This article was downloaded by: [NUS National University of Singapore]
On: 29 June 2015, At: 18:57
Publisher: Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,
37-41 Mortimer Street, London W1T 3JH, UK
Computer Methods in Biomechanics and Biomedical
Engineering
Publication details, including instructions for authors and subscription information:
http://www.tandfonline.com/loi/gcmb20
A review of numerical methods for red blood cell flow
simulation
a
a
a
a
ab
Meongkeun Ju , Swe Soe Ye , Bumseok Namgung , Seungkwan Cho , Hong Tong Low , Hwa
a
Liang Leo & Sangho Kim
ac
a
Department of Bioengineering, National University of Singapore, Singapore
b
Department of Mechanical Engineering, National University of Singapore, Singapore
c
Department of Surgery, National University of Singapore, Singapore
Published online: 14 Apr 2013.
Click for updates
To cite this article: Meongkeun Ju, Swe Soe Ye, Bumseok Namgung, Seungkwan Cho, Hong Tong Low, Hwa Liang Leo &
Sangho Kim (2015) A review of numerical methods for red blood cell flow simulation, Computer Methods in Biomechanics and
Biomedical Engineering, 18:2, 130-140, DOI: 10.1080/10255842.2013.783574
To link to this article: http://dx.doi.org/10.1080/10255842.2013.783574
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained
in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no
representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the
Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and
are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and
should be independently verified with primary sources of information. Taylor and Francis shall not be liable for
any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever
or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of
the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematic
reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any
form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://
www.tandfonline.com/page/terms-and-conditions
Computer Methods in Biomechanics and Biomedical Engineering, 2015
Vol. 18, No. 2, 130–140, http://dx.doi.org/10.1080/10255842.2013.783574
A review of numerical methods for red blood cell flow simulation
a
Meongkeun Ju , Swe Soe Yea, Bumseok Namgunga, Seungkwan Choa, Hong Tong Lowa,b, Hwa Liang Leoa and
Sangho Kima,c*
a
Department of Bioengineering, National University of Singapore, Singapore; bDepartment of Mechanical Engineering, National
University of Singapore, Singapore; cDepartment of Surgery, National University of Singapore, Singapore
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
(Received 4 September 2012; final version received 5 March 2013)
In this review, we provide an overview of the simulation techniques employed for modelling the flow of red blood cells
(RBCs) in blood plasma. The scope of this review omits the fluid modelling aspect while focusing on other key components
in the RBC –plasma model such as (1) describing the RBC deformation with shell-based and spring-based RBC models, (2)
constitutive models for RBC aggregation based on bridging theory and depletion theory and (3) additional strategies
required for completing the RBC – plasma flow model. These include topics such as modelling fluid – structure interaction
with the immersed boundary method and boundary integral method, and updating the variations in multiphase fluid property
through the employment of index field methods. Lastly, we summarily discuss the current state and aims of RBC modelling
and suggest some research directions for the further development of this field of modelling.
Keywords: RBC aggregation modelling; RBC deformation modelling; haemodynamics; blood rheology
Introduction
Red blood cells (RBCs) are the most important component
of blood for its biological functions and its direct effect on
haemodynamics. The major function of RBCs is to deliver
oxygen, and this gas exchange process occurs mainly in
the microvascular network of arterioles, venules and
capillaries. It has recently been demonstrated that RBCs
can also synthesise nitric oxide (NO) enzymatically just as
endothelial cells do. Exposure of RBCs to physiological
levels of shear stress activates the NO synthases and the
export of NO from the RBCs into the blood vessel, which
may contribute to the regulation of vascular tonus (Ulker
et al. 2009).
RBCs in the human circulatory system constitute
around 35– 50% of the blood volume and are highly
deformable partly due to their circular biconcave shapes.
Geometrically, undeformed RBCs are 6 –8 mm in diameter
on their circular discoid plane and 2 mm in average
thickness on the biconcave radial plane. They have an
average volume of 90 fL and a surface area of about
136 mm2 (Mchedlishvili and Maeda 2001; Popel and
Johnson 2005). As a result of their major constituency in
blood, RBCs affect the blood flow through their
viscoelastic rheological properties (Maeda 1996) and by
their volume fraction. Furthermore, the features of blood
flow and the importance of RBC to RBC interactions in
describing overall blood flow may also vary greatly with
the vessel diameter. In vessels with diameters larger than
200 mm, the size effect of the RBCs in relation to the
vessel diameter can be neglected. In these larger vessels,
*Corresponding author. Email: [email protected]
q 2013 Taylor & Francis
blood can be modelled as a homogeneous non-Newtonian
fluid using a continuum description (Popel and Johnson
2005). However, in vessels with smaller diameters, such as
arterioles, venules and capillaries, the size of the RBCs is
comparable to the vessel’s internal diameter and a twophase description of blood as a suspension of RBCs in
plasma becomes substantial. Therefore, the detailed
quantitative understanding of blood flow in microcirculation requires explicit modelling of RBCs. Such explicit
RBC modelling for microcirculation is necessary for
describing many physiological processes such as the
haemodynamic resistance and its regulation in the
microcirculation, transport of oxygen and nutrients, and
the body’s immunological and inflammatory responses.
A vital parameter for simulating RBC flow in
microcirculation is the mechanostructural characteristics
of the RBC. Over the past decades, many researchers have
attempted to describe the micromechanics of RBCs, and
their studies have generated several mathematical and
numerical models. These models are constructed with
various degrees of physical relevance, idealisation and
sophistication with regard to the cell constitution,
geometrical configuration and membrane properties
(Pozrikidis 2003). Some of these models utilise a
continuum description (Evans and Skalak 1980; Fung
1993; Pozrikidis 2001), whereas others employ discrete
RBC representations at the spectrin molecular level
(Discher et al. 1998; Li et al. 2005) or at the mesoscopic
scale (Dzwinel et al. 2003; Noguchi and Gompper 2005;
Tsubota et al. 2006b; Dupin et al. 2007; Pivkin and
Karniadakis 2008).
