DescriptionDrug Patch Design

Mass Transfer Processes

Dr. Manolis Tomadakis

Taleb Alsaffar

Report Submitted: 03/2/2021

I.

Project Description:

The primary aim of this project is to design a drug patch that provides Scopolamine, to

assist astronauts, that prevents motion sickness by reducing the muscle memory activity. After

some research, it has been decided that the problem will be solved via a Scopolamine drug patch.

Since the astronauts will not be able to adjust the patch regularly, the patch is supposed to

provide them with enough Scopolamine for a 64-hour duration.

The nerve fibers responsible of a person’s balance are located deep within the ear; they

become very active when a person is in motion. The increased activity could cause dizziness,

nausea, and/or vomiting. Scopolamine inhibits the activity of certain nerve fibers, thus

alleviating the symptoms.

Drug patches mostly consist of three layers: a backing layer, a drug reservoir, and a

membrane. The backing layer is impermeable to the outside, i.e. it only allows the drug to diffuse

in one direction, which is towards the skin. In the drug reservoir, especially for Scopolamine

patches, the drug is present in a dilute solution.

Backing Layer

Membrane

Skin

Blood

Figure 1: Elastration of a drug patch and its components

Diffusional Direction

Drug Reservoir

According to Fick’s law, the mass transfer happens because of a concentration gradient,

where it travels from a high concentration to a low concentration region. In the case of the patch,

the drug leaves the reservoir and diffuses across the membrane and the skin to reach to the blood.

The diffusivity and thickness of both the skin and the membrane differ from one individual to the

next. The aim is to create a patch that works well on all skin types, different skin thickness (l2)

and diffusivity (D2), by manipulating the diffusivity of the membrane (D1), the membrane

thickness (l1), and the concentration of Scopolamine in the reservoir (Cp).

II.

Pseudo-steady State Model

The simulator employs a simplified version of the design known as the pseudo-steady

state model. First by deriving the equations of the system using Fick’s Law:

•

Membrane: = − 1

•

Skin: = − 2

Where is the molar flux, is the change in concentration with respect to the z-axis.

We find that in both the membrane and the skin only the diffusion term of Fick’s Law is present.

This is because Scopolamine is present in a dilute solution; it is then presumed that the bulk flow

effect is negligible.

The drug reservoir is assumed to be well mixed (i.e. no concentration gradient within the

reservoir) and the temperature effects are neglected. From the mole balance, the molar flux is

found to be constant through both the membrane and the skin at steady state:

•

= 1 = 2 =

In order to solve for the overall flux, the boundary conditions must be defined; note that the

Scopolamine concentration in the blood can be assumed zero because it is reported in

micrograms/liter, which is order of 5 magnitudes smaller than the smallest patch concentration:

•

Membrane: at = 0 → = , = 1 → =

•

Skin: at = 1 → = , = 1 + 2 → = 0

Where z is the length from the drug reservoir, Cp is the concentration of the patch and Ci is the

concentration of patch at the interface.

Z

Figure 2: System Schematic

Now we can rewrite equations in terms of the overall flux (Nao) and solve for (Nao):

•

Membrane:

•

Skin:

−

2

−

1

1 + 2

∫ 1

1

∫0 = ∫ → − = − 1

1

0

= ∫ → 0 − = − 2

2

Solving both equations and rearranging for (Nao) we obtain the following:

•

= 1

[

1

2

+ 2]

From equation , the flux through multiple resistances can be obtained; the term in between the

brackets is the overall resistance of the system, which is a linear sum of the resistance of the

membrane and the skin.

The previous parameters can be related to obtain the following O.D.E:

•

×

( ) = ( 0 ) − ∫

0

Where Na is:

•

Membrane: = − 1

•

Skin: = − 2 (2)

With the following boundary conditions:

o At = 0 → = ( )

o At = 1 → =

o At = 1 + 2 → = 0

The initial conditions for equation then are:

o at = 0, = 0 → ( ) =

o at = 0, > 0 → ( ) =

Where Css is the concentration profile from the steady state solution.

The O.D.E equation then integrated in time increments to give linear concentration profile

evolutions and flux as follow:

∆

•

( ) = ( 0 ) (1 − )

•

0

(1 −

( ) =

( )

∆

)

Where = 1 + 2 , and is the thickness of the patch

1

III.

2

Rigorous Unsteady State Model

The multidimensional unsteady state model for drug diffusion into the blood was

simplified in the simulator via numerical techniques, several assumptions, and the pseudo-steady

state model. The multidimensional unsteady state model consists of three partial derivatives that

are all first order in time and second order in direction. The general form is as following:

•

2

2

= 2 + 2

The neglect of the y-axis was based on the presumption of symmetry in the astronaut’s hand. The

three partial derivative equations with twelve boundary conditions and three initial conditions

can be solved to obtain the concentration profiles, i.e. the concentration profile of the patch, Cp(z,

x, t), the concentration profile of the membrane, Ca1(z, x, t), and the concentration profile of the

skin, Ca2(z, x, t). The flux profiles, Np(z, x, t), Na1(z, x, t), Na2(z, x, t) can be then obtained from

the concentration profiles.

