BACKGROUND: In microcirculation, the non-Newtonian behavior of blood and the complexity of the microvessel network are responsible for the high flow resistance and the large reduction of the blood pressure. Red blood cell aggregation along with inward radial migration are two significant mechanisms determining the former. Yet, their impact on hemodynamics in non-straight vessels is not well understood.
OBJECTIVE: In this study, the steady state blood flow in stenotic rigid vessels is examined, employing a sophisticated non-homogeneous constitutive law. The effect of red blood cells migration on the hydrodynamics is quantified and the constitutive model’s accuracy is evaluated.
METHODS: A numerical algorithm based on the two-dimensional mixed finite element method and the EVSS/SUPG technique for a stable discretization of the mass and momentum conservation equations in addition to the constitutive model is employed.
RESULTS: The numerical simulations show that a cell-depleted layer develops along the vessel wall with an almost constant thickness for slow flow conditions. This causes the reduction of the drag force and the increase of the pressure gradient as the constriction ratio decreases.
CONCLUSIONS: Viscoelastic effects in blood flow were found to be responsible for steeper decreases of tube and discharge hematocrits as decreasing function of constriction ratio.
PopelASJohnsonPC. Microcirculation and hemorheology. Annu Rev Fluid Mech. 2005;37:43–69.
2.
LipowskyHH. Microvascular rheology and hemodynamics. Microcirculation. 2005;12:5–15.
3.
FreundJB. Numerical simulation of flowing blood cells. Annu Rev Fluid Mech. 2014;46:67–95.
4.
SharanMPopelAS. A two-phase model for flow of blood in narrow tubes with increased effective viscosity near the wall. Biorheology. 2001;38:415–428.
5.
BaskurtOK. Handbook of hemorheology and hemodynamics. Amsterdam, The Netherlands: IOS Press; 2007.
6.
ChienS. Biophysical behavior of red blood cells in suspension. In: SurgenorDM, editor. The red blood cell, Vol. II. New York: Academic Press; 1975. p. 1031–1133.
7.
ChienSUsamiSTaylorHMLundbergJLGregersenMI. Effects of hematocrit and plasma proteins on human blood rheology at low shear rates. J Appl Physiol. 1966;21:81–87.
8.
MeiselmanH. Red blood cell aggregation: 45 years being curious. Biorheology. 2009;46:1–19.
9.
ThurstonGB. Frequency and shear rate dependence of viscoelasticity of human blood. Biorheology. 1973;10:375–381.
10.
ThurstonGB. Elastic effects in pulsatile blood flow. Microvasc Res. 1975;9:145–157.
11.
EvansEAHochmuthRM. Membrane viscoelasticity. Biophysical J. 1976;16(1):1–11.
12.
ApostolidisABerisA. Modeling of the blood rheology in steady-state shear flows. J Rheol. 2014;58:607–633. doi:10.1122/1.4866296.
13.
QuemadaD. A non-linear Maxwell model of biofluids: Application to normal human blood. Biorheology. 1993;30:253–265.
14.
WilliamsMCRosenblattJSSoaneDS. Theory of blood rheology based on a statistical mechanics treatment of rouleaux, and comparisons with data. Int J Polym Mater. 1993;21:57–63.
15.
VlastosGLercheDKochB. The superposition of steady on oscillatory shear and its effect on the viscoelasticity of human blood and a blood-like model fluid. Biorheology. 1997;34:19–36.
16.
DimakopoulosYBogaerdsAAndersonPHulsenMBaaijensFPT. Direct numerical simulation of a 2D idealized aortic heart valve at physiological flow rates. Comp Meth Biomech and Biomed Engrg. 2012;15(11):1157–1179.
17.
AnandMRajagopalKR. A shear-thinning viscoelastic fluid model for describing the flow of blood. Int J of Cardiovascular Medicine and Science. 2004;4(2):59–68.
18.
RajagopalKRSrinivasaAR. A thermodynamic frame work for rate type fluid models. J Non-Newtonian Fluid Mech. 2000;80:207–227.
19.
BodnarTSequeiraA. Numerical study of the significance of the non-Newtonian nature of blood in steady flow through a stenosed vessel. In: RannacherRSequeiraA, editors. Advances in mathematical fluid mechanics. Springer; 2010. p. 83–104.
20.
ChakrabortyDBajajbMYeobLFriendbJPasqualiMPrakashJR. Viscoelastic flow in a two-dimensional collapsible channel. J Non-Newtonian Fluid Mech. 2010;165:1204–1218.
21.
Moyers-GonzalezMAOwensRGFangJ. A non-homogeneous constitutive model for human blood. Part I. Model derivation and steady flow. J Fluid Mech. 2008;617:327–354.
22.
OwensRG. A new microstructure-based constitutive model for human blood. J Non-Newtonian Fluid Mech. 2006;140:57–70.
23.
BureauMHealyJCBourgoinDJolyM. Rheological hysteresis of blood at low shear rate. Biorheology. 1980;17:191–203.
24.
FangJOwensRG. Numerical simulations of pulsatile blood flow using a new constitutive model. Biorheology. 2006;43:637–660.
25.
ThurstonGB. Rheological parameters for the viscosity, viscoelasticity and thixotropy of blood. Biorheology. 1979;16:149–162.
26.
IolovAKaneASBourgaultYOwensRGFortinA. A FEM for a microstructure-based model of blood. Intern J Numerical Meth in Biomedical Eng. 2011;27(9):1321–1349.