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
Computer Methods in Biomechanics and Biomedical Engineering
In addition to the structural properties of RBCs
affecting blood flow in the microcirculation, RBC
aggregation also plays a key role in many important
biological processes. In human blood (and many other
mammalian species likewise), RBCs aggregate at low
shear rates, forming linear or fractal-like structures called
‘rouleaux’ (Baumler et al. 1999; Stoltz et al. 1999; Popel
and Johnson 2005). Many previous studies have revealed
that the aggregation of blood is influenced by parameters
such as the shear rate, haematocrit and the concentration of
macromolecules in the plasma or suspending medium
(Ben Ami et al. 2001; Rampling et al. 2004). From a
pathophysiological understanding, an altered state of RBC
aggregation is commonly observed in patients with
peripheral vascular disease, where, for instance, elevated
aggregation levels are often associated with a higher risk
of cardiovascular disease. Elevated RBC aggregation
levels also occur after myocardial infarction, ischaemic
brain infarcts, in diabetes, and during sepsis. Furthermore,
the RBC aggregation tendency can be affected by the
cell’s deformability which itself can vary in both
physiological and pathological states. The RBC deformability varies between the younger and older RBCs under
physiological conditions and can be significantly changed
under pathological conditions such as parasitic infection,
sepsis, diabetes and genetic defects in the RBCs. In
consideration of the aforementioned observations and
points, it is clear that understanding the mechanics of RBC
aggregation may lead to a more complete understanding of
cardiovascular diseases and the pathology of blood.
Apart from the pathophysiological sensitivities of the
RBC aggregation phenomenon, the general haemodynamic
relevance of RBC aggregation to microcirculatory blood
flow is that it leads to an increased blood viscosity, and
hence an elevated resistance to blood flow. Although the
physiological and pathological importance of RBC
aggregation has been realised and extensive experimental
investigations have been carried out (Chien et al. 1977;
Baumler et al. 1999; Kounov and Petrov 1999; Stoltz et al.
1999; Popel and Johnson 2005; Kim et al. 2006), the
underlying mechanisms of the RBC aggregation are still
subjects of investigation. The RBC aggregation in blood
flow is initiated when the cells are drawn close together by
the hydrodynamic forces governing the flow of plasma. If the
shearing forces by the fluid motion are small, the cells tend
to adhere to one another and form aggregates. Presently,
there exist two theories that describe the mechanism of
aggregation: bridging between cells by cross-linking
molecules (Chien and Jan 1973), and the balance of osmotic
forces generated by the depletion of molecules in the
intercellular space (Neu and Meiselman 2002).
In the bridging model, RBC aggregation is proposed to
occur when the bridging forces due to the adsorption of
macromolecules onto adjacent cell surfaces exceed
disaggregation forces due to electrostatic repulsion,
131
membrane strain and mechanical shearing (Brooks 1973;
Chien and Jan 1973). The depletion model, however,
proposes that RBC cell aggregation occurs as a result of
lower localised protein or polymer concentrations near the
cell surface as compared to the suspending medium
(i.e. relative depletion near the cell surface). This exclusion
of macromolecules near the cell surface produces an
osmotic gradient and thus a depletion interaction (Baumler
et al. 1996). Both the bridging and the depletion models
have specific limitations but are generally very useful
models for describing aggregation phenomena (Armstrong
et al. 1999; Baumler et al. 1999). In general, both models
assume that the attractive interaction between RBC
surfaces occurs when the surfaces are within a close
enough range and repulsive interaction occurs when the
separation distance becomes sufficiently small. The
repulsive interaction represents the steric forces due to the
glycocalyx and electrostatic repulsion from the negative
charges on the pairing RBC surfaces (Liu et al. 2004).
In this study, we have summarised the methods for the
simulation of blood flow in microcirculation. It consists of
three parts: (a) methods for RBC modelling, (b) methods
for aggregation of RBCs and (c) other supplementary
methods required for simulating RBC flow. We hope that
this review proves helpful for researchers who intend to
embark on the numerical analysis of RBC and blood flow
in microcirculation.
RBC deformation model
Shell-based membrane models
One of the representative models based on a continuum
description of the RBC is the model developed by
Pozrikidis (2001). In his approach, the membrane of an
RBC is represented by a highly deformable twodimensional (2D) shell without thickness. During deformation and membrane displacement, the velocity across
the RBC membrane is continuous thereby satisfying the
non-slip condition. However, there exists a jump in the
~ across the membrane which is
interfacial tension DF
presented in the form:
~ ¼ DF n n~ þ DF t~t ¼ 2
DF
dT~
d
¼2
t~t þ q~n ;
dl
dl
ð1Þ
where T~ is the membrane tension. The membrane tension
can be decomposed into the in-plane tension t and
transverse shear tension q as shown in Figure 1, where n~
and ~t are unit vectors taken in the directions normal and
tangential to the membrane surface, respectively.
Pozrikidis’s formulation can be used in conjunction
with any constitutive law or laws that describe the in-plane
tension t and transverse shear tension q. Accordingly,
these various constitutive laws are employed to satisfy the
132
M. Ju et al.
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
Figure 1. Schematic diagram of the shell-based RBC model
showing decomposition of the membrane tension into the
in-plane tension t and transverse shear tension q components.
different requirements of the studied physical phenomenon. In the case of in-plane tension t, the neo-Hookean
model is the most widely utilised membrane constitutive
law because of its simplicity (Bagchi et al. 2005; Bagchi
2007; Zhang et al. 2007, 2008, 2009). Despite being a
simple model, it is however sufficient for taking
deformability into account (Barthes-Biesel et al. 2002).
In this model, the constitutive law is expressed by the
strain energy function (Ramanujan and Pozrikidis 1998) as
follows:
1
1
W ¼ ES I 1 2 logðI 2 þ 1Þ þ log 2 ðI 2 þ 1Þ ;
ð2Þ
6
2
where ES is the shear elastic modulus of the RBC
membrane and I 1 and I 2 are the first and second strain
invariants, given by the relations I 1 ¼ l21 þ l22 2 2 and
I 2 ¼ ðl1 l2 Þ2 2 1. Terms l1 and l2 are the principal
strains. Equation (2) can also be expressed in terms of
principal stretch ratios 11 and 12 as follows:
22
W ¼ ES ð121 þ 122 þ 122
1 þ 12 Þ:
ð3Þ
Tensions t1 and t2 in the principal directions are given by
(Barthes-Biesel et al. 2002)
ES
1
t1 ¼
121 2
ð4Þ
11 12
ð11 12 Þ2
and
t2 ¼
ES
1
122 2
:
11 12
ð11 12 Þ2
ES 3
ð1 2 1Þ;
1 3=2
where t ¼ t1 and 1 ¼ 11 .
q¼
dm d
¼ ½EB {kðlÞ 2 k0 ðlÞ};
dl
dl
ð7Þ
where EB is the bending modulus, kðlÞ the instantaneous
membrane curvature and k0 ðlÞ is the position-dependent
mean curvature of the resting shape. An alternate model
for predicting the bending resistance is presented by
Helfrich’s formulation (Zhongcan and Helfrich 1989)
which is as follows:
DF n n~ ¼ ½EB ð2k þ c0 Þð2k 2 2 2kg 2 c0 kÞ
þ 2EB DLB k~n;
ð8Þ
where k is the mean curvature, kg the Gaussian curvature,
c0 the spontaneous curvature and DLB is the Laplace –
Beltrami operator.
