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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: N.W.Bressloff@soton.ac.uk

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Keywords: Non-Newtonian, microvessel, cell-depletion, Quemada model, shear-

induced 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.

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The radial migration of red blood cells (and rouleaux) leads to the formation of a cell-

depleted 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

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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 self-

diffusivity 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 cell-

free 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

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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.

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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 self-

diffusion 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

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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,

0V (1)

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pVVt

V (2)

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

D)( (3)

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

i j

jiijDD2

1 (4)

and,

z

w

z

v

y

w

2

1

x

w

z

u

2

1

z

v

y

w

2

1

y

v

x

v

y

u

2

1

x

w

z

u

2

1

x

v

y

u

2

1

x

u

D (5)

The shear rate is, therefore, calculated in Cartesian coordinates as

222x/wx/vx/u2

22x/vy/ux/wz/u

2/12

y/wz/v (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

2pkH5.01

1 (7)

where p denotes the plasma viscosity.

The rheological model proposed by Quemada considers blood as a structured fluid,

depletio

n

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wherein the state of RBC aggregation is described by the structural parameter k,

which characterizes the average number of RBCs in an aggregate,

c

c0

/1

/kkk (8)

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

N)H(Vt

H (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

NNN c (13)

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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

HHHaKN 22

cc (14)

22 a

HKN (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

01

HKaHHHKaHV 222

c

2 (16)

or

1HKaHKaHHKaHV 222

c

2

c

2 (17)

The second term on the right-hand-side of Eq. 17 is obtained through the chain rule as

d

d1HH

dH

d1HKa

1HKa 22222 (18)

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

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HdH

d1HKa

1HKa 2222 (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

dH

dHKaHK.aH

dH

dHKaHKa.H.V 222

c

222

c

2

(20)

where dH

dHKaHKa 22

c

2 represents the diffusion coefficient and

dH

dHKHKa c

222 . 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

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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 ( 0N.n , 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

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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 0N.n 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/a

2.

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

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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

82.1

m

pH

H1 (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,

1n

a

0bT

Texpm (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

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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.

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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.

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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 best-

fit ratio of the tunable parameters is

1142.0H*01042.0K

K c (23)

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

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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

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blood flow. The flux in the x-direction is small compared with that in the y and z-

directions.

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. (24)

A 100 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

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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

t

cBH

)R

r(Exp*A

K (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).

21

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 cell-

depletion 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,

R

rExp*H0115.16Exp*0731.0

R

rK tc (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

22

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 non-

Newtonian 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 non-

Newtonian 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, R

rExp*H0115.16Exp*0731.0

R

rK tc . A comparison

between the results obtained for the cell-depletion layer thickness and those

23

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,

24

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 shear-

induced particle migration, Phys. Fluids., vol. 4 (1), (1992), 30-40

[14] D. Quemada, Rheology of concentrated disperse systems: A model for non-

Newtonian 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,

25

Chem. Eng. Comm., Vol. 189(1) (2002), 1-22.

[16] D. Saintillan, E. S. G. Shaqfeh and E. Darve, Effect of flexibility on the shear-

induced 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).

26

[23] I. E. Zarraga, D. T. Leighton, Measurement of an unexpectedly large shear-

induced self diffusivity in a dilute suspension of spheres, Physics of Fluids,

vol.14 (7), (2002), 2194-2201.

27

Vessel Diameter, (μm) Discharge Haematocrit, % Tube Haematocrit, % Mean Velocity,

(mm/s)

40 20 13.5 13

60 20 14.7 8.5

80 20 15.7 7.5

100 20 16.5 8

40 45 35 9.5

60 45 36.7 8

80 45 38.4 5

100 45 39.6 6

Table (1). Mean blood velocity, vessel diameters, discharge haematocrit and tube

haematocrit for all the simulations based on the data in Bagchi [1].

28

Kc Kμ Kc/ Kμ

Power Law model 0.32 0.65 0.49

Cross model 0.33 0.65 0.51

Newtonian model 0.41 0.62 0.66

Table (2). Kc and Kμ values for Newtonian and different non-Newtonian models based

on Lam et. al. [11]

29

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

1

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.00 0.20 0.40 0.60 0.80 1.00

Dimensionless radius

Hae

mat

ocr

it

Computed results

Weert results

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.00 0.20 0.40 0.60 0.80 1.00

Dimensionless radius

Dim

ensi

on

less

vel

oci

ty

Computed results

Weert results

(b)

Fig. 1.

