# A Two-Species Finite Volume Scalar Model for Modeling the Diffusion of Poly(lactic-co-glycolic acid) into a Coronary Arterial Wall from a Single Half-Embedded Drug Eluting Stent Strut

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- A theoretical methodology for computational modeling of the diffusion of PLGA into a coronary arterial wall from a single half-embedded drug eluting stent strut.
- A computational drug diffusion model that considers the pharmaco-kinetic reactions in the arterial wall as equilibrium reversible binding reaction source terms for the free and bound-drug.
- Validation of the reported computational model via simulation-based results from a finite difference model developed from methods reported in previous works.
- A computational drug diffusion model that provides an understanding of the relationship between drug physicochemical properties and the local transport environment which is crucial to the success of new stent designs.
- The model reported in this work is the second reported model in literature that successfully uses an ANSYS FLUENT user-defined scalar (UDS) model to model the diffusion of the free and bound drug in the arterial wall with reversible binding source terms. Additionally, this is the first reported model to use a UDS model to incorporate the polymer layer in the computational domain.

## 2. Material and Methods

#### 2.1. Model Development

#### 2.2. Boundary Conditions and Meshing

**Figure 2.**Model diagram of the half-embedded stented arterial model: (

**a**) Surface model created in ANSYS SpaceClaim and (

**b**) mesh computational domain (model used in the simulations is a finer meshed model). P

_{1}is the location point of interest for evaluating the concentration profiles over time). At the PLGA coating and artery wall interface, the following flux condition is applied:

_{wp}is the mass transfer resistance, C

_{w}, is the drug concentration on the arterial wall side of the interface, is the partition coefficient, and C

_{p}is the perivascular drug concentration. The right and left sides of the arterial zone domain are treated as symmetrical boundary conditions, as shown in Figure 2.

#### 2.3. Plasma Flow

^{3,}and the dynamic viscosity is 0.0035 Pa∙s [2,3,4,5,6,7] at the standard body temperature (37 °C) Whale et al. [40] examined the effects of aging and pressure on the Darcy permeabilities of human aortic walls. A representative value of 2.0 × 10

^{−18}m

^{2}was implemented for this work. Equation (2) is subject to the incompressibility constraint. As described above, the vessel lumen is not a part of the computational domain. This introduces an additional assumption because luminal flow decreases axial non-uniformity of the drug in the artery wall [37,41]. The degree of non-uniformity was observed to increase with increasing the aspect ratio of the stent strut [41]. The impact of this assumption is therefore minimized in the case of square struts and/or stents with an abluminal coating. The next sections discuss the drug transport modeling methodology.

#### 2.4. Drug Transport in the PLGA Coating and Arterial Domains

_{b}is formed by associating the free drug C

_{f}with the available binding sites S

_{0}. The bound drug is immobilized, and only the free drug can diffuse. The reversible binding process, however, does not provide a mechanism for drug consumption (e.g., drug uptake by tissue cells), which can be characterized by drug internalization. This work did not take into consideration the internalization of the drug.

**Drug Binding:**

**Free Drug in the PLGA Coating Domain:**

**Free Drug in the Arterial Domain:**

**Bound Drug in the Arterial Domain:**

**Drug Transport Boundary Conditions:**

_{a}and k

_{d}are the rates of association and dissociation constants, respectively. S

_{0}is the net tissue binding capacity. D

_{T}is the is the true diffusivity of the free drug diffusing into the arterial wall and is expressed as:

_{free}and D

_{eff}(Equations (9) and (10)) are the coefficients of free and effective diffusivity, respectively. R

_{d}= (k

_{d}/k

_{a}) is the equilibrium dissociation constant.

_{b}= 1.0 × 10

^{−7}D

_{u}. The 1.0 × 10

^{−7}pre-factor was adopted from the study of Horner et al. [25] and was used to decrease the true drug diffusivity until the bound drug distribution became independent of the diffusivity results. A cartesian coordinate system was used to specify the components of the diffusion tensor D in the x and y directions, corresponding to D

_{xx}and D

_{yy}, respectively. Both D

_{u}and D

_{b}have two independent components:

_{yy}is larger than D

_{xx}.

**User Defined Scalar and Numerical Modelling**

_{k}. When ϕ

_{k}is fixed in a given cell, the UDS scalar transport is not solved, and the cell is not included when the residual sum is computed. For the present work, the value of the initial drug concentration, C

_{0,}was fixed, and the coating diffusivity was allowed to vary as a function of ϕ, time, and molecular weight, also allowed to vary with time. For the bound drug transport equation, the mass transport was deselected, which allowed the convection term to be neglected, thus making the bound drug immobile. The same was done for the internalized drug transport equation. The source terms S

_{ϕk}include the reversible binding reactions in Equations (6) and (7).

