See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/318894176 Multiscale Boundary Conditions for Non-Fickian Diffusion Applied to Drug- Eluting Stents Article  in  SIAM Journal on Applied Mathematics · January 2017 DOI: 10.1137/16M1107577 CITATION 1 READS 107 3 authors: Some of the authors of this publication are also working on these related projects: Modélisation mathématique des écoulements sanguins avec applications à l'athérosclérose et autre maladies View project Asymptotics and Simulation of Complex Fluids View project Elias Gudiño Universidade Federal do Paraná 13 PUBLICATIONS   80 CITATIONS    SEE PROFILE C.M. Oishi São Paulo State University 46 PUBLICATIONS   270 CITATIONS    SEE PROFILE Adélia Sequeira Technical University of Lisbon 146 PUBLICATIONS   1,377 CITATIONS    SEE PROFILE All content following this page was uploaded by C.M. Oishi on 14 August 2017. 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APPL. MATH. c© 2017 Society for Industrial and Applied Mathematics Vol. 77, No. 4, pp. 1184–1203 MULTISCALE BOUNDARY CONDITIONS FOR NON-FICKIAN DIFFUSION APPLIED TO DRUG-ELUTING STENTS∗ ELÍAS GUDIÑO† , CASSIO M. OISHI‡ , AND ADÉLIA SEQUEIRA§ Abstract. We propose a multiscale model for the study of blood flow, plasma filtration, and non-Fickian mass transport by a coronary drug-eluting stent. We derive an analytic solution of a one- dimensional model for non-Fickian diffusion and replace the problem of drug transport in the coating of the stent with suitable boundary conditions. In this way, the computational cost of numerical simulations in realistic three-dimensional stent geometries is reduced. Numerical experiments to show the effectiveness of the method are also presented. Key words. drug-eluting stents, drug delivery, non-Fickian diffusion, multiscale models AMS subject classifications. 65M60, 35Q30, 35A20 DOI. 10.1137/16M1107577 1. Introduction. When an artery becomes obstructed due to the presence of an atherosclerotic plaque, it is necessary to perform an endovascular medical procedure called angioplasty. During this procedure, a medical device consisting of a small wire mesh tube, called a stent, is inserted and guided to the place of the obstruction with the aid of a catheter. Then, the catheter is removed and the stent stays, permanently acting as a supporting scaffold (Figure 1). New generation drug-eluting stents (DES) are coated with a polymeric membrane containing antiproliferative drugs (paclitaxel, rapamycin, everolimus, etc.) in order to interrupt the cell cycle of smooth muscle cells (SMC) [31]. In this way the proliferation of SMC can be regulated in situ and avoid the renarrowing of the lumen (restenosis) due to the accumulation of tissue around the stent. In order to account for drug diffusion from the polymeric coating of the DES into the arterial wall, many mathematical models have been proposed [22]. These models provide a low-cost alternative to research done in laboratories, where many experi- ments must be carried out to understand the different variables that can influence the process of the drug delivery, i.e., the geometrical design of the device, the type of polymer used in the coating, the type of drug embedded in the polymer, the initial loading of drug, etc. For brief reviews on models for drug deposition on the arterial wall and for three-dimensional models for drug transport in stents, we refer the reader to [1] and [17], respectively. ∗Received by the editors December 13, 2016; accepted for publication (in revised form) April 25, 2017; published electronically August 3, 2017. http://www.siam.org/journals/siap/77-4/M110757.html Funding: This work was partially supported by the Coordenação de Aperfeiçoamento de Pes- soal de Nı́vel Superior (CAPES), funded by the Brazilian Government through the Ministério da Educação, and by the project “PHYSIOMATH—Mathematical and Computational Modeling of Hu- man Physiology” EXCL/MAT-NAN/0114/2012, funded by the Portuguese Government through the FCT-Fundação para a Ciência e a Tecnologia. The second author’s work was supported by Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico (CNPq) grant 307459/2016-0 and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) grant 2013/07375-0. †Department of Mathematics, Universidade Federal do Paraná, Curitiba, Paraná 81530, Brazil (egudino@ufpr.br). ‡Department of Mathematics and Computer Science, Faculdade de ciências e tecnologia, Univer- sidade Estadual Paulista (UNESP), Pres. Prudente, Sau Paulo 19060, Brazil (oishi@fct.unesp.br). §Department of Mathematics and CEMAT-IST, Instituto Superior Técnico, Universidade de Lis- boa, Lisboa 1049-001, Portugal (asequeir@math.tecnico.ulisboa.pt). 1184 http://www.siam.org/journals/siap/77-4/M110757.html mailto:egudino@ufpr.br mailto:oishi@fct.unesp.br mailto:asequeir@math.tecnico.ulisboa.pt BOUNDARY CONDITIONS FOR NON-FICKIAN DIFFUSION 1185 Fig. 1. Stent use in a clogged artery. 1. The stent is inserted. 2. The balloon is inflated. 