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Segregation of Flowing Blood: Mathematical Description

Published online by Cambridge University Press:  10 August 2011

A. Tokarev ,
G. Panasenko  and
F. Ataullakhanov
A. Tokarev*
Affiliation:
National Research Center for Hematology, Russian Academy of Medical Sciences Novii Zykovskii proezd, 4a, Moscow, Russia, 125167
G. Panasenko
Affiliation:
University Jean Monnet, 23 rue Dr. Paul Michelon, 42023 Saint-Etienne, France
F. Ataullakhanov
Affiliation:
National Research Center for Hematology, Russian Academy of Medical Sciences Novii Zykovskii proezd, 4a, Moscow, Russia, 125167
*
⋆ Corresponding author. E-mail: alexey.tokarev@mail.ru
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Abstract

Blood rheology is completely determined by its major corpuscles which are erythrocytes, or red blood cells (RBCs). That is why understanding and correct mathematical description of RBCs behavior in blood is a critical step in modelling the blood dynamics. Various phenomena provided by RBCs such as aggregation, deformation, shear-induced diffusion and non-uniform radial distribution affect the passage of blood through the vessels. Hence, they have to be taken into account while modelling the blood dynamics. Other important blood corpuscles are platelets, which are crucial for blood clotting. RBCs strongly affect the platelet transport in blood expelling them to the vessel walls and increasing their dispersion, which has to be considered in models of clotting. In this article we give a brief review of basic modern approaches in mathematical description of these phenomena, discuss their applicability to real flow conditions and propose further pathways for developing the theory of blood flow.

Keywords

blooderythrocytesplateletssuspension dynamics
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 5: Complex Fluids , 2011 , pp. 281 - 319
Copyright
© EDP Sciences, 2011

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References

C. G. Caro, T. J. Pedley, R. C. Schroter, W. A. Seed. The mechanics of the circulation (russian edition). Mir, Moscow, 1981.
A. M. Chernukh, P. N. Aleksandrov, O. V. Alekseev. Microcirculation. Medicina, Moscow, 1984.
Goldsmith, H. L., Skalak, R.. Hemodynamics. Annual Review of Fluid Mechanics, 7 (1975), 213-247. CrossRefGoogle Scholar
Goldsmith, H. L., Turitto, V. T.. Rheological aspects of thrombosis and haemostasis: basic principles and applications. ICTH-Report–Subcommittee on Rheology of the International Committee on Thrombosis and Haemostasis. Thromb. Haemost., 55 (1986), No. 3, 415-435. Google ScholarPubMed
Goldsmith, H. L.. The Microcirculatory Society Eugene M. Landis Award lecture. The microrheology of human blood. Microvasc. Res., 31 (1986), No. 2, 121-142. CrossRefGoogle Scholar
Popel, A. S., Johnson, P. C.. Microcirculation and hemorheology. Annu. Rev. Fluid Mech., 37 (2005), 43-69. CrossRefGoogle ScholarPubMed
Lipowsky, H. H.. Microvascular rheology and hemodynamics. Microcirculation, 12 (2005), 5-15. CrossRefGoogle ScholarPubMed
Cokelet, G. R.. Viscometric, in vitro and in vivo blood viscosity relationships: how are they related? (Poiseuille Award Lecture). Biorheology, 36 (1999), 343-358. Google Scholar
Quarteroni, A. M., Tuveri, M., Veneziani, A.. Computational vascular fluid dynamics: problems, models and methods. Computing and Visualization in Science, 2 (2000), 163-197. CrossRefGoogle Scholar
Kim, S., Ong, P. K., Yalcin, O., Intaglietta, M., Johnson, P. C.. The cell-free layer in microvascular blood flow. Biorheology, 46 (2009), 181-189. Google ScholarPubMed
