Research Papers

Temporal and Spatial Variations of Wall Shear Stress in the Entrance Region of Microvessels

[+] Author and Article Information
Othmane Oulaid

Bharti School of Engineering,
Laurentian University,
935 Ramsey Lake Road,
Sudbury, ON P3E 2C6, Canada

Junfeng Zhang

Bharti School of Engineering,
Laurentian University,
935 Ramsey Lake Road,
Sudbury, ON P3E 2C6, Canada
e-mail: jzhang@laurentian.ca

1Corresponding author.

Manuscript received November 4, 2014; final manuscript received March 6, 2015; published online April 6, 2015. Assoc. Editor: Tim David.

J Biomech Eng 137(6), 061008 (Jun 01, 2015) (9 pages) Paper No: BIO-14-1548; doi: 10.1115/1.4030055 History: Received November 04, 2014; Revised March 06, 2015; Online April 06, 2015

Using a simplified two-dimensional divider-channel setup, we simulate the development process of red blood cell (RBC) flows in the entrance region of microvessels to study the wall shear stress (WSS) behaviors. Significant temporal and spatial variation in WSS is noticed. The maximum WSS magnitude and the strongest variation are observed at the channel inlet due to the close cell-wall contact. From the channel inlet, both the mean WSS and variation magnitude decrease, with a abrupt drop in the close vicinity near the inlet and then a slow relaxation over a relatively long distance; and a relative stable state with approximately constant mean and variation is established when the flow is well developed. The correlations between the WSS variation features and the cell free layer (CFL) structure are explored, and the effects of several hemodynamic parameters on the WSS variation are examined. In spite of the model limitations, the qualitative information revealed in this study could be useful for better understanding relevant processes and phenomena in the microcirculation.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Ando, J., and Yamamoto, K., 2009, “Vascular Mechanobiology: Endothelial Cell Responses to Fluid Shear Stress,” Circ. J., 73(11), pp. 1983–1992. [CrossRef] [PubMed]
Szymanski, M. P., Metaxa, E., Meng, H., and Kolega, J., 2008, “Endothelial Cell Layer Subjected to Impinging Flow Mimicking the Apex of an Arterial Bifurcation,” Ann. Biomed. Eng., 36(10), pp. 1681–1689. [CrossRef] [PubMed]
Chiu, J.-J., and Chien, S., 2011, “Effects of Disturbed Flow on Vascular Endothelium: Pathophysiological Basis and Clinical Perspectives,” Physiol. Rev., 91(1), pp. 327–387. [CrossRef] [PubMed]
Resnick, N., Yahav, H., Shay-Salit, A., Shushy, M., Schubert, S., Zilberman, L. C. M., and Wofovitz, E., 2003, “Fluid Shear Stress and the Vascular Endothelium: for Better and for Worse,” Prog. Biophys. Mol. Biol., 81(3), pp. 177–199. [CrossRef] [PubMed]
Stroka, K., and Aranda-Espinoza, H., 2010, “A Biophysical View of the Interplay Between Mechanical Forces and Signaling Pathways During Transendothelial Cell Migration,” FEBS J., 277(5), pp. 1145–1158. [CrossRef] [PubMed]
Himburg, H. A., Grzybowski, D. M., Hazel, A. L., LaMack, J. A., Li, X. M., and Friedman, M. H., 2004, “Spatial Comparison Between Wall Shear Stress Measures and Porcine Arterial Endothelial Permeability,” Am. J. Physiol.: Heart Circ. Physiol., 286(5), pp. H1916–H1922. [CrossRef] [PubMed]
Lehoux, S., and Tedgui, A., 1998, “Signal Transduction of Mechanical Stresses in the Vascular Wall,” Hypertension, 32(2), pp. 338–345. [CrossRef] [PubMed]