Boundary conditions are:

o The cover material of the patch is impervious to the drug:

=0

o The drug will only diffuse in the z-direction, into the blood:

1 2

=

=

=0

o At the reservoir-membrane interface:

= 1 &

= 1

1

o At the membrane-skin interface:

1 = 2 & 1

1

= 2

2

o At the skin-blood interface, we assume instantaneous absorption: 2 = 0

The three initial conditions are:

o The initial concentration of the drug in the reservoir is equal to the initial concentration of

the scopolamine in the patch: at 0 → = ( 0 )

o

The initial concentration of scopolamine in the membrane is zero: at 0 → 1 ( 0 ) = 0

o The initial concentration of scopolamine in the skin is zero: at 0 → 2 ( 0 ) = 0

In order to solve the unsteady state model, the pseudo-steady state approximation with

some other assumptions were used. We assumed that the reservoir is well mixed, and that

Scopolamine is uniformly spread within the reservoir. Then we assume that the concentration

will decrease uniformly in the patch, i.e. concentration will vary with time in the patch. The

parameters that will determine the patch concentration at any time, t, are the following:

➢ Initial concentration, Cp (t0)

➢ Area of the Patch, A

➢ Flux, Na, at all previous times

The previous parameters can be related to obtain the following O.D.E:

2

2

= 2 + 2

1

2

= 1

2 1

2

2

+ 1

2 1

2

2

2

= 2 2

2 + 2 2

IV.

Data Collection Process

The module includes a data simulator that produces a set of values and graphs to help

determine the best fit for the design. When accessing the simulator, a set of criteria is given

which includes a flux range, a time frame, the skin diffusivity, D2, and thickness, l2, with their

corresponding errors. The simulator then provides a set of three membrane diffusivity, D1,

values, three membrane thicknesses, l1, and seven patch concentrations, Cp, to choose from for

the design. Using the data collection sheet provided by Dr. Tomadakis the data was first

collected for the fast skin by using the highest skin diffusivity value, D2 + error, and the lowest

skin thickness, l2 – error. Then choosing the first membrane diffusivity, D1, and thickness, l1, and

matching it with the best concentration value, Cp, that gives a flux value between the flux range

and a time greater than, or equal to, the time frame. Following that, we vary the thickness, l1, and

match it again with the best concentration value, Cp, later on we proceed to the next value of

membrane diffusivity, D1, and do the same to all membrane diffusivity values.

Having finished the fast skin, we proceed to the slow skin which has the lowest skin

diffusivity, D2 – error, and highest skin thickness, l2 + error. We then use the best set of D1, l1,

and Cp to find the one that gives a flux value between the flux range and a time greater than, or

equal to, the time frame. Finally, the values that are in the range of, or have a flux that is

equivalent to or greater than the time frame in the slow and fast skin set are selected

V.

Data Collection Sheet

Table 1: Data from Simulator

VI.

Flux and Concentration Plots

These next two plots are obtained from the simulator’s best design parameters; they both worked

for the fast and the slow skin.

∆t=124 min

Ip=0.022 cm

R=4.93e+04 s/cm

Cp=1.83e-7 mol/cm^3

Figure 3: Flux vs. Time plot

It can be noticed from the graph that the pseudo-steady state steps of flux decrease with time

until it reaches the time frame of 64 hr (3480 mins).

∆t=124 min

Ip=0.022 cm

R=4.93e+04 s/cm

Cp=1.83e-7 mol/cm^3

Figure 4: Concentration vs. Time plot

VII.

Discussion and Results

The simulator led us certain criteria for the drug patch design. The patch had to be

effective for at least 64-hour, delivering Scopolamine at a rate in the range of (2.56 ± 0.57)×10-12

mol/cm2 for astronauts with a skin diffusivity between (5.2 ± 0.08) ×10-7 cm2/s and a skin

thickness between 0.01 ± 0.0019 cm. Only one value was found to be a good enough patch for

the design, but it worked only for fast skin case. Unfortunately, the modelling did not yield any

good combinations for slow skin. The parameters of the best design are the following: D1 = 1.

1.17e-6 cm2/s, l1 = 0.04 cm, Cp (t0) = 1.83e-7 mol/cm3. It is important to keep in mind that these

constants are estimated using the pseudo-steady state approximation.

VIII.

Appendix

•

Data used for generating the plots

IX.

References

1. M. Mass Transfer Projects, Modules, Patch Simulator.

2. M. Mass Transfer Notes, Projects, Drug Patch Module Project.

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