Abstract
The modeling of blood flows confined in micro-channels or micro-capillary beds depends on the interactions between the cell-phase, plasma and the complex geometry of the network. In the case of capillaries or channels having a high aspect ratio (their longitudinal size is much larger than their transverse one), this modeling is much simplified from the use of a continuous description of fluid viscosity as previously proposed in the literature. Phase separation or plasma skimming effect is a supplementary mechanism responsible for the relative distribution of the red blood cell’s volume density in each branch of a given bifurcation. Different models have already been proposed to connect this effect to the various hydrodynamics and geometrical parameters at each bifurcation. We discuss the advantages and drawbacks of these models and compare them to an alternative approach for modeling phase distribution in complex channels networks. The main novelty of this new formulation is to show that albeit all the previous approaches seek for a local origin of the phase segregation phenomenon, it can arise from a global non-local and nonlinear structuration of the flow inside the network. This new approach describes how elementary conservation laws are sufficient principles (rather than the complex parametric models previously proposed) to provide non local phase separation. Spatial variations of the hematocrit field thus result from the topological complexity of the network as well as nonlinearities arising from solving a new free boundary problem associated with the flux and mass conservation. This network model approach could apply to model blood flow distribution either on artificial micro-models, micro-fluidic networks, or realistic reconstruction of biological micro-vascular networks.
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Abbreviations
- μ 0 :
-
Plasma dynamic viscosity
- u :
-
Velocity field
- p :
-
Lubrication pressure field
- ρ 0 :
-
Plasma density
- ν 0 :
-
Plasma kinematic viscosity
- Δc :
-
Radial contribution of the Laplacian in cylindrical coordinates
- W :
-
Womersley dimensionless number
- T :
-
Time-scale
- R :
-
Tube radius
- θ :
-
Local relative volume of red blood cell
- Q :
-
Longitudinal flow rate
- H :
-
Discharge hematocrit
- D :
-
Tube diameter
- μ a :
-
Apparent viscosity of blood
- μ :
-
Relative apparent viscosity of blood
- H t :
-
Tube hematocrit
- L :
-
Tube length
- J :
-
Junction of three links i
- i :
-
Link index
- C :
-
Hydraulic conductance
- μ c :
-
Core apparent viscosity for a large tube in microns (Kiani and Hudetz 1991)
- δ :
-
Width of marginal layer in microns (Kiani and Hudetz 1991)
- d m :
-
Effective diameter of one red blood cell in a small tube (Kiani and Hudetz 1991)
- μ 0.45 :
-
Relative apparent viscosity for H = 0.45 (Pries et al. 1990)
- c :
-
Parameter of Pries et al.’s (1990) model
- n i :
-
Number of input nodes
- n o :
-
Number of output nodes
- \({n_{\rm int}^{\rm I}}\) :
-
Number of internal inflow nodes
- \({n_{\rm int}^{\rm O}}\) :
-
Number of internal outflow nodes
- \({n_{\rm int}^{\rm U}}\) :
-
Number of internal undetermined nodes
- f :
-
Subscript for the incoming branch of an inflow junction
- α :
-
Subscript for an outgoing branch of an inflow junction
- β :
-
Subscript for an outgoing branch of an inflow junction
- κ :
-
Generic subscript for α or β
- γ κ :
-
Fractional blood flow entering κ-branch at the inflow junction
- θ κ :
-
Fractional discharge hematocrit entering κ-branch at the inflow junction
- m :
-
Empiric parameter for Dellimore et al.’s (1983) phase separation model
- a :
-
Plasma skimming parameter (Fenton et al. 1985a,b)
- d c :
-
Size of red blood cell Fenton et al.’s model (1985a; 1985b)
- I:
-
Internal inflow node
- O:
-
Internal outflow node
- U:
-
Internal undetermined node
- H N :
-
Node hematocrit
- L i :
-
Set of internal nodes linked to input nodes n i
- h 0 :
-
Input hematocrit
References
Barenblatt G.I., Entov V.M., Ryzhik V.M.: Theory of Fluid Flows Through Natural Rocks. Kluwer, Dordrecht (1990)
Carr R.T., Lacoin M.: Nonlinear dynamics of microvascular blood flow. Ann. Biomed. Eng. 28, 641–652 (2000)
Carr R.T., Geddes J.B., Wu F.: Oscillations in a simple microvascular network. Ann. Biomed. Eng. 33(6), 764–771 (2005)
Chien S., Tvetenctrand C.D., Farrel Epstein M.A., Schmid-Schönbein G.W.: Model studies on distributions of blood cells at microvascular bifurcations. Am. J. Physiol. Heart Circ. Physiol. 248, 568–576 (1985)
Cokelet G.R.: Viscometric, in vitro and in vivo blood viscosity relationships: how are they related?. Biorheology 36, 343–358 (1999)
Damiano E.R., Duling B.R., Ley K., Skalak T.C.: Axisymmetric pressure-driven flow of rigid pellets through a cylindrical tube lined with a deformable porous wall layer. J. Fluid Mech. 314, 163–189 (1996)
Dellimore J.W., Dunlop M.J., Canham P.B.: Ratio of cells and plasma in blood flowing past branches in small plastic channels. Am. J. Physiol. Heart Circ. Physiol. 244, 635–643 (1983)
