Transdermal Patch Simulation Using the Lattice Boltzmann
Method with Active Diffusion in the Cell and Lipid
Pathways
Angell Cheea, Calvin Chenga, Steven Zhua, and Jiaolong Jianga
a
Materials and Science Engineering Department, Stony Brook University, Room 314 Old
Engineering, Stony Brook, New York 11794, United States
Abstract
The transdermal patch is continuing to grow in popularity as a drug delivery system that
shows promising results. Since transdermal patch testing on humans is expensive, time consuming,
and limited due to ethical constraints, a simulation of this system would provide a relatively
cheaper option. An existing model simulates this system but only accounts for drug diffusion in
the skin lipid layer. Our model expanded on this simulation to also involve the drug diffusion
through the skin cell layer. Through this simulation, it was found that the cell layer introduces a
much more rapid drug diffusion through the stratum corneum (SC). A sensitivity test was done on
the lipid diffusion coefficient, cell diffusion coefficient, and partition coefficient to see their effect
on the overall properties of the transdermal diffusion. From these tests, the cell layer was
concluded to be the most important factor in transdermal diffusion. The cell diffusion coefficient
affected the drug diffusion the most while the lipid diffusion coefficient and the partition
coefficient showed negligible changes.
Keywords: Transdermal Patch, Lattice Boltzmann, Stratum Corneum, Drug Delivery, Brick and
Mortar
1. Introduction
Computer modeling of transport
processes provided a relatively inexpensive
means to observe how interactions of
molecules on the atomic level affect a system
as a whole. It encompassed computation and
theoretical knowledge in order to create a
model that can simulate real life situations.
Numerical methods have been developed to
model various situations and help in
processing the simulation [1].
In the paper, we used a lattice
Boltzmann model to study the diffusion of
drugs from a drug patch through the human
skin. Getting data from experimentation can
be expensive and are prone to human error.
This, combined with human experimentation
can make it very difficult for scientists to test
for efficient transdermal patches [2]. By
creating a model, scientists will be able to
modify different parameters to find the key
parameters affecting the drug flux or peak
times in the patch. By running the parameters
through numerous simulations, the accuracy
of the human trials conducted can be greatly
improved.
The simulation will involve the
diffusion of drugs from a transdermal patch
to the outermost layer of the human skin, the
stratum corneum (SC). The stratum corneum
is the first barrier any foreign substance will
encounter when trying to enter the human
body. It has proved to be very efficient in
keeping these substances out [3]. In fact, this
barrier is often considered the rate-limiting
step of the drug diffusion.
50
Figure 1. Brick and mortar representation of the
stratum corneum layer
In order to create this model, we first
needed to create a structure that mimicked the
stratum corneum. The brick and mortar
model was is usually employed to model this
skin layer because of the resemblance [4].
When observed under the microscope, the
stratum corneum has many layers of skin
cells stack atop of each other, staggered in
various ways [5][6]. The brick and mortar
model created a similar structure, where the
bricks represented the skin cell while the
mortar represented the lipid layer in between
the skin cells.
To model the movement of the drug
within each layer, we used the lattice
Boltzmann method. The Lattice Boltzmann
Method (LBM) was developed relatively
recently as a mesoscopic method that could
bring together some properties from both
macroscopic and mesoscopic modeling
techniques [7,8,9,10]. This method was often
used to solve incompressible, time dependent
Navier Stokes equations. The advantage was
that it can simulate complex systems such as
multiphase flows and chemical reactions
between different fluids and their
surroundings. Instead of focusing on the
microscopic behavior of the molecules, the
LBM acted on the averaged microscopic
motion. Using the lattice Boltzmann method
and the brick and mortar model, a simulation
can be created to predict the drug flow.
Previous works have already created a
simulation using both lattice Boltzmann and
the brick and mortar model [11,12]. The
model created by Sun Ning only accounted
for the diffusion of the drug through the lipid
layer. Drug flow through the cell layer was
not computed by the simulation. Therefore,
this model was not completely accurate since
it does not represent the whole system.
In our simulation, the drug will be
allowed to flow through both lipid and cell
layer, while also including the diffusion in
between those layers. The data collected from
our simulation will be compared to the data
from the simulation that only includes the
lipid layer. The overall drug flow into the
stratum corneum should increase when the
drug is allowed to diffuse through the cell and
lipid layer as opposed to when the drug is
only allowed to diffuse through the lipid
layer.