27.
FahraeusR. The suspension stability of the blood. Physiol Rev. 1929;9:241–279.
28.
FahraeusRLindqvistT. The viscosity of blood in narrow capillary tubes. Am J Physiol. 1931;96:562–568.
29.
BerisANMavrantzasVG. On the compatibility between various macroscopic formalisms for the concentration and flow of dilute polymer solutions. J Rheol. 1994;38:1235–1250.
30.
BhaveAVArmstrongRCBrownRA. Kinetic theory and rheology of dilute, nonhomogeneous polymer solutions. J Chem Phys. 1991;95:2988–3000.
31.
CookLPRossiLF. Slippage and migration in models of dilute wormlike micellar solutions and polymeric fluids. J Non-Newtonian Fluid Mech. 2004;116(2,3):347–369.
32.
RossiLFMcKinleyGCookLP. Slippage and migration in Taylor–Couette flow of a model for dilute wormlike micellar solutions. J Non-Newtonian Fluid Mech. 2006;136:79–92.
33.
PriesARNeuhausDGaehtgensP. Blood viscosity in tube flow: Dependence on diameter and hematocrit. Am J Physiol Heart Circ Physiol. 1992;263:H1770–H1778.
34.
SuteraSPSeshadriVCrocePAHochmuthRM. Capillary blood flow: II. Deformable model cells in tube flow. Microvasc Res. 1970;2:420–433.
35.
CaroCGPedleyTJSchroterRC. Atheroma and arterial wall shear: Observation, correlation and proposal of shear dependent mass transfer mechanism for atherogenesis. Proc Roy Soc London B. 1971;177:109–159.
36.
ZydneyALColtonCK. Augmented solute transport in the shear-flow of a concentrated suspension. PCH PhysicoChem Hydrodynamics. 1988;10:77–96.
37.
BishopJJPopelASIntagliettaMJohnsonPC. Effect of aggregation and shear rate on the dispersion of red blood cells flowing in venules. Am J Physiol Heart Circ Physiol. 2002;283(5):H1985–H1996.
38.
AcrivosA. Shear-induced particle diffusion in concentrated suspensions of noncolloidal particles. J Rheol. 1995;39:813–826.
39.
PranayPHenriquez-RiveraRGGrahamMD. Depletion layer formation in suspensions of elastic capsules in Newtonian and viscoelastic fluids. Phys Fluids. 2012;2(4):061902.
40.
MaHGrahamMD. Theory of shear-induced migration in dilute polymer solutions near solid boundaries. Phys Fluids. 2005;17:083103.
41.
ChienSUsamiSDellenbackRJGregersenMI. Shear-dependent deformation of erythrocytes in rheology of human blood. Am J Physiol. 1970;219:136–142.
42.
LacEPelekasisNBarthes-BieselDTsamopoulosJ. Spherical capsules in three-dimensional unbounded Stokes flows: Effect of the membrane constitutive law and onset of buckling. J Fluid Mech. 2004;516:303–334.
43.
KramersHA. Het gedrag van macromoleculen in een stroomende vloeistof. Physica. 1944;11:1–19.
44.
TsoukaSDimakopoulosYMavrantzasVTsamopoulosJ. Stress-gradient induced migration of polymers in corrugated channels. J Rheol. 2014;58(4):911–947.
45.
DimakopoulosYTsamopoulosJ. A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations. J Computational Phys. 2003;192:494–522.
46.
RajagopalanDArmstrongRCBrownRA. Finite element methods for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity. J Non-Newtonian Fluid Mech. 1990;36:159–192.
47.
BrownRASzadyMJNortheyPJArmstrongRC. On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations. Theoretical Comp Fluid Dynamics. 1993;5:77–106.
48.
BrooksANHughesTJR. Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comp Methods Applied Mech Eng. 1982;32:199–259.
49.
AmestoyPRDuffISKosterJL’ExcellentJ-Y. A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J Matrix Anal Applic. 2001;23(1):15–41.
50.
AmestoyPRGuermoucheAL’ExcellentJ-YPraletS. Hybrid scheduling for the parallel solution of linear systems. Paral Comp. 2006;32(2):136–156.
51.
Moyers-GonzalezMAOwensRGFangJ. A non-homogeneous constitutive model for human blood. Part III. Oscillatory flow. J Non-Newtonian Fluid Mech. 2008;155:161–173.
52.
MatsumotoTKajiyaF. Coronary microcirculation: Physiology and mechanics. Fluid Dynamics Research. 2005;37:60–81.
53.
DamianoERLongDSSmithML. Estimation of viscosity profiles using velocimetry data from parallel flows of linearly viscous fluids: Application to microvascular haemodynamics. J Fluid Mech. 2004;512:1–19.
54.
LeiHFedosovDACaswellBKarniadakisGE. Blood flow in small tubes: Quantifying the transition to the non-continuum regime. J Fluid Mech. 2013;722:214–239.
55.
LipowskyHHUsamiSChienS. In vivo measurements of “apparent viscosity” and microvessel hematocrit in the mesentery of the cat. Microvasc Res. 1980;19:297–319.
56.
MurataTSecombTW. Effects of shear rate on rouleaux formation in simple shear flow. Biorheology. 1988;25:113–122.
57.
ShigaTImaizumiKHaradaNSekiyaM. Kinetics of rouleaux formation using TV image analyzer. I. Human erythrocytes. Am J Physiol. 1983;245:H252–H258.
