Red Blood Cell Migration in Microvessels
Mohamed H. Mansour a,b, Neil W. Bressloff a, and Cliff P. Shearmanc
a
School of Engineering Sciences, University of Southampton,
Southampton, U.K.
b
Mechanical Engineering Department, Faculty of Engineering,
Mansoura University, Mansoura, Egypt.
c
Department of Vascular Surgery, Southampton General Hospital,
Southampton, U.K.
Abstract Red blood cell (RBC) migration effects and RBC–plasma interactions
occurring in microvessel blood flow have been investigated numerically using a
shear-induced particle migration model. The mathematical model is based on the
momentum and continuity equations for the suspension flow and a constitutive
equation accounting for the effects of shear-induced RBC migration in concentrated
suspensions. The model couples a non-Newtonian stress/shear rate relationship with a
shear-induced migration model of the suspended particles in which the viscosity is
dependent on the haematocrit and the shear rate (Quemada model). The focus of this
paper is on the determination of the two phenomenological parameters, Kc and K , in
a diffusive flux model when using the non-Newtonian Quemada model and assuming
deformable particles. Previous use of the diffusive flux model has assumed constant
values for the diffusion coefficients which serve as tuning parameters in the
phenomenological equation. Here, previous data [1 and 16] is used to develop a new
model in which the diffusion coefficients depend upon the tube haematocrit and the
dimensionless vessel radius for initially uniform suspensions. This model is validated
through previous publications and close agreement is obtained.
Corresponding author: Tel.: ++44 (0)2380595473; Fax: ++44 (0)2380594813; Email:
[email protected]
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Keywords: Non-Newtonian, microvessel, cell-depletion, Quemada model, shearinduced migration.
1. Introduction
In microvessels (20-500 microns in diameter), there are two key processes influencing
the flow of blood. The first concerns the aggregation of red blood cells (RBCs)
resulting from shear rates that are small enough to enable RBCs to form aggregate
structures of varying sizes and shapes. One explanation often advanced for
aggregation is
the bridging hypothesis, which postulates
that long-chain
macromolecules such as fibrinogen and dextrans of high molecular weight may be
adsorbed onto the surface of more than one cell, leading to a bridging effect between
cells. It has been proposed by other investigators that the reduced concentration of
macromolecules in the vicinity of RBCs lowers local osmotic forces causing fluid to
move away and increasing the tendency for adjacent cells to come together.
According to both the bridging and the depletion theories, the total adherent force
between two cells is maximal when the cells are oriented en face. Thus it is not
uncommon to observe cells arranged in rouleaux [2].
The second key process influencing microvessel blood flow concerns the inward
migration of erythrocytes and rouleaux resulting from the effect of shear rate
gradients on individual and groups of deformable cells. Forces are created that
counteract dispersion forces and tend to move red cells and aggregates away from the
vessel wall [2]. Furthermore, cells initially on the axis of a tube continue to move
along the axis, whereas cells initially released away from the axis tend to deform and
move towards the axis.
2
The radial migration of red blood cells (and rouleaux) leads to the formation of a celldepleted layer at the vessel wall.Thus, a two-layer flow is established of an inner core
of erythrocytes and rouleaux surrounded by a cell-depleted peripheral layer, [1, 3, 5,
6, 7, 9, 17 and 18]. The formation of this layer is known to be accompanied by a
decrease in hydrodynamic resistance to flow. Furthermore, the size and distribution of
the aggregates affect the flow impedance in a way that may be characterized by an
"apparent viscosity". Effectively, when aggregates migrate to the centre of the vessel,
particle size is greater in the centre than near the wall, and so the effective viscosity is
greater in the centre as well. Thus, the net effect of aggregation on effective blood
viscosity creates two opposing tendencies, increased viscosity in the centre due to
increased particle size and decreased viscosity near the wall due to reduced
haematocrit. In the presence of red blood cell aggregation, velocity profiles become
blunted.
To study this migration process, a number of researchers have attempted to
numerically and experimentally measure the effects of shear-induced self-diffusion.
Much of the work has been performed in order to determine a self-diffusion transport
equation and the corresponding diffusion coefficient which can be used to model the
flow.
Experimental studies have been performed to verify the mechanisms of shear-induced
particle self-diffusion and viscous resuspension. Tirumkudulu et. al. [21] qualitatively
observed the effects of shear-induced diffusion in a horizontally rotating cylinder.
Although they could not mathematically formulate a model to predict the behavior,
the resulting particle distributions were attributed to shear-induced particle diffusion.
Rao et. al. [15] studied the shear-induced migration of particles during the slow flow
3
of suspensions of neutrally buoyant spheres, at 50% particle volume fraction, in a
shear-thinning suspending fluid. Nuclear magnetic resonance (NMR) imaging
demonstrated that the movement of particles away from the high shear rate region is
more pronounced than for a Newtonian suspending liquid. Also, they tested a
continuum constitutive model for the evolution of particle concentration in a flowing
suspension proposed by Phillips et. al. [13]. The model captured many of the trends
found in the experimental data, but did not agree quantitatively. They concluded that
the quantitative agreement with a diffusive flux constitutive equation would be
impossible without the addition of another fitting parameter that may depend on the
shear-thinning nature of the suspending fluid.
Zarraga and Leighton [23] obtained an unexpectedly large shear-induced selfdiffusivity from their measurements in a concentric cylinder Couette apparatus. Their
results showed that for two particle irreversible interactions, the diffusivity scaled
proportionately with concentration. However, the asymmetry of three-particle
interactions caused the diffusivity to scale with the square of the concentration.
