Mass diffusion through two-layer porous media: an ... - ResearchGate

International Journal of Heat and Mass Transfer 50 (2007) 3658–3669www.elsevier.com/locate/ijhmtMass diffusion through two-layer porous media:an application to the drug-eluting stentGiuseppe Pontrelli a, *, Filippo de Monte ba Istituto per le Applicazioni del Calcolo – CNR, Viale del Policlinico, 137 – 00161 Roma, Italyb Dipartimento di Ingegneria Meccanica, Energetica e Gestionale, University of L’Aquila, Località Monteluco, 67040 Roio Poggio (AQ), ItalyReceived 2 June 2006Available online 2 January 2007AbstractA mathematical model for the diffusion–transport of a substance between two porous homogeneous media of different properties anddimensions is presented. A strong analogy with the one-dimensional transient heat conduction process across two-layered slabs is shownand a similar methodology of solution is proposed. Separation of variables leads to a Sturm–Liouville problem with discontinuous coefficientsand an exact analytical solution is given in the form of an infinite series expansion. The model points out the role of four nondimensionalparameters which control the diffusion mechanism across the two porous layers. The drug-eluting stent constitutes the mainapplication of the present model. Drug concentration profiles at various times are given and analyzed. Also, qualitative considerationsand a quantitative description to evaluate feasibility of new drug delivery strategies are provided, and some indicators, such as the emptyingtime, useful to optimize the drug-eluting stent design are discussed.Ó 2006 Elsevier Ltd. All rights reserved.Keywords: Mass diffusion; Multi-layered porous media; Advection–diffusion equation; Sturm–Liouville problem; Drug delivery; Pharmacokinetics1. IntroductionIn last years, polymeric gels are widely used as drug carriersdevices in many biotechnological applications, such astissue engineering [1]. A matrix of gel is filled with a specificdrug which is subsequently released into the living tissue.To have a desired therapeutic effect, the concentration ofthe drug in the tissue should lie within a given range. Ifbelow that, the therapy results ineffective. On the otherhand, if the concentration is above a threshold value, thedrug can produce a toxic effect [2].A recent clinical application of such a technology are thedrug-eluting stents: these are complex medical devicesinserted in the arterial wall aimed to widen the lumen of stenosedarteries, to prevent occlusion, and to restore the* Corresponding author. Tel.: +39 6 884 70251; fax: +39 6 440 4306.E-mail addresses: pontrelli@iac.rm.cnr.it (G. Pontrelli), demonte@ing.univaq.it (F. de Monte).blood flow perfusion to the downstream tissues [2–4](Fig. 1). Drug-eluting stents combine the mechanical supportwith local drug delivery to the arterial tissue to fightearly restenosis caused by proliferation of smooth musclecells. To this aim, a polymeric matrix (gel) is added to themetallic struts and loaded with a drug which is released afterimplantation. The stent struts can be uniformly covered bythe polymer (coated stent) or it can contain honeycombedstrut elements with an inlaid polymer (stent with drug reservoirs)(Fig. 2) [5]. It is recognized that the time and rate ofthe drug release is crucial for the therapy. This process isinfluenced by many factors, such as properties of the drug(hydrophily), coating (structure and material parameters)as well as the transport characteristics of the arterial wall.Although intravascular stents are designed to operate in staticregimes, they act under complex stress conditions, varyingin time and it is difficult to forecast their performanceand efficiency over long times. Mathematical models predictingthe dynamics of solute concentration and mass flux0017-9310/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2006.11.003

G. Pontrelli, F. de Monte / International Journal of Heat and Mass Transfer 50 (2007) 3658–3669 3659Nomenclaturec imass volume-averaged concentration in the ithlayerc 0 i fluid-phase concentration in the ith layerD i drug diffusive coefficient in the ith layerJ mass flux at the interface x =0(Fig. 3)k partition coefficientL thickness ratio, L 1L 2.L i thickness of the ith layerM i dimensionless mass per unit of area in the ithlayerP membrane permeability coefficientS concentration jump at the interface x =0(Fig. 3)t timet ExX iemptying time of drugspace coordinateeigenfunction of the ith layerGreek symbolsc diffusivity ratio, D 1D 2 medium porosityk i eigenvalue of the ith layerr material ratio, k 1 1k 2 2/ nondimensional permeability,Subscripts1 first layer (coating)2 second layer (wall)PL 2D 2 k 2 2Fig. 1. Sketch of a stented artery.are of interest for biomedical engineers and clinicians, asthey offer a simple tool for optimizing the drug deliverydesign and technology.A number of recent studies has been devoted to the masstransfer process in the arterial wall, modelled as a multilayeredmedium, coupled