In this paper we describe a theoretical mathematical model of dual drug delivery from a durable

polymer coated medical device. We demonstrate how the release rate of each drug may in principle

be controlled by altering the initial loading configuration of the two drugs. By varying the underlying

microstructure of polymer coating, further control may be obtained, providing the opportunity to

tailor the release profile of each drug for the given application

McGinty, Sean

Pontrelli, Giuseppe

English

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5th International Conference on Computational and Mathematical Biomedical Engineering - CMBE201710–12 April 2017, United StatesP. Nithiarasu, A.M. Robertson (Eds.)MATHEMATICAL MODELLING OF VARIABLE POROSITYCOATINGS FOR DUAL DRUG DELIVERYSean McGinty1 and Giuseppe Pontrelli21Division of Biomedical Engineering, University of Glasgow, Glasgow, UK.2Istituto per le Applicazioni del Calcolo - CNR, Roma, Italy.SUMMARYIn this paper we describe a theoretical mathematical model of dual drug delivery from a durablepolymer coated medical device. We demonstrate how the release rate of each drug may in principlebe controlled by altering the initial loading configuration of the two drugs. By varying the underlyingmicrostructure of polymer coating, further control may be obtained, providing the opportunity totailor the release profile of each drug for the given application.Key words: Drug delivery, variable porosity, coupled partial differential equations1 INTRODUCTIONThe topic of controlled drug delivery has received much attention in recent years, for example in thedesign of tablets [1] and local drug delivery devices such as stents [2], transdermal patches [3], contactlenses [4] and orthopaedic implants [5]. A key predictor of performance is the drug release profile.Drug delivery is usually initially tested in an in-vitro environment to assess the range of releaseprofiles that can be obtained and to test the repeatability of the drug coating process, before conductinganimal or human trials. The disadvantage of a purely experimental approach is that the experimentsneed to be repeated whenever system parameters are changed. A mathematical model, on the otherhand, can potentially provide useful insight into the effect of varying system parameters. In thispaper we focus on dual drug delivery, i.e. the delivery of two drugs from a tablet/device. Dependingon the particular application in question, it may be desirable for the drugs to be released at similarrates, or perhaps one of the drugs released rapidly with the other being eluted over a longer periodof time. In the case of drug-eluting stents, for example, devices which release an anti-proliferativeand a ‘pro-healing’ drug have been proposed, whilst a combination of two of the early drug-elutingstent drugs - Paclitaxel and sirolimus - has also been suggested. Motivated by today’s advances inmicro and nanotechnology, we propose variable porosity multi-layer coatings as an additional meansof controlling the dual drug delivery and tailoring the release profile to the desired application.2 METHODOLOGY2.1 Dual drug deliveryIn Figure 1 we display the situation that we wish to model: either a single durable polymer coatinglayer containing two drugs mixed together, or two coating layers with the two drugs loaded separatelyin each layer. Since the coating thickness (L) is typically considerably smaller than the lateral dimen-sions, we restrict our attention to a one-dimensional model, with x representing the single spatialvariable. In this preliminary analysis, we assume that the polymer/drug coating has been designedsuch that diffusion is the principal release mechanism. We further assume that both drug species aredilute, i.e. the presence of one drug does not influence the transport of the other. Under this as-sumption, the transport of both drugs may be treated separately and we therefore have two diffusionequations for the concentrations, c1 and c2, of drugs 1 and 2, respectively:184Figure 1: Sketch of two possible initial configurations of a dual drug delivery system. At initial time, the drugsmay be uniformly mixed (left, Case 1) in a single layer of thickness L or loaded separately over two layers(right, Case 2), with δ the thickness of the first layer. x = 0 and x = L correspond to the interface with thedevice backing and release medium, respectively. Figure not to scale.