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Transient Euler-Lagrange/DEM simulation of stent thrombosis

  • Michael Stiehm EMAIL logo , Carolin Wüstenhagen , Niels Grabow and Klaus-Peter Schmitz
Published/Copyright: September 30, 2016
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Current Directions in Biomedical Engineering
From the journal Current Directions in Biomedical Engineering Volume 2 Issue 1

Abstract

Stent implantation is the treatment of choice for cardiovascular diseases. By introduction of biodegradable thick strut stents investigations of thrombosis formation is one focus of research. This study deals with a transient Euler-Lagrange/DEM approach to simulate the flow field, platelet movement and clotting. The recirculation zones prolong particle residence time. As a result, the vicinity of stent struts shown a particularly higher risk for stent thrombosis.

Keywords: CFD; Lagrangian; particles; thrombosis

1 Introduction

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 [1].

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 [2]. 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 [3]. 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 [4]. 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 [5].

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 [6]. 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 [7].

The transient Euler-Lagrange/DEM model presented in this paper is based on the Lagrangian particle solver, which is implemented in the OpenFOAM framework [8].

2 Methods

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.

Figure 1: Design parameters of the stent.
Figure 1:

Design parameters of the stent.

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 [7]. 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.

  1. The Navier-Stokes equation (NSE) is solved for the whole domain in the Eulerian frame:

    ∇⋅U=0ρf⁢∂ U∂ t+ρf⁢(U⋅∇ U)=−∇ p+μf⁢∇2 U−SP

    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.

    For the inlet velocity condition a transient block profile is used. The mean Reynolds number is about Re = 250. The waveform was adopted from [9], see Figure 2.

  2. 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:

    FD=−12ρfCDUrel2AP

    where Urel = |U−Up|is the relative velocity and CD is the drag coefficient:

    CD=24Re

    Considering Strokes regime, which means the particle Reynolds number is below 0.1. The particle Reynolds number can be calculated from:

    ReP=ρf⁢DP⁢Urelμf

    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.

Figure 2: Waveform used as inlet velocity condition based on [9].
Figure 2:

Waveform used as inlet velocity condition based on [9].

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 [10]. 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.

Figure 3: Spring damper model used for particle collision.
Figure 3:

Spring damper model used for particle collision.

Table 1:

Simulation parameters and boundary conditions.

PropertyValue or condition
Artery radius or channel height1.35 mm
Strut height150 μm
Strut width150 μm
Mean fluid velocity0.31 m/s
Fluid density1060 kg/m3
Dynamic viscosity0.00355 kg/ms
Mean Reynolds number250
Inlet boundary conditionTransient velocity block profile
Outlet boundary conditionZero-pressure
Solid wall boundary conditionNo-slip
Centerline boundary conditionSymmetry
Number of particles1000 particles every Δt = 0.04 s
Particle diameter2.4 μm
Particle density1060 kg/m3

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.

Figure 4 Velocity distribution around two stent struts during systole (t = 0.4 s).
Figure 4

Velocity distribution around two stent struts during systole (t = 0.4 s).

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.

Figure 5 Timeline of thrombosis formation upstream of a stent strut.
Figure 5

Timeline of thrombosis formation upstream of a stent strut.

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 [11].

Figure 6 Particle position in the vicinity of stent struts, particles are coloured by their age.
Figure 6

Particle position in the vicinity of stent struts, particles are coloured by their age.

4 Conclusion

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.

Acknowledgement

Financial support by the Federal Ministry of Education and Research (BMBF) within RESPONSE “Partnership for Innovation in Implant Technology” is gratefully acknowledged.

Author’s Statement

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.

