 2b, 2c. Cartoon of Hydrogel Matrix Phases. Each image represents one
of the three stages that the hydrogel experienced in the modeled 24 hours of
dissolution. The first phase (2a) is when the tablet is initially dry; the second phase
(2b) occurs when the tablet begins to absorb water; and the third phase (2c) occurs
when the erosion of the tablet becomes more significant than the swelling.
Therefore, Figures 1 illustrates the initial geometry for COMSOL. Additionally, Figures
2a, 2b, and 2c explain how this geometry will undergo phase changes once the tablet is consumed.
6.0
Governing Equations
To model the hydrogel in COMSOL, we used the Transport of Concentrated Species
equation. We could not model the hydrogel using Transport of Dilute Species because our model
uses three species that are dependent on the others.
The following equation defines the governing equation for Transport of Concentrated
Species, which relates Water Diffusion In and Drug Diffusion Out:
𝜌
𝜕𝜔𝑖
𝜕𝑡
= ∇ ∙ (𝜌𝐷𝑖 ∇𝜔𝑖 + 𝜌𝐷𝑖
𝜔𝑖
𝑀
∇𝑀)
𝑖 = 1, 2
(1)
where ρ is density, ωi is weight fraction of species i, Di is diffusion of species i, and M is molecular
weight. Water, drug, and polymer represent i equal to 1, 2, and 3, respectively.
6.1
Deformation Geometry
COMSOL used the Arbitrary Lagrangian-Eulerian Method to describe the domain
deformation which was computed from the swelling and erosion velocities. Additionally, the
Laplacian Smoothing Equations were used to redefine the mesh with changing geometry.
7
6.2
Diffusion Coefficient Equation
To complete our COMSOL model, the Diffusion Coefficients for water and drug were
calculated.
The Diffusion Coefficients for water and drug were defined as:
𝐷𝑖 = 𝐷𝑖,𝑒𝑞 exp [−𝛽𝑖 (1 −
𝜔1
)]
𝜔1,𝑒𝑞
(2)
where β is Fujita-type equation coefficient.
6.2
Velocity Equations
The Swelling and Erosion velocities were also necessary to determine the overall boundary
velocities along boundaries three and four in Figure 1. These velocities were found from the fluxes
resulting from the governing equation, Equation 1.
The Swelling and Erosion Velocities were defined as:
v𝑠𝑤𝑒 =
(𝑱1 + 𝑱2 )
J3
=−
𝜌𝜔3
𝜌𝜔3
|v𝑒𝑟 | = −𝑘𝑒𝑟
(3)
(4)
where vswe is swelling velocity, J is flux, ver is erosion velocity, and ker is erosion constant.
7.0
Boundary and Initial Conditions
Before fully implementing our model in COMSOL, we determined our boundary and initial
conditions.
7.1
Boundary Conditions
The following four boundary conditions characterize what occurred at the boundary layers
during the swelling and dissolving of the hydrogel. The numerical boundaries are visualized in
Figure 1.

Mass Fractions kept constant and System Density defined
𝜔3 = 1 − 𝜔1 − 𝜔2
(5)
8
1 𝜔1 𝜔2 𝜔3
=
+
+
𝜌 𝜌1 𝜌2 𝜌3



(6)
Flux is 0, dr is equal to 0, and dz is equal to 0 at boundary 1 and two throughout
Mass fractions of water, drug, and polymer were equal to the equilibrium mass fractions at
boundary three and four throughout experiment
The changes in z and r with respect to time were equal to the sum of the swelling velocity
and the eroding velocity
These boundary conditions relate how the hydrogel interacts with its surrounding
environment. The boundary conditions were chosen to represent the environment of the hydrogel
most accurately.
7.2
Initial Conditions
The following initial conditions represent what was happening to the hydrogel at the first
time step, 0 hours.