^{−6}for the momentum equations. The drug transport problem was solved using a transient solver, with the velocity field fixed for all time steps. A first-order implicit time integrator was used along with the QUICK up-winding scheme for spatial discretization of the scalar transport equations. Smooth convergence was observed when using the default URFs of 1.0 for both transport equations. The convergence criterion for concentration at each time. All plasma flow and drug concentration simulations were conducted with a time step of 1 picosecond and resulted in a simulation run time of at least 15 days. All simulations were conducted on an ASUS ROG STRIX desktop computer (ASUS ROG; Taiwan) with 12 cores and an NVIDIA GeForce GTX 1660 TI graphics card. All simulations were conducted in parallel with 11 CPU cores and the NVIDIA graphics card.

#### 2.5. Non-Dimensional Pre-Analyses

_{y}is the transmural filtration velocity, Pe

_{T}= [V

_{y}δ/(D

_{T})], and Da = [(k

_{a}S

_{0}δ

^{2})/(D

_{T})] are the Peclet and DamKöhler numbers in the tissue. Here, ε

_{1}= (R

_{d}/C

_{0}), ε

_{2}= (C

_{0}/S

_{0}), and ε

_{3}= (R

_{d}/S

_{0}) are three scaling parameters. Pe

_{c}= [V

_{y}(h

^{2}/δ)]/D

_{c}is the Peclet number in the coating of the strut, and h is the thickness of the coating of the strut.

_{1}, τ

_{2}, and τ

_{3}, corresponding to diffusion coating, transmural diffusion, and the binding reaction. The characteristic time’s scales are shown below:

_{1}, 10

^{3}–10

^{5}s, τ

_{2}, 10

^{3}–10

^{5}s, and τ

_{3}10

^{2}s, which indicate that reversible binding is very fast compared to diffusion. The relative significance of diffusion and reversible binding in the wall is also implied by their corresponding dimensionless groups DamKohler and Peclet numbers. Compared with the coefficient of the transmural diffusion component (which is one), the reaction components have very large DamKöhler numbers on the order of 10

^{2}–10

^{4}, which also implies that the binding reactions perform a very strong role in the spatiotemporal dynamics. The non-dimensional analysis is provided in the Appendix A of this paper.

#### 2.6. Grid Independence Analysis, Modelling Parameters, and Validation and Verification

_{t}= 0.00001 s was used for the sake of computational power and time.

_{T}= 2, Da = 40, and ε

_{2}= 100. As shown in the plot in Figure 3b, the overall trend in the growth of the plots is similar; however, the finite volume model has a higher concentration profile (approximately 10 percent higher). This could be attributed to the fine mesh used in the finite volume model, the implementation of the PGLA layer, the application of a diffusion pre-factor, and the application of the diffusive tensor. The key takeaway is that the free drug concentration trend behaves as expected when compared to a finite difference model that was developed based on work reported in the literature. The next section discusses the results of this work.

## 3. Results

_{T}and Da numbers over the course of 400 h (16.67 days). The next section discusses the initial diffusion flow modeling results.

#### 3.1. Interstitial Flow into the Arterial Wall

#### 3.2. Free and Bound Drug Concentration Profiles with Erosion and Interstitial Flow

#### 3.3. Free and Bound Drug Concentration Profiles with Erosion and Convection

_{1}= (R

_{d}/C

_{0}) are presented in Figure 9 and Figure 10, respectively, and the same for different values of ε

_{2}= (C

_{0}/B

_{M}), which are shown in Figure 11 and Figure 12, respectively. The value of the scaling parameter ε

_{1}, decreases with a decrease in the dissociation rate constant k

_{d}and with an increase in the association rate constant k

_{a}depending on R

_{d}= (k

_{d}/k

_{a}). Additionally, ε

_{2}increases with decreasing S

_{0}(while keeping c

_{0}fixed).

**Figure 8.**Contours of drug diffusion into the arterial wall at: (

**a**) 4 days, (

**b**) 8 days, and (

**c**) 16.67 days.

_{1}for Pe

_{T}= 2, Da = 40, ε

_{2}= 100, up to a certain time and, thereafter, no significant changes occurred. It can be concluded and justified that, as ε

_{1}decreases, the rate of reversible binding (k

_{d}) decreases and/or the rate of forward binding increases, which lowers the mean concentration of the free drug.

**Figure 9.**Distribution of normalized mean bound drug concentration for different values of ε

_{1}at Pe

_{T}= 2, Da = 40, ε

_{2}= 100.

