Some design considerations for polymer-free drug-eluting stents: a mathematical approach

In this paper we provide the ﬁrst model of drug elution from polymer-free arterial drug-eluting stents. The generalised model is capable of predicting the drug release from a number of polymer-free systems including those that exhibit nanoporous, nanotubular and smooth surfaces. We derive analytical solutions which allow us to easily determine the important parameters that control the drug release. Drug release proﬁles are provided, and we oﬀer design recommendations so that the release proﬁle may be tailored to achieve the desired outcome. The models presented here are not speciﬁc to drug-eluting stents and may also be applied to other biomedical implants that use nanoporous surfaces to release a drug.


Background
Drug-eluting stents (DES) have significantly improved the treatment of coronary heart disease (CHD) and are the current gold standard in percutaneous coronary interventions (PCIs).These small drug-containing mesh-like devices are now routinely inserted into arteries which have become dangerously narrowed due to a condition known as atherosclerosis.Their role is to increase the diameter of the diseased lumen, so that adequate blood flow can be restored.Their predecessor, the so-called bare metal stents, while revolutionary at the time, were soon found to be inadequate due to the common occurrence of restenosis (the re-narrowing of the lumen).Subsequent stent designs included an antiproliferative drug designed to prevent smooth muscle cell proliferation and migration which is thought to contribute to restenosis: these are the drug-eluting stents [1].The drug was typically contained within a polymer coating on the surface of the metal stent.To date there have been several generations of DES, each with design features aimed at improving clinical results.These include multi-layer polymer coatings to help control the release, thinner struts to reduce damage to the arterial tissue and more biocompatible polymer coatings and metal alloys [2].However, several studies have raised concerns that the permanent presence of a polymer may trigger an allergic reaction and possibly a local vascular inflammatory response in some patients [3], [4].Moreover, several early studies reported evidence of delayed healing of the endothelial cell layer of the arterial wall following DES use, in comparison to bare metal stents [5], [6].These unwanted effects have been associated with the occurrence of late stent thrombosis and sudden cardiac death.With this in mind, cardiologists have recommended that anti-platelet therapy is continued for a full twelve months after stent implantation [1].Driven by a desire to improve clinical outcomes, newer generation DES have focussed on biodegradable polymers, where the polymer carries and controls the drug release   [20].A nanoporous drug-infused layer covering a steel strut acts as a reservoir for the prolonged release of drug.The pore diameters here are of the order of 10 nm.(b) A schematic representation of part of a ceramic-coated tacrolimus-eluting stent.The pore diameters here are again of the order of 10 nm.Nanotubular systems of this kind have also been investigated in the context of orthopedic and dental drug releasing implants.(c) An image of the drug-coated surface of an Amazonia Pax stent [21].In this polymer-free system, a layer of semi-crystalline paclitaxel of thickness ≈ 5 µm covers the the stent strut surface.
To date, several polymer-free stents have been designed and some of them have reached the market.There have, however, been many challenges for the stent manufacturers.For example, they have had to address how the drug release can be controlled with no polymer, how the drug can be adequately adhered to the stent surface and have had to consider carefully the stent platform material to ensure that it is biocompatible.The stent manufacturers have adopted different approaches in designing these stents, which can be roughly separated into four categories [20]: macroporous, microporous, nanoporous and smooth surface.Macroporous DES utilize precise manufacturing processes to accurately inlay the drug into holes or slits in the body of the stent.Some examples include the Janus Flex (Sorin Group), Conor Stent (Conor Medsystems), CoStar (Conor Medsystems), and Nevo (Cordis) ( [20], [22]).Microporous polymer-free DES contain a modified surface of pits and holes whose size is of the order of microns.The drug is then coated directly on the rough surface, resulting in the micropores being filled and a nominal layer of drug on top of the surface.The purpose of the micropores is to act as a reservoir for the drug and also to aid adhesion to the stent surface.The rough surface may be created by, for example, a sandblasting technique (Yukon stent, Translumina), or by a microabrasion process (BioFreedom stent, Biosensors Inc.) [23], [22].The VESTAsync stent (MIV Therapeutics) uses a hydroxyapatite surface coating [22].Hydroxyapatite is an organic porous material that makes up bone mineral and the matrix of teeth and is widely used as a bone substitute.Nanoporous DES are distinguishable from microporous DES by the nature and size of their pores.They exhibit a bulk porous layer (cf. a surface porous layer) and the pores are of the order of nanometres (cf.microns).This layer may be obtained electrochemically (Ceramic-coated TES, Jomed International) or through sputter coating techniques (Setagon stent, Setagon Inc).These stents have the advantage of allowing for a higher drug loading capacity.The Optima stent (CID) contains nanopores too, but the pores are arranged in a regular slotted tubular fashion [23].Nanoporous drug-eluting coatings have also been investigated for other implantable biomedical devices, notably for orthopedic and dental implants ( [24], [25], [26], [27]).Figure 1(a) displays a nanoporous polymer-free stent while 1(b) displays a nanotubular polymer-free stent.Perhaps the most simple polymer-free design is where the drug is coated directly onto the unmodified (relatively) smooth surface of the metal stent.An example of this type of polymer-free DES is the Amazonia Pax (MINVASYS) [22] where a semi-crystalline paclitaxel coating is applied directly to the chromium cobalt stent (see Figure 1(c)).With no polymer or pores to control the release, it appears that the release rate is determined solely by the solubility and diffusion coefficient of the drug in the release medium and by the thickness of the coating.