ð5Þ
For a 2D simulation, the 2D RBC is the equivalent of a
three-dimensional (3D) cell subject to stretching in one
direction. This leads to the reduction in stress terms:
t1 – 0, t2 ¼ 0 where the subscript ‘1’ indicates the inplane direction along the membrane and ‘2’ indicates the
out-of-plane direction. The deformation 12 in the out-ofplane direction is not zero but can be expressed in terms of
11 through equation (5) since t2 ¼ 0. Consequently, the
formulation for a 2D cell model is given by
t¼
In addition to elastic deformation, the RBC membrane
has been observed to demonstrate area incompressibility
in experiments (Hochmuth and Mohandas 1972). The
membrane incompressibility can be represented by
employing the modulus of area dilatation. The contribution of the modulus of area dilatation can be achieved
through two general approaches. The first approach
includes the area dilatation term in the strain energy
function as proposed by Evans and Skalak (1980).
However, Eggleton and Popel (1998) have reported that
the Evans – Skalak model requires a large modulus of area
dilatation to achieve membrane incompressibility and this
resulted in numerical instability. The second approach
includes the area dilatation term directly in the stress
calculation as demonstrated by Pozrikidis (2003).
As mentioned earlier, Pozrikidis’s approach requires
explicit modelling of the two principal tensions. In
addition to a constitutive model for the in-plane tension
t (Equations (2) – (6)), a constitutive model is required for
the transverse shear tension q. The transverse tension is
expressed in terms of bending moment m:
ð6Þ
Spring-based membrane models
Apart from the shell-based models employed by
Pozrikidis, the spring-based membrane network model
has been developed by various researchers. An example of
a 3D implementation of the spring network approach is the
worm-like chain (WLC) model. In this method, the RBC
membrane is discretised by a set of vertex points {~xi } to
form a 2D triangulated network as shown in Figure 2. The
deformation of the membrane is governed by the potential
energy of the system which is defined as follows:
Vð{~xi }Þ ¼ V in – plane þ V bending þ V area þ V volume :
ð9Þ
The in-plane elastic energy V inplane represents the energy
from the elastic deformations of the spectrin network. It is
Computer Methods in Biomechanics and Biomedical Engineering
133
of the lipid bilayer and is defined as follows:
V bending ¼
X
kb ½1 2 cosðuj 2 u0 Þ;
ð11Þ
j
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
where kb is the bending constant, uj the instantaneous
angle between two adjacent triangles sharing the common
edge j and u0 is the spontaneous angle. The area and
volume conservation constraints are represented through
the third and fourth potentials V area and V volume . They
account for area incompressibility of the lipid bilayer and
incompressibility of the inner cytosol, respectively. Their
constitutive relations are expressed as follows:
V area ¼
2
X kd ðAj 2 A0 Þ2
ka ðA 2 Atot
0 Þ
þ
;
tot
2A0
2A0
j
V volume ¼
Figure 2. Schematic diagram of the WLC RBC membrane
model with bending response: (a) 2D triangulated network of
springs for modelling the membrane shear response. (b)
Determination of the instantaneous angle between two adjacent
triangles sharing the common edge j for modelling the membrane
bending response.
given by
V in – plane ¼
X kB Tlm ð3x2j 2 2x3j Þ
j
4pð1 2 xj Þ
þ
XC
;
Aa
a
ð10Þ
where lj is the length of the spring j, lm the maximum
spring extension, xj the spring length ratio given by
xj ¼ lj =lm , p the persistence length, kB T the energy unit, C
a constant and Aa is the area of a given element
triangulated from the spring network. The first summation
term is the attractive WLC potential for individual links.
The second summation term is the elastic energy stored in
the RBC membrane (Li et al. 2005), and it represents a
repulsive potential that balances the WLC term to achieve
a non-zero equilibrium spring length for the RBC
membrane at a relaxed state (Fedosov et al. 2010). The
bending energy V bending represents the bending resistance
2
kv ðV 2 V tot
0 Þ
;
tot
2V 0
ð12Þ
ð13Þ
where ka , kd and kv are the global area, local area and
volume constraint coefficients, respectively. Terms A and
V are the total area and volume of RBC, respectively,
tot
whereas terms Atot
0 and V 0 are the specified total area and
volume, respectively. Detailed description of the RBC
model can be found in Fedosov (2010).
This approach originated from Marko and Siggia
(1995) where a simple interpolating model was applied
between ideal chain behaviour at very small chain
extension and divergent-tension behaviour at large
extension. Discher et al. (1998) adopted this model in
their 3D RBC micropipette aspiration simulation and Li
et al. (2005) utilised this model in their optical tweezers
RBC stretching simulation. This method has also been
adopted for the RBC membrane model in the dissipative
particle dynamics (DPD) simulations by Pivkin and
Karniadakis (2008) and Fedosov (2010). The WLC spring
model is general enough and can be used together with
other simulation methods, such as the lattice Boltzmann
and immersed boundary method (IBM).
A different spring-based constitutive model has also
been developed by Secomb et al. (2007) where the RBC
and its mechanical behaviour are represented by a
collection of spring and damper interconnections between
nodes on a 2D RBC. The nodes are located along the
perimeter of the RBC, with an additional internal node
located in the centre of the cell as shown in Figure 3. The
outer line segments (external elements) connecting the
perimeter nodes to their adjacent nodes are viscoelastic,
and are employed to model the RBC membrane’s shear
elasticity, bending elasticity and in-plane viscosity. The
interior line segments (internal elements) of the cell
connecting the perimeter nodes to the internal node are
viscous and are employed to represent the viscous
134
M. Ju et al.
of the enclosed cell area as follows:
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
p int ¼ kp ð1 2 A=Aref Þ;
Figure 3. Schematic diagram of the 2D RBC model employed
by Secomb et al. (2007). White rectangles are viscous elements
that represent internal viscosity of the RBC, whereas grey
rectangles are viscoelastic elements.
resistance to both cytoplasmic flow and out-of-plane
membrane deformation.
The perimeter nodes have coordinates given by ~xi ¼
ðxi ; yi Þ with i ¼ 1 – n, where n is the number of perimeter
nodes used to discretise the RBC membrane boundary. The
interior node coordinates are given as ~x0 ¼ ðx0 ; y0 Þ.