Manjunatha, M., Singh, M.. Digital blood flow analysis from microscopic images of mesenteric microvessel with multiple branching. Clin. Hemorheol. Microcirc., 27 (2002), 91-106. Google ScholarPubMed
Manjunatha, M., Singh, S. S., Singh, M.. Blood flow analysis in mesenteric microvascular network by image velocimetry and axial tomography. Microvascular Research, 65 (2003), 49-55. CrossRefGoogle ScholarPubMed
Palmer, A. A., Betts, W. H.. The axial drift of fresh and acetaldehyde-hardened erythrocytes in 25 mum capillary slits of various lengths. Biorheology, 12 (1975), No. 5, 283-293. Google ScholarPubMed
Ellsworth, M. L., Pittman, R. N.. Evaluation of photometric methods for quantifying convective mass transport in microvessels. Am. J. Physiol., 251 (1986), H869-H879. Google ScholarPubMed
Pries, A. R., Ley, K., Claassen, M., Gaehtgens, P.. Red Cell Distribution at Microvascular Bifurcations. Microvasc. Res., 38 (1989), 81-101. CrossRefGoogle ScholarPubMed
Phibbs, R. H.. Distribution of leukocytes in blood flowing through arteries. Am. J. Physiol., 210 (1966), No. 5, 919-925. Google ScholarPubMed
Tangelder, G. J., Teirlinck, H. C., Slaaf, D. W., Reneman, R. S.. Distribution of blood platelets flowing in arterioles. Am. J. Physiol., 248 (1985), H318-H323. Google ScholarPubMed
Woldhuis, B., Tangelder, G. J., Slaaf, D. W., Reneman, R. S.. Concentration profile of blood platelets differs in arterioles and venules. Am. J. Physiol., 262 (1992), H1217-H1223. Google ScholarPubMed
Aarts, P. A., van den Broek, S. A., Prins, G. W., Kuiken, G. D., Sixma, J. J., Heethaar, R. M.. Blood platelets are concentrated near the wall and red blood cells, in the center in flowing blood. Arteriosclerosis, 8 (1988), No. 6, 819-824. CrossRefGoogle Scholar
Goldsmith, H. L.. Red cell motions and wall interactions in tube flow. Fed. Proc., 30 (1971), No. 5, 1578-1590. Google ScholarPubMed
Segré, G., Silberberg, A.. Radial particle displacements in poiseuille flow of suspensions. Nature, 189 (1961), 209-210. CrossRefGoogle Scholar
Segré, G., Silberberg, A.. Behaviour of macroscopic rigid spheres in Poiseuille flow Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams. Journal of Fluid Mechanics, 14 (1962), 115-135. CrossRefGoogle Scholar
Segré, G., Silberberg, A.. Behaviour of macroscopic rigid spheres in Poiseuille flow Part 2. Experimental results and interpretation. Journal of Fluid Mechanics, 14 (1962), 136-157. CrossRefGoogle Scholar
Oliver, D. R.. Influence of particle rotation on radial migration in the Poiseuille flow of suspensions. Nature, 194 (1962), 1269-1271. CrossRefGoogle Scholar
Takano, M., Goldsmith, H. L., Mason, S. G.. The flow of suspensions through tubes VIII. Radial Migration of Particles in Pulsatile Flow. Journal of Colloid and lnterface Science, 27 (1968), No. 2, 253-267. CrossRefGoogle Scholar
Leal, L. G.. Particle motions in a viscous fluid. Annu. Rev. Fluid Mech., 12 (1980), 435-476. CrossRefGoogle Scholar
Wang, S. K., Hwang, N. H. C.. On transport of suspended particulates in tube flow. Biorheology, 29 (1992), 353-377. Google ScholarPubMed
Brenner, H., Bungay, P. M.. Rigid-particle and liquid-droplet models of red cell motion in capillary tubes. Fed. Proc., 30 (1971), No. 5, 1565-1577. Google ScholarPubMed
Goldsmith, H. L., Mason, S. G.. The flow of suspensions through tubes I. Single spheres, rods and discs. Journal of Colloid Science, 17 (1962), 448-476. CrossRefGoogle Scholar
Tam, C. K. W., Hyman, W. A.. Transverse motion of an elastic sphere in a shear field. Journal of Fluid Mechanics, 59 (1973), No. part 1, 177-185. CrossRefGoogle Scholar
