A Two-Species Finite Volume Scalar Model for Modeling the Diffusion of Poly(lactic-co-glycolic acid) into a Coronary Arterial Wall from a Single Half-Embedded Drug Eluting Stent Strut

Abstract

This paper outlines the methodology and results for a two-species finite volume scalar computational drug transport model developed for simulating the mass transport of Poly(lactic-co-glycolic acid (PLGA)) from a half-embedded single strut implanted in a coronary arterial vessel wall. The mathematical drug transport model incorporates the convection-diffusion equation in scalar form (dimensionless) with a two-species (free-drug and bound-drug) mass transport setup, including reversible equilibrium reaction source terms for the free and bound-drug states to account for the pharmaco-kinetic reactions in the arterial wall. The relative reaction rates of the added source terms control the interconversion of the drug between the free and bound-drug states. The model is solved by a 2D finite-volume method for discretizing and solving the free and bound drug transport equations with anisotropic vascular drug diffusivities. This model is an improvement over previously developed models using the finite-difference and finite element method. A dimensionless characteristic scaling pre-analysis was conducted a priori to evaluate the significance of implementing the reaction source terms in the transport equations. This paper reports the findings of an investigation of the interstitial flow profile into the arterial wall and the free and bound drug diffusion profiles with a parametric study of varying the polymer drug concentration (low and high), tortuosity, porosity, and Peclet and DamKöhler numbers over the course of 400 h (16.67 days). The results also reveal how a single species drug delivery model that neglects both a reversible binding reaction source term and the porosity and tortuosity of the arterial wall cannot accurately predict the distribution of both the free and bound drug.

Full text

Citation: Hewlin, R.L., Jr.;Edwards, M.;
Kizito, J.P. A Two-Species Finite
Volume Scalar Model for Modeling the
Diffusion of Poly(lactic-co-glycolic
acid) into a Coronary Arterial Wall
from a Single Half-Embedded Drug
Eluting Stent Strut. Biophysica 2023,3,
385–408. https://doi.org/10.3390/
biophysica3020026
Academic Editor: Javier Sancho
Received: 30 April 2023
Revised: 25 May 2023
Accepted: 9 June 2023
Published: 15 June 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
biophysica
Article
A Two-Species Finite Volume Scalar Model for Modeling
the Diffusion of Poly(lactic-co-glycolic acid) into a Coronary
Arterial Wall from a Single Half-Embedded Drug Eluting
Stent Strut
Rodward L. Hewlin, Jr. 1,* , Maegan Edwards 1,2 and John P. Kizito 3
1Center for Biomedical Engineering and Science (CBES), Department of Engineering Technology and
Construction Management (ETCM), University of North Carolina at Charlotte, Charlotte, NC 28223, USA;
medwar79@uncc.edu
2Applied Energy and Electromechanical Engineering (AEES), Department of Engineering Technology and
Construction Management (ETCM), University of North Carolina at Charlotte, Charlotte, NC 28223, USA
3Department of Mechanical Engineering, North Carolina Agricultural & Technical State University,
Greensboro, NC 27411, USA; jpkizito@ncat.edu
*Correspondence: rhewlin@uncc.edu
Abstract:
This paper outlines the methodology and results for a two-species finite volume scalar
computational drug transport model developed for simulating the mass transport of Poly(lactic-
co-glycolic acid (PLGA)) from a half-embedded single strut implanted in a coronary arterial vessel
wall. The mathematical drug transport model incorporates the convection-diffusion equation in
scalar form (dimensionless) with a two-species (free-drug and bound-drug) mass transport setup,
including reversible equilibrium reaction source terms for the free and bound-drug states to account
for the pharmaco-kinetic reactions in the arterial wall. The relative reaction rates of the added
source terms control the interconversion of the drug between the free and bound-drug states. The
model is solved by a 2D finite-volume method for discretizing and solving the free and bound drug
transport equations with anisotropic vascular drug diffusivities. This model is an improvement over
previously developed models using the finite-difference and finite element method. A dimensionless
characteristic scaling pre-analysis was conducted a priori to evaluate the significance of implementing
the reaction source terms in the transport equations. This paper reports the findings of an investigation
of the interstitial flow profile into the arterial wall and the free and bound drug diffusion profiles with
a parametric study of varying the polymer drug concentration (low and high), tortuosity, porosity,
and Peclet and DamKöhler numbers over the course of 400 h (16.67 days). The results also reveal
how a single species drug delivery model that neglects both a reversible binding reaction source term
and the porosity and tortuosity of the arterial wall cannot accurately predict the distribution of both
the free and bound drug.
Keywords:
arterial vessel; bound drug; DamKöhler number; diffusivity; finite volume; free drug;
internalized drug; stent; Pecklet number; poly(lactic-co-glycolic acid); porosity; scalar; species;
tortuosity; transport
1. Introduction
Cardiovascular disease remains to be the leading cause of death worldwide [
1
–
8
]. Drug
eluting stents have demonstrated exceptional benefits in reducing in-stent restenosis [
9
,
10
].
These stents are commonly used in coronary angioplasty procedures to provide structural
support and release drug molecules locally at the implanted arterial site to prevent adverse
outcomes (such as in-stent restenosis) in patients. Although drug-eluting stents are now
the main choice of treatment in coronary interventions, questions regarding their longevity
and safety are still prominent [
11
]. In the United States, present drug-eluting stent designs
Biophysica 2023,3, 385–408. https://doi.org/10.3390/biophysica3020026 https://www.mdpi.com/journal/biophysica

Biophysica 2023,3386
incorporate sirolimus and paclitaxel and release these drugs into the arterial wall from
the eluting struts [
12
–
14
]. Both sirolimus and paclitaxel eluting stents appear to have
comparable clinical benefits.
Initial drug eluting stent treatments were prone to washout by transmural plasma
flow, which lowered the drug residence time in the arterial vessel wall. This was a major
hindrance since these implants were designed to provide local drug delivery to the diseased
site. Hydrophobic drugs, such as sirolimus and paclitaxel, were reported to have higher
retention times as compared to other drugs because they can bind to structural elements
and intracellular targets in the vessel wall [
15
,
16
]. Hydrophobic drugs, such as these, exist
in both bound and unbound states within the vessel wall. These states are in equilibrium,
and the binding is reversible. Consequently, the diffusion of a hydrophobic drug into the
arterial wall from a stent cannot be modeled without interaction of both the bound and
free drug forms in the vessel wall.
Several experimental and numerical investigations have been carried out recently with
the aim of quantifying the capability of this device to reduce in-stent restenosis after stent
implantation. Lovich et al. [
17
–
19
] studied the behavior of heparin in implanted arteries
and concluded that the presence of binding sites changes along the transmural direction,
being higher in the endothelium and lower in the adventitia. Lovich and Edelman studied
the effects of specific binding sites inside the arterial wall on drug uptake [
20
], where
the presence of specific binding site action was modeled using the reversible chemical
reaction. Sakharov et al. [
21
] disregarded the convective effects on the transport of free
drugs.
Hwang et al. [22]
predicted the free and bound drug concentrations by solving for
the distribution of the free drug, then using a multiplicative factor (partition approach)
to predict the concentration of the bound drug. Migliavacca et al. [
23
] studied the drug
release pattern in the vascular wall from drug-eluting stents using a single species ap-
proach in addition to a partition coefficient approach to relate the free and the bound drug
concentrations.
Borghi et al. [24]
stated that the inclusion of reversible binding leads to
delayed release and that the erosion of the polymer affects the drug release from a single
strut. Horner et al. [
25
] considered a two-species drug delivery model including reversible
binding sites, and their model predicted that a single species drug delivery model cannot
accurately predict the distribution of bound drugs. They also concluded that a two-species
approach that includes reversible binding is the way forward for future stent-based drug
delivery systems.
Following Tzafriri et al. [
26
], a second-order dynamic model that describes a saturating
reversible binding process by treating the bound drug as a dynamic variable has been
taken into account to explore drug interaction with cells of the arterial wall. In most of the
studies cited above, transient drug release has been modeled as a uniform release, which
is unrealistic and not representative of actual stent-based delivery. Instead, a simple time-
dependent Dirichlet boundary condition is often applied on the surface of the struts [
27
–
29
].
Arterial properties, such as porosity and tortuosity, dictate the transport of drugs within the
arterial tissue. When an endovascular drug-eluting stent is implanted, it has a major impact
on the structure of the arterial wall, eventually influencing the overall rates of diffusion
through tissues [
30
]. For diffusion in a porous medium, the effective diffusion coefficient
is assumed to depend on two factors: porosity (a dimensionless parameter, which is the
ratio of pore volume to the total material volume) and diffusion path tortuosity (ratio of
the actual pore length to the distance between its ends; i.e., arc-chord ratio) [
31
]—these
parameters change the free diffusivity of the drug eluted from a pair of struts [32].
The goal of this work is to develop a two-dimensional two-species scalar finite volume
computational model that can model the reversible binding characteristics of poly(lactic-
co-glycolic acid) (PLGA) released into a coronary artery wall from a single drug eluting
strut. The model described in this work is an improvement over previous works and
considers the integrated process of the drug release in the PLGA coating, the free and
bound drug diffusion profiles with varying polymer drug concentration (low and high),
vascular diffusivities, tortuosity, porosity, and Peclet and Dahmokoler numbers over the