_{1}, which is attributed to the increase in the rate of forward binding and/or to the decrease in the rate of reversible binding.

**Figure 10.**Distribution of normalized weighted average free drug concentration for different values of ε

_{1}at Pe

_{T}= 2, Da = 40, ε

_{2}= 100.

_{2}(i.e., net tissue binding potential on the mean concentrations of free and bound drug) are displayed in Figure 11 and Figure 12, respectively. As previously mentioned, ε

_{2}increases with decreasing binding potential.

**Figure 11.**Distribution of normalized weighted averaged bound drug concentration for different values of ε

_{2}at Pe

_{T}= 2, Da = 40, ε

_{1}= 0.001.

_{2}= 100, but the concentration reaches a quasi-steady state for weaker binding capacity (ε

_{2}= 1000) as compared to the other cases.

**Figure 12.**Distribution of normalized weighted averaged free drug concentration for different values of ε

_{2}at Pe

_{T}= 2, Da = 40, ε

_{1}= 0.001.

_{2}≤ 100, the quasi-equilibrium is not fully established until approximately 17 days, while the PLGA coating has eroded significantly. On the other hand, in the case that bound drug for ε

_{2}≥ 10, the quasi-equilibrium is attained very rapidly.

#### 3.4. Average Weighted Concentration Results for Varying Tortuosity

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

C_{f} | Free drug |

C_{b} | Bound drug |

C_{p} | Perivascular drug concentration |

C_{w} | Wall drug concentration |

C_{0} | Initial drug concentration |

D | Outer diameter of the artery |

D_{C} | Coefficient of the coating diffusion |

D_{a} | Dimensionless DamKöhler number in the tissue |

D_{free} | Coefficient of free diffusivity |

D_{eff} | Coefficient of effective diffusivity |

D_{T} | True diffusivity of the free drug |

J_{wp} | PLGA flux parameter |

L_{x} | Arterial domain length |

L_{y} | Arterial domain wall thickness |

L_{sx} | Stent length |

L_{sy} | Stent thickness |

k_{a} | Tissue binding capacity |

k_{d} | Dissociation rate constant |

Pe_{C} | Dimensionless Peclet number in the coating |

Pe_{T} | Dimensionless Peclet number in the tissue |

R_{d} | Equilibrium dissociation constant |

R_{wp} | Mass transfer resistance |

S_{0} | Available binding sites |

T | Time |

V_{y} | Transmural filtration velocity |

x | x-coordinate |

y | y-coordinate |

δ | Strut dimension |

ε | Porosity of the arterial wall |

ε_{1} and ε_{2} | Dimensionless scaling parameters |

τ | Tortuosity of the arterial wall |

τ_{1}, τ_{2}, and τ_{2} | Characteristic time scales |

PLGA | Poly(lactic-co-glycolic acid) |

UDS | User defined scalar |

## Appendix A

**Free-drug in the PLGA Coating Domain:**$$\frac{\partial {C}_{f}}{\partial t}={D}_{C}(\frac{{\partial}^{2}{C}_{f}}{\partial {x}^{2}}+\frac{{\partial}^{2}{C}_{f}}{\partial {y}^{2}}),$$**Free-drug in the Arterial Domain:**$$\frac{\partial {C}_{f}}{\partial t}=-v\left(\frac{{\partial}^{}{C}_{f}}{\partial {x}^{}}+\frac{\partial {C}_{f}}{\partial {y}^{}}\right)+{D}_{T}\left(\frac{{\partial}^{2}{C}_{f}}{\partial {x}^{2}}+\frac{{\partial}^{2}{C}_{f}}{\partial {y}^{2}}\right)-\left[{k}_{a}{C}_{b}\left({S}_{0}-{C}_{b}\right)-{k}_{d}{C}_{b}\right],$$**Bound-drug in the Arterial Domain:**$$\frac{\partial {C}_{b}}{\partial t}=\left[{k}_{a}{C}_{f}\left({S}_{0}-{C}_{b}\right)-{k}_{d}{C}_{b}\right],$$