Correspondingly, line elements are spatially described by
their two end-nodes. Quantities associated with an external
element are denoted by i whereby the end-nodes are given
as ðxi ; yi Þ and ðxiþ1 ; yiþ1 Þ. Internal elements are described
by their end-nodes ðx0 ; y0 Þ and ðxi ; yi Þ. Equations (14) –
(16) present the average tension in an external element, the
bending moment acting on an external node and the
average tension in an internal element, respectively, which
are as follows:
li
1 dli
;
2 1 þ mm
li dt
l0
ð14Þ
mi ¼ 2
k b ai
;
l0
ð15Þ
T i ¼ m0m
1 dLi
:
Li dt
ð16Þ
ti ¼ kt
In the formulation, li is defined to be the length of the ith
external element, l0 a reference length, kt the elastic
modulus, mm the viscosity of the external elements, kb the
bending modulus at the perimeter nodes, ai the angle
between two adjacent external elements, Li the length of
the ith internal element and m0m the viscosity of the internal
element. The cell interior is assumed to exert a uniform
pressure on the membrane in order to restrict the variation
ð17Þ
where A is the area of the cell cross section, Aref the
reference area in the undeformed state or initialised area at
the start of the simulation and kp is a modelled constant.
The value of kp should scale appropriately with other
forces in order to retain an approximately constant cell
area.
In Secomb’s model, the employment of a 2D approach
greatly reduced the computational difficulty of the
problem. The inclusion of internal viscous elements
provides an approximate representation of 3D aspects of
cell deformation. Predicted RBC shapes match surprisingly well with the observed shapes despite the limitations
of a 2D model. However, this model is expected to be less
effective in many practical flow situations where 3D
kinematics cannot be overlooked.
Separately, another 2D elastic spring model has been
developed by Tsubota et al. (2006a, 2006b) to describe the
deformable behaviour of the RBCs. Similarly in this
model, the RBC membrane is represented as a series of
spring interconnections between adjacent membrane
nodes. However, unlike the Secomb model, the viscoelastic property of the RBC interior (cytoplasm) is not
described in the RBC model. This is because the
cytoplasm is discretised as part of the fluid domain and
it is solved by the moving particle semi-implicit fluid
solver.
Membrane viscoelasticity and thermal fluctuation
The RBC membrane exhibits viscoelastic behaviour
during deformation and it can affect the RBC flow
dynamics (Mohandas et al. 1980). Puig-De-MoralesMarinkovic et al. (2007) investigated the RBC deformation under oscillatory excitation using optical magnetic
twisting cytometry and concluded that elastic stresses
dominate at low frequency. Conversely, the viscoelastic
behaviour of the RBC membrane becomes apparent at
high frequency (. 30 Hz). Examples of viscoelasticity
implementation for the membrane model can be found in
the works presented by Dao et al. (2006), Secomb et al.
(2007) and Fedosov et al. (2010).
Apart from viscoelasticity, recent developments by
various researchers have also included membrane thermal
fluctuations in their numerical models (Noguchi and
Gompper 2005; Fedosov et al. 2010). The flickering of the
RBC membrane at rest state has been observed by Browicz
(1890) and it has been suggested to be a result of Brownian
motion or thermal agitation by Parpart and Hoffman
(1956). Subsequently, the flickering phenomenon has been
referred to as thermal fluctuation (Brochard and Lennon
1975). Recent experiments conducted by Strey and
Computer Methods in Biomechanics and Biomedical Engineering
Peterson (1995) and Popescu et al. (2007) quantified the
membrane thermal fluctuation by using phase contrast
microscopy and diffraction phase microscopy, respectively. In correspondence to the experiments, results from
Noguchi and Gompper (2005) and Fedosov et al. (2010)
indicate that the membrane thermal fluctuation can alter
the RBC deformation.
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
Other RBC models
Alternatively, other models that describe RBC mechanical
behaviour exist and do not fall under the classification of
either shell-based or spring-based RBC membrane models.
Svetina and Ziherl (2008) used area-difference-elasticity
theory to describe the deformation of RBCs. Although it
does not explicitly include the shear elasticity of the
erythrocyte cytoskeleton, this approach describes the main
features of large-scale cell deformations rather well
(Ziherl and Svetina 2005). Separately, MacMeccan et al.
(2009) have employed the linear finite element method
(FEM) to describe the deformation of RBCs. In their
model, they simulated several hundreds of 3D RBCs in
high volume fraction flows but have not considered the
interactions between RBCs. In addition, the FEM
approach is limited by a significant increase in
computational time when incorporating non-linear
RBC properties associated with large membrane
deformations.
Separately, several researchers have studied large
systems with many RBCs through employing lower
fidelity models. Janoschek et al. (2010) and Melchionna
(2011) have excluded the deformable characteristic of
RBCs by modelling RBCs as rigid bodies suspended in a
fluid medium. Similarly, Pan et al. (2011) reduced the
complexity of their simulated system by coarse-graining
the model used by Fedosov (2010). In these low-fidelity
approaches, the relative apparent viscosity values
predicted were in reasonable agreement with those
reported by Pries et al. (1992).
RBC aggregation model
135
obtain the minimum distance between the cell membrane
segment and its neighbouring cells. If the minimum
distance between the cell segment and its neighbouring
cell segment is less than a threshold length, then this
segment pair is given an opportunity to form bonds.
Assuming that both cells contribute an equal number of
molecules to bond formation, the reaction equation for
bond density nb is given by
›nb
nb 2
2
¼ 2 kþ n 2
2k2 nb ;
ð18Þ
›t
2
where n is the density of the cross-linking molecules on
each cell, and kþ , k2 are the forward and reverse reaction
rate coefficients, respectively. Note that the above equation
is slightly different from that of a receptor – ligand
interaction, but consistent with the models of Zhu
(1991). The reaction rates are computed as proposed by
Dembo et al. (1988)
kts ðl 2 l0 Þ2
kþ ¼ k0þ exp 2
;
2K B T
k2 ¼
k02 exp
if j~xj ¼ l , lt ;
ðkb 2 kts Þðl 2 l0 Þ2
2
;
2K B T
ð19Þ
ð20Þ
where j~xj is the distance between the two pairing segments
of the two cells, lt the threshold distance below which the
bond formation is initiated, l ¼ j~xj and l0 are the stretched
and unstretched bond length, respectively, kb is the spring
constant (force per stretched length), kts is the transition
spring constant, K B is the Boltzmann constant, T is the
absolute temperature and k0þ and k02 are the rate
coefficients in equilibrium. The bonds behave like
stretched springs, and the force per bond is given by
f b ¼ kb ðl 2 l0 Þ:
ð21Þ
The aggregation force ~fm per unit length of the cell
membrane is then obtained by the relation:
~fm ¼ f b nb x~ :
j~xj
ð22Þ
Bridging hypothesis model
As introduced earlier, the first model proposed for
describing the RBC aggregation dynamics is the bridging
hypothesis model. As proponents of the model, Bagchi
et al. (2005) adopted the formalism of a ligand – receptor
dynamics according to the bridging hypotheses theory.