with the transport in the lumen[6,7]. Some work has been done to correlate the numberand the location of the metallic net structure of a stent withthe extension of the perfused area [8]. Other mathematicalmodels have been developed to predict the release of a substancein a tissue and the influence of the physical propertiesof the drug. However, computational difficulties incoupling different geometrical scales are reported [5,9].The present paper provides a fundamental study of thetransient mass elution between two homogeneous porousmedia having different thickness and material properties.The treatment is quite general and can be applied to severalmass transfer diffusion dominated problems in compositematerials. The main application is the drug-eluting stent,where the whole drug is initially in a polymeric matrix coatingthe metallic structure and is subsequently released intothe arterial wall. Being interested in the pharmacokineticsFig. 2. Section of a stented artery with the struts (in black) embedded intothe wall (courtesy of P. Zunino).only, any structural analysis as well as all mechanicaleffects of the stent are neglected. The chemistry and thebinding of the drug to the tissue are beyond the scope ofthe present work and will not be considered here. An outlineof the paper is now given.Section 2 presents the time–space diffusion–transportequation for the dynamics of a substance across a doublelayer structured material in the general case. Due to a prevalentflux direction, the problem is written in one dimensionalcase in terms of nondimensional variables. Thisresults in a coupled system of partial differential equationsin two domains with an interface condition. Despite thesurprising analogy between mass transfer and heat conductionproblems, solutions of identical form are admittedprovided that not only the differential equations, but alsothe initial and boundary conditions match [10]. In thecurrent case, however, the boundary conditions at the

3660 G. Pontrelli, F. de Monte / International Journal of Heat and Mass Transfer 50 (2007) 3658–3669interface do not have a counterpart in the heat diffusionproblem. Differently that in the case of two-layer heat conduction,the mass flux is continuous, but a concentrationjump may occur at the interface. This follows from theequilibrium conditions with regard to mass transfer (equalityof the chemical potential of each component in eachphase, see [11, chapter XI]). As a consequence, the solutionto the transient two-layer heat conduction problem proposedin [16] cannot entirely be transferred to the analogousmass diffusion process of interest here. However, thesame theoretical procedure based on separation of variablesfor solving the differential problem may be reformulated,as shown in Section 3. Thus the correspondentSturm–Liouville problem is analyzed and the correspondenteigenvalue equation is solved. Finally, the concentrationsolution is expressed in the form of a Fourier series(Section 3).Compared to a fully numerical method, the analyticalapproach provides a greater insight into the physical senseof the drug delivery process. As a matter of fact, the presentone-dimensional model is shown to catch most of therelevant aspects of the drug dynamics. By showing relationshipsamong the relevant variables and material parameters,it can be used to identify simple indexes orclinical indicators of biomechanical significance. Moreoverthe methodology can be easily extended to a multi-layeredcoating and wall structure. The model enables the effect ofimportant factors such as drug diffusivity, coating thickness,and membrane permeability to be analyzed. Tunedin optimal way, they can be used to design novel releasemechanisms, as well as to improve drug delivery protocolsused in therapy and diagnostics.2. Formulation of the problemA drug-eluting stent (DES) is a metallic prosthesis(strut) implanted into the arterial wall and coated with athin layer of biocompatible polymeric gel that encapsulatesa therapeutic drug (coating). Such a drug, released in a controlledmanner through a permeable membrane, is aimed athealing the vascular tissues or at preventing a possiblerestenosis by virtue of its anti-proliferative action againstsmooth muscle cells. In the present work we are interestedin the mechanism of drug elution into the arterial tissue. Asa matter of fact any other effects, such as the drug metabolismin the living tissue and a possible decompositionof the polymeric matrix, are neglected.Let us consider a stent coated by a thin layer (of thicknessL 1 ) of gel containing a drug and embedded into thearterial wall (of thickness L 2 ), as illustrated in Fig. 3. Thecomplex multi-layered structure of the arterial wall has beendisregarded and a homogeneous material with averagedproperties has been considered for simplicity (fluid-wallmodel) as in Refs. [9,13]. Both the coating and the arterialwall are treated as porous media [14]. Because most of themass transport process occurs