**Scaled free-drug concentration time derivative:**$$\frac{\partial {C}_{f}}{\partial t}=\frac{\partial \left({C}_{f}^{*}{C}_{0}\right)}{\partial ({t}^{*}\delta /{V}_{y})}=\frac{{C}_{0}{V}_{y}}{\delta}\frac{\partial {C}_{f}^{*}}{\partial {t}^{*}}$$**Scaled bound-drug concentration time derivative:**$$\frac{\partial {C}_{b}}{\partial t}=\frac{\partial \left({C}_{b}^{*}{S}_{0}\right)}{\partial ({t}^{*}\delta /{V}_{y})}=\frac{{S}_{0}{V}_{y}}{\delta}\frac{\partial {C}_{b}^{*}}{\partial {t}^{*}}$$**Scaled free-drug concentration first-order x-direction derivative:**$$\frac{\partial {C}_{f}}{\partial x}=\frac{\partial \left({C}_{f}^{*}{C}_{0}\right)}{\partial ({x}^{*}\delta )}=\frac{{C}_{0}}{\delta}\frac{\partial {C}_{f}^{*}}{\partial {x}^{*}}$$**Scaled free-drug concentration first-order y-direction derivative:**$$\frac{\partial {C}_{f}}{\partial y}=\frac{\partial \left({C}_{f}^{*}{C}_{0}\right)}{\partial ({y}^{*}\delta )}=\frac{{C}_{0}}{\delta}\frac{\partial {C}_{f}^{*}}{\partial {y}^{*}}$$

**Scaled free-drug concentration second-order x-direction derivative:**$$\frac{{\partial}^{2}{C}_{f}}{\partial {x}^{2}}=\frac{\partial}{\partial x}\frac{\partial {C}_{f}}{\partial x}=\frac{\partial}{\partial \left({x}^{*}\delta \right)}\frac{\partial \left({C}_{f}^{*}{C}_{0}\right)}{\partial \left({x}^{*}\delta \right)}=\frac{{C}_{0}}{{\delta}^{2}}\frac{{\partial}^{2}{C}_{f}}{\partial {x}^{*2}}$$**Scaled free-drug concentration second-order y-direction derivative:**$$\frac{{\partial}^{2}{C}_{f}}{\partial {y}^{2}}=\frac{\partial}{\partial y}\frac{\partial {C}_{f}}{\partial y}=\frac{\partial}{\partial \left({y}^{*}\delta \right)}\frac{\partial \left({C}_{f}^{*}{C}_{0}\right)}{\partial \left({y}^{*}\delta \right)}=\frac{{C}_{0}}{{\delta}^{2}}\frac{{\partial}^{2}{C}_{f}}{\partial {y}^{*2}}$$

_{0}V

_{y}/δ yields the following:

_{C}= [V

_{y}δ/(D

_{C})]. The scaled free-drug concentration derivatives (Equations (A5) and (A7) are now substituted into the free drug transport equation into the arterial wall domain:

_{0}V

_{y}/δ, factoring out the first- and second-order derivative constants, and substituting the equilibrium constant R

_{d}= (k

_{d}/k

_{a}) yields the following:

_{T}= [V

_{y}δ/(D

_{T}) and Da = [(k

_{a}S

_{0}δ

^{2})/(D

_{T})] are in the tissue and that ε

_{1}= (R

_{d}/C

_{0}) is an additional scaling parameter and yields the final non-dimensional form of the free-drug transport equation in the arterial domain as shown below:

_{0}Vy/δ and entering the relations for ε

_{2}, Da, and Pe yields.