C. Crowe, M. Sommerfield, Y. Tsuji. Multiphase flows with drops and particles. CRC Press, 1998.
Cherukat, P., McLaughlin, J. B., Dandy, D. S.. A computational study of the inertial lift on a sphere in a linear shear fow field. International Journal of Multiphase Flow, 25 (1999), 15-33. CrossRefGoogle Scholar
Matas, J.-P., Morris, J. F., Guazzelli, E.. Inertial migration of rigid spherical particles in Poiseuille flow. Journal of Fluid Mechanics, 515 (2004), 171-195. CrossRefGoogle Scholar
Munn, L. L., Dupin, M. M.. Blood cell interactions and segregation in flow. Ann. Biomed. Eng, 36 (2008), No. 4, 534-544. CrossRefGoogle ScholarPubMed
Michaelides, E. E.. Hydrodynamic Force and Heat-Mass Transfer From Particles. Journal of Fluids Engineering, 125 (2003), 209-238. CrossRefGoogle Scholar
Rubinow, S. I., Keller, J. B.. The transverse force on a spinning sphere moving in a viscous fluid. Journal of Fluid Mechanics, 11 (1961), 447-459. CrossRefGoogle Scholar
Cox, R. G., Brenner, H.. The lateral migration of solid particles in Poiseuille flow - I theory. Chemical Engineering Science, 23 (1968), 147-173. CrossRefGoogle Scholar
Saffman, P. G.. The lift on a small sphere in a slow shear flow. Journal of Fluid Mechanics, 22 (1965), No. part 2, 385-400. CrossRefGoogle Scholar
McLaughlin, J. B.. Inertial migration of a small sphere in linear shear flows. Journal of Fluid Mechanics, 224 (1991), 261-274. CrossRefGoogle Scholar
Cherukat, P., McLaughlin, J. B., Graham, A. L.. The inertial lift on a rigid sphere translating in a linear shear flow field. International Journal of Multiphase Flow, 20 (1994), No. 2, 339-353. CrossRefGoogle Scholar
Vasseur, P. V., Cox, R. G.. The lateral migration of a spherical particle in two-dimensional shear flows. Journal of Fluid Mechanics, 78 (1976), 385-413. CrossRefGoogle Scholar
Cox, R. G., Hsu, S. K.. The lateral migration of solid particles in a laminar flow near a plane. International Journal of Multiphase Flow, 3 (1977), 201-222. CrossRefGoogle Scholar
Cherukat, P., Oliver, D. R.. The inertial lift on a rigid sphere in a linear shear flow field near a flat wall. Journal of Fluid Mechanics, 263 (1994), 1-18. CrossRefGoogle Scholar
Longest, P. W., Kleinstreuer, C.. Comparison of blood particle deposition models for non-parallel fow domains. Journal of Biomechanics, 36 (2003), 421-430. CrossRefGoogle Scholar
Longest, P. W., Kleinstreuer, C.. Numerical Simulation of Wall Shear Stress Conditions and Platelet Localization in Realistic End-to-Side Arterial Anastomoses. Journal of Biomechanical Engineering, 125 (2003), 671-681. CrossRefGoogle ScholarPubMed
Longest, P. W., Kleinstreuer, C., Buchanan, J. R.. Efficient computation of micro-particle dynamics including wall effects. Computers & Fluids, 33 (2004), 577-601. CrossRefGoogle Scholar
Goldsmith, H. L., Mason, S. G.. Axial migration of particles in Poiseuille Flow. Nature, 190 (1961), 1095-1096. CrossRefGoogle Scholar
Abkarian, M., Viallat, A.. Vesicles and red blood cells in shear flow. Soft Matter, 4 (2008), 653-657. CrossRefGoogle Scholar
Coulliette, C., Pozrikidis, C.. Motion of an array of drops through a cylindrical tube. Journal of Fluid Mechanics, 358 (1998), 1-28. CrossRefGoogle Scholar
Mortazavi, S., Tryggvason, G.. A numerical study of the motion of drops in Poiseuille flow. Part 1.Lateral migration of one drop. Journal of Fluid Mechanics, 411 (2000), 325-350. CrossRefGoogle Scholar
Pozrikidis, C.. Numerical Simulation of Cell Motion in Tube Flow. Ann. Biomed. Eng, 33 (2005), No. 2, 165-178. CrossRefGoogle ScholarPubMed
B. Kaoui, G. Biros, C. Masbah. Why Do Red Blood Cells Have Asymmetric Shapes Even in a Symmetric Flow? Physical Review Letters, 103 (2009), No. 18, 188101(1)-188101(4).
Chaffey, C. E., Brenner, H., Mason, S. G.. Particle motions in sheared suspensions XVIII. Wall Migration (Theoretical). Rheologica Acta, 4 (1965), No. 1, 64-72. CrossRefGoogle Scholar
Chaffey, C. E., Brenner, H., Mason, S. G.. Correction of the paper Particle motions in sheared suspensions XVIII. Wall Migration (Theoretical). Rheologica Acta, 6 (1967), No. 1, 100. CrossRefGoogle Scholar
Wohl, P. R., Rubinow, S. I.. The transverse force on a drop in an unbounded parabolic flow. Journal of Fluid Mechanics, 62 (1974), No. part 1, 185-207. CrossRefGoogle Scholar
Chan, P. C. H., Leal, L. G.. The motion of a deformable drop in a second-order fluid. Journal of Fluid Mechanics, 92 (1979), No. part 1, 131-170. CrossRefGoogle Scholar
Uijttewaal, W. S. J., Nijhof, E.-J., Heethaar, R. M.. Droplet migration, deformation, and orientation in the presence of a plane wall: A numerical study compared with analytical theories. Phys. Fluids A, 5 (1993), No. 4, 819-825. CrossRefGoogle Scholar
Hudson, S. D.. Wall migration and shear-induced diffusion of fluid droplets in emulsions. Physics of Fluids, 15 (2003), No. 5, 1106-1113. CrossRefGoogle Scholar
King, M. R., Leighton, D. T. Jr. Measurement of shear-induced dispersion in a dilute emulsion. Physics of Fluids, 13 (2001), No. 2, 397-406. CrossRefGoogle Scholar