2. Methods/Procedure
Palabos is an open source
computation fluid dynamics (CFD) solver
that has a heavy reliance on the lattice
Boltzmann methods. Palabos deals with a
large range of topics, some of which are flow
through a porous media, and multi-phase
flow. By having references to alleviate
complex calculations Palabos is the perfect
software to help model a transdermal patch
diffusion.
We created two lattices to help
simulate the drug diffusion through the cell
and lipid layer. These layers are effectively
mirror images of each other. The first lattice
is the cell lattice, where the cells, represented
by brick-like shapes, are given a basic
dynamic for which the drug is allowed to
travel through. Most of the lipid pathways in
this lattice is set to have no dynamics,
meaning the simulation will ignore any
particles in this range and not compute them.
The lipid layer that immediately surrounding
the brick is given a bounce back dynamic.
From this setup, drugs in the cell will stay in
the cell. If they were to travel to the edge of
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the cell, they will hit a bounce back layer that
will keep it in the cell and away from the lipid
layer.
interface layer. This interface layer allowed
the transport of drugs from the cell lattice to
the lipid lattice and vice versa. Using this
method, we modeled the transdermal drug
diffusion with two lattices that solely
computed the diffusion through their
respective layer, and an interface processor
that modeled the drug diffusion between
them.
Figure 2. Location of the types of dynamics
located in each lattice
The lipid lattice is similar except that
the lipid is given the basic dynamic and most
of the cell layer is given no dynamic. The
edge of the brick layer, however, is also
treated as a lipid layer and given normal
dynamics. The cell layer directly under this
edge layer is set to have a bounce back
dynamic. From this setup, the drug in the
lipid layer will stay in the lipid layer, but is
also allowed to transfer into the edge of the
brick layer. This edge is utilized as an
interface which will be used to model the
transport of drugs from lipid to cell layer.
From the edge of the brick layer, the drug will
not be able to transverse any deeper into the
cell since it will be immediately met with the
bounce back dynamic that is given to the
layer immediately under this edge.
The creation of two lattices is optimal
because it breaks a complex problem into
smaller problems, which are significantly
easier to solve. By separating the two lattices
with
their
respective
conditions,
understanding the formation of the model and
managing each of its subparts becomes much
easier.
The resultant interface layer was
created from the two lattices. In both lattices,
the edge of the brick layer is given regular
dynamics. Because of this, a drug density will
exist in both lattices. All the points on the
model where the drug density exist for both
the cell and lipid lattice was regarded as an
Figure 3: Interface Layout. Interface layer is
shown in black.
A sensitivity test was performed on
the new model to determine which factors
affected the drug flux the most. Three
parameters were tested: diffusion coefficient
of drug through the lipid layer, diffusion
coefficient of drug through the cell layer, and
the partition coefficient of the drug between
the lipid and cell layer. In our simulation we
used coefficients that simulated the drug
fentanyl. A standard control case utilized a
lipid and cell coefficient of 1.2 * 10-7 cm2/s
[13]. The partition coefficient used for the
base case was 0.14. The partition coefficient
controls the amount of drug that travels
between the lipid and cell layers. The lipid
diffusion coefficient and partition coefficient
are the same values used in Sun’s Thesis [11].
We used the same diffusion coefficient for
the cell and lipid pathway for this sensitivity
test.
The total drug amount that was
measured in the patch and SC is recorded in
equivalent lattice units (ELU). This unit
52
represents the total amount of individual
lattices units in the simulation that contains
the drug. It is difficult to translate this drug
amount into physical units. This data was
calculated with the following equation:
𝐸𝐿𝑈𝑝𝑎𝑡𝑐ℎ = 𝜌𝑝𝑎𝑡𝑐ℎ ∗ 𝑊𝑝𝑎𝑡𝑐ℎ ∗ 𝐻𝑝𝑎𝑡𝑐ℎ
𝐸𝐿𝑈𝑆𝐶 = 𝜌𝑆𝐶 ∗ 𝑊𝑆𝐶 ∗ 𝐻𝑆𝐶
(1)
(2)
In the above equations, ρ represents
the average drug density in the structure. W
and H represents the width and height
respectively of the structure. Together, they
represent the total area in which the ELU is
being evaluated over.