along the direction normalto the two layers (radial direction), we restrict our studyto a simplified 1D model. In particular, we consider a radialline crossing the metallic strut, the coating and the arterialwall and pointing outwards and, being the wall thicknessvery small with respect to the arterial radius, a Cartesiancoordinate system x is used along it (Fig. 3).At the initial time (t = 0), the drug is contained only inthe coating and it is uniformly distributed at a maximumconcentration C 1 and, subsequently, it is released into thewall. Here, and throughout this paper, a mass volumeaveragedconcentration c(x,t) (mg/ml) is considered. Sincethe strut is impermeable to the drug, no mass flux passesthrough the boundary surface x = L 1 . Moreover, it isassumed that the plasma does not penetrate the surfaceof the stent. Thus, the dynamics of the drug in the coating(first layer) is described by the following 1D differentialequation:oc 1ot(a)–L1(b) (c)(d)c 1lumenwall0c2Fig. 3. Cross-section of a stented artery with a zoomed area near the wallthat shows the metallic mesh and the two-layer medium at the adventitialside described by the model (2.1) and (2.2): (a) stent strut, (b) coating, (c)topcoat, (d) arterial wall. Due to an initial difference of concentration,drug is diffusing in the arterial wall from (b) to (d) through the permeablemembrane (c). An analogous two-layer pattern is present on the oppositeside of the strut, referring to the drug diffusion towards the lumen(lumenal side).D 1o 2 c 1ox 2 ¼ 0 in ½ L 1; 0Š;oc 1D 1ox ¼ 0 at x ¼ L 1;c 1 ¼ C 1 at t ¼ 0;where D 1 is the drug diffusivity in the coating.L2xð2:1Þ

Let us now consider the drug dynamics in the wall (secondlayer). In principle, drug diffusion occurs from thecoating towards both the lumen and the adventitia. Thereforean inner double layer system (lumenal side (lum)) andan outer double layer system (adventitial side (adv)) have tobe distinguished. The relative importance of them dependson the penetration depth of the stent, but the adventitialside is generally larger in size and deemed more relevantfrom a clinical point of view (Fig. 3). Even though theresults in the sequel of the paper are mainly referred toit, the following analysis is general and applies to bothinner and outer sides.At adventitial side, being L 2 L 1 , the concentration c 2at x = L 2 does not change as the time increases and hence itremains equal to its initial value, namely zero. Similarly, inthe case of the lumenal side, concentration is nearly zero,because of the extremely high mass transfer coefficient ofthe blood stream (washout). Therefore, in both cases, weend up with an homogeneous boundary condition of thefirst kind at x = L 2 leading to a fraction of drug lost inthe tissues adjacent to the adventitia and a fraction dispersedin the lumen.In general, mass transport in the wall is not governed bydiffusion only, but convection is equally important and anadvective transport term is added into the model. Thus, wehave the following equations:oc 2ot þ o oxa 2 u 2 oc 2c 2 D 2 2 ox¼ 0 in ½0; L 2 Š;c 2 ¼ 0 at x ¼ L 2 ;c 2 ¼ 0 at t ¼ 0;ð2:2Þwhere D 2 is the drug diffusion coefficient in the wall and 2 itsporosity. The coefficient a 2 is the so-called hindrance coefficientin the x-direction [9]. Finally, u 2 (x,t) denotes the volume-averagedfiltration velocity of the plasma over a crosssection. Such a variable is not known a priori and constitutesa strong coupling term with the fluid-dynamics. In the presentcase, however, it may simply be estimated as follows.2.1. Plasma convectionG. Pontrelli, F. de Monte / International Journal of Heat and Mass Transfer 50 (2007) 3658–3669 3661For the unidirectional flow of an isothermal and incompressiblefluid in a porous medium the velocity is time independent.The mass conservation gives u 2 (x) =U 2 = const in[0,L 2 ]. Assuming the plasma viscosity l 2 and the wall permeabilityK 2 as constants, integration of Darcy equationover [0,L 2 ] gives [14]:U 2 ¼Dpð2:3Þl 2 L 2 =K 2In human medium sized arteries, the pressure differencebetween lumen and adventitia, at normal conditions, doesnot exceed 100 mmHg and hence the numerator of Eq.(2.3) is less than that. Consequently U 2 10 6 cm/s. By ascaling analysis, the orders of magnitude of the convectiveand diffusive terms in (2.2) are estimated [10]:a 2 U 2 c 2oc 2¼ OðC 1 U 2 Þ; D 2 2 ox ¼ O C 1D 2: ð2:4ÞL 2In the present application D 2 /L 2 is one order (resp. threeorders) of magnitude larger than U 2 at the adventitial side(resp. lumenal side) and the convective term remains negligiblein comparison with the diffusive one. This result isalso confirmed by Zunino in Ref. [9], where it is shown thatafter 6 h up to 76% of heparin is lost into the blood for thecase of finite filtration velocity, versus a 72% lost in the casewithout it.2.2. Interface conditionsTo close the previous system of Eqs. (2.1) and (2.2),the conditions at the interface x = 0 (the so-called innerboundary conditions) have