Betamethasone Phosphate Drug Delivery
by Biodegradable Ocular Implant
Keywords: Intrascleral, Ocular, Implant, Biodegradable, Bulk Erosion, Computational Modeling, Drug
Delivery
Ⓒ May 2017 Tara Chari, Catherine Li, Rebecca Varghese, Qiuwei Yang
Cornell University
BEE 4530
Computer-Aided Engineering: Applications to Biomedical Processes
Department of Biological and Environmental Engineering
Table of Contents
I.
A.
B.
C.
D.
Introduction to Ocular Drug Delivery Mechanisms
Background
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Problem Statement
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Design Objectives
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Problem Schematic
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A.
B.
C.
D.
Methods
Governing Equation
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Boundary Conditions .
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Initial Conditions
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Bulk Erosion of PLA Implant and its Effects on Drug Diffusivity .
A.
B.
C.
D.
E.
F.
G.
H.
Results & Discussion
Examining Drug Diffusion in the Human Eye with Original Implant Dimensions
Optimization of Intrascleral Ocular Implant
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Discussion of the Concentration Profile of the Optimized Implant
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Mesh Convergence
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Validation of COMSOL Model .
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Sensitivity Analysis
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Discussion of Simplifications and Assumptions .
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Applications in Clinical Treatment
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II.
III.
IV.
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Conclusions and Design Recommendations
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References
Appendix
A. Input Parameters
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B. Computational Models .
C. Objective Function Values
D. Bulk Erosion Plots
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E. Mesh Convergence
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F. CPU and Memory
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Page 1 of 15
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15
Betamethasone Phosphate Drug Delivery
by Biodegradable Ocular Implant
Tara Chari, Catherine Li, Rebecca Varghese, Qiuwei
Yang
Biodegradable implants have been steadily
gaining popularity, offering safer, more efficient
alternatives to common therapeutics. Currently,
most ocular implants that are used for sustainable
drug delivery are non-biodegradable. In addition to
the eye being a sensitive area, the likelihood of
complications increases when multiple surgeries
are performed to remove the implant as well as
insert it. Biodegradability of the implant can
eliminate these removal complications.
Using COMSOL® Multiphysics, we modeled the
delivery of betamethasone phosphate (BP) from a
biodegradable implant that is placed in the scleral
layer of a human eye. As a glucocorticoid, BP is used
to reduce inflammation in the tissue layers
surrounding the sclera. We used a 2D axisymmetric
geometry to represent the eye with the implant and
focused on the delivery of BP to the retina.
The implant’s bulk erosion and mass transfer
dynamics resulted in an initial spike of BP release
followed by a slow, gradual depletion of the
remaining drug. The initial size and location of the
implant in our model were taken from those of
similar implants tested in rabbit eyes. Upon finding
that BP concentration exceeded the desired range in
the retina, we optimized the location and size of our
implant for drug delivery in the human eye. After
optimization, an effective concentration in the
retina was maintained for 34 days. The model was
validated with experimental data for BP release
from implants in the rabbit eye, which is an accurate
model for drug pharmacokinetics in the human eye.
Our results suggest that this biodegradable ocular
implant is a viable option for drug delivery in the
human eye.
I. Introduction to Ocular Drug Delivery Mechanisms
A.
Background
Many topical and systemic administration of drugs to the
eye can be ineffective. The presence of barriers at the
anterior of the eye, such as the corneal epithelium, inhibit
the diffusion of drugs from topical applicators to the back
of the eye, making these types of devices ineffective for
treating posterior eye diseases [1]. Intravitreal injections
have been implemented to treat back-of-the-eye diseases to
deliver higher drug concentrations, but drugs with a low
molecular weight, such as corticosteroids, have low halflives and thus require frequent injections [2].
Recently, research has been done on sustained drug
delivery implants for posterior eye diseases. Implants allow
sustained release without repeated injections [3], which
have secondary complications, and biodegradable implants
can eliminate the need for a second operation to remove the
implant, thus reducing the risk of ocular toxicity [4].
However, the blood-retinal barrier presents problems in
drug delivery at the posterior segment of the eye. The tight
junctions created by retinal pigment epithelial (RPE) cells
and retinal capillaries limit the delivery of drug to the retina
[5].