Also, some approaches have been developed to study the migration of RBCs
numerically. Sharan and Popel [17] described the two-phase model for blood flow in
narrow tubes. Their model consists of a central core of suspended erythrocytes and a
cell-free layer surrounding the core. This discrete model has been developed for
multiple rigid particles in a circular tube. They assumed that the viscosity in the cellfree layer differs from that of plasma as a result of dissipation of energy near the wall
from the core due to the roughness of the surface between the core and the cell-free
plasma layer. A consistent system of nonlinear equations is solved numerically to
estimate the cell-free layer thickness and good agreement with experiment is obtained
4
for tubes of diameter in the range 20 ≤ D ≤ 300 μm. However, the application of this
method to complex geometries is very difficult.
Another notable work by Bagchi [1] presented two-dimensional computational
simulation of blood flow in vessels of size 20–300 μm, taking into consideration the
particulate nature of blood and cell deformation. This numerical model is based on the
immersed boundary method, and the red blood cells are modeled as liquid capsules. A
large RBC population comprising of as many as 2500 cells were simulated. Migration
of the cells normal to the wall of the vessel and the formation of the cell-free layer
were studied. Bagchi’s computational results were compared with those by Sharan
and Popel [17] and experimental results for the cell-free layer thickness. Interestingly,
Bagchi’s model doesn’t take into account the aggregation of RBCs.
Doddi and Bagchi [8] presented a three-dimensional computational model of multiple
deformable cells flowing in microvessels. They calculated the width of the cell-free
layer, the apparent blood viscosity and the haematocrit distribution. Also, they
developed a three-layer model by taking into consideration the smooth variation in
viscosity and haematocrit across the interface of the cell-free layer and the core.
Whilst the immersed boundary method has been successfully extended to model
individual RBCs flowing through relatively simple three-dimensional blood vessels
[8], it is likely that application to more complex geometry and high haematocrit
values would be prohibitively expensive and time consuming because this method
considers the motion of each RBC individually.
In light of these methods, a mathematical model is described here to capture the
effects described above and which could be sensibly applied to microvessel networks.
5
Cell depletion effects are simulated by a two-layer model in which the haematocrit
and viscosity are tightly coupled.
In the present study, the Phillips model [13] has been chosen to represent the
behaviour of RBC transport in laminar flows. The Phillips model is an extension of
the shear induced particle diffusion model originally proposed by Leighton and
Acrivos [12]. It is only applicable to laminar flow (Couette and Poisseuille) where
inertial effects of the particles can be ignored. The model is phenomenological and
derived by generalizing the simple scaling arguments based on the shear-induced selfdiffusion theory of Leighton and Acrivos [12]. The model accounts for the fact that
particles in a shear flow will not remain stationary, but will migrate to different
regions of the flow depending on the variation in local shear rate, concentration and
viscosity. The resulting model equation accounts for all of the self-diffusion effects
discovered by previous researchers combined into a single, scalar transport equation.
The Phillips model is a constitutive model that does not account for all mechanisms of
particle transport. Rather, it only considers those that occur in concentrated
suspensions from particle-particle (or two-body interactions) interactions. In
concentrated systems there are a variety of interparticle interactions including
hydrodynamic and electrostatic [13].
The particle fluxes can be attributed to two effects, resulting from the interaction of
two particles [22]. These concern the spatial variation of interaction frequency and
viscosity. The two phenomenological parameters that account for these two effects are
the diffusive parameters, Kc and Kμ, respectively.
The accurate determination of these proportionality constants is not straightforward.
In addition, it is not easy to accurately determine the haematocrit profile. As such, a
6
new formula for the determination of the values of these two parameters is proposed
by comparing previous numerical and experimental results with the results obtained
from this model. Also, a Quemada model, that describes the viscosity behaviour of
blood, is introduced to the Phillips model [13].
2. Quemada model for two-layer blood flow
A number of constitutive models have been proposed to describe the bulk rheological
behavior of blood. Among these, the Casson model has been most widely used.
Quemada [14] extended the Casson model using first physical principles and
explicitly described the kinetics of RBC aggregation; this model includes a structural
parameter, which is related to the size of RBC aggregates. As a result, the viscosity is
not infinite at zero shear rates in the Quemada model making it attractive for
modeling blood flow in microvessels. Also, the Quemada model accurately fits
experimental viscometric data for small diameter vessels (above 12 m diameter) [6].
The viscosity results for this model lie between the in vitro and in vivo values. The
Quemada model is thus used here to model RBC suspensions.
3. Mathematical model
3.1. Governing equations
A suspension of RBCs in plasma solution is considered in a three dimensional
circular microvessel.. The flow of the concentrated suspension is assumed to be
steady, incompressible and laminar. The two-phase suspension of both the particles
and the fluid is modelled as a single continuum. The continuity and the momentum
equations for the three-dimensional suspension flow are given by,
V
0
(1)
7
V
t
V
V
(2)
p
where, ρ, V, p, τ denote the bulk blood density, the velocity field, the pressure and the
deviatoric stress tensor, respectively. This tensor is related to the viscosity and the
shear rate tensor according to the relation
(3)
( )D
where the viscosity,
, is a function of the shear rate
and the shear rate tensor D.
The relation between the shear rate and the shear rate tensor is expressed as
1
2
D ijD ji
i
(4)
j
and,
u
x
D
1
2
1
2
1
2
u
y
u
z
v
x
w
x
u
y
v
x
v
y
1
2
w
y
1
2
1
2
u
w
z
x
w
v
y
z
w
z
v
z
(5)
The shear rate is, therefore, calculated in Cartesian coordinates as
2
u/ x
v/ z
2
v/ x
w/ y
2
w/ x
2
u/ z
2 1/ 2
w/ x
2
u/ y
v/ x
2
(6)
The viscosity is related to the structural parameter (k) and the blood local haematocrit
H, according to the Quemada model [14], in the form
1
p
where
1 0.5kH
p denotes
(7)
2
the plasma viscosity.