3. The catheter is removed. Lower-dimensional models can provide some idea of drug kinetics in the DES and drug deposition in the arterial wall, but the fluid dynamics of the blood around the stent is fully three-dimensional [8, 9, 33]; therefore, in order to capture more accurately the interactions between drug diffusion and hemodynamics, it is important to set up mathematical models in three dimensions, considering realistic stent geometries. The computational cost of numerical simulations for a three-dimensional stent is very high, since it is necessary to consider many elements in order to generate a mesh that correctly represents the coating of the stent. Thus, a large number of degrees of freedom is needed to numerically solve the system of partial differential equations (PDEs). Motivated by the fact that the polymeric coating of the DES is very thin, it is reasonable to approximate its surface by a uniform slab of finite thickness. Then, by calculating an analytical solution of the one-dimensional diffusion problem in the slab, it is possible to replace the drug diffusion problem in the coating of the stent with a Robyn-type boundary condition. Some multiscale models have been proposed in the literature [8, 9, 10, 27, 32], but to the best of our knowledge all of them neglect the influence of the viscoelastic properties of the polymer in the transport problem. The aim of this work is to deduce suitable boundary conditions for a mathematical multiscale model of drug delivery in a coronary DES on complex three-dimensional geometries that takes into consideration the viscoelastic properties of the polymeric coating of the stent. Therefore, our main focus is on DES with coatings where the polymer can affect the transport process. In section 2 we introduce the complete model for the study of drug release in the coating of the DES, the lumen, and the arterial wall. A semianalytic solution of a one-dimensional problem for non-Fickian drug diffusion is described in section 3, while in section 4 a multiscale boundary condition to approximate the outgoing flux of drug into the lumen and the wall is presented. In section 5 we introduce the model for blood flow and plasma filtration. The complete multiscale model for drug transport is introduced in section 6. In section 7 we present the numerical scheme used to 1186 ELÍAS GUDIÑO, CASSIO M. OISHI, AND ADÉLIA SEQUEIRA Fig. 2. Structure of the arterial wall. Fig. 3. Transversal section of the stented coronary artery. approximate the solution of the multiscale model. Numerical results are presented in section 8, including some plots to exhibit the behavior of the mathematical model. Finally, section 9 is devoted to discussion and conclusions. 2. Modeling drug delivery. A healthy arterial wall is a complex structure with different homogeneous layers (Figure 2). When atherosclerosis occurs, a lipid fibrous plaque is formed in the arterial wall due to the accumulation of a high level of low-density lipoprotein (LDL) cholesterol in the lumen and to a series of biochem- ical reactions that ends with the formation of the atheroma. This lipid accumulation causes the intima to thicken and change its physical properties, which may influence the drug deposition in the arterial wall [26]. In this work, we consider as regions of interest the intima with the presence of a soft lipid plaque and the media, sepa- rated by the internal elastic lamina (IEL). For brief reviews on the modeling of LDL accumulation on the arterial wall, we refer the reader to [20, 21]. Figure 3 shows a truncated portion of a stented coronary artery. In this domain we denote by Ωw = Ωin ∪ Ωp ∪ Ωme the arterial wall, where Ωin is the subregion of the wall that represents the intima layer, Ωp the atherosclerotic plaque, and Ωme the BOUNDARY CONDITIONS FOR NON-FICKIAN DIFFUSION 1187 Fig. 4. Mechanical model. media. By Ωl we denote the lumen, and by Ωc the polymeric coating of the stent. The arterial wall boundary is represented by ∂Ωw = Γl,w ∪ Γw ∪ Γc,w ∪ Γp,w ∪ Γiel ∪ Γadv, where Γl,w is the interface between the lumen and the wall, Γw is the artificial wall boundary, Γc,w is the interface between the wall and the coating of the stent, Γp,w is the interface between the plaque and the wall, Γiel represents the IEL, and Γadv is the interface between Ωw and the adventitia. Analogously, the lumen boundary is defined as ∂Ωl = Γl,w ∪ Γc,l ∪ Γin ∪ Γout, where Γc,l is the interface between the coating of the stent and the lumen, Γin is the inflow boundary, and Γout is the outflow. The boundary of the polymeric coating of the stent is represented as ∂Ωc = Γc,l ∪ Γc,w ∪ Γstr, where Γstr is the interface between the polymeric matrix and the stent strut. The exterior unit normal vectors on ∂Ωw, ∂Ωl, Ωp, and ∂Ωc are denoted, respectively, by ηw, ηl, ηp, and ηc. We neglect the presence of the endothelium as a physical barrier at the interface between the lumen and the wall. This is motivated by the fact that after the im- plantation of the stent the endothelium is injured and loses its ability to regulate the transport of molecules. In order to model the drug diffusion problem we make the following assumptions: 1. The coating of the DES is made of a porous polymeric matrix that encapsu- lates a therapeutic drug. 2. The initial concentration of the drug exceeds the drug solubility; therefore the entire drug is present from the start in the dissolved state. 