Recent literature outlines the development of transscleral
and intrascleral implants for drug delivery that target
diseases of the choroid and outer retina [3], such as retinal
degenerative diseases. Most of the available scientific
literature on in vivo biodegradable intrascleral implants for
sustained drug delivery are conducted in rabbit models, not
humans, due to ethical restrictions [6]. Thus, the goal of this
project was to provide evidence, via our model, of the
effectiveness of intrascleral implants in humans that can
guide future design of biodegradable ocular implants.
We want to determine the effectiveness of biodegradable
intrascleral implants in treating retinal disease and
achieving the desired drug release profile. In the work by
Okabe et. al. [7], an implant made of poly(DL-lactide), or
PLA, containing the drug BP, 0.5 mm thick and 4 mm in
diameter, was tested in a rabbit eye. We
Page 2 of 15
The objectives accomplished were
• Optimization of the size of implant for the human
eye, as the current implant dimensions are for the
rabbit eye
• Optimization of the location of implant
• Modeling the degradation of the implant as bulk
erosion
will model this same setup in the human eye using
COMSOL® to study the implant’s drug release profile [7].
B.
Problem Statement
Though several biodegradable ocular implants have
been developed, modeling of their drug delivery abilities is
relatively unexplored and many have only been tested in
situ in rabbit eyes. Thus, we performed a transient analysis
of the transport of BP in the retina-choroid by drug
diffusion and release via an intrascleral disc-shaped
implant. The implant consists of BP loaded onto a matrix
made of PLA such that the total weight per unit volume of
BP in the PLA matrix was 25% [7].
The sclera, choroid, retina, and vitreous chamber of the
eye were modeled as 2D axisymmetric concentric
semicircles. Following the work done by Okabe et. al., the
implant was initially modeled as a rectangle of a thickness
of 0.5 mm and a radius of 2 mm and inserted halfway into
the sclera [7].
Even though the implant is biodegradable, the implant
size was kept constant throughout the simulation because
most of the drug diffused out of the implant in less than 40
days, while the PLA matrix took a full 60 days to fully
biodegrade [7]. Instead of deforming the implant mesh to
model biodegradation of the PLA matrix of the implant, the
diffusivity of BP through the implant was changed as a
function of time.
The entire eye was modeled as a sphere with a total
radius of 11.25 mm [8]. The sclera-choroid boundary was
at a radius of 10.75 mm, the retina-choroid boundary was
at a radius of 10.5 mm from the center of the eye, and the
retina-vitreous humor boundary was at a radius of 10.4 mm
from the center of eye [8].
C.
D.
Problem Schematic
The human eye was modeled as 2D axisymmetric layers,
with the implant placed fully inside the scleral layer and
modeled as a 2D axisymmetric disk (Figure 1). As shown
in Figure 2, BP diffuses out of the scleral implant as water
moves in to hydrolyze the PLA matrix. The BP diffusion
was modeled through each of the eye layers, including the
vitreous chamber.
Figure 1: Representation of the eye and the layers modeled
through COMSOL
Design Objectives
The goal of this project is to optimize the design of a
biodegradable, intrascleral, disc-shaped implant that
sustains the delivery of BP to the retina-choroid boundary
within 150 to 4000 mg/m3 of eye [7], the range for effective
suppression of inflammatory properties.
Page 3 of 15
− D A( ∂c∂rA ) = 0
Figure 2: Schematic of BP Diffusion.
II. Methods
A.
Governing Equations
The rate of BP transport in the eye is governed by the
equation
This assumption was made based on papers discussing
the permeability of Timolol [10]. Timolol is nonpolar and
less than 1 kDa, making it a small molecule, just like BP,
the drug of interest. The permeability of timolol in the
conjunctiva was found to be substantially lower than its
permeability in the sclera [10]. Thus, we assumed that the
sclera-conjunctiva boundary was impermeable to BP and
the flux out of the sclera would be zero.
Additionally, we added partition coefficients between
the layers of the eye, as shown in Table 1. Typically, the
partition coefficient between two solids is approximated to
be one, and the partition coefficient at the sclera-choroid
boundary was one. However, the partition coefficient
between the choroid and the retina was 0.75 due to the
presence of the blood-retinal barrier, which prevents rapid
diffusion of drugs from the choroid to the retina [11]. The
partition coefficient between the retina and the vitreous
humor was 0.1 because the vitreous humor is a gel-like
substance [11].