M. Scott. 2005. The modeling of Blood Rheology in small vessels. University of Waterloo, Waterloo, Ontario, Canada.
Olla, P.. The lift on a tank-treading ellipsoidal cell in a shear flow. Journal de Physique II, 7 (1997), No. 10, 1533-1540. CrossRefGoogle Scholar
Faivre, M., Abkarian, M., Bickraj, K., Stone, H. A.. Geometrical focusing of cells in a microfluidic device: An approach to separate blood plasma. Biorheology, 43 (2006), 147-159. Google Scholar
Blackshear, P. L. Jr., Forstrom, R. J., Dorman, F. D., Voss, G. O.. Effect of flow on cells near walls. Fed. Proc., 30 (1971), No. 5, 1600-1609. Google ScholarPubMed
Eggleton, C. D., Popel, A. S.. Large deformation of red blood cell ghosts in a simple shear flow. Physics of Fluids, 10 (1998), No. 8, 1834-1845. CrossRefGoogle Scholar
Korin, N., Bransky, A., Dinnar, U.. Theoretical model and experimental study of red blood cell (RBC) deformation in microchannels. Journal of Biomechanics, 40 (2007), 2088-2095. CrossRefGoogle ScholarPubMed
Nott, P. R., Brady, J. F.. Pressure-driven flow of suspensions: simulation and theory. Journal of Fluid Mechanics, 275 (1994), 157-199. CrossRefGoogle Scholar
Morris, J. F., Brady, J. F.. Pressure-driven flow in a suspension: buoyancy effects. International Journal of Multiphase Flow, 24 (1998), No. 1, 105-130. CrossRefGoogle Scholar
Tsubota, K., Wada, S., Kamada, H., Kitagawa, Y., Lima, R., Yamaguchi, T.. A Particle Method for Blood Flow Simulation -Application to Flowing Red Blood Cells and Platelets. Journal of the Earth Simulator, 5 (2006), 2-7. Google Scholar
Chen, S., Doolen, G. D.. Lattice Boltzmann Method for Fluid Flows. Annu. Rev. Fluid Mech., 30 (1998), 329-364. CrossRefGoogle Scholar
Dupin, M. M., Halliday, I., Care, C. M., Alboul, L., Munn, L. L.. Modeling the flow of dense suspensions of deformable particles in three dimensions. Physical Review E., 066707 (2007), 1-17. Google Scholar
L. M. Crowl, A. L. Fogelson. Computational model of whole blood exhibiting lateral platelet motion induced by red blood cells. Commun. Numer. Meth. Engng, (2009).
Sun, C., Migliorini, C., Munn, L. L.. Red Blood Cells Initiate Leukocyte Rolling in Postcapillary Expansions: A Lattice Boltzmann Analysis. Biophysical Journal, 85 (2003), 208-222. CrossRefGoogle ScholarPubMed
Bagchi, P.. Mesoscale Simulation of Blood Flow in Small Vessels. Biophysical Journal, 92 (2007), 1858-1877. CrossRefGoogle ScholarPubMed
Bagchi, P., Johnson, P. C., Popel, A. S.. Computational Fluid Dynamic Simulation of Aggregation of Deformable Cells in a Shear Flow. Transactions of the ASME, 127 (2005), 1070-1080. Google Scholar
Zhang, J., Johnson, P. C., Popel, A. S.. Red blood cell aggregation and dissociation in shear flows simulated by lattice Boltzmann method. Journal of Biomechanics, 41 (2008), 47-55. CrossRefGoogle ScholarPubMed
Zhang, J., Johnson, P. C., Popel, A. S.. Effects of erythrocyte deformability and aggregation on the cell free layer and apparent viscosity of microscopic blood flows. Microvasc. Res., 77 (2009), 265-272. CrossRefGoogle ScholarPubMed
Svetina, S., Ziherl, P.. Morphology of small aggregates of red blood cells. Bioelectrochemistry, 73 (2008), No. 2, 84-91. CrossRefGoogle ScholarPubMed
Fogelson, A. L.. A Mathematical Model and Numerical Method for Studying Platelet Adhesion and Aggregation during Blood Clotting. Journal of Computational Physics, 56 (1984), 111-134. CrossRefGoogle Scholar
Pivkin, I. V., Richardson, P. D., Karniadakis, G.. Blood flow velocity effects and role of activation delay time on growth and form of platelet thrombi. PNAS, 103 (2006), No. 46, 17164-17169. CrossRefGoogle ScholarPubMed
Miyazaki, H., Yamaguchi, T.. Formation and destruction of primary thrombi under the influence of blood flow and von Willebrand factor analyzed by a discrete element method. Biorheology, 40 (2003), 265-272. Google ScholarPubMed
K. Yano, K. Tsubota, S. Wada, T. Yamaguchi. 2003. Computational mechanical simulation of the aggregation and adhesion of platelets in the blood flow. In Summer Bioengineering Conference. Sonesta Beach Resort in Key Biscayne, Florida. 0613-0614.
