Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-16T19:55:07.456Z Has data issue: false hasContentIssue false
  • English
  • Français

Cellular flow in a small blood vessel

Published online by Cambridge University Press:  18 February 2011

JONATHAN B. FREUND  and
M. M. ORESCANIN
JONATHAN B. FREUND*
Affiliation:
Departments of Mechanical Science and Engineering and Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
M. M. ORESCANIN
Affiliation:
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
*
†Email address for correspondence: jbfreund@illinois.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In the smallest capillaries, or in tubes with diameter D ≲ 8 μm, flowing red blood cells are well known to organize into single-file trains, with each cell deformed into an approximately static bullet-like shape. Detailed high-fidelity simulations are used to investigate flow in a model blood vessel with diameter slightly larger than this: D = 11.3 μm. In this case, the cells deviate from this single-file arrangement, deforming continuously and significantly. At the higher shear rates simulated (mean velocity divided by diameter U/D ≳ 50s−1), the effective tube viscosity is shear-rate insensitive with μeff/μplasma = 1.21. This matches well with the value μeff/μplasma = 1.19 predicted for the same 30% cell volume fraction by an established empirical fit of high-shear-rate in vitro experimental data. At lower shear rates, the effective viscosity increases, reaching μeff/μplasma ≈ 1.65 at the lowest shear rate simulated (U/D ≈ 3.7s−1). Because of the continuous deformations, the cell-interior viscosity is potentially important for vessels of this size. However, most results for simulations with cell interior viscosity five times that of the plasma (λ = 5), which is thought to be close to physiological conditions, closely match results for cases with λ = 1. The cell-free layer that forms along the vessel walls thickens from 0.3 μm for U/D = 3.7s−1 up to 1.2 μm for U/D ≳ 100s−1, in reasonable agreement with reported experimental results. The thickness of this cell-free layer is the key factor governing the overall flow resistance, and this in turn is shown to follow a trend expected for lubrication lift forces for shear rates between U/D ≈ 8s−1 and U/D ≈ 100s−1. Only in this same range are the cells near the vessel wall on average inclined relative to the wall, as might be expected for a lubrication mechanism. Metrics are developed to quantify the kinematics of this dense cellular flow in terms of the well-known tank-treading and tumbling behaviours often observed for isolated cells in shear flows. One notable effect of λ = 5 versus λ = 1 is that it suppresses treading rotation rates by a factor of about 2. The treading rate is found to scale with the velocity difference across the cell-rich core and is thus significantly slower than the overall shear rate in the flow, which is presumably why the flow is otherwise insensitive to λ. The cells in all cases also have a similarly slow mean tumbling motion, which is insensitive to cell-interior viscosity and decreases monotonically with increasing U/D.