Pahakis, M., Kosky, J., Dull, R. O., and Tarbell, J. M., 2007, “The Role of Endothelial Glycocalyx Components in Mechanotransduction of Fluid Shear Stress,” Biochem. Biophys. Res. Commun., 355(1), pp. 228–233. [CrossRef] [PubMed]
Feaver, R. E., Gelfand, B. D., and Blackman, B. R., 2013, “Human Haemodynamic Frequency Harmonics Regulate the Inflammatory Phenotype of Vascular Endothelial Cells,” Nat. Commun., 4, Art. No. 1529. [CrossRef]
Uzarski, J. S., Scott, E. W., and McFetridge, P. S., 2013, “Adaptation of Endothelial Cells to Physiologically-Modeled, Variable Shear Stress,” PLoS One, 8(2), p. e57004. [CrossRef] [PubMed]
Dolan, J. M., Meng, H., Singh, S., Paluch, R., and Kolega, J., 2011, “High Fluid Shear Stress and Spatial Shear Stress Gradients Affect Endothelial Proliferation, Survival, and Alignment,” Ann. Biomed. Eng., 39(6), pp. 1620–1631. [CrossRef] [PubMed]
Xiong, W., and Zhang, J., 2010, “Shear Stress Variation Induced by Red Blood Cell Motion in Microvessel,” Ann. Biomed. Eng., 38(8), pp. 2649–2659. [CrossRef] [PubMed]
Yin, X., and Zhang, J., 2012, “Cell-Free Layer and Wall Shear Stress Variation in Microvessels,” Biorheology, 49(4), pp. 261–270. [CrossRef] [PubMed]
Freund, J. B., and Vermot, J., 2014, “The Wall-Stress Footprint of Blood Cells Flowing in Microvessels,” Biophys. J., 106(3), pp. 752–762. [CrossRef] [PubMed]
Pries, A. R., Ley, K., Claassen, M., and Gaehtgens, P., 1989, “Red Cell Distribution at Microvascular Bifurcations,” Microvasc. Res., 38(1), pp. 81–101. [CrossRef] [PubMed]
Ong, P. K., and Kim, S., 2013, “Effect of Erythrocyte Aggregation on Spatiotemporal Variations in Cell-Free Layer Formation Near on Arteriolar Bifurcation,” Microcirculation, 20(5), pp. 440–453. [CrossRef] [PubMed]
Ong, P. K., Jain, S., and Kim, S., 2012, “Spatio-Temporal Variations in Cell-Free Layer Formation Near Bifurcations of Small Arterioles,” Microvasc. Res., 83(2), pp. 118–125. [CrossRef] [PubMed]
Barber, J. O., Alberding, J. P., Restrepo, J. M., and Secomb, T. W., 2008, “Simulated Two-Dimensional Red Blood Cell Motion, Deformation, and Partitioning in Microvessel Bifurcations,” Ann. Biomed. Eng., 36(10), pp. 1690–1698. [CrossRef] [PubMed]
Xiong, W., and Zhang, J., 2012, “Two-Dimensional Lattice Boltzmann Study of Red Blood Cell Motion Through Microvascular Bifurcation: Cell Deformability and Suspending Viscosity Effects,” Biomech. Model. Mechanobiol., 11(3–4), pp. 575–583. [CrossRef] [PubMed]
Yin, X., Thomas, T., and Zhang, J., 2013, “Multiple Red Blood Cell Flows Through Microvascular Bifurcations: Cell Free Layer, Cell Trajectory, and Hematocrit Separation,” Microvasc. Res., 89, pp. 47–56. [CrossRef] [PubMed]
Ishikawa, T., Fujiwara, H., Matsuki, N., Yoshimoto, T., Imai, Y., Ueno, H., and Yamaguchi, T., 2011, “Asymmetry of Blood Flow and Cancer Cell Adhesion in a Microchannel With Symmetric Bifurcation and Confluence,” Biomed. Microdevices, 13(1), pp. 159–167. [CrossRef] [PubMed]
Oulaid, O., and Zhang, J., 2015, “Cell Free Layer Development Process in the Entrance Region of Microvessels,” Biomech. Model. Mechanobiol (in press). [CrossRef]
Zhang, J., Johnson, P. C., and Popel, A. S., 2009, “Effects of Erythrocyte Deformability and Aggregation on the Cell Free Layer and Apparent Viscosity of Microscopic Blood Flows,” Microvasc. Res., 77(3), pp. 265–272. [CrossRef] [PubMed]
Zhang, J., Johnson, P. C., and Popel, A. S., 2007, “An Immersed Boundary Lattice Boltzmann Approach to Simulate Deformable Liquid Capsules and Its Application to Microscopic Blood Flows,” Phys. Biol., 4(4), pp. 285–295. [CrossRef] [PubMed]
