Skip to main content
Log in

Red blood cell distribution in a microvascular network with successive bifurcations

  • Original Paper
  • Published:
Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript

Abstract

Nonproportional RBC distribution is an important characteristic in microvascular networks, which can result in heterogeneity of oxygen supply that may cause ischemic death in severe cases. In this paper, we perform three-dimensional numerical simulations of a large number of RBCs in a microvascular network, by using a hybrid method of smoothed dissipative particle dynamic and immersed boundary method. The distribution of multiple RBCs in a T-bifurcation is first simulated as a validation study, and a reasonable agreement is observed both qualitatively and quantitatively on the RBC flux between our results and the previously published numerical and empirical results. Next, the distribution of a large number of RBCs in a microvascular network is investigated, including the effects of cell deformability, aggregation and tube hematocrit. The simulation results indicate that decreased deformability and increased aggregation strength have a similar effect on the RBC distribution: the large RBC flux becomes larger, but the small becomes smaller. A high hematocrit also causes a similar phenomenon that the RBCs are more apt to flow into a high RBC-flux branch, because they are arranged compactly into a rouleaux and difficultly broken up at a high hematocrit. These results imply that lower cell deformability, stronger aggregation or higher tube hematocrit would be conducive to the phase separation of hematocrit and plasma skimming processes in microcirculation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Balogh P, Bagchi P (2017a) A computational approach to modeling cellular-scale blood flow in complex geometry. J Comput Phys 334:280–307

    MathSciNet  Google Scholar 

  • Balogh P, Bagchi P (2017b) Direct numerical simulation of cellular-scale blood flow in 3D microvascular networks. Biophys J 113:2815–2826

    Google Scholar 

  • Balogh P, Bagchi P (2018) Analysis of red blood cell partitioning at bifurcations in simulated microvascular networks. Phys Fluids 30:051902

    Google Scholar 

  • Barber JO, Alberding JP, Restrepo JM, Secomb TW (2008) Simulated two-dimensional red blood cell motion, deformation, and partitioning in microvessel bifurcations. Ann Biomed Eng 36:1690–1698

    Google Scholar 

  • Barber JO, Restrepo JM, Secomb TW (2011) Simulated red blood cell motion in microvessel bifurcations: effects of cell–cell interactions on cell partitioning. Cardiovasc Eng Technol 2:349–360

    Google Scholar 

  • Bian X, Litvinov S, Qian R, Ellero M, Adams NA (2012) Multiscale modeling of particle in suspension with smoothed dissipative particle dynamics. Phys Fluids 24:012002

    Google Scholar 

  • Catita J, López-Luppo M, Ramos D, Nacher V, Navarro M, Carretero A, Sánchez-Chardi A, Mendes-Jorge L, Rodriguez-Baeza A, Ruberte J (2015) Imaging of cellular aging in human retinal blood vessels. Exp Eye Res 135:14–25

    Google Scholar 

  • Chien S, Tvetenstrand CD, Epstein MA, Schmid-Schönbein GW (1985) Model studies on distributions of blood cells at microvascular bifurcations. Am J Physiol 248:568–576

    Google Scholar 

  • Chung B, Johnson PC, Popel AS (2007) Application of Chimera grid to modelling cell motion and aggregation in a narrow tube. Int J Numer Methods Fluids 53:105–128

    MathSciNet  MATH  Google Scholar 

  • Ellero M, Español P (2018) Everything you always wanted to know about SDPD (but were afraid to ask). Appl Math Mech Engl Ed 39:103–124

    MathSciNet  Google Scholar 

  • Español P, Revenga M (2003) Smoothed dissipative particle dynamics. Phys Rev E 67:026705

    Google Scholar 

  • Fedosov DA, Caswell B, Karniadakis GE (2010a) A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys J 98:2215–2225

    Google Scholar 

  • Fedosov DA, Caswell B, Karniadakis GE (2010b) Systematic coarse-graining of spectrin-level red blood cell models. Comput Methods Appl Mech Eng 199:1937–1948

    MathSciNet  MATH  Google Scholar 

  • Fenton BM, Carr RT, Cokelet GR (1985) Nonuniform red cell distribution in 20 to 100 micrometers bifurcations. Microvasc Res 29:103–126

    Google Scholar 

  • Francis C, Frederic L, Cline F, Steffen P, Valerie LC (2006) A novel three-dimensional computer-assisted method for a quantitative study of microvascular networks of the human cerebral cortex. Microcirculation 13:1–18

    Google Scholar 

  • Freund JB (2007) Leukocyte margination in a model microvessel. Phys Fluids 19:023301

    MATH  Google Scholar 

  • Freund JB (2014) Numerical simulation of flowing blood cells. Ann Rev Fluid Mech 46:67–95

    MathSciNet  MATH  Google Scholar 

  • Gompper G, Fedosov DA (2016) Modeling microcirculatory blood flow: current state and future perspectives. WIREs Syst Biol Med 8:157–168

    Google Scholar 

  • Groot RD, Warren PB (1997) Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 107:4423–4435

