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A blockage of blood vessels resulting from thrombus or plaque deposit causes serious cardiovascular diseases. This study developed a computational model of blood flow and drug transport to investigate the effectiveness of drug delivery to the stenotic sites. A three-dimensional (3D) model of the curved stenotic right coronary artery (RCA) was reconstructed based on the clinical angiogram image. Then, blood flow and drug transport with the flexible RCA wall were simulated using the fluid structure interaction (FSI) analysis and compared with the rigid RCA wall. Results showed that the maximal total displacement and von Mises stress of the flexible RCA model are 2.14 mm and 92.06 kPa. In addition, the effective injecting time point for the best performance of drug delivery was found to be between 0 s and 0.15 s ( i.e., the fluid acceleration region) for both rigid and flexible RCA models. However, there was no notable difference in the ratio of particle deposition to the stenotic areas between the rigid and flexible RCA models. This study will be significantly useful to the design of a drug delivery system for the treatment of the stenotic arteries by targeting drugs selectively to the stenotic sites.

Cardiovascular disease is a leading cause of mortality and morbidity in developed countries [

So far, there have been numerous studies to investigate the hemodynamics of the stenotic arteries for the treatment of the related diseases. Significant advances in the understanding of the relationship between hemodynamics, mechanical factors, and atherosclerotic changes have been made over the past few decades. Many of the studies have revealed that the wall pressure, wall shear stress (WSS), or pressure drop of the stenotic blood vessel (between the inlet and outlet) become much larger than the normal blood vessel [

Still, there are several challenges in the blood flow and drug delivery of the blood vessels. First, the dose of drugs as a therapeutic strategy should be performed with caution. This is because long-term infusion of drugs can cause the potential risk of bleeding [

Motivated by these facts, the aim of this study was to: 1) develop a computational model of blood flow for the stenotic right coronary artery (RCA), 2) investigate the effect of wall compliance on the drug transport in the vessel using fluid-structure interaction (FSI), and 3) determine the injecting time point for the best effectiveness of the drug delivery to the stenotic sites. First, the effect of Newtonian and non-Newtonian viscosity models on velocity profiles in the five major locations was examined. Second, the area-averaged pressure and velocity profiles between rigid and flexible RCAs at the throat of the stenosis were compared. Third, the particle transport was simulated using rigid and flexible RCAs for a range of times, and the effective injecting time point was discussed. Finally, limitations of this study were addressed.

A three-dimensional (3D) stenotic RCA was reconstructed based on a 62-year-old patient as reported as shown in

The stenosis severity is estimated to be approximately 80% by the following form [

where A_{s} and A_{n} are the area of the stenotic cross section (occluded at the throat of the stenosis) and normal cross section of the RCA, respectively.

The hybrids composed of hexahedral and tetrahedral mesh were generated. In addition, since the numerical results should be independent of the number of mesh, the mesh independence test was performed for five cases as shown in

Mesh density [elem./mm^{3}] | Velocity [m/s] | Pressure [Pa] |
---|---|---|

16.77 | 1.721 | 108.3 |

27.42 | 1.720 | 105.3 |

50.31 | 1.725 | 106.3 |

109.67 | 1.752 | 106.7 |

219.34 | 1.712 | 106.6 |

The Newtonian fluid model was considered and the flow was assumed to be transient, isothermal, incompressible and laminar due to low Reynolds number (Re = 2678) at the peak flow. The Womersley number (α = R(w/ν)^{0.5}) is approximately between 2 and 5 depending on vessel diameter, viscosity model, and blood density. For fluid mechanics computation, the Navier-Stokes equations with arbitrary Lagrangian-Eulerian formulation were employed as follows:

where ρ_{f} is the blood density (1050 kg/m^{3}) [_{g} is the moving coordinate velocity and µ is the dynamic viscosity. Thus, u − u_{g} is the relative velocity of the fluid with respect to the moving coordinatevelocity.

It has been reported that the arterial wall, composed of layered collagen, is dynamic and compliant during the cardiac cycle [

where c_{10} = 0.07 MPa, c_{20} = 3.2 MPa, c_{21} = 0.0716 MPa, and c_{ij} = 0 MPa. Ī_{1} and Ī_{1} are the first and second deviatoric strain invariants, d_{m} (2/K, K is bulk modulus) is the material incompressibility parameter (10^{−5} Pa^{−1}), and J is the determinant of the elastic deformation gradient tensor.

The momentum equation with the ignoring gravity for the wall segment of the RCA based on Lagrangian coordinate system is given by Equation (5).

σ_{s} is the solid stress tensor, ρ_{s} is the arterial wall density (1366 kg/m^{3}) [_{s} is the solid displacement.

As for the boundary conditions, the time-varying velocity and pressure wave were applied at the inlet and outlet, as previously reported [

At the FSI interface, the following boundary conditions are applied: 1) displacements of the fluid and solid domain are equal, 2) traction forces are compatible, and 3) the non-slip condition is chosen for the fluid:

whered, σ, and n are displacements, stress tensors, and normal vectors with the subscripts f and s for fluid and solid, respectively. The inlet and outlet ends were set to be fixed.

To describe the drug transport in the RCA, the Lagrangian particle transport equation was employed:

where m_{p}, d_{p}, and C_{D} are the particle mass, particle diameter, and drag coefficient. The used particle, composed of aggregates of multiple nanoparticles, was taken to be micro-sized PLGA with 3.8 µm of diameter and 1.34 g/cm^{3} of density, respectively [

100 at each four-time point (0 s right before acceleration, 0.15 s for inlet peak velocity in systole, 0.4 s for outlet peak pressure in mid-cycle, and 0.7 s in diastole) during the cardiac cycle.