The inter-membrane force ~fm due to molecular cross
linking is governed by reaction equations and their
reaction rates which are functions of the local distance
between the membranes. In the simulation model, the
reaction term for a specific segment of a cell membrane is
computed by visiting and querying all the other cells to
Note that in this formulation, the bond density, rate
coefficients, bond length and the intermolecular force are
not constant and vary along the contact area of the cells.
However, these parameters are not available from
experiments. Therefore, this model is difficult to use in a
complex problem.
Depletion theory models
A representative approach for Depletion Theory Aggregation models has been given by Neu and Meiselman (2002).
136
M. Ju et al.
In their study, they proposed a theoretical model for
depletion-mediated RBC aggregation in polymer solutions.
In their model, the total interaction energy W T per unit
surface between two infinite plane surfaces is measured.
The plane surfaces representing RBC surfaces brought into
close contact by the use of polymers such as dextran or
polyethylene glycol and the interaction energy W T is given
by the sum of the depletion attractive and electrostatic
repulsive energies, with negligible van der Waals
interactions
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
WT ¼ WD þ WE:
ð23Þ
In Equation (23), W D and W E are the depletion interaction
and electrostatic energies per unit surface, respectively.
The depletion energy is modelled in the form:
d
W D ¼ 22P D 2 þ d 2 P ;
2
ð24aÞ
d
2 d þ P , D;
when
2
d
2 d þ P . D;
W D ¼ 0; when
ð24bÞ
2
where P, D, d, d and P are the osmotic pressure term,
depletion thickness, intercellular distance, RBC glycocalyx
thickness (5 nm) and penetration depth, respectively. The
osmotic pressure term P and the depletion thickness D are
functions of the molecular weight of the polymer and of the
polymer concentration Cp . The penetration depth P is a
function of the polymer concentration calculated as
follows:
P ¼ dð1 2 eð2Cp =Cb Þ Þ;
ð25Þ
where Cb is the penetration constant of the polymer in
solution.
The electrostatic energy is expressed in the form:
WE ¼
s2
sinhðkdÞðeðkd2kdÞ 2 e2kd Þ;
d 2 110 k 3
ð26aÞ
when d . 2d;
WE ¼
s2
ð2kd 2 kdÞ 2 ðe2kd þ 1Þsinhðkd 2 kdÞ
d 2 110 k 3
ð26bÞ
2 sinhðkdÞe2kd ; when d , 2d;
where s, 1, 10 and k are the surface charge density of RBC,
the relative permittivity of the solvent, the permittivity of
the vacuum and the Debye –Huckel length, respectively.
The above formulation predicts an optimal polymer
concentration for the interaction energy: the interaction
energy initially increases with an increase in concen-
tration, reaches a maximum and then decreases as the
concentration increases. Furthermore, the interaction
energy gradually increases to reach a maximum and then
decreases to zero as the two surfaces approach. When the
surfaces approach a distance equal to the sum of their
glycocalyx thicknesses, they experience a strong repulsive
force.
Chung et al. (2007) used Derjaguin’s integral
approximation to extend this formulation to surfaces of
arbitrary shapes. The result of this approach shows that the
attractive forces do not vary significantly across the region
in which the interaction energies are effective. This is
because the force is a derivative of energy with respect
to distance and the energy tail is a linear function of
distance.
As mentioned above, the total interaction energy
predicted by the depletion theory model is both a function
of distance between membranes and polymer concentration. Due to the complexity of total interaction energy
computation, Liu et al. (2004) have proposed employing
the Morse-type potential energy function to simplify the
formulation. In their method, the intercellular interaction
energy F is given by
FðrÞ ¼ De exp{2bðr 0 2 rÞ} 2 2exp{bðr 0 2 rÞ} ; ð27Þ
where r is the surface separation, r 0 and De are,
respectively, the zero force separation and surface energy
and b is a scaling factor controlling the interaction decay
behaviour. The total interaction force from such a potential
is its negative derivative, i.e. f ðrÞ ¼ 2›F=›r. The
aggregation strength or interaction force between RBCs
is represented by amount of surface energy De in this
model. However, since no experimental value of surface
energy is available, one must find suitable value of De for
the purpose of the simulation.
Methods for multiphase RBC flow
Fluid – structure interaction scheme
In the multiphase description of the RBC flow, the RBC
membrane and the suspending fluid (plasma) need to be
discretised into separate regions in the computational
mesh. Most commonly, an Eulerian mesh is used for
describing the fluid domain, whereas a Lagrangian mesh is
used to describe the structure domain. The coupling
between the Eulerian and Lagrangian meshes is essential
in an FSI. Pozrikidis (2003) utilised the boundary integral
method to simulate the 3D motion of a liquid capsule
enclosed by a membrane in shear flow. Using the boundary
integral formulation for Stokes flow (Pozrikidis 1992), he
presented that the velocity at a point ~x0 that lies in the cell
Computer Methods in Biomechanics and Biomedical Engineering
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
membrane satisfies the integral equation:
ð
2
1
1
uj ð~x0 Þ ¼
u ð~x0 Þ 2
Df i ð~xÞGij ð~x; ~x0 ÞdSð~xÞ
1þl j
8pm D
ð
1 2 l PV
þ
ui ð~xÞT ijk ð~x; ~x0 Þnk ð~xÞdSð~xÞ ;
ð28Þ
8p D
where D stands for the membrane, u~ 1 is the unperturbed
velocity prevailing in the absence of the cell, m1 and m2
are, respectively, the viscosity of the ambient and cell fluid,
l ¼ m2 =m1 is the viscosity ratio, Gij and T ijk are the Stokes
flow Green’s functions for the velocity and stress
corresponding to the type of flow under consideration, n~
is the unit vector normal to the membrane pointing into the
ambient fluid and PV denotes the principal value. The first
integral on the right-hand side of Equation (26) expresses
the flow due to an interfacial distribution of point forces
known as the single-layer Stokes flow potential, and the
second integral expresses the flow due to an interfacial
distribution of point sources with zero net strength
accompanied by point-force dipoles, known as the doublelayer Stokes flow potential (Pozrikidis 1992).
The strength density of the single-layer potential, D~f,
is the jump in the hydrodynamic traction across the
membrane, defined as
D~f ; ðs~ 1 2 s~ 2 Þ·~n;
ð29Þ
where s~ 1 and s~ 2 are the hydrodynamic stress tensors in
the ambient and interior cell fluid (Pozrikidis 1992). In the
absence of significant membrane inertia, a differential
force balance requires
~
D~f ¼ 2L;
ð30Þ
~ is the membrane load given in terms of the inwhere L
plane and transverse membrane tensions.