    Google Scholar 

  • Hochmuth RM (1987) Erythrocyte membrane elasticity and viscosity. Ann Rev Physiol 49:209–219

    Google Scholar 

  • Hu J, Cao Y, Wu T, Li D, Lu H (2014) High-resolution three-dimensional visualization of the rat spinal cord microvasculature by synchrotron radiation micro-CT. Med Phys 41:101904

    Google Scholar 

  • Hyakutake T, Nagai S (2015) Numerical simulation of red blood cell distributions in three-dimensional microvascular bifurcations. Microvasc Res 97:115–123

    Google Scholar 

  • Hyakutake T, Matsumoto T, Yanase S (2006) Lattice Boltzmann simulation of blood cell behavior at microvascular bifurcations. Math Comput Simul 72:134–140

    MathSciNet  MATH  Google Scholar 

  • Hyakutake T, Tominaga S, Matsumoto T, Yanase S (2008) Numerical study on flows of red blood cells with liposome-encapsulated hemoglobin at microvascular bifurcation. J Biomech Eng 130:011014

    Google Scholar 

  • Kaliviotis E, Sherwood JM, Balabani S (2017) Partitioning of red blood cell aggregates in bifurcating microscale flows. Sci Rep 7:44563

    Google Scholar 

  • Lei H, Fedosov DA, Karniadakis GE (2011) Time-dependent and outflow boundary conditions for dissipative particle dynamics. J Comput Phys 230:3765–3779

    MathSciNet  MATH  Google Scholar 

  • Li X, Popel AS, Karniadakis GE (2012) Blood-plasma separation in Y-shaped bifurcating microfluidic channels: a dissipative particle dynamics simulation study. Phys Biol 9(2):026010

    Google Scholar 

  • Liu YL, Liu WK (2006) Rheology of red blood cell aggregation by computer simulation. J Comput Phys 220:139–154

    MathSciNet  MATH  Google Scholar 

  • Liu Y, Zhang L, Wang X, Liu W (2004) Coupling of Navier–Stokes equations with protein molecular dynamics and its application to hemodynamics. Int J Numer Methods Fluids 46:1237–1252

    MathSciNet  MATH  Google Scholar 

  • Liu MB, Shao JR, Chang JZ (2012) On the treatment of solid boundary in smoothed particle hydrodynamics. Sci China Technol Sci 55:244–254

    MATH  Google Scholar 

  • Lykov K, Li X, Lei H, Pivkin IV, Karniadakis GE (2015) Inflow/outflow boundary conditions for particle-based blood flow simulations: application to arterial bifurcations and trees. PLoS Comput Biol 11:e1004410

    Google Scholar 

  • Maeda N (1996) Erythrocyte rheology in microcirculation. Jpn J Physiol 46:1–14

    Google Scholar 

  • Noguchi H, Gompper G (2005) Vesicle dynamics in shear and capillary flows. J Phys 17:S3439–S3444

    Google Scholar 

  • Peskin CS (2002) The immersed boundary method. Acta Numer 11:1–39

    MathSciNet  MATH  Google Scholar 

  • Pivkin IV, Karniadakis GE (2008) Accurate coarse-grained modeling of red blood cells. Phys Rev Lett 101:118105

    Google Scholar 

  • Popel AS, Johnson PC (2005) Microcirculation and hemorheology. Ann Rev Fluid Mech 37:43–69

    MathSciNet  MATH  Google Scholar 

  • Pries AR, Secomb TW (2003) Rheology of the microcirculation. Clin Hemorheol Microcirc 29:143–148

    Google Scholar 

  • Pries AR, Secomb TW (2005) Microvascular blood viscosity in vivo and the endothelial surface layer. Am J Physiol 289:2657–2664

    Google Scholar 

  • Pries AR, Ley K, Claassen M, Gaehtgens P (1989) Red cell distribution at microvascular bifurcations. Microvasc Res 38:81–101

    Google Scholar 

  • Pries AR, Secomb TW, Gaehtgens P, Gross JF (1990) Blood flow in microvascular networks: experiments and simulation. Circ Res 67:826–834

    Google Scholar 

  • Schmid-Schönbein GW, Skalak R, Usami S, Chien S (1980) Cell distribution in capillary networks. Microvasc Res 19:18–44

    Google Scholar 

  • Secomb TW (2017) Blood flow in the microcirculation. Ann Rev Fluid Mech 49:443–461

    MathSciNet  MATH  Google Scholar 

  • Secomb TW, Styp-Rekowska B, Pries AR (2007) Two-dimensional simulation of red blood cell deformation and lateral migration in microvessels. Ann Biomed Eng 35:755–765

    Google Scholar 

  • Secomb TW, Barber JO, Alberding JP, Restrepo JM (2008) Motion of red blood cells in microvessels and bifurcations: computational simulations. Biorheology 45:35–36

    Google Scholar 

  • Skorczewski T, Erickson LC, Fogelson AL (2013) Platelet motion near a vessel wall or thrombus surface in two-dimensional whole blood simulations. Biophys J 104:1764–1772