3. The transport of drug out of the polymer occurs by non-Fickian diffusion. 4. Drug transfer does not affect blood and plasma flows. 5. The transport of drug in the tissue occurs by diffusion, advection, and sat- urable binding. In most polymers the plasticizing effect produced by a penetrant causes a change in its internal structure that may affect the transport process [6]. In this case, many authors agree [6, 7, 11, 16] that the classical Fick’s law does not include this phe- nomenon. Therefore, a modified flux must be considered, given by ∂c ∂t = −∇ · (JF (c) + JNF (σ)), where c denotes the concentration of the penetrant, JF (c) = −(D∇c) represents the Fickian part of the flux, JNF (σ) = −(Dv∇σ) represents the non-Fickian part of the flux, and σ stands for the stress. The parameters D and Dv represent respectively the Fickian diffusion coefficient and the non-Fickian diffusion coefficient. In this work we consider that the non-Fickian diffusion coefficient Dv is constant. But, as shown in [13], a functional expression in terms of the concentration can be obtained for Dv when the non-Fickian flux is interpreted as a convective field. In order to study the viscoelastic behavior of the polymeric coating of the DES in the drug desorption process, we consider a mechanical model (Figure 4) which consists of a Maxwell fluid model in parallel with a free spring. The relaxation modulus can 1188 ELÍAS GUDIÑO, CASSIO M. OISHI, AND ADÉLIA SEQUEIRA be calculated as [4, Chapter 5] E(t) = E0 + E1e −t/τ1 ,(1) where E1 and E0 denote the Young modulus of the spring elements of the Maxwell fluid arm and the free spring, respectively. The parameter τ1 = μ1 E1 is the relaxation time of the Maxwell fluid arm, where μ1 represents the viscosity of the dashpot element. From a practical point of view, this simple model is well adapted to simulating a realistic polymer behavior by fitting the parameters to the values obtained from experimental tests. Let cc denote the concentration of drug in the coating. The stress associated to the desorption and exerted by the polymer is defined as σc(cc) = α (E0 + E1) cc − α E1 τ1 ∫ t 0 e− t−s τ1 cc(s)ds, where α > 0 is the proportionality constant between the strain of the polymer and the concentration, i.e, ε(t) = αcc(t). For a more complex model for non-Fickian transport in a stent, we refer the reader to [17]. The evolution of drug diffusion in the polymeric matrix is described by the fol- lowing equation: ∂cc ∂t = ∇ · (Dc∇cc + Dv∇σc(cc)) in Ωc × (0, T ),(2) where Dc is the constant diffusion coefficient of the drug. The concentration of the drug in the lumen, denoted by cl, can be tracked by the following equation: (3) ∂cl ∂t = ∇ · (Dl∇cl − ulcl) , where Dl denotes the diffusion coefficient of the drug in the lumen and ul is the velocity vector field in the lumen. It is known that drug binding to suitable sites of the extracellular matrix may strongly influence the deposition of the drug in the arterial wall [1, 3, 18, 23, 24, 29]. Therefore, in order to include this phenomenon in the model, we consider a nonlinear saturable reversible binding model [29]. Let b denote the concentration of drug which is bound to nonspecific general extracellular matrix sites, and cw the concentration of drug which is free to diffuse and is transported by plasma. Then, the equations for drug diffusion in the arterial wall are given by ∂cw ∂t = ∇ · (Dw∇cw − uwcw) − kf,wcw(bmax,w − bw) + kb,wbw,(4) ∂bw ∂t = kf,wcw(bmax,w − bw) − kb,wbw,(5) where Dw denotes the diffusion coefficients of the drug in the wall; uw the velocity vector field in the wall; kf,w and kb,w, respectively, the forward and backward rate constants; and bmax,w the local density of binding sites. The concentration of free drug cp and bound drug bp in the plaque are described BOUNDARY CONDITIONS FOR NON-FICKIAN DIFFUSION 1189 by the following system of equations: ∂cp ∂t = ∇ · (Dp∇cp − upcp) − kf,pcp(bmax,p − bp) + kb,pbp,(6) ∂bp ∂t = kf,pcp(bmax,p − bp) − kb,pbp,(7) where Dp, up, kf,p, kb,p, and bmax,p are the same parameters defined above but now applied in the plaque. The previous system of equations is closed with initial conditions cc(0) = c0 in Ωc,(8) cw(0) = 0 in Ωw,(9) bw(0) = 0 in Ωw,(10) cp(0) = 0 in Ωp,(11) bp(0) = 0 in Ωp,(12) cl(0) = 0 in Ωl,(13) and boundary conditions −(Dc∇cc + Dv∇σ(cc)) · ηc = 0 on Γstr,(14) (−Dl∇cl + ulcl) · ηl = 0 on Γin,(15) −Dl∇cl · ηl = 0 on Γout,(16) (−Dw∇cw + uwcw) · ηw = 0 on Γw,(17) −Dw∇cw · ηw = 0 on Γadv.