Filipovic, N., Ravnic, D., Kojic, M., Mentzer, S. J., Haber, S., Tsuda, A.. Interactions of blood cell constituents: Experimental investigation and computational modeling by discrete particle dynamics algorithm. Microvasc. Res., 75 (2008), 279-284. CrossRefGoogle ScholarPubMed
Mori, D., Yano, K., Tsubota, K., Ishikawa, T., Wada, S., Yamaguchi, T.. Simulation of platelet adhesion and aggregation regulated by fibrinogen and von Willebrand factor. Thromb. Haemost., 99 (2008), No. 1, 108-115. Google ScholarPubMed
Almomani, T., Udaykumar, H. S., Marshall, J. S., Chandran, K. B.. Micro-scale dynamic simulation of erythrocyte-platelet interaction in blood flow. Ann. Biomed. Eng, 36 (2008), No. 6, 905-920. CrossRefGoogle ScholarPubMed
Miller, R. M., Morris, J. F.. Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions. Journal of Non-Newtonian Fluid Mechanics, 135 (2006), 149-165. CrossRefGoogle Scholar
L. G. Loitzanskii. Mechanics of Fluid and Gas. Nauka, Moscow, 1978.
A. Sequeira, J. Janela. An overview of some mathematical models of blood rheology. In A Portrait of State-of-the-Art Research at the Technical University of Lisbon. M. S. Pereira, editor. Springer, 2007. pp. 65-87.
A. M. Robertson, A. Sequeira, M. V. Kameneva. Hemorheology. In Hemodynamical Flows: Modeling, Analysis and Simulation (Oberwolfach Seminars). Birkhauser Basel, 2008. pp. 63-120.
G. R. Cokelet. The Rheology and Tube Flow of Blood. In Handbook of Bioengineering. R. Skalak, S. Chen, editors. McGraw-Hill, New York, 1987.
Wolters, B. J. B. M., Rutten, M. C. M., Schurink, G. W. H., Kose, U., Hart, J. d., v. d. Vosse, F. N.. A patient-specific computational model of fluid-structure interaction in abdominal aortic aneurysms. Medical Engineering & Physics, 27 (2005), 871-883. CrossRefGoogle ScholarPubMed
Jung, J., Hassenein, A., Lyczkowski, R. W.. Hemodynamic Computation Using Multiphase Flow Dynamics in a Right Coronary Artery. Ann. Biomed. Eng, 34 (2006), No. 3, 393-407. CrossRefGoogle Scholar
Jung, J., Lyczkowski, R. W., Panchal, C. B., Hassenein, A.. Multiphase hemodynamic simulation of pulsatile flow in a coronary artery. Journal of Biomechanics, 39 (2006), 2064-2073. CrossRefGoogle Scholar
Jung, J., Hassenein, A.. Three-phase CFD analytical modeling of blood flow. Medical Engineering & Physics, 30 (2008), 91-103. CrossRefGoogle ScholarPubMed
Quemada, D., Berli, C.. Energy of interaction in colloids and its implications in rheological modeling. Advances in Colloid and Interface Science, 98 (2002), 51-85. CrossRefGoogle ScholarPubMed
Marcinkowska-Gapinska, A., Gapinski, J., Elikowski, W., Jaroszyk, F., Kubisz, L.. Comparison of three rheological models of shear flow behavior studied on blood samples from post-infarction patients. Medical and Biological Engineering and Computing, 45 (2007), No. 9, 837-844. CrossRefGoogle ScholarPubMed
Das, B., Johnson, P. C., Popel, A. S.. Effect of nonaxisimmetric hematoctit distribution on non-newtonian blood flow in small tubes. Biorheology, 35 (1998), No. 1, 69-87. CrossRefGoogle ScholarPubMed
Buchanan, J. R. Jr., Kleinstreuer, C., Comer, J. K.. Rheological effects on pulsatile hemodynamics in a stenosed tube. Computers & Fluids, 29 (2000), 695-724. CrossRefGoogle Scholar
Das, B., Enden, G., Popel, A. S.. Stratified multiphase model for blood flow in a venular bifurcation. Annals of Biomedical Engineering, 25 (1997), 135-153. CrossRefGoogle Scholar
Popel, A. S., Enden, G.. An analytical solution for steady flow of a Quemada fluid in a circular tube. Rheologica Acta, 32 (1993), 422-426. CrossRefGoogle Scholar
Berli, C. L., Quemada, D.. Aggregation behavior of red blood cells in shear flow. A theoretical interpretation of simultaneous rheo-optical and viscometric measurements. Biorheology, 38 (2001), No. 1, 27-38. Google ScholarPubMed
Quemada, D.. Rheological modelling of complex fluids. I. The concept of effective volume fraction revisited. The European Physical J. AP, 1 (1998), 119-127. CrossRefGoogle Scholar
