**2. Coronary artery disease**

Atherosclerosis is a degenerative disease that affects coronary, carotid and other peripheral arteries in the body. Disease formation can occur as early as childhood with the development of fatty streaks within the artery wall. As the aging process progresses, these fatty streaks accumulate to become larger lipid deposits within the artery. This gradual propagation of plaque can be detrimental to the smooth operation of the vasculature. Occlusions ensuing from aggressive atherosclerotic plaque progression can often culminate in an ischemic attack, such as a stroke or a heart attack. CAD pertains to a blockage or narrowing of the coronary arteries that provide oxygen and nutrients vital to the smooth operation of the heart muscle. Once identified as such, there are a number of interventional procedures available to the cardiologist but the successful emergence of stents, and more recently DES, has seen them become the preferred choice of treatment for CAD, so much so that by the beginning of 2006 more than 8 out of 10 coronary stents were DES (Head et al., 2007) at a cost of between \$4 and \$5 billion annually (Kaul et al., 2007).

CAD has been intrinsically linked to atherosclerosis since the early 20th century (Chen et al., 2005) and refers to the localisation of disease within the coronary arteries. It is generally asymptomatic and those afflicted often only realise they have the condition when it manifests itself in the form of a heart attack. For this reason CAD is the foremost cause of mortality in the world's industrialised nations (Khakpour &Vafai, 2008). CAD alone is reportedly responsible for approximately 700,000 deaths in the United States of America annually (Kaazempur-Mofrad et al., 2005). Lifestyle choices made by an individual, such as not smoking, regular exercise and a balanced diet, have been shown to influence CAD development but it is the concentration of lipid rich cholesterol in the blood that is considered the most important factor (Sun et al., 2006).

### **2.1 Traditional interventional procedures**

Numerous ways exist to alleviate a stenosis in a coronary artery once detected. The first such interventional procedure commonly practiced was coronary artery bypass graft (CABG) surgery. CABG surgery was successfully performed first by Robert H. Goetz and his team in 1960 at the Albert Einstein College of Medicine (Haller & Olearchyk, 2002). Prior to surgery an angiogram is conducted in order to locate the occlusion within the artery, after which a median sternotomy is performed, which exposes the heart and enables the blocked coronary arteries to be bypassed. This procedure is traumatic for the patient with extensive recovery times and significant scarring to the chest. Furthermore, the long term patency rates of these grafts were moderate and several efforts have been made in vain to optimise downstream graft artery junctions. Despite this, CABG remained the gold standard in the treatment of CAD until 1977 when Andreas Gruntzig first performed percutaneous transluminal coronary angioplasty (PTCA) (Kukreja et al., 2008). CABG is a procedure still used today as not every patient is eligible for minimally invasive surgeries such as PTCA due to highly tortuous or extensively blocked arteries.

Initially PTCA was welcomed by the clinical community due to its minimally invasive approach to arterial stenosis alleviation. A balloon catheter is introduced through an incision in the femoral artery and is manoeuvred through the vasculature until it reaches the stenosis. Once inflated, the balloon pushes the plaque back against the artery wall and enables blood flow to recommence after it has been deflated. The initial success of PTCA was short lived as investigators soon discovered that a substantial percentage of patients, reported to be anywhere between 30% and 60%, experienced recurrent ischemia due to the re-blocking of the artery (restenosis) within the first 6 months. This was attributed to mechanical injury caused by over dilating a device within the vessel (Head et al., 2007).

The next major advance in the field of minimally invasive interventional cardiology came in the early 1990's with the advent of the coronary artery stent (CAS). Prior to surgery a cylindrical metallic scaffold, or stent, is placed on the end of an existing balloon catheter and deployed in the same way as traditional PTCA. Initially, these stents were mounted on the balloon catheter by the physician, however, in more recent times the manufacturers supply the catheters with a stent already in situ. After deployment the stent remains within the artery in an attempt to retain arterial patency. CAS reduced failure rates tobetween 10% and 40% (Duraiswamy et al., 2007; Mongrain et al., 2007) through the elimination of elastic recoil and negative remodelling of the artery associated with PTCA (Costa and Simon, 2005).

operation of the heart muscle. Once identified as such, there are a number of interventional procedures available to the cardiologist but the successful emergence of stents, and more recently DES, has seen them become the preferred choice of treatment for CAD, so much so that by the beginning of 2006 more than 8 out of 10 coronary stents were DES (Head et al.,

CAD has been intrinsically linked to atherosclerosis since the early 20th century (Chen et al., 2005) and refers to the localisation of disease within the coronary arteries. It is generally asymptomatic and those afflicted often only realise they have the condition when it manifests itself in the form of a heart attack. For this reason CAD is the foremost cause of mortality in the world's industrialised nations (Khakpour &Vafai, 2008). CAD alone is reportedly responsible for approximately 700,000 deaths in the United States of America annually (Kaazempur-Mofrad et al., 2005). Lifestyle choices made by an individual, such as not smoking, regular exercise and a balanced diet, have been shown to influence CAD development but it is the concentration of lipid rich cholesterol in the blood that is