Biophysica 2023,3387
course of 400 h (16.67 days) [
33
]. The mechanism of diffusion in the PLGA is adapted from
the work of Zhu and Braatz [
34
] and couples the drug diffusion to degradation and erosion
along with the drug pharmacokinetics taking place in the arterial wall from the work of
Saha and Mandal [35]. The main contributions of the proposed work include:
•
A theoretical methodology for computational modeling of the diffusion of PLGA into
a coronary arterial wall from a single half-embedded drug eluting stent strut.
•
A computational drug diffusion model that considers the pharmaco-kinetic reactions
in the arterial wall as equilibrium reversible binding reaction source terms for the free
and bound-drug.
•
Validation of the reported computational model via simulation-based results from a
finite difference model developed from methods reported in previous works.
•
A computational drug diffusion model that provides an understanding of the relation-
ship between drug physicochemical properties and the local transport environment
which is crucial to the success of new stent designs.
•
The model reported in this work is the second reported model in literature that
successfully uses an ANSYS FLUENT user-defined scalar (UDS) model to model the
diffusion of the free and bound drug in the arterial wall with reversible binding source
terms. Additionally, this is the first reported model to use a UDS model to incorporate
the polymer layer in the computational domain.
The next section presents the methodology of this work.
2. Material and Methods
2.1. Model Development
In this work, an implanted drug eluting coronary stent (as shown in Figure 1a) is
analyzed in the coronary artery where the stent struts are evenly placed and half-embedded
in the cross-section of the lumen (as shown in Figure 1b).
Biophysica 2023, 3, FOR PEER REVIEW 3
and considers the integrated process of the drug release in the PLGA coating, the free and
bound drug diffusion profiles with varying polymer drug concentration (low and high),
vascular diffusivities, tortuosity, porosity, and Peclet and Dahmokoler numbers over the
course of 400 h (16.67 days) [33]. The mechanism of diffusion in the PLGA is adapted from
the work of Zhu and Braa [34] and couples the drug diffusion to degradation and erosion
along with the drug pharmacokinetics taking place in the arterial wall from the work of
Saha and Mandal [35]. The main contributions of the proposed work include:
• A theoretical methodology for computational modeling of the diffusion of PLGA into
a coronary arterial wall from a single half-embedded drug eluting stent strut.
• A computational drug diffusion model that considers the pharmaco-kinetic reactions
in the arterial wall as equilibrium reversible binding reaction source terms for the
free and bound-drug.
• Validation of the reported computational model via simulation-based results from a
finite difference model developed from methods reported in previous works.
• A computational drug diffusion model that provides an understanding of the rela-
tionship between drug physicochemical properties and the local transport environ-
ment which is crucial to the success of new stent designs.
• The model reported in this work is the second reported model in literature that suc-
cessfully uses an ANSYS FLUENT user-defined scalar (UDS) model to model the dif-
fusion of the free and bound drug in the arterial wall with reversible binding source
terms. Additionally, this is the first reported model to use a UDS model to incorporate
the polymer layer in the computational domain.
The next section presents the methodology of this work.
2. Material and Methods
2.1. Model Development
In this work, an implanted drug eluting coronary stent (as shown in Figure 1a) is
analyzed in the coronary artery where the stent struts are evenly placed and half-embed-
ded in the cross-section of the lumen (as shown in Figure 1b).
Figure 1. Cross-sectional view diagram of the arterial stented model: (a) Schematic of a single PLGA
coated half-embedded stent strut implanted into the arterial wall and (b) the full stented (all stent
struts included) arterial model.
The strut and arterial wall configuration is based on a previous study by Xiaoxiang
and Braa [36] involving a bio-durable polymer coating and is common for drug eluting
Figure 1.
Cross-sectional view diagram of the arterial stented model: (
a
) Schematic of a single PLGA
coated half-embedded stent strut implanted into the arterial wall and (
b
) the full stented (all stent
struts included) arterial model.
The strut and arterial wall configuration is based on a previous study by Xiaoxiang
and Braatz [
36
] involving a bio-durable polymer coating and is common for drug eluting
stent diffusion analysis applications. The blood flow is moving in the direction of the
paper plane, as labeled in Figure 1b. Standard square-shaped stent struts are considered

Biophysica 2023,3388
in this work [
37
–
39
]. Due to symmetry, a single stent strut with its surrounding arterial
wall domain is extracted for the study to simplify the computational domain and reduce
computational costs. The model was developed in ANSYS SpaceClaim (2022) and deployed
in ANSYS Meshing (2022) for meshing.
The extracted model domain is illustrated in Figure 1a, where half of the stent strut is
embedded into the arterial wall. Distinct from previous works, here, the curvature of the
arterial wall is kept intact, and the computational domain consists of a cartesian coordinate
system (observed as x and y). The mathematical models for describing the drug delivery
process are described in Sections 2.2 and 2.3. The model for describing drug transport and
pharmacokinetics in the arterial wall was developed based on the works of Xiaoxiang and
Braatz [
36
] and Saha and Mandal [
35
]. The next section describes the boundary conditions
for the developed domain model.
2.2. Boundary Conditions and Meshing
The names for each boundary zone are provided in Figure 2. The “inlet” zone repre-
sents the exposed inner surface of the artery where plasma flow enters the arterial domain.
The “stent surface” represents the location where the stent is in contact with the vessel wall.
Biophysica 2023, 3, FOR PEER REVIEW 4
stent diffusion analysis applications. The blood flow is moving in the direction of the pa-
per plane, as labeled in Figure 1b. Standard square-shaped stent struts are considered in
this work [37–39]. Due to symmetry, a single stent strut with its surrounding arterial wall
domain is extracted for the study to simplify the computational domain and reduce com-
putational costs. The model was developed in ANSYS SpaceClaim (2022) and deployed in
ANSYS Meshing (2022) for meshing.
The extracted model domain is illustrated in Figure 1a, where half of the stent strut
is embedded into the arterial wall. Distinct from previous works, here, the curvature of
the arterial wall is kept intact, and the computational domain consists of a cartesian coor-
dinate system (observed as x and y). The mathematical models for describing the drug
delivery process are described in Sections 2.2 and 2.3. The model for describing drug
transport and pharmacokinetics in the arterial wall was developed based on the works of
Xiaoxiang and Braa [36] and Saha and Mandal [35]. The next section describes the
boundary conditions for the developed domain model.
2.2. Boundary Conditions and Meshing
The names for each boundary zone are provided in Figure 2. The “inlet” zone repre-
sents the exposed inner surface of the artery where plasma flow enters the arterial domain.
The “stent surface” represents the location where the stent is in contact with the vessel
wall.
Figure 2. Model diagram of the half-embedded stented arterial model: (a) Surface model created in
ANSYS SpaceClaim and (b) mesh computational domain (model used in the simulations is a finer
meshed model). P1 is the location point of interest for evaluating the concentration profiles over
time). At the PLGA coating and artery wall interface, the following flux condition is applied:
Figure 2.
Model diagram of the half-embedded stented arterial model: (
a
) Surface model created in
ANSYS SpaceClaim and (
b
) mesh computational domain (model used in the simulations is a finer
meshed model). P
1
is the location point of interest for evaluating the concentration profiles over
time). At the PLGA coating and artery wall interface, the following flux condition is applied:

Biophysica 2023,3389
Jwp =1
Rwp Cw
κwp
−Cp(1)
where R
wp
is the mass transfer resistance, C
w
, is the drug concentration on the arterial
wall side of the interface, is the partition coefficient, and C
p
is the perivascular drug
concentration. The right and left sides of the arterial zone domain are treated as symmetrical
boundary conditions, as shown in Figure 2.
ANSYS Meshing (2022) was used for meshing the computational domain. The meshing
scheme used is a tetrahedral cell mesh type which is applied to all surfaces with the surface
size meshed based on the edge spacing selection and an inflation scheme applied to rectify
meshing irregularities. The next section describes the plasma flow modeling methodology.
2.3. Plasma Flow
In this work, ANSYS FLUENT (2022 R1 (ver. 21.1)) computational fluid dynamic (CFD)
software was used to model both the fluid flow (plasma flow) and the convection-diffusion
of the free, bound, and internalized drug. For plasma flow in the arterial domain, a pressure
drop filtration is implemented to simulate the steady flow of plasma through the domain.
The arterial domain tissue is assumed to behave as a porous media. The Darcy Law model
was used to solve the plasma flow field. FLUENT allows implementation of the Darcy Law
equation as a source term in the Navier–Stokes equations as shown below in Equation (2):
ρ∂v
∂t+v· ∇v=−∇P+µ∇2v−vµ
K(2)
where vis the velocity vector, Pis the pressure, Kis the permeability of the vessel wall,
and
ρ
and
µ
are the density and dynamic viscosity of plasma, respectively. The density
of plasma is 1020 kg/m
3,
and the dynamic viscosity is 0.0035 Pa
·
s [
2
–
7
] at the standard
body temperature (37
◦
C) Whale et al. [
40
] examined the effects of aging and pressure on
the Darcy permeabilities of human aortic walls. A representative value of 2.0
×
10
−18
m
2
was implemented for this work. Equation (2) is subject to the incompressibility constraint.
As described above, the vessel lumen is not a part of the computational domain. This
introduces an additional assumption because luminal flow decreases axial non-uniformity
of the drug in the artery wall [
37
,
41
]. The degree of non-uniformity was observed to increase
with increasing the aspect ratio of the stent strut [
41
]. The impact of this assumption is
therefore minimized in the case of square struts and/or stents with an abluminal coating.
The next sections discuss the drug transport modeling methodology.
2.4. Drug Transport in the PLGA Coating and Arterial Domains
When the drug is released into the arterial wall, the drug molecules are exposed and
interact with the physiological environment. Various drug-tissue interactions occur that
affect the arterial wall drug transport, distribution, and drug uptake. The drug-arterial
wall interaction has been commonly modeled as a reversible binding reaction of the drug
molecules with binding sites present in the arterial wall. During this process, the bound
drug C
b
is formed by associating the free drug C
f
with the available binding sites S
0
. The
bound drug is immobilized, and only the free drug can diffuse. The reversible binding
process, however, does not provide a mechanism for drug consumption (e.g., drug uptake
by tissue cells), which can be characterized by drug internalization. This work did not take
into consideration the internalization of the drug.
Drug Binding:
CF+S0
ka
Biophysica 2023, 3, FOR PEER REVIEW 6
0
a
d
k
FB
k
C S C+
(3)
Free Drug in the PLGA Coating Domain:
22
22
,
f f f
C
C C C
D
t x y

  
=+


  

(4)
Free Drug in the Arterial Domain:
( )
22
0
22 ,
f f f f f
T a f b d b
C C C C C
v D k C S C k C
t x y x y

    
 
= − + + + − − −

 

    


(5)
Bound Drug in the Arterial Domain:
( )
0,
ba f b d b i b
Ck C S C k C k C
t

= − − −


(6)
Drug Transport Boundary Conditions:
0,
fb
CC
xx

==

(7)
0
f
C=
(8)
2
0
() c
b
DC
Jt t

=
(9)
Where x and y are coordinates along the horizontal and vertical directions, respectively,
and ka and kd are the rates of association and dissociation constants, respectively. S0 is the
net tissue binding capacity. DT is the is the true diusivity of the free drug diusing into
the arterial wall and is expressed as:
0
1,
T eff
d
S
DD
R