to be assigned. One of them isobtained by imposing continuity of the mass flux:oc 1D 1ox ¼ D oc 22 at x ¼ 0: ð2:5ÞoxAlso, to slow down the drug release rate, a permeablemembrane (called topcoat) of permeability P (cm/s) islocated at the interface (x = 0) between the coating andthe arterial wall. A continuous mass flux passes throughit orthogonally to the coating film with a possible concentrationjump. In the present case, the mass transfer throughthe topcoat can be described using the second Kedem–Katchalskyequation [9,13,15]. Thus, the continuous flux ofmass passing across the membrane normally to the coatingis expressed byoc 1D 1ox ¼ Pðc0 1c 0 2Þ at x ¼ 0; ð2:6Þor, alternatively,oc 2D 2ox ¼ Pðc0 1c 0 2Þ at x ¼ 0; ð2:7ÞIn Eqs. (2.6) and (2.7) the fluid-phase concentration c 0 isused. This is related to the volume-averaged concentrationc through the formula c 0 ¼ c , where k is the partition coefficient[3,9]. As one of the last three equations is redundant,kwe can choose any two of them, for example Eqs. (2.5) and(2.7).Note that, when P ? 1, Eqs. (2.6) and (2.7) reduce toc 0 1 ¼ c0 2. This equality contrasts with what occurs at theinterface of two-layer heat conduction problems wherethe temperatures across the interface are always equal, ifthe two bodies are in perfect thermal contact. In mass diffusion,instead, because the media are porous and drug ispartly dissolved in the fluid and partly bound with thepolymeric matrix in solid phase, only the fluid-phase concentrationenters to the flux dynamics.2.3. Dimensionless formLet us define the following nondimensional variablesand constants:

a 1 ¼ r tan k 2 þ k 2b 2 ;/ð3:17Þb 1 ¼ p 1 ffiffi b 2 ;ð3:18Þca 2 ¼ ðtan k 2 Þb 2 ; ð3:19Þwhere the multiplicative constant b 2 will be determinedthrough the initial condition (see below). Let us note thatu depends on the four parameters r, /, L, c, it is an oddfunction and its roots (eigenvalues) are infinite, real anddistinct (see Appendix A.3). We also remark that u hasan infinite number of singularity points, sayl j ¼ p 1 ;2 þ j m j ¼ p p ffiffi cL12 þ j; j ¼ 0; 1; 2; ...ð3:20ÞDifferently than in heat transfer models [18] – due to thepresent interface condition (3.9) – the function u is notmonotone and the property that each root is locatedexactly between any two adjacent asymptotes is notsatisfied. In correspondence of each eigenvalue k 2m(m = 1,2,...) solution of Eq. (3.16), an associated k 1m isfound (see Eqs. (3.10)). Subsequently, the constants a 1m ,b 1m and a 2m are obtained from (3.17), (3.18) and (3.19)respectively, and thus the corresponding eigenfunctionsX 1m and X 2m defined in Eqs. (3.11) may be written asX 1m ¼ b 2meX 1m ¼ b 2m r tanðk 2m Þþ k 2mcosðk 1m xÞþ 1 pffiffisinðk 1m xÞ ;/cð3:21ÞX 2m ¼ b 2meX 2m¼ b 2m ½ tanðk 2m Þ cosðk 2m xÞþsinðk 2m xÞŠ: ð3:22ÞFurthermore, the corresponding time-variable functionsG 1m and G 2m defined by Eqs. (3.3) are computed, using(3.10), asG 1m ¼ G 2m ¼ e k2 2m t :ð3:23ÞFinally, the complete solution of the problem is given bya linear superposition of the fundamental solutions (3.21)and (3.22) in the formc 1 ðx; tÞ ¼ X1m¼1c 2 ðx; tÞ ¼ X1m¼1with A m :=b 2m .A meX 1m ðxÞe k2 2m t ;A meX 2m ðxÞe k2 2m t ;G. Pontrelli, F. de Monte / International Journal of Heat and Mass Transfer 50 (2007) 3658–3669 3663ð3:24ÞZ 0LZ 10XAm eX 1meX 1n dx ¼Z 0LSimilarly in the interval [0,1], we haveeX 1n dx; n ¼ 1; 2; ... ð3:25ÞXAm eX 2meX 2n dx ¼ 0; n ¼ 1; 2; ... ð3:26ÞBy combining Eqs. (3.25) and (3.26) and by using theorthogonality property (A.1) we haveZ 1eX 2 1m dx þ r eX 2 2m dx ¼ eX 1m dx; ð3:27ÞA mZ 0L0Z 0Lwhere the term in brackets on the l.h.s. is the normeN m ¼ N mb 2 2mwith N m given in Eq. (A.2). Bearing in mindEq. (3.21) and integrating from L to 0, we haver tan k 2m þ k 2msinðk 1m LÞþp 1 ffiffi ðcosðk 1m LÞ 1Þ/cA m ¼;eN m k 1mm ¼ 1; 2; ...ð3:28Þ3.2. Physical quantities and operational conditionsFrom Eq. (3.24) it is possible to compute the dimensionlessdrug mass (per unit of area) in both coating and walllayers as function of time asZ 0 Z 1M 1 ðtÞ ¼ c 1 ðx; tÞdx; M 2 ðtÞ ¼ c 2 ðx; tÞdx; ð3:29ÞL0obtainingM 1 ðtÞ ¼ X1A 2 e mN m e k2 2m t ;ð3:30Þm¼1M 2 ðtÞ ¼ X1m¼1tanðk 2m Þ sinðk 2m Þ cosðk 2m Þþ1A m e k2 2m t :k 2mð3:31ÞIn particular M 1 (0) = L and M 2 (0) = 0.Similarly, the dimensionless mass flux and jump ofconcentration at interface are respectivelyJðtÞ¼ c oc 1ox ð0;tÞ¼ oc 2ox ð0;tÞ¼SðtÞ¼ðc 1c 2 Þð0;tÞ¼ X1m¼1X1m¼1A m k 2m e k2 2m tð3:32ÞA m ð1 rÞtanðk 2m Þ r k 2me k2 2m t :/ð3:33ÞIn particular J(0) ? 1 and S(0) = 1.All the previous variables characterize the performanceof the drug elution process and can be used for itsoptimization.3.1. Application of the initial conditionBy evaluating c 1 in (3.24) att = 0 and multiplying it byeX 1n , after integration we get4. Computational resultsThe following physical parameters are considered forcomputational experiments:

3664 G. Pontrelli, F. de Monte / International Journal of Heat and Mass Transfer 50 (2007) 3658–3669L 1 ¼ 5 10 4 cm; L 2 ¼ 10 2 cm;P ¼ 10 6 cm=s; D 1 ¼ 10 10 cm 2 =s; D 2 ¼ 7 10 8 cm 2 =sk 1 ¼ 1; k 2 ¼ 1; 1 ¼ 0:1; 2 ¼ 0:61:These parameters have been chosen according to a physicalbasis and in agreement with the typical scales in DES anddata in literature for the arterial wall and heparin drug inthe coating layer [2,3,19].The physical problem apparently depends on a largenumber of parameters, each of them may vary in a finiterange, and there is a variety of different limiting cases. Asa matter of fact, they cannot be chosen independently fromeach other, but they are related by some compatibility conditionto give rise a well-posed model. However the problemdepends only on the four nondimensional operationalparameters defined by Eq. (2.8). In the present case, theirvalues are/ ¼ 0:234; r ¼ 0:164; L ¼ 0:05; c ¼ 0:0014: ð4:1ÞThe four ratios /, r, L, c are the only intrinsic parametersof the problem and they are used as mean values for a set ofnumerical simulations.First of all, Eq. (3.16) is solved numerically with a bisection-likemethod by using intervals limited by the points l jand m j , j = 0,1,2,...(see Eq. (3.20)) and midpoints betweenthem, as starting guess (routine fzero of MATLAB,with accuracyof 10 8 ). In particular, the first roots of Eq. (3.16) aregiven in Table 1. The damping factor, defined as d m ðtÞ ¼expð ck 2 1mtÞ, measures the attenuation of the various termsin summations (3.24) and is listed in the same table. Its valuesindicate a fast (exponential) convergence of the solu-Table 1First eigenvalues k 2m of Eq. (3.16), and corresponding damping factors d(t) at three different timesm k 2m d m (0.01) d m (0.1) d m (1) max x jA meX 1m j max x jA meX 2m j1 1.153 0.986 0.875 0.264 1.2485 0.10642 1.580 0.975 0.778 0.082 0.0232 0.07553 3.490 0.885 0.295 5.098 10 6 0.4206 0.01684 4.712 0.800 0.108 2.262 10 10 0.0001 0.00075 5.823 0.712 0.033 1.864 10 15 0.2503 0.01046 7.841 0.540 2.134 10 3

G. Pontrelli, F. de Monte / International Journal of Heat and Mass Transfer 50 (2007) 3658–3669 3665tion. Because of that, the series (3.24) will be truncated at afinite number of terms. Such a number can be found inaccordance with the accuracy desired at the time of interest.Being max x jA meX im j < 1 for any i = 1,2, m > 1 (seeTable 1), to reach an accuracy of 10 r , it is sufficient toconsider a finite series summation up to the index M t >1such thatrffiffiffiffiffiffiffiffiffiffiffiffir ln 10k 2Mt > ;tand the series is truncated at the first M t terms. M t is adaptivelychanged during computations: its values drop fromM t = 40 at the shorter times, down to M t = 5 at highertimes.The concentration profiles for three values of time aredisplayed in Fig. 4: drug is eluting from coating to the wall,with concentration decaying in time. Because of the differentmaterial properties of the substrates, at first instants c 1and c 2 exhibit a steep boundary layer near x = 0, which disappearsat later times.Fig. 5 shows that drug mass in the coating layer M 1 ismonotonically decreasing, while mass in the wall M 2 , firstincreasing to a maximum M 2 at time t* , decreases to zerowith the same rate of M 1 . Since drug is absorbed atx = 1, the total mass is not preserved and tends to zeroat time large enough.The mass flux J(t) and the concentration jump S(t) atthe interface are depicted in Fig. 6. Both quantities evidencea sudden change at first instants and then an asymptoticvanishing. There is a finite instant where S reachesa minimum, namely the same time where M 2 reaches itsmaximum M 2 .4.1. Validation of the resultsComparison with numerical results of other models in3D complex geometries has shown a good agreement. Inparticular, a certain discrepancy in the values of concentrationsoccurs at small times (t 6 0.05), since the boundarylayer at the interface is not sufficiently resolved. For example,at x = 0 and t = 0.01 the maximum absolute error is0.06 for c 1 and 0.0018 for c 2 [5].We also compare the fraction of drug mass lost partly atthe lumen (h lum ) and partly external to the adventitia (h adv )[9]. They are defined respectively as:h lum ðtÞ ¼ ½M 1ð0Þ ðM 1 ðtÞþM 2 ðtÞÞŠ lum;M tot ð0Þh adv ðtÞ ¼ ½M 1ð0Þ ½ðM 1 ðtÞþM 2 ðtÞÞŠ adv;M tot ð0Þwhere M tot (0) is the total (lumenal + adventitial side) initialmass. It is shown how the mass lost in the lumen isstrongly influenced by the penetration depth (L lum2) of thestent, when all the other parameters are kept constant(Table 2). Note that h adv is absent in model [9], since a zeromass flux is imposed as external boundary condition.0.050.0450.040.035drug mass0.030.0250.02M 1(t * ,M 2* )time0.0150.01M 2M 1+M 20.00500 0.5 1 1.5 2 2.5 3Fig. 5. Dimensionless drug mass in the coating (-), in the wall (---), and total mass (—) as function of time. In the coating, drug mass M 1 ismonotonically decreasing, while in the wall there is a characteristic time t * at which the drug reaches a maximum peak M 2. Due to the absorption at thewall, drug is not preserved and vanishes at a time large enough.