At the start of the experiment, the mass fractions of water, drug, and polymer were all equal
to their initial mass fraction values, as defined in Appendix A;
At the start of the experiment, r is equal to ro and z is equal to zo, as defined in Appendix
A.
Because of the transient nature of the hydrogel study, the above initial conditions were
used to represent what was occurring in or on the hydrogel at time zero. They set the initial mass
fractions and the initial size of the hydrogel.
8.0
Parameter Values
Parameter values are an essential part of our model to list various constants. The full list
of parameters is available in Appendix A. Some important parameters are the initial tablet radius,
r0, and thickness, z0, which are 6.5 mm and 1 mm, respectively. Additionally, the initial water
mass fraction, 10, drug mass fraction, 20, and water mass fraction, 30, are 0, 0.5, and 0.5,
respectively. Finally, the erosion constant, ker, is the only parameter value that affects the erosion
of the tablet boundaries.
We obtained some of these values from the published works of Caccavo (2015, IJP). We
estimated the rest of them based on these publications. It should be noted that Caccavo used the
simplest possible values for the diffusive fluxes (2015, IJP). This choice is to develop a simple,
yet highly-functional model that allows flexibility to deal with the system's complexities.
9
9.0
Results
Once we successfully imported the geometry, boundary and initial conditions, and input
parameter values, we were able to create a full model of the hydrogel over 24 hours. First, we
completed a mesh convergence to determine the best mesh for our geometry. Then, we created a
full solution that showed the effects of swelling over 24 hours. Additionally, we compared our
results with the experimental and computational results of published works. Finally, we conducted
a sensitivity analysis.
9.1
Mesh Convergence
To determine the best mesh for our geometry, we conducted a mesh convergence analysis.
We analyzed the changing water mass fraction at one point at one time. We saw the graph begin
to converge around 1500 elements and ultimately choose a 20 by 80 element mesh.
Mass Fraction of Water
Mesh Convergence
0.78
0.76
0.74
0.72
0.7
0.68
0
500
1000
1500
2000
2500
Number of Elements
Figure 3. Mesh Convergence Plot. To produce it, we examined the changing mass
fraction of water at a specific point and time while increasing the number of mesh
elements. The point selected was at x equal to 0.005 and y equal to 0.0009 at a time
of 8 hours.
Figure 3 shows that the mass fraction of water begins to converge around 1500 elements.
Thus, the mesh converges around values of xdiv equal to 80, ydiv equal to 20, and the number of
elements equal to 1600. We realize that 1600 elements are not where the graph is perfectly
converged, but to minimize our computation time, we accepted the small error assumed with a
slightly coarser mesh.
10
This 1600 element mapped mesh was implemented and maintained throughout the
remainder of the experiments. The mesh contained 20 by 80 elements.
Figure 4. Final Mesh. This final mesh used 20 elements for the axial height and 80
elements along the radial axis. This mesh totaled to 1600 elements.
The mesh shown in Figure 4 was a relatively course mesh. We tried to create a finer mesh
but received errors above roughly 2500 elements. This error is also explained why the mesh
convergence in Figure 3 ends around 2500 elements. While the mesh was relatively course, we
accepted the small errors that this mesh may have caused.
9.2
Complete Solution
As explained in Section 7.2, the initial water mass fraction of the tablet was equal to zero.
Therefore, once the tablet was consumed, it would begin to absorb water. This water absorption
would cause the tablet to swell. We examined this swelling front by plotting water mass fractions
at different times.
Figure 5a, 5b, 5c, 5d. Changing Water Mass Fraction Contour Plots. The contour
plots show the water mass fraction at time equal to 3 hours (5a), 6 hours (5b), 18
11
hours (5c), and 24 hours (5d). These sequence of contour plots exhibit the swelling
due to water absorption over time.
Figure 5a, 5b, 5c, and 5d were obtained through COMSOL computation. These figure show
domain deformation and mass dissolution are occurring simultaneously. In Figure 5c, we begin to
observe a slight error in the orange region of the tablet. This discrepancy is likely due to using a
coarser mesh. This error, however, does not significantly affect our results.
