This document describes a computational model for simulating transdermal drug delivery from binary mixtures of immiscible polymers. The model considers diffusion of fentanyl from a mixture with a hexagonal array of polyacrylate spheres in a polysiloxane matrix. Simulation results show that transdermal flux increases with polyacrylate volume fraction and is insensitive to drug diffusivity in polyacrylate. The model allows formulation of mixtures that produce flux similar to pure polyacrylate but with reduced drug content and enhanced utilization.

1. Computational Analysis of Transdermal Delivery From Binary Mixtures of Immiscible Polymers
PM Pinsky1
, J Audett2
and WW van Osdol 2
1
Department of Mechanical Engineering, Stanford University, Palo Alto, CA 94305
2
ALZA Corporation, Mountain View, CA 94043
bvanosdo@alzus.jnj.com
Abstract Summary
We have developed a computational model to
simulate the transdermal delivery of small molecules
from patches composed of binary mixtures of immiscible
polymers. The model is used to assess the sensitivity of
transdermal flux to characteristics of the polymeric
formulation and the permeating molecular species.
Introduction
Because of their ease of manufacture and relatively
simple delivery characteristics, the formulation of
transdermal patches for passive delivery has recently
favored drug-in-adhesive designs, which do perform well
in the case of potent compounds with reasonable
epidermal permeability (e.g. synthetic opioids and some
steroids). However, in the case of compounds for which
higher doses must be delivered or for which chemical
permeation enhancement is necessary, severe demands
are placed on the adhesive to dissolve all formulation
components to the required concentrations, while
maintaining acceptable adhesion and rheology. One
approach to resolving such difficulties involves tailoring
adhesive properties to match formulation requirements.
This can be done synthetically – the properties of
polyacrylates can be tuned by judicious choice of hard
and soft segment monomers, a wide variety of which are
available. It can also be accomplished via binary or higher
order mixtures of adhesives or adhesives and polymers1
.
In this work, we study mathematically the transdermal
delivery characteristics of binary mixtures of immiscible
polymers that differ strongly in their physical chemical
properties. In particular, we compare the performance of
the mixtures to that of the pure polymers.
Mathematical Methods
We consider the diffusive transdermal permeation of a
representative compound (fentanyl) from a binary mixture
of immiscible polymers, in which the minor phase is
distributed as a hexagonal array of spheres within the
major phase. This approximates the physically realizable
case of certain polyacrylate-polysiloxane (PA-PS) blends.
Such a system is characterized by the volume fraction of
the minor phase, the radius of the spherical domains, and
the solubility, S, and diffusivity, D, of the compound in
each polymer.
The problem is stated over the 3-D domain
, where ΩPSPA
Ω∪Ω=Ω PA
and ΩPS
are the union of
disjoint PA spheres and the simply connected PS region,
respectively. The diffusion equation that we solve over Ω
is
( ) ( ) ( ) 0=








∂
∂
∂
∂
−
∂
∂
j
ij
i x
tc
D
xt
tc x,
x
x,
where the spatial indices i and j range from 1 to 3. The
diffusivity Dij assumes different values according to
( )




Ω∈
Ω∈
= PSPS
ij
PAPA
ij
ij
D
D
D
x
x
x
The different material characteristics of the two phases
result in a partitioning of the diffusing compound at the
interface , giving rise to a discontinuity
in the concentration on Γ. For mass conservation, the
normal flux at the interface Γ is continuous. This requires
the following jump conditions to be satisfied on Γ
PSPA
Ω∂∩Ω∂=Γ
( )
( )
0
x
x,
x
=





Γ=
K
tc and ( ) ( ) 0
x,
x
x
=








∂
∂
Γ=
i
j
ij n
x
tc
D
where K is the partition coefficient between the respective
phases.
A finite element formulation for this problem has been
developed and implemented in the code Tensus902
.
Exploiting the symmetry of the hexagonal array of PS
spheres allows analysis of a representative volume, shown
in Figure 1, which has symmetry boundary conditions
imposed on the vertical “cut” faces.
Fig. 1 Geometry and finite element mesh for a volume
element of the binary mixture: 20% polyacrylate (v/v)

