Skip to main content

Local hemodynamic analysis after coronary stent implantation based on Euler-Lagrange method


Coronary stents are deployed to treat the coronary artery disease (CAD) by reopening stenotic regions in arteries to restore blood flow, but the risk of the in-stent restenosis (ISR) is high after stent implantation. One of the reasons is that stent implantation induces changes in local hemodynamic environment, so it is of vital importance to study the blood flow in stented arteries. Based on regarding the red blood cell (RBC) as a rigid solid particle and regarding the blood (including RBCs and plasma) as particle suspensions, a non-Newtonian particle suspensions model is proposed to simulate the realistic blood flow in this work. It considers the blood’s flow pattern and non-Newtonian characteristic, the blood cell-cell interactions, and the additional effects owing to the bi-concave shape and rotation of the RBC. Then, it is compared with other four common hemodynamic models (Newtonian single-phase flow model, Newtonian Eulerian two-phase flow model, non-Newtonian single-phase flow model, non-Newtonian Eulerian two-phase flow model), and the comparison results indicate that the models with the non-Newtonian characteristic are more suitable to describe the realistic blood flow. Afterwards, based on the non-Newtonian particle suspensions model, the local hemodynamic environment in stented arteries is investigated. The result shows that the stent strut protrusion into the flow stream would be likely to produce the flow stagnation zone. And the stent implantation can make the pressure gradient distribution uneven. Besides, the wall shear stress (WSS) of the region adjacent to every stent strut is lower than 0.5 Pa, and along the flow direction, the low-WSS zone near the strut behind is larger than that near the front strut. What’s more, in the regions near the struts in the proximal of the stent, the RBC particle stagnation zone is easy to be formed, and the erosion and deposition of RBCs are prone to occur. These hemodynamic analyses illustrate that the risk of ISR is high in the regions adjacent to the struts in the proximal and the distal ends of the stent when compared with struts in other positions of the stent. So the research can provide a suggestion on the stent design, which indicates that the strut structure in these positions of a stent should be optimized further.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17


τ :

Shear stress (F·m−2)

μ :

Dynamic viscosity of fluid (Pa·s)

u :

Velocity (m·s−1)

ρ mix :

Density of blood fluid (kg·m−3)

ρ plasma :

Density of plasma fluid (kg·m−3)

ρ RBC :

Density of red blood cell fluid (kg·m−3)

ε plasma :

Volume fraction of plasma

ε RBC :

Volume fraction of RBCs

η mix :

Kinematic viscosity of blood fluid (kg·m−1·s−1)

η plasma :

Kinematic viscosity of plasma fluid (kg·m−1·s−1)

η RBC :

Kinematic viscosity of RBC fluid (kg·m−1·s−1)

β kl :

Momentum exchange coefficient

C d :

Drag coefficient


Reynolds number

d RBC :

Diameter of the RBCs (m)

ϕ :

Shape factor

a :

Carreau-Yasuda fluid parameter

γ :

Scalar measure of rate deformation (s−1)

λ :

Relaxation time (s)

\( \overset{\rightharpoonup }{F} \) :

Additional acceleration term (m·s−2)

τ r :

Relaxation time (s)

α :

Impact angle of particle path and wall

μ :

Infinite shear rate viscosity (Pa·s)

μ 0 :

Zero shear rate viscosity (Pa·s)

K :

Spring constant (n·m−1)

f loss :

Loss factor

F friction :

Tangential force between particles (N)

F normal :

Normal force between particles (N)

μ(v r):

Friction coefficient

μ stick :

Sticking friction coefficient

μ glide :

Gliding friction coefficient

μ limit :

High-velocity limit friction coefficient

C w :

Rotational drag coefficient

C D :

Non-spherical drag coefficient

A face :

Unit surface area on wall (m2)

ν :

Relative particle velocity (m·s−1)

R erosion :

Erosion rate (kg·m2·s−1)

R accresion :

Deposition rate (kg·m2·s−1)


Coronary artery disease


In-stent restenosis


Wall shear stress


Neointimal thickness


Computational fluid dynamics


Red blood cell


Lattice Boltzmann method


Immersed boundary method


Discrete phase model


Discrete element method


  1. 1.