C.
where DA is the diffusion coefficient, cA is the concentration
of the BP, and RA is the consumption of the drug. Diffusion
occurs along the r and z axes. The drug also undergoes first
order degradation in the eye. We assumed that there was no
convection because mass transport of BP through the layers
of the eye primarily involves diffusion and the vitreous can
be modeled as stagnant for drugs with a small molecular
weight, such as BP [9].
B.
Boundary Conditions
Our 2D representation of the eye contains one boundary
between the sclera and the conjunctiva, the thin membrane
that covers the eye. The flux of drug out of the sclera and
into the conjunctiva was assumed to be zero, where DA was
the diffusion coefficient (Figure 2).
(Eq 2)
Initial Conditions
At the beginning of the simulation, it was assumed that
the concentration of the drug along all layers of the eye was
zero.
D.
Bulk Erosion of PLA Implant and its Effects on
Drug Diffusivity
Degradation can occur through surface or bulk erosion.
The small size and thickness of this PLA implant allows
water to diffuse entirely and homogeneously through the
implant before the surface begins to erode, thus causing
bulk rather than surface erosion [12]. The mechanism of
degradation involves hydrolysis of the PLA polymer into
lactic acid oligomers and monomers [13], which changes
the implant from a solid to aqueous phase and results in a
higher diffusivity of the drug through the implant. Thus, we
modeled the process of bulk erosion in COMSOL by
changing the implant diffusivity as a function of time. The
Page 4 of 15
diffusivity was based on another study of a poly(lactic-coglycolic), or PLGA, stent coating, where drug diffusion
through the PLGA changes as the coating degrades [14].
The effective drug diffusivity, D1,e through the PLGA
coating was given as
D1,e =
(1−φ)Ds0(M w /M w0)−a+κφDl0
1−φ+κφ
(Eq 3) [14]
where Mw denotes the molecular weight of PLGA, which
is changing from the solid to aqueous phase
M w = M w0e−kw t
(Eq 4) [14]
B.
and Φ denotes the changing porosity of the coating
φ = φ0 + (1 − φ0)(1 + e−2knt − 2e−knt)
Optimization of Intrascleral Ocular Implant
The radius, the thickness, and the location of the
(Eq 5) [14]
We can model our PLA implant based on these equations
for PLGA because the latter is a copolymer of PLA and
glycolic acid. As the ratio of PLA to glycolic acid increases,
PLGA displays similar properties for solubility and
degradation as PLA [15].
III. Results and Discussion
A.
Figure 3: Concentration profile at the retina-choroid boundary
with the original implant geometry. The upper and lower bounds
of the effective concentration are denoted by the dashed lines.
Examining Drug Diffusion in the Human Eye with
Original Implant Dimensions
The diffusion of BP out of the implant through the eye
was modeled using COMSOL, and importance was placed
on the concentration of BP at the retina. Since the initial
size and location of the implant was tested in a rabbit eye,
we wished to observe the delivery of BP to the retina using
the dimensions of a human eye. The concentration profile
(Figure 3) shows that the drug exceeds the upper boundary
of the effective range for the first 20 days after
implantation.
implant were changed in order to optimize the diffusion of
BP. The objective function (Eq 6) was minimized to
determine the optimal implant properties.
J = Fretina(cretina) + Fchoroid(cchoroid) + Fsclera(csclera) +
Fvitreous(cvitreous)
(Eq 6)
The functions for the vitreous, retina, choroid, and sclera
layers are detailed in Table 3 in the appendix.
The optimization function increases linearly with
concentration outside of the effective dosage range for BP.
One criteria for our optimization was that the concentration
of BP should never exceed 4000 mg/m3 eye in all four layers
because this would put a patient at risk of an overdose [7].
Additionally, we minimized the length of time at which the
concentration of BP in the retina was lower than 150 mg/m3
eye because such concentrations are too low to be effective
[7]. This condition was not implemented for the vitreous,
choroid, and sclera because the purpose of the implant was
to target inflammation in the retina. Thus, it was not
important for the concentration of BP to reside above the
lower bound of the effective concentration range for eye
layers other than retina.
The optimized implant had a thickness of 0.1 mm and a
radius of 2 mm, and the optimal eye location was 10.8 mm
from the center of the eye. As the thickness of the implant
Page 5 of 15
decreased, the objective function was minimized (Figure
4A).