Outline
In this paper, rather than focusing on one or more commercially available stents, we provide a generalised model to describe the elution of drug from a range of polymer-free DES.Since the majority of macroporous stents and several microporous stents now actually include a polymer-drug formulation within the pores, we neglect these classes of DES in the analysis that follows.We start by writing down a model for drug release from a nanoporous system.By making some reasonable assumptions, we derive an analytical solution to the model which allows for immediate calculation of the drug release profile.We then proceed to demonstrate how slotted tubular nanoporous DES and smooth surface DES can be shown to be special cases of the generalised model.We provide a model for the two-stage release of drug from a stent which has a pure drug layer on the surface and a nanoporous drug layer below the surface.Finally, we determine the important parameters in the system that govern the drug release, and provide design recommendations so that the release profile may be tailored to achieve the desired outcome.We wish to stress that the models presented here are for drug release in an in-vitro environment.While the in-vivo release profile will undoubtedly be different (due to a multitude of factors including pulsatile flowing blood, drug binding to proteins and tissue and wound healing), we believe that it is essential to try to understand the release of drug in a controlled in-vitro environment before embarking on the complex in-vivo case.Furthermore, for many cases, in-vitro drug release profiles have been shown to be good predictors of in-vivo profiles.In addition to this, stent manufactures routinely test the release of drug from their stents in an in-vitro environment since it allows them to compare the release profile between different designs [28].

The general model for an unstirred release medium
We now formulate a mathematical model for a nanoporous DES system, which includes smooth surface DES and slotted nanotubular DES as special cases.Figure 2 schematically depicts the nanoporous system, as well as the smooth surface and nanotubular subcases.We suppose that the system is one-dimensional, and denote by x the spatial variable as shown in Figure 3.We denote by c p (x, t) and c w (x, t) the concentration of drug in the liquid-filled pores and in the aqueous medium, respectively, where t is the time.The drug may become bound in a region close to the walls of the pores, and we denote by c b (x, t) the concentration of immobile drug in this region.The volume fraction of this region is denoted by φ s , and the volume fraction of the pores (the porosity) is denoted by φ.