= + 


(10)
where
𝐷e =𝜀
𝜏×𝐷free
(11)
ε and τ are the porosity and the tortuosity of the wall material, respectively. Dfree and
De (Equations (9) and (10)) are the coecients of free and eective diusivity, respec-
tively. Rd = (kd/ka) is the equilibrium dissociation constant.
As mentioned in Section 2.2, symmetry boundary conditions for both the free and
bound drug are imposed at the proximal and distal walls of the computational domain.
An impermeable boundary condition for the bound drug is also imposed at the perivascu-
lar wall, lumen-tissue, and strut-tissue interfaces (Equation (7)). For the free drug, a per-
fect sink condition is imposed at the perivascular end (Equation (8)). In this work, we con-
sidered two situations, either that owing blood is extremely ecient at washing out the
mural-adhered drug, modeled as a zero-concentration interface condition [42], or that mu-
ral-adhered drug is insensitive to owing blood, modeled as a zero-ux boundary condi-
tion (Equation (6)). As a substitute to modeling the uniform release of drug from a single
kd
CB(3)

Biophysica 2023,3390
Free Drug in the PLGA Coating Domain:
∂Cf
∂t=DC ∂2Cf
∂x2+∂2Cf
∂y2!, (4)
Free Drug in the Arterial Domain:
∂Cf
∂t=−v∂Cf
∂x+∂Cf
∂y+DT ∂2Cf
∂x2+∂2Cf
∂y2!−hkaCf(S0−Cb)−kdCbi, (5)
Bound Drug in the Arterial Domain:
∂Cb
∂t=hkaCf(S0−Cb)−kdCb−kiCbi, (6)
Drug Transport Boundary Conditions:
∂Cf
∂x=∂Cb
∂x=0, (7)
Cf=0 (8)
Jb(t) = rDcC02
πt(9)
where xand yare coordinates along the horizontal and vertical directions, respectively, and
k
a
and k
d
are the rates of association and dissociation constants, respectively. S
0
is the net
tissue binding capacity. D
T
is the is the true diffusivity of the free drug diffusing into the
arterial wall and is expressed as:
DT=1+S0
Rd×De f f , (10)
where
De f f =ε
τ×Df ree (11)
ε
and
τ
are the porosity and the tortuosity of the wall material, respectively. D
free
and
D
eff
(Equations (9) and (10)) are the coefficients of free and effective diffusivity, respectively.
Rd= (kd/ka) is the equilibrium dissociation constant.
As mentioned in Section 2.2, symmetry boundary conditions for both the free and
bound drug are imposed at the proximal and distal walls of the computational domain. An
impermeable boundary condition for the bound drug is also imposed at the perivascular
wall, lumen-tissue, and strut-tissue interfaces (Equation (7)). For the free drug, a perfect sink
condition is imposed at the perivascular end (Equation (8)). In this work, we considered two
situations, either that flowing blood is extremely efficient at washing out the mural-adhered
drug, modeled as a zero-concentration interface condition [
42
], or that mural-adhered
drug is insensitive to flowing blood, modeled as a zero-flux boundary condition (Equation
(6)). As a substitute to modeling the uniform release of drug from a single strut, a simple
time-dependent release kinetics with a flux condition (Equation (9)) is assumed at the strut
eluting surface.
In this work, the contribution of the true diffusion term was minimized by setting
Db= 1.0 ×10−7Du
. The 1.0
×
10
−7
pre-factor was adopted from the study of
Horner et al. [25]
and was used to decrease the true drug diffusivity until the bound drug distribution became
independent of the diffusivity results. A cartesian coordinate system was used to specify

Biophysica 2023,3391
the components of the diffusion tensor Din the x and y directions, corresponding to D
xx
and Dyy, respectively. Both Duand Dbhave two independent components:
D=Dxx 0
0Dyy(12)
PLGA tends to localize within elastic sheathes in the vessel wall. Hwang and Edelman [
22
]
have proven this experimentally. In our study, we assume that Dyy is larger than Dxx.
User Defined Scalar and Numerical Modelling
In this work, we implemented the user-defined scalar (UDS) model available in ANSYS
FLUENT for solving Equations (5)–(7). A fluent UDS model allows a user to define up
to fifty UDS transport equations in a single computational model. The general (UDS)
transport equation is shown below in Equation (12) with the four terms (transient, flux,
diffusivity, and source terms) that can be customized. The UDS model allows the user to set
boundary conditions for the variables within cells of a fluid or solid zone for a particular
scalar equation. This is done by fixing the value of
φk
. When
φk
is fixed in a given cell, the
UDS scalar transport is not solved, and the cell is not included when the residual sum is
computed. For the present work, the value of the initial drug concentration, C
0,
was fixed,
and the coating diffusivity was allowed to vary as a function of
φ
,time, and molecular
weight, also allowed to vary with time. For the bound drug transport equation, the mass
transport was deselected, which allowed the convection term to be neglected, thus making
the bound drug immobile. The same was done for the internalized drug transport equation.
The source terms Sφkinclude the reversible binding reactions in Equations (6) and (7).
∂φk
∂t
|{z}
unsteady
+∂
∂xi
(Fiφk
|{z}
convection
−Γk
∂φk
∂xi
| {z }
diffusion
) = Sφk
|{z}
sources
(13)
For the drug transport and plasma flow simulations, the drug concentration was
assumed to be low enough that it does not affect the plasma velocity field. Therefore,
the velocity and scalar transport equations were decoupled and solved sequentially. The
velocity field in the tissue was solved using a steady-state formulation. FLUENT’s pressure-
based segregated solver was used with the pressure-implicit with splitting of operators
(PISO) scheme to couple pressure and velocity degrees of freedom. The standard pressure
interpolation scheme was used along with second order up-winding for discretizing the
velocity degrees of freedom. The default under-relaxation factor (URF) for pressure was
increased to 0.5, and the URF for momentum was lowered to 0.4.
The convergence criterion for the steady fluid flow problem was 10
−6
for the mo-
mentum equations. The drug transport problem was solved using a transient solver, with
the velocity field fixed for all time steps. A first-order implicit time integrator was used
along with the QUICK up-winding scheme for spatial discretization of the scalar transport
equations. Smooth convergence was observed when using the default URFs of 1.0 for both
transport equations. The convergence criterion for concentration at each time. All plasma
flow and drug concentration simulations were conducted with a time step of 1 picosecond
and resulted in a simulation run time of at least 15 days. All simulations were conducted
on an ASUS ROG STRIX desktop computer (ASUS ROG; Taiwan) with 12 cores and an
NVIDIA GeForce GTX 1660 TI graphics card. All simulations were conducted in parallel
with 11 CPU cores and the NVIDIA graphics card.
2.5. Non-Dimensional Pre-Analyses
Similar to our previous work, we began by performing a dimensionless characteristic
scaling analysis to gain insight into the dominant mechanisms of transport throughout the

Biophysica 2023,3392
arterial wall. The dimensionless scaling parameters for scaling Equations (2) and (3) are
shown below:
x∗=x
δ,y∗=y
δ,t∗=tVy
δ,C∗
f=Cf
C0
,C∗
b=Cb
S0
Using these characteristic dimensionless terms, the drug transport equations take the
following form:
∂C∗
f
∂t∗=1
PeC"∂2C∗
f
∂x∗2+∂2C∗
f
∂y∗#(14)
∂C∗
f
∂t∗=1
PeT"∂2C∗
f
∂x∗2+∂2C∗
f
∂y∗2#−Da
PeThC∗
f(1−C∗
b)−ε1C∗
bi(15)
∂C∗
b
∂t∗=ε2Da
PeThC∗
f(1−C∗
b)−ε1C∗
bi(16)
Jb(t) = s1
Pecπt(17)
where V
y
is the transmural filtration velocity, Pe
T
=[V
yδ
/(D
T
)],and Da = [(k
a
S
0δ2
)/(D
T
)]
are the Peclet and DamKöhler numbers in the tissue. Here,
ε1
=(R
d
/C
0
),
ε2
=(C
0
/S
0
), and
ε3
=(R
d
/S
0
) are three scaling parameters. Pe
c
=[V
y
(h
2
/
δ
)]/D
c
is the Peclet number in the
coating of the strut, and his the thickness of the coating of the strut.
In these dimensionless equations, three characteristic time scales appear,
τ1
,
τ2
, and
τ3
, corresponding to diffusion coating, transmural diffusion, and the binding reaction. The
characteristic time’s scales are shown below:
τ1=δ2
DT
,
τ2=x2
DT
,
and
τ3=1
kd
,
The evaluation of the magnitude of the three groups gives
τ1
, 10
3
–10
5
s,
τ2
, 10
3
–10
5
s,
and
τ3
10
2
s, which indicate that reversible binding is very fast compared to diffusion. The
relative significance of diffusion and reversible binding in the wall is also implied by their
corresponding dimensionless groups DamKohler and Peclet numbers. Compared with the
coefficient of the transmural diffusion component (which is one),the reaction components
have very large DamKöhler numbers on the order of 10
2
–10
4
, which also implies that
the binding reactions perform a very strong role in the spatiotemporal dynamics. The
non-dimensional analysis is provided in the Appendix Aof this paper.
2.6. Grid Independence Analysis, Modelling Parameters, and Validation and Verification
A grid independence analysis was conducted on the developed computational do-
mains for obtaining a mesh that produces results independent of the mesh size for model
simulations. The grid independent analysis results are shown in Table 1. The grid inde-
pendence analysis was performed with constant drug diffusivities in the coating and in
the arterial wall, and the relative error was calculated for the drug release profile. The
reference mesh uses a 0.1
µ
m element size for the coating and a 5
µ
m element size for the
arterial wall. During the analysis, the relative errors were similar and remained under 5%
error for different mesh sizes of the arterial wall domain, while the mesh size of the coating
remained the same at 0.1
µ
m. Although not shown, the final mesh yielded a relative error
of less than 5% and contained 372,125 cells. The chosen mesh was approximately 3.3 times