(a)
The optimal radius of the implant was determined to be
2 mm even though the corresponding objective function
exhibited a larger initial peak than that for a radius of 1 mm
(Figure 4B). However, at longer times in the retina, the
values for the objective function for a radius of 2 mm was
smaller than those for a radius of 1 mm (Figure 5). Since
the purpose of the implant was for targeted drug delivery to
the retina, we decided that an implant radius of 2 mm was
better since the objective function was lower for a longer
period of time.
While there was no significant difference in the
objective functions after moving the implant closer to the
sclera-choroid boundary (Figure 4C), the objective
function at 10.8 mm from the center of the eye had the
smallest initial peak. Thus, the optimal eye location was
determined to be 10.8 mm from the center of the eye.
(b)
Figure 5: Optimization of the implant radius in the retina. A
radius of 3 mm exhibits a large peak at approximately two
days. A radius of 2 mm had an objective value function that
was lower than than of 1 mm.
(c)
Figure 4: Optimization of the implant thickness, implant
location, and implant radius in the all the layers of the eye. (a) As
the thickness of the implant increases, the objection function
decreases. (b) The objective function in the eye decreases as the
distance from the retina decreases. (c) As the radius increases, the
objective function decreases significantly. From the data, the
optimal thickness was 0.1 mm and the optimal radius for the
implant was 2 mm. The optimal location is 10.8 mm from the
center of the eye.
Figure 6: Optimization function of the implant for all layers of
the eye. Optimized implant dimensions are as follows: thickness
of 0.1 mm, radius of 2 mm at a location of 10.8 mm from the
center of the eye. Original implant dimensions are as follows:
thickness of 0.5 mm, radius of 2 mm at a location of 11 mm from
the center of the eye.
Page 6 of 15
A comparison of the new and original implant geometry
shows that the new, optimized dimensions minimize the
objective function while the original geometry results in
considerably large values for the objective function (Figure
6). Thus, the optimized implant size and location exhibited
a more effective release of BP than the original size and
location of the implant, and the new geometry optimized
for drug delivery of BP to the retina-choroid boundary
within the effective concentration range.
implantation, there is a burst phase of drug release caused
by diffusion of drug from the surface of the implant [1],
which correlates with the initial burst of BP seen in the first
five days in Figure 7. Afterwards, water molecules diffuse
deeper into the implant, causing random cleavage of the
implant matrix [1]. The cleavage of the matrix results in a
steady release of drug from the implant [1], which is
illustrated by days 6 to 60 in Figure 7.
D.
C.
Discussion of the Concentration Profile of the
Optimized Implant
The concentration profile of BP in the implant (Figure
7) shows that there was an initial burst of BP in the first five
days that was caused by the drug at the surface of the
implant diffusing into the eye. This was followed by a
period of steady concentration decrease from days 6 to 60
as the drug within the implant was steadily released into the
eye. After approximately 34 days, the concentration in the
retina falls below the lower bound of the effective range for
BP.
Figure 7: Concentration profile of BP at (0.00104m, 0.0106m) at
the retina-choroid boundary after optimization of implant
dimensions, with the upper and lower bounds of the effective
concentration included.
The drug release profile of BP given in Figure 7 is
typical of other concentration profiles for drugs released
from a biodegradable implant [1]. Right after
Mesh Convergence
Figure 8 shows that the mesh elements are finest
around the boundaries of each eye layer. The
concentration of BP at the retina-choroid boundary was
plotted as a function of the number of element in the mesh
to determine when the mesh converged. The concentration
stopped changing when the mesh contained 5258 elements
(Figure 8), which is the number of elements in the “finer”
mesh constructed by COMSOL. Thus, we chose the
“finer” mesh to study drug delivery in our model. For a
mesh convergence plot of the concentration profile, refer
to Figure 14 in the Appendix.
Figure 8. The mesh for the optimized implant geometry with
a plot of the concentration at a point on the retina-choroid
boundary over the number of elements in the mesh. It was
important to perform mesh convergence on the first day
because due to the initial burst of BP, the greatest changes in
concentration at the retina-choroid boundary occurred on the
first day. Mesh convergence was also performed at 28 days
because that was when most of the drug had diffused from the
eye [7].