Quemada, D., Berli, C.. Energy of interaction in colloids and its implications in rheological modeling. Adv. Colloid Interface Sci., 98 (2002), No. 1, 51-85. CrossRefGoogle ScholarPubMed
Neofytou, P.. Comparison of blood rheological models for physiological flow simulation. Biorheology, 41 (2004), No. 6, 693-714. Google ScholarPubMed
Cokelet, G. R., Goldsmith, H. L.. Decreased hydrodynamic resistance in the two-phase flow of blood through small vertical tubes at low flow rates. Circ. Res., 68 (1991), No. 1, 1-17. CrossRefGoogle ScholarPubMed
Buchanan, J. R. Jr., Kleinstreuer, C.. Simulation of particle-hemodynamics in a partially occluded artery segment with implications to the initiation of microemboli and secondary stenoses. J. Biomech. Eng, 120 (1998), No. 4, 446-454. CrossRefGoogle Scholar
S. A. Regirer. Lections on Biological Mechanics [in russian]. Izdatelstvo MGU, Moscow, 1980.
Sharan, M., Popel, A. S.. A two-phase model for flow of blood in narrow tubes with increased effective viscosity near the wall. Biorheology, 38 (2001), 415-428. Google Scholar
Ware, J. H., Sorrell, F. Y., Felder, R. M.. A model of steady blood flow. Biorheology, 11 (1974), 97-109. Google ScholarPubMed
Das, B., Johnson, P. C., Popel, A. S.. Computational fluid dynamic studies of leukocyte adhesion effects. Biorheology, 37 (2000), 239-258. Google ScholarPubMed
Perkkio, J., Keskinen, R.. On the effect of the concentration profile of red cells on blood flow in the artery with stenosis. Bull. Math. Biol., 45 (1983), No. 2, 259-267. CrossRefGoogle ScholarPubMed
D. Lerche. Modelling hemodynamics in small tubes (hollow fibers) considering . In Biomechanical Transport Processes. F. e. al. Mosora, editor. Plenum, New York, 1990. pp. 243-250.
Carr, R. T., Lacoin, M.. Nonlinear Dynamics of Microvascular Blood Flow. Annals of Biomedical Engineering, 28 (2000), 641-652. CrossRefGoogle ScholarPubMed
Brunn, P.. The general solution to the equations of creeping motion of a micropolar fluid and its application. International Journal of Engineering Science, 20 (1982), 575-585. CrossRefGoogle Scholar
Stokes, V. K.. Couple stress in fluids. The Physics of Fluids, 9 (1966), No. 9, 1709-1715. CrossRefGoogle Scholar
Eringen, A. C.. Theory of Micropolar Fluids. Journal of Mathematics and Mechanics, 16 (1966), No. 1, 1-18. Google Scholar
Askar, A., Cakmak, A. S.. A structural model of a micropolar continuum. International Journal of Engineering Science, 6 (1968), 583-589. CrossRefGoogle Scholar
Ariman, T.. Microcontinuum fluid mechanics - a review. International Journal of Engineering Science, 11 (1973), 905-930. CrossRefGoogle Scholar
Ariman, T., Turk, M. A., Sylvester, N. D.. Application of microcontinuum fluid mechanics. International Journal of Engineering Science, 12 (1974), 273-293. CrossRefGoogle Scholar
Kline, K. A.. Predictions from Polar Fluid Theory Which Are Independent of Spin Boundary Condition. Transactions of the society of rheology, 19 (1975), No. 1, 139-145. CrossRefGoogle Scholar
Cowin, S. C.. A Note on the Predictions from Polar Fluid Theory Which Are Independent of the Spin Boundary Condition. Transactions of the society of rheology, 20 (1976), No. 2, 195-202. CrossRefGoogle Scholar
Hogan, H. A., Henriksen, M.. An evaluation of a micropolar model for blood flow through an idealized stenosis. Journal of Biomechanics, 22 (1989), No. 3, 211-218. CrossRefGoogle ScholarPubMed
Pralhad, R. N., Schultz, D. H.. Modeling of arterial stenosis and its applications to blood diseases. Mathematical Biosciences, 190 (2004), 203-220. CrossRefGoogle ScholarPubMed
Akay, G., Kaye, A.. Numerical solution of time dependent stratified two-phase flow of micropolar fluids and its application to flow of blood through fine capillaries. International Journal of Engineering Science, 23 (1985), No. 3, 265-276. CrossRefGoogle Scholar
Ikbal, Md. A., Chakravarty, S., Mandal, P. K.. Two-layered micropolar fluid flow through stenosed artery: Effect of peripheral layer thickness. Computers and Mathematics with Applications, 58 (2009), 1328-1339. CrossRefGoogle Scholar