Another well-known and popular method for FSI
treatment in blood flow simulations is the IBM. It was
developed by Peskin (1977) to simulate flexible membranes in fluid flows. The membrane – fluid interaction is
accomplished by distributing membrane forces as local
fluid forces and subsequently updating the membrane
configuration according to the new local flow velocity.
The membrane force consists of structural forces
generated in the membrane in response to deformation
and intercellular forces due to membrane – membrane
~ m ð~xm Þ at nodes
interaction. In IBM, the membrane force F
on membrane ~xm induced by membrane deformation is
distributed to the nearby fluid nodes ~xf through a local fluid
~ f ð~xf Þ in the form:
force F
X
~ f ð~xf Þ ¼
~ m ð~xm Þ;
F
Dð~xf 2 ~xm ÞF
ð31Þ
m
where D ~x is a discrete delta function which is chosen
to represent the properties of the Dirac delta function
137
(N’dri et al. 2003). In a 2D lattice system, D ~x is given as
follows:
Dð~xÞ ¼
1 px
py
1 þ cos
1 þ cos
;
2
4h
2h
2h
when jxj # 2h and jyj # 2h;
Dð~xÞ ¼ 0;
ð32Þ
when otherwise;
where h is the lattice unit. The membrane velocity u~m ð~xm Þ
can be updated in a similar way according to the local flow
field:
X
u~m ð~xm Þ ¼
Dð~xf 2 ~xm Þ~uð~xf Þ:
ð33Þ
f
In this formulation, there will be no relative velocity
between the membrane and the local fluid; therefore, the
general no-slip boundary condition is satisfied and no mass
transfer through the membrane can occur.
The IBM was originally applied to Cartesian grids but
has recently been further extended in immersed FEM,
where a Lagrangian finite element solid mesh is described
to be moving on top of an Eulerian finite element mesh.
The utilisation of finite element meshes for the fluid
domain allows for non-uniform meshes with arbitrary
geometries and boundary conditions (Sui et al. 2007;
Zhang et al. 2007), hence enabling better discretisation and
achieving higher accuracy for the simulation model.
Mesoscopic approach
The FSI treatment based on the popular Eulerian –
Lagrangian approach for blood flow simulation often
incurs high computational expense due to a non-trivial
coupling between the structural deformation and fluid flow.
Alternatively, a Lagrangian – Lagrangian-based mesoscopic approach can lead to sufficient accuracy at an
affordable computational cost. Accordingly, several
mesoscopic RBC models have recently been developed.
Most of them employ a similar idea whereby the RBC
cytoskeleton and membrane are represented by a network
of springs in combination with bending rigidity and
constraints for surface area and volume conservation.
Dupin et al. (2007) coupled a discrete RBC to a fluid
modelled by the lattice Boltzmann method. Noguchi and
Gompper (2005) modelled RBCs and vesicles within the
multiparticle collision dynamics framework (Malevanets
and Kapral 1999). Pivkin and Karniadakis (2008)
employed DPD (Hoogerbrugge and Koelman 1992) for a
multiscale RBC model, whereas Fedosov et al. (2010) have
extended this model to accurately capture the viscoelastic
properties of the RBC membrane and to incorporate the
external/internal fluid viscosity contrast – features which
were not taken into account in most of the previous models.
138
M. Ju et al.
Despite the current developments of RBC models, all
of the methods above still suffer from high computational
expense when several thousand RBCs are required to be
modelled. As a consequence, there have been relatively
few mesoscopic simulations (Bagchi 2007; Zhang et al.
2009) of blood flow in large vessels (D ¼ 100 mm);
because of the computational expense even these few
studies are limited to the application of 2D RBC models.
In addition, most literature does not include the
aggregation of RBCs which will significantly affect the
transport behaviour of blood.
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
Fluid property update scheme
The fluid phases separated by the RBC membrane can
have different properties, such as density and viscosity.
The densities of cytoplasm in the RBC interior and plasma
are close enough for the difference to be neglected for low
inertia flow states at low Reynolds numbers. However, the
difference in viscosity cannot be ignored: a healthy RBC
on average contains cytoplasm with a viscosity that is five
times that of the suspending plasma. As the membrane
moves with the fluid flow, the fluid properties (here the
viscosity) in the fluid domain should be adjusted
accordingly. This process seems straightforward; however, it is not trivial. Typically, an index field is introduced
to identify the relative position of a fluid node to the
membrane. Tryggvason et al. (2001) solved the index field
through a Poisson equation over the entire domain at each
time step (the so-called front-tracking method), but the
available information from the explicitly tracked membrane is not fully employed (Udaykumar et al. 1997).
Alternatively, Shyy and coworkers (Udaykumar et al.
1997; Francois and Shyy 2003; N’dri et al. 2003) have
proposed to update the fluid properties directly in relation
to the normal distance d from the membrane surface by
employing the Heaviside function as follows:
HðdÞ ¼ 0:0 when d , 22h;
d
1
pd
HðdÞ ¼ 0:5 1 þ þ sin
2h p
2h
ð34aÞ
ð34bÞ
when 2 2h # d # 2h;
HðdÞ ¼ 1:0
when d . 2h:
ð34cÞ
A varying fluid property a can be related to the index d by
að~xÞ ¼ ain þ ðaex 2 ain ÞHðdð~xÞÞ;
ð35Þ
where the subscripts in and ex indicate interior domain and
exterior domain, respectively. It should be noted that the
originally proposed formulations of the Heaviside function
are coordinate dependent. Zhang et al. (2008) modified
this approach by simply defining an index d of a fluid node
as the shortest distance to the membrane. In comparison to
solving the Poisson equation over the entire fluid domain,
the Heaviside function approach has proven to be
significantly more efficient.
Discussion of numerical modelling
In summary, we have surveyed and presented some of the
methods employed in the simulation of RBC flow.
Modelling components essential to capturing the haemodynamics of the RBC flow are CFD models, RBC
mechanical deformation and aggregation models, fluid and
membrane coupling schemes and a property index
updating scheme for distinguishing the separate phases
in the blood mixture. These components provide a basic
set of tools capable of studying and addressing a breadth of
problems related to haemodynamics.
However, as with any numerical approach, the topic of
computational efficiency and fidelity ought to be
considered. This sensitivity is particularly pertinent to
the study of RBC transport behaviour, where feature sizes
such as RBC diameters are typically on the micrometer
scale. Correspondingly, the scale of discretisation for the
numerical model can be on the order of nanometres in
order to preserve the accuracy and fidelity of the
simulation. This is problematic for studies that are
essentially multiscale in nature, such as the study of a
capillary network or an organ – the modelled domain in its
entirety is several orders larger than the discretisation scale
required to capture reasonably correct flow physics.