Page 7 of 15
E.
Validation of COMSOL Model
We validated the accuracy of our results by comparing
the cumulative percent drug released from the original
implant geometry with that obtained experimentally in the
rabbit eye by Okabe et. al in 2003 [7]. Although our model
is of the human eye, we can make this comparison because
the rabbit eye has been found to be a valid model for drug
pharmacokinetics in the human eye [6], [16].
Another study was conducted by the same lab measuring
cumulative BP release from a 85:15 (PLA:glycolic acid)
PLGA rod-shaped implant (Figure 8) [17]. The diffusivity
of BP through the implant in our model was interpolated
from drug diffusivities through a 53:47 PLGA implant from
another study [18]. This could explain why our model’s
release profile more closely resembles that of the PLGA
implant (seen in Figure 9).
Figure 10: Sensitivity analysis wherein the effect of changing
the magnitude of the diffusivities and drug reaction terms on
the concentration at the retina-choroid boundary at 28 days
was determined.
Figure 11: Sensitivity analysis wherein the effect of
increasing all partition coefficients by 10%, decreasing them
by 10%, or setting them equal to 1 on the concentration at the
retina-choroid boundary at 28 days was determined.
Figure 9. The cumulative percent of BP released by the
implants over time.
G.
F.
Discussion of Simplifications and Assumptions
Sensitivity Analysis
The sensitivity analysis shows that changing the
magnitude of the diffusivities has the greatest influence on
the concentration at the retina-choroid boundary (Figure
10). This is most likely because the diffusivity affects how
fast the drug can move through the eye layers and thus how
much drug is accumulated at the retina-choroid boundary at
28 days. It can be seen that the partition coefficients have a
negligible effect on the concentration, as increasing or
decreasing them by 10% results in no change at all and
setting them equal to 1 results in a 0.06% change (Figure
11).
The geometry of the model was simplified when created
on COMSOL. Typically, the thickness of the different layers
of the eye are not constant. However, since there was no
available and complete model of the eye available in online
databases, a simplified, 2D axisymmetric circular geometry
of the eye was constructed using COMSOL with uniform
thickness for each layer. It was also assumed that the initial
concentration of the BP was evenly distributed throughout
the implant. Furthermore, it was assumed that convection in
the vitreous was negligible at the retina-vitreous boundary
since BP is considered a small molecule [9] and has a high
diffusivity
Page 8 of 15
[19]. Some parameters that were used for BP are from
studies using dexamethasone.
The diffusivity of BP in PLA was interpolated from
diffusivities of other drugs with similar molecular weights
in PLGA [17]. Here, we assumed that the movement of BP
through PLA would be similar to that of these other drugs
through PLGA. The diffusivities of BP in the sclera,
choroid, and retina given in Table 1 are taken from general
values valid for small drug molecules, or molecules that are
less than 1 kDa [11], and were found from data measured
in vivo for similarly sized drugs. The diffusivity of BP
through the vitreous humor was approximated to be the
same as that of dexamethasone [9].
H.
of the matrix of of the implant, along with additional
chemical coatings and scaffolds, can be changed in order
to produce the desirable drug release profile. Thus, further
COMSOL research focused on optimizing the polymers
used in biodegradable implants or the composition of the
implant matrix would be necessary.
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Our results show that a biodegradable intrascleral
implant is a viable option for short-term sustained drug
delivery in posterior areas of the eye. The concentration of
betamethasone phosphate in the retina was within the
effective concentration range of the drug for about thirty
days. This is ideal for ocular implants where drug is only
needed in the short term and when it would not be optimal
to perform another surgery to extract the implant, such as
corticosteroids.
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From the values given for the implant in the rabbit eye,
we were able to optimize the size of the implant for the
human eye. The most significant change to the implant for
the human eye was the thickness, which was changed from
0.5 mm to 0.1 mm. In addition to changing the size of the
implant, we modeled the implant degradation with bulk
erosion. According to the concentration profile, with both
the bulk erosion and the new implant size, the drug should
be within the effective range at the retinal-choroid
boundary for about 35 days.
Further research could be conducted to improve upon
the model. The focus of this paper was on optimizing the
physical components of the implant, such as its size and
location in the eye. However, there is another field of
research focused on the implant matrix. The composition
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Bruce, J. F. Griffith and L. W. Sanders, "Human and rabbit eye
responses to chemical insult," Toxicological Sciences, vol. 7, no.