The rheological model proposed by Quemada considers blood as a structured fluid,
8
wherein the state of RBC aggregation is described by the structural parameter k,
which characterizes the average number of RBCs in an aggregate,
k0
k
k
/
1
/
c
(8)
c
where, k0 and k∞ are the intrinsic viscosities at zero and infinity shear rates,
respectively, of the flow particles which predominate at those shear rates. γc signifies
the critical shear rate, which can be considered to be the inverse of the relaxation time
for the dominant structural unit causing the suspension to be non-Newtonian. Here,
k0, k∞ and γc are functions of H [4].
k0 = exp(3.8740 + H(-10.41 + H(13.8 – 6.738H)))
(9)
k∞ = exp(1.3435 + H(-2.803 + H(2.711 - 0.6479H)))
(10)
γc = exp(-6.1508 + H(27.923 + H(-25.6 + 3.697H)))
(11)
The migration of haematocrit is simulated by a conservation equation describing
the transport of RBCs and encapsulating the associated behaviour through the
microvessel.
The haematocrit is governed by an evolution equation
H
t
V
(H)
N
(12)
which represents a balance between stored particles, the convected particle flux and
diffusive particle flux N. The momentum equation (Eq. 2) and the concentration
equation (Eq. 12) are coupled through the velocity field and Quemada viscosity
equation (Eq. 7). The diffusivity flux of the RBCs is given by
N
Nc
N
(13)
9
where Nc is the flux contribution due to hydrodynamic particle interactions and it
incorporates the effect of particle migration in the direction of decreasing interaction
frequency. Nμ denotes the flux contribution due to spatial variation in viscosity, which
causes a resistance to motion after a two-particle collision. Effectively, both particles
are displaced in a direction of lower viscosity relative to their position in the case of
no viscosity gradient. Based on the scaling arguments of Leighton and Acrivos [12],
Phillips et al. [13] proposed that
Nc
K ca 2 H 2
N
K
H2
H
(14)
H
a2
(15)
where a is the RBC radius and
is the local shear rate.
Rearranging Eq. 12, assuming steady state flow and using Eqs. 14 and 15
V
H a 2Kc
H2
H
H
a 2K
a 2Kc
H2
H2
1
(16)
0
or
V
H
a 2KcH
H
a 2K
H2
1
(17)
The second term on the right-hand-side of Eq. 17 is obtained through the chain rule as
a 2K
H2
1
a 2K
H2
1d
H
dH
H2
1d
(18)
d
It is found in the literature [15] that the model follows experimental trends much more
accurately if a simplified gradient of the logarithm of the viscosity is used that does
not contain the derivative with respect to shear rate. Then the second term on the
right-hand-side of Eq. 17 can be written as
10
a 2K
H2
1
a 2K
H2
1d
H
dH
(19)
Thus, in this work, predictions using a viscosity gradient with respect to haematocrit
only and with respect to both the haematocrit and the shear rate have been examined.
When using a viscosity gradient with respect to haematocrit only, this implies that the
migration behavior of the particles is only dependent on the viscosity insofar as the
viscosity is dependent on particle concentration, (i.e. much of the viscosity effect is a
lumped concentration effect).
By using Eq. 18, the shear-induced migration equation can be written as
V. H
.
a 2KcH
a 2K
H2
d
dH
H
a2 .
KcH2
a 2K
H2
d
dH
(20)
where
a2 .
a 2KcH
Kc H 2
a 2K
K
H2
H2
d
dH
d
dH
represents the diffusion coefficient and
represents a source term.
4. Numerical procedure
The solution algorithm is coded as a set of user defined functions in Fluent
(Ansys), which are applied in two stages in order to obtain convergence. At first,
constant initial values for the velocity field, haematocrit, diffusion coefficient of the
scalar conservation equation and the viscosity are applied. Then, the continuity and
momentum equations are solved followed by the scalar conservation equation for H to
yield new values of the velocity field and H. These values are used again to calculate
11
more accurate values of V and H. This procedure is repeated for 20 iterations since it
was found that, without this step, µ could not be suitably initialised.
In the second step of the solution, the viscosity is evaluated from K and H, first by
calculating k from Eq. 8. Then, the source term is applied to the scalar conservation
equation. This iterative procedure is repeated until convergence is achieved (when the
mean residuals of the continuity, momentum and haematocrit equations reach stable
constant values). At convergence, the haematocrit distribution is assessed to evaluate
the cell-depletion layer thickness.
The setup in Fluent comprises the implicit pressure based solver with second order
velocity and pressure interpolation, and the Green-Gauss cell based method for
gradients.
The mesh was generated by using Harpoon (Sharc Ltd). A mesh dependence study
was performed on the microvessel to determine a suitable grid resolution. The
velocity distribution was observed at the vessel outlet for different cell sizes: 1, 2 and
4 m. There was only a small difference between the two finest meshes. Consequenly,
the grid was choosen to be between 1 and 2 m depending on the vessel diameter to
limit the cell count to less than two million cells.