References

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Published Online: 2016-9-30
Published in Print: 2016-9-1

©2016 Michael Stiehm et al., licensee De Gruyter.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/cdbme-2016-0066
Keywords for this article
CFD; Lagrangian; particles; thrombosis
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BY-NC-ND 4.0

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  105. Numerical evaluation of image parameters of ETR-1
  106. Fabrication of a compliant phantom of the human aortic arch for use in Particle Image Velocimetry (PIV) experimentation
  107. Effect of the number of electrodes on the reconstructed lung shape in electrical impedance tomography
  108. Hardware dependencies of GPU-accelerated beamformer performances for microwave breast cancer detection
  109. Computer assisted assessment of progressing osteoradionecrosis of the jaw for clinical diagnosis and treatment
  110. Evaluation of reconstruction parameters of electrical impedance tomography on aorta detection during saline bolus injection
  111. Evaluation of open-source software for the lung segmentation
  112. Automatic determination of lung features of CF patients in CT scans
  113. Image analysis of self-organized multicellular patterns
  114. Effect of key parameters on synthesis of superparamagnetic nanoparticles (SPIONs)
  115. Radiopacity assessment of neurovascular implants
  116. Development of a desiccant based dielectric for monitoring humidity conditions in miniaturized hermetic implantable packages
  117. Development of an artifact-free aneurysm clip
  118. Enhancing the regeneration of bone defects by alkalizing the peri-implant zone – an in vitro approach
  119. Rapid prototyping of replica knee implants for in vitro testing
  120. Protecting ultra- and hyperhydrophilic implant surfaces in dry state from loss of wettability
  121. Advanced wettability analysis of implant surfaces
  122. Patient-specific hip prostheses designed by surgeons
  123. Plasma treatment on novel carbon fiber reinforced PEEK cages to enhance bioactivity
  124. Wear of a total intervertebral disc prosthesis
  125. Digital health and digital biomarkers – enabling value chains on health data
  126. Usability in the lifecycle of medical software development
  127. Influence of different test gases in a non-destructive 100% quality control system for medical devices
  128. Device development guided by user satisfaction survey on auricular vagus nerve stimulation
  129. Empirical assessment of the time course of innovation in biomedical engineering: first results of a comparative approach
  130. Effect of left atrial hypertrophy on P-wave morphology in a computational model
  131. Simulation of intracardiac electrograms around acute ablation lesions
  132. Parametrization of activation based cardiac electrophysiology models using bidomain model simulations
  133. Assessment of nasal resistance using computational fluid dynamics
  134. Resistance in a non-linear autoregressive model of pulmonary mechanics
  135. Inspiratory and expiratory elastance in a non-linear autoregressive model of pulmonary mechanics
  136. Determination of regional lung function in cystic fibrosis using electrical impedance tomography
  137. Development of parietal bone surrogates for parietal graft lift training
  138. Numerical simulation of mechanically stimulated bone remodelling
  139. Conversion of engineering stresses to Cauchy stresses in tensile and compression tests of thermoplastic polymers
  140. Numerical examinations of simplified spondylodesis models concerning energy absorption in magnetic resonance imaging
  141. Principle study on the signal connection at transabdominal fetal pulse oximetry
  142. Influence of Siluron® insertion on model drug distribution in the simulated vitreous body
  143. Evaluating different approaches to identify a three parameter gas exchange model
  144. Effects of fibrosis on the extracellular potential based on 3D reconstructions from histological sections of heart tissue
  145. From imaging to hemodynamics – how reconstruction kernels influence the blood flow predictions in intracranial aneurysms
  146. Flow optimised design of a novel point-of-care diagnostic device for the detection of disease specific biomarkers
  147. Improved FPGA controlled artificial vascular system for plethysmographic measurements
  148. Minimally spaced electrode positions for multi-functional chest sensors: ECG and respiratory signal estimation
  149. Automated detection of alveolar arches for nasoalveolar molding in cleft lip and palate treatment
  150. Control scheme selection in human-machine- interfaces by analysis of activity signals
  151. Event-based sampling for reducing communication load in realtime human motion analysis by wireless inertial sensor networks
  152. Automatic pairing of inertial sensors to lower limb segments – a plug-and-play approach
  153. Contactless respiratory monitoring system for magnetic resonance imaging applications using a laser range sensor
  154. Interactive monitoring system for visual respiratory biofeedback
  155. Development of a low-cost senor based aid for visually impaired people
  156. Patient assistive system for the shoulder joint
  157. A passive beating heart setup for interventional cardiology training
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