Understandably, the computational cost for such studies
will be high. Therefore, strategies such as parallel
computing techniques for large multiscale RBC simulations need to be developed in order to study the many
practical biological systems. From the perspective of
applied models, many studies are trending towards
tackling large multiscale problems. Furthermore, the
popularity of multicore computing provides a huge
potential for more efficient computational algorithms for
solving mathematical RBC models numerically.
Another attention-worthy topic in RBC modelling
relates back to the essential physiological function of
RBCs in blood flow. The gas transport role of discrete
RBCs in blood has not been well investigated in the
existing literature (Beard and Bassingthwaighte 2001;
Chen et al. 2006; Deonikar and Kavdia 2010; Ong et al.
2011). Most gas convection– diffusion transport models
thus far have excluded two potentially key attributes of the
transport phenomena: first, the discrete representation of
advecting RBCs as sources/sinks for gaseous species and
second, the non-homogeneous radial distribution of
gaseous species in blood. Invariably, the underlying
haemodynamics predicted by RBC models constructed
from the techniques discussed in this review can
significantly affect the gas transport physics. For example,
Computer Methods in Biomechanics and Biomedical Engineering
the local wall shear stress can vary significantly with the
dynamics of discrete RBCs and this in turn affects the NO
production rate in the endothelium cells. A gas transport
model coupled with the haemodynamics model may
provide an RBC model with sufficient sophistication to
investigate and perhaps even explain several physiological
mechanisms for gas exchanges in the microcirculation.
Acknowledgements
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
This work was supported by NMRC/CBRG/0019/2012. The
authors wish to thank Dr Junfeng Zhang, Laurentian University,
for discussion on his immersed boundary lattice Boltzmann
approach.
References
Armstrong JK, Meiselman HJ, Fisher TC. 1999. Evidence against
macromolecular “bridging” as the mechanism of red blood
cell aggregation induced by nonionic polymers. Biorheology.
36(5 – 6):433– 437.
Bagchi P. 2007. Mesoscale simulation of blood flow in small
vessels. Biophys J. 92(6):1858– 1877.
Bagchi P, Johnson PC, Popel AS. 2005. Computational fluid
dynamic simulation of aggregation of deformable cells in a
shear flow. J Biomech Eng-Trans ASME. 127(7):1070– 1080.
Barthes-Biesel D, Diaz A, Dhenin E. 2002. Effect of constitutive
laws for two-dimensional membranes on flow-induced
capsule deformation. J Fluid Mech. 460:211 – 222.
Baumler H, Donath E, Krabi A, Knippel W, Budde A,
Kiesewetter H. 1996. Electrophoresis of human red blood
cells and platelets. Evidence for depletion of dextran.
Biorheology. 33(4 – 5):333– 351.
Baumler H, Neu B, Donath E, Kiesewetter H. 1999. Basic
phenomena of red blood cell rouleaux formation. Biorheology. 36(5 – 6):439– 442.
Beard DA, Bassingthwaighte JB. 2001. Modeling advection and
diffusion of oxygen in complex vascular networks. Ann
Biomed Eng. 29(4):298– 310.
Ben Ami R, Barshtein G, Zeltser D, Goldberg Y, Shapira I, Roth A,
Keren G, Miller H, Prochorov V, Eldor A, et al., 2001.
Parameters of red blood cell aggregation as correlates of
the inflammatory state. Am J Physiol-Heart C. 280(5):
H1982–H1988.
Brochard F, Lennon JF. 1975. Frequency spectrum of flicker
phenomenon in erythrocytes. J Phys-Paris. 36(11):
1035– 1047.
Brooks DE. 1973. Effect of neutral polymers on electrokinetic
potential of cells and other charged-particles. IV. Electrostatic effects in dextran-mediated cellular interactions. J
Colloid Interf Sci. 43(3):714– 726.
Browicz T. 1890. Further observation of motion phenomena on
red blood cells in pathological states. Zbl med Wissen.
28:625 – 627.
Chen XW, Jaron D, Barbee KA, Buerk DG. 2006. The influence
of radial RBC distribution, blood velocity profiles, and
glycocalyx on coupled NO/O-2 transport. J Appl Physiol.
100(2):482– 492.
Chien S, Jan KM. 1973. Ultrastructural basis of mechanism of
rouleaux formation. Microvasc Res. 5(2):155– 166.
Chien S, Simchon S, Abbott RE, Jan KM. 1977. Surface
adsorption of dextrans on human red-cell membrane. J
Colloid Interf Sci. 62(3):461– 470.
139
Chung B, Johnson PC, Popel AS. 2007. Application of Chimera
grid to modelling cell motion and aggregation in a narrow
tube. Int J Numer Meth Fl. 53(1):105– 128.
Dao M, Li J, Suresh S. 2006. Molecularly based analysis of
deformation of spectrin network and human erythrocyte. Mat
Sci Eng C-Bio S. 26(8):1232– 1244.
Dembo M, Torney DC, Saxman K, Hammer D. 1988. The
reaction-limited kinetics of membrane-to-surface adhesion
and detachment. P R Soc B-Bio. 234(1274):55– 83.
Deonikar P, Kavdia M. 2010. A computational model for nitric
oxide, nitrite and nitrate biotransport in the microcirculation:
effect of reduced nitric oxide consumption by red blood cells
and blood velocity. Microvasc Res. 80(3):464– 476.
Discher DE, Boal DH, Boey SK. 1998. Simulations of the
erythrocyte cytoskeleton at large deformation. II. Micropipette aspiration. Biophys J. 75(3):1584 –1597.
Dupin MM, Halliday I, Care CM, Alboul L, Munn LL. 2007.
Modeling the flow of dense suspensions of deformable
particles in three dimensions. Phys Rev E. 75(6):066707.
Dzwinel W, Boryczko K, Yuen DA. 2003. A discrete-particle
model of blood dynamics in capillary vessels. J Colloid Interf
Sci. 258(1):163– 173.
Eggleton CD, Popel AS. 1998. Large deformation of red blood
cell ghosts in a simple shear flow. Phys Fluids.
10(8):1834– 1845.
Evans EA, Skalak R. 1980. Mechanics and thermodynamics of
biomembranes. Boca Raton, FL: CRC Press.
Fedosov AE. 2010. Multiscale modeling of blood flow and soft
matter. USA: PhD. Brown.
Fedosov DA, Caswell B, Karniadakis GE. 2010. A multiscale red
blood cell model with accurate mechanics, rheology, and
dynamics. Biophys J. 98(10):2215 –2225.
Francois M, Shyy W. 2003. Computations of drop dynamics with
the immersed boundary method, part 2: drop impact and heat
transfer. Numer Heat Tr B-Fund. 44(2):119– 143.
Fung YC. 1993. Biomechanics: mechanical properties of living
tissues. 2nd ed. New York: Springer-Verlag.