5. Boundary conditions
The governing equations are subjected to no-slip boundary conditions (V=0) and a
zero-flux boundary condition at solid boundaries ( n .N
0 , where N=Nc+Nμ and n is
the normal outward unit vector on the boundary). At the inlet, the flow velocity and
the haematocrit are prescribed and they vary according to the geometry and the flow
12
under consideration. At the outflow section, the normal components of the gradients
of the shear rate and the haematocrit vanish and the zero flux condition n .N
0 is
satisfied. It is worth mentioning that the length of the computational domain has been
chosen appropriately to ensure that the outlet conditions given above are satisfied.
The mean blood velocity, vessel diameters, discharge haematocrit and tube
haematocrit for all the simulations are presented in Table 1. All the investigated
microvessels have the same length of 2 mm, which is the average arteriole length as
found in the literature.
6. Validity assessment
The investigation for fully developed steady-state flow of a concentrated suspension
in a straight vessel has been carried out to validate the code. For Poiseuille flow of
concentrated suspensions (Hbulk=0.3 and 0.45), the concentration and velocity profiles
reach fully developed forms. The velocity field develops faster than the concentration,
and the entrance length (the length required for the corresponding variable to become
fully developed) depends on the vessel radius (R) and the particle radius (a).
Experiments have indicated that the entrance length for the velocity field is
considerably less than that for the concentration, which is in turn considerably less
than the estimated value R3/a2.
For the Poiseuille flow of concentrated suspension, the inlet conditions correspond to
a constant mass flow rate with an average velocity of uav=6.5 mm/s and a uniform
concentration at the inlet. The radius and length of the vessel are 50 μm and 1 mm,
respectively. The steady solutions have been computed and compared with that
presented in Weert [22]. The computations have been carried out for Hinlet=0.3 and
13
0.45. In these simulations the value of the constants Kc and K are 0.41 and 0.62,
respectively as stated by Weert [22]. Here, a suspension viscosity proposed by Krieger
[10] was employed to describe the effective suspension viscosity
p
H
1
Hm
1.82
(21)
Figs. 1a and 1b show, respectively, the computed fully developed particle
concentration profile and the velocity profile along with the Weert computed results
for inlet haematocrit of 0.45. The dimensionless radius in these figures is the local
radius divided by the vessel radius. The numerical results demonstrate strong particle
migration towards the centre of the channel and an increasing blunting of the velocity
profiles with increase in initial particle concentration, which is in close agreement
with Weert [22]. The computed values of H at the centre are 0.6228 and 0.658 for
Hinlet of 0.3 and 0.45, respectively.
Also, comparisons are made between the experimental results of Tan et. al. [19] and
numerical predictions based on a continuum diffusive-flux model for shear-induced
particle migration of nickel (Ni-171) particles in an Ethylene Vinyl Acetate (EVA460)
as a binder to study the non-Newtonian effect of the binder. The density of the nickel
is 3600 kg/m3 and the power-law model is employed to describe the binder viscosity,
b
m 0 exp
Ta
T
n 1
(22)
where m0 and n are material constants, T and Ta are the testing temperature for
capillary rheological measurement and the temperature dependent material constant
for the binder, respectively. For EVA460, the fluid exhibits a Newtonian plateau value
at low shear rates and shear thinning at high shear rates. In the present investigation,
the constants of the rheological model m0, n and Ta are determined to be 392.9, 0.385
14
and 6702.5, respectively [19]. In these simulations the value of the constants Kc and
K are 0.2 and 0.58, respectively as stated by Tan et. al. [19].
It is shown in Fig. 2 that there is a good agreement between experimental
observations and numerical predictions based on the shear-induced migration model
above. Furthermore, changing the values of the constants Kc and K from that used in
Weert [22] and introducing the power-law viscosity model, the concentration profile
becomes different from a cusp-like profile of concentration predicted for Newtonian
suspension fluids as described above.
7. Adjustable parameters
The two parameters, Kc and Kμ, presented in Eqs. 14 and 15, exist to account for the
pseudo-diffusive nature of the Phillips model. They are proportionality constants
determined from fitting model simulations to experimental results. These parameters
represent different material properties, particle shape, size distribution and surface
roughness, as they play an important role in irreversible particle collisions. They are
of order unity.
Phillips et. al. [13] found from a comparison with their experimental results that a
ratio of Kc/Kμ equal to 0.66 provided a best fit to the experimental data under a
number of flow geometries. Values for Kc of approximately 0.43 and Kμ of 0.65
provided an excellent fit to their experimental concentration profiles in concentrated
Couette flow. It was also reasoned that the ratio of Kc to Kμ can never exceed 1. This
ensures particles always migrate down a shear rate gradient. Increasing the ratio has
the effect of dramatically increasing the steady concentration gradient across the
domain.
15
Ideally both parameters would be independent of a, H and . However, they should
also be independent of the flow geometry and particle density. Other researchers [11
and 15] have shown that they are not completely independent of the particle volume
fraction. Phillips et. al. [13] admit that due to the sensitivity of the results to the ratio
of Kc to Kμ that the parameters may in fact be weak functions of local concentration.
Rao et. al. [15] investigated the effects of neutrally buoyant particles in a slow
flowing, shear thinning (Carreau model) fluid. Particle migration was due to gradients
in shear rate, concentration and viscosity, and they suggested a normal stress
correction for non-Newtonian fluids (by using Eq. 18 instead of Eq. 19) when using
the Phillips model because of the anisotropy of non-Newtonian flows. Their results
led them to conclude that the Phillips model without normal stress corrections may be
fundamentally inadequate for simulating flow in non-Newtonian fluids.
Lam et. al. [11] investigated particle migration in Poiseuille flow of nickel powder
injection moldings. They also investigated the effects of a shear thinning carrier fluid
which they fit with the non-Newtonian Cross model. Their resulting best fit values for
the Phillips model adjustable parameters are shown in Table (2).