(18) As interface conditions, we consider the continuity of the drug concentrations and the continuity of the normal components of the fluxes, cw = cl, −(Dw∇cw − uwcw) · ηw = (Dl∇cl − ulcl) · ηl on Γl,w,(19) cw = cp, −(Dw∇cw − uwcw) · ηw = (Dp∇cp − upcp) · ηp on Γp,w,(20) cc = cw, −(Dc∇cc + Dv∇σ(cc)) · ηc = (Dw∇cw − uwcw) · ηw on Γc,w,(21) cc = cl, −(Dc∇cc + Dv∇σ(cc)) · ηc = (Dl∇cl − ulcl) · ηl on Γc,l.(22) For the transport across the IEL we impose the continuity of the fluxes and allow for a possible concentration jump, (−Din∇cin + uwcin) · ηIEL = (−Dme∇cme + uwcme) · ηIEL,(23) (−Dme∇cme + uwcme) · ηIEL = PIEL(cme − cin),(24) where ηIEL is the exterior unit normal to the IEL, cin the concentration of free drug in the intima, Din the diffusion coefficient in the intima, cme the concentration of free drug in the media, Dme the diffusion coefficient in the media, and PIEL the permeability of the IEL. Assuming that most of the drug delivery occurs in the direction normal to the stent coating, we can calculate an analytical solution for a one-dimensional diffusion problem in the coating and obtain a Robyn boundary condition to approximate the problem (2), (8) with interface conditions (21) and (22). In this way, we significantly reduce the computational cost of modeling the drug delivery process over realistic three-dimensional stent geometries, as will be shown in section 8. 1190 ELÍAS GUDIÑO, CASSIO M. OISHI, AND ADÉLIA SEQUEIRA 3. Analytical solution of the one-dimensional problem. In what follows we calculate an analytical solution for a general non-Fickian desorption problem. We use Laplace transforms along with the residue theorem, following an approach similar to that in [25]. Let us denote by c1 the concentration of drug; then the evolution of non-Fickian drug diffusion in a polymeric membrane of thickness L can be described by ∂c1 ∂t (x, t) = A1 ∂2c1 ∂x2 (x, t) + A2 ∫ t 0 e− t−s τ1 ∂2c1 ∂x2 (x, s)ds in [0, L] × (0, T ),(25) where A1 = Dc + Dvα(E0 + E1) and A2 = −Dvα E1 τ1 . We close the system with initial conditions c1(x, 0) = c0 for x ∈ [0, L],(26) and boundary conditions − ( A1 ∂c1 ∂x (0, t) + A2 ∫ t 0 e− t−s τ1 ∂c1 ∂x (0, s)ds ) = 0 for t ∈ (0, T ],(27) c1(L, t) = cout for t ∈ (0, T ],(28) where cout is a constant that represents the drug concentration of the external medium. The solution of (25)–(28) is detailed in Appendix A and is given by c1(x, t) = cout + 2π(cout − c0) L2 × 2∑ j=1 +∞∑ n=1 (−1)n−1(n − 1 2 ) cos ( π(n − 1 2 ) x L ) ( A1(sjn + τ−1 1 ) + A2 )2 esjnt sjn ( A1(sjn + τ−1 1 )2 + A2(2sjn + τ−1 1 ) ) ,(29) where s1n = − 1 2 ( τ−1 1 + A1π 2 L2 ( n − 1 2 )) + 1 2 √√√√(τ−1 1 − A1π2 L2 ( n − 1 2 )2 )2 − 4A2 π2 L2 ( n − 1 2 )2 ,(30) s2n = − 1 2 ( τ−1 1 + A1π 2 L2 ( n − 1 2 )2 ) − 1 2 √√√√(τ−1 1 − A1π2 L2 ( n − 1 2 )2 )2 − 4A2 π2 L2 ( n − 1 2 )2 .(31) The time variation of mass of drug inside the membrane can be calculated as M1(t) = ∫ L 0 c1(x, t)dx; BOUNDARY CONDITIONS FOR NON-FICKIAN DIFFUSION 1191 then from (29) we get M1(t) = coutL + 2(cout − c0) L 2∑ j=1 +∞∑ n=1 ( A1(sjn + τ−1 1 ) + A2 )2 esjnt sjn ( A1(sjn + τ−1 1 )2 + A2(2sjn + τ−1 1 ) ) .(32) 4. Multiscale boundary conditions. We note that the concentration c1 de- fined in the previous section does not exactly satisfy the problem for the concentration cc defined in section 2, but motivated by the fact that there is a big difference of the spatial scales between the coating of the stent and the release media, the time evolu- tion of the diffusion problem for cl and cw is much slower than that for cc [10, 32]. Then it is reasonable to use the expression (29) to replace the subproblem of drug diffusion in the coating of the stent ((2), with initial condition (8) and boundary conditions (21) and (22)) with a Robyn boundary condition. In order to couple the coating with the lumen and the wall, we use the continuity of the fluxes to replace the original problem for cc by the boundary conditions −(Dl∇cl − ulcl) · ηl = Jl(t; cl) on Γc,l, −(Dw∇cw − uwcw) · ηw = Jw(t; cw) on Γc,w, where Ji(t; ci) = −A1 ∂c1 ∂x (L, t) − A2 ∫ t 0 e− t−s τ1 ∂c1 ∂x (L, s) ds(33) for i = l, w. We consider that cout is identified with cl on Γc,l and with cw on Γc,w. From (29) and (33) the flux of outgoing drug of the stent can be calculated as Ji(t; ci) = 2π2(ci(t)|Γc,i − c0) L3 2∑ j=1 +∞∑ n=1 (n − 1 2 )2 ( A1(sjn + τ−1 1 ) + A2 )2 sjn ( A1(sjn + τ−1 1 )2 + A2(2sjn + τ−1 1 ) ) × ( A1e sjnt + A2 (sjn + τ−1 1 ) (esjnt − e − t τ1 ) ) (34) for i = l, w. 5. Modeling the blood flow. Since blood is considered an isotropic Newtonian and laminar flow in the coronary arteries where the drug-eluting stents are typically implanted [14, 9], we will assume the steady Navier–Stokes equations to model blood flow in the lumen. In fact, as discussed, e.g., in [2, 28], the relevance of non-Newtonian effects in midsize arteries is negligible. For plasma filtration through the wall, assum- ing that both the intima and the media layers are porous media with isotropic and uniform permeability, we assume the Darcy equation. Let us denote by pi the pressure field in each subdomain i = l, w. Then, the model is defined by the following system of equations: μbΔul − ρb(ul · ∇)ul − ∇pl = 0 in Ωl × (0, T ),(35) ∇ · ul = 0 in Ωl × (0, T ),(36) uw + κw μp ∇pw = 0 in Ωw × (0, T ),(37) ∇ · uw = 0 in Ωw × (0, T ),(38) 1192 ELÍAS GUDIÑO, CASSIO M. OISHI, AND ADÉLIA SEQUEIRA where μb and μp are the dynamic viscosities of the blood and the plasma, respectively, ρb the density of the blood, and κw the intrinsic permeability of the arterial wall. The previous system of equations is closed with boundary conditions ul − uin = 0 on Γin,(39) ul = 0 on Γc,l,(40) (plI − μb∇ul) ηl − p0ηl = 0 on Γout,(41) uw · ηw = 0 on Γw ∪ Γc,w,(42) pw = 0 on Γadv,(43) where uin : R 2 → R 2 is a given function and p0 ∈ R is a constant. As interface conditions, we consider the continuity of the normal components of the stresses and the continuity of the normal components of the velocity, and the tangential component of the velocity in the lumen is set to zero, (plI − μb∇ul) ηl − pwηl = 0 on Γl,w,(44) ul · ηl + uw · ηw = 0 on Γl,w,(45) ηw × ul = 0 on Γl,w.(46) In order to numerically approximate the solution of problem (35)–(46), we employ a finite element method in combination with a Picard’s iteration to linearize the nonlinear advective term in (35). We approximate the plasma filtration problem in Ωw in terms of the pressure pw by quadratic Lagrange elements, recovering a posteriori the velocity field uw. The problem of blood flow is approximated with quadratic and linear Lagrange elements for ul and pl, respectively. 6. A multiscale model for drug delivery. Before presenting the numerical scheme for the drug delivery problem, we briefly outline the model for drug release in the coating and drug transport in the lumen and the wall. The multiscale model is defined by ∂cl ∂t = ∇ · (Dl∇cl − ulcl) in Ωl × (0, T ),(47) ∂cw ∂t = ∇ · (Dw∇cw − uwcw) − kf,wcw(bmax,w − bw) + kb,wbw in Ωw × (0, T ),(48) ∂bw ∂t = kf,wcw(bmax,w − bw) − kb,wbw in Ωw × (0, T ),(49) ∂cp ∂t = ∇ · (Dp∇cp − upcp) − kf,pcp(bmax,p − bp) + kb,pbp in Ωp × (0, T ),(50) ∂bp ∂t = kf,pcp(bmax,p − bp) − kb,pbp in Ωp × (0, T ),(51) closed with initial conditions cw(0) = 0 in Ωw,(52) bw(0) = 0 in Ωw,(53) cp(0) = 0 in Ωp,(54) bp(0) = 0 in Ωp,(55) cl(0) = 0 in Ωl,(56) BOUNDARY CONDITIONS FOR NON-FICKIAN DIFFUSION 1193 and boundary conditions (−Dl∇cl + ulcl) · ηl = 0 on Γin,(57) −Dl∇cl · ηl = 0 on Γout,(58) (−Dw(cw)∇cw + uwcw) · ηw = 0 on Γw,(59) −Dw(cw)∇cw · ηw = 0 on Γadv,(60) cw = cl on Γl,w,(61) −(Dw∇cw − uwcw) · ηw = (Dl∇cl − ulcl) · ηl on Γl,w,(62) cw = cp on Γp,w,(63) −(Dw∇cw − uwcw) · ηw = (Dp∇cp − upcp) · ηp on Γp,w,(64) −(Dw∇cw − uwcw) · ηw = Jw(t; cw) on Γc,w,(65) −(Dl∇cl − ulcl) · ηl = Jl(t; cl) on Γc,l.(66) 7. Overview of the numerical scheme. In order to numerically approximate the solution of problem (47)–(66), we propose a coupled implicit-explicit (IMEX) finite element method, since the problem of drug transport in the blood flow is advection dominated. We combine the IMEX method with the SUPG-method introduced in [5]. We will assume that each subdomain Ωi is a convex polygonal domain, equipped with a family of quasi-uniform triangulations Th,i made of affine simplexes K that are conforming on Γl,w, Γc,l, Γp,w, and Γc,w. On each subdomain we consider finite element spaces defined by Vh,i = {vh,i ∈ C(Ωi) : vh,i|K ∈ P 1(K) ∀K ∈ Th,i} for i = l, w, p. In [0, T ] we consider a uniform grid In = {tn, n = 0, 1, . . . , N}, with t0 = 0, tN = T , and tn − tn−1 = Δt for all n = 1, 2, . . . , N . For the numerical solution of the diffusion problems in the lumen, in the wall, and in the plaque we define the following bilinear forms: al(ch,l, vh,l) = ∫ Ωl (ch,l Δt vh,l + Dl∇ch,l · ∇vh,l + uh,l · ∇ch,lvh,l ) , dl(ch,l, vh,l) = ∫ Ωl (ch,l Δt − DlΔch,l + uh,l · ∇ch,l ) k̂l uh,l · ∇vh,l ‖uh,l‖2 , aw(ch,w, vh,w) = ∫ Ωw (ch,w Δt vh,w + Dw∇ch,w · ∇vh,w + uh,w · ∇ch,wvh,w + ( kf,wcn−1 h,w (bmax,w − bn−1 h,w ) − kb,wbn−1 h,w ) vh,w ) , ab(bh,w, vh,w) = ∫ Ωw ( bh,w Δt − kf,wcn−1 h,w (bmax,w − bn−1 h,w ) + kb,wbn−1 h,w ) vh,w, ap(ch,p, vh,p) = ∫ Ωp (ch,p Δt vh,p + Dp∇ch,p · ∇vh,p + uh,p · ∇ch,pvh,p + ( kf,pc n−1 h,p (bmax,p − bn−1 h,p ) − kb,pb n−1 h,p ) vh,p ) , abp(bh,p, vh,p) = ∫ Ωp ( bh,p Δt − kf,pc n−1 h,p (bmax,p − bn−1 h,p ) + kb,pb n−1 h,p ) vh,p, 1194 ELÍAS GUDIÑO, CASSIO M. OISHI, AND ADÉLIA SEQUEIRA where cn−1 h,i (for i = l, w, p), bn−1 h,w , and bn−1 h,p denote the known concentrations from the previous iteration. The parameter k̂l is defined as k̂l = ul,ehe 2 ( coth(P e e ) − 1 P e e ) , where ul,e denotes the characteristic element velocity, he the characteristic element length, and P e e the local Péclet number given by P e e = ul,ehe 2Dl . The mass transport problems in Ωl, Ωw, and Ωp can be summarized as: find at each time step the functions (ch,l, ch,w, ch,p) ∈ Vh,l × Vh,w × Vh,p such that ch,w = cn−1 h,l on Γl,w, ch,p = cn−1 h,w on Γp,w, and al(ch,l, vh,l) + dl(ch,l, vh,l) = ∫ Ωl ( vh,l + k̂l uh,l · ∇vh,l ‖uh,l‖2 ) cn−1 h,l Δt − ∫ Γc,l Jl(tn; cn−1 h,l )vh,l + ∫ Γl,w (−Dw∇cn−1 h,w + uh,wcn−1 h,w ) · ηlvh,l,(67) aw(ch,w, vh,w) = ∫ Ωw cn−1 h,w Δt vh,w + ∫ Γp,w (−Dp∇cn−1 h,p + uh,pc n−1 h,p ) · ηpvh,w − ∫ Γc,w Jw(tn; cn−1 w,l )vh,w,(68) ap(ch,p, vh,p) = ∫ Ωp cn−1 h,p Δt vh,p(69) for all (vh,l, vh,w, vh,p) ∈ Vh,l × Vh,w × Vh,p. We note that in the IMEX method (67)–(69) we use the information from the previous iteration as boundary data. Therefore, it is possible to decouple the solution of the global discrete problem and separately solve each local subproblems on Ωl, Ωw, and Ωp. 8. Numerical results. In what follows we consider different idealized stent ge- ometries (Figure 5) for our numerical simulations. We approximated each geometry using the information available in the literature, which do not represent faithful copies of the original models. Figure 5(a) corresponds to a generic mesh stent, while Fig- ure 5(b) is an approximation of the Promus element stent, manufactured by Boston Scientific Corporation. The Dynalink-E stent, manufactured by Abbott Vascular, is represented in Figure 5(c), and the Nobori stent, manufactured by the Terumo Corporation, is depicted in Figure 5(d). In each case we consider a 2×14 mm stent with rectangular struts of size 0.15×0.19 mm and a coating thickness of 0.014 mm. The embedment of the stent struts into the arterial wall is assumed at 50% for all the simulations. The viscosity and the density of the blood are set, respectively, to μb = 3.5×10−3 Pa·s and ρb = 1050 kg/m3. The viscosity of the plasma is set to μp = 1.7 × 10−3 BOUNDARY CONDITIONS FOR NON-FICKIAN DIFFUSION 1195 (a) Generic mesh stent. (b) Promus element stent. (c) Dynalink-E stent. (d) Nobori stent. Fig. 5. Idealized stent geometries. Fig. 6. Physical domain for the arterial wall. Pa·s, and the density to ρp = 1025 kg/m3. The permeability of the wall is set to κw = 2 × 10−18 m2. At the inflow of the artery we consider a parabolic velocity profile with vmax = 0.2 m/s. At the outflow of the artery we consider a fixed pressure of 13330 Pa. These parameters correspond to standard physiological data available in the literature [14, Chapter 1], [27, 30] For the arterial wall domain we consider a typical left anterior descending coronary artery. We set the thickness of the healthy part of the intima to 0.2 mm, and the diseased part to 0.65 mm [15]. Since the media is not affected significantly by the disease process, we set its thickness to 0.35 mm for both the healthy and the diseased parts [19]. The thickness of the plaque is set to 0.15 mm (see Figure 6). In this study, the release of the drug Sirolimus is considered. The drug diffusion 1196 ELÍAS GUDIÑO, CASSIO M. OISHI, AND ADÉLIA SEQUEIRA Fig. 7. Time variation of the mass of bound drug in the media layer for the complete model versus the multiscale model. coefficients in the media, lumen, and polymer are set to Dme = 2.5 × 10−10 m2/s, Dl = 1.5×10−9 m2/s, and Dc = 1.2×10−16 m2/s, respectively [23, 24]. The diffusion in the intima is in general approximated to be one hundred times faster than that in the media [19], and therefore we set the drug diffusion coefficient in the intima to Din = 2.5 × 10−8 m2/s. The diffusion in the plaque can be approximately 85 times faster than in the media [19]; therefore, we set the diffusion coefficient of free drug in the plaque to Dp = 2.94 × 10−12 m2/s. The parameters considered for drug binding in the arterial wall correspond to data from [23, 24], set as kf,w = 2 m3/(kg· s), kr,w = 5.2 × 10−3 1/s, and bmax,w = 0.363 mol/m3. The permeability of the IEL is set to PIEL = 1 × 10−9 m/s, as estimated in [19] from the transport of albumin through the IEL. The initial concentration of drug loaded in the stent is set to c0 = 1000 mol/m3. The parameters considered for the viscoelastic behavior of the polymer correspond to data from [12], set as E0 = 1 × 103 Pa·s, E1 = 9 × 103 Pa, and μ1 = 2.25 × 105 Pa·s, resulting in the relaxation time τ1 = 25 s. By Mci , we denote total mass of drug released at time t for the concentration ci, defined as (70) Mci(t) = ∫ Ωi ci(t). In the case of the mesh stent (Figure 5(a)) we were able to generate an appropriate computational mesh for the coating and solve the original model described in section 2. In Figure 7 we present the time variation of the mass of bound drug in the media layer for the original complete model and the proposed multiscale model, normalized with respect to the maximum value of the mass for the complete model. We observe that the multiscale model is a good approximation of the complete model, with a maximum underestimation of approximately 15% for the first three days of the simulation. After this stage, multiscale and complete models produce similar qualitative results, with a maximum overestimation of approximately 10%. The next results were obtained using the proposed multiscale model. Figure 8 shows the concentration of free drug in the arterial wall after 1–4 days, for the Dynalink-E stent (Figure 5(c)). According to this figure, we can observe that during the first days of the release there is a higher accumulation of drug in the intima be- cause the diffusion coefficient in the intima is higher than the diffusion coefficient in BOUNDARY CONDITIONS FOR NON-FICKIAN