Numerous ways exist to alleviate a stenosis in a coronary artery once detected. The first such interventional procedure commonly practiced was coronary artery bypass graft (CABG) surgery. CABG surgery was successfully performed first by Robert H. Goetz and his team in 1960 at the Albert Einstein College of Medicine (Haller & Olearchyk, 2002). Prior to surgery an angiogram is conducted in order to locate the occlusion within the artery, after which a median sternotomy is performed, which exposes the heart and enables the blocked coronary arteries to be bypassed. This procedure is traumatic for the patient with extensive recovery times and significant scarring to the chest. Furthermore, the long term patency rates of these grafts were moderate and several efforts have been made in vain to optimise downstream graft artery junctions. Despite this, CABG remained the gold standard in the treatment of CAD until 1977 when Andreas Gruntzig first performed percutaneous transluminal coronary angioplasty (PTCA) (Kukreja et al., 2008). CABG is a procedure still used today as not every patient is eligible for minimally invasive surgeries such as PTCA

Initially PTCA was welcomed by the clinical community due to its minimally invasive approach to arterial stenosis alleviation. A balloon catheter is introduced through an incision in the femoral artery and is manoeuvred through the vasculature until it reaches the stenosis. Once inflated, the balloon pushes the plaque back against the artery wall and enables blood flow to recommence after it has been deflated. The initial success of PTCA was short lived as investigators soon discovered that a substantial percentage of patients, reported to be anywhere between 30% and 60%, experienced recurrent ischemia due to the re-blocking of the artery (restenosis) within the first 6 months. This was attributed to mechanical injury caused by over dilating a device within the vessel (Head et al., 2007). The next major advance in the field of minimally invasive interventional cardiology came in the early 1990's with the advent of the coronary artery stent (CAS). Prior to surgery a cylindrical metallic scaffold, or stent, is placed on the end of an existing balloon catheter and deployed in the same way as traditional PTCA. Initially, these stents were mounted on the balloon catheter by the physician, however, in more recent times the manufacturers supply the catheters with a stent already in situ. After deployment the stent remains within the artery in an attempt to retain arterial patency. CAS reduced failure rates tobetween 10% and 40% (Duraiswamy et al., 2007; Mongrain et al., 2007) through the elimination of elastic recoil and negative remodelling of the artery associated with PTCA (Costa and Simon, 2005).

2007) at a cost of between \$4 and \$5 billion annually (Kaul et al., 2007).

considered the most important factor (Sun et al., 2006).

due to highly tortuous or extensively blocked arteries.

**2.1 Traditional interventional procedures** 

### **2.2 Restenosis and the advent of the drug eluting stent era**

Restenosis can best be described as an overly aggressive inflammatory healing response in the artery wall due to the mechanical injury inflicted by balloon/stent expansion. It can be quantified by the reduction of lumen size after an intravascular interventional procedure. The development of restenosis can be described by three processes after PTCA; 1) elastic recoil, 2) arterial negative remodelling and 3) neointimal hyperplasia (Rajagopal&Rockson, 2003). Elastic recoil can occur within an hour of PTCA and is due to passive recoil of the elastic medial layer of the artery. Arterial remodelling on the other hand can be both positive (vessel enlargement) or negative (vessel shrinking) and is characterised as such by a change in vascular dimension. Investigators report contrasting views on the mechanisms behind negative remodelling but whatever the underlying pathology behind vascular remodelling, it is believed to be virtually eliminated when angioplasty is used in conjunction with a stent.

Over inflation of a balloon catheter can result in the fracture of atherosclerotic plaque and in some cases can cause partial fracture of the artery wall (Schwartz et al., 2004). The same crushing/fracturing effect is witnessed when a stent is used in conjunction with an over inflated balloon. However, a stent can also cause excessive injury by penetrating the media which in turn increases neointimal formation. In some extreme cases stents have been known to penetrate as deep as the adventitial layer of the artery (Costa and Simon, 2005). The introduction of the DES to market has gone someway to alleviating the issue of arterial restenosis and excessive vessel injury via stent expansion. Variations of anti-restenotic drugs have been used to coat the stent in order to prevent post-operative in stent restenosis (ISR) and these modern stents can have strut profiles in the order of 80*µ*m which would minimise the possibility of adverse artery wall penetration.

It is generally accepted that one of the main causes of restenosis following BMS implantation is SMC proliferation from the medial artery layer to the injured site. Attempts at systemic drug delivery to inhibit restenosis after stenting failed because effective dosing levels had a toxic effect and could not be tolerated by the patients (Waksman, 2002). Therefore the concept of local drug delivery was developed to redress the issue through the application of a drug eluting coating to the stent platform. This enables site specific local delivery of drugs that can be applied to the injured vessel at the exact location and time that damage occurs. The anti-restenotic coating on DES inhibits the formation of neointimal hyperplasia via suppression of the inflammatory reaction, platelet activation and SMC proliferation, curbing the overly aggressive healing response. Most of the early drugs explored originally were those used as agents for anti-transplant rejection or immunosuppressive drugs (van der Hoeven et al., 2004).