4, pp. 626-634, 1986.
[17] N. Kunou, Y. Ogura, Y. Honda, S. Hyon and Y. Ikada,
"Biodegradable scleral implant for controlled intraocular delivery
of betamethasone phosphate," Journal of Biomedical Materials
Research, vol. 51, no. 4, pp. 635-641, 2000.
Page 10 of 15
Appendix
A. Input Parameters
Table 1. Input Parameters for Bulk Erosion of PLA Implant
Variable
Description
Expression
Units
Source
Ds0
Initial diffusivity of BP through the
implant matrix
6.39x10-14
m2/s
Interpolated from
[18]
a
Constant derived from the power
law model
1.714
--
[14]
�
Drug partitioning between the
liquid-filled pores and solid PLGA
phase
10-4
--
[14]
kw
Reaction rate constant of PLGA
7.5x10-7
s-1
[14]
Φ0
Initial porosity of the implant
0
kn
Degradation rate constant
determined from the half-time of
PLGA
2.5x10-7
s-1
[14]
Mw0
Initial molecular weight of BP
60,000
kDa
[20]
Dl0
Diffusivity of dexamethasone in the
liquid-filled pores
7.7x10-10
m2/s
[21]
Page 11 of 15
[14]
B. Computational Parameters
Table 2. Parameters needed to model the diffusion and degradation of BP through the various layers of the posterior portion of
the eye.
Variable
Description
Expression
Units
Source
RA
Pseudo-first order degradation of
drug
-kc
mg/ (m3s)
[1]
k
Rate constant of the reaction of
BP (Sclera, Choroid, Retina)
3x10-10
s-1
[11]
using first
order kinetics
kv
Rate constant of the reaction of BP
(Vitreous humor)
8x10-11
s-1
[11]
c0
Initial concentration of drug in
implant
2.5x108
mg/m3
[18]
DR
Diffusivity of drug in retina
1.17x10-11
m2/s
[11]
DS
Diffusivity of drug in sclera
4x10-11
m2/s
[11]
DC
Diffusivity of drug in choroid
1x10-11
m2/s
[11]
DPLA
Diffusivity of drug in PLA implant
6.39x10-14
m2/s
[18]
PSC
Partition coefficient for sclera and
choroid
1
[11]
PCR
Partition coefficient for choroid
and retina
0.75
[11]
PRV
Partition coefficient for retina and
vitreous humor
0.1
[11]
DV
Diffusion coefficient of
dexamethasone in vitreous humor
1.8 x10-9
Page 12 of 15
m2/s
[9]
C. Objective Functions
Table 3. Objective Functions in the Sclera, Choroid, Retina, and Vitreous
3
Fretina
3
cretina - 4000 mg/m eye
cretina > 4000 mg/m eye
0
150 mg/m3 eye ≤ c
retina
cretina < 150 mg/m eye
3
3
csclera - 4000 mg/m eye
csclera > 4000 mg/m eye
0
c
sclera
≤ 4000 mg/m3 eye
3
Fchoroid
3
cchoroid - 4000 mg/m eye
cchoroid> 4000 mg/m eye
0
c
choroid
≤ 4000 mg/m3 eye
3
Fvitreous
≤ 4000 mg/m3 eye
3
150 mg/m3 eye- c
Fsclera
retina
3
cvitreous - 4000 mg/m eye
cvitreous > 4000 mg/m eye
0
c
vitreous
≤ 4000 mg/m3 eye
Page 13 of 15
D. Bulk Erosion Plots
Figure 12: Comparison of how the properties of the implant change over time due to bulk erosion. The values were rescaled
to be within a range of 0 to 1 so that all three parameters could be plotted on the same graph.
Figure 13. Change in diffusivity of drug through the implant as a function of molecular weight.
Page 14 of 15
E. Mesh Convergence
Figure 14. Plot of the mesh convergence of the concentration at a point on the retina-choroid boundary. A parametric sweep was
performed on the mesh properties of the nine mesh types in COMSOL. The zoomed-in portion shows the mesh convergence in
more detail.
F. CPU and Memory
Our model takes 7 seconds to run. It uses 1.15 GB of physical memory and 1.24 GB of virtual memory.
Page 15 of 15