    Sakamotoa, A., Yu, S.: Risk prediction of in-stent restenosis among patients with coronary drug-eluting stents: current clinical approaches and challenges. Expert. Rev. Cardiovasc. Ther. (2021).

  2. 2.

    Sanmartin, M., Goicolea, J.: Influence of shear stress on in-stent restenosis: in vivo study using 3D reconstruction and computational fluid dynamics. Rev. Esp. Cardiol. 59(1), 20–27 (2006)

    Article  Google Scholar 

  3. 3.

    Bukač, M., Čanić, S., Muha, B.: A nonlinear fluid-structure interaction problem in compliant arteries treated with vascular stents. Appl. Math. Optim. 73(3), 433–473 (2016)

  4. 4.

    Jiang, Y.F., Zhang, J.: Influence of strut cross-section of stents on local hemodynamics in stented arteries. Chin. J. Mech. Eng. 29(3), 624–632 (2016)

    Article  Google Scholar 

  5. 5.

    Liu, P.F., Zhao, K.: Influence of thickness and width of bioabsorbable vascular stent on local hemodynamics. Chin. J. Med. Instrum. 44(2), 118–121 (2020) (in Chinese)

    Google Scholar 

  6. 6.

    Tsubota, K.I., Wada, S.: Particle method for computer simulation of red blood cell motion in blood flow. Comput. Methods Prog. Biomed. 83(2), 139–146 (2006)

    Article  Google Scholar 

  7. 7.

    Passos, A., Sherwood, J.M.: The effect of deformability on the microscale flow behavior of red blood cell suspensions. Phys. Fluids (2019).

  8. 8.

    Bishop, J.J., Nance, P.R.: Effect of erythrocyte aggregation on velocity profiles in venules. Am. J. Physiol. Heart Circ. 280, 222–236 (2001)

    Article  Google Scholar 

  9. 9.

    Sui, Y., Chew, Y.T.: Dynamic motion of red blood cells in simple shear flow. Phys. Fluids (2008).

  10. 10.

    Jung, J.H., Hassanein, A.: Hemodynamic computation using multiphase flow dynamics in a right coronary artery. Ann. Biomed. Eng. 34(3), 393–407 (2006)

    Article  Google Scholar 

  11. 11.

    Kim, Y.H., VandeVord, P.J.: Multiphase non-Newtonian effects on pulsatile hemodynamics in a coronary artery. Int. J. Numer. Methods Fluids 58(7), 803–825 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Yilmaz, F., Kutlar, A.I.: Analysis of drag effects on pulsatile blood flow in a right coronary artery by using Eulerian multiphase model. Korea Aust. Rheol. J. 23(2), 89–103 (2011)

    Article  Google Scholar 

  13. 13.

    Meongkeun, J., Ye, S.S.: A review of numerical methods for RBC flow simulation. Comput. Methods Biomech. Biomed. Eng 18(2), 130–140 (2015)

    Article  Google Scholar 

  14. 14.

    Chien, S., Jan, K.M.: Ultrastructural basis of the mechanism of rouleaux formation. Microvasc. Res. 5, 155–166 (1973)

    Article  Google Scholar 

  15. 15.

    Neu, B., Meiselman, H.J.: Depletion-mediated red blood cell aggregation in polymer solutions. Biophys. J. 83(5), 2482–2490 (2002)

    ADS  Article  Google Scholar 

  16. 16.

    Ye, T., Nhan, P.T.: Numerical modelling of a healthy/malaria-infected erythrocyte in shear flow using dissipative particle dynamics method. J. Appl. Phys. (2014).

  17. 17.

    Jeongho, K., James, F.A.: Computational study of blood flow in microchannels. J. Comput. Appl. Math. 292(15), 174–187 (2016).

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Tatsumi, K., Noguchi, S.: Particle and rigidized red blood cell concentration distributions in microchannel flows. Phys. Fluids (2019).

  19. 19.

    Dill, D.B., Costill, D.L.: Calculation of percentage changes in volumes of blood, plasma, and red cells in dehydration. J. Appl. Physiol. 37(2), 247–248 (1974)

    Article  Google Scholar 

  20. 20.

