Simulating Drug-eluting Stents
Progress Made and the Way Forward
Sean McGinty
1
, Christopher McCormick
2
, Sean McKee
1
, Marcus Wheel
3
, Simon Kennedy
4
and Keith Oldroyd
5
1
Department of Mathematics and Statistics, University of Strathclyde, Glasgow, U.K.
2
Biomedical Engineering and Strathclyde Institute of Pharmacy and Biomedical Sciences, University of Strathclyde,
Glasgow, U.K.
3
Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow, U.K.
4
Institute of Cardiovascular and Medical Sciences, University of Glasgow, Glasgow, U.K.
5
West of Scotland Regional Heart and Lung Centre, Golden Jubilee National Hospital, Glasgow, U.K.
Keywords:
Drug-eluting Stents, Mathematical Modelling, Drug Release, Binding.
Abstract:
Drug-eluting stents have significantly improved the treatment of coronary artery disease. Compared with their
bare metal predecessors, they offer reduced rates of restenosis and thus represent the current gold standard
in percutaneous coronary interventions. Drug-eluting stents have been around for over a decade, and while
progress is continually being made, they are not suitable in all patients and lesion types. Furthermore there are
still real concerns over incomplete healing and late stent thrombosis. In this paper, some modelling approaches
are reviewed and the future of modelling and simulation in this field is discussed.
1 INTRODUCTION
Coronary heart disease (CHD) is the main cause of
death in developed countries (Murray and Lopez,
1997). Such is the extent of the condition, CHD is
responsible for 18% of all deaths in the United States
annually (Lloyde-Jones, 2010). In simple terms, CHD
is caused by a blockage or interruption to blood flow
and often results in heart attack. It is generally con-
sidered that this problem is the result of fatty deposits
accumulating and lining the arterial walls over a pe-
riod of many years (Beers, 2004). The fatty deposits
are called atheroma and the process during which the
atheroma accumulates is termed atherosclerosis (Lu-
sis, 2000). If left untreated, this leads to episodes
of angina. As well as restricting blood flow, the
atherosclerotic plaque is also vulnerable to rupture,
ultimately resulting in a heart attack.
Traditionally, by-pass surgery was the only avail-
able treatment option. However, in the majority of
cases this has now been replaced by minimally in-
vasive procedures such as the insertion of a small
metallic cage called a stent into the occluded artery.
When a stent is implanted into an artery, the endothe-
lium is severely damaged. This provokes an inflam-
matory response which leads to excessive prolifera-
tion and migration of smooth muscle cells (SMCs)
towards the lumen. The result is restenosis - the re-
narrowing of the arterial wall. The introduction of
drug-eluting stents (DESs) has significantly reduced
the occurrence of in-stent restenosis (ISR), by releas-
ing a drug to inhibit SMC proliferation.
Drug-eluting stents have undoubtedly signifi-
cantly improved the treatment of coronary artery dis-
ease. Compared with their bare metal predecessors,
they offer reduced rates of restenosis and thus repre-
sent the current gold standard in percutaneous coro-
nary interventions. But despite being around for over
a decade, and while progress is continually being
made, they are still not suitable in all patients or lesion
types. Furthermore, there are still concerns over in-
complete healing of the artery and late stent thrombo-
sis (LST), resulting in anti-platelet therapy being rec-
ommended for at least 12 months after implantation
(Stefanini and Holmes, 2013). Substantial efforts are
now dedicated towards the development of enhanced
DESs and in this paper we summarise the modelling
approaches currently adopted and suggest important
considerations for future modelling and simulation in
this field.
664
McGinty S., McCormick C., McKee S., Wheel M., Kennedy S. and Oldroyd K..
Simulating Drug-eluting Stents - Progress Made and the Way Forward.
DOI: 10.5220/0004622706640672
In Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (BIOMED-2013), pages
664-672
ISBN: 978-989-8565-69-3
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
2 THE CHICKEN OR THE EGG
DILEMMA
What is the logical sequence of events? Stent design
and then modelling of drug release and uptake into the
arterial wall? Or is it the other way round? Presently,
the stent manufacturers are predominantly concerned
with mechanical integrity of the device and as such
the stent design is usually the first consideration. The
stent must be flexible and expandable and stay in situ
after deployment. During the expansion process the
stent should undergo minimum shortening and after
implantation should conform to the natural geome-
try of the vessel without any unnatural straightening
(Khan et al., 2012). Radial strength is another key
component; without this the stent will collapse under
the strain of the artery. Furthermore, the materials
used must be biocompatible and must not fracture.