DIFFUSION 1197 (a) After 1 day. (b) After 2 days. (c) After 3 days. (d) After 4 days. Fig. 8. Concentration of free drug. Fig. 9. Mass of bound drug in the arterial wall layers and in the plaque. the media, and due to the presence of the plaque and the IEL acting as a barrier to diffusion from one layer to the other. In order to investigate the time variation of the mass of bound drug for the Dynalink-E stent (Figure 5(c)), we have plotted in Figure 9 the results considering the intima and the media layers and the plaque, normalized with respect to the maximum value of the mass in the intima. We observe that in each case, first the mass increases rapidly to an upper bound and then it decreases asymptotically. We also note from Figure 9 that during the first hours of drug release the amount of mass 1198 ELÍAS GUDIÑO, CASSIO M. OISHI, AND ADÉLIA SEQUEIRA Fig. 10. Mass of free drug in the media for different stent geometries. Fig. 11. Mass of free drug in the media layer as a function of Dv. in the intima is higher than in the other areas. Also as expected, the amount of mass in the plaque is smaller than in the other areas for the complete time domain. Figure 10 displays the mass of free drug in the media for the different stent models, normalized with respect to the maximum value of the mass for the Dynalink-E stent (Figure 5(c)). We can see from Figure 10 that each stent has a different quantitative release profile. These variations account for the fact that the physical barrier to the flow in each case generates a different velocity field in both the lumen and the wall, influencing the mass transport. The effect of the Dv on the time variation of the mass of free drug in the media for the Nobori stent (Figure 5(d)) is analyzed in Figure 11. In this figure, the mass is normalized with respect to the maximum value of the mass when Dv = 9 × 10−19 s. The case Dv = 0 corresponds to Fickian diffusion. We observe that the mass is an increasing function of Dv. We note also that an increase in Dv amplifies the effect of the relaxing flux term JNF . Figure 12 shows the time variation of the mass of free drug in the media as a function of E0, for the Nobori stent (Figure 5(d)), normalized with respect to the maximum value of the mass when E0 = 50000 Pa. We observe that the mass is an increasing function of E0. Like in the previous case, an increase in E0 amplifies the effect of the relaxing flux term JNF . In order to study the influence of the plaque in the transport problem, in Fig- ure 13 we show the time variation of the mass of bound drug in the media for three BOUNDARY CONDITIONS FOR NON-FICKIAN DIFFUSION 1199 Fig. 12. Mass of free drug in the media layer as a function of E0. Fig. 13. Mass of bound drug in the media layer (plaque vs. no plaque). different cases, considering the mesh stent (Figure 5(a)), normalized with respect to the maximum value of the mass in the absence of the plaque in the intima. The case of the solid calcified plaque is represented by excluding from the computational domain of the arterial wall Ωw the domain of the plaque Ωp. We observe that the presence of the plaque is important for the study of the drug deposition in the arterial wall. In particular, we point out from Figure 13 that there is a considerable difference in the quantitative behavior during the time domain considered. In the previous figure we note a slight difference of the drug accumulation between the solid and soft plaques. We believe that this is motivated by the fact that we are considering the same values for the parameters associated to binding in the media, the intima, and the plaque. It is known that a lipid accumulation can alter the microviscosity of the tissue and local binding properties [19]. However, to the authors’ knowledge, there is not enough information in the literature on how to estimate the parameters for binding in the plaque. Finally, to supply further evidence concerning the time variation of the mass of bound drug in the media, we have investigated the influence of the forward rate constant kf,p; see Figure 14. To obtain these results, the Boston stent was considered, and the mass was normalized with respect to the maximum value of the mass when kf,p = 0.1. We observe that the mass is a decreasing function of kf,p. We note that the changes in the parameter kf,p can influence the deposition of drug in the media. Therefore, some effort must be made in the future in order to understand how to 1200 ELÍAS GUDIÑO, CASSIO M. OISHI, AND ADÉLIA SEQUEIRA Fig. 14. Mass of bound drug in the media layer as a function of kf,p. estimate the parameters for binding in the plaque. 