## References

- Edwards, M.; Hewlin, R.L., Jr.; Smith, M. A 2-D Transient Computational Multi-Physics Model for Analyzing Magnetic and Non-Magnetic (Red Blood Cells and E. Coli Bacteria) Particle Dynamics in a Travelling Wave Ferro-Magnetic Microfluidic Device. ASME J. Eng. Sci. Med. Ther. Diagn.
**2023**, 1–47. [Google Scholar] [CrossRef] - Hewlin, R.L., Jr.; Edwards, M. Continuous Flow Separation of Red Blood Cells and Platelets in a Y-Microfluidic Channel Device with Saw-Tooth Profile Elec-trodes via Low Voltage Dielectrophoresis. Curr. Issues Mol. Biol.
**2023**, 45, 3048–3067. [Google Scholar] [CrossRef] [PubMed] - Hewlin, R.L., Jr.; Ciero, A.; Kizito, J.P. Development of a Two-Way Coupled Eulerian-Lagrangian Computational Magnetic Nanoparticle Targeting Model for Pulsatile Flow in a Patient-Specific Diseased Left Carotid Bifurcation Artery. Cardiovasc. Eng. Technol.
**2019**, 10, 299–313. [Google Scholar] [CrossRef] [PubMed] - Stanley, N.; Ciero, A.; Timms, W.; Hewlin, R.L., Jr. A 3-D Printed Optically Clear Rigid Diseased Carotid Bifurcation Arterial Mock Vessel Model for Particle Image Velocimetry Analysis in Pulsatile Flow. ASME Open J. Eng. ASME
**2023**, 2, 021010. [Google Scholar] [CrossRef] - Hewlin, R.L., Jr.; Tindall, J.M. Computational Assessment of Magnetic Nanoparticle Targeting Efficiency in a Simplified Circle of Willis Arterial Model. Int. J. Mol. Sci.
**2023**, 24, 2545. [Google Scholar] [CrossRef] [PubMed] - Hewlin, R.L., Jr.; Kizito, J.P. Comparison of Carotid Bifurcation Hemodynamics in Patient-Specific Geometries at Rest and During Exercise. In Proceedings of the Fluids Engineering Division Summer Meeting. American Society of Mechanical Engineers, Volume 1A, Symposia: Advances in Fluids Engineering Education; Advances in Numerical Modeling for Turbomachinery Flow Optimization; Applications in CFD; Bio-Inspired Fluid Mechanics; CFD Verification and Validation; Development and Applications of Immersed Boundary Methods; DNS, LES, and Hybrid RANS/LES Methods, Incline Village, NV, USA, 7–11 July 2013; p. V01AT04A001. [Google Scholar] [CrossRef]
- Stanley, N.; Ciero, A.; Timms, W.; Hewlin, R.L., Jr. Development of 3-D Printed Optically Clear Rigid Anatomical Vessels for Particle Image Velocimetry Analysis in Cardiovascular Flow. In Proceedings of the ASME International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, Volume 7: Fluids Engineering, Salt Lake City, UT, USA, 11–14 November 2019; p. V007T08A004. [Google Scholar] [CrossRef]
- Hewlin, R.L., Jr.; Kizito, J.P. Development of an Experimental and Digital Cardiovascular Arterial Model for Transient Hemodynamic and Postural Change Studies: “A Preliminary Framework Analysis”. Cardiovasc. Eng. Tech.
**2018**, 9, 1–31. [Google Scholar] [CrossRef] - Daemen, J.; Serruys, P.W. Drug-Eluting Stent Update 2007 Part I: A Survey of Current and Future Generation Drug-Eluting Stents: Meaningful Advances or More of the Same. Circulation
**2007**, 116, 316–328. [Google Scholar] [CrossRef] [Green Version] - Beshchasna, N.; Saqib, M.; Kraskiewicz, H.; Wasyluk, Ł.; Kuzmin, O.; Duta, O.C.; Ficai, D.; Ghizdavet, Z.; Marin, A.; Ficai, A.; et al. Recent Advances in Manufacturing Innovative Stents. Pharnaceutics
**2020**, 12, 349. [Google Scholar] [CrossRef] [Green Version] - Conway, C. Clinical Evidence vs. the Testing Paradigm. Cardiovasc. Eng. Technol.
**2018**, 9, 752–760. [Google Scholar] [CrossRef] - Beckman, J.A.; White, C.J. Paclitaxel-Coated Balloons and Eluting Stents: Is There a Mortality Risk in Patients with Peripheral Artery Disease. Circulation
**2019**, 140, 1342–1351. [Google Scholar] [CrossRef] - Capell, W.H.; Bonaca, M.P.; Nehler, M.R.; Kittelson, J.M.; Anand, S.S.; Berkowitz, S.D.; Debus, E.S.; Fanelli, F.; Haskell, L.; Patel, M.R.; et al. Rationale and Desitn for the Vascular Outcomes Study of ASA Along with Rivaroxaban in Endovascular or Surgical Limb Revascularization for Peripheral Artery Disease. Am. Heart J.
**2018**, 199, 83–91. [Google Scholar] [CrossRef] - US Food and Drug Administration. Update: Treatment of Peripheral Arterial Disease with Paclitaxel-Coated Ballons and Paclitaxel-Eluting Stents Potentiall Associated with Increased Mortality: Letter to Health Care Providers. Available online: https://www.fda.gov/medical-devices/letters-health-care-providers/update-treatment-peripheral-arterial-disease-paclitaxel-coated-balloons-and-paclitaxel-eluting (accessed on 5 May 2022).
- Levin, A.; Jonas, M.; Hwang, C.W.; Edelman, E. Local and Systemic Drug Competition in Drug-Eluting Stent Tissue Deposition Properties. J. Control. Release
