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A New Approach to Model Confined Suspensions Flows in Complex Networks: Application to Blood Flow

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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

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Authors and Affiliations

  1. Université de Toulouse, INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, 31400, Toulouse, France

    R. Guibert & F. Plouraboué

  2. CNRS, IMFT, 31400, Toulouse, France

    R. Guibert & F. Plouraboué

  3. Université de Toulouse, UPS, Centre de Recherche Cerveau et Cognition, Toulouse, France

    R. Guibert & C. Fonta

  4. CNRS, CerCo, Toulouse, France

    R. Guibert & C. Fonta

Authors
  1. R. Guibert
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  2. C. Fonta
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  3. F. Plouraboué
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Correspondence to F. Plouraboué.

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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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