In April 2003 the first DES to gain commercial approval from the Food and Drug Administration (FDA) in the United States was the Cypher stent, which was developed by Cordis Corporation (Miami, FL. USA). The drug used on the Cypher stent is called sirolimus. Boston Scientific's (Natik, MA. USA) TAXUS family of stents were the second DES platform approved by the FDA in March of the following year. The drug employed on the TAXUS stent is called paclitaxel (Venkatraman and Boey, 2007).The first generation DES had a profound effect on reducing restenosis rates compared to bare metal stent (BMS) models. Clinical trials carried out on the Cypher stent (SIRIUS-1) showed restenosis rates of 8.9% after 8 months compared to 36.6% for BMS in the same study. Likewise the TAXUS IV trials heralded a dramatic reduction in restenosis rates when compared to BMS after 9 months, 7.9% versus 26.6% respectively (Venkatraman and Boey, 2007).

A successful DES procedure is defined by its ability to transport the right amount of an appropriate drug within the correct timeframe that will ultimately deem the operation a success through the prevention of ISR. There are aspects of the performance of a DES that can be controlled by the manufacturer which has led to considerable reductions in the instances of ISR, such as the type of drug used and the characteristics of the coating that the drug is stored in. Even the stent shape, thickness and width of the struts can all influence the manner in which the drug is transported. However, the ability for drug uptake within the artery wall is also governed by its interaction with the patient specific arterial environment, making it near impossible to completely eradicate ISR. The degree of initial stenosis, the presence of luminal and abluminal thrombus on the stent and even the advent of reendothelialisation will all contribute to the DES ability to transport drugs throughout the artery wall.

### **3. Mass transport theory of drug eluting stents**

Mass transport refers to the movement of mass, i.e. the species of interest which is drugs in the case of a DES, within a defined system. This transport of species may be provoked by concentration gradients between two points, but quite often in systems, especially in the vasculature, overpowering complex flow dynamics will ultimately be responsible for the mass transport outcome. In the absence of a free flowing system the presence of these concentration gradients induces diffusion, e.g. between the DES and the artery wall. Mass transport can be broken up into two types within the human vasculature. Firstly blood side mass transport (BSMT) refers to species transport within the vessel lumen and is subject to the haemodynamics therein. Often evanescent due to haemodynamic washout, BSMT can only be effective in transporting anti-proliferative agents to the wall in regions of high recirculation.

The second, and most important, mode of mass transport is in relation to transport within the wall of the artery, referred to as wall side mass transport (WSMT). Along with the properties of the species being transported within the artery wall, WSMT depends on the structural condition of the wall itself, whereby a damaged intimal layer could facilitate accelerated mass transport through to the medial layer. WSMT can be governed by two transport forces, a pressure driven convective force and a diffusive force. The Peclet number (Pe), see equation 11, is a dimensionless parameter that can be used to determine the relative influences of these two forces. A small Pe (i.e. Pe<1) is representative of transport which is dominated by diffusion, while a higher Pe (i.e. Pe>1) indicates convection dominated mass transport (Friedman, 2008).

### **3.1 Governing equations**

Computational Fluid Dynamics has emerged as one of the most powerful numerical tools for engineers, scientists and mathematicians alike. Its foundations are based on theoretical analysis drawn from experimental observations over various branches of physics. The starting point for any computational analysis is the appropriate allocation of the governing equations. These equations are then substituted with equivalent numerical descriptions that are then solved using appropriate mathematical techniques. There are a number of numerical techniques available that will return a solution to a specified problem. Two of the more popular methods are the Finite Volume Method and the Finite Element Method. The assumptions generally applied when modelling fluid flow problems of this nature are as follows:


A successful DES procedure is defined by its ability to transport the right amount of an appropriate drug within the correct timeframe that will ultimately deem the operation a success through the prevention of ISR. There are aspects of the performance of a DES that can be controlled by the manufacturer which has led to considerable reductions in the instances of ISR, such as the type of drug used and the characteristics of the coating that the drug is stored in. Even the stent shape, thickness and width of the struts can all influence the manner in which the drug is transported. However, the ability for drug uptake within the artery wall is also governed by its interaction with the patient specific arterial environment, making it near impossible to completely eradicate ISR. The degree of initial stenosis, the presence of luminal and abluminal thrombus on the stent and even the advent of reendothelialisation will all contribute to the DES ability to transport drugs throughout the