**2005**, 109, 226–243. [Google Scholar] [CrossRef] [PubMed] - Granada, J.F.; Stenoien, M.; Buszman, P.P.; Tellez, A.; Langanki, D.; Kaluza, G.L.; Leon, M.B.; Gray, W.; Jaff, M.R.; Schwartz, R.S. Mechanisms of Tissue Uptake and Retention of Paclitaxel-Coated Balloons: Impact on Neointimal Proliferation and Healing. Open Heart
**2014**, 1, e000117. [Google Scholar] [CrossRef] [Green Version] - Lovich, M.A.; Philbrook, M.; Sawyer, S.; Weselcouch, E.; Edelman, E.R. Arterial Heparin Deposition: Role of Diffusion, Convection, and Extravascular Space. Am. J. Physiol.
**1998**, 275, H2236–H2242. [Google Scholar] [CrossRef] - Moses, J.W.; Stone, G.W.; Nikolsky, E.; Mintz, G.S.; Dangas, G.; Grube, E. Drug-eluting Stents in the Treatment of Intermediate Lesions: Pooled Analysis from four Randomized Trials. J. Am. Coll. Cardiol.
**2006**, 47, 2164–2171. [Google Scholar] [CrossRef] [Green Version] - Serruys, P.W.; Sianos, G.; Abizaid, A.; Aoki, J.; den Heijer, P.; Bonnier, H. The Effect of Variable Dose and Release Kinetics on Neointimal Hyperplasia using a Novel Paclitaxel-eluting Stent Platform: The Paclitaxel In-Stent Controlled Elution Study (Pices). J. Am. Coll. Cardiol.
**2005**, 46, 253–260. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lovich, M.A.; Edelman, E.R. Computational Simulations of Local Vascular Heparin Deposition and Distribution. Am. J. Physiol.
**1996**, 271, H2014–H2024. [Google Scholar] [CrossRef] - Sakharov, D.V.; Kalachev, L.V.; Rijken, D.C. Numerical Simulation of Local Pharmcokinetics of a Drug after Intervascular Delivery with an Eluting Stent. J. Drug Target.
**2002**, 10, 507–513. [Google Scholar] [CrossRef] [PubMed] - Hwang, C.; Wu, D.; Edelman, E. Physiological Transport Forces Govern Drug Distribution for Stent-Based Delivery. Circulation
**2001**, 104, 600–605. [Google Scholar] [CrossRef] [Green Version] - Migiliavacca, F.; Gervaso, F.; Prosi, M.; Zunino, P.; Minisini, S.; Formaggia, L. Expansion and Drug Elution Model of a Coronary Stent. Comput. Methods Biomech. Biomed. Eng.
**2007**, 10, 63–73. [Google Scholar] [CrossRef] - Borghi, A.; Foa, E.; Balossino, R.; Migliavacca, F.; Dubini, G. Modelling Drug Elutiogn from Stents: Effects of Reversible Binding in the Vascular Wall and Degradable Polymeric Matrix. Comput. Methods Biomech. Biomed. Eng.
**2008**, 11, 367–377. [Google Scholar] [CrossRef] - Horner, M.; Joshi, S.; Dhruva, V.; Sett, S.; Stewart, S.F.C. A Two-Species Drug Delivery Model is Required to Predict Deposition from Drug-Eluting Stents. Cardiovasc. Eng. Technol.
**2010**, 1, 225–234. [Google Scholar] [CrossRef] - Tzafriri, A.R.; Levin, A.D.; Edelman, E.R. Diffusion-limited Binding Explains Binary Dose Response for Local Arterial and Tumor Drug Delivery. Cell Prolif.
**2009**, 42, 348–363. [Google Scholar] [CrossRef] [Green Version] - Higuchi, T. Theoretical Analysis of Rate of Release of Solid Drugs Dispersed in Solid Matrices. J. Pharm. Sci.
**1963**, 52, 1145–1149. [Google Scholar] [CrossRef] [PubMed] - Mandal, A.P.; Mandal, P.K. An Unsteady Analysis of Arterial Drug Transport from Half-Embedded Drug Eluting Stent. Appl. Math. Comput.
**2015**, 266, 968–981. [Google Scholar] [CrossRef] - Mandal, A.P.; Mandal, P.K. Computational Modelling of Three-Phase Stent-Based Delivery. J. Explor. Res. Pharm.
**2017**, 2, 31–40. [Google Scholar] [CrossRef] [Green Version] - Saltzman, W.M. Drug Delivery: Engineering Principles for Drug Therapy; Oxford University Press: Oxford, UK, 2001. [Google Scholar]
- Sarifuddin; Mandal, P.K. Effect of Diffusivity on the Transport of Drug Eluted from Drug Eluting Stent. Int. J. Appl. Comput. Math.