2. Results and Discussion
We first consider how transdermal flux varies as the
volume fraction of the spherical PA domains increases at
fixed domain number. The volume increase thus occurs
by increasing the radii of the spheres. The solubility and
diffusivity of fentanyl in the PA and PS are 8% (wt/wt)
and 1·10-9
cm2
/s, and 0.3% and 1·10-7
cm2
/s, respectively3
.
The thickness of the adhesive layer is 50.8 µm. Results
are plotted in Figure 2. As expected, flux at each time
point increases monotonically with PA volume fraction,
as does the patch’s ability to sustain near-peak levels of
flux. The time required to reach peak flux also increases
monotonically.
Fig. 2 Variation of transdermal flux with volume fraction
of the polyacrylate
Integrating the flux curves, one finds that the binary
mixture at 35% PA (v/v) produces 82% of the cumulative
flux of the pure PA patch. This suggests that one can
develop a binary adhesive formulation that produces
almost the same flux as the pure PA, while having
reduced drug content and enhanced drug utilization. The
physical basis for this is that high drug activity in the PS
can be maintained via resupply from the PA.
We consider next the effect of drug diffusivity in the
PA minor phase. Transdermal flux was calculated with
the parameter values listed above at a 20% fraction of PA
(v/v) as the diffusivity ranged over five decades. Results
are shown in Figure 3. The flux is insensitive to any
change of drug diffusivity in the PA from its baseline
value: a 1000-fold reduction is necessary to effect a 10%
reduction in peak flux.
In part, this insensitivity reflects a balance among the
parameters of the system, as they set the length and time
scales for transfer of fentanyl from the PA through the PS.
It also reflects the fact that epidermal transport is the rate-
limiting step in the delivery process: calculation reveals
that the epidermis is 12-fold less permeable than the
adhesive4
. Our observation that transdermal flux displays
a similar, though less severe, insensitivity to changes in
drug diffusivity and solubility in the PS further supports
this perspective. Thus, we expect significantly greater
dependence of transdermal flux on adhesive permeability
when that is rate limiting or the adhesive and skin
permeabilities are roughly equal.
0
0.3
0.6
0.9
1.2
1.5
0 6 12 18 24 30 36
Time (hours)
Flux(ug/cm
2
/hr)
1.E-09
1.E-10
1.E-11
1.E-12
1.E-13
D (cm
2
/s)
Fig. 3 Variation of transdermal flux with drug diffusivity
in the polyacrylate
0
0.4
0.8
1.2
1.6
2
0 6 12 18 24 30 36
Time (hours)
Flux(µg/cm
2
/hr)
100%
35%
20%
10%
4.3%
1.3%
Vol Fraction
Conclusions
We have developed a computational model that allows
us to simulate transdermal delivery from binary mixtures
of immiscible polymers, and used the model to study how
transdermal flux of a particular compound depends on the
physical properties of the mixture components. Such a
capability facilitates the optimal formulation and design
of such delivery systems, which provide an alternative to
single adhesive monoliths, when high drug and
permeation enhancer loadings are required, and which
may be drug sparing.
Our model is currently based on a hexagonal lattice of
spheres of the minor phase, but the mathematical methods
can, in principle, treat regular arrays of arbitrary
geometries. Thus, they permit exploration of the rich
phase behavior of polymer mixtures and block
copolymers with respect to their use in the design of
transdermal delivery devices.
Notes and References
(1) U.S. patents 6235306, 6221383, 6024976 and
5958446, assigned to Noven Pharmaceuticals, Miami, FL
(2) Tensus90 was written by Dr. Pinsky.
(3) Values were measured experimentally for a
polyacrylate currently used for transdermal formulation
and estimated for a polysiloxane used in marketed
transdermal products
(4) We calculate permeability, P, by P=D·S, using a
volume-fraction weighted average for the adhesive
mixture, and experimentally determined values of D
and S for epidermis.