Mass transport refers to the movement of mass, i.e. the species of interest which is drugs in the case of a DES, within a defined system. This transport of species may be provoked by concentration gradients between two points, but quite often in systems, especially in the vasculature, overpowering complex flow dynamics will ultimately be responsible for the mass transport outcome. In the absence of a free flowing system the presence of these concentration gradients induces diffusion, e.g. between the DES and the artery wall. Mass transport can be broken up into two types within the human vasculature. Firstly blood side mass transport (BSMT) refers to species transport within the vessel lumen and is subject to the haemodynamics therein. Often evanescent due to haemodynamic washout, BSMT can only be effective in transporting anti-proliferative agents to the wall in regions of high recirculation. The second, and most important, mode of mass transport is in relation to transport within the wall of the artery, referred to as wall side mass transport (WSMT). Along with the properties of the species being transported within the artery wall, WSMT depends on the structural condition of the wall itself, whereby a damaged intimal layer could facilitate accelerated mass transport through to the medial layer. WSMT can be governed by two transport forces, a pressure driven convective force and a diffusive force. The Peclet number (Pe), see equation 11, is a dimensionless parameter that can be used to determine the relative influences of these two forces. A small Pe (i.e. Pe<1) is representative of transport which is dominated by diffusion, while a higher Pe (i.e. Pe>1) indicates convection dominated mass

Computational Fluid Dynamics has emerged as one of the most powerful numerical tools for engineers, scientists and mathematicians alike. Its foundations are based on theoretical analysis drawn from experimental observations over various branches of physics. The starting point for any computational analysis is the appropriate allocation of the governing equations. These equations are then substituted with equivalent numerical descriptions that are then solved using appropriate mathematical techniques. There are a number of numerical techniques available that will return a solution to a specified problem. Two of the more popular methods are the Finite Volume Method and the Finite Element Method. The assumptions generally applied when modelling fluid flow problems of this nature are as

artery wall.

transport (Friedman, 2008).

**3.1 Governing equations** 

follows:

**3. Mass transport theory of drug eluting stents** 

### **3.1.1 Conservation of mass: Continuity equation**

The conservation of mass is a form of continuity equation which states the net mass flow into a control volume is equal to the rate at which mass leaves the control volume. That is providing there are no sinks or sources within the control volume. The differential form of the equation can be obtained by simply considering the flow into and out of elementary control volume. For the Cartesian co-ordinate system, having coordinates x, y, z referenced to a stationary frame with the corresponding velocity components u, v, w (m/s), the continuity equation can be written as:

$$\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0 \tag{1}$$

Where the density (*ρ*, kg/m3) is a constant, as is the case of incompressible flow, this reduces further to a volume continuity equation.

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \tag{2}$$

### **3.1.2 Balance of momentum: Navier-Stokes Equations**

The balance of momentum is derived from Newton's second law of motion, which states that the rate of change of momentum of a fluid particle is equal to the sum of the forces on the particle. The Navier-Stokes Equations describe the full three dimensional, viscous nature of fluid motion in a control system:

$$\begin{aligned} \rho \frac{\partial u}{\partial t} &= -\frac{\partial P}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \\\\ \rho \frac{\partial v}{\partial t} &= -\frac{\partial P}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) \\\\ \rho \frac{\partial w}{\partial t} &= -\frac{\partial P}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) \end{aligned} \tag{3}$$

Where *µ*(Pa s) is dynamic viscosity and *P*(Pa) is pressure. Due to the porous nature of the artery wall, flow within it must consider the influence of the tissues permeability. Therefore the flow within the wall is assumed to follow Darcy's law and is demonstrated (in the x direction) by the inclusion of the permeability term in equation 4, where *K*(m2) is the permeability of the arterial tissue.

$$
\rho \frac{\partial \mathbf{u}}{\partial t} = -\frac{\partial P}{\partial \mathbf{x}} + \mu \left( \frac{\partial^2 \mathbf{u}}{\partial \mathbf{x}^2} + \frac{\partial^2 \mathbf{u}}{\partial y^2} + \frac{\partial^2 \mathbf{u}}{\partial z^2} \right) - \left( \frac{\mu}{K} \right) \mathbf{u} \tag{4}
$$

### **3.1.3 Fick's laws of diffusion**

Species transport via diffusion is a process driven by concentration gradients between two locations. Fick's first law can be used to describe the diffusional flux (*Jx*, mol/m2s) of such species, shown in 1D in equation 5, where *D*(m2/s) is diffusivity and *c* is concentration (mol/m3):

$$J\_{\chi} = -D \frac{\partial \mathbf{c}}{\partial \mathbf{x}}\tag{5}$$

The negative term in equation 5 indicates that the flux is positive in the presence of a negative concentration gradient. Biological mass transport often requires the application of a time-dependent mass transport process that can predict variations in concentration over time. Fick's second law (equation 6) can provide such a relationship and is defined here in one dimension:

$$\frac{\partial \mathbf{c}}{\partial t} = D \frac{\partial^2 \mathbf{c}}{\partial \mathbf{x}^2} \tag{6}$$

### **3.1.4 Conservation of species: Convection-diffusion equation**

The addition of a convective term, equal to the product of the fluid velocity and the local concentration, to equation 6 demonstrates the 3D transport of species in a flowing solution. This is known as the convection-diffusion equation.