JFM classification

Biological Fluid Dynamics: Blood flow Biological Fluid Dynamics: Capsule/cell dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 671 , 25 March 2011 , pp. 466 - 490
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abkarian, M., Faivre, M. & Viallat, A. 2007 Swinging of red blood cells under shear flow. Phys. Rev. Lett. 98, 188302.CrossRefGoogle ScholarPubMed
Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K. & Walter, P. 2008 Molecular Biology of the Cell, 5th edn. Garland Science.Google Scholar
Alonso, C., Pries, A. R., Kiesslich, O., Lerche, D. & Gaehtgens, P. 1995 Transient rheological behavior of blood in low-shear tube flow: velocity profiles and effective viscosity. Am. J. Physiol. Heart Circ. Physiol. 268, H25–H32.CrossRefGoogle ScholarPubMed
Bachmann, L., Schmitt-Fumain, W. W., Hammel, R. & Lederer, K. 1973 Size and shape of fibrinogen. Part 1. Electron microscopy of the hydrated molecule. Die Makromol. Chem. 176 (9), 2603–2618.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1987 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Chien, S., Usami, S., Dellenback, R. J. & Gregersen, M. I. 1967 Blood viscosity: influence of erythrocyte aggregation. Science 157 (3790), 829–831.CrossRefGoogle ScholarPubMed
Cokelet, G. R. & Goldsmith, H. L. 1991 Decreased hydrodynamic resistance in the two-phase flow of blood through small vertical tubes at low flow rates. Circ. Res. 68, 1–17.CrossRefGoogle ScholarPubMed
Dao, M., Li, J. & Suresh, S. 2006 Molecularly based analysis of deformation of spectrin network and human erythrocyte. Mater. Sci. Engng C 26, 1232–1244.CrossRefGoogle Scholar
Dintenfass, L. 1968 Internal viscosity of the red cell and a blood viscosity equation. Nature 219, 956–958.CrossRefGoogle Scholar
Doddi, S. K. & Bagchi, P. 2008 Lateral migration of a capsule in a plane Poiseuille flow in a channel. Intl J. Multiphase Flow 34, 966–986.CrossRefGoogle Scholar
Doddi, S. K. & Bagchi, P. 2009 Three-dimensional computational modeling of multiple deformable cells flowing in microvessels. Phys. Rev. E 79, 046318.CrossRefGoogle ScholarPubMed
Dupin, M. M., Halliday, I., Care, C. M., Alboul, L. & Munn, L. L. 2007 Modeling the flow of dense suspensions of deformable particles in three dimensions. Phys. Rev. E 75, 066707.CrossRefGoogle ScholarPubMed
Essemann, U., Perera, L., Berkowitz, M. L., Darden, T., Lee, H. & Pedersen, L. G. 1995 A smooth particle mesh Ewald method. J. Chem. Phys. 103 (19), 8577–8593.CrossRefGoogle Scholar
Evans, E. A. 1983 Bending elastic modulus of red blood cell membrane derived from buckling instability in micropipet aspiration tests. Biophys. J. 43, 27–30.CrossRefGoogle ScholarPubMed
Evans, E. A. & Hochmuth, R. M. 1976 Membrane viscoelasticity. Biophys. J. 16, 1–11.CrossRefGoogle ScholarPubMed
Fåhræus, R. & Lindqvist, T. 1931 The viscosity of the blood in narrow capillary tubes. Am. J. Physiol. 96, 562–568.CrossRefGoogle Scholar
Fedosov, D. A., Caswell, B. & Karniadakis, G. E. 2010 a A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys. J. 98, 2215–2225.CrossRefGoogle ScholarPubMed
Fedosov, D. A., Caswell, B. & Karniadakis, G. E. 2010 b Systematic coarse-graining of spectrin-level red blood cell models. Comput. Meth. Appl. Mech. Engng 199 (29–32), 1937–1948.CrossRefGoogle ScholarPubMed
Freund, J. B. 2007 Leukocyte margination in a model microvessel. Phys. Fluids 19 (3), 023301.CrossRefGoogle Scholar