Stent strut
Solid drug The drug concentration in the pores is assumed to be governed by where D e is the effective diffusion coefficient for the drug in the porous medium (see below), and k a , k d are the rate constants for drug absorption and desorption, respectively.The moving boundary x = s(t) locates the interface between the undissolved drug and the aqueous release medium, and will be determined as part of the solution to an initial boundary value problem; see Figure 3.If L d is the initial thickness of the drug layer through the porous medium, then c p (x, t) = c 0 for −L d < x < s(t), t > 0, where c 0 is the constant concentration of the undissolved drug.
The effective diffusivity D e in (1) incorporates a number of effects that can arise for diffusion in porous media.If D w is the free aqueous diffusion coefficient for the drug, then where φ t is the effective porosity of the medium and τ is its tortuosity [29].Both of these parameters are dimensionless.The effective porosity φ t refers to the porosity that can contribute to solute transport through the medium, and this can be smaller than the overall porosity φ.
For example, φ t < φ if the medium contains small pores that the solute cannot access.The tortuosity τ takes account of the fact that the particles may have to travel through an increased path length due to the circuitous nature of the pores.Tortuosities usually range in value between two and six, and have an average value of about three [29].Tortuosity provides another degree of freedom to modulate the drug release rate.It is clear that the nanotubular system depicted in Figure 2(b) will have a smaller tortuosity than a system for which the pores are randomly oriented (see Figure 2(a)).
The concentration of bound drug in the region close to the pore walls, c b (x, t), satisfies Adding ( 1) to (3) gives the following evolution equation for the total drug in the porous medium Assuming that the desorption and absorption rates are fast compared to the diffusion rate, we can replace (3) by the equilibrium expression In the current study, all of the analytical results presented will be for this case.Numerical solutions based on a front-tracking finite difference scheme have been calculated for the nonequilibrium case, and show excellent agreement with the analytical results for those cases where diffusion is the slowest process.Substituting ( 5) into (4) gives where is referred to as the apparent diffusion coefficient, to distinguish it from the effective diffusion coefficient.This single parameter takes account of the effects of porosity, absorption, desorption, tortuosity and constrictivity.The concentration of drug in the release medium is assumed to be governed by the diffusion equation Equation ( 8) does not contain a convection term because we are considering the case of an unstirred fluid here.It is also noteworthy here that the release medium has been taken to be infinite.This is a reasonable assumption given that a typical lengthscale for the release medium is in the centimetre range while a typical thickness for the drug-infused porous layer is at least two orders of magnitude smaller.
The case of a well-stirred release medium will also be considered and will be modelled by imposing an infinite sink boundary condition for the drug at x = 0; see below.While a consensus panel of experts [30] recommend conducting release experiments under infinite sink conditions, it has also been noted by Seidlitz et al. [31] that sink conditions "do not necessarily exist at a particular in-vivo site."Consequently, we have chosen in the current study to consider both the unstirred and infinite sink situations.It may be argued that the consideration of both unstirred and infinite sink conditions corresponds to a treatment of the worst-and best-case scenarios, respectively, from the point of view of speed of drug dissolution.In view of this, the consideration of the two cases is a worthwhile exercise given the uncertainty associated with the in-vivo situation.

Boundary and initial conditions for an unstirred release medium
We now supplement the governing equations with appropriate boundary and initial conditions.The initial conditions are chosen to be Since c p has been taken to measure the concentration of drug in the fluid fraction of the porous medium, we impose continuity in drug concentration at the interface between the porous medium and the release medium, so that At the interface between the dissolved and undissolved drug, x = s(t), we impose where c s is the solubility of the drug in aqueous medium.The solubility gives the maximum concentration of drug that may be dissolved in the medium.We also require conditions for the drug flux on x = 0 and x = s(t).These conditions are motivated using the assumption that the total amount of drug in the system is conserved.If A is the planar surface area of the porous medium, then the total amount of drug in the system at time t is Imposing dm(t)/dt = 0 then leads to where we have imposed ∂c w /∂x → 0 as x → ∞.Inspecting this last equation leads to the following choice for the boundary conditions and This last equation gives a so-called Stefan condition to determine the motion of the moving boundary.The final boundary condition is Parameter Range considered Default value References Table 1: Values for the parameters appearing in the mathematical model of nanoporous drug release.

Boundary and initial conditions for a well-stirred release medium
For a well-stirred release medium, we assume a very low concentration for the drug in the medium, and replace equation (10) by so that x = 0 acts as a perfect sink for dissolved drug in the liquid-filled pores.The boundary and initial conditions ( 11) and ( 9) remain unchanged.It should be emphasised here that the well-stirred case is of particular importance because many release experiments are conducted under well-stirred conditions.Perfect sink boundary conditions can also be appropriate if the release medium is regularly replaced in the experiments.