9.3
Accuracy Check
To validate our results, we compared our COMSOL model with experimental and
computational data from Caccavo’s published papers (2015, MP). We compared contour plots,
radius and thickness of a swollen tablet, and the drug release profile.
Figure 6a, 6b. Comparison of Contour Plots at 24 Hours. It should be noted that
the color scales are slightly different. The red area of our model (5a) is comparable
to the gray area of the published model (5b). The contour plots look almost
identical, with slight error along the radial axis.
The compared contour plots in Figure 6a and 6b appear to be similar. The most notable
difference occurred where the contours fall on the radial axis. Our mass fraction within the orange
region, between 0.8 and 0.9 on the scale, ends at radius equal to 7 mm. On the published contour
plot, the red region ends at radius equal to 5.5 mm. We hypothesize that this discrepancy is caused
by a mesh deformation error. Caccavo’s work used 279,125 free quadrilateral elements (2015,
MP). As mentioned in Section 9.1, we used only 1600 mapped mesh elements.
12
Next, we validated our model by comparing the radius and thickness of a swollen tablet
with respect to time. The experimental data points were obtained in the literature by Caccavo
(2015, MP). By comparing their values and trend, we were able to confirm the accuracy of our
model.
Radius and Thickness of Swollen Tablet
Radius/Thickness (cm)
1.2
Experiment
Radius
1
(Caccavo, 2015)
Model
0.8
0.6
Thickness
0.4
0.2
0
0
5
10
15
20
25
30
Time (h)
Figure 7. Radius and Thickness of Swollen Tablet vs. Time. The experimental data
points from Caccavo’s experiment exist as points on the graph (2015, MP). Our
COMSOL data is represented by the curved line. The figure exhibits a close fit
between experimental data and our COMSOL model.
In Figure 7, we compared the axial and radial sizes of the swollen tablet over time. An even
distribution of experimental points above and below the lines existed, which indicates that the
curve is a good fit to the data. This close fit shows that the swelling and eroding effects on the
tablet match those discovered experimentally.
Finally, we validated our model by looking at the drug release profile with respect to time.
Again, the experimental data points were obtained in the literature by Caccavo (2015, MP).
13
Fractional Drug Release (/)
Drug Release Profile
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Experiment
(Caccavo, 2015)
0
5
10
15
20
25
30
Time (h)
Figure 8. Comparison of Drug Release Profile vs. Time. The experimental data
points from Caccavo’s experiment exist as points on the graph (2015, MP). Our
COMSOL data is represented by the curved line. While our model does not fit the
data perfectly, the trend is maintained.
Figure 8 shows the fractional drug release as it varies with time. As is apparent, the fit was
not as close that seen in Figure 7. Our drug was released more rapidly than the experimental data
shows; with half of our drug released at approximately 6.5 hours, while the experimental results
show half of the drug released at nearer to 10 hours. However, the overall trend was well
maintained. This difference is likely attributed to our coarser mesh.
9.4
Sensitivity Analysis
Finally, a sensitivity analysis was performed to determine the significance of small changes
in the equilibrium diffusivities and erosion constant. The results are displayed in Figure 9.
14
Figure 9. Sensitivity Analysis at 12 hours. The fractional drug release was
examined when the water and drug diffusivities, respectively D1,eq and D2,eq, were
increased and decreased by 5%. Additionally, the percent change in volume was
examined when the erosion constant, ker, was changed by 10%.
We monitored changes in the equilibrium diffusivities of both water, D1,eq, and drug, D2,eq
by their effects on the fraction of drug released after 12 hours. Figure 9 shows that a 5% change in
the equilibrium diffusivity resulted in a 4-6% change in the fraction of drug released. Logically,
an increase in the equilibrium diffusivity of water or a decrease in the equilibrium diffusivity of
drug cause a decrease in the fraction of drug released after 12 hours. This result shows a relatively
substantial change. Thus it is crucial that the equilibrium coefficients be exact in this model. We
used those discovered by Caccavo (2015, MP).