But it is no good having a stent which is mechan-
ically sound but does not elute drug in a favourable
fashion. The release of the drug must be controlled
so that it elutes over a defined period of time and, fur-
thermore, the drug concentration in the arterial wall
should ideally be maintained between therapeutic and
toxic levels over and beyond the period of release.
Taking this into account, it would seem that the drug
release and uptake is intrinsically linked to the stent
design and so a fair argument could be made either
way.
Ideally the stent should be optimised, both in
terms of the mechanical design (material used, num-
ber and pattern of struts) as well as drug loading (type
and mass of drug, coating technique) so that the re-
quired clinical outcomes are realised. This optimisa-
tion is further complicated by the fact that every pa-
tient is different; the lesions vary in size and compo-
sition as well as location in the arterial tree. Further-
more, some patients have other complications such
as diabetes or hypertension. Thus a single optimised
stent design is simply unrealistic, but it may well be
possible to develop an optimised stent for a set of dif-
ferent situations.
Realising the importance of stent design, a num-
ber of authors have investigated various aspects. The
influence of stent geometry on restenosis was inves-
tigated (Garasic et al., 2000) while the distribution
of the stent struts has been experimentally studied
(Hwang et al., 2001), with the authors concluding
that the mere proximity of delivery device to tissue
does not ensure adequate drug targeting. The effect of
the number of struts and the ratio between the coated
area was researched by Delfour et al., and they at-
tempted to optimize the effect of the dose (Delfour et
al., 2005). A mathematical model for the study of the
mechanical properties of endovascular stents in their
expanded state has also been proposed (Tambaca et
al., 2010). Three-dimensional models of stent expan-
sion have been presented by, among others, (Zunino
et al., 2009) and (Horner et al., 2010).
3 THE EVOLUTION OF THE
DRUG-ELUTING STENT
Over the past decade DESs have evolved, and al-
ready third-generation DESs have been developed.
Despite their differences and improvements, these de-
vices typically all have three main components; the
stent platform, the coating and the drug.
3.1 First Generation DESs
The first-generation DESs Cypher (sirolimus-eluting
stent; Cordis Corporation) and Taxus (paclitaxel-
eluting stent; Boston Scientific Corporation) com-
prised a stainless steel platform with a drug contain-
ing polymer coating attached to the stent struts ((Ste-
fanini and Holmes, 2013), (Tzafriri et al., 2012)). The
philosophy behind this design was to allow the drug
to be released gradually so as to avoid toxic levels
of drug initially, but also to permit sustained deliv-
ery over many weeks. The Cypher stent actually con-
sists of three distinct layers; a base coat, a drug-filled
middle layer and a drug-free topcoat. This design en-
hances the controlled nature of the release. While the
polymers used in these first generation DESs are dif-
ferent, they are both non-erodible. The drugs used
(sirolimus and paclitaxel) are both lipophilic and are
able to inhibit SMC proliferation and migration.
3.2 Second Generation DESs
The second-generation DESs Endeavor (zotarolimus-
eluting; Medtronic), ZoMaxx (zotarolimus-eluting;
Abbott Laboratories), Promus (everolimus-eluting;
Boston Scientific Corporation) and Xience V
(everolimus eluting; Abbott Laboratories) attempted
to improve the biocompatibility and reduce the
incidence of thrombosis rate which was associ-
ated with first-generation DES. These stents were
generally designed with thinner struts and utilised
cobalt-chromium and platinum chromium platforms.
A variety of multi-layer polymer combinations were
used on these stents to attempt to control the release.
Generally these stents have been shown to exhibit
lower thrombosis rates compared with first generation
DES (Khan et al., 2012).
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3.3 Third Generation DESs
Since the polymer coating in the earlier DES has been
associated with local vascular inflammatory reaction
and potentially inducing late stent thrombosis, newer
generation stents have focussed on biodegradable
polymers (BioMatrix, Biosensors Inc and NEVO,
Cordis Corporation, Johnson & Johnson), where the
polymer carries and controls the drug release and then
erodes or vanishes, and also coatings which do not
contain any polymer at all (Yukon, Translumina and
BioFreedom, Biosensors Inc), with the drug being
contained on a modified surface of the stent.
4 MODELLING THE RELEASE
OF DRUG FROM STENTS
Most of the modelling of drug release from stents in
the literature has thus far been concerned with first
generation DESs. Drug release from these stents has
been modelled as a diffusion dominated process (see
for example (McGinty, 2011) and (Pontrelli and de
Monte, 2007)), with the drug concentration in the
polymer C
p
satisfying a diffusion equation with drug
diffusion coefficient D
p
. In one dimension this is sim-
ply
C
p
t
= D
p
2
C
p
x
2
. (1)
For the case of in-vitro drug release a zero flux bound-
ary condition is normally assumed at the impermeable
stent and either an infinite sink or Robin type bound-
ary condition at the interface with the release medium.