    Google Scholar 

  • Tian T, Zhai Q, Zhang R (2018) A new modified weak Galerkin finite element scheme for solving the stationary stokes equations. J Comput Appl Math 329:268–279

    MathSciNet  MATH  Google Scholar 

  • Vahidkhah K, Diamond S, Bagchi P (2014) Platelet dynamics in three-dimensional simulation of whole blood. Biophys J 106:2529–2540

    Google Scholar 

  • Vázquez-Quesada A, Ellero M, Español P (2009) Smoothed particle hydrodynamic model for viscoelastic fluids with thermal fluctuations. J Chem Phys 130:034901

    Google Scholar 

  • Wang T, Rongin U, Xing ZX (2016) A micro-scale simulation of red blood cell passage through symmetric and asymmetric bifurcated vessels. Sci Rep 6:20262

    Google Scholar 

  • Xiao LL, Liu Y, Chen S, Fu BM (2017) Effects of flowing RBCs on adhesion of a circulating tumor cell in microvessels. Biomech Model Mechanobiol 16:1–14

    Google Scholar 

  • Xiong WJ, Zhang J (2012) Two-dimensional lattice Boltzmann study of red blood cell motion through microvascular bifurcation: cell deformability and suspending viscosity effects. Biomech Mod Mechanobiol 11:575–583

    Google Scholar 

  • Xu YQ, Tian FB, Li HJ, Deng YL (2012) Red blood cell partitioning and blood flux redistribution in microvascular bifurcation. Theor Appl Mech Lett 2:024001

    Google Scholar 

  • Yang S, Undar A, Zahn JD (2006) A microfluidic device for continuous, real time blood plasma separation. Lab Chip 6:871–880

    Google Scholar 

  • Ye T, Peng LN (2019) Motion, deformation and aggregation of multiple red blood cells in three-dimensional microvessel bifurcations. Phys Fluids 31:021903

    Google Scholar 

  • Ye T, Phan-Thien N, Khoo BC, Lim CT (2014) A file of red blood cells in tube flow: a three-dimensional numerical study. J Appl Phys 116:124703

    Google Scholar 

  • Ye T, Phan-Thien N, Lim CT (2016) Particle-based simulations of red blood cells—a review. J Biomech 49:2255–2266

    Google Scholar 

  • Ye T, Phan-Thien N, Lim CT, Peng LN, Shi HX (2017a) Hybrid smoothed dissipative particle dynamics and immersed boundary method for simulation of red blood cells in flows. Phys Rev E 95:063314

    MathSciNet  Google Scholar 

  • Ye T, Shi HX, Peng LN, Li Y (2017b) Numerical studies of a red blood cell in rectangular microchannels. J Appl Phys 122:084701

    Google Scholar 

  • Ye T, Phan-Thien N, Khoo BC, Li Y (2018a) Flow patterns and red blood cell dynamics in a U-bend. J Appl Phys 124:124701

    Google Scholar 

  • Ye T, Shi HX, Phan-Thien N, Lim CT, Li Y (2018b) Relationship between transit time and mechanical properties of a cell through a stenosed microchannel. Soft Matter 14:533–545

    Google Scholar 

  • Ye T, Pan DY, Huang C, Liu MB (2019) Smoothed particle hydrodynamics (SPH) for complex fluid flows: recent developments in methodology and applications. Phys Fluids 31:011301

    Google Scholar 

  • Yin XW, Thomas T, Zhang J (2013) Multiple red blood cell flows through microvascular bifurcations: cell free layer, cell trajectory, and hematocrit separation. Microvasc Res 89:47–56

    Google Scholar 

  • Zhang J (2011a) Effect of suspending viscosity on red blood cell dynamics and blood flows in microvessels. Microcirculation 18:562–573

    Google Scholar 

  • Zhang J (2011b) Lattice Boltzmann method for microfluidics: models and applications. Microfluid Nanofluid 10:1–28

    Google Scholar 

  • Zhang J, Johnson PC, Popel AS (2008) Red blood cell aggregation and dissociation in shear flows simulated by lattice Boltzmann method. J Biomech 41:47–55

    Google Scholar 

  • Zhang J, Johnson PC, Popel AS (2009) Effects of erythrocyte deformability and aggregation on the cell free layer and apparent viscosity of microscopic blood flows. Microvasc Res 77:265–272

    Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China under Project No. 11502094.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ting Ye.

Additional information

Publisher's Note

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

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 399 KB)

Supplementary material 2 (SLDPRT 724 KB)

Supplementary material 3 (avi 15288 KB)

Supplementary material 4 (avi 17467 KB)

Supplementary material 5 (avi 15986 KB)

Supplementary material 6 (avi 16890 KB)

Supplementary material 7 (avi 14407 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ye, T., Peng, L. & Li, G. Red blood cell distribution in a microvascular network with successive bifurcations. Biomech Model Mechanobiol 18, 1821–1835 (2019). https://doi.org/10.1007/s10237-019-01179-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-019-01179-5

Keywords

Navigation