Evans, E. A., and Fung, Y. C., 1972, “Improved Measurements of the Erythrocyte Geometry,” Microvasc. Res., 4(4), pp. 335–347. [CrossRef] [PubMed]
Bagchi, P., Johnson, P. C., and Popel, A. S., 2005, “Computational Fluid Dynamic Simulation of Aggregation of Deformable Cells in a Shear Flow,” ASME J. Biomech. Eng., 127(7), pp. 1070–1080. [CrossRef]
Pozrikidis, C., 2001, “Effect of Membrane Bending Stiffness on the Deformation of Capsules in Simple Shear Flow,” J. Fluid Mech., 440, pp. 269–291. [CrossRef]
Liu, Y., and Liu, W. K., 2006, “Rheology of Red Blood Cell Aggregation by Computer Simulation,” J. Comput. Phys., 220(1), pp. 139–154. [CrossRef]
Neu, B., and Meiselman, H. J., 2002, “Depletion-Mediated Red Blood Cell Aggregation in Polymer Solutions,” Biophys. J., 83(5), pp. 2482–2490. [CrossRef] [PubMed]
Succi, S., 2001, The Lattice Boltzmann Equation, Oxford University, Oxford.
Zhang, J., 2011, “Lattice Boltzmann Method for Microfluidics: Models and Applications,” Microfluid. Nanofluid., 10(1), pp. 1–28. [CrossRef]
Peskin, C. S., 1977, “Numerical Analysis of Blood Flow in the Heart,” J. Comput. Phys., 25(3), pp. 220–252. [CrossRef]
Zhang, J., 2011, “Effect of Suspending Viscosity on Red Blood Cell Dynamics and Blood Flows in Microvessels,” Microcirculation, 18(17), pp. 562–573. [CrossRef] [PubMed]
Zhang, J., Johnson, P. C., and Popel, A. S., 2008, “Red Blood Cell Aggregation and Dissociation in Shear Flows Simulated by Lattice Boltzmann Method,” J. Biomech., 41(1), pp. 47–55. [CrossRef] [PubMed]
Skalak, R., and Chien, S., 1987, Handbook of Bioengineering, McGraw-Hill, New York.
Bronzino, J. D., 2006, Biomedical Engineering Fundamentals, 3rd ed., CRC, Boca Raton.
Tan, Y., Sun, D., Wang, J., and Huang, W., 2010, “Mechanical Characterization of Human Red Blood Cells Under Different Osmotic Conditions by Robotic Manipulation With Optical Tweezers,” IEEE Trans. Biomed. Eng., 57(7), pp. 1816–1825. [CrossRef] [PubMed]
Dao, M., Lim, C. T., and Suresh, S., 2005, “Erratum: Mechanics of the Human Red Blood Cell Deformed by Optical Tweezers [Journal of the Mechanics and Physics of Solids, 51 (2003) 2259–2280],” J. Mech. Phys. Solids, 53, pp. 493–494. [CrossRef]
Lim, C., Dao, M., Suresh, S., Sow, C., and Chew, K., 2004, “Large Deformation of Living Cells Using Laser Traps,” Acta Mater., 52(7), pp. 1837–1845. [CrossRef]
Breyiannis, G., and Pozrikidis, C., 2000, “Simple Shear Flow of Suspensions of Elastic Capsules,” Theor. Comput. Fluid Dyn., 13(5), pp. 327–347. [CrossRef]
Ye, S. S., Ng, Y. C., Tan, J., Leo, H. L., and Kim, S., 2014, “Two-Dimensional Strain-Hardening Membrane Model for Large Deformation Behavior of Multiple Red Blood Cells in High Shear Conditions,” Theor. Biol. Med. Modell., 11(19), pp. 1–21. [CrossRef]
Yin, X., and Zhang, J., 2012, “An Improved Bounce-Back Scheme for Complex Boundary Conditions in Lattice Boltzmann Method,” J. Comput. Phys., 231(11), pp. 4295–4303. [CrossRef]
Pries, A. R., Ley, K., and Gaehtgens, P., 1986, “Generalization of the Fahraeus Principle for Microvessel Networks,” Am. J. Physiol.: Heart Circ. Physiol., 20, pp. Hl324–Hl332. [CrossRef]
Lipowsky, H. H., 2005, “Microvascular Rheology and Hemodynamics,” Microcirculation, 12(1), pp. 5–15. [CrossRef] [PubMed]
Popel, A. S., and Johnson, P. C., 2005, “Microcirculation and Hemorheology,” Annu. Rev. Fluid Mech., 37, pp. 43–69. [CrossRef] [PubMed]