Parameter values
Given that nanoporous DES are a relatively new technology which have yet to be refined, we consider a range of parameter values (see Table 1) rather than trying to model an existing system.In any case, a complete data set is not readily available for any of the currently available systems.We choose our range of values of L d based on the strut thickness of currently available stents and the thickness of typical DES polymer coatings.The drug diffusion coefficients are representative of the aqueous diffusivities of molecules of similar size to those coated on DES.Since these molecules are known to be poorly soluble we allow for high ratios of c 0 /c s .There would appear that no data exists in the literature on the drug absorption and desorption rates so we consider a range of values for this parameter spanning four orders of magnitude.Finally we choose values of φ, φ t and φ s such that 0 < φ s < φ t ≤ φ < 1.

General solution for the unstirred case
For convenience, we gather together here the equations constituting the initial boundary value problem for the unstirred case This problem is self-similar in the Boltzmann variable x/ √ t, and we may write where θ is a constant, to obtain Integrating the equations for F and G twice gives where A, B, C, D are constants whose values are fixed using the boundary conditions in (18).A straightforward calculation shows that where θ is determined by solving In the original variables, this solution is given by Figure 4 displays normalized concentration profiles of drug within the pores and in the release medium, as calculated from equations (21).We denote by t d the time it takes for the moving boundary to reach the bottom of the porous layer, so that s(t d ) = −L d , and This quantity gives a sensible measure for the drug release lifetime of the device.From (20), it is clear that s(t) has the functional form for some function H, and then it follows that t d has the structure where K ≡ 1/H 2 .We denote by M (t) the amount of drug that has dissolved in the drug-filled Figure 4: Normalized concentration profiles of drug in liquid-filled pores, cp/cs (x < 0), and in the release medium, cw/cs (x > 0) at four different times.The plot on the right is simply a magnification of the release medium drug concentration profiles that are displayed in the plot on the left.The parameter values used in the generation of this plot are displayed in Table 1.
pores by time t.This is simply the volume of the liquid-filled pores multiplied by c 0 , so that and where A again denotes the planar surface area of the porous medium.The fraction M (t)/M (∞) gives the ratio of the drug that was originally in the pores that has dissolved by time t, and is given by In the current study, plots of M % (= 100 × M (t)/M (∞)) versus t will be referred to as release profiles.Technically, t d is the time to dissolution and thus at time t d some drug will still be contained within the pores (both bound and unbound).Thus M % is likely to underestimate what we usually mean by release profile: the percentage of drug that has left the device and has entered the release medium.The drug release rate is defined to be the rate of change of M (t)/M (∞), and is given here by We now highlight some notable special cases and extensions of the above solution.

A nanotubular system
The nanotubular system depicted in Figure 1(b) is modelled by simply letting τ → 1 in the preceding results, so that D e → φ t D w and D a →

A smooth surface system
Here we consider the case depicted in Figure 2(c), in which a layer of pure drug of thickness L d overlays a stent strut with a smooth surface.The solution for this case can be extracted from the solution displayed in Section 3.1 by letting φ, φ t → 1, k a /k d → 0, to obtain (c p becomes c w here) A comparison between the cumulative percentage of drug released (M % ) for a nanoporous, nanotubular and smooth surface system.We observe that a nanotubular system results in quicker release of drug than a nanoporous system due to less tortuous pores.The smooth surface system results in a dramatic increase in the rate of drug release, since there are no pores to hinder the release of drug.Two additional cases have been added to demonstrate one way in which the nanoporous system can be tuned to achieve quicker release profiles.Nanoporous* and Nanoporous** have the reduced porosities φ = φt = 0.3 and φ = φt = 0.1, respectively.The default parameter values used in the generation of this plot are displayed in Table 1.
where θ is determined by solving Figure 5 displays a comparison between the release profiles of nanoporous, nanotubular and smooth surface systems.