$$\frac{\partial \mathbf{c}}{\partial t} + u \frac{\partial \mathbf{c}}{\partial \mathbf{x}} + v \frac{\partial \mathbf{c}}{\partial y} + w \frac{\partial \mathbf{c}}{\partial z} = D \left( \frac{\partial^2 \mathbf{c}}{\partial \mathbf{x}^2} + \frac{\partial^2 \mathbf{c}}{\partial y^2} + \frac{\partial^2 \mathbf{c}}{\partial w^2} \right) \tag{7}$$

### **3.1.5 Ratio of convective to diffusive forces**

The Peclet number (Pe) is a dimensionless number that determines the relative contribution of convective and diffusive forces to species transport within a defined system. It can be defined as a product of the Reynolds number (Re) and the Schmidt number (Sc).

$$Pe = Re.Sc \tag{8}$$

The Reynolds number is a non-dimensional parameter concerning fluid forces due to viscosity and inertia and is essentially used to determine whether a flow is laminar, transitional or turbulent in nature. For example a Reynolds number of approximately 90 can be obtained for a mean arterial velocity (*u*) of 0.1m/s in an artery with a diameter (*a*) of 3mm. When considering transmural flow through the porous artery wall the value *a*would represent the thickness of the porous wall.

$$Re = \frac{\rho ua}{\mu} \tag{9}$$

The Schmidt number (Sc) is defined as the ratio of kinematic viscosity (*υ*, m2/s) to diffusivity (*D*).

$$Sc = \frac{v}{D} = \frac{\mu}{\rho D} \tag{10}$$

Substituting equations 9 and 10 into 8 describes how convective and diffusive forces can influence the outcome of the Peclet number.

$$Pe = \frac{\mu a}{D} \tag{11}$$

### **3.1.6 Diffusion in porous materials**

306 Biomedical Science, Engineering and Technology

Species transport via diffusion is a process driven by concentration gradients between two locations. Fick's first law can be used to describe the diffusional flux (*Jx*, mol/m2s) of such species, shown in 1D in equation 5, where *D*(m2/s) is diffusivity and *c* is concentration

The negative term in equation 5 indicates that the flux is positive in the presence of a negative concentration gradient. Biological mass transport often requires the application of a time-dependent mass transport process that can predict variations in concentration over time. Fick's second law (equation 6) can provide such a relationship and is defined here in

The addition of a convective term, equal to the product of the fluid velocity and the local concentration, to equation 6 demonstrates the 3D transport of species in a flowing solution.

> *ccc c ccc uvwD txy z xyw* ∂∂∂ ∂ ∂∂∂ ⎛ ⎞ + + + = ++ ⎜ ⎟ ∂∂∂ ∂ ∂∂∂ ⎝ ⎠

The Peclet number (Pe) is a dimensionless number that determines the relative contribution of convective and diffusive forces to species transport within a defined system. It can be defined as a product of the Reynolds number (Re) and the Schmidt

The Reynolds number is a non-dimensional parameter concerning fluid forces due to viscosity and inertia and is essentially used to determine whether a flow is laminar, transitional or turbulent in nature. For example a Reynolds number of approximately 90 can be obtained for a mean arterial velocity (*u*) of 0.1m/s in an artery with a diameter (*a*) of 3mm. When considering transmural flow through the porous artery wall the value *a*would

> *ua Re* ρ

The Schmidt number (Sc) is defined as the ratio of kinematic viscosity (*υ*, m2/s) to diffusivity

*<sup>v</sup> Sc*

μ

*D D* μ

ρ

*x*

2 2 *c c <sup>D</sup> t x* <sup>∂</sup> <sup>∂</sup> <sup>=</sup> <sup>∂</sup> <sup>∂</sup>

> 22 2 22 2

*Pe Re Sc* = . (8)

<sup>=</sup> (9)

= = (10)

<sup>∂</sup> = − <sup>∂</sup> (5)

(6)

(7)

*x <sup>c</sup> J D*

**3.1.4 Conservation of species: Convection-diffusion equation** 

This is known as the convection-diffusion equation.

**3.1.5 Ratio of convective to diffusive forces** 

represent the thickness of the porous wall.

**3.1.3 Fick's laws of diffusion** 

(mol/m3):

one dimension:

number (Sc).

(*D*).

When considering diffusion in a fluid saturated porous media, as is the case with the artery wall, diffusion takes place over a tortuous path. Because these pores are not straight, the distance over which diffusion takes place becomes effectively longer than for a homogenous material of the same thickness. The effective diffusivity (*Deff*) can therefore be deduced by considering the impact of the materials structure on the species free diffusivity (*Dfree*). The effective diffusivity of a porous material is a function of its porosity (*ε*) and tortuosity (*τ*).

$$D\_{\rm eff} = \frac{\mathcal{E}}{\pi} D\_{\rm free} \tag{12}$$

One of the more common ways to determine the free diffusivity of a species in a solvent is to use the Stokes-Einstein equation (13), where *k*(J/K) is the Boltzmann constant, *T*(K) is temperature and *R*(m) is the radius of the solute. For the purpose of diffusion in the artery wall, this solvent is considered to be plasma.