**2016**, 2, 291–301. [Google Scholar] [CrossRef] [Green Version] - O’Connel, B.M.; Walsh, M.T. Demonstrating the Influence of Compression on Artery Wall Mass Transport. Ann. Biomed. Eng.
**2010**, 38, 1354–1366. [Google Scholar] [CrossRef] - Edwards, M.; Kizito, J.P.; Hewlin, R.L., Jr. A Time-Dependent Two Species Explicit Finite Difference Computational Model for Analyzing Diffusion in a Drug Eluting Stented Coronary Artery Wall: A Phase I Study. In Proceedings of the ASME International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, Volume 4: Biomedical and Biotechnology; Design, Systems, and Complexity, Columbus, OH, USA, 30 October–3 November 2022; p. V004T05A009. [Google Scholar] [CrossRef]
- Zhu, X.; Pack, D.W.; Braatz, R.D. Modelling Intravascular Delivery from Drug-Eluting Stents with Biodegradable Coating: Investigation of Anisotropic Vascular Drug Diffusivity and Arterial Drug Distribution. Comput. Methods Biomech. Biomed. Eng.
**2014**, 17, 187–198. [Google Scholar] [CrossRef] [Green Version] - Saha, R.; Mandal, P.K. Modelling Time-dependent Release Kinetics in Stent-based Delivery. J. Explor. Res. Pharmacol.
**2018**, 3, 61–70. [Google Scholar] [CrossRef] [Green Version] - Zhu, X.; Braatz, R.D. Modelling and Analysis of Drug-Eluting Stents With Biodegradable PLGS Coating: Consequences on Instravascular Drug Delivery. J. Biomech. Eng.
**2015**, 136, 111004. [Google Scholar] [CrossRef] [Green Version] - Balakrishnan, B.; Tzafriri, A.R.; Arifin, D.Y.; Edelman, E.R. Intravascular Drug Release Kinetics Dictate Arterial Drug Deposition, Rentention and Distribution. J. Control. Release
**2007**, 123, 100–108. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Vairo, G.; Cioffi, M.; Cottone, R.; Dubini, G.; Migliavacca, F. Drug Release From Coronary Eluting Stents: A Multidomain Approach. J. Biomech. Eng.
**2010**, 43, 1580–1589. [Google Scholar] [CrossRef] - Hewlin, R.L., Jr. Transient Cardiovascular Hemodynamics in a Patient-Specific Arterial System. Ph.D. Thesis, North Carolina Agricultural and Technical State University, Greensboro, NC, USA, 2015. [Google Scholar]
- Whale, M.; Grodzinsky, A.; Johnson, M. The Effect of Aging and Pressure on the Specific Hydraulic Conductivity of the Aortic Wall. Biorheology
**1996**, 33, 17–44. [Google Scholar] [CrossRef] - Kolachalama, V.; Tzafriri, A.; Arifin, D.; Edelman, E. Luminal Flow Patterns Dictate Arterial Drug Deposition in Stent-Based Delivery. J. Control. Release
**2009**, 133, 24–30. [Google Scholar] [CrossRef] [Green Version] - Baldwin, A.L.; Wilson, I.; Gardus-Pizlo, R.; Wilensky, R.; March, K. Effect of Atherosclerosis on Transmural Convection and Arterial Ultrastructure: Implications for Loacl Vascualr Drug Delivery. Aterioscler. Thromb. Vasc. Biol.
**1997**, 17, 3365–3375. [Google Scholar] [CrossRef] - Wang, D.; Tarbell, J. Modelling Interstitial Flow in an Artery Wall Allows Estimation of Wall Shear Stress on Smooth Muscle Cells. ASME J. Biomech. Eng.
**1995**, 117, 358–363. [Google Scholar] [CrossRef] - Santin, M.; Colombo, P.; Brushci, G. Interfacial Biology of In-Stent Restenosis. Expert Rev. Med. Dev.
**2005**, 2, 429–443. [Google Scholar] [CrossRef] - Balakrishnan, B.; Tzafriri, A.R.; Seifert, P.; Groothuis, A.; Rogers, C.; Edelman, E.R. Strut Position, Blood Flow, and Drug Deposition-Implications for Single and Overlapping Drug-Eluting Stents. Circulation
**2005**, 111, 2958–2965. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wang, X.T.; Venkatraman, S.S.; Boey, F.Y.C.; Loo, J.S.C.; Tan, L.P. Controlled Release of Sirolimus from a Multilayered PLGA Stent Matrix. Biomaterials
**2006**, 27, 5588–5595. [Google Scholar] [CrossRef] [PubMed] - Castellot, J.J.; Wong, K.; Herman, B.; Hoover, R.L.; Albertini, D.F.; Wright, T.C.; Caleb, B.L.; Karnovsky, M.J. Binding and Internalization of Heparin by Vascular Smooth Muscle Cells. J. Cell. Physiol.
**1985**, 124, 13–20. [Google Scholar] [CrossRef] [PubMed] - Deux, J.-F.; Meddahi-Pelle, A.; Le Blanche, A.F.; Feldman, L.J.; Colliec-Jouault, S.; Brée, F.; Boudghène, F.; Michel, J.-B.; Letourneur, D. Low Molecular Weight Fucoidan Prevents Neointimal Hyperplasia in Rabbit Iliac Artery In-Stent Restenosis Model. Arter. Thromb. Vasc. Biol.
**2002**, 22, 1604–1609. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Cross-sectional view diagram of the arterial stented model: (