9. Conclusions. In this paper we presented a multiscale model for non-Fickian drug delivery in a coronary DES. Assuming that most of the drug delivery occurs in the direction normal to the coating, we approximated the diffusion problem inside the coating by a Robin boundary condition. We observed numerically that the reduced model is a good approximation of the complete model. The main advantage of our approach is that it allows the incorporation of ex- perimental rheological information about the polymer-solvent system into the trans- port problem, and it will significantly reduce the computational cost of simulations over realistic three-dimensional stent geometries. Finally, we remark that the Robin boundary condition for the outgoing flux of drug deduced in this work is not restricted to stents; in fact, it can be relevant in other applications where a solvent is slowly released from a thin polymeric coating. Appendix A. Derivation of the solution. In this appendix we calculate the solution of the problem (25)–(28) presented in section 3. Let c̄1 represent the Laplace transform of c1; then from (25) and (26) we deduce ∂2c̄1 ∂x2 (x, s) − λ2(s)c̄1(x, s) = −c0λ 2(s) s ,(71) where λ2(s) = s(s + τ−1 1 ) (A1(s + τ−1 1 ) + A2) .(72) Solving (71) using the method of variation of parameters yields c̄1(x, s) = C1(s)e−λ(s)x + C2(s)eλ(s)x + c0 s . From the boundary condition (27) we get that c̄1(x, s) = C1(s) cosh (λ(s)x) + c0 s ,(73) while from (28) it follows that C1(s) = (cout s − c0 s )( 1 cosh (λ(s)L) ) . BOUNDARY CONDITIONS FOR NON-FICKIAN DIFFUSION 1201 Substituting the previous expression for C1(s) into (73) gives c̄1(x, s) = cout cosh (λ(s)x) s cosh (λ(s)L) + c0(cosh (λ(s)L) − cosh (λ(s)x)) s cosh (λ(s)L) .(74) In order to calculate the inverse Laplace transform of (74) with the aid of the residue theorem, we begin by considering the first term of (74). There is a singularity at s = 0, and its residue can be calculated as lim s→0 cout cosh (λ(s)x) est cosh (λ(s)L) = cout.(75) All the other singularities can be found by solving cosh (λ(s)L) = 0, which leads to λ(sn)L = iπ ( n − 1 2 ) for n = ±(1, 2, 3, . . .), where i represents the imaginary unit. Then, from (72) we get that the roots must satisfy the equation s2 n + ( τ−1 1 + A1π 2 L2 ( n − 1 2 )2 ) sn + π2 L2 ( n − 1 2 )2 (A1τ −1 1 + A2) = 0, which leads to (30) and (31). The residues at s = sjn for j = 1, 2 can be calculated as( cout cosh (λ(sjn)x) esjnt sjn ) lim s→sjn ( (s − sjn) cosh (λ(s)L) ) = 2cout cosh (λ(sjn)x) esjnt ( A1(sjn + τ−1 1 ) + A2 )2 λ(sjn) Lsjn sinh (λ(sjn)L) ( A1(sjn + τ−1 1 )2 + A2(2sjn + τ−1 1 ) ) .(76) From (75) and (76) we deduce L−1 [ cout cosh (λ(s)x) s (cosh (λ(s)L)) ] = cout + 2πcout L2 × 2∑ j=1 +∞∑ n=1 (−1)n−1(n − 1 2 ) cos ( π(n − 1 2 ) x L ) ( A1(sjn + τ−1 1 ) + A2 )2 esjnt sjn ( A1(sjn + τ−1 1 )2 + A2(2sjn + τ−1 1 ) ) .(77) Concerning the second term of (74), we note that the singularity at s = 0 is removable. For the singularities at s = sjn for j = 1, 2, the residues can be calculated as ( c0(cosh (λ(sjn)L) − cosh (λ(sjn)x))esjnt sjn ) lim s→sjn ( (s − sjn) cosh (λ(s)L) ) = − 2πc0(−1)n−1(n − 1 2 ) cos ( π(n − 1 2 ) x L ) ( A1(sjn + τ−1 1 ) + A2 )2 esjnt L2sjn ( A1(sjn + τ−1 1 )2 + A2(2sjn + τ−1 1 ) ) .(78) 1202 ELÍAS GUDIÑO, CASSIO M. OISHI, AND ADÉLIA SEQUEIRA Clearly s2n ≤ 0 for all n = 1, 2, 3, . . . . For the s1n, if we impose the condition A1 + A2 ≥ 0,(79) then s1n ≤ 0 for all n = 1, 2, 3, . . . . We note that the condition (79) is a reasonable assumption. Indeed, from the mathematical point of view it means that the parabolic part of (25) has to dominate over the hyperbolic part of the equation. From the physical point of view it means that the non-Fickian part of (25) cannot dominate the equation because it would lead to a negative total flux. The roots s1n and s2n all lie on the left of s = 0, provided that (79) holds. 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View publication statsView publication stats https://doi.org/10.1016/S0002-9149(02)02993-4 https://doi.org/10.2174/1567201052772906 https://doi.org/10.1016/j.apm.2017.01.057 https://doi.org/10.1007/s13239-010-0016-4 https://doi.org/10.1007/s00466-011-0633-2 https://doi.org/10.1007/s00466-011-0633-2 https://doi.org/10.1016/j.jbiomech.2016.03.017 https://doi.org/10.1016/j.jbiomech.2015.12.015 https://doi.org/10.1016/j.jbiomech.2015.12.015 https://doi.org/10.1016/j.mbs.2014.06.016 https://doi.org/10.1016/j.jconrel.2015.09.025 https://doi.org/10.1016/j.jconrel.2015.09.025 https://doi.org/10.1007/s10910-016-0618-7 https://doi.org/10.1007/s10910-016-0618-7 https://doi.org/10.1016/j.ijheatmasstransfer.2015.06.061 https://doi.org/10.1007/s10439-015-1456-7 https://doi.org/10.1080/10255840601071087 https://doi.org/10.1007/s13239-016-0269-7 https://doi.org/10.1111/j.1365-2184.2009.00602.x https://doi.org/10.1016/j.jbiomech.2010.01.033 https://doi.org/10.1016/j.ijcard.2004.01.021 https://doi.org/10.1137/07070214X https://doi.org/10.1137/07070214X https://doi.org/10.1016/j.cma.2008.07.019 https://doi.org/10.1016/j.cma.2008.07.019 https://www.researchgate.net/publication/318894176 Introduction Modeling drug delivery Analytical solution of the one-dimensional problem Multiscale boundary conditions Modeling the blood flow A multiscale model for drug delivery Overview of the numerical scheme Numerical results Conclusions Appendix A. 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