$$D\_{free} = \frac{kT}{6\pi\mu R} \tag{13}$$

The radius of the solute can be calculated from equation 14 assuming that the particle is spherical in shape, where *M*(Kg/mol) is the solute molecular weight and *Na*(mol-1) is Avogadro's number.

$$R = \left(\frac{\Im M}{4\pi\rho Na}\right)^{\aleph\_3} \tag{14}$$

The structure of the porous medium is defined by the tortuosity(*τ*) of its porous network (15) and by the porosity(*ε*) (16) of the material itself.

$$
\tau = \frac{L}{X} \tag{15}
$$

Where *L* = pore path length and *X* = distance between beginning and end of the pore path.

$$
\varepsilon = \frac{\text{Power Volume}}{\text{Total Volume}} \tag{16}
$$

### **4. Problem classification**

In reality the classification of problems of this nature are inherently patient specific and as such no one representation of the problem is correct. However, there are innate similarities between patients. Blood flow within the vasculature is a highly complex 3D process to model given the pulsatile nature of arterial haemodynamics. Coupled with this pulsatile process, the coronary arteries are situated on the surface of the heart and as such are subject to cyclic motion due to the beating of the organ. Therefore the modelling of drug transport from a DES in these arteries is multifaceted in nature, comprising of both luminal and artery wall mass transport, the latter of which may also be subject to a reaction giving that some drug may bind to the arterial tissue. The introduction of a multi-layered artery wall to the model increases the complexity of the domain even further. So to what extent does one go to when modelling DES mass transport?

Comprehensively modelling the behaviour of a DES computationally over a given time period would require the application of the following *in vivo* conditions experience by the device.


In light of the computational requirements to undertake such a model it is possible, and almost necessary, to make certain assumptions in order to simplify both BSMT and WSMT models whilst retaining enough detail of the actual model to draw relevant conclusions from the analysis.

The implementation of an arterial pulse and a beating heart are neglected by most researchers. Often the artery is modelled as rigid in space in order to analyse mass transport post DES deployment. This is an effective assumption but one must consider the deformation of the artery wall due to the dynamic expansion of the stent, as this can have an impact on the mass transport outcome due to the porous nature of the wall and the compression it incurs upon stent expansion. As for the application of laminar blood flow, it can be seen that the majority of drug that enters the artery wall from the DES does so via physical contact with the wall and the drugs emanating from the areas of the stent exposed to flow, be it laminar or pulsatile, are predominantly carried downstream.

### **4.1 Artery wall classification**

Arteries transport oxygen rich blood around the body providing essential nutrients to vital organs. The artery wall consists of a complex multilayer porous substructure with an interstitial phase comprising predominantly of plasma. In a healthy artery this substructure (Figure 1) is comprised of three concentric layers; the tunica intima, the tunica media and the tunica adventitia. The tunica intima is the innermost layer, consisting of a single layer of endothelial cells and a subendothelial layer mainly consisting of delicate connective tissues and collagen fibres. The outer boundary of the tunica intima is surrounded by an elastic tissue with fenestral pores known as the internal elastic lamina (IEL). The medial layer consists primarily of concentric sheets of smooth muscle cells (SMC) within a loose connective tissue framework. This configuration of SMC enables the artery wall to contract and relax. The tunica media and the tunica adventitia are separated by another thin band of elastic fibres known as the external elastic lamina (EEL). The outermost layer of the artery, the tunica adventitia, is comprised of connective tissue fibres and some capillaries. These

process, the coronary arteries are situated on the surface of the heart and as such are subject to cyclic motion due to the beating of the organ. Therefore the modelling of drug transport from a DES in these arteries is multifaceted in nature, comprising of both luminal and artery wall mass transport, the latter of which may also be subject to a reaction giving that some drug may bind to the arterial tissue. The introduction of a multi-layered artery wall to the model increases the complexity of the domain even further. So to what extent does one go to

Comprehensively modelling the behaviour of a DES computationally over a given time period would require the application of the following *in vivo* conditions experience by the device.

• The real time occurrence of thrombus formation and re-endothelialisation under

• The structural deformation of the artery wall due to DES deployment would need to be

• Also the movement of the vessel in space due to its location on the surface of the beating heart and how this may alter depending on the extent of the patient's physical

In light of the computational requirements to undertake such a model it is possible, and almost necessary, to make certain assumptions in order to simplify both BSMT and WSMT models whilst retaining enough detail of the actual model to draw relevant conclusions

The implementation of an arterial pulse and a beating heart are neglected by most researchers. Often the artery is modelled as rigid in space in order to analyse mass transport post DES deployment. This is an effective assumption but one must consider the deformation of the artery wall due to the dynamic expansion of the stent, as this can have an impact on the mass transport outcome due to the porous nature of the wall and the compression it incurs upon stent expansion. As for the application of laminar blood flow, it can be seen that the majority of drug that enters the artery wall from the DES does so via physical contact with the wall and the drugs emanating from the areas of the stent exposed