All of the above coefficients are in close agreement with what Phillips et. al. [13]
determined from their experimental study. Their simulations produced concentration
profiles for pressure driven flows, where solid particle migration was from the vessel
walls to the vessel center. They found that the non-Newtonian, shear thinning
behaviour enhanced particle migration from regions of high shear rate to regions of
low shear rate.
16
Tetlow and Graham [20] performed experiments and modeling on particle migration
in Newtonian fluids for creeping flows in the annular space of a wide gap Couette,
concentric cylinder apparatus. They determined the optimum tuning coefficients for
their numerical model based on experimental data. They found that the coefficients Kc
and K should not be constant but rather slight functions of concentration. Their bestfit ratio of the tunable parameters is
Kc
K
(23)
0.01042 * H 0.1142
8. Results
From the previous discussion, it is observed that the value of Kc lies between 0.2 and
0.4, and the value of Kμ lies between 0.5 and 0.7. So, calculations were performed
using these values to obtain the haematocrit distribution at the vessel outlet to
calculate the cell-depletion layer thickness.
At first, for a discharge haematocrit of 0.3 in a 100 m diameter vessel, values of 0.41
and 0.62 were used for Kc and K , respectively, and a suspension viscosity proposed
by Krieger [10] was employed to describe the effective suspension viscosity (Eq. 21).
A viscosity gradient with respect to the haematocrit is employed by substituting with
Eq. 19 in Eq. 17. The results of the haematocrit, shear rate, velocity and viscosity
distribution were investigated. The numerical results of the haematocrit, Fig. 3, show
a cusp-like concentration profile for this Newtonian concentrated suspension during
pressure-driven tube flow. This profile is different to the expected plug-like profile, as
a result of using the Newtonian concentrated suspension viscosity equation. The shear
rate value is high at the wall and decreases until it reaches zero at the center. Due to
the aggregation of the red blood cells in the vessel core, the velocity profile has a
17
blunted shape rather than a parabolic shape as shown in Fig. 4. Also, the viscosity at
the outlet has its smallest value at the wall and then increases until it reaches its
highest value close to the center of the vessel.
Due to the advantages of the Quemada model over other non-Newtonian models, a
Quemada model has been introduced to the particle shear-induced migration model.
In the Quemada model, the viscosity is a function of the shear rate, the haematocrit
and the structural parameter. As discussed before, the value of the parameters Kc and
Kμ are different for the Quemada model relative to other non- Newtonian models, so,
different values of Kc and Kμ are examined to determine approximate values of these
parameters. Fig. 5 shows the haematocrit distribution for two different values of Kc
and Kμ. A viscosity gradient with respect to the haematocrit only is employed by
substituting Eq. 19 in Eq. 17 and a viscosity gradient with respect to the haematocrit
and the shear rate is employed by substituting Eq. 18 in Eq. 17. The results of the
haematocrit distribution are shown in Figs. 5 and 6 and better symmetry in the
haematocrit distribution inside the vessel is found by using the viscosity gradient with
respect to the haematocrit and the shear rate. Moreover, the solution converges rapidly
and becomes more stable. So, in all the subsequent simulations, the viscosity gradient
is taken with respect to both the haematocrit and the shear rate.
To study the effect of the flux contribution due to hydrodynamic particle interactions
and the flux contribution due to varying viscosity, a UDF has been created to calculate
the components of these fluxes in three dimensions (x, y and z). The results showed
that the flux in the y-direction has the same magnitude as that in the z-direction. Fig.
7a shows the flux contribution due to hydrodynamic particle interactions and that due
to varying viscosity in the x-direction (axial direction) for non-Newtonian Quemada
18
blood flow. The flux in the x-direction is small compared with that in the y and zdirections.
Also, a comparison has been performed between the two fluxes in the z-direction for
the non-Newtonian Quemada blood flow. Fig. 7b shows that the flux contribution due
to hydrodynamic particle interactions is higher than that due to varying viscosity. This
means that the effect of the migration due to hydrodynamic particle interactions is
more pronounced than that due to that of varying viscosity.
From these figures, it is clear that Kc has a far more significant effect on radial
migration than Kμ. Also, for these values of Kc and Kμ, both the migration flux and the
thickness of the cell-depletion layer are very high. So, a new trial was performed by
using a range of values of Kc (2e-2, 1e-2, 7e-3, 2e-3, 2e-4 and 2e-6) and the following
relation for K ,
K = 1.52 Kc.
A 100
(24)
m diameter vessel was used with discharge haematocrit, HD, equal to 0.2
(which is equivalent to tube haematocrit Ht=0.165). For these six runs, the cell
depletion layer was calculated (it is defined as the thickness of the layer beside the
wall where the haematocrit profile increases until it reaches the tube haematocrit
value, [16]) and compared to the thickness that is listed in Sharan and Popel [17]
(which is 6.5 m for this case).
Fig. 8 shows the different haematocrit profiles for HD=0.2 for different values of Kc.
With increasing Kc, the migration flux and the thickness of the depletion cell layer
both increase. It is found that the most appropriate value of Kc lies between 1e-2 and
2e-4. So, further simulations were performed for Kc = 8e-4, 2e-3 and 5e-2 with HD=0.2
19
and for different vessel diameters (40, 60, 80, 100 m). A comparison of the results of
the particle shear-induced migration model using these values of Kc with the results
from Sharan and Popel [17] results are shown in Fig. 9. It is clear that the trend of the
computed cell-depletion layer thickness curves is different to that of Sharan and Popel
[17].