Biophysica 2023,3393
the size of the previous mesh of 113,114 cells in which the results were well under a 5%
difference which signifies grid independence. A mesh inflation was also applied to the final
mesh to create high-quality geometry-aligned elements within the computational domain
and along the boundaries.
Table 1. Table of grid independent study results.
Element Number Average Weighted Concentration Average Velocity
74,212 1.121327 13.72 ×10−6
82,458 1.057641 12.68 ×10−6
91,621 1.034241 10.23 ×10−6
101,802 0.983541 9.83 ×10−6
113,114 0.977732 9.77 ×10−6
372,125 0.977654 9.76 ×10−6
The physiological and pharmacokinetic parameters modeled in Equations (1)–(20) are
listed in Table 2. These values were obtained from other relevant works [35,36].
Table 2. Model description table. Parameters from this work are obtained from prior works [35,36].
Description Parameter Value
Outer diameter of the artery, mm D3
Artery wall thickness, µmLy200
Strut dimension, m δ0.00014
Transmural filtration velocity, m/s Vy4×10−8
Porosity of the arterial wall ε0.787
Tortuosity of the arterial wall τ1.333
Coating drug diffusivity, m2/s Dc1.0 ×10−12
Coefficient of free diffusivity, m2/s Dfree 3.65 ×10−12
Coefficient of effective diffusivity, m2/s Deff 2.15 ×10−12
True diffusivity of the free drug, m2/s DT24 ×10−12
Initial drug concentration in the coating, mol/m
3C00.01
Tissue binding capacity, mol/m3ka10
Dissociation rate constant kd0.01
Equilibrium dissociation constant, mol/m3Rd0.001
Dimensionless Peclet number in the coating PeC100
Dimensionless Peclet number in the tissue PeT2
Dimensionless DamKöhler number in the tissue Da40
Dimensionless scaling parameter 1 ε10.001
Dimensionless scaling parameter 2 ε2100
For validation and verification of the developed finite volume scalar model, we
compared the free drug concentration profiles in the arterial domain of the developed finite
volume scalar model with a MATLAB finite difference code developed in our previous
work [
33
]. Similar to the finite volume scalar model, the MATLAB finite difference model
uses a cartesian coordinate system that describes the arterial domain, including the square-
shaped stent strut. A rectangular domain is used as opposed to a curvature domain, as
shown in Figure 3a. The same dimensions for the arterial domain are used for the finite
difference and finite volume simulations. The same boundary conditions are imposed
in the MATLAB finite difference code. Additionally, the same dimensionless diffusion
equations are implemented in the MATLAB finite difference code. The PLGA coating is not
modeled in the finite difference code. For the finite difference solution, a numerical grid
with a size of 400
×
200 and an initial time step value of
δt
= 0.00001 s was used for the
sake of computational power and time.

Biophysica 2023,3394
Biophysica 2023, 3, FOR PEER REVIEW 10
is not modeled in the finite difference code. For the finite difference solution, a numerical
grid with a size of 400 × 200 and an initial time step value of δt = 0.00001 s was used for the
sake of computational power and time.
(a)
(b)
Figure 3. Validation results for the finite volume model using a finite difference model developed
in reference to the work of Saha and Mandal [35]: (a,b) Distribution of normalized mean bound drug
concentration for values of: PeT = 2, Da = 40, and ε2 = 100.
Figure 3b shows a comparison of the finite difference and finite volume free drug
concentration solution within the arterial domain at point 1 with values of PeT = 2, Da = 40,
and ε2 = 100. As shown in the plot in Figure 3b, the overall trend in the growth of the plots
is similar; however, the finite volume model has a higher concentration profile (approxi-
mately 10 percent higher). This could be aributed to the fine mesh used in the finite vol-
ume model, the implementation of the PGLA layer, the application of a diffusion pre-
factor, and the application of the diffusive tensor. The key takeaway is that the free drug
concentration trend behaves as expected when compared to a finite difference model that
was developed based on work reported in the literature. The next section discusses the
results of this work.
3. Results
This section of the paper presents the results of the interstitial plasma flow profile
into the arterial wall, the initial diffusion flow modeling results using an eroding polymer
coating (free and bound drug concentration profile with the interstitial flow), and the par-
ametric study results of varying polymer drug concentration (low and high), tortuosity,
Figure 3.
Validation results for the finite volume model using a finite difference model developed in
reference to the work of Saha and Mandal [
35
]: (
a
,
b
) Distribution of normalized mean bound drug
concentration for values of: PeT= 2, Da = 40, and ε2= 100.
Figure 3b shows a comparison of the finite difference and finite volume free drug
concentration solution within the arterial domain at point 1 with values of Pe
T
= 2,
Da = 40
,
and
ε2
= 100. As shown in the plot in Figure 3b, the overall trend in the growth of the plots is
similar; however, the finite volume model has a higher concentration profile (approximately
10 percent higher). This could be attributed to the fine mesh used in the finite volume
model, the implementation of the PGLA layer, the application of a diffusion pre-factor, and
the application of the diffusive tensor. The key takeaway is that the free drug concentration
trend behaves as expected when compared to a finite difference model that was developed
based on work reported in the literature. The next section discusses the results of this work.
3. Results
This section of the paper presents the results of the interstitial plasma flow profile
into the arterial wall, the initial diffusion flow modeling results using an eroding polymer
coating (free and bound drug concentration profile with the interstitial flow), and the
parametric study results of varying polymer drug concentration (low and high), tortuosity,
porosity, and Pe
T
and Da numbers over the course of 400 h (16.67 days). The next section
discusses the initial diffusion flow modeling results.

Biophysica 2023,3395
3.1. Interstitial Flow into the Arterial Wall
The steady flow of plasma through the cross-section of the coronary arterial vessel
wall is shown in Figure 4. The stent strut obstructs the plasma flow due to the no-slip
boundary condition being applied at the boundaries of the polymer coating. There are two
small regions of high velocity due to the energy loss incurred by the sharp regions on the
top edges of the strut. The flow magnitude dampens out away from the top and middle
region of the strut. There were three drug concentration analyses that were conducted in
this work: (1) a plasma flow and drug concentration analysis conducted without the no-slip
condition applied at the polymer and arterial wall interface (polymer erosion analysis),
(2) a plasma flow and drug concentration analysis conducted with the no-slip condition,
and (3) a drug concentration analysis conducted without plasma flow.
Biophysica 2023, 3, FOR PEER REVIEW 11
porosity, and PeT and Da numbers over the course of 400 h (16.67 days). The next section
discusses the initial diffusion flow modeling results.
3.1. Interstitial Flow into the Arterial Wall
The steady flow of plasma through the cross-section of the coronary arterial vessel
wall is shown in Figure 4. The stent strut obstructs the plasma flow due to the no-slip
boundary condition being applied at the boundaries of the polymer coating. There are
two small regions of high velocity due to the energy loss incurred by the sharp regions on
the top edges of the strut. The flow magnitude dampens out away from the top and mid-
dle region of the strut. There were three drug concentration analyses that were conducted
in this work: (1) a plasma flow and drug concentration analysis conducted without the
no-slip condition applied at the polymer and arterial wall interface (polymer erosion anal-
ysis), (2) a plasma flow and drug concentration analysis conducted with the no-slip con-
dition, and (3) a drug concentration analysis conducted without plasma flow.
Figure 4. Interstitial flow profile into the half-embedded strut arterial wall. Black arrows represent
the velocity vectors.
The next section discusses the free and bound drug concentration with erosion and
interstitial flow results.
3.2. Free and Bound Drug Concentration Profiles with Erosion and Interstitial Flow
Figure 5a,b show the free and bound drug concentration profiles with erosion and
interstitial plasma flow. The interstitial flow within the arterial wall is induced by the
pressure difference between the lumen and the perivascular space and is typically very
small (in the range of 0.01–0.1 µm/s [43]) in reference to the centerline pulsatile flow and
the convective transport term for the arterial wall is often left out in the drug transport
models of drug-eluting stents [44,45]. In this scenario, the no-slip condition is not applied
at the boundaries of the stent, and the plasma flow is allowed to flow through the polymer,
which is modeled as a porous medium. In this case, the average free and bound drug
concentrations in the arterial wall are significantly impacted by the presence of convection
and cause the polymer region to erode, as shown in both Figure 5a,b. The peaking of the
average drug concentrations also suggests an early expected resident time, as the transient
time of the bound drug is within 2 days. This is again due to the high convection due to
plasma flow and the eroding effect of the polymer.
Figure 4.
Interstitial flow profile into the half-embedded strut arterial wall. Black arrows represent
the velocity vectors.
The next section discusses the free and bound drug concentration with erosion and
interstitial flow results.
3.2. Free and Bound Drug Concentration Profiles with Erosion and Interstitial Flow
Figure 5a,b show the free and bound drug concentration profiles with erosion and
interstitial plasma flow. The interstitial flow within the arterial wall is induced by the
pressure difference between the lumen and the perivascular space and is typically very
small (in the range of 0.01–0.1
µ
m/s [
43
]) in reference to the centerline pulsatile flow and
the convective transport term for the arterial wall is often left out in the drug transport
models of drug-eluting stents [44,45]. In this scenario, the no-slip condition is not applied
at the boundaries of the stent, and the plasma flow is allowed to flow through the polymer,
which is modeled as a porous medium. In this case, the average free and bound drug
concentrations in the arterial wall are significantly impacted by the presence of convection
and cause the polymer region to erode, as shown in both Figure 5a,b. The peaking of the
average drug concentrations also suggests an early expected resident time, as the transient
time of the bound drug is within 2 days. This is again due to the high convection due to
plasma flow and the eroding effect of the polymer.