3. Results
Figure 4a, 4b and 4c showed the
effect fentanyl diffusing through the model at
different cell diffusion coefficients.
The cell diffusion coefficient
variation did not change the diffusion
through the patch but had a difference in the
amount of drug diffused through the SC
layer. At a higher cell diffusion coefficient,
the drug traveled through the system much
quicker. Because of this, the SC contained a
much lower overall concentration of the drug.
The cumulative flux of the patch did
not correlate very well with the cell diffusion
coefficient. A max peak was observed at a
two times base case, but the other cases
showed no changes. More cases needed to be
simulated regarding cell diffusion coefficient
on the overall flux before a conclusion can be
determined.
In Figure 5a, 5b, and 5c, the lipid
diffusion coefficient was another parameter
tested in the simulation.
Figure 4a. Drug distribution in the patch with
different cell coefficients
Figure 4b. Drug distribution in the SC with
different cell coefficients
Figure 4c. Flux for different cell coefficients
53
Figure 5a. Drug distribution in the patch with
different lipid coefficients
variation in the cell coefficient. This may be
because the SC is mostly made up of cell
layer, so a change in the lesser lipid layer
would have an overall lesser effect.
The cumulative overall flux of the
drug decreased as the lipid diffusion
coefficient increased. It did follow the same
overall pattern of spiking and a resultant
gradual decrease, but the peak of this spike
seems to be controlled by the lipid
coefficient. The highest peak occurred when
the coefficient was the smallest and
decreased as the coefficient is increased.
Figure 6a, 6b, and 6c showed the
effects of varying the partition coefficient.
Figure 5b. Drug distribution in the SC with
different lipid coefficients
Figure 6a. Drug distribution in the patch with
different partition coefficients
Figure 5c. Flux for different lipid coefficients
Similar to the cell coefficient test, the
patch concentration was largely unaffected
by lipid coefficients variations.
An
increasing lipid coefficient decreased the
total drug amount in the SC, which was the
exact same trend observed in the cell
coefficient test. The variation of the lipid
coefficient however did not have as great of
an effect on the total drug in the SC as did the
Figure 6b. Drug distribution in the SC with
different partition coefficients
54
Figure 6c.
coefficients
Flux
for
different
partition
A higher partition coefficient should
allow the drugs to more easily move between
the lipid and cell layers. This will then free
more room in the SC for the drug to diffuse
into from the patch. Unlike the previous two
tests, varying the partition coefficient seemed
to have more of an effect on the drug amount
in the patch.
Similar to the lipid coefficient test,
the overall drug amount in the SC showed
little change when the partition coefficient
was varied. This was an interesting result
since the drug amount in the patch showed
some variation.
A lower partition coefficient caused
the drug to diffuse out the patch more slowly.
Also, a lower partition coefficient caused the
drug amount to peak at a lower value than the
other base conditions. This was apparent
since a lower partition coefficient did not
allow as much drug to diffuse to and from the
lipid and cell and vise-versa.
Cumulative overall flux and partition
coefficient seemed to have a positive
correlation. Similar to the analysis of
cumulative overall flux and lipid partition
coefficient, the overall flux followed the
same general pattern with a sharp peak and a
gradual decline. Unlike the negative
correlation observed in 5c however, an
increase in partition coefficient resulted in an
increase in the peak of the cumulative overall
flux.
4. Discussion
The human SC is primarily made up
of cells, which is also true in our model. The
“bricks” make up a large portion of our
structure. The lipid layer only existed in the
small surroundings between each cell.
Because of this, the cell layer dynamics will
probably cause the most drastic changes in
the results. Variation of the lipid coefficient
in figure 5a and 5b shows that the lipid
coefficient have minimal effects on the total
drug amount in SC and the patch. As seen in
figure 6a and 6b, the partition coefficient also
has minimal changes in the outcome. Since
the partition coefficient controlled the
movement of drugs between lipid and cell
layer, it should have more effect on the total
drug amount than the lipid coefficient, but
less effect than the cell coefficient. Figure 4b
showed that the cell coefficient did indeed
affect the total drug amount in SC by much
more than the lipid coefficient and the
partition coefficient. As a result, the amount
of drug present on the SC was largely
dependent on the diffusion of drugs through
the cell layers in the SC.