3666 G. Pontrelli, F. de Monte / International Journal of Heat and Mass Transfer 50 (2007) 3658–36690.250.2S(t) at x = 0J(t) at x = 00.150.10.050–0.050 0.5 1 1.5 2 2.5 3timeFig. 6. Mass flux (continuous line) and concentration jump (dashed line) at the interface x = 0 as function of time.Table 2Percentage of drug mass lost at lumenal (h lum ) and advential (h adv ) sides atnominal (L lum2¼ 10 lm) and at two off-nominal configurations. Mass arecomputed in dimensional form and compared with those in the model ofZunino [9], showing a strong influence of the penetration depth (L lum2)onthe drug elution and dispersionTime Present work Ref. [9]L lum2ðlmÞh lum (%) h adv (%) h lum (%) h adv (%)3000 s (50 min) 10 34.9 8.3 53 05 91.7 4.32.5 94 2.210,000 s (2 h 45 min) 10 64 9.1 66 05 95.2 4.72.5 97.5 2.420,000 s (5 h 30 min) 10 81.5 9.1 72 05 95.2 4.82.5 97.6 2.44.2. Sensitivity analysisDesign parameters governing the drug release can bescreened by analyzing the solution dependence and thesensitivity on the groups L, c, / and r when they are variedin a limited realistic range one at a time around the meanvalues in Eq. (4.1), with the others fixed. Results fromnumerical simulations are shortly reported below. Dependence on L: Increasing L, higher values for theconcentrations c 1 and c 2 at the interface are obtainedat larger times. A slower drug release is exhibited in bothlayers at larger L. The solution is shown very sensitive toL. Dependence on c: The drug is released faster with c inboth layers. The mass M 1 is decaying with c, while M 2shows an increased value of the peak, followed by afaster decay. The solution is quite sensitive to c. Dependence on r: Increasing r we have a raising ofc 1 and a reduction of c 2 at the interface for initialtimes. The mass M 1 is slightly increased at larger times,while the peak for M 2 is reduced and occurs at laterinstants. Dependence on /: Associated with a high /, a reductionof c 1 and an increase of c 2 at the interface are reportedonly at first times. When / is decreased, the peak ofM 2 lowers and is attained at a larger time. Consequently,the drug release is slower at lower /. For valuesof / > 1 the solution does not change significantly.4.3. Drug-delivery indicatorsFor novel coating design and comparative purposes, itis desirable to search for simple quantitative indicators ofthe drug dynamics. Several scalar indexes can be defined,for example:1. Emptying time t E , namely the time at which the drugmass drops below a given amount in the coating layer(1) with respect to the initial mass (in the simulationsM 1 (t E ) = 0.001M 1 (0) = 0.001L, where the expressionfor M 1 (t) is given by Eq. (3.30)).2. Mass M 2 at a given time.

G. Pontrelli, F. de Monte / International Journal of Heat and Mass Transfer 50 (2007) 3658–3669 3667876t E= 1726 L 2 + 13.56 L + 0.10276t E= 0.001 γ –5/4 + 1.184emptying time t E5432emptying time t E5431200 0.01 0.02 0.03 0.04 0.05 0.06L10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1γ6.66.410emptying time t E6.265.85.65.45.25t E= 1.753 σ + 4.75emptying time t E9876t E= 0.0524 φ –1 + 4.7944.854.60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1σ0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4φFig. 7. Variation of the indicator (1) (emptying time in coating) with c, L, r and / in a typical range. In each plot, only one parameter is changed, and theother three hold as in Eq. (4.1). Starred points are results from simulation, continuous lines are fitting curves.3. Magnitude (m 2 ) and uniformity (s 2 ) of the wall concentrationc 2 .4. Location and value of the peak in the wall layer (2).Time t * and maximum value M 2are possible pointersof drug release kinetics.Depending on the specific optimizing purposes and technologicallimitations, one or a combination of some ofthem may provide significant and useful hints.The indicators (3) are computed by first averagingc 2 (and its derivative) over the wall length and thenconsidering their maximum values in time. The variationof both indexes with each of the four parameters is monotone,and has the same trend. In particular, a reduction ofone order of magnitude on / (resp. c) implies a reductionof 25% (resp. 55%) on s 2 and of 23% (resp. 67%) on m 2 .The indicator (1) seems to be more promising and itssensitivity is fully investigated. The dependence of the emptyingtime on c, L, / and r is shown in Fig. 7 with oneparameter varied in a convenient interval and the otherthree fixed as in Eq. (4.1): in all cases