The Phillips model is usually applied to study the migration of rigid particles and it is
well known that the RBCs are deformable. So, Kc and Kμ should not be constants and
could be functions of the tube haematocrit, Ht, [20], and position. It is shown in Fig. 8
that the value of the haematocrit at the wall is relatively higher than expected and
there is a peak value at the center of the vessel. To avoid this, the migration flux
should be high near the wall and small at the core region. A new formula for Kc and
Kμ is applied to the Phillips model to address these issues. This formula is an
exponential function to give high Kc at the wall and small values in the core. The ratio
between Kc and Kμ remains constant and equal to 1.66. The general form of this
equation is
Kc
r
A * Exp ( )
R
BH t
(25)
where A and B are constants calculated from comparing the results with previous
studies, r and R are the local position and the vessel radius, respectively. Three values
of A (5e-3, 1e-3 and 5e-2) and three values of B (5, 10 and 15) are examined and the
cell-depletion layer thicknesses are calculated and compared with that of Bagchi [1],
and Sharan and Popel [17] to determine the most appropriate values of A and B for
two discharge haematocrit values (0.2 and 0.45).
20
The parameters of the Quemada model were determined from experimental data of
blood at 37 °C [4]. The blood in this study consists of plasma and deformable red
blood cells. In the case of the migration of solid particles, Kc in the RBC conservation
equation should be a function of the tube haematocrit only as proposed by Tetlow
[20]. In the case of the deformable particles, Kc should be a function of the tube
haematocrit and the tube radius (r).. The RBC radius changes with position in the
vessel, due to its deformability, so it is a function of the local position, r. Therefore,
the effect of the RBC deformability is implied in the Quemada model and in the
diffusion coefficient Kc.
Figs. 10a, 10b and 10c show the dimensionless cell-depletion layer thickness at
discharge haematocrit equal to 0.2 for different A and B values. It is shown that with
increasing A at fixed B, Kc increases and as a result of this the thickness of the celldepletion increases too. The best result is obtained for A = 0.01 and B = 10.
The same simulations are repeated but for a discharge haematocrit of 0.45. The results
are shown in Figs. 11a, 11b and 11c. It is clear from Bagchi [1], and Sharan and Popel
[17] that the cell-depletion layer thickness is relatively smaller for HD=0.45 compared
with that at HD=0.2. This leads to A = 0.001 and B = 15 as the best values.
Finally, since it is potentially useful to have only one equation for Kc for different
discharge haematocrit values, an interpolation for these results has been performed
and a new equation has been derived for calculating Kc,
Kc
r
0.0731* Exp
R
16.0115H t * Exp
r
R
(26)
The dimensionless cell-depletion layer thickness has been calculated by using Kc
values obtained from Eq. 26 for a discharge haematocrit of 0.2 and 0.45. Fig. 12
21
shows that the results obtained by using Eq. 26 are very close to those obtained by
Bagchi [1], and Sharan and Popel [17].
By way of example, the haematocrit, velocity and viscosity profiles at the outlet
section are calculated for HD equal to 45% in a 40 μm diameter vessel as shown in
Figs. 13a, 13b and 13c. The haematocrit profile is a plug-like profile at the core due to
the migration and aggregation of RBCs. Also, the viscosity has increased in the core
to have its maximum value (0.0036 Pa.s) at the center due to the aggregation of
RBCs. The velocity profile is blunted (not parabolic) as a result of using the nonNewtonian Quemada model and the migration of RBCs (which increases the viscosity
at the core region).
9. Conclusion
A theoretical cell-depletion model has been developed to simulate blood flow through
microvessels. The Quemada model that takes into account the dependence of the
viscosity of the blood on the structure parameter and the haematocrit has been
introduced to this model. RBC migration is calculated by using the shear-induced
particle migration model (Phillips model). The parameter Kc in the Phillips model
requires a smaller value relative to its usual value due to the introduction of the nonNewtonian Quemada model. Also, using the Quemada model with fixed Kc values
leads to relatively higher haematocrit value than that expected at the wall and creates
a sharp peak value at the vessel center. So, a new expression for calculating the
parameter Kc is investigated as a function of the dimensionless local radius and the
tube haematocrit, K c
r
0.0731* Exp
R
16.0115H t * Exp
r
. A comparison
R
between the results obtained for the cell-depletion layer thickness and those
22
previously published leads to a new single method for the accurate prediction of
haematocrit under the simulated conditions.
.
References
[1] P. Bagchi, P., Mesoscale simulation of blood flow in small vessels, Biophysical
Journal. 92 (2007), 1858-1877.
[2] J. J. Bishop, A. S. Popel, M. Intaglietta and P. C. Johnson, Rheological effects of
red blood cell aggregation in the venous network: A review of recent studies,
Biorheology, 38 (2001), 263–274.
[3] X. Chen, D. Jaron, K. A. Barbee, and D. G. Buerk, The influence of radial RBC
distribution, blood velocity profiles and coupled NO/O2 transport, J. Appl.
Physiol. 100 (2006), 482–492.
[4] G. R. Cokelet, The rheology and tube flow of blood. in: Handbook of
bioengineering, edited by R. Skalak and S. Chien. New York: McGraw Hill.
(1987) 14.1-14.17.
[5] G. R. Cokelet and H. L. Goldsmith, Decreased hydrodynamic resistance in the
two-phase flow of blood through small vertical tubes at low flow rates, Circ. Res.
68 (1991), 1–17.
[6] B. Das, G.. Enden and A. S. Popel, Stratified multiphase model for blood flow in a
venular bifurcation, Annals of Biomedical Engineering. 25 (1997), 135–153.