∂c1∂t= D1∂2c1∂x2,∂c2∂t= D2∂2c2∂x2, 0 < x < L, t > 0. (1)Here,D1 andD2 are the effective diffusion coefficients of the respective drugs: these may incorporatemany other physical and microstructural parameters, such as porosity and tortuosity (see Section 2.2)Since our focus is to model in-vitro drug release, we impose an impermeable condition at x = 0 andzero concentration at x = L, where it is assumed that sink conditions are maintained:−D1∂c1∂x= −D2∂c2∂x= 0, x = 0, t > 0, (2)c1 = c2 = 0, x = L, t > 0. (3)We model three sets of initial conditions, describing Case 1 and Case 2 above (Figure 1) as well as amore general case (Case 3).Case 1: drugs are uniformly distributed in the coating at initial constant concentrations c01 and c02:c1(x, 0) = c01, c2(x, 0) = c02, 0 ≤ x ≤ L. (4)Case 2: drugs are initially contained within two separate layers at constant concentrations:c1(x, 0) ={c01, 0 ≤ x ≤ δ0, δ < x ≤ L, c2(x, 0) ={0, 0 ≤ x ≤ δc02, δ < x ≤ L.(5)Case 3: Drugs are assigned general spatially varying initial concentration profiles f(x) and g(x):c1(x, 0) = f(x), c2(x, 0) = g(x), 0 ≤ x ≤ L. (6)In the assumption of dilute concentrations, c1 and c2 may be solved for independently. By separationof variables, concentrations and masses may be expressed analytically as Fourier expansions.2.2 Varying the underlying microstructureThe effective diffusion coefficients D1 and D2 are not only drug-dependent, but they will also varydepending on the underlying microstructure of the coating. These are related to the porosity φ andtortuosity τ of the coating, and to the free diffusion coefficient of the drug (Df1 and Df2 ) via:D1 =φDf1τ, D2 =φDf2τ.Therefore, it is clear that by simulating release profiles for various values of D1 and D2 we cannot only capture the effect of different drugs, but also different coating microstructural properties,providing further flexibility for tailoring the release profiles for a given application. In addition, thedrugs may be released from multi-layer coatings, with varying microstructure between layers (e.g.Figure 2).185Figure 2: Two adjacent polymer coatings with different microstructural properties.3 RESULTS AND CONCLUSIONSFor the purposes of this study, we are interested in assessing the effect of the loading configurationon the release profiles of each drug, when the initial total mass of drug per unit area (M01 and M02 )are assigned fixed quantities:∫ L0c1(x, 0)dx = M01 ,∫ L0c2(x, 0)dx = M02 . (7)In Figure 3 we consider two uniformly mixed drugs (Case 1) and demonstrate that markedly differentrelease profiles may be obtained simply by varying the effective diffusion coefficients and initialloading masses of the drugs. In Figure 4 we consider two drugs loaded separately in two adjacent0 5 10 15 200  10 20 30 40 50 time (hours)Cumulative mass of drug released (%)M1M2Figure 3: Release profiles of drug 1 and 2 uniformly mixed (Case 1).layers (Case 2) and demonstrate that we can also in this case obtain very different release profiles. Inparticular, simply by varying the relative thickness of the layers, the release of the drug in the firstlayer may be delayed, which could be desirable in certain applications. With this relatively simplemathematical model, it is straightforward to assess the effect of varying the drug (via the diffusioncoefficient) and initial loading mass and configuration on the resulting release profile. Moreover, byvarying the effective diffusion coefficients, the effect of changes in underlying porosity and tortuosityon the release profiles may be investigated.1860 5 10 15 200  10 20 30 40 50 time (hours)Cumulative mass of drug released (%)M1M20 5 10 15 200  10 20 30 40 50 time (hours)Cumulative mass of drug released (%)M1M2Figure 4: Release profiles of drug 1 and 2 (Case 2). Left: δ = L/4. Right: δ = 3L/4.REFERENCES[1] A. Streubel, J. Siepmann, N.A. Peppas, R. Bodmeier, Bimodal drug release achieved with multi-layer matrix tablets: transport mechanisms and device design, J. Contr. Release, 69:455-468,2000.[2] S. McGinty, A decade of modelling drug release from arterial stents, Math. Biosci. 257: 80–90,2014[3] M.R. Prausnitz, R. Langer, Transdermal drug delivery, Nat. Biotechnol., 26(11): 1261–1268,2008[4] W.E. Zhang, M.R. Prausnitz, A.U. Edwards, Model of transient drug diffusion across cornea, J.Control. Release, 99: 241–258, 2004[5] J.A. Lyndon, B.J. Boyd, N. Birbilis, Metallic implant drug/device combinations for controlleddrug release in orthopaedic applications, J. Control. Release 179: 63–75,2014.187

Mathematical Modelling of Variable Porosity Coatings for Dual Drug Delivery

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Mathematical Modelling of Variable Porosity Coatings for Dual Drug Delivery

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