D. Biswas. Blood Flow Models: A Comparative Study. Mittal Publications , 2002.
Kang, C. K., Eringen, A. C.. The effect of microstructure on the rheological properties of blood. Bull. Math. Biol., 38 (1976), 135-159. CrossRefGoogle Scholar
Popel, A. S., Regirer, S. A.. Ob osnovnih uravneniyah hydrodinamiki krovi. Nauchnie trudi instituta mechaniki MGU, 1 (1970), 3-20. Google Scholar
Popel, A. S., Regirer, S. A., Usick, P. I.. A Continuum Model of Blood Flow. Biorheology, 11 (1974), 427-437. Google ScholarPubMed
Popel, A. S.. O hydrodynamike suspensii. Mechanika zjidkosti i gaza, 4 (1969), 24. Google Scholar
A. C. Eringen. Microcontinuum Field Theories II: Fluent media. Springer-Verlag, 2001.
Levtov, V. A., Regirer, S. A., Shadrina, N. Kh.. Aggregation and diffusion of Erythrocites. Sovremennie problemi biomekhaniki, 9 (1994), 5-41. Google Scholar
Kolpashchikov, V. L., Migun, N. P., Prokhorenko, P. P.. Experimental determination of material micropolar fluid constants. International Journal of Engineering Science, 21 (1983), No. 4, 405-411. CrossRefGoogle Scholar
Kirwan, A. D. J.. Boundary conditions for micropolar fluids. International Journal of Engineering Science, 24 (1986), No. 7, 1237-1242. CrossRefGoogle Scholar
Goldsmith, H. L., Marlow, J. C.. Flow Behavior of Erythrocytes II. Particle Motions in Concentrated Suspensions of Ghost Cells. Journal of Colloid and lnterface Science, 71 (1979), No. 2, 383-407. CrossRefGoogle Scholar
V. A. Levtov, S. A. Regirer, N. Kh. Shadrina. Rheology of Blood. Medicina, Moscow, 1982.
Ahmadi, G.. A continuum theory of blood flow. Scientia Sinica, 24 (1981), No. 10, 1465-1473. Google ScholarPubMed
Ahmadi, G.. A Continuum Theory for Two Phase Media. Acta Mechanica, 44 (1982), 299-317. CrossRefGoogle Scholar
Jung, J., Gidaspow, D., Gamwo, I. K.. Bubble Computation, Granular Temperatures, and Reynolds Stresses. Chem. Eng. Comm., 193 (2006), 946-975. CrossRefGoogle Scholar
Eckstein, E. C., Bailey, D. G., Shapiro, A. H.. Self-diffusion of particles in shear flow of a suspension. Journal of Fluid Mechanics, 79 (1977), No. part 1, 191-208. CrossRefGoogle Scholar
Leighton, D., Acrivos, A.. The shear-induced migration of particles in concentrated suspensions. Journal of Fluid Mechanics, 181 (1987), 415-439. CrossRefGoogle Scholar
Aref, H., Jones, S. W.. Enhanced separation of diffusing particles by chaotic advection. Phys. Fluids A, 1 (1989), No. 3, 470-474. CrossRefGoogle Scholar
Koh, C. J., Hookham, P., Leal, L. G.. An experimental investigation of concentrated suspension flows in a rectangular channel. Journal of Fluid Mechanics, 266 (1994), 1-32. CrossRefGoogle Scholar
Lyon, M. K., Leal, L. G.. An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 1. Monodisperse systems. Journal of Fluid Mechanics, 363 (1998), 25-56. CrossRefGoogle Scholar
Phillips, R. J., Armstrong, R. C., Brown, R. A.. A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys. Fluids A, 4 (1992), No. 1, 30-40. CrossRefGoogle Scholar
Butler, J. E., Bonnecaze, R. T.. Imaging of particle shear migration with electrical impedance tomography. Physics of Fluids, 11 (1999), No. 8, 1982-1994. CrossRefGoogle Scholar
Hofer, M., Perctold, K.. Computer simulation of concentrated fluid-perticle suspension flows in axisimmetric geometries. Biorheology, 54 (1997), No. 4/5, 261-279. CrossRefGoogle Scholar
Lyon, M. K., Leal, L. G.. An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 2. Bidisperse systems. Journal of Fluid Mechanics, 363 (1998), 57-77. CrossRefGoogle Scholar
Morris, J. F., Boulay, F.. Curvilinear flows of noncolloidal suspensions: The role of normal stresses. Journal of Rheology, 43 (1999), No. 5, 1213-1236. CrossRefGoogle Scholar