the variation is monotoneand the maximum value is reached at one end of theinterval. The computed data are fitted with appropriatefunctions. In particular, t E shows a linear rising with rand a quadratic behaviour with L. On the other hand, t Edecreases as / 1 and as c 5 4. As a consequence, large valuesof L and r and small values of c and /, compatible withtechnological requirements and implantation limits, guaranteethe best drug release performance.5. ConclusionsThe release of a substance in a living tissue for therapeuticpurposes is becoming quite common in medicine nowadays,through drug delivery devices. Drug-eluting stentsare revealed a promising technique for healing the vascularwall and for the treatment of atherosclerosis and restenosis.However, the mechanism of release is quite complex anddepends on many concurrent biochemical, physical andindividual factors. The model presented here, althoughsome simplifying assumptions, is able to simulate and predictthe dynamics of a drug through a two-layered mediumand to estimate the dose absorption rate and is solved in aclosed form. An advantage of the methodology is that theprocedure is analytical and the only computational costconsists in searching the roots of the eigenvalue equation.The model can be easily extended to a multi-layered structure,including both a more realistic wall configuration and

3668 G. Pontrelli, F. de Monte / International Journal of Heat and Mass Transfer 50 (2007) 3658–3669a novel design for multi-coating DES. In addition, the Subtracting Eq. (A.4) from Eq. (A.3) we haveZ 0 Z 0k 2 1nX 1n X 1m dx ¼ X 001n X By using the first of Eqs. (3.11) and Eq. (3.12), the r.h.s1m dxcan be easily manipulated to getLLZ 0 Z 0¼ ½X 0 1n X 1mŠ 0 L þ X 0 1n X 0 1m dx: ðA:4Þ X 2 1m dx ¼ LLL 2 ða2 1m þ b2 1m Þ a 1m b 1m: ðA:14Þ2k 1mmathematical formulation is able to incorporate the drugconsumption effect due to the tissue cell binding. The applicationof the dimensional analysis to the governing equationshas indicated that the dynamics of a drug throughðk 2 1mZ 0k 2 1n Þ X 1n X 1m dx ¼L½X 0 1m X 1nŠ 0 L þ½X 0 1n X 1mŠ 0 L : ðA:5Þan eluting stent is fully controlled by only four dimensionlessoperational parameters. Numerical experiments haveBy replacing the expression of the first of Eqs. (3.11) ofthe eigenfunctions and by using Eq. (3.12) we obtainbeen extensively carried out over several typical configurations.Results have shown the influence of the solution oneach single parameter, in particular the ratios betweenðk 2 1mZ 0k 2 1n Þ X 1n X 1m dx ¼ a 1m b 1n k 1nLa 1n b 1m k 1m : ðA:6Þthickness and drug diffusivities of the two layers. Also,some biomechanical indicators, such as the emptying timeRepeating a similar procedure for the eigenvalues k 2m ,k 2n and for the eigenfunctions X 2m , X 2n , we getof the coating, are suggested to obtain an optimal drugelutionand a desired tissue concentration.ðk 2 2mZ 1k 2 2n Þ X 2n X 2m dx ¼0½X 0 2m X 2nŠ 1 0 þ½X 0 2n X 2mŠ 1 0ðA:7ÞAcknowledgementsand by using the second of Eqs. (3.11) and Eq. (3.13) wehaveThe authors wish to thank Dr. M. Prosi and Dr. P. Zuninofor their valuable suggestion and criticism in the ðk 2 2mZ 1k 2 2n Þ X 2n X 2m dx ¼ a 2n b 2m k 2m0development of this work.a 2m b 2n k 2n : ðA:8ÞBy summing up Eq. (A.6) multiplied by c and Eq. (A.8)Appendix Amultiplied by r, and by using Eqs. (3.10), (3.18) and (3.15)we haveZ A.1. Orthogonality0 Z 1ðk 2 2mk 2 2n Þ X 1n X 1m dx þ r X 2n X 2m dx ¼ 0;L 0We shall prove that the eigenfunction system (X 1 ,X 2 )satisfies the following orthogonality property:which immediately proves Eq. (A.1).ðA:9ÞZ 0 Z 10 for m 6¼ n;X 1n X 1m dx þ r X 2n X 2m dx ¼L0N m for m ¼ n;A.2. Normalization integralðA:1Þ We shall now prove that the norm N m has the expressiongiven by Eq. (A.2).with the expression for the norm N m given byMultiplying Eq. (3.4) by X 1m and integrating we have( " 2 N m ¼ b 2 1 a 1m2mþ b ! #2 Z1m a 1m b 0 Z 0 Z 01mL2 b 2m b 2m b 2 2m k k 2 1mX 2 1m dx ¼ X 001m X 1m dx ¼ ½X 0 1m X 1mŠ 0 L þ1mLLLðX 0 1m Þ2 dx:"þ r #)2a 2mþ 1!þa ðA:10Þ2mðA:2Þ2 b 2m b 2m k 2mMultiplying the first of Eqs. (3.11) by k 1m and summingits square with the square of(see subsection A.2 for the proof). Note that the all ratiosin the square brackets are expressible in terms of eigenvaluesX 0 1m ¼ a 1mk 1m sinðk 1m xÞþb 1m k 1m cosðk 1m xÞ;through Eqs. (3.17), (3.18) and (3.19).we getLet us consider two different eigenvalues k 1m and k 1n andthe corresponding eigenfunctions X 1m , X 1n . Multiplyingðk 1m X 1m Þ 2 þðX 0 1m Þ2 ¼ k 2 1m ða2 1m þ b2 1m Þ:ðA:11ÞEq. (3.4) by X 1n and integrating:Integration of both side of the above equation leads toZ 0k 2 1mX 1m X 1n dx ¼LZ 0X 001m X 1n dxLZ 0 Z 0k 2 1mX 2 1m dx þLLðX 0 1m Þ2 dx ¼ k 2 1m ða2 1m þ b2 1mÞL: ðA:12ÞZ 0¼ ½X 0 1m X 1nŠ 0 L þ X 0 1m X 0 Summation of Eqs. (A.10) and (A.12) gives1ndx: ðA:3ÞZL 02k 2 1mX 2 1mSimilarly, for the eigenvalue k 1m ða2 1m þ b2 1m ÞL ½X 0 1m X 1mŠ 0 L : ðA:13Þ1nL