Arteries transport oxygen rich blood around the body providing essential nutrients to vital organs. The artery wall consists of a complex multilayer porous substructure with an interstitial phase comprising predominantly of plasma. In a healthy artery this substructure (Figure 1) is comprised of three concentric layers; the tunica intima, the tunica media and the tunica adventitia. The tunica intima is the innermost layer, consisting of a single layer of endothelial cells and a subendothelial layer mainly consisting of delicate connective tissues and collagen fibres. The outer boundary of the tunica intima is surrounded by an elastic tissue with fenestral pores known as the internal elastic lamina (IEL). The medial layer consists primarily of concentric sheets of smooth muscle cells (SMC) within a loose connective tissue framework. This configuration of SMC enables the artery wall to contract and relax. The tunica media and the tunica adventitia are separated by another thin band of elastic fibres known as the external elastic lamina (EEL). The outermost layer of the artery, the tunica adventitia, is comprised of connective tissue fibres and some capillaries. These

• The DES would have to be placed in multi-layered diseased artery.

to flow, be it laminar or pulsatile, are predominantly carried downstream.

• Both BSMT and WSMT would have to be considered.

activeness would have to be considered.

pulsating flow conditions would need to be modelled.

when modelling DES mass transport?

taken into account.

**4.1 Artery wall classification** 

from the analysis.

fibres blend into the surrounding connective tissues and aid in stabilising the arteries within the body (Khakpour and Vafai, 2007).

The target layer for the anti-restenotic drugs is the tunica media, where the SMC reside, and quite often computational studies will consider just this arterial layer not only because of this fact but also due to the possible erosion of the tunica intima upon stent deployment. Regardless of the level of complexity modelled, the artery wall is porous in composition and drug transport is facilitated through the surrounding plasma not only via diffusion but there is also the presence of a transmural velocity due to a pressure gradient observed across the artery wall. However, the presence of arterial plaque will reduce the magnitude of this transmural velocity and can even stem it altogether. As DES are deployed in highly occluded arteries it is reasonable to reduce the complexity of the problem by neglecting convection in the wall. Equation 12 gives us an indication of how arterial properties such as porosity, tortuosity and free diffusivity can influence the transport of drugs within the respective artery wall layers. The compression of these layers will alter these properties which in turn may inhibit the transport of species as governed by the mass transport equations. The compression of a porous structure not only reduces the materials porosity but it results in the creation of a more arduous pore path over which mass transport would normally occur. The combination of a reduced porosity with an increased tortuosity, when the artery wall has been compressed, has a net effect of reducing the effective diffusivity thus hindering mass transport within the vessel.

Fig. 1. Illustration of the cross-sectional structure of a healthy artery wall.

### **4.2 Influence of thrombus**

Hwang et al. (2005) were among the first to explore the influence of thrombus height, width and type on the arterial drug uptake. Stents can be deployed at sites of thrombus and after implantation a clot will inevitably develop once the struts become covered with plasma proteins. In most cases this will not be angiographically present or clinically evident but even a fine layer of clotting blood deposited on the surface of the DES can alter drug distribution within the wall. The presence of clot alters the local environment of the stent strut and the physiological transport forces that regulate arterial uptake and retention. Balakrishnan et al.(2008) reported that drug eluting stents clot at a rate of 0.6% each year after implantation for up to 3 years. Strut position within a clot also has a major influence on the arterial uptake. The greater the volume of clot covering the strut, and the closer the strut is to the wall, creates improved conditions for greater drug delivery. Hwang et al. (2005) discovered that in this configuration concentration distribution in the wall can be 30 fold higher than situations where no clot is present. Similarly thrombus or plaque between the strut and the artery wall act as a buffer layer and reduce wall concentrations. Clot diffusivities higher than that of the artery wall will result in drug transfer to the blood at a rate faster than can be absorbed by the wall. Clots with diffusivities equal to or lower than the artery wall can transport drugs to the wall at a rate where the wall can effectively absorb the drugs, thus reducing drug loss to the bloodstream.

Fig. 2. An illustration of how Focal (A) and Diffuse (B) thrombus, surrounding a DES strut, contribute to drug mass transport within the artery wall.

In 2008, Balakrishnan et al. demonstrated how variations in thrombus size and distribution contribute to fluctuating arterial drug intensities (Figure 2). Their simulations indicate that thrombus cannot influence the slow rate of drug release from the stent because the polymer resistance to drug transport is significantly greater than that of the thrombus, which is consistent with *in vivo* experimental drug release. Focal thrombus, with a thickness of 0.1mm, increased peak average drug concentration by 80%. Greater clot formation between stent struts will also have an effect on arterial concentrations because once the species is transported within the clot it effectively increases the surface area from which the artery wall can absorb drugs. The formation of this interstrut thrombus, described as diffuse thrombus, acts as a shield from drug washout and culminates in an increase in arterial concentration by up to 3.5-fold.