The low solubility limit, c s /c 0 ≪ 1
We now consider the behaviour of the solution displayed in Section 3.1 in the limit of low drug solubility in the aqueous medium, c s /c 0 ≪ 1.This case is important because two of the most common drugs that have been used on DES, paclitaxel and sirolimus, are known to be very poorly soluble in water [35].For c s /c 0 ≪ 1, it follows from (20) that θ/ √ D a ≪ 1, and making explicit the dependence of the dissolution rate on the porosity, binding properties, and drug solubility.It follows that It is noteworthy here that the time it takes for the dissolution front to reach the bottom of the drug layer, t d , is proportional to the square of the large parameter c 0 /c s , so that the behaviour is strongly dependent on the solubility.This is clearly seen by the large variation in drug release profiles when c 0 /c s is increased from 2 to 20 in Figure 6.

Two stage release
In this section we consider the case of drug release from a system where the release is in two distinct stages.The initial phase is release from a pure drug layer on the surface of the stent, as Figure 6: In this plot we investigate the effect on the cumulative percentage of drug released (M % ) with increasing ratio of initial drug concentration to drug solubility, c0/cs, for the nanoporous system.We observe that this parameter has a dramatic influence on the rate of release of drug.Increasing this parameter (by either increasing the initial drug concentration or reducing the solubility of the drug in the release medium) can result in a prolonged duration of release.The implication is that if release is to be maintained then drugs which exhibit a low solubility in the release medium are to be preferred.This is supported by clinical data, which shows that DES (almost) exclusively use lipophilic compounds (limited water solubility).The default parameter values used in the generation of this plot are displayed in Table 1.

Stent strut
Drug in porous layer

Pure drug layer
Figure 7: The system discussed in Section 3.1.4.A pure layer of drug overlays a drug-infused nanoporous medium.This system is capable of exhibiting a two stage release behaviour since the drug release from the pure drug layer is more rapid than that from the porous medium.The system is shown here in its initial configuration.
described by the equations presented in Section 3.1.2.When all of the surface coated drug has dissolved, drug stored within the nanopores is then released.Here we denote by L p the initial thickness of the pure drug layer and by L d the total thickness of the drug layer and the porous medium (see Figure 7).
This system does not correspond to a special case of the solution (21), and must be considered separately.The time for the pure drug layer to dissolve is given by t p , where t p = L 2 p /θ 2 and θ is determined by solving (26).For 0 < t < t p , the pure drug dissolution phase, the solution is as described in Section 3.1.2.For t > t p , when the drug is released from the nanopores, the governing equations are  This problem does not admit a similarity reduction, and is not solved here analytically.However, analytical progress can be made by considering the asymptotic limit D a /D w → 0, with k a /k d and all of the other dimensionless parameters being held O( 1).This corresponds to the case of slow drug diffusion in the porous medium compared to that in the release medium.We omit almost all of the asymptotic details here, confining ourselves instead to quoting the result that is of most interest from the point of view of applications.Writing ε = D a /D w and taking the limit ε → 0, it is found that the total amount of drug released by time t is given by where t d is the dissolution time, θ = θ/ √ D w , and ρ satisfies As would be expected, equation (29) predicts a two stage release rate with relatively rapid release for t < t p when the pure drug is dissolving, and slow for t p < t < t d when the drug is releasing from the porous medium.At leading order, the solution for t > t p has the same form as the solution for release into a well-stirred medium, and this case is considered in the next section.
The problem (28) has also been solved numerically using a front-tracking method, and a description of the numerical method used can be found in the supplementary material.In Figure 8, we display the cumulative fraction of drug released from a two-stage release system for various values of D a corresponding to varying the ratio k a /k d .