Biophysica 2023,3396
Biophysica 2023, 3, FOR PEER REVIEW 12
Figure 5. Drug concentration contours at 2 days: (a) free drug and (b) bound drug. With an initial
concentration of C0 = 0.01. The light blue lines incorporate the stent strut boundaries.
Figure 6 shows the free and bound drug concentration contours at 8 days. Similar to
the results shown in Figure 5a,b at 2 h, the high convection due to plasma flow and the
eroding effect of the polymer has a significant effect on the transit time and diffusion pro-
file. It also appears that when modeling the polymer boundary as porous media without
the no-slip condition, washing out of the polymer tends to lower the concentration mag-
nitudes. It is evident that the presence of interstitial flow increases the transport in the
transmural direction and leads to faster drug clearance at the perivascular interface using
this modeling method. In an effort to compare other works, we continued this study by
applying the no-slip condition at the boundaries of the polymer and arterial wall interface
and not incorporating interstitial flow with plasma flow through the inlet. Convection is
modeled with the tissue Peclet number.
Figure 5.
Drug concentration contours at 2 days: (
a
) free drug and (
b
) bound drug. With an initial
concentration of C0= 0.01. The light blue lines incorporate the stent strut boundaries.
Figure 6shows the free and bound drug concentration contours at 8 days. Similar to
the results shown in Figure 5a,b at 2 h, the high convection due to plasma flow and the
eroding effect of the polymer has a significant effect on the transit time and diffusion profile.
It also appears that when modeling the polymer boundary as porous media without the
no-slip condition, washing out of the polymer tends to lower the concentration magnitudes.
It is evident that the presence of interstitial flow increases the transport in the transmural
direction and leads to faster drug clearance at the perivascular interface using this modeling
method. In an effort to compare other works, we continued this study by applying the
no-slip condition at the boundaries of the polymer and arterial wall interface and not
incorporating interstitial flow with plasma flow through the inlet. Convection is modeled
with the tissue Peclet number.

Biophysica 2023,3397
Biophysica 2023, 3, FOR PEER REVIEW 13
Figure 6. Drug concentration contours at 8 days: (a) free drug and (b) bound drug. The light blue
lines incorporate the stent strut boundaries.
3.3. Free and Bound Drug Concentration Profiles with Erosion and Convection
As mentioned previously, the results shown in this section are results from the sim-
ulation by applying the no-slip condition at the boundaries of the polymer and the arterial
wall interface and neglecting the interstitial flow with plasma flow through the inlet. Con-
vection, in this case is modelled with the tissue Peclet number. Figure 7a,b show the free
drug diffusion contours in the arterial wall with convection modeled with the tissue Peclet
number. The drug release contour profiles have similar release rates in the first 4 to 8 days
when the PLGA diffusion, degradation, and erosion are insignificant. In Figure 7b, a lower
concentration of the free drug in the polymer coating is observed in the case of time-de-
pendent release of the drug from the coating. This is due to the time-dependent boundary
condition of Equation (21). The effect of release kinetics on the spatial distribution of the
free drug can be visualized clearly in Figure 8. In this case, the heterogeneous distribution
and retention of the drug are found to be observed throughout the domain.
The characteristics of the release profiles in the intravascular delivery reported here
are in good correspondence to what was reported for in vitro release in previously re-
ported works [46]. In the simulation comparison, significant drug release is achieved in
the PLGA coating at around day 17. The arterial bound drug distributions are shown in
Figure 8 for the PLGA coating on day 17, shortly after the drug levels have peaked in the
arterial wall. The bound drug distribution is close to uniform in the circumferential direc-
tion, whereas in the transmural direction, a gradient is clearly observed closer to the peri-
vascular interface. Improved uniformity in the circumferential direction is expected with
the anisotropic drug diffusivity, which results in fast drug diffusion in the circumferential
direction. This is an improvement over the results shown previously in Figures 5 and 6.
In Figure 7, the observed arterial drug distribution paern for the PLGA coating case is
similar to previous studies of a bio-durable coating [34].
Figure 6.
Drug concentration contours at 8 days: (
a
) free drug and (
b
) bound drug. The light blue
lines incorporate the stent strut boundaries.
3.3. Free and Bound Drug Concentration Profiles with Erosion and Convection
As mentioned previously, the results shown in this section are results from the simula-
tion by applying the no-slip condition at the boundaries of the polymer and the arterial wall
interface and neglecting the interstitial flow with plasma flow through the inlet. Convection,
in this case is modelled with the tissue Peclet number. Figure 7a,b show the free drug diffu-
sion contours in the arterial wall with convection modeled with the tissue Peclet number.
The drug release contour profiles have similar release rates in the first 4 to 8 days when the
PLGA diffusion, degradation, and erosion are insignificant. In Figure 7b, a lower concen-
tration of the free drug in the polymer coating is observed in the case of time-dependent
release of the drug from the coating. This is due to the time-dependent boundary condition
of Equation (21). The effect of release kinetics on the spatial distribution of the free drug can
be visualized clearly in Figure 8. In this case, the heterogeneous distribution and retention
of the drug are found to be observed throughout the domain.
The characteristics of the release profiles in the intravascular delivery reported here
are in good correspondence to what was reported for
in vitro
release in previously reported
works [
46
]. In the simulation comparison, significant drug release is achieved in the PLGA
coating at around day 17. The arterial bound drug distributions are shown in Figure 8for
the PLGA coating on day 17, shortly after the drug levels have peaked in the arterial wall.
The bound drug distribution is close to uniform in the circumferential direction, whereas in
the transmural direction, a gradient is clearly observed closer to the perivascular interface.

Biophysica 2023,3398
Improved uniformity in the circumferential direction is expected with the anisotropic drug
diffusivity, which results in fast drug diffusion in the circumferential direction. This is
an improvement over the results shown previously in Figures 5and 6. In Figure 7, the
observed arterial drug distribution pattern for the PLGA coating case is similar to previous
studies of a bio-durable coating [34].
Biophysica 2023, 3, FOR PEER REVIEW 14
Figure 7. Contours of the free drug diffusion into the arterial wall at: (a) 4 days and (b) 8 days.
Although the free and bound drug concentration cases are shown here, the internal-
ized drug is neglected. Although not modeled in this work, drug internalization describes
the cellular uptake of drug molecules after they associate with the binding sites. This is an
important mechanism for drug metabolism in the physiological environment [47,48]. Only
limited studies have considered the impact of the internalization process on stent-based
drug delivery. While the drug internalization rate may vary for the different drugs, and
such data are lacking in the literature, the model discussed in this paper only considers
the free and bound drug case. Future work will involve examining the internalized drug.
The distributions of the average weighted free and bound drug concentrations for
varying values of the scaling parameter ε1 = (Rd/C0) are presented in Figures 9 and 10,
respectively, and the same for different values of ε2 = (C0/BM), which are shown in Figures
11 and 12, respectively. The value of the scaling parameter ε1, decreases with a decrease
in the dissociation rate constant kd and with an increase in the association rate constant ka
depending on Rd = (kd/ka). Additionally, ε2 increases with decreasing S0 (while keeping c0
fixed).
Figure 7. Contours of the free drug diffusion into the arterial wall at: (a) 4 days and (b) 8 days.
Although the free and bound drug concentration cases are shown here, the internalized
drug is neglected. Although not modeled in this work, drug internalization describes the
cellular uptake of drug molecules after they associate with the binding sites. This is an
important mechanism for drug metabolism in the physiological environment [
47
,
48
]. Only
limited studies have considered the impact of the internalization process on stent-based
drug delivery. While the drug internalization rate may vary for the different drugs, and
such data are lacking in the literature, the model discussed in this paper only considers the
free and bound drug case. Future work will involve examining the internalized drug.
The distributions of the average weighted free and bound drug concentrations for vary-
ing values of the scaling parameter
ε1
=(R
d
/C
0
) are presented in Figures 9and 10, respectively,
and the same for different values of
ε2
=(C
0
/B
M
), which are shown in
Figures 11 and 12
,
respectively. The value of the scaling parameter
ε1
, decreases with a decrease in the dissoci-
ation rate constant k
d
and with an increase in the association rate constant k
a
depending on
Rd= (kd/ka). Additionally, ε2increases with decreasing S0(while keeping c0fixed).

Biophysica 2023,3399
Biophysica 2023, 3, FOR PEER REVIEW 15
Figure 8. Contours of drug diffusion into the arterial wall at: (a) 4 days, (b) 8 days, and (c) 16.67
days.
Figure 4a shows that the normalized mean free drug concentration decreases with
decreasing ε1 for PeT = 2, Da = 40, ε2 = 100, up to a certain time and, thereafter, no significant
changes occurred. It can be concluded and justified that, as ε1 decreases, the rate of re-
versible binding (kd) decreases and/or the rate of forward binding increases, which lowers
the mean concentration of the free drug.
Figure 8.
Contours of drug diffusion into the arterial wall at: (
a
) 4 days, (
b
) 8 days, and (
c
) 16.67 days.
Figure 4a shows that the normalized mean free drug concentration decreases with
decreasing
ε1
for Pe
T
= 2, Da = 40,
ε2
= 100, up to a certain time and, thereafter, no
significant changes occurred. It can be concluded and justified that, as
ε1
decreases, the rate
of reversible binding (k
d
) decreases and/or the rate of forward binding increases, which
lowers the mean concentration of the free drug.

Biophysica 2023,3400
Biophysica 2023, 3, FOR PEER REVIEW 16
Figure 9. Distribution of normalized mean bound drug concentration for different values of ε1 at PeT
= 2, Da = 40, ε2 = 100.
Figure 10 shows how the rates of forward and reversible binding affect the average
weighted concentration of bound drug within the arterial tissue. It can be concluded that
the average weighted concentration is increased with the decrease in ε1, which is at-
tributed to the increase in the rate of forward binding and/or to the decrease in the rate of
reversible binding.
Figure 10. Distribution of normalized weighted average free drug concentration for different values
of ε1 at PeT = 2, Da = 40, ε2 = 100.
The effects of ε2 (i.e., net tissue binding potential on the mean concentrations of free
and bound drug) are displayed in Figures 11 and 12, respectively. As previously men-
tioned, ε2 increases with decreasing binding potential.
Figure 9.
Distribution of normalized mean bound drug concentration for different values of
ε1
at
PeT= 2, Da = 40, ε2= 100.
Figure 10 shows how the rates of forward and reversible binding affect the average
weighted concentration of bound drug within the arterial tissue. It can be concluded that the
average weighted concentration is increased with the decrease in
ε1
, which is attributed to the
increase in the rate of forward binding and/or to the decrease in the rate of reversible binding.
Biophysica 2023, 3, FOR PEER REVIEW 16
Figure 9. Distribution of normalized mean bound drug concentration for different values of ε1 at PeT
= 2, Da = 40, ε2 = 100.
Figure 10 shows how the rates of forward and reversible binding affect the average
weighted concentration of bound drug within the arterial tissue. It can be concluded that
the average weighted concentration is increased with the decrease in ε1, which is at-
tributed to the increase in the rate of forward binding and/or to the decrease in the rate of
reversible binding.
Figure 10. Distribution of normalized weighted average free drug concentration for different values
of ε1 at PeT = 2, Da = 40, ε2 = 100.
The effects of ε2 (i.e., net tissue binding potential on the mean concentrations of free
and bound drug) are displayed in Figures 11 and 12, respectively. As previously men-
tioned, ε2 increases with decreasing binding potential.
Figure 10.
Distribution of normalized weighted average free drug concentration for different values
of ε1at PeT= 2, Da = 40, ε2= 100.
The effects of
ε2
(i.e., net tissue binding potential on the mean concentrations of free
and bound drug) are displayed in Figures 11 and 12, respectively. As previously mentioned,
ε2increases with decreasing binding potential.