Grabowski, E. F., L.I., Friedman, Leonard, E. F.. Effects of Shear Rate on the Diffusion and Adhesion of Blood Platelets to a Foreign Surface. Ind. Eng. Chem. Fundamen., 11 (1972), No. 2, 224-232. CrossRefGoogle Scholar
Strong, A. B., Stubley, G. D., Chang, G., Absolom, D. R.. Theoretical and experimental analysis of cellular adhesion to polymer surfaces. J. Biomed. Mater. Res., 21 (1987), No. 8, 1039-1055. CrossRefGoogle ScholarPubMed
Stubley, G. D., Strong, A. B., Hale, W. E., Absolom, D. R.. A review of mathematical models for the prediction of blood cell adhesion. PCH PhysicoChem. Hydrodynics, 8 (1987), No. 2, 221-235. Google Scholar
Wootton, D. M., Markou, C. P., Hanson, S. R., Ku, D. N.. A mechanistic model of acute platelet accumulation in thrombogenic stenoses. Ann. Biomed. Eng, 29 (2001), No. 4, 321-329. CrossRefGoogle ScholarPubMed
Sorensen, E. N., Burgreen, G. W., Wagner, W. R., Antaki, J. F.. Computational simulation of platelet deposition and activation: II. Results for Poiseuille flow over collagen. Ann. Biomed. Eng, 27 (1999), No. 4, 449-458. CrossRefGoogle ScholarPubMed
Sorensen, E. N., Burgreen, G. W., Wagner, W. R., Antaki, J. F.. Computational simulation of platelet deposition and activation: I. Model development and properties. Ann. Biomed. Eng, 27 (1999), No. 4, 436-448. CrossRefGoogle ScholarPubMed
David, T., Walker, P. G.. Activation and extinction models for platelet adhesion. Biorheology, 39 (2002), 293-298. Google ScholarPubMed
Anand, M., Rajagopal, K., Rajagopal, K. R.. A Model Incorporating some of the Mechanical and Biochemical Factors Underlying Clot Formation and Dissolution in Flowing Blood. Computational and Mathematical Methods in Medicine, 5 (2003), No. 3&4, 183-218. Google Scholar
Fogelson, A. L., Guy, R. D.. Platelet-wall interactions in continuum models of platelet thrombosis: formulation and numerical solution. Math. Med. Biol., 21 (2004), No. 4, 293-334. CrossRefGoogle ScholarPubMed
Wang, N.-T., Fogelson, A. L.. Computational methods for continuum models of platelet aggregation. Journal of Computational Physics, 151 (1999), 649-675. CrossRefGoogle Scholar
Fogelson, A. L.. Continuum models of platelet aggregation: formulation and mechanical properties. SIAM J. Appl. Math., 52 (1992), No. 4, 1089-1110. CrossRefGoogle Scholar
Jordan, A., David, T., Homer-Vanniasinkam, S., Graham, A., Walker, P.. The effects of margination and red cell augmented platelet diffusivity on platelet adhesion in complex flow. Biorheology, 41 (2004), 641-653. Google ScholarPubMed
Eckstein, E. C., Bilsker, D. L., Waters, C. M., Kippenhan, J. S., Tilles, A. W.. Transport of platelets in flowing blood. Ann. N. Y. Acad. Sci., 516 (1987), 442-452. CrossRefGoogle ScholarPubMed
Eckstein, E. C., Belgacem, F.. Model of platelet transport in flowing blood with drift and diffusion terms. Biophys. J., 60 (1991), No. 1, 53-69. CrossRefGoogle ScholarPubMed
Yeh, C., Calvez, A. C., Eckstein, E. C.. An estimated shape function for drift in a platelet-transport model. Biophys. J., 67 (1994), No. 3, 1252-1259. CrossRefGoogle Scholar
Zydney, A. L., Colton, C. K.. Augmented solute transport in the shear flow of a concentrated suspension. PCH PhysicoChem. Hydrodynamics, 10 (1988), No. 1, 77-96. Google Scholar
S. N. Antontsev, A. V. Kazhikhov, V. N. Monakov. Boundary Value Problems in the Mechanics of Heterogeneous Fluids, Novosibirsk, Nauka, 1983.
J. Málek, J. Nečas, M. Pokyta, M. Ruzička. Weak and Measure-valued Solutions to Evolutionary DPEs. Chapman and Hall, London, 1996.
G. P. Galdi, R. Rannacher, A. H. Robertson, S. Turek. Hemodynamical Flows Modeling: Analysis and Simulation. Oberwolfach Seminar, Birkhauser, Basel, Boston, Berlin, 2008.