[7] B. Das, P. C. Johnson and A. S. Popel, Computational fluid dynamic studies of
leukocyte adhesion effects on non-Newtonian blood flow through microvessels,
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Biorheology. 37 (2000), 239–258.
[8] S. K. Doddi and P. Bagchi, Three-dimensional computational modeling of multiple
deformable cells flowing in microvessels, Physical Review E 79, 046318(2009),
1–14.
[9] S. Kim, R. L. Kong, A. S. Popel, M. Intaglietta and P. C. Johnson, Temporal and
spatial variations of cell-free layer width in arterioles, Am. J. Physiol Heart Circ.
Physiol. 239 (2007), H1526–H1535.
[10] I. M. Krieger, Rheology of monodisperse lattice, Adv. Colloid Interface Sci., vol.
3 (1972),111–136.
[11] Y. C. Lam, X. Chen, K. W. Tan, J. C. Chai and S. C. M. Yu, Numerical
investigation of particle migration in poiseuille flow of composite system,
Composites Science and Technology, vol. 64 (2004), 1001–1010.
[12] D. T. Leighton and A. Acrivos, The shear-induced migration of particles in
concentrated suspension, J. Fluid Mech., (1987), 415-439.
[13] R. J. Phillips, R. C. Armstrong, R. A. Brown, A. L. Graham and J. R. Abbott, A
constitutive equation for concentrated suspensions that accounts for shearinduced particle migration, Phys. Fluids., vol. 4 (1), (1992), 30-40
[14] D. Quemada, Rheology of concentrated disperse systems: A model for nonNewtonian shear viscosity in steady flows, Rheol. Acta (17), (1978), 632-642.
[15] R. Rao, L. A. Mondy, T. A. Baer, S. A. Altobelli, and T. S. Stephens, NMR
measurements and simulations of particle migration in non-Newtonian fluids,
24
Chem. Eng. Comm., Vol. 189(1) (2002), 1-22.
[16] D. Saintillan, E. S. G. Shaqfeh and E. Darve, Effect of flexibility on the shearinduced migration of short-chain polymers in parabolic channel flow, J. Fluid
Mech. (2006) 557, 297-306.
[17] M. Sharan, and A. S. Popel, Two-phase model for flow of blood in narrow tubes
with increased effective viscosity near the wall, Biorheology. 38 (2001), 415–
428.
[18] V. P. A. Srivastava, Theoretical model for blood flow in small vessels.
Applications and Applied Mathematics, (AAM). 2 (1) (2007), 51–65.
[19] K. W. Tan, X. Chen, Y. C. Lam, J. Ma and K. C. Tam, experimental
investigation of shear-induced particle migration in steady-state isothermic
extrusion, Journal of Society of Reology, vol.31 (3) (2003), 165-173.
[20] N. Tetlow, and A. L. Graham, Particle migration in a couette apparatus:
experiment and modeling, J. Rheol., vol. 42(2) (1998), 307-327.
[21] M. Tirumkudulu, A. Tripathi, and A. Acrivos, Particle segregation in
monodisperse sheared suspensions, PHYSICS OF FLUIDS, vol. 11(3), (1999),
507-509.
[22] K. V. Weert, Numerical and experimental analysis of shear-induced migration in
suspension flow, A thesis for the degree of master, Eindhoven University (2005).
25
[23] I. E. Zarraga, D. T. Leighton, Measurement of an unexpectedly large shearinduced self diffusivity in a dilute suspension of spheres, Physics of Fluids,
vol.14 (7), (2002), 2194-2201.
26
Vessel Diameter, (μm)
40
60
80
100
40
60
80
100
Discharge Haematocrit, %
Tube Haematocrit, %
20
20
20
20
45
45
45
45
13.5
14.7
15.7
16.5
35
36.7
38.4
39.6
Mean Velocity,
(mm/s)
13
8.5
7.5
8
9.5
8
5
6
Table (1). Mean blood velocity, vessel diameters, discharge haematocrit and tube
haematocrit for all the simulations based on the data in Bagchi [1].
27
Power Law model
Cross model
Newtonian model
Kc
0.32
0.33
0.41
Kμ
0.65
0.65
0.62
Kc/ Kμ
0.49
0.51
0.66
Table (2). Kc and Kμ values for Newtonian and different non-Newtonian models based
on Lam et. al. [11]
28
Figure captions
Fig. 1. Comparison of computational results and Weert [22] results for Hinlet=0.45 for:
(a) Haematocrit
(b) Velocity
Fig. 2. Outlet concentration distribution comparison of numerical results and
experimental results of Tan et. al. [19] for Hinlet=0.3.
Fig. 3. Outlet concentration distribution for Hinlet=0.3.
Fig. 4. Outlet velocity distribution for Hinlet=0.3.
Fig. 5. Outlet Haematocrit distribution for HD=0.2 using different values of Kc and K
and the viscosity gradient is calculated with respect to the haematocrit only.
Fig. 6. Outlet Haematocrit distribution for HD=0.2 using different values of Kc and K
and the viscosity gradient is calculated with respect to the haematocrit and the shear
rate.
Fig. 7. Flux contribution due to hydrodynamic particle interactions and the flux
contribution due to varying viscosity for the non-Newtonian Quemada flow in:
(a) x-direction
(b) z-direction
Fig. 8. Outlet Haematocrit distribution for HD=0.2 using different small values of Kc
and K and the viscosity gradient is calculated with respect to the haematocrit and the
shear rate.
Fig. 9. Dimensionless cell-depletion layer δ/r for the particle shear-induced model
compared of that of Sharan and Popel [17].