Models such as these can admit analytical solutions
and have shown favourable results when compared
with in-vitro experimental data. Figure 1 displays a
comparison between the model predicted cumulative
percentage of drug released and the average of four
experimental in-vitro release profiles from the Cypher
stent (for more details see Section 6). These analyti-
cal solutions also allow the drug diffusion coefficient
to be estimated via a best fitting process.
Several simplifying assumptions are usually
made. Firstly, it is assumed that the device geometry
is that of a thin film with no edge effects so that the
modelling may be restricted to one dimension. The
diffusion of the drug in the polymer is thus consid-
ered to be isotropic and it is usually assumed that the
diffusion coefficient is independent of time, space and
concentration. Furthermore, the initial drug concen-
tration is usually taken to be uniform. However, in
reality this is not always the case. For example, the
polymer coating on the Cypher stent consists of three
distinct layers; a base coat, a drug-filled middle layer
0 10 20 30 40 50 60
0
10
20
30
40
50
60
70
80
90
100
Cypher release under infinite sink conditions
Release Period (days)
Percentage of Rapamycin released
Model, Dp=6.3x10
17
EXPT(Average)
Figure 1: Comparison between in-vitro experimental data
and model of Cypher release.
and a drug-free topcoat. However, the good fit be-
tween the experimental data and the simple model de-
scribed above suggests that these modelling assump-
tions are reasonable and capture the release of drug
from the Cypher stent, albeit with a layer averaged, or
effective diffusion coefficient. A tighter fit is antici-
pated if the polymer is modelled as a tri-layer system
with the drug contained only within the middle layer
initially. Actually, it has been shown that the drug can
be found in the top coat prior to implantation, sug-
gesting that the intended drug-free top coat may not
always be realised. However it is less clear how the
in-vivo situation may be modelled. This is considered
in the following section.
Newer generation DESs which have focussed on
biodegradable polymers and polymer-free modified
surface designs have received less attention in the lit-
erature in terms of modelling their release. In the case
of biodegradable polymeric stents, the drug release
is likely governed by diffusion, erosion and possi-
bly dissolution and/or swelling ((Siepmann and Siep-
mann, 2008); (Fredenberg et al., 2011)). However, in
the case of polymer free stents, it is less clear how
the sustained release is obtained and how this may be
modelled. Furthermore, it even less clear how in-vivo
release may be modelled. Realising this, Tzafriri et
al. chose to write down a two-part equation to de-
scribe drug release from the Cypher and NEVO stents
and, using a best fit process, found the values of the
parameters of the model (Tzafriri et al., 2012). Their
equation assumes that these stents contain two pools
of dispersed drug, one that is surface-connected and
elutes through a percolating network of drug filled
pores, and another that is embedded within the ma-
trix and diffuses more slowly through the percolating
polymer phase. Thus they utilised the following equa-
tion for the mass of drug released from the Cypher and
NEVO stents:
M
stent
(0) M
stent
(t) = M
f
0
1 e
K
f
0
t
+ Q
sus
t,
(2)
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where M
stent
(0), M
f
0
, K
f
0
and Q
sus
denote, respec-
tively, the initial load of drug, the initial pool of first
order eluting drug, rate constant and Higuchi rate con-
stant. This equation, however, does not satisfy mass
conservation principles: the mass of drug released
eventually tends to infinity as time increases. As a re-
sult, while their data is well-fitted to this equation for
this set of experiments, it is unlikely that their model
may be used in a predictive capacity. A further warn-
ing must be issued when attempting to identify the re-
lease mechanism: it may be possible to fit experimen-
tal data to a given model with a variety of different
parameter combinations, even if the values of some
of the parameters are unlikely (Sirianni et al., 2010).
5 MODELLING UPTAKE OF
DRUG INTO THE ARTERIAL
WALL
While the stent manufacturer may be primarily con-
cerned with stent design and the drug release profile,
clinicians are more interested in the drug concentra-
tion profile across the arterial wall, and in particular
in the target SMCs, as well as the therapeutic dura-
tion. Thus an understanding of what happens to the
drug in the arterial wall is essential. The arterial wall
is porous and plasma flows through the extracellular
matrix as a result of the transmural pressure gradi-
ent. This pressure gradient induces convection so that
drug transport across the arterial wall is governed by
a combination of convection, molecular diffusion and
binding. A drug with anti-proliferative properties is
usually chosen so that when it is uptaken by SMCs it
suppresses their proliferation. Depending on the par-
ticular properties of the drug, it may also bind to sites
within the extracellular matrix.
5.1 One-dimensional Models
Early models of drug transport through the arterial
wall tended to include only convection and diffusion
((Pontrelli and de Monte, 2010), (Zunino, 2004)) and
neglected drug uptake/binding. Furthermore, they re-
stricted the number of dimensions and reduced the
problem to a single layer in the arterial wall. One of
the first models which encompassed convection, dif-
fusion and uptake into SMC within the porous media
was presented by McGinty et al.:
φ
C
E
t
+ v
C
E
x
= D
m
2
C
E
x
2
α
C
E
C
I
K
(3)
(1 φ)
C
I
t
= α
C
E
C
I
K
, (4)
where C
E
and C
I
denote the volume averaged con-
centration of drug in the extracellular and cellular re-
gions, respectively (McGinty, 2011). The parameters
φ, v, D
m
, α and K denote the porosity, magnitude
of transmural convection, drug diffusion coefficient
in the media, drug uptake rate constant and partition
coefficient. Equation 4 expresses the rate of uptake
of drug by the cells: it is initially proportional to C
E
but that proportionality diminishes with increasing C
I
until the carrying capacity (or partition coefficient) of
the drug is reached at which point the uptake becomes
zero. This system of equations allows for an exchange
of drug between the extracellular phase and the cells
which is dependent on the concentration in the ex-
tracellular phase. They considered the coupled poly-
mer/media system (equations 1,3 and 4) with continu-
ity of drug concentration and continuity of the relative
fluxes expressed as boundary conditions at the inter-
face:
C
p
= C
E
(5)
D
p
C
p
x
= D
m
C
E
x
+ vC
E
. (6)
They assumed that the flux of drug out of the media
was proportional to the concentration at the interface
between the media and adventitia to provide the final
boundary condition. Their model was also extended
to include the adventitia region (where fibroblast cells
were modelled in a similar way to SMCs), a topcoat
of polymer to slow the release of the drug, and one of
the first models of atherosclerotic plaque (modelled
using an equilibrium model in the same way as SMCs
uptake).
McGinty et al. simulated the problem using a fi-
nite difference scheme and conducted a thorough sen-
sitivity analysis which allowed them to infer the im-
portance of the parameters in their model. They found
that the results were particularly sensitive to fluctua-
tions in the magnitude of the transmural velocity, and
to changes in the drug uptake rate and partition coef-
ficient. Their simple model of plaque suggested that
the plaque could act as a reservoir for the drug, ensur-
ing that patients with a higher degree of atherosclero-
sis may receive therapeutic levels of drug for longer
than those with a lesser degree of plaque. This find-
ing is in contrast to an experimental study in the litera-
ture (Tzafriri et al., 2010) where it was concluded that
drug concentration was inversely correlated to lipid
concentrations. The McGinty et al. model did, how-
ever, neglect the intimal region of the arterial wall and
the endothelium layer of cells. Their justification for
this is that the endothelium is severely damaged when
a stent is inserted and in some cases is completely re-
moved; and indeed the properties of the intimal may
not be too different from those in the media.
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667
Despite considering two layers and accounting
for convection, diffusion and uptake/binding in each,
their model makes several simplifying assumptions.
Firstly, the geometry is restricted to one dimension
and as such cannot account for the anisotropic nature
of diffusion within the tissue ((Levin et al., 2004),
(Hwang et al., 2001)). Furthermore, the 1D model
clearly has its limitations in approximating 3D geom-
etry. The strut is considered to be in contact with
the arterial wall, when in reality it is likely embed-
ded within it. Finally, the flow interaction problem
between the blood and the struts is not accounted for,
leading to the potential for under-prediction of drug
lost to the lumen, an aspect which has also been in-
vestigated in the literature (Zunino, 2004).
Pontrelli and de Monte (Pontrelli and de Monte,
2007) proposed a similar model to that of McGinty
et al. which allowed for a diffusion controlled re-
lease from the stent as well as convection-diffusion-
reaction in the arterial wall. Their most sophisticated
model has the benefit of being multi-layered, but is
unable to model the drug concentration in the SMCs.
They do, however, model drug consumption via a lin-
ear reaction:
φ
C
i
t
+ 2γ
i
C
i
x
= D
i
2
C
i
x
2
β
i
C
i
, (7)
where the subscript i indicates the ith layer and 2γ
i
represents a constant characteristic convection param-
eter. Pontrelli and de Monte’s model has the advan-
tage of admitting an analytical solution.
The convective and diffusive element of the drug
transport is well established, as evidenced by the
above models and countless others. However, the is-
sue of uptake/binding is more controversial. Some
authors have assumed equilibrium models ((McGinty,
2011), (Horner et al., 2010), (Abraham et al., 2013)),
while others have considered simple loss terms (Pon-
trelli and de Monte, 2007). More recently, a sec-
ond order reaction model which allows for saturable
reversible binding of sirolimus to specific receptors
and general extracellular matrix (ECM) sites has been
proposed (Tzafriri et al., 2012). The equations for the
rate of uptake of sirolimus to ECM sites and receptors
are:
b
ECM
t
= k
ECM
on
c(b
ECM,max
b
ECM
)
k
ECM
on
k
ECM
d
b
ECM
, (8)
b
REC
t
= k
REC
on
c(b
REC,max
b
REC
)
k
REC
on
k
REC
d
b
REC
. (9)
Here, c is the molar concentration of free drug per
unit tissue volume, b
ECM
and b
REC
are the molar con-
centrations of ECM-bound and receptor-bound drug,
respectively. The parameters b
ECM,max
and b
REC,max
denote the local molar concentration of ECM and re-
ceptor drug binding sites, k
ECM
on
and k
REC
on
are the re-
spective binding on-rate constants and k
ECM
d
and k
REC
d
are the respective equilibrium dissociation constants.
It is certainly true that the drug will bind to bind-
ing sites in the tissue and in the cells ((Levin et al.,
2005), (Tzafriri et al., 2012), (Bierer et al., 1990))
although the strength of the affinity will likely vary
substantially with the particular drug under consider-
ation. Furthermore, it is not clear how the density of
the binding sites could easily be determined. Thus it
may be that this binding model is specific to a partic-
ular class of drugs and not suitable for more general
compounds. A greater understanding of the binding
process would undoubtedly assist with model devel-
opment.
5.2 Higher-dimensional Models
Despite providing useful, and in some cases, counter-
intuitive physiological insights, one-dimensional
models are inadequate for accurately resolving quan-
titative aspects. When the dimension of the model
is increased, numerical approaches are necessarily re-
quired.
Two-dimensional models in simplified geometries
were computed by (Hwang et al., 2001), (Grassi et
al., 2009) and (Zunino, 2004) among others. A num-
ber of three-dimensional models have also been de-
vised. (Weiler et al., 2012) provided a broad general-
ization of the works of (Mongrain, 2007), (Zunino et
al., 2009) and (Vairo et al., 2010); a three-dimensional
model of drug transport in the lumen and the arterial
wall. Laminar steady flow was assumed in the lumen
and the steady diffusion equation (no convection) in
the arterial wall. Through numerical simulation using
commercial finite element software, they found that
the highest rates of mass transfer occurred at the for-
ward portion of the stent and the rate of drug delivery
to the lumen was greater than that to the tissue.
(Horner et al., 2010) appear to be one of the
first authors to provide a three-dimensional reaction-
diffusion-convection model in a realistic geometry.
They stress the importance of considering two phases
of the drug (bound and unbound) and use a first or-
der reaction kinetics model to describe the transfer of
drug between the two phases. They utilise ABAQUS
to obtain a realistic geometry of a deformed stent and
vessel wall and then utilise FLUENT to solve their
transport equations. Their three-dimensional setting
allows for the consideration of anisotropic diffusion
in the arterial wall. They do, however, make three
significant simplifications. Firstly, they model the ar-
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terial wall as a linear homogeneous solid and do not
distinguish between the intima, media and adventi-
tia. Secondly, despite calculating the transmural ve-
locity field, they assume this is fixed when solving
the transport equations. Perhaps the most unrealistic
assumption is that the drug concentration on the stent
remains constant and does not deplete. They find that
deposition patterns tend to follow the pattern of the
stent struts and that the drug is able to penetrate deep
into the arterial wall. The pattern of bound drug be-
comes less uniform as the Peclet number is increased,
eventually becoming restricted to areas adjacent to the
struts, as convection dominates over diffusion.
6 EXPERIMENTAL VALIDATION
An important aspect of modelling and simulation is
validation. The accuracy of the model results can
only ever be as good as the quality of the inputs, espe-
cially when the model is sensitive to changes in one
or more of the parameters. At present, before a DES
is approved for use in humans it must undergo in-vivo
testing in an animal model (for example porcine coro-
nary artery); this is a very costly exercise and also
raises ethical questions. However, a series of in-vitro
and ex-vivo experiments can be carried out at various
stages of the modelling process to verify certain as-
pects and to suggest improvements/modifications to
the model.
One such example of this is in the estimation of
the model parameters. In our laboratory we have
been performing in-vitro DES release experiments
which have allowed us to estimate the diffusion co-
efficient of the drug in polymer-coated stents based
on a least squares analysis and utilising analytical so-
lutions (Figure 1). The experiments consisted of plac-
ing Cypher DESs in a sealed glass vial containing
physiological release medium (phosphate buffered
saline:ethanol (90:10)). At several time points up to
60 days, the stent was removed and placed in a sep-
arate vial containing fresh release medium, with the
mass of drug in the original solution subsequently
quantified using UV-spectroscopy. These simple ex-
periments give confidence in the modelling and pro-
vide reliable estimates of the drug diffusion coeffi-
cient in the polymer, which can feed into more sophis-
ticated models. Alternatively, a diffusion cell contain-
ing a membrane made from the polymer under study
can be used. The diffusion coefficients in each layer
of the arterial wall can be measured in a similar way.
The porosity of each layer of the arterial wall may be
measured from histological sections by quantitative
microscopy. Drug uptake/binding parameters, such
as the partition coefficient K and uptake rate constant
α in equations (3-4) may be estimated by quantifying
drug uptaken by cells grown in culture plates at dif-
ferent time points. For estimation of the parameters
in the second order reaction model (equations 8-9),
the reader is referred to (Tzafriri et al., 2012) and ref-
erences therein.
But the use of experiments to inform the mod-
elling is not a one-way process. Indeed, our group
have utilised the modelling to design experiments
which in turn have fed back into the model. For ex-
ample, recognising that one of the important features
of in-vivo drug release from DESs is transport by con-
vection, in our laboratory we are developing ex-vivo
perfusion circuit experiments which will allow us to
control the intraluminal pressure (and thus vary con-
vection) across the arterial wall. Thus, experimental
validation should not be seen as the final step, only to
be performed once the model has been built and the
results simulated. Instead, the modelling and exper-
imentation should go hand-in-hand, complementing
each other.
7 MODELLING
CONSIDERATIONS FOR THE
FUTURE
Whilst significant progress has been made in simulat-
ing various aspects of drug-eluting stents, there is still
a need to better understand the drug elution process
and the drug transport in the lumen and arterial wall.
Here we indicate some possible modelling consider-
ations for future research in this field so that the aim
of achieving an ‘optimal’ drug-eluting stent may be
realised.
7.1 Simulating Drug Release from New
Generation DESs
There is a real need for models to be developed
for drug release from newer generation biodegrad-
able polymer coated stents and polymer-free surface-
modified stents. It may be that the dominant release
mechanism in these stents is not diffusion and so
models which assume purely diffusion may not cap-
ture the release kinetics. Any new models should be
experimentally validated in-vitro to verify the phys-
ical processes governing the release have been cap-
tured. It is anticipated that for most DESs the in-vivo
release profile may be significantly different due to
the complex biological processes involved, but if the
in-vitro release can be well modelled then this should
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669
shed some light as to how to model the in-vivo situa-
tion. Comparison of the in-vitro drug release profile
of different stent platforms may indicate the release
mechanism(s) which give rise to the most favourable
release profile. This, in turn, may allow for better de-
sign of DESs in the future.
7.2 Inclusion of the Endothelium,
Intima and Plaque
The inner layer of the the arterial wall, the intima, is a
thin layer comprising the endothelium as well as the
elastic lamina. It is well established that the endothe-
lium is damaged during the stent insertion process and
in some cases even removed. As a result, many cur-
rent models neglect the intima region.
A more complete model of the arterial wall should
include the intima region as well as the endothelium
and internal elastic lamina. The endothelium is im-
portant in vasoconstriction and vasodilation and reg-
ulates the uptake of plasma into the arterial wall.
While diseased endothelium is dysfunctional (this is
the starting point of atherosclerosis) it will neverthe-
less have an effect on the relative importance of con-
vection to diffusion in the arterial wall. It is antici-
pated that drug transport through the initma will oc-
cur via similar processes to that in the media, namely
diffusion (albeit with a different diffusion coefficient
from the media), convection and possibly binding.
Another possible modelling consideration is
atherosclerotic plaque; the presence of the plaque
is the very reason that the stent is inserted and yet
it has received very little attention in the literature.
McGinty et al. seem to be the only authors who
have attempted to model the plaque to date, although
(Tzafriri et al., 2010) have experimentally examined
the effect of the plaque. The plaque is known to con-
tain a fibrous cap of variable thickness as well as a
necrotic core made up of cellular debris, cholesterol
cleft and cell membranes. Furthermore, the plaque
also contains macrophages and SMCs as well as a
lipid pool containing lipid dispersed in a collagen
matrix. A More sophisticated model of plaque, tak-
ing into account its various components, may provide
more insight into the effect of plaque on tissue drug
concentrations.
7.3 Modelling Lumenal Blood Flow and
Stent Interaction
In simulations where the blood flow is taken into ac-
count, it is common for the blood flow to be modelled
as steady Poiseuille flow. Of course, in reality blood
flow near the heart is pulsatile and the artery is contin-
ually contracting and expanding. The presence of the
stent interrupts the flow (Peacock et al., 1995) and the
effect this has on the drug transport should be simu-
lated.
In addition to simply considering some of the pa-
rameters to be time and space dependent, it may be
necessary to consider the proliferation of SMCs and
neointima growth as a wound healing problem.
7.4 Inclusion of Complex
Three-dimensional Geometry
While simplified one-dimensional models can pro-
vide useful insights into this problem, ultimately
three-dimensional models which capture the full com-
plex geometry of the stent and the arterial wall are
required. The idea of numerically simulating such a
complex problem may have seemed impossible not
so long ago, but with the accelerating advances in
computational power and numerical techniques it is
now possible. The existing three-dimensional models
in the literature all make certain simplifying assump-
tions, whether it be in idealising the stent geometry,
or in neglecting convection, diffusion or binding, or
in considering only single or bi-layer arterial walls.
Thus there is an opportunity to increase the sophisti-
cation of the three-dimensional models, whether it be
incrementally or in one fell swoop. However, caution
must be exercised to ensure that the results of the sim-
ulations are not subject to high uncertainty, in which
case the fidelity of the results may be called into ques-
tion.
8 CONCLUSIONS
Drug-eluting stents have significantly improved the
treatment of coronary artery disease. Despite real
progress being made in the past decade, a complete
understanding of DESs is still some way off. The
process of in-vitro drug release from newer genera-
tion DESs is still not fully understood, let alone the
complex in-vivo situation where flowing blood, pul-
satility, wound healing, proliferation and migration
of SMCs and complex uptake/binding no doubt all
play some part. Future research should include the
modelling of drug release from biodegradable and
polymer-free modified surface stents, more accurately
modelling lumenal blood flow and stent interaction,
including the endothelium, the intima and atheroscle-
rotic plaque. In order to be able to use simulations in a
predictive capacity, three-dimensional models which
encompass the full complex geometry are necessarily
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required. But care should be taken to verify the cor-
rectness of the numerical results. Finally, the mod-
elling should be complemented by appropriate exper-
iments to validate the resulting simulations and im-
prove on the model.
ACKNOWLEDGEMENTS
We would like to acknowledge the funding provided
by EPSRC under grant number EP/J007242/1. The
first author would also like to acknowledge the receipt
of a Carnegie Scholarship.
REFERENCES
Abraham, J. P., Gorman, J. M., Sparrow, E. M., Stark, J. R.,
and Kohler, R. E. (2013). A mass transfer model of
temporal drug deposition in artery walls. Int. J. Heat
Mass Trans., 58:632–638.
Beers, M. H. (2004). The Merck Manual of Health & Aging.
Elsevier Health Sciences, London.
Bierer, B. E., Patilla, P. S., Standaert, R. F., Herzenberg,
L. A., Burakoff, S. J., Crabtree, G., and Schreiber,
S. (1990). Two distinct signal transmission pathways
in t lymphocytes are inhibited by complexes formed
between an immunophilin and either fk605 or ra-
pamycin. Proc. Natl. Acad. Sci. USA, 87:9231–9235.
Delfour, M. C., Garon, A., and Longo, V. (2005). Modeling
and design of coated stents to optimize the effect of
the dose. SIAM J. Appl. Math.., 65(3):858–881.
Fredenberg, S., Wahlgren, M., Reslow, M., and Axels-
son, A. (2011). The mechanisms of drug release in
poly(lactic-co-glycolic acid)-based drug delivery sys-
tems - a review. Int. J. Pharmaceutics, 415:34–52.
Garasic, J. M., Edelman, E. R., Squire, J. C., Seifert, P.,
Williams, M. S., and Rogers, C. (2000). Stent and
artery geometry determine intimal thickening inde-
pendent of arterial injury. Circulation, 101(7):812–
818.
Grassi, M., Pontrelli, G., Teresi, L., Grassi, G., Comel, L.,
Ferluga, A., and Galasso, L. (2009). Novel design of
drug delivery in stented arteries: a numerical compar-
ative study. Math. Biosci. Eng., 6(3):493–508.
Horner, M., Joshi, S., Dhruva, V., Sett, S., and Stewart, S.
F. C. (2010). A two-species drug delivery model is re-
quired to predict deposition from drug-eluting stents.