G. Pontrelli, F. de Monte / International Journal of Heat and Mass Transfer 50 (2007) 3658–3669 3669Z 1Repeating the same procedure for an eigenfunction X 2mX 2 2m dx ¼ 10 2 ða2 2m þ b2 2m Þþa 2mb 2m: ðA:15Þ2k 2mSubstitution of Eqs. (A.14) and (A.15) into Eq. (A.1) form = n gives the result.A.3. Reality of the eigenvaluesWe first prove that the Eq. (3.16) cannot have a complexroot of the form k 2 = p ± jq, where p and q are positive realnumbers, and j is the imaginary unit. In fact, if this werepossible, we would have two conjugate roots k 2m = p + jqand k 2n = p jq. Substituting these two complex roots inEq. (3.11), after some algebraic manipulations, the eigenfunctionsmay be separated into their real and imaginaryparts and rewritten asX 1r ðxÞ ¼R 1 ðxÞjS 1 ðxÞ;r ¼ m; nX 2r ðxÞ ¼R 2 ðxÞjS 2 ðxÞ;Applying the orthogonality property (A.1) with m 6¼ n,we would haveZ 0 Z 1LðR 2 1 þ S2 1 Þdx þ r ðR 2 2 þ S2 2Þdx ¼ 0;ðA:16Þ0which is impossible as both terms on the l.h.s. of Eq. (A.16)are positive. With a similar argument, we demonstrate thatEq. (3.16) cannot admit purely imaginary roots of the formk 2 =±jq. In such a case Eq. (A.16) would hold withR 1 = R 2 =0.References[1] P.A. Netti, F. Travascio, R.K. Jain, Coupled macromoleculartransport and gel mechanics: poroviscoelastic approach, AiChE J.49 (6) (2003) 1580–1596.[2] D.V. Sakharov, L.V. Kalachev, D.C. Rijken, Numerical simulation oflocal pharmacokinetics of a drug after intravascular delivery with aneluting stent, J Drug Target 10 (6) (2002) 507–513.[3] C. Creel, M. Lovich, E. Edelman, Arterial paclitaxel distribution anddeposition, Circ. Res. 86 (8) (2000) 879–884.[4] C.D.K. Rogers, Drug-eluting stents: role of stent design deliveryvehicle and drug selection, Rev. Cardiov. Med. 3 (supp. 5) (2002)S10–S15.[5] M. Prosi, F. Gervaso, P. Zunino, S. Minisini, L. Formaggia,Numerical simulations of drug release from stents, in: Proc. 7th Int.Symp. on Comp. Meth. in Biomech. and Biomed. Eng., Antibes (FR),22–25 March, 2006.[6] N. Yang, K. Vafai, Modelling of low-density lipoprotein (LDL)transport in the artery – effect of hypertension, Int. J. Heat MassTransf. (2006) 850–867.[7] L. Ai, K. Vafai, A coupling model for macromolecules transport ina stenosed arterial wall, Int. J. Heat Mass Transf. (2006)1568–1591.[8] M.C. Delfour, A. Garon, V. Longo, Modeling and design of coatedstents to optimize the effect of the dose, SIAM J. App. Math. 65 (3)(2005) 858–881.[9] P. Zunino, Multidimensional pharmacokinetic models applied to thedesign of drug-eluting stents, Cardiov. Eng.: Int. J. 4 (2) (2004).[10] H.D. Baehr, K. Stephan, Heat Mass Transf., Springer Verlag, Berlin,1998.[11] S.R. deGroot, P. Mazur, Non-Equilibrium Thermodynamics, DoverPub, New York, 1984.[12] F. de Monte, Transient heat conduction in one-dimensional compositeslab. A natural analytical approach, Int. J. Heat Mass Transf. 43(2000) 3607–3619.[13] M. Prosi, P. Zunino, K. Perktold, A. Quarteroni, Mathematical andnumerical models for transfer of low-density lipoproteins through thearterial walls: a new methodology for the model set up withapplications o the study of disturbed lumenal flow, J. Biomech. 38(2005) 903–917.[14] A.-R.A. Khaled, K. Vafai, The role of porous media in modeling flowand heat transfer in biological tissues, Int. J. Heat Mass Transf. 46(2003) 4989–5003.[15] A. Kargol, M. Kargol, S. Przestalski, The Kedem–Katchalskyequations as applied for describing substance transport acrossbiological membranes, Cell. Mol. Biol. Lett. 2 (1996) 117–124.[16] J.V. Beck, K.D. Cole, A. Haji-Sheikh, B. Litkouhi, Heat Conductionusing Greens functions, Hemisphere Publ. Corp., Washinghton, 1992.[17] C.W. Tittle, Boundary value problems in composite media: quasiorthogonal functions, J. Appl. Phys. 36 (4) (1965) 1486–1488.[18] F. de Monte, Multi-layer transient heat conduction using transitiontime scales, Int. J. Thermal Sci. 45 (2006) 882–892.[19] C. Hwang, D. Wu, E.R. Edelman, Physiological transport forcesgovern drug distribution for stent-based delivery, Circulation 104 (5)(2001) 600–605.