Fig. 10. Dimensionless cell-depletion layer thickness δ/r for the particle shear-induced
migration model for discharge haematocrit=0.2 at:
(a) B=5
(b) B=10
(c) B=15
Fig. 11. Dimensionless cell-depletion layer thickness δ/r for the particle shear-induced
migration model for discharge haematocrit=0.45 at:
(a) B=5
(b) B=10
(c) B=15
Fig. 12. Dimensionless cell-depletion layer thickness δ/r for the particle shear-induced
migration model by using fitting function for Kc for discharge haematocrit equal to
0.2 and 0.45.
Fig. 13. The distribution at the vessel outlet for HD equal to 45% in a 40 μm diameter
vessel for:
(a) haematocrit
(b) viscosity
(c) velocity
29
0.7
0.65
Haematocrit
0.6
0.55
Computed results
Weert results
0.5
0.45
0.4
0.35
0.00
0.20
0.40
0.60
0.80
1.00
Dimensionless radius
(a)
1
Dimensionless velocity
0.9
0.8
0.7
0.6
0.5
Computed results
Weert results
0.4
0.3
0.2
0.1
0
0.00
0.20
0.40
0.60
Dimensionless radius
(b)
Fig. 1.
1
0.80
1.00
0.35
Concentration
0.32
0.29
Numerical Results
Experimental Results
0.26
0.23
0.2
0
0.2
0.4
0.6
Dimensionless radius
Fig. 2.
2
0.8
1
0.8
Haematocrit
0.6
0.4
0.2
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Dimensionless radius
Fig. 3.
3
0.4
0.6
0.8
1
0.35
Velocity, mm/s
0.3
0.25
0.2
0.15
0.1
0.05
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Dimensionless radius
Fig. 4.
4
0.4
0.6
0.8
1
0.35
Haematocrit
0.3
0.25
0.2
0.15
0.1
0.05
0
-50
-40
-30
Kc=0.2-Kμ=0.5
-20
-10
0
10
20
Radius, μm
Kc=0.2-Kμ=0.7
Kc=0.4-Kμ=0.5
Fig. 5.
5
30
40
50
Kc=0.4-Kμ=0.7
0.35
Haematocrit
0.3
0.25
0.2
0.15
0.1
0.05
0
-50
-40
-30
-20
-10
Kc=0.2-K=μ0.5
0
Radius, μm
10
Kc=0.2-Kμ=0.7
Fig. 6.
6
20
30
40
Kc=0.4-Kμ=0.5
50
7
Flux, kg/(m2.s) * 1e-5
6
5
4
3
2
1
0
-1
-50
-40
-30
-20
-10
0
10
20
30
40
50
-2
-3
Radius, m
(a)
25
Flux, kg/(m2.s) *1e-5
20
15
10
5
0
-5 -50
-40
-30
-20
-10
0
10
20
30
40
-10
-15
-20
-25
Radius, m
Flux due to hydrodynamic interactions
(b)
Fig. 7.
7
Flux due to varying viscosity
50
0.18
0.17
Haematocrit
0.16
0.15
0.14
0.13
0.12
-50
-40
Kc=0.02
-30
-20
Kc=1e-2
-10
0
Radius, μm
Kc=7e-3
Fig. 8.
8
10
Kc=2e-3
20
30
Kc=2e-4
40
50
Kc=2e-6
0.5
Kc=8e-4
Kc=2e-3
0.4
Kc=5e-2
Sharan&Popel
/R
0.3
0.2
0.1
0
20
30
40
50
60
70
Diameter, μm
Fig. 9.
9
80
90
100
110
0.5
0.4
/R
0.3
0.2
0.1
0
20
30
40
50
60
70
Diameter, μm
80
90
100
110
(a)
0.5
0.4
/R
0.3
0.2
0.1
0
20
30
40
50
60
70
80
90
100
110
80
90
100
110
Diameter, μm
(b)
0.5
0.4
/R
0.3
0.2
0.1
0
20
30
40
Sharan&popel
50
60
70
Diameter, μm
A=0.005
A=0.01
(c)
Fig. 10.
10
A=0.05
Bagchi
0.5
0.4
/R
0.3
0.2
0.1
0
20
30
40
50
60
70
Diameter, μm
80
90
100
110
80
90
100
110
80
90
100
110
(a)
0.5
0.4
/R
0.3
0.2
0.1
0
20
30
40
50
60
70
Diameter, μm
(b)
0.5
0.4
/R
0.3
0.2
0.1
0
20
30
40
50
60
70
Diameter, μm
Sharan&popel
A=0.005
A=0.01
(c)
Fig. 11.
11
A=0.05
Bagchi
0.5
Sharan&Popel H=0.2
Bagchi H=0.2
Shear-induced H=0.2
Sharan&Popel H=0.45
Bagch H=0.45
shear-induced H=0.45
0.4
/R
0.3
0.2
0.1
0
20
30
40
50
60
70
Diameter, μm
Fig. 12.
12
80
90
100
110
0.37
Haematocrit
0.35
0.33
0.31
0.29
0.27
0.25
-20
-15
-10
-5
0
5
10
15
20
10
15
20
15
20
Radius, μm
(a)
0.0038
0.0036
Viscosity, Pa.s
0.0034
0.0032
0.003
0.0028
0.0026
0.0024
0.0022
0.002
-20
-15
-10
-5
0
5
Radius, μm
(b)
20
18
Velocity, mm/s
16
14
12
10
8
6
4
2
0
-20
-15
-10
-5
0
Radius, μm
(c)
Fig. 13.
13
5
10