A 10% change in the erosion constant, ker, results in a 0.7% inverse change to the volume
at time 12 hours. As such, decreasing the ker value resulted in a 0.66% increase of the volume after
12 hours. When we increased ker, we saw a 0.70% decrease of the volume. We deemed this
relationship to be less influential, and thus the exactness of the erosion coefficient to be
inconsequential to our final result.
10.0
Conclusion
In conclusion, we successfully developed a model that aligns with experimental data and
can be used for various matrix combinations. To achieve our goal, we met all of our design
objectives. First, we modeled the swelling and erosion of a hydrogel matrix over time. Next, we
modeled the release of a drug out of the hydrogel drug-matrix. Third, we successfully solved the
first and second design objectives in COMSOL. Finally, we were able to model a realistic drug
delivery.
15
As seen in our results and analysis above, our model was successful in simulating the trends
of drug dissolution from a swelling hydrogel tablet. As seen in Figure 4, our model was quite
effective in modeling the swelling and erosion of the tablet with time. However, our drug was
released sooner than that observed in the literature. As seen in Figure 6, we reach a fractional drug
release of 0.5 at approximately 6.5 hours, while the experimental results show this to occur at
nearer to 10 hours. This slight error, as well as the radial contour error observed in Figure 11, are
likely due to our coarse mesh. Due to time constraint and difficulty implementing the mesh
deformation, we used a mesh with 16,000 elements while the published paper implemented a free
quadrilateral mesh of nearly 300,000 elements.
11.0
Discussion
Our results simulate the drug release profile from a hydrogel matrix. This type of
simulation is significant because it can be used and modified to show drug dissolution from various
polymer-drug combinations. Pharmaceutical companies will be able to perform preliminary design
testing through our simulation. With our current and future refined models, the drug, geometry,
and even surrounding medium can be user-input variables in COMSOL. Furthermore, our
computational model showed the amount of drug delivered at a specific time, such as at 12 hours.
This concept could be very influential in the design of drugs that must be maintained between very
specific blood concentration levels over extended periods. Scientists can use this data to determine
dose frequency to deliver the desired amount within a given time frame. Therefore, the extended
release of hydrogels is an innovative way to decrease dosage frequency while maintaining drug
levels in the body. Furthermore, hydrogel tablets' extended release capabilities will relieve patients
of the nuisance of frequent administration of low dose tablets.
The next step in research will be: correcting for a finer deforming mesh; examining
different geometries; and considering drug combinations dissolving simultaneously. In our model,
we faced challenges with our deforming geometry and would be interested in developing a more
refined model. Additionally, we did not have time to address different geometries or various drugpolymer combination. However, all three will be substantial in the realistic modeling of hydrogelsuspended drug administration.
In summary, our simulation supports the viability of computer-aided engineering
techniques in the design of pharmaceutical capsules. Computational models will allow scientists
and engineers to test different designs without the cost of creating and experimentally testing each
model first. The drug industry is certain to move towards such modeling techniques in the future.
Models such as our own are not only able to capture all of the facets of hydrogel drug delivery:
swelling, erosion, and dissolution, but also enable design engineers to manipulate each of the input
parameters individually.