Biophysica 2023,3401
Biophysica 2023, 3, FOR PEER REVIEW 17
Figure 11. Distribution of normalized weighted averaged bound drug concentration for different
values of ε2 at PeT = 2, Da = 40, ε1 = 0.001.
The results of these figures indicate that the average weighted concentration of free
drug increases with decreasing binding potential up to ε2 = 100, but the concentration
reaches a quasi-steady state for weaker binding capacity (ε2 = 1000) as compared to the
other cases.
Figure 12. Distribution of normalized weighted averaged free drug concentration for different val-
ues of ε2 at PeT = 2, Da = 40, ε1 = 0.001.
In the case of the free drug for ε2 ≤ 100, the quasi-equilibrium is not fully established
until approximately 17 days, while the PLGA coating has eroded significantly. On the
other hand, in the case that bound drug for ε2 ≥ 10, the quasi-equilibrium is aained very
rapidly.
Figure 11.
Distribution of normalized weighted averaged bound drug concentration for different
values of ε2at PeT= 2, Da = 40, ε1= 0.001.
The results of these figures indicate that the average weighted concentration of free
drug increases with decreasing binding potential up to
ε2
= 100, but the concentration
reaches a quasi-steady state for weaker binding capacity (
ε2
= 1000) as compared to the
other cases.
Biophysica 2023, 3, FOR PEER REVIEW 17
Figure 11. Distribution of normalized weighted averaged bound drug concentration for different
values of ε2 at PeT = 2, Da = 40, ε1 = 0.001.
The results of these figures indicate that the average weighted concentration of free
drug increases with decreasing binding potential up to ε2 = 100, but the concentration
reaches a quasi-steady state for weaker binding capacity (ε2 = 1000) as compared to the
other cases.
Figure 12. Distribution of normalized weighted averaged free drug concentration for different val-
ues of ε2 at PeT = 2, Da = 40, ε1 = 0.001.
In the case of the free drug for ε2 ≤ 100, the quasi-equilibrium is not fully established
until approximately 17 days, while the PLGA coating has eroded significantly. On the
other hand, in the case that bound drug for ε2 ≥ 10, the quasi-equilibrium is aained very
rapidly.
Figure 12.
Distribution of normalized weighted averaged free drug concentration for different values
of ε2at PeT= 2, Da = 40, ε1= 0.001.
In the case of the free drug for
ε2≤
100, the quasi-equilibrium is not fully established
until approximately 17 days, while the PLGA coating has eroded significantly. On the other
hand, in the case that bound drug for ε2≥10, the quasi-equilibrium is attained very rapidly.

Biophysica 2023,3402
3.4. Average Weighted Concentration Results for Varying Tortuosity
We also conducted a simple analysis to demonstrate the effect of tortuosity (
τ
as listed
in Equation (11)) on the average weighted concentration. We observed that a decrease in
the mean concentration of free drug took place with while increasing tortuosity (i.e., an
inverse relationship between free drug concentration and tortuosity is revealed) as shown
in Figure 13. A similar pattern is also observed for bound drug in Figure 14.
Biophysica 2023, 3, FOR PEER REVIEW 18
3.4. Average Weighted Concentration Results for Varying Tortuosity
We also conducted a simple analysis to demonstrate the effect of tortuosity (τ as listed
in Equation (11)) on the average weighted concentration. We observed that a decrease in
the mean concentration of free drug took place with while increasing tortuosity (i.e., an
inverse relationship between free drug concentration and tortuosity is revealed) as shown
in Figure 13. A similar paern is also observed for bound drug in Figure 14.
Figure 13. Normalized average weighted free drug concentration for varying tortuosity (τ).
The above observation may be justified in the sense that as the tortuosity increases so
too does the effective distance over which diffusion has to take place (i.e., the progression
of diffusion eventually lowering the mean concentration of both drug forms is impeded).
Figure 14. Normalized average weighted bound drug concentration for varying tortuosity (τ).
4. Conclusions
This paper reports the findings of an investigation of the interstitial flow profile into
the arterial wall and the free and bound drug diffusion profiles with a parametric study
of varying polymer drug concentration (low and high), tortuosity, porosity, and Peclet
and DamKöhler numbers over the course of 400 h (16.67 days). Acquiring an understand-
ing of the relationship between drug physicochemical properties and the local transport
environment is crucial to the success of new stent designs. Computational studies can
Figure 13. Normalized average weighted free drug concentration for varying tortuosity (τ).
The above observation may be justified in the sense that as the tortuosity increases so
too does the effective distance over which diffusion has to take place (i.e., the progression
of diffusion eventually lowering the mean concentration of both drug forms is impeded).
Biophysica 2023, 3, FOR PEER REVIEW 18
3.4. Average Weighted Concentration Results for Varying Tortuosity
We also conducted a simple analysis to demonstrate the effect of tortuosity (τ as listed
in Equation (11)) on the average weighted concentration. We observed that a decrease in
the mean concentration of free drug took place with while increasing tortuosity (i.e., an
inverse relationship between free drug concentration and tortuosity is revealed) as shown
in Figure 13. A similar paern is also observed for bound drug in Figure 14.
Figure 13. Normalized average weighted free drug concentration for varying tortuosity (τ).
The above observation may be justified in the sense that as the tortuosity increases so
too does the effective distance over which diffusion has to take place (i.e., the progression
of diffusion eventually lowering the mean concentration of both drug forms is impeded).
Figure 14. Normalized average weighted bound drug concentration for varying tortuosity (τ).
4. Conclusions
This paper reports the findings of an investigation of the interstitial flow profile into
the arterial wall and the free and bound drug diffusion profiles with a parametric study
of varying polymer drug concentration (low and high), tortuosity, porosity, and Peclet
and DamKöhler numbers over the course of 400 h (16.67 days). Acquiring an understand-
ing of the relationship between drug physicochemical properties and the local transport
environment is crucial to the success of new stent designs. Computational studies can
Figure 14. Normalized average weighted bound drug concentration for varying tortuosity (τ).
4. Conclusions
This paper reports the findings of an investigation of the interstitial flow profile
into the arterial wall and the free and bound drug diffusion profiles with a parametric
study of varying polymer drug concentration (low and high), tortuosity, porosity, and
Peclet and DamKöhler numbers over the course of 400 h (16.67 days). Acquiring an
understanding of the relationship between drug physicochemical properties and the local
transport environment is crucial to the success of new stent designs. Computational studies
can provide highly detailed predictions of the drug distribution in the vessel wall over time.

Biophysica 2023,3403
Most computational investigations of drug delivery include only one drug form. This has
the drawback of not accounting for binding and convective diffusive transport directly. The
developed mathematical model discussed in this paper provides the basis for evaluating
and studying diffusion characteristics for drug-eluting stent applications.
Future work will be carried out to enhance this model to characterize the internalized
drug, evaluate further the eluting behavior of the PGLA coating, compare PLGA to other
bio durable coatings, describe the anisotropic behavior of the diffusion coefficient within the
arterial wall with the ease of adaptation to more sophisticated scenarios (e.g., consideration
of more pathological conditions) and compare the effect of stent position on drug diffusion
profiles (i.e., half, full, and partial embedment). Simulations using the presented model can
help provide insight into the drug release and distribution by a stent with PLGA coating,
and the potential impacts of various factors that can affect the efficacy of drug delivery.
With the developed preliminary model, optimization of the model parameters, such as
different stent strut geometries and coating thickness, can also be performed for exploration
on the design of drug-eluting stents.
Author Contributions:
Conceptualization, R.L.H.J., M.E. and J.P.K.; methodology, R.L.H.J. and
J.P.K.; software, R.L.H.J.; validation, R.L.H.J. and M.E.; formal analysis, R.L.H.J., M.E. and J.P.K.;
investigation, R.L.H.J., M.E. and J.P.K.; resources, R.L.H.J.; data curation, R.L.H.J.; writing—original
draft preparation, R.L.H.J.; writing—review and editing, R.L.H.J.; visualization, R.L.H.J.; supervision,
R.L.H.J.; project administration, R.L.H.J.; funding acquisition, R.L.H.J. All authors have read and
agreed to the published version of the manuscript.
Funding:
This research was funded by the University of North Carolina at Charlotte (UNC-C) Faculty
Research Grant (FRG).
Data Availability Statement:
The data from this work will be shared and made available upon
request to the corresponding author.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
CfFree drug
CbBound drug
CpPerivascular drug concentration
CwWall drug concentration
C0Initial drug concentration
DOuter diameter of the artery
DCCoefficient of the coating diffusion
DaDimensionless DamKöhler number in the tissue
Dfree Coefficient of free diffusivity
Deff Coefficient of effective diffusivity
DTTrue diffusivity of the free drug
Jwp PLGA flux parameter
LxArterial domain length
LyArterial domain wall thickness
Lsx Stent length
Lsy Stent thickness
kaTissue binding capacity
kdDissociation rate constant
PeCDimensionless Peclet number in the coating
PeTDimensionless Peclet number in the tissue
RdEquilibrium dissociation constant
Rwp Mass transfer resistance
S0Available binding sites
TTime