Grahic Jump Location
Fig. 1

Schematic drawings (a) for the RBC separation at a bifurcation and flow development in branch microvessels (arrows for the flow directions, different colors/shades for the concentrate RBC region and the CFL, and an elliptical circle for the flow development region to be simulated in this study), (b) for the divider-channel setup utilized in this study to mimic the flow development process in the entrance region (the channel length is shortened to fit the page width, and the cells are not plotted for clarity), and (c) for the lattice nodes and RBC membrane representation

Grahic Jump Location
Fig. 2

The RBC distribution and configuration (shaded patches in (b)), flow field (arrows in (b)), and instantaneous WSS distributions on the top (a) and bottom (c) walls at t = 55×105Δt. The x location has been shifted to the starting point of the flat part of the channel walls at x0 = 300Δx.

Grahic Jump Location
Fig. 3

The temporal variations of cell-wall gap distance (top panels) and WSS (bottom panels) during the simulation period 89-94 × 105Δt on the bottom wall at x = 300 (a), 320 (b), 400 (c), 500 (d), 1000 (e), and 3000 Δx (f). The nominal CFL thickness δ¯ at each location is provided in number and plotted as a dashed line in the top panels; and the mean and SD values for the temporal WSS variation at each location are also shown in the bottom panel as numbers and dashed lines (top: mean + SD; middle: mean; bottom: mean − SD).

Grahic Jump Location
Fig. 4

The distribution profiles of mean WSS τ¯w (a), WSS SD στ (b), and SD/mean ratio (c) along the channel. Mean and SD values calculated on both the top and bottom walls, as well as the average of them, are displayed in (a) and (b) with different colors and line styles, and the SD/mean ratio in (c) is obtained using the average mean and SD curves in (a) and (b) only. The curves are plotted over the entire channel length (x-x0 = 0-3160 Δx) for completeness; however, the right ends of these curves (x-x0 = 3000-3160 Δx) should be excluded for consideration of the exit effect.

Grahic Jump Location
Fig. 5

The distribution profiles of mean WSS τ¯w (a) and WSS SD στ (b) for the five cases considered in this study. Short horizontal bars with corresponding colors and line styles on the left figure edges are plotted to indicate the maximum values at x = x0 = 300 Δx of these curves (also available in Table 2), which will otherwise be undistinguishable. The curves are plotted over the entire channel length (x-x0 = 0-3160 Δx) for completeness; however, the right ends of these curves (x-x0 = 3000-3160 Δx) should be excluded for consideration of the exit effect.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In