16
12.0
References
1. Caccavo, D., Cascone, S., Lamberti, G., Barba, A.A. (2015) Controlled drug release from
hydrogel-based matrices: Experiments and modeling. International Journal of
Pharmaceutics, 486(1-2), pp. 144-152. doi:10.1016/j.ijpharm.2015.03.054
2. Caccavo, D., Cascone, S., Lamberti, G., Barba, A.A. (2015) Modeling the Drug Release
from Hydrogel-Based Matrices. Mol. Pharmaceutics, 12(2), pp. 474-483.
doi:10.1021/mp500563n
3. Hoare, T. R., & Kohane, D. S. (2008). Hydrogels in drug delivery: Progress and challenges.
Polymer,49(8), 1993-2007. doi:10.1016/j.polymer.2008.01.027
4. Kopeček, J. (2007). Hydrogel biomaterials: A smart future? Biomaterials,28(34), 51855192. doi:10.1016/j.biomaterials.2007.07.044
5. Langer, R., & Peppas, N. A. (2003). Advances in biomaterials, drug delivery, and
bionanotechnology. AIChE Journal,49(12), 2990-3006. doi:10.1002/aic.690491202
17
13.0
Appendices
13.1
Appendix A: Input Parameters
Parameter values are an essential part of our model to list various constants. Some of the
notable parameters are the initial tablet radius, r0, and thickness, z0, which are 6.5 mm and 1 mm,
respectively. Additionally, the initial water mass fraction, 10, drug mass fraction, 20, and water
mass fraction, 30, are 0, 0.5, and 0.5, respectively. Finally, the erosion constant, ker, is the only
parameter value that affects the erosion of the tablet boundaries.
Table 1. Input Parameters
Parameter
Symbol
Tablet Properties
Initial tablet radius
r0
Initial tablet thickness
z0
Erosion constant
ker
Water
Density
1
Molecular weight
M1
Initial mass fraction
10
Equilibrium mass fraction
1,eq
Effective diffusivity
D1,eq
Fujita-type equation coefficient
1
Drug
Density
2
Molecular weight
M2
Initial mass fraction
20
Equilibrium mass fraction
2,eq
Effective diffusivity
D2,eq
Fujita-type equation coefficient
2
Polymer
Density
3
Molecular weight
M3
Initial mass fraction
30
Value
Unites
Source
6.5
1
5 x 10-9
mm
mm
m s-1
(Caccavo et al., 2015)
(Caccavo et al., 2015)
(Caccavo et al., 2015)
1000
18
0
0.97
2.2 x 10-9
3.53
kg m-3
g mol-1
m2 s-1
-
(Caccavo et al., 2015)
(Caccavo et al., 2015)
(Caccavo et al., 2015)
(Caccavo et al., 2015)
(Caccavo et al., 2015)
(Caccavo et al., 2015)
1200
180.16
0.5
0.0001
1.5 x 10-10
4
kg m-3
g mol-1
m2 s-1
-
(Caccavo et al., 2015)
(Caccavo et al., 2015)
(Caccavo et al., 2015)
(Caccavo et al., 2015)
(Caccavo et al., 2015)
(Caccavo et al., 2015)
1200
120,000
0.5
kg m-3
g mol-1
-
(Caccavo et al., 2015)
(Caccavo et al., 2015)
(Caccavo et al., 2015)
We obtained some of these values from the published works of Caccavo (2015, MP). The
rest we estimated from these publications. It should be noted that Caccavo used the simplest values
for the diffusive fluxes (2015, MP). This choice is in part due to the develop a simple, but highly
functional model that allows flexibility to deal with the system's complexities.
18
13.2
Appendix B: Computational Method
The model, using finite element method (FEM), was solved on COMSOL Multiphysics 5.2
and 5.2a, a university licensed software. We implemented the transport of concentrated species
equation in COMSOL to model our species flux (Eq. 1). We used this implementation rather than
that of a dilute species because the mass fractions of each of our species, water, drug, and polymer,
are comparable at some point in the simulation. Also, we used an unusual feature on COMSOL to
simulate the geometry deformation. For this, we implemented the Arbitrary Lagrangian-Eulerian
Method in COMSOL, which was computed from the swelling and erosion velocities. Finally, the
Laplacian Smoothing Equations were used to redefine the mesh while the geometry changes.
Using a standard university computer, our model took about 2 minutes to run. Additionally, it used
1.37 GB of physical memory and 1.49 GB of virtual memory.
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