Devor A., Hillman E.M.C., Tian P., Waeber C., Teng I.C., Ruvinskaya L., Shalinsky M.H., Zhu H., Haslinger R.H., Narayanan S.N., Ulbert I., Dunn A.K., Lo E.H., Rosen B.R., Dale A.M., Kleinfeld D., Boas D.A.: Stimulus-induced changes in blood flow and 2-deoxyglucose uptake dissociate in ipsilateral somatosensory cortex. J. Neurosci. 28(53), 14347–14357 (2008)
El-Kareh A.W., Secomb T.W.: A model for red blood cell motion in bifurcating microvessels. Int. J. Multiph. Flow 26, 1545–1564 (2000)
Enden G., Popel A.S.: A numerical study of plasma skimming in small vascular bifurcations. J. Biomech. Eng. 116, 79–88 (1994)
Entov V.M., Rozhkov A. N.: Elastic effects in the flow of polymer solutions in channels of variable cross section and a porous medium. J. Eng. Phys. Thermophys. 49(3), 1032–1038 (1985)
Fåhræus R., Lindquist T.: The viscosity of blood in narrow capillary tubes. Am. J. Physiol. 96, 562–568 (1931)
Faivre M., Abkarian M., Bickra K., Stone H.A.: Geometrical focusing of cells in a microfluidic device: an approach to separate blood plasma. Biorheology 43(2), 147–155 (2006)
Feng J., Weinbaum S.: Lubrication theory in highly compressible porous media: the mechanics of skiing, from red cells to humans. J. Fluid Mech. 422, 281–317 (2000)
Fenton B.M., Carr R.T., Cokelet G.R.: Nonuniform red cell distribution in 20 to 100 μm bifurcations. Microvasc. Res. 29, 103–126 (1985a)
Fenton B.M., Wilson D.W., Cokelet G.R.: Analysis of the effects of mearsured white blood cell entrance times on hemodynamics in a computer model of microvascular bed. Pflüg. Arch. Eur. J. Physiol 403, 396–401 (1985b)
Fung Y-.C.: Stochastic flow in capillary blood vessels. Microvasc. Res. 5, 34–48 (1973)
Kiani M.F., Hudetz A.G.: A semi-empirical model of apparent blood viscosity as a function of vessel diameter and discharge hematocrit. Biorheology 28, 65–73 (1991)
Kiani M.F., Pries A.R., Hsu L.L., Sarelius I.H., Cokelet G.R.: Fluctuations in microvascular blood flow parameters caused by hemodynamic mechanisms. Am. J. Physiol. Heart Circ. Physiol. 266, 1822–1828 (1994)
Kleinfeld D., Mitra P.P., Helmchen F., Denk W.: Fluctuations and stimulus-induced changes in blood flow observed in individual capillaries in layers 2 through 4 of rat neocortex. Proc. Natl Acad. Sci. USA 95, 15741–15746 (1998)
Leal L.G.: Laminar flow and convective transport processes. Butterworth-Heinemann Ltd, Boston (1992)
Lee J., Smith N.P.: Theoretical modeling in hemodynamics of microcirculation. Microcirculation 15, 699–714 (2008)
Lighthill M.J.: Pressure-forcing of tightly fitting pellets along fluid-filled elestic tubes. J. Fluid Mech. 34(1), 113–143 (1968)
McWhirter J.L., Noguchia H., Gomppera G.: Flow-induced clustering and alignment of vesicles and red blood cells in microcapillaries. Proc. Natl Acad. Sci. 106(15), 6039–6043 (2009)
Plouraboué F., Flukiger F., Prat M., Crispel P.: Geodesic network method for flows between two rough surfaces in contact. Phys. Rev. E 73, 036305 (2006)
Pop S.R., Richardson G., Waters S.L., Jensen O.E.: Shock formation and non-linear dispersion in a microvascular capillary network. Math. Med. Biol. 24, 379–400 (2007)
Popel A.S., Johnson P.C.: Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37, 43–69 (2005)
Popel A.S., Liu A., Dawant B., Koller A., Johnson P.C.: Distribution of vascular resistance in terminal arteriolar networks of cat sartorius muscle. Am. J. Physiol. Heart Circ. Physiol. 254, 1149–1156 (1988)
Pries A.R., Ley K., Gaehtgens P.: Generalization of the Fåhræus principle for microvessel networks. Am. J. Physiol. Heart Circ. Physiol. 251, 1324–1332 (1986)
Pries A.R., Ley K., Claassen M., Gaehtgens P.: Red cell distribution at microvascular bifurcations. Microvasc. Res. 38(1), 81–101 (1989)
Pries A.R., Secomb T.W., Gaehtgens P., Gross J.F.: Blood flow in microvascular networks—experiments and simulation. Circ. Res. 67, 826–834 (1990)
Pries A. R., Neuhaus D., Gaehtgens P.: Blood viscosity in tube flow: dependence on diameter and hematocrit. Am. J. Physiol. Heart Circ. Physiol. 263, 1770–1778 (1992)
Pries A.R., Secomb T.W., Gaehtgens P.: Biophysical aspects of blood flow in the microvasculature. Cardiovasc. Res. 32, 654–667 (1996)
Quéguiner C., Barthès-Biesel D.: Axisymmetric motion of capsules through cylindrical channels. J. Fluid Mech. 348, 349–376 (1997)
Risser L., Plouraboué F., Cloetens P., Fonta C.: A 3D investigation shows that angiogenesis in primate cerebral cortex mainly occurs at capillary level. Int. J. Dev. Neurosci. 27(2), 185–196 (2009)
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Guibert, R., Fonta, C. & Plouraboué, F. A New Approach to Model Confined Suspensions Flows in Complex Networks: Application to Blood Flow. Transp Porous Med 83, 171–194 (2010). https://doi.org/10.1007/s11242-009-9492-0
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DOI: https://doi.org/10.1007/s11242-009-9492-0