Biophysica 2023,3404
VyTransmural filtration velocity
xx-coordinate
yy-coordinate
δStrut dimension
εPorosity of the arterial wall
ε1and ε2Dimensionless scaling parameters
τTortuosity of the arterial wall
τ1,τ2, and τ2Characteristic time scales
PLGA Poly(lactic-co-glycolic acid)
UDS User defined scalar
Appendix A
This section provides an overview of the characteristic scaling methodology imple-
mented to dimensionless Equations (4)–(6). In this method, we begin with stating the free
and bound drug transport equations as mentioned previously:
Free-drug in the PLGA Coating Domain:
∂Cf
∂t=DC ∂2Cf
∂x2+∂2Cf
∂y2!, (A1)
Free-drug in the Arterial Domain:
∂Cf
∂t=−v∂Cf
∂x+∂Cf
∂y+DT ∂2Cf
∂x2+∂2Cf
∂y2!−[kaCb(S0−Cb)−kdCb], (A2)
Bound-drug in the Arterial Domain:
∂Cb
∂t=hkaCf(S0−Cb)−kdCbi, (A3)
The dimensionless scaling parameters used for scaling Equation (A1) through (A3) are
shown below:
x∗=x
δ,y∗=y
δ,t∗=tVy
δ,C∗
f=Cf
C0
,C∗
b=Cb
S0
(A4)
The first order free and bound drug concentration derivatives are first non-dimensionalized
using the characteristic dimensionless parameters as shown below:
Scaled free-drug concentration time derivative:
∂Cf
∂t=∂C∗
fC0
∂(t∗δ/Vy)=C0Vy
δ
∂C∗
f
∂t∗(A5)
Scaled bound-drug concentration time derivative:
∂Cb
∂t=∂C∗
bS0
∂(t∗δ/Vy)=S0Vy
δ
∂C∗
b
∂t∗(A6)
Scaled free-drug concentration first-order x-direction derivative:
∂Cf
∂x=∂C∗
fC0
∂(x∗δ)=C0
δ
∂C∗
f
∂x∗(A7)

Biophysica 2023,3405
Scaled free-drug concentration first-order y-direction derivative:
∂Cf
∂y=∂C∗
fC0
∂(y∗δ)=C0
δ
∂C∗
f
∂y∗(A8)
The second-order derivatives are scaled as shown below:
Scaled free-drug concentration second-order x-direction derivative:
∂2Cf
∂x2=∂
∂x
∂Cf
∂x=∂
∂(x∗δ)
∂C∗
fC0
∂(x∗δ)=C0
δ2
∂2Cf
∂x∗2(A9)
Scaled free-drug concentration second-order y-direction derivative:
∂2Cf
∂y2=∂
∂y
∂Cf
∂y=∂
∂(y∗δ)
∂C∗
fC0
∂(y∗δ)=C0
δ2
∂2Cf
∂y∗2(A10)
The dimensionless derivatives are now substituted into Equation (A1) as shown below:
C0Vy
δ
∂C∗
f
∂t∗=DC C0
δ2
∂2Cf
∂x∗2+C0
δ2
∂2Cf
∂y∗2!, (A11)
Dividing Equation (A11) through on both sides by C0Vy/δyields the following:
∂C∗
f
∂t∗=DC
C0
C0Vy
1
δ ∂2Cf
∂x∗2+∂2Cf
∂x∗2!, (A12)
The finalized dimensionless form of the free-drug transport into the PLGA coating is
shown below: ∂C∗
f
∂t∗=1
PeC ∂2Cf
∂x∗2+∂2Cf
∂x∗2!, (A13)
where Pe
C
= [V
yδ
/(D
C
)]. The scaled free-drug concentration derivatives (
Equations (A5) and (A7)
are now substituted into the free drug transport equation into the arterial wall domain:
C0Vy
δ
∂C∗
f
∂t∗=−vyC0
δ∂C∗
f
∂x∗+∂C∗
f
∂y∗+DTC0
δ2∂2Cf
∂x∗2+∂2Cf
∂y∗2−
kaS0C0hC∗
f(1−Cb∗)−kd
ka
1
C0Cb∗i,
(A14)
Dividing Equation (A14) through on both sides by C
0
V
y
/
δ
, factoring out the first- and
second-order derivative constants, and substituting the equilibrium constant R
d
= (k
d
/k
a
)
yields the following:
∂C∗
f
∂t∗=−VyC0δ
VyδC0∂C∗
f
∂x∗+∂C∗
f
∂y∗+DTδ
C0Vy
C0
δ2∂2Cf
∂x∗2+∂2Cf
∂y∗2−
kaS0C0δ
C0Vy
DT
DT
δ
δhC∗
f1−C∗
b−Rd
C0C∗
bi,
(A15)
Recognizing that the Peclet and DamKöhler numbers are Pe
T
= [V
yδ
/(D
T
)and
Da = [(kaS0δ2)/(DT)]
are in the tissue and that
ε1
= (R
d
/C
0
)is an additional scaling pa-
rameter and yields the final non-dimensional form of the free-drug transport equation in
the arterial domain as shown below:
∂C∗
f
∂t∗=−∂C∗
f
∂x∗+∂C∗
f
∂y∗+1
PeT∂2Cf
∂x∗2+∂2Cf
∂y∗2−
Da
Pe hC∗
f1−C∗
b−ε1C∗
bi,
(A16)

Biophysica 2023,3406
The scaled bound-drug concentration derivative (Equation (A6) and the scaling pa-
rameters are now substituted into the free drug transport equation into the arterial wall
domain: S0Vy
δ
∂C∗
b
∂t∗=kaS0C0
DT
DT
δ
δC∗
f(1−C∗
b)−Rd
C0
C∗
b, (A17)
Dividing through by S0Vy/δand entering the relations for ε2,Da, and Pe yields.
∂C∗
b
∂t∗=ε2Da
Pe hC∗
f(1−C∗
b)−ε1C∗
bi, (A18)
References
1.
Edwards, M.; Hewlin, R.L., Jr.; Smith, M. A 2-D Transient Computational Multi-Physics Model for Analyzing Magnetic and
Non-Magnetic (Red Blood Cells and E. Coli Bacteria) Particle Dynamics in a Travelling Wave Ferro-Magnetic Microfluidic Device.
ASME J. Eng. Sci. Med. Ther. Diagn. 2023, 1–47. [CrossRef]
2.
Hewlin, R.L., Jr.; Edwards, M. Continuous Flow Separation of Red Blood Cells and Platelets in a Y-Microfluidic Channel Device
with Saw-Tooth Profile Elec-trodes via Low Voltage Dielectrophoresis. Curr. Issues Mol. Biol.
2023
,45, 3048–3067. [CrossRef]
[PubMed]
3.
Hewlin, R.L., Jr.; Ciero, A.; Kizito, J.P. Development of a Two-Way Coupled Eulerian-Lagrangian Computational Magnetic
Nanoparticle Targeting Model for Pulsatile Flow in a Patient-Specific Diseased Left Carotid Bifurcation Artery. Cardiovasc. Eng.
Technol. 2019,10, 299–313. [CrossRef] [PubMed]
4.
Stanley, N.; Ciero, A.; Timms, W.; Hewlin, R.L., Jr. A 3-D Printed Optically Clear Rigid Diseased Carotid Bifurcation Arterial Mock
Vessel Model for Particle Image Velocimetry Analysis in Pulsatile Flow. ASME Open J. Eng. ASME 2023,2, 021010. [CrossRef]
5.
Hewlin, R.L., Jr.; Tindall, J.M. Computational Assessment of Magnetic Nanoparticle Targeting Efficiency in a Simplified Circle of
Willis Arterial Model. Int. J. Mol. Sci. 2023,24, 2545. [CrossRef] [PubMed]
6.
Hewlin, R.L., Jr.; Kizito, J.P. Comparison of Carotid Bifurcation Hemodynamics in Patient-Specific Geometries at Rest and During
Exercise. In Proceedings of the Fluids Engineering Division Summer Meeting. American Society of Mechanical Engineers,
Volume 1A
,
Symposia: Advances in Fluids Engineering Education; Advances in Numerical Modeling for Turbomachinery Flow Optimization;
Applications in CFD; Bio-Inspired Fluid Mechanics; CFD Verification and Validation; Development and Applications of Immersed
Boundary Methods; DNS, LES, and Hybrid RANS/LES Methods, Incline Village, NV, USA, 7–11 July 2013; p. V01AT04A001.
[CrossRef]
7.
Stanley, N.; Ciero, A.; Timms, W.; Hewlin, R.L., Jr. Development of 3-D Printed Optically Clear Rigid Anatomical Vessels for
Particle Image Velocimetry Analysis in Cardiovascular Flow. In Proceedings of the ASME International Mechanical Engineering
Congress and Exposition. American Society of Mechanical Engineers, Volume 7: Fluids Engineering, Salt Lake City, UT, USA,
11–14 November 2019; p. V007T08A004. [CrossRef]
8.
Hewlin, R.L., Jr.; Kizito, J.P. Development of an Experimental and Digital Cardiovascular Arterial Model for Transient Hemody-
namic and Postural Change Studies: “A Preliminary Framework Analysis”. Cardiovasc. Eng. Tech. 2018,9, 1–31. [CrossRef]
9.
Daemen, J.; Serruys, P.W. Drug-Eluting Stent Update 2007 Part I: A Survey of Current and Future Generation Drug-Eluting
Stents: Meaningful Advances or More of the Same. Circulation 2007,116, 316–328. [CrossRef]
10.
Beshchasna, N.; Saqib, M.; Kraskiewicz, H.; Wasyluk, Ł.; Kuzmin, O.; Duta, O.C.; Ficai, D.; Ghizdavet, Z.; Marin, A.;
Ficai, A.; et al.
Recent Advances in Manufacturing Innovative Stents. Pharnaceutics 2020,12, 349. [CrossRef]
11. Conway, C. Clinical Evidence vs. the Testing Paradigm. Cardiovasc. Eng. Technol. 2018,9, 752–760. [CrossRef]
12.
Beckman, J.A.; White, C.J. Paclitaxel-Coated Balloons and Eluting Stents: Is There a Mortality Risk in Patients with Peripheral
Artery Disease. Circulation 2019,140, 1342–1351. [CrossRef]
13.
Capell, W.H.; Bonaca, M.P.; Nehler, M.R.; Kittelson, J.M.; Anand, S.S.; Berkowitz, S.D.; Debus, E.S.; Fanelli, F.; Haskell, L.;
Patel, M.R.; et al.
Rationale and Desitn for the Vascular Outcomes Study of ASA Along with Rivaroxaban in Endovascular or
Surgical Limb Revascularization for Peripheral Artery Disease. Am. Heart J. 2018,199, 83–91. [CrossRef]
14.
US Food and Drug Administration. Update: Treatment of Peripheral Arterial Disease with Paclitaxel-Coated Ballons and
Paclitaxel-Eluting Stents Potentiall Associated with Increased Mortality: Letter to Health Care Providers. Available online:
https://www.fda.gov/medical-devices/letters-health-care-providers/update-treatment-peripheral-arterial-disease-paclitaxel-
coated-balloons-and-paclitaxel-eluting (accessed on 5 May 2022).
15.
Levin, A.; Jonas, M.; Hwang, C.W.; Edelman, E. Local and Systemic Drug Competition in Drug-Eluting Stent Tissue Deposition
Properties. J. Control. Release 2005,109, 226–243. [CrossRef] [PubMed]
16.
Granada, J.F.; Stenoien, M.; Buszman, P.P.; Tellez, A.; Langanki, D.; Kaluza, G.L.; Leon, M.B.; Gray, W.; Jaff, M.R.; Schwartz, R.S.
Mechanisms of Tissue Uptake and Retention of Paclitaxel-Coated Balloons: Impact on Neointimal Proliferation and Healing.
Open Heart 2014,1, e000117. [CrossRef]
17.
Lovich, M.A.; Philbrook, M.; Sawyer, S.; Weselcouch, E.; Edelman, E.R. Arterial Heparin Deposition: Role of Diffusion, Convection,
and Extravascular Space. Am. J. Physiol. 1998,275, H2236–H2242. [CrossRef]