General solution for the well-stirred case
For the case of a well-stirred release medium, the perfect sink condition ( 16) replaces ( 10), and we need only solve for c p (x, t) and s(t).The solution has the similarity structure c p = c p (x/ √ t), s(t) = −θ √ t, and we find that Figure 9: A comparison between the cumulative percentage of drug released (M % ) from the nanoporous system in unstirred and well-stirred release media.Also on display are the approximate solutions for the case of poorly soluble drugs.In each case we have cs/c0 = 0.1 < 1.We observe that drug release is significantly sped up in the well-stirred release medium, as one would expect.Furthermore, the respective approximate solutions for poorly soluble drugs are providing good agreement with the full solutions, even for cs/c0 = 0.1.The agreement improves as cs/c0 is reduced further.The default parameter values used in the generation of this plot are displayed in Table 1.
where θ is determined from For the case of poorly soluble drugs, c s /c 0 ≪ 1, (31) gives that θ ≈ 2D a (c s /c 0 ), so that s(t) ≈ − 2D a (c s /c 0 )t and It is instructive to compare this result with the corresponding expression in (27) for the unstirred case.Note that t d in (27) depends on the square of the large parameter c 0 /c s , while (32) depends only on its first power.Hence, for poorly soluble drugs, dissolution is an order of magnitude faster in a well-stirred medium than for an unstirred medium.Of course, faster drug dissolution in the well-stirred medium conforms to our intuitive expectations.In Figure 9 we display a comparison between the release profiles of drug released from a nanoporous system into unstirred and well-stirred release media.The solution (30), (31) cannot be used to describe the dissolution of a pure drug layer in a well-stirred release medium since the boundary condition c p = 0 on x = 0 is not appropriate for this case -there is now no porous medium to maintain the validity of the perfect sink boundary condition at x = 0. Drug dissolution in a well-stirred release medium has been the subject of numerous previous studies, and was recently reviewed by Siepmann & Siepmann [36].We now use these well-established ideas to model this case.

A pure drug layer in a well-stirred release medium
The system is as depicted in Figure 2(c).The moving boundary x = s(t), initially located at x = 0, separates the release medium from the pure drug layer, and the initial thickness of the drug layer is L d .For this case, we shall take the release medium to be finite in extent, and to initially occupy 0 < x < L w .When the medium is well-stirred, a boundary layer of poorly stirred fluid forms close to the surface of the dissolving drug layer.This layer is taken to be of thickness h, with the size of h depending on the degree of agitation in the fluid bulk.This is admittedly a somewhat crude characterisation of the behaviour, but it is commonly used, and does lead to a useable theory that gives satisfactory agreement with experimental results ( [36]).In the current analysis, we take h ≪ L d ≪ L w , and denote by c T (t) the uniform drug concentration in the well-stirred medium at time t.In Figure 10, we schematically depict the The special case of a pure drug layer in a well-stirred release medium.This case cannot be described by the solution (30), (31) and must be considered separately.A narrow boundary layer of poorly stirred fluid forms near the surface of the solid drug, and it is assumed that the flux of drug from the dissolving surface is proportional to the difference between the drug concentration at the surface and the drug concentration in the well-stirred release medium.
various regions and drug concentrations arising.
The key assumption of the Noyes-Whitney and the Nernst-Brunner dissolution models ( [37], [38]) is that the flux of drug from the surface of the dissolving drug layer is proportional to the difference between the concentration of drug in the release medium and the concentration of drug at the surface.Denoting by j| x=s(t) the flux of drug from x = s(t), we write where c s , the solubility, is the drug concentration on x = s(t), and D w is the diffusivity of the drug in the release medium.Equation ( 33) is a statement of Fick's first law, with the right hand side of (33) playing the role of −D w ∂c/∂x in the more familiar statement of the law.In view of ( 14), the appropriate equation for the speed of the front is now Equating the amount of drug that has dissolved with the amount in the release medium, and using the fact that h ≪ L d ≪ L w , we have Combining (34) and ( 35) leads to which is solved subject to s(0) = 0 to give where t ′ = hL w D w (1 − c s /c 0 ) /D w determines the time scale for dissolution.It follows that For t ≪ t ′ , we have s(t) ≈ −c s L w t/(c 0 t ′ ), which clearly distinguishes this case from the t 1/2 behaviour of the unstirred case.
The time for the drug to dissolve, t d , is determined from s(t d ) = −L d , which gives and this quantity is clearly only defined for L d c 0 < L w c s .This is as expected since for L d c 0 > L w c s there is sufficient drug to completely saturate the release medium, and the drug cannot then fully dissolve.For L d c 0 < L w c s , the release profile is given by