Freund, J. B. & Zhao, H. 2010 A fast high-resolution boundary integral method for multiple interacting blood cells. In Computational Hydrodynamics of Capsules and Biological Cells (ed. Pozrikidis, C.), pp. 71–111. Chapman & Hall/CRC.CrossRefGoogle Scholar
Gaehtgens, P., Duhrssen, C. & Albrecht, K. H. 1980 Motion, deformation, and interaction of blood cells and plasma during flow through narrow capillary tubes. Blood Cells 6, 799–812.Google ScholarPubMed
Gaehtgens, P. & Schmid-Schönbein, H. 1982 Mechanisms of dynamic flow adaptation of mammalian erythrocytes. Naturwissenschaften 69, 294–296.CrossRefGoogle ScholarPubMed
Gidaspow, D. & Huang, J. 2009 Kinetic theory based model for blood flow and its viscosity. Ann. Biomed. Engng 37, 1534–1545.CrossRefGoogle ScholarPubMed
Gupta, B. B., Nigam, K. M. & Jaffrin, M. Y. 1982 A three-layer semi-empirical model for flow of blood and other particulate suspensions through narrow tubes. J. Biomech. Engng 104, 129–135.CrossRefGoogle ScholarPubMed
Hernández-Ortiz, J. P., de Pablo, J. J. & Graham, M. D. 2007 Fast computation of many-particle hydrodynamic and electrostatic interactions in a confined geometry. Phys. Rev. Lett. 98, 140602.CrossRefGoogle Scholar
Hochmuth, R. M. & Waugh, R. E. 1987 Erythrocyte membrane elasticity and viscosity. Annu. Rev. Phys. 49, 209–219.CrossRefGoogle ScholarPubMed
Hsu, R. & Secomb, T. W. 1989 Motion of nonaxisymmetric red blood cells in cylindrical capillaries. J. Biomech. Engng 111, 147–151.CrossRefGoogle ScholarPubMed
Kamm, R. D. 2002 Cellular fluid mechanics. Annu. Rev. Fluid Mech. 34, 211–32.CrossRefGoogle ScholarPubMed
Kim, S., Kong, R. L., Popel, A. S., Intaglietta, M. & Johnson, P. C. 2007 Temporal and spatial variations of cell-free layer with arterioles. Am. J. Physiol. Heart Circ. Physiol. 293, H1526–H1535.CrossRefGoogle ScholarPubMed
Kim, S., Ong, P. K., Yalcin, O., Intaglietta, M. & Johnson, P. C. 2009 The cell-free layer in microvascular blood flow. Biorheology 46, 181–189.CrossRefGoogle ScholarPubMed
Li, J., Dao, M., Lim, C. T. & Suresh, S. 2005 Spectrin-level modeling of the cytoskeleton and optical tweezers stretching of the erythrocyte. Biophys. J. 88, 3707–3719.CrossRefGoogle ScholarPubMed
Long, D. S. 2004 Microviscometric analysis of microvascular hemodynamics in vivo. PhD thesis, University of Illinois at Urbana-Champaign, Urbana, Illinois.Google Scholar
Marton, Z., Kesmarky, G., Vekasi, J., Cser, A., Russai, R., Horvath, B. & Toth, K. 2001 Red blood cell aggregation measurements in whole blood and in fibrinogen solutions by different methods. Clin. Hemorheol. Microcirc. 24 (2), 75–83.Google ScholarPubMed
McWhirter, J. L., Noguchi, H. & Gompper, G. 2009 Flow-induced clustering and alignment of vesicles and red blood cells in microcapillaries. Proc. Natl Acad. Sci. 106, 6039–6043.CrossRefGoogle ScholarPubMed
Messlinger, S., Schmidt, B., Noguchi, H. & Gompper, G. 2009 Dynamical regimes and hydrodynamic lift of viscous vesicles under shear. Phys. Rev. E 80, 011901.CrossRefGoogle ScholarPubMed
Nair, P. K., Hellums, J. D. & Olson, J. S. 1989 Prediction of oxygen transport rates in blood flowing in large capillaries. Microvasc. Res. 38, 269–285.CrossRefGoogle ScholarPubMed
Noguchi, H. & Gompper, G. 2005 Shape transitions of fluid vesicles and red blood cells in capillary flows. Proc. Natl Acad. Sci. 102 (40), 14159–14164.CrossRefGoogle ScholarPubMed