2

0.2

0.23

0.26

0.29

0.32

0.35

0 0.2 0.4 0.6 0.8 1

Dimensionless radius

Co

nce

ntr

atio

n

Numerical Results

Experimental Results

Fig. 2.

3

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Dimensionless radius

Hae

mat

ocr

it

Fig. 3.

4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Dimensionless radius

Vel

oci

ty,

mm

/s

Fig. 4.

5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-50 -40 -30 -20 -10 0 10 20 30 40 50Radius, μm

Hae

mat

ocr

it

Kc=0.2-Kμ=0.5 Kc=0.2-Kμ=0.7 Kc=0.4-Kμ=0.5 Kc=0.4-Kμ=0.7

Fig. 5.

6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-50 -40 -30 -20 -10 0 10 20 30 40 50

Radius, μm

Hae

mat

ocr

it

Kc=0.2-K=μ0.5 Kc=0.2-Kμ=0.7 Kc=0.4-Kμ=0.5

Fig. 6.

7

-3

-2

-1

0

1

2

3

4

5

6

7

-50 -40 -30 -20 -10 0 10 20 30 40 50

* 1

e-5

Radius, m

Flu

x,

kg/

(m2.s

)

(a)

-25

-20

-15

-10

-5

0

5

10

15

20

25

-50 -40 -30 -20 -10 0 10 20 30 40 50

*1

e-5

Radius, m

Flu

x,

kg/

(m2.s

)

Flux due to hydrodynamic interactions Flux due to varying viscosity

(b)

Fig. 7.

8

0.12

0.13

0.14

0.15

0.16

0.17

0.18

-50 -40 -30 -20 -10 0 10 20 30 40 50Radius, μm

Hae

mat

ocr

it

Kc=0.02 Kc=1e-2 Kc=7e-3 Kc=2e-3 Kc=2e-4 Kc=2e-6

Fig. 8.

9

0

0.1

0.2

0.3

0.4

0.5

20 30 40 50 60 70 80 90 100 110

Diameter, μm

/R

Kc=8e-4

Kc=2e-3

Kc=5e-2

Sharan&Popel

Fig. 9.

10

0

0.1

0.2

0.3

0.4

0.5

20 30 40 50 60 70 80 90 100 110Diameter, μm

/R

(a)

0

0.1

0.2

0.3

0.4

0.5

20 30 40 50 60 70 80 90 100 110

Diameter, μm

/R

(b)

0

0.1

0.2

0.3

0.4

0.5

20 30 40 50 60 70 80 90 100 110

Diameter, μm

/R

Sharan&popel A=0.005 A=0.01 A=0.05 Bagchi

(c)

Fig. 10.

11

0

0.1

0.2

0.3

0.4

0.5

20 30 40 50 60 70 80 90 100 110Diameter, μm

/R

(a)

0

0.1

0.2

0.3

0.4

0.5

20 30 40 50 60 70 80 90 100 110

Diameter, μm

/R

(b)

0

0.1

0.2

0.3

0.4

0.5

20 30 40 50 60 70 80 90 100 110

Diameter, μm

/R

Sharan&popel A=0.005 A=0.01 A=0.05 Bagchi

(c) Fig. 11.

12

0

0.1

0.2

0.3

0.4

0.5

20 30 40 50 60 70 80 90 100 110Diameter, μm

/R

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

Fig. 12.

13

0.25

0.27

0.29

0.31

0.33

0.35

0.37

-20 -15 -10 -5 0 5 10 15 20

Radius, μm

Hae

mat

ocr

it

(a)

0.002

0.0022

0.0024

0.0026

0.0028

0.003

0.0032

0.0034

0.0036

0.0038

-20 -15 -10 -5 0 5 10 15 20

Radius, μm

Vis

cosi

ty,

Pa.

s

(b)

0

2

4

6

8

10

12

14

16

18

20

-20 -15 -10 -5 0 5 10 15 20

Radius, μm

Vel

oci

ty,

mm

/s

(c)

Fig. 13.