The fully coupled solid and fluidmodels were solved interactively with 0.01 s of the time step and 1 s of the total time (i.e., one cardiac cycle) based on a heart rate of 60 beats per minute using a commercial program ANSYS for structural mechanics analysis and ANSYS-CFX for fluid mechanics analysis. The fluid pressure at the fluid- solid interface can be applied as a load for the structural analysis, and the resulting displacement, velocity, or acceleration obtained in the structural analysis can be passed back for fluid analysis. As a solver scheme, a high resolution for advection and second order backward Euler for the transientscheme were utilized, and the convergence criterion was set as 10^{−4}. Iteration between the structural analysis solution and fluid mechanics solution continues until overall equilibrium is reached.

In reality, the blood has been known to be a non-Newtonian fluid where the blood viscosity changes with the hematocrit and macroglobulin concentration [^{−1}) [

Blood Model | Effective Viscosity, µ [Pa×s] | Parameters |
---|---|---|

Plasma [ | 0.00122 | |

Blood [ | 0.00345 | |

Power Law [ | m = 0.035, n_{p} = 0.6 | |

Carreau [ | λ = 3.313, n_{c} = 0.3568, µ_{0} = 0.056, and µ_{c} = 0.00345 | |

Casson [ | τ_{y} = (0.625H)^{3}, _{0} = 0.012, H = 0.4 | |

Walburn-Schneck [ | C_{1} = 0.00797, C_{2} = 0.0608, C_{3} = 0.00499, C_{4} = 14.585, H = 40, TPMA = 25.9 |

For the comparison of the viscosity models, the Casson model predictsslightly higher velocity magnitudes than others, especially for L1, L2, and L4, whereas the plasma model predicts relatively lower values at the same locations. Among non-Newtonian models, the Walburn-Schneck model tends to underpredict the velocity magnitudes, particularly for L1, L2, and L4 compared to others. Overall, only subtle difference in velocity mag- nitudes among the viscosity models―except for the plasma model and Walburn-Schneck―was observed for all locations. It should be noted that the Newtonian blood model in this RCA model presents relatively similar velocity patterns and values for all times and locations to other non-Newtonian models. For example, for the comparison of the blood Newtonian model to non-Newtonian models except for the Walburn-Schneck model, the maximum difference is found to be within 4%. In light of the result, it is believed that applying the Newtonian blood model to FSI analysis (i.e., flexible wall) does not have a significant impact on the hemodynamic results, compared to non-Newtonian models.

found to be highest at the stenotic throat. The maximum displacements for 0.15 s, 0.4 s, and 0.7 s were shown to be about 1.67 mm, 2.14 mm, and 1.46 mm, respectively. These values relatively agree with data reported in the other study by showing the same order of the magnitude [

The maximal von Mises stress was found to be 92.06 kPa at the downstream away from the throat. This is less than 10% of the breaking strength of the RCA wall, which has been known to be approximately 1 MPa [

_{in} = 0 s, 0.15 s, 0.4 s and 0.7 s). The red color indicates trajectories of deposited particles and the green indicates trajectories of particles exiting from the RCA. This qualitative result clearly shows that the number of particles exiting (green) is higher than deposited particles (red) within the RCA.

Such a trend was shown in quantitative results (

Interestingly, for the time period up to 0.15 s as shown in

In this study, the numerical simulations of the curved stenotic RCA were quantitatively performed in terms of velocity profiles, wall displacement, maximal von Mises stress, and particle deposition in the rigid and flexible RCAs. I employed the real pulse wave for the inlet and outlet condition to make the simulation closer to the actual physiological phenomena. In addition, I justified the utilization of the Newtonian blood viscosity model by comparing velocity profiles in five major areas of the RCAs with non-Newtonian viscosity models.

Since the characteristics of blood vessel are highly complicated, modeling a 3D realistic blood vessel is essential for accurate surgical prediction. Complex flow such as turbulent flow cannot be precisely described in two- dimension (2D). In contrast, our 3D model and simulation of deformable blood vessel and drug transport enable to describe more realistic phenomena, and thereby can give exact information and feedback to the patient. In addition, it can reduce the amount of time for model validation.

Nevertheless, several limitations still exist that need to be addressed. First, only a specific particle size (3.8 µm, microparticle composed of aggregates of nanoparticles) was adopted as reported [

I have developed a computational model with FSI for blood flow and drug delivery of the curved RCA with stenosis. The key points in this research are as follows:

1) The Newtonian blood viscosity model used in this study shows similar flow patterns and velocity magnitudes to non-Newtonian models.

2) The maximal total displacement and von Mises stress of the RCA wall are 2.14 mm and 92.06 kPa, which are both on the same order of the magnitude reported in the other study.

3) There is no notable difference in the particle sedimentation to the stenotic areas between the rigid and flexible RCA models.

4) The effective time point for the drug injection is found to be between 0 s and 0.15 s (i.e., fluid acceleration region).

I believe that this information will significantly contribute to the design of a drug delivery system for the treatment of the stenotic arteries by targeting drugs selectively to the stenotic sites.

This work was done in the mechanical engineering computer laboratory (MECL) at Purdue University.

Seungman Park, (2016) Fluid-Structure Interaction Analysis for Drug Transport in a Curved Stenotic Right Coronary Artery. Journal of Biosciences and Medicines,04,105-115. doi: 10.4236/jbm.2016.45011