Phillips, R. J., Armstrong, R. C., Brown, R. A., Graham, A. L. & Abbott, J. R. 1992 A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys. Fluids A 4 (1), 30–40.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 2003 Numerical simulation of the flow-induced deformation of red blood cells. Ann. Biomed. Engng 31, 1194–1205.CrossRefGoogle ScholarPubMed
Pozrikidis, C. 2005 Axisymmetric motion of a file of red blood cells through capillaries. Phys. Fluids 17, 031503.CrossRefGoogle Scholar
Pries, A. R., Neuhaus, D. & Gaehtgens, P. 1992 Blood viscosity in tube flow: dependence on diameter and hematocrit. Am. J. Physiol. Heart Circ. Physiol. 263, H1770–H1778.CrossRefGoogle ScholarPubMed
Pries, A. R. & Secomb, T. W. 2003 Rheology of the microcirculation. Clin. Hemorheol. Microcirc. 29, 143–148.Google ScholarPubMed
Pries, A. R., Secomb, T. W., Gessner, T., Sperandio, M. B., Gross, J. F. & Gaehtgens, P. 1994 Resistance to blood flow in microvessels in vivo. Circ. Res. 75, 904–915.CrossRefGoogle ScholarPubMed
Reinke, W., Gaehtgens, P. & Johnson, P. C. 1987 Blood viscosity in small tubes: effect of shear rate, aggregation and sedimentation. Heart Circ. Physiol. 253, H540–H547.CrossRefGoogle ScholarPubMed
Reinke, W., Johnson, P. C. & Gaehtgens, P. 1986 Effect of shear rate variation on apparent viscosity of human blood in tubes of 29 to 94μm diameter. Circ. Res. 59, 124–132.CrossRefGoogle Scholar
Reynolds, O. 1886 On the theory of lubrication and its application to Mr Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil. Proc. R. Soc. Lond. 177, 157–234.Google Scholar
Saintillan, D., Darve, E. & Shaqfeh, E. S. G. 2005 A smooth particle-mesh Ewald algorithm for Stokes suspension simulations: the sedimentation of fibers. Phys. Fluids 17, 033301.CrossRefGoogle Scholar
Secomb, T. W. 1987 Flow-dependent rheological properties of blood in capillaries. Microvasc. Res. 34, 46–58.CrossRefGoogle ScholarPubMed
Secomb, T. W. 2003 Mechanics of red blood cells and blood flow in narrow tubes. In Modeling and Simulation of Capsules and Biological Cells (ed. Pozrikidis, C.), pp. 163–196. Chapman & Hall/CRC.Google Scholar
Sharan, M. & Popel, A. S. 2001 A two-phase model for flow of blood in narrow tubes with increased effective viscosity near the wall. Biorheology 38, 415–428.Google Scholar
Skalak, R. & Branemark, P.-I. 1969 Deformation of red blood cells in capillaries. Science 164, 717–719.CrossRefGoogle ScholarPubMed
Skotheim, J. M. & Secomb, T. W. 2007 Red blood cells and other nonspherical capsules in shear flow: oscillatory dynamics and tank-treading-to-tumbling transition. Phys. Rev. Lett. 98, 078301.CrossRefGoogle ScholarPubMed
Soutani, M., Suzuki, Y., Tateishi, N. & Maeda, N. 1995 Quantitative evaluation of flow dynamics of erythrocytes in microvessels: influence of erythrocyte aggregation. Am. J. Physiol. Heart Circ. Physiol. 268, H1959–H1965.CrossRefGoogle ScholarPubMed
Sui, Y., Chew, Y. T., Roy, P., Cheng, Y. P. & Low, H. T. 2008 Dynamic motion of red blood cells in simple shear flow. Phys. Fluids 20, 1–10.CrossRefGoogle Scholar
Whitmore, R. L. 1968 Rheology of the Circulation. Pergamon.Google Scholar
Yamaguchi, S., Yamakawa, T. & Niimi, H. 1992 Cell-free plasma layer in cerebral microvessels. Biorheology 29, 251–260.CrossRefGoogle ScholarPubMed
Zhao, H., Isfahani, A. H. G., Olson, L. & Freund, J. B. 2010 A spectral boundary integral method for flowing blood cells. J. Comput. Phys. 229 (10), 3726–2744.CrossRefGoogle Scholar