The flux of the drugs however, seems
to follow a different trend. According to the
figure 5c, the lipid diffusion coefficient had
the most effect on the flux. The highest lipid
diffusion coefficient resulted in the lowest
peak flux while the lowest coefficient
resulted in the highest peak. With a high lipid
diffusion coefficient, the drug is able to pass
through the lipid layer in much faster. This
gives the drug less time to diffuse into the cell
layer. The diffusion of the drug through the
cell layer is predicted to be the main pathway
for which the drug diffuses. Because of this,
a high lipid diffusion coefficient can actually
be detrimental to the flux because it decreases
the amount of drug that can enter this main
diffusion pathway, the cell layer. The
partition coefficient in figure 6c showed
negligible changes in flux for the range that
we tested. The cell diffusion coefficient data
in figure 4c remains inconclusive. There was
55
no general pattern that can be observed. The
base case, 0.5x base case, and 4x base case all
have the same peak height, but the the 2x base
case observed a sudden 50% jump in peak
height. This could be an optimal point where
the lipid diffusion coefficient and cell
diffusion coefficient may experience some
resonance effect resulting in more flux.
Further testing will need to be done with
these parameters.
Ning Sun’s model of transdermal
drug diffusion only accounted for the
diffusion of drug through the lipid layer. His
results showed that the cumulative flux
increases sharply, then steadily drops off.
Our results in figure 4c, 5c, and 6c all showed
a similar trend observed in Sun’s results [2].
Our results were similar in the sense that we
also have a large initial spike in flux. Adding
a cell layer however, seems to have given our
results a less steep increase when compared
to Sun’s results. The overall flux after this
peak also decreased more rapidly than Sun’s
results. Sun’s results showed the overall flux
to have a steady linear decrease after the
spike while our results in figure 4c, 5c, and
6c all showed a decrease that's more
comparative to an exponential decay.
The cumulative overall flux was
observed to have a positive correlation with
partition coefficient and a negative
correlation with lipid diffusion coefficient.
The lipid diffusion coefficient and
cumulative
overall
flux’s
negative
correlation made sense when looking at the
data in 5b. The highest flux peak correlated
with the highest drug concentration in the SC.
Higher drug concentration in the SC will lead
to a higher flux since there was a greater drug
concentration that would want to diffuse out
of the SC. This pattern should ideally also be
observed in the sensitivity test for cell
coefficient, but that data is inconclusive as is.
Similarly, the partition coefficient
and cumulative overall flux’s positive
correlation agreed with the data in figure 6b.
The highest cumulative overall flux was the
four times base case, which is also where the
SC observed the highest peak in drug
concentration. All these cases followed the
same peak height order for concentration in
SC and cumulative overall flux. A higher
partition coefficient will allow more drugs to
pass from layer to layer. A higher transfer in
between these layers will allow the drug to
spend less time in the SC and therefore exit
the SC in greater numbers. This will speed up
the rate of drug transfer which can explain the
positive correlation observed between the
cumulative overall flux and the partition
coefficient.
5. Conclusion
By adding the cell layer into the
transdermal patch simulation, the resultant
drug in SC, patch, and accumulative flux
changed drastically. Diffusion of drugs
through the lipid layer only seemed to lead to
a steady and slow flushing of the drug from
the SC. The addition of the cell layer
increased this rate dramatically. Ultimately,
the addition of the cell layer in the
transdermal patch sped up the diffusion of
drugs through the system and lead to a steeper
decreases in overall cumulative flux. Our
various sensitivity tests have concluded that
the cell layer did indeed play the largest role
in the diffusion of the drugs. Variation in the
partition coefficient and the lipid coefficient
showed negligible changes when compared
to variations in the cell coefficient. The
diffusion of drugs from a transdermal patch
and through the stratum corneum was
ultimately most dependent on the cell
diffusion coefficient and least dependent on
the lipid coefficient and partition coefficient.
Acknowledgements
This research was supported Stony
Brook University. We would like to thank
Dr. Dilip Gersappe for providing us
mentorship throughout the project. We
56
would also like to thank our graduate
advisor, Jialong Jiang, for guiding us
through the simulation process. We would
also like to thank Miriam Rafailovich and
Clement Marmorat for guiding us through
the research.
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