Hochmuth RM, Mohandas N. 1972. Uniaxial loading of red-cell
membrane. J Biomech. 5(5):501 – 509.
Hoogerbrugge PJ, Koelman JMVA. 1992. Simulating microscopic hydrodynamic phenomena with dissipative particle
dynamics. Europhys Lett. 19(3):155– 160.
Janoschek F, Toschi F, Harting J. 2010. Simplified particulate
model for coarse-grained hemodynamics simulations. Phys
Rev E. 82(5):056710.
Kim S, Popel AS, Intaglietta M, Johnson PC. 2006. Effect of
erythrocyte aggregation at normal human levels on
functional capillary density in rat spinotrapezius muscle.
Am J Physiol-Heart C. 290(3):H941 – H947.
Kounov NB, Petrov VG. 1999. Determination of erythrocyte
aggregation. Math Biosci. 157(1 – 2):345– 356.
Li J, Dao M, Lim CT, Suresh S. 2005. Spectrin-level modeling of
the cytoskeleton and optical tweezers stretching of the
erythrocyte. Biophys J. 88(5):3707– 3719.
Liu YL, Zhang L, Wang XD, Liu WK. 2004. Coupling of NavierStokes equations with protein molecular dynamics and its
application to hemodynamics. Int J Numer Meth Fl.
46(12):1237– 1252.
MacMeccan RM, Clausen JR, Neitzel GP, Aidun CK. 2009.
Simulating deformable particle suspensions using a coupled
lattice-Boltzmann and finite-element method. J Fluid Mech.
618:13 – 39.
Maeda N. 1996. Erythrocyte rheology in microcirculation. Jpn J
Physiol. 46(1):1 – 14.
Downloaded by [NUS National University of Singapore] at 18:57 29 June 2015
140
M. Ju et al.
Malevanets A, Kapral R. 1999. Mesoscopic model for solvent
dynamics. J Chem Phys. 110(17):8605– 8613.
Marko JF, Siggia ED. 1995. Stretching DNA. Macromolecules.
28(26):8759– 8770.
Mchedlishvili G, Maeda N. 2001. Blood flow structure related to
red cell flow: a determinant of blood fluidity in narrow
microvessels. Jpn J Physiol. 51(1):19– 30.
Melchionna S. 2011. A model for red blood cells in simulations
of large-scale blood flows. Macromol Theory Simul.
20(7):548– 561.
Mohandas N, Clark MR, Jacobs MS, Shohet SB. 1980. Analysis
of factors regulating erythrocyte deformability. J Clin Invest.
66(3):563– 573.
N’dri NA, Shyy W, Tran-Soy-Tay R. 2003. Computational
modeling of cell adhesion and movement using a continuumkinetics approach. Biophys J. 85(4):2273– 2286.
Neu B, Meiselman HJ. 2002. Depletion-mediated red blood
cell aggregation in polymer solutions. Biophys J. 83(5):
2482– 2490.
Noguchi H, Gompper G. 2005. Shape transitions of fluid vesicles
and red blood cells in capillary flows. P Natl Acad Sci USA.
102(40):14159 –14164.
Ong PK, Jain S, Kim S. 2011. Temporal variations of the cell-free
layer width may enhance NO bioavailability in small
arterioles: effects of erythrocyte aggregation. Microvasc Res.
81(3):303– 312.
Pan WX, Fedosov DA, Caswell B, Karniadakis GE. 2011.
Predicting dynamics and rheology of blood flow: a
comparative study of multiscale and low-dimensional
models of red blood cells. Microvasc Res. 82(2):163– 170.
Parpart AK, Hoffman JF. 1956. Flicker in erythrocytes –
Vibratory movements in the cytoplasm. J Cell Compar Physl.
47(2):295– 303.
Peskin CS. 1977. Numerical-analysis of blood-flow in heart. J
Comput Phys. 25(3):220– 252.
Pivkin IV, Karniadakis GE. 2008. Accurate coarse-grained
modeling of red blood cells. Phys Rev Lett. 101(11):118105.
Popel AS, Johnson PC. 2005. Microcirculation and hemorheology. Ann Rev Fluid Mech. 37:43– 69.
Popescu G, Park Y, Dasari RR, Badizadegan K, Feld MS. 2007.
Coherence properties of red blood cell membrane motions.
Phys Rev E. 76(3):031902.
Pozrikidis C. 1992. Boundary integral and singularity methods
for linearized viscous flow. Cambridge: Cambridge University Press.
Pozrikidis C. 2001. Effect of membrane bending stiffness on the
deformation of capsules in simple shear flow. J Fluid Mech.
440:269 – 291.
Pozrikidis C. 2003. Numerical simulation of the flow-induced
deformation of red blood cells. Ann Biomed Eng.
31(10):1194– 1205.
Pries AR, Neuhaus D, Gaehtgens P. 1992. Blood-viscosity in
tube flow - Dependence on diameter and hematocrit. Am J
Physiol. 263(6):H1770– H1778.
Puig-De-Morales-Marinkovic M, Turner KT, Butler JP, Fredberg
JJ, Suresh S. 2007. Viscoelasticity of the human red blood
cell. Am J Physiol-Cell Ph. 293(2):C597– C605.
Ramanujan S, Pozrikidis C. 1998. Deformation of liquid capsules
enclosed by elastic membranes in simple shear flow: large
deformations and the effect of fluid viscosities. J Fluid Mech.
361:117 – 143.
Rampling MW, Meiselman HJ, Neu B, Baskurt OK. 2004.
Influence of cell-specific factors on red blood cell aggregation. Biorheology. 41(2):91 –112.
Secomb TW, Styp-Rekowska B, Pries AR. 2007. Twodimensional simulation of red blood cell deformation and
lateral migration in microvessels. Ann Biomed Eng.
35(5):755– 765.
Stoltz JF, Singh M, Riha P. 1999. Hemorheology in practice.
Amsterdam: IOS Press.
Strey H, Peterson M. 1995. Measurement of erythrocytemembrane elasticity by flicker eigenmode decomposition.
Biophys J. 69(2):478–488.
Sui Y, Chew YT, Roy P, Low HT. 2007. A hybrid immersedboundary and multi-block lattice Boltzmann method for
simulating fluid and moving-boundaries interactions. Int J
Numer Meth Fl. 53(11):1727– 1754.
Svetina S, Ziherl P. 2008. Morphology of small aggregates of red
blood cells. Bioelectrochemistry. 73(2):84 – 91.
Tryggvason G, Bunner B, Esmaeeli A, Juric D, Al-Rawahi N,
Tauber W, Han J, Nas S, Jan YJ. 2001. A front-tracking
method for the computations of multiphase flow. J Comput
Phys. 169(