The treatment of choice for cardiovascular diseases initiated by atherosclerosis is the implantation of coronary stents. These implants open the affected artery to obtain the necessary blood flow rate.
Unfortunately the stent struts induce unphysiological flow conditions like recirculation zones, stasis and vortices resulting in post-operative complications. The most fatal one is the stent thrombosis which leads to myocardial infarction with high mortality rates. Especially with the introduction of biodegradable stents consisting of thick struts made of polymers or magnesium alloys thrombus formation is one focus of research again .
By means of numerical fluid dynamic simulations the understanding of thrombosis formation based on flow disturbances increases. Simulations give detailed information about flow distribution and platelet transport.
Blood is a suspension of viscoelastic particles like red blood cells and platelets in a Newtonian fluid, the blood plasma. The influence of the particles on fluid mechanical behaviour is often approximated by a non-Newtonian approach like the Carreau-Yasuda model . With these models the shear-thinning behaviour of the whole-blood can be taken into account. Especially in regions with a shear rate below 100 s−1 this effect plays a significant role . Nevertheless, the assumption of an ideal mixed and homogenous fluid neglects the transport mechanisms of thrombogenious particles and the adhesion of blood components.
Therefore Feng et al. simulated the viscous behaviour of blood in a stenosed two dimensional vessel with an advanced discrete particle dynamic (DPD) approach . This model consists of multiple discrete particles interacting with each other. The results have shown a very good agreement of the DPD model and conventional computational fluid dynamic (CFD) method. Freund presented a review of further numerical models which also simulate the deformation of the red blood cells .
But all these discrete particle models request huge computational costs and so they are yet limited to capillary flow or simple cases.
Another numerical model, the so called Euler-Lagrange/DEM approach (discrete element model), combines the DPD and CFD method . Therein the suspension is separated in a fluid and a solid phase. The fluid phase is treated as a homogeneous fluid with Newtonian or non-Newtonian flow behaviour and is calculated in the Eulerian framework as usual in common CFD simulations.
The solid phase consists of discrete zero dimensional particles and the movement is calculated in the Lagrangian framework based on Newton’s law. The interaction between both phases is considered by the source term in the momentum equation and the particle-particle interaction is often solved by a spring damper collision model. This kind of numerical model is successfully used by Chesnutt and Hanto simulate the process of stent thrombosis formation under steady state flow conditions .
The transient Euler-Lagrange/DEM model presented in this paper is based on the Lagrangian particle solver, which is implemented in the OpenFOAM framework .
The vessel is approximated by a rigid straight 2D channel with a height of h = 1.35 mm and an overall length of 10 h. The stent is modelled by 4 square struts with an edge length of 150 μm. Detailed information of stent design parameters can be seen in Figure 1.
The particles are seeded in the red dotted box (Figure 1) every Δt = 0.04 s. In summation around 20,000 particles were modelled in one pulse (70 bpm). The particle count is 300,000 mm−3 . This results in an average particle distance of 15 μm. The particle diameter is d = 2.4 μm.
The procedure of the Euler-Lagrange/DEM approach can be divided into two main steps.
The Navier-Stokes equation (NSE) is solved for the whole domain in the Eulerian frame:
where U is the velocity, ρ the density, p the pressure and μ the dynamic viscosity of the homogenous phase. The source term SP represents the forces on the solid particles and connects the Eulerian with the Lagrangian phase. Subscripts refer variables either to fluid phase or particles.
The equation of motion is solved for the inserted particles. The particle force is a result of the difference of the particle momentum in every control volume. The drag force FD influences the particle motion. By assuming a spherical particle, the drag force is calculated by:
where Urel = |U−Up|is the relative velocity and CD is the drag coefficient:
Considering Strokes regime, which means the particle Reynolds number is below 0.1. The particle Reynolds number can be calculated from:
These two steps for flow field and particle simulation can be done separately for a steady state flow or in one time step to regard a transient flow field as we used here.
In this study only one way coupling is considered, so the source term in the momentum equation for the fluid phase is neglected. The particles are affected by the flow field via drag force but the particles do not influence the flow field.
The collision model implemented in the OpenFOAM solver is a non-linear viscoelastic model proposed by . It’s based on the Hertz theory which means that particles are considered as elastic bodies, see Figure 3. Relevant simulation parameters can be found in Table 1.
3 Results and discussion
3.1 Flow field
Numerous papers deal with the flow field in stented vessels. Therefore, only a short description of the velocity field is given here. As expected, recirculation zones can be identified up and down stream of the stent struts. Between both struts there exists one continuous recirculation zone, see Figure 4.
3.2 Formation of stent thrombosis
The thrombosis formation is initiated by the contact of separate particles due to spatial flow disturbances triggered by stent struts. At that moment two or more particles collide and merge the developed clot starts to tumble. Also the clot inertia causes a difference in velocity between separate particles and the clot, whereby more particles get in contact with the clot and fuse, see Figure 5.
The process of thrombosis formation in the flow field reinforces itself after ignition by the stent struts.
Additionally, the particle residence time increases enormous in the recirculation zones and so the risk of thrombosis increases. Figure 6 depicts particle positions and particle ages in the vicinity of stent struts. It can be seen, that a clot consists of particles with different ages. So, the particle laden flow isn’t layered anymore but mixed by the stent. Several computational and in vivo studies have shown that the vicinity of stent struts is particularly at risk .
In this paper a transient Euler-Lagrange/DEM approach for simulation of thrombosis formation is introduced. This approach allows simulations of transient flow field and discrete particle interaction in one time step. Particle collision is taken into account by a non-linear viscoelastic collision model. Biochemical mechanisms like platelet activation are not considered.
Stent struts induce flow disturbances like recirculation zones promoting platelet collision. As a result thrombosis formation can be initiated. Furthermore, particle residence time is prolonged in recirculation zones up and down stream of stent struts leading to higher risk of thrombosis formation.
This approach goes beyond common CFD simulation and is beneficial for thrombosis forecast in stented coronary vessels. Transient simulations of three dimensional cases are planned for further research.
Financial support by the Federal Ministry of Education and Research (BMBF) within RESPONSE “Partnership for Innovation in Implant Technology” is gratefully acknowledged.
Research funding: The author state no funding involved. Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent is not applicable. Ethical approval: The conducted research is not related to either human or animal use.
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About the article
Published Online: 2016-09-30
Published in Print: 2016-09-01
Citation Information: Current Directions in Biomedical Engineering, Volume 2, Issue 1, Pages 297–300, ISSN (Online) 2364-5504, DOI: https://doi.org/10.1515/cdbme-2016-0066.
©2016 Michael Stiehm et al., licensee De Gruyter.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0