Sensitivity analysis of the design parameters
The models we have presented include several parameters which may in principle be altered during the manufacturing of the device.It is therefore of interest to consider the sensitivity of the release profiles to changes in these parameters.Nanoporous systems appear to have the best potential for controlling the drug release since they exhibit the highest number of tunable parameters: porosity, absorption, desorption, tortuosity, constrictivity, thickness of the porous layer, drug diffusivity in the release medium and the ratio of drug solubility to initial drug concentration.Nanotubular systems have tortuosity set to 1 and therefore provide faster drug release.Smooth surface systems result in significantly quicker drug release with the rate of release being controlled purely by the thickness of the drug layer, the ratio of drug solubility to initial drug concentration and the diffusivity of the drug in the release medium.In all of these systems, the thickness of the drug layer L d is an important parameter since from (22) it is clear that the release time varies as the square of L d .Thus, in all cases, increasing the value of L d will result in an increase in the duration of the drug release.Taking the example of the nanoporous one-layer system, if we reduce L d from 10 −4 m to 5 × 10 −5 m and then 10 −5 m (using the default parameter values in Table 1), the release time decreases significantly from 3.6 days to 21.4 hours to 51.5 minutes, respectively.Of course, there will undoubtedly be constraints on the value that L d can take, both from the manufacturing side and also from the physiological viewpoint.
From equation (7) we observe that the single parameter D a takes account of the effects of porosity, absorption, desorption, tortuosity and constrictivity and thus it is of key importance in determining the speed of drug release.Depending on the values of τ , k a /k d , φ t , and φ, the parameter D a can vary by several orders of magnitude and as a consequence the release time can be significantly delayed.In Table 2 we present calculated release times when these parameters are varied.The parameter k a /k d not only has the largest impact on the value of D a (spanning two orders of magnitude for the values considered), but also has the greatest influence on the release times which increase from 79 minutes to 274 days when k a /k d is increased from 0.1 to 100.The parameter τ has less of an effect with release times varying by only half a day when τ is increased from 1 to 6.The effect of reducing the ratio φ t /φ by reducing φ t or increasing φ is to increase the release time since in the first case it is more difficult for drug to pass through the pores, and in the second we are effectively increasing the initial mass of drug in the porous layer and so it takes longer to dissolve.In reality, it is likely that the drug coating process will result in a thin layer of drug on the surface of the stent in addition to the drug that is contained within the porous region.If this layer is sufficiently thin then it will have little effect on the release time since the surface layer will be dissolved rapidly.But if the layer is of a non-negligible thickness then two distinct phases of release are observed.The first is a fast phase corresponding to pure drug dissolving on the surface and the second is a slow phase corresponding to drug contained within the nanopores being eluted.By varying the surface layer thickness we can therefore add another degree of tunability.For example, if a large burst of drug is desired initially followed by a slow zero-order release then this can be achieved by tweaking the value of L p .Thus by varying the parameter values of the system, a stent may in principle be designed to deliver a given Here we plot the numerical solution for the default parameter values in Table 1.
amount of drug rapidly, over a defined time period, and then the remainder of the drug slowly, over a longer defined period of time.In Figure 11, the cumulative fraction of drug released from a two-stage release system is plotted for various values of L p .When we analyse the case of nanoporous (or nanotubular) drug release into a well-stirred release medium, we see that the drug release depends purely on the value of D a and the ratio c s /c 0 .Thus, by varying these parameters as above we can also tailor the release profile for the well-stirred case.As described in Section 3.2, drug dissolution is significantly faster in a well-stirred release medium.This is evidenced in Figure 9 which compares the release profiles from the nanoporous system in unstirred and well-stirred release media.