**a**) Schematic of a single PLGA coated half-embedded stent strut implanted into the arterial wall and (

**b**) the full stented (all stent struts included) arterial model.

**Figure 3.**Validation results for the finite volume model using a finite difference model developed in reference to the work of Saha and Mandal [35]: (

**a**,

**b**) Distribution of normalized mean bound drug concentration for values of: Pe

_{T}= 2, Da = 40, and ε

_{2}= 100.

**Figure 4.**Interstitial flow profile into the half-embedded strut arterial wall. Black arrows represent the velocity vectors.

**Figure 5.**Drug concentration contours at 2 days: (

**a**) free drug and (

**b**) bound drug. With an initial concentration of C

_{0}= 0.01. The light blue lines incorporate the stent strut boundaries.

**Figure 6.**Drug concentration contours at 8 days: (

**a**) free drug and (

**b**) bound drug. The light blue lines incorporate the stent strut boundaries.

Element Number | Average Weighted Concentration | Average Velocity |
---|---|---|

74,212 | 1.121327 | 13.72 × 10^{−6} |

82,458 | 1.057641 | 12.68 × 10^{−6} |

91,621 | 1.034241 | 10.23 × 10^{−6} |

101,802 | 0.983541 | 9.83 × 10^{−6} |

113,114 | 0.977732 | 9.77 × 10^{−6} |

372,125 | 0.977654 | 9.76 × 10^{−6} |

Description | Parameter | Value |
---|---|---|

Outer diameter of the artery, mm | D | 3 |

Artery wall thickness, µm | L_{y} | 200 |

Strut dimension, m | δ | 0.00014 |

Transmural filtration velocity, m/s | V_{y} | 4 × 10^{−8} |

Porosity of the arterial wall | ε | 0.787 |

Tortuosity of the arterial wall | τ | 1.333 |

Coating drug diffusivity, m^{2}/s
| D_{c} | 1.0 × 10^{−12} |

Coefficient of free diffusivity, m^{2}/s
| D_{free} | 3.65 × 10^{−12} |

Coefficient of effective diffusivity, m^{2}/s
| D_{eff} | 2.15 × 10^{−12} |

True diffusivity of the free drug, m^{2}/s
| D_{T} | 24 × 10^{−12} |

Initial drug concentration in the coating, mol/m^{3} | C_{0} | 0.01 |

Tissue binding capacity, mol/m^{3} | k_{a} | 10 |

Dissociation rate constant | k_{d} | 0.01 |

Equilibrium dissociation constant, mol/m^{3} | R_{d} | 0.001 |

Dimensionless Peclet number in the coating | Pe_{C} | 100 |

Dimensionless Peclet number in the tissue | Pe_{T} | 2 |

Dimensionless DamKöhler number in the tissue | D_{a} | 40 |

Dimensionless scaling parameter 1 | ε_{1} | 0.001 |

Dimensionless scaling parameter 2 | ε_{2} | 100 |

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## Share and Cite

**MDPI and ACS Style**

Hewlin, R.L., Jr.; Edwards, M.; Kizito, J.P.
A Two-Species Finite Volume Scalar Model for Modeling the Diffusion of Poly(lactic-co-glycolic acid) into a Coronary Arterial Wall from a Single Half-Embedded Drug Eluting Stent Strut. *Biophysica* **2023**, *3*, 385-408.
https://doi.org/10.3390/biophysica3020026

**AMA Style**

Hewlin RL Jr., Edwards M, Kizito JP.
A Two-Species Finite Volume Scalar Model for Modeling the Diffusion of Poly(lactic-co-glycolic acid) into a Coronary Arterial Wall from a Single Half-Embedded Drug Eluting Stent Strut. *Biophysica*. 2023; 3(2):385-408.
https://doi.org/10.3390/biophysica3020026

**Chicago/Turabian Style**

Hewlin, Rodward L., Jr., Maegan Edwards, and John P. Kizito.
2023. "A Two-Species Finite Volume Scalar Model for Modeling the Diffusion of Poly(lactic-co-glycolic acid) into a Coronary Arterial Wall from a Single Half-Embedded Drug Eluting Stent Strut" *Biophysica* 3, no. 2: 385-408.
https://doi.org/10.3390/biophysica3020026