    Chaichana, T., Sun, Z., Jewkes, J.: Computation of hemodynamics in the left coronary artery with variable angulations. J. Biomech. 44(10), 1869–1878 (2011)

    Article  Google Scholar 

  21. 21.

    Gijsen, F.J., Allanic, E., Van, F.N.: The influence of the non-Newtonian properties of blood on the flow in large arteries: unsteady flow in a 90 degrees curved tube. J. Biomech. 32(7), 705–713 (1999)

    Article  Google Scholar 

  22. 22.

    Razavi, A., Shirani, E.: Numerical simulation of blood pulsatile flow in a stenosed carotid artery using different rheological models. J. Biomech. 44(11), 2021–2030 (2011)

    Article  Google Scholar 

  23. 23.

    Chen, Z.S., Zhan, F.: A new stent with streamlined cross-section can suppress monocyte cell adhesion in the flow disturbance zones of the endovascular stent. Comput. Methods Biomech. Biomed. Eng. 19(1), 60–66 (2016)

    Article  Google Scholar 

  24. 24.

    Chen, Z.S., Fan, Y.B.: A new way to reduce flow disturbance in endovascular stents: a numerical study. Artif. Organs 35(4), 392–397 (2011)

    Article  Google Scholar 

  25. 25.

    Peng, L.S., Lan, L.: The clinical value of the change on red blood cell aggregation in patients with coronary heart disease. Chin. J. Thromb. Hemost. 1, 27–28 (2005) (in Chinese)

    Google Scholar 

  26. 26.

    Cundall, P.A., Strack, O.D.L.: A discrete numerical model for granular assemblies. Geotechnique. 29, 47–65 (1979)

    Article  Google Scholar 

  27. 27.

    Dennis, S.C.R., Singh, S.N., Ingham, D.B.: The steady flow due to a rotating sphere at low and moderate Reynolds numbers. J. Fluid Mech. 101, 257–279 (1980)

    ADS  Article  Google Scholar 

  28. 28.

    Haider, A., Levenspiel, O.: Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Technol. 58, 63–70 (1989)

    Article  Google Scholar 

  29. 29.

    Li, X.: Application of blood test red blood cell parameters in differential diagnosis of anemia. Guide China Med. 33(18), 108–109 (2020) (in Chinese)

    Google Scholar 

  30. 30.

    Carvalho, V., Rodrigues, N.: Hemodynamic study in 3D printed stenotic coronary artery models: experimental validation and transient simulation. Comput. Methods Biomech. Biomed. Eng. (2020).

  31. 31.

    Liu, Y., Zhang, D.F., Yin, Y.F.: Numerical analysis of unsteady blood flow model of fluid-solid interaction in carotid artery. J. Interv. Radiol. 10, 885–889 (2015).

    Article  Google Scholar 

  32. 32.

    Moore, J.E., Berry, J.L.: Fluid and solid mechanical implications of vascular stenting. Ann. Biomed. Eng. 30(4), 498–508 (2002)

    Article  Google Scholar 

  33. 33.

    Xiang, Y.F., Yin, J.F.: Three-dimensional simulation of blood flow in human thoracic aorta. Acad. J. Second Mil. Univ. 31(5), 516–520 (2010)

    Article  Google Scholar 

  34. 34.

    Van der Heiden, K., Gijsen, F.J.H.: The effects of stenting on shear stress: relevance to endothelial injury and repair. Cardiovasc. Res. 99(2), 269–275 (2013)

    Article  Google Scholar 

Download references


This work was financially supported by the National Natural Science Foundation of China under Grant No. 81672187 and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant No. 51721004.

Author information



Corresponding author

Correspondence to Min Zeng.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


1. A non-Newtonian particle suspensions model is proposed to investigate blood flow at stented areas.

2. The hemodynamic model with the non-Newtonian characteristic is more suitable to describe the realistic blood flow.

3. The risk of the in-stent restenosis is high in the regions adjacent to the struts in the proximal and the distal ends of the stent.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Zhan, J., Bian, W. et al. Local hemodynamic analysis after coronary stent implantation based on Euler-Lagrange method. J Biol Phys 47, 143–170 (2021).

Download citation


  • Blood flow
  • Red blood cells
  • Coronary stent
  • Non-Newtonian fluid
  • In-stent restenosis