Conclusions
In this paper we have presented the first model of drug elution from polymer-free drugeluting stents.Our generalised model is capable of predicting the drug release from a number of systems including nanoporous, nanotubular and smooth surface systems.We have identified the key parameters of the system which may be tuned at the manufacturing stage to achieve the desired drug release profile.In particular, we observed that the duration of release is particularly sensitive to the thickness of the drug layer, the ratio of drug absorption to desorption and ratio of drug solubility to initial drug concentration.While the first two of these may be manipulated during the manufacture of the stent material, the latter will depend on the properties of the particular drug considered (for example whether it is in an amorphous or crystalline phase).Other parameters such as the porosity and tortuosity can be utilised to fine tune a particular release profile.We have also demonstrated that a two-layer system (comprising a pure drug layer on the surface above a porous drug layer) can provide additional flexibility in tuning the release profile.Two distinct phases of release can be obtained, with the duration of each phase and the amount of drug delivered during each phase able to be varied by adjusting the model parameters accordingly.
One of the main advantages of the models presented in this work is the ability to achieve analytical solutions.These solutions not only allow for release profiles to be rapidly calculated, but they also clearly show the dependence of the various parameters of the system on the release profiles.This is of great benefit when considering the required design parameters to achieve a particular release profile.We would like to emphasise that the models presented in this paper are for drug release in an in-vitro environment since we believe that it is essential to understand the drug release dynamics in a controlled environment before embarking on the in-vivo environment.We acknowledge that while our work may suggest that a system designed with certain parameter values will give rise to a particular in-vitro release profile, the in-vivo release profile may be quite different.Furthermore, we acknowledge that there may be biological and mechanical constraints on the design of the device which are not considered here.

Figure 1 :
Figure1: Some of the polymer-free drug-eluting stent systems modelled in the current study.(a) A scanning electron microscope image of the Setagon stent[20].A nanoporous drug-infused layer covering a steel strut acts as a reservoir for the prolonged release of drug.The pore diameters here are of the order of 10 nm.(b) A schematic representation of part of a ceramic-coated tacrolimus-eluting stent.The pore diameters here are again of the order of 10 nm.Nanotubular systems of this kind have also been investigated in the context of orthopedic and dental drug releasing implants.(c) An image of the drug-coated surface of an Amazonia Pax stent[21].In this polymer-free system, a layer of semi-crystalline paclitaxel of thickness ≈ 5 µm covers the the stent strut surface.

Figure 3 :
Figure 3: A diagram introducing some of the notation used.Here x = 0 locates the interface between the porous medium and the release medium, and the boundary x = s(t) gives the position of a moving dissolution front separating the undissolved and dissolved drug.

Figure 5 :
Figure5: A comparison between the cumulative percentage of drug released (M % ) for a nanoporous, nanotubular and smooth surface system.We observe that a nanotubular system results in quicker release of drug than a nanoporous system due to less tortuous pores.The smooth surface system results in a dramatic increase in the rate of drug release, since there are no pores to hinder the release of drug.Two additional cases have been added to demonstrate one way in which the nanoporous system can be tuned to achieve quicker release profiles.Nanoporous* and Nanoporous** have the reduced porosities φ = φt = 0.3 and φ = φt = 0.1, respectively.The default parameter values used in the generation of this plot are displayed in Table1.

Figure 8 :
Figure 8: Plot of the cumulative fraction of drug released from a two-stage release system for various Da/Dw corresponding to varying the ratio ka/k d .Here we plot the numerical solution for the default parameter values in Table 1 except that Lp = 5 × 10 −5 m and L d = 10 −4 m.
Figure10: The special case of a pure drug layer in a well-stirred release medium.This case cannot be described by the solution(30),(31) and must be considered separately.A narrow boundary layer of poorly stirred fluid forms near the surface of the solid drug, and it is assumed that the flux of drug from the dissolving surface is proportional to the difference between the drug concentration at the surface and the drug concentration in the well-stirred release medium.

Figure 11 :
Figure11: Plot of the cumulative fraction of drug released from a two-stage release system for various Lp/L d .Here we plot the numerical solution for the default parameter values in Table1.

Table 2 :
Parameter Varied Value of parameter Value of Da (m 2 s −1 ) Release time (days) Parameter values used in the sensitivity analysis.