Cardiovasc. Eng. Technol., 1(3):225–234.
Hwang, W.,Wu, D., and Edelman, E. R. (2001). Physiolog-
ical transport forces govern drug-distribution for stent
based delivery. Circulation, 104(7):600–605.
Khan, W., Farah, S., and Domb, A. J. (2012). Drug eluting
stents: Developments and current status. J. Controlled
Release., 161:703–712.
Levin, A. D., Jonas, M., Hwang, C. W., and Edelman, E.
R. (2005). Local and systemic drug competition in
drug-eluting stent tissue deposition properties. J. Con-
trolled Release, 109:236–243.
Levin, A. D., Vukmirovic, N., Hwang, C. W., and Edel-
man, E. R. (2004). Specific binding to intracellular
proteins determines arterial transport properties for ra-
pamycin and paclitaxel. Proc. Natl. Acad. Sci. USA,
101(25):9463–9467.
Lloyde-Jones, D. (2010). Heart disease and stroke statistics-
2010 update: A report from the american heart asso-
ciation. Circulation, 121:e46–e215.
Lusis, A. (2000). Atherosclerosis. Nature, 407:233–241.
McGinty, S., McKee, S., Wadsworth, R. M., and Mc-
Cormick, C. (2011). Modelling drug-eluting stents.
Math. Med. Biol., 28:1–29.
Mongrain, R., Faik, I., Leask, R., Rodes-Cabau, J., Larose,
E., and Bertrand, O. (2007). Effects of diffusion co-
efficients and struts apposition using numerical simu-
lations for drug eluting coronary stents. J. Biomech.
Eng., 129:733–742.
Murray, C. and Lopez, A. (1997). Alternative projections of
mortality and disability by cause 1990-2020: Global
burden of disease study. The Lancet, 349(9064):1498–
1504.
Peacock, J., Hankins, S., Jones, T., and Lutz, R. (1995).
Flow instabilities induced by coronary artery stents:
Assessment with an in vitro pulse duplicator. J.
Biomech, 28:17–26.
Pontrelli, G. and de Monte, F. (2007). Mass diffusion
through two-layer porous media: an application to the
drug-eluting stent. Int J. Heat Mass Trans., 50:3658–
3669.
Pontrelli, G. and de Monte, F. (2010). A multi-layer porous
wall model for coronary drug-eluting stents. Int J.
Heat Mass Trans., 53:13629–3627.
Siepmann, J. and Siepmann, F. (2008). Mathematical
modelling of drug delivery. Int. J. Pharmaceutics,
364:328–343.
Sirianni, R. W., Jang, E.-H., Miller, K. M., and Saltzman,
W. M. (2010). Parameter estimation methodology in
a model of hydrophobic drug release from a polymer
coating. SIAM J. Appl. Math., 142):474–482.
Stefanini, G. G. and Holmes, D. R. (2013). Drug-eluting
coronary artery stents. N. Engl. J.Med., 368:254–265.
Tambaca, J., Kosor, M., Canic, S., and Paniagua, D. (2010).
Mathematical modeling of vascular stents. SIAM J.
Appl. Math., 70(6):1922–1952.
Tzafriri, A., Vukmirovic, N., Kolachalama, V., Astafieve, I.,
and Edelman, E. R. (2010). Lesion complexity deter-
mines arterial drug distribution after local drug deliv-
ery. J. Controlled Release, 142(3):332–338.
Tzafriri, A. R., Groothuis, A., Price, G. S., and Edelman,
E. R. (2012). Stent elution rate determines drug de-
position and receptor-mediated effects. J. Controlled
Release, 161:918–926.
Vairo, G., Cioffi, M., Cottone, R., Dubini, G., and Migli-
avacca, F. (2010). Drug release from coronary
artery stents: a multidomain approach. J. Biomech.,
43:1580–1589.
Weiler, J. M., Sparrow, E. M., and Ramazani, R. (2012).
Mass transfer by advection and diffusion from a
drugeluting stent. J. Heat Mass Transfer, 55:1–7.
SimulatingDrug-elutingStents-ProgressMadeandtheWayForward
671
Zunino, P. (2004). Multidimensional pharmacokinetic mod-
els applied to the deign of drug-eluting stents. Car-
diov. Eng.: Int. J., 4(2):181–191.
Zunino, P., DAngelo, C., Petrini, L., Vergara, C., Capelli,
C., and Migliavacca, F. (2009). Numerical simula-
tion of drug eluting coronary stents: mechanics, fluid
dynamics and drug release. Comput. Methods Appl.
Mech. Eng.., 198:3633–3644.
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