Biophysica 2023,3407
18.
Moses, J.W.; Stone, G.W.; Nikolsky, E.; Mintz, G.S.; Dangas, G.; Grube, E. Drug-eluting Stents in the Treatment of Intermediate
Lesions: Pooled Analysis from four Randomized Trials. J. Am. Coll. Cardiol. 2006,47, 2164–2171. [CrossRef]
19.
Serruys, P.W.; Sianos, G.; Abizaid, A.; Aoki, J.; den Heijer, P.; Bonnier, H. The Effect of Variable Dose and Release Kinetics on
Neointimal Hyperplasia using a Novel Paclitaxel-eluting Stent Platform: The Paclitaxel In-Stent Controlled Elution Study (Pices).
J. Am. Coll. Cardiol. 2005,46, 253–260. [CrossRef] [PubMed]
20. Lovich, M.A.; Edelman, E.R. Computational Simulations of Local Vascular Heparin Deposition and Distribution. Am. J. Physiol.
1996,271, H2014–H2024. [CrossRef]
21.
Sakharov, D.V.; Kalachev, L.V.; Rijken, D.C. Numerical Simulation of Local Pharmcokinetics of a Drug after Intervascular Delivery
with an Eluting Stent. J. Drug Target. 2002,10, 507–513. [CrossRef] [PubMed]
22.
Hwang, C.; Wu, D.; Edelman, E. Physiological Transport Forces Govern Drug Distribution for Stent-Based Delivery. Circulation
2001,104, 600–605. [CrossRef]
23.
Migiliavacca, F.; Gervaso, F.; Prosi, M.; Zunino, P.; Minisini, S.; Formaggia, L. Expansion and Drug Elution Model of a Coronary
Stent. Comput. Methods Biomech. Biomed. Eng. 2007,10, 63–73. [CrossRef]
24.
Borghi, A.; Foa, E.; Balossino, R.; Migliavacca, F.; Dubini, G. Modelling Drug Elutiogn from Stents: Effects of Reversible Binding
in the Vascular Wall and Degradable Polymeric Matrix. Comput. Methods Biomech. Biomed. Eng. 2008,11, 367–377. [CrossRef]
25.
Horner, M.; Joshi, S.; Dhruva, V.; Sett, S.; Stewart, S.F.C. A Two-Species Drug Delivery Model is Required to Predict Deposition
from Drug-Eluting Stents. Cardiovasc. Eng. Technol. 2010,1, 225–234. [CrossRef]
26.
Tzafriri, A.R.; Levin, A.D.; Edelman, E.R. Diffusion-limited Binding Explains Binary Dose Response for Local Arterial and Tumor
Drug Delivery. Cell Prolif. 2009,42, 348–363. [CrossRef]
27.
Higuchi, T. Theoretical Analysis of Rate of Release of Solid Drugs Dispersed in Solid Matrices. J. Pharm. Sci.
1963
,52, 1145–1149.
[CrossRef] [PubMed]
28.
Mandal, A.P.; Mandal, P.K. An Unsteady Analysis of Arterial Drug Transport from Half-Embedded Drug Eluting Stent. Appl.
Math. Comput. 2015,266, 968–981. [CrossRef]
29.
Mandal, A.P.; Mandal, P.K. Computational Modelling of Three-Phase Stent-Based Delivery. J. Explor. Res. Pharm.
2017
,2, 31–40.
[CrossRef]
30. Saltzman, W.M. Drug Delivery: Engineering Principles for Drug Therapy; Oxford University Press: Oxford, UK, 2001.
31.
Sarifuddin; Mandal, P.K. Effect of Diffusivity on the Transport of Drug Eluted from Drug Eluting Stent. Int. J. Appl. Comput. Math.
2016,2, 291–301. [CrossRef]
32.
O’Connel, B.M.; Walsh, M.T. Demonstrating the Influence of Compression on Artery Wall Mass Transport. Ann. Biomed. Eng.
2010,38, 1354–1366. [CrossRef]
33.
Edwards, M.; Kizito, J.P.; Hewlin, R.L., Jr. A Time-Dependent Two Species Explicit Finite Difference Computational Model for
Analyzing Diffusion in a Drug Eluting Stented Coronary Artery Wall: A Phase I Study. In Proceedings of the ASME International
Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, Volume 4: Biomedical and
Biotechnology; Design, Systems, and Complexity, Columbus, OH, USA, 30 October–3 November 2022; p. V004T05A009.
[CrossRef]
34.
Zhu, X.; Pack, D.W.; Braatz, R.D. Modelling Intravascular Delivery from Drug-Eluting Stents with Biodegradable Coating: Inves-
tigation of Anisotropic Vascular Drug Diffusivity and Arterial Drug Distribution. Comput. Methods Biomech. Biomed. Eng.
2014
,17,
187–198. [CrossRef]
35.
Saha, R.; Mandal, P.K. Modelling Time-dependent Release Kinetics in Stent-based Delivery. J. Explor. Res. Pharmacol.
2018
,3, 61–70.
[CrossRef]
36.
Zhu, X.; Braatz, R.D. Modelling and Analysis of Drug-Eluting Stents With Biodegradable PLGS Coating: Consequences on
Instravascular Drug Delivery. J. Biomech. Eng. 2015,136, 111004. [CrossRef]
37.
Balakrishnan, B.; Tzafriri, A.R.; Arifin, D.Y.; Edelman, E.R. Intravascular Drug Release Kinetics Dictate Arterial Drug Deposition,
Rentention and Distribution. J. Control. Release 2007,123, 100–108. [CrossRef] [PubMed]
38.
Vairo, G.; Cioffi, M.; Cottone, R.; Dubini, G.; Migliavacca, F. Drug Release From Coronary Eluting Stents: A Multidomain
Approach. J. Biomech. Eng. 2010,43, 1580–1589. [CrossRef]
39.
Hewlin, R.L., Jr. Transient Cardiovascular Hemodynamics in a Patient-Specific Arterial System. Ph.D. Thesis, North Carolina
Agricultural and Technical State University, Greensboro, NC, USA, 2015.
40.
Whale, M.; Grodzinsky, A.; Johnson, M. The Effect of Aging and Pressure on the Specific Hydraulic Conductivity of the Aortic
Wall. Biorheology 1996,33, 17–44. [CrossRef]
41.
Kolachalama, V.; Tzafriri, A.; Arifin, D.; Edelman, E. Luminal Flow Patterns Dictate Arterial Drug Deposition in Stent-Based
Delivery. J. Control. Release 2009,133, 24–30. [CrossRef]
42.
Baldwin, A.L.; Wilson, I.; Gardus-Pizlo, R.; Wilensky, R.; March, K. Effect of Atherosclerosis on Transmural Convection and
Arterial Ultrastructure: Implications for Loacl Vascualr Drug Delivery. Aterioscler. Thromb. Vasc. Biol.
1997
,17, 3365–3375.
[CrossRef]
43.
Wang, D.; Tarbell, J. Modelling Interstitial Flow in an Artery Wall Allows Estimation of Wall Shear Stress on Smooth Muscle Cells.
ASME J. Biomech. Eng. 1995,117, 358–363. [CrossRef]
44.
Santin, M.; Colombo, P.; Brushci, G. Interfacial Biology of In-Stent Restenosis. Expert Rev. Med. Dev.
2005
,2, 429–443. [CrossRef]

Biophysica 2023,3408
45.
Balakrishnan, B.; Tzafriri, A.R.; Seifert, P.; Groothuis, A.; Rogers, C.; Edelman, E.R. Strut Position, Blood Flow, and Drug
Deposition-Implications for Single and Overlapping Drug-Eluting Stents. Circulation
2005
,111, 2958–2965. [CrossRef] [PubMed]
46.
Wang, X.T.; Venkatraman, S.S.; Boey, F.Y.C.; Loo, J.S.C.; Tan, L.P. Controlled Release of Sirolimus from a Multilayered PLGA Stent
Matrix. Biomaterials 2006,27, 5588–5595. [CrossRef] [PubMed]
47.
Castellot, J.J.; Wong, K.; Herman, B.; Hoover, R.L.; Albertini, D.F.; Wright, T.C.; Caleb, B.L.; Karnovsky, M.J. Binding and
Internalization of Heparin by Vascular Smooth Muscle Cells. J. Cell. Physiol. 1985,124, 13–20. [CrossRef] [PubMed]
48.
Deux, J.-F.; Meddahi-Pelle, A.; Le Blanche, A.F.; Feldman, L.J.; Colliec-Jouault, S.; Brée, F.; Boudghène, F.; Michel, J.-B.;
Letourneur, D.
Low Molecular Weight Fucoidan Prevents Neointimal Hyperplasia in Rabbit Iliac Artery In-Stent Restenosis
Model. Arter. Thromb. Vasc. Biol. 2002,22, 1604–1609. [CrossRef] [PubMed]
Disclaimer/Publisher’s Note:
The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.