The variability of thrombus can have a major impact on arterial drug concentrations. It can aid in drug uptake and retention within the wall when it covers the stent, but too much thrombus will effectively block the artery, thus creating a problem that the DES aimed to alleviate. Also it can act as a barrier in preventing drugs from reaching the wall when it is located between the strut and the wall. The likely scenario following DES implantation within the vasculature is that at some location along the length of the stent each of these situations will be present. In addition to issues arising from the development of thrombus, the presence of plaque in a freshly stented artery will have a similar effect on mass transport but does so immediately upon implantation, as opposed to thrombus which develops with time. The initial presence of the plaque coupled with the time dependent formation of thrombus, and even the occurrence of re-endothelialisation, would create a realistic stenting scenario if modelled but the volume and formation of these will be different for each person. So to what extent does one model this?

### **4.3 Polymer and drug characterisation**

310 Biomedical Science, Engineering and Technology

implantation a clot will inevitably develop once the struts become covered with plasma proteins. In most cases this will not be angiographically present or clinically evident but even a fine layer of clotting blood deposited on the surface of the DES can alter drug distribution within the wall. The presence of clot alters the local environment of the stent strut and the physiological transport forces that regulate arterial uptake and retention. Balakrishnan et al.(2008) reported that drug eluting stents clot at a rate of 0.6% each year after implantation for up to 3 years. Strut position within a clot also has a major influence on the arterial uptake. The greater the volume of clot covering the strut, and the closer the strut is to the wall, creates improved conditions for greater drug delivery. Hwang et al. (2005) discovered that in this configuration concentration distribution in the wall can be 30 fold higher than situations where no clot is present. Similarly thrombus or plaque between the strut and the artery wall act as a buffer layer and reduce wall concentrations. Clot diffusivities higher than that of the artery wall will result in drug transfer to the blood at a rate faster than can be absorbed by the wall. Clots with diffusivities equal to or lower than the artery wall can transport drugs to the wall at a rate where the wall can effectively absorb

Fig. 2. An illustration of how Focal (A) and Diffuse (B) thrombus, surrounding a DES strut,

In 2008, Balakrishnan et al. demonstrated how variations in thrombus size and distribution contribute to fluctuating arterial drug intensities (Figure 2). Their simulations indicate that thrombus cannot influence the slow rate of drug release from the stent because the polymer resistance to drug transport is significantly greater than that of the thrombus, which is consistent with *in vivo* experimental drug release. Focal thrombus, with a thickness of 0.1mm, increased peak average drug concentration by 80%. Greater clot formation between stent struts will also have an effect on arterial concentrations because once the species is transported within the clot it effectively increases the surface area from which the artery wall can absorb drugs. The formation of this interstrut thrombus, described as diffuse thrombus, acts as a shield from drug washout and culminates in an increase in arterial

the drugs, thus reducing drug loss to the bloodstream.

contribute to drug mass transport within the artery wall.

concentration by up to 3.5-fold.

Stent coatings play a vital role in the regulation of drug release from the stent. Careful consideration must be observed when allocating a polymer as a drug carrier for a DES. If the polymer is not biocompatible an inflammatory response ensues. It has been shown that the development of neointimal hyperplasia can be doubled when certain polymer configurations are used compared with a controlled substance (Granada et al., 2003). Each biologically viable polymer must be able to endure the stresses exerted when stents are deployed, resist cracking, peeling and maintain their physiochemical properties after sterilisation. Drug release rates can be altered with the addition of an extra layer of polymer coating to modulate between slow and fast release formulations. This adds an extra degree of complexity when designing functional DES and indeed analysing them computationally.

Sirolimus, everolimus and paclitaxel were among the first anti-restenotic agents used in DES. Sirolimus, or rapamycin, is a naturally occurring macrocyclic lactone. It was approved by the FDA in 1999 for the prophylaxis of renal transplant rejection because of its potent immunosuppressive properties (Costa & Simon, 2005). Essentially sirolimus inhibits the activation of multiple kinases, associated with cell proliferation, resulting in the cessation of cellular division between the G1/S phases and is said to be cytostatic in nature (Burt & Hunter, 2006). Everolimus is an analogue of the sirolimus immunosuppressant which binds to cytosolic immunophillin FKBP12. It is very similar to sirolimus in that it prevents the cellular division between the G1/S phases of the cell cycle therefore inhibiting SMC proliferation (Stone et al., 2008).

Paclitaxel is a naturally occurring drug that was originally extracted from the pacific yew tree Taxusbrevifolia. It was initially used to treat several types of cancer such as breast and ovarian cancer (Tanabe et al., 2004). Like sirolimus and everolimus, paclitaxel is effective in reducing restenosis but it does so in a manner that results in cell death, suggesting that it works through a cytotoxic mechanism (Parry et al., 2005). Paclitaxel achieves its antiproliferative effects by binding to aminoterminus β-tubulin thus disrupting microtubule dynamics. This results in the arrest of cells at the M stage and even G2 stages of the cell cycle, leading to cell death (Hara et al., 2006). Due to its lipophilic properties paclitaxel has been known to be loaded directly onto stents without a polymer coating but there could be potential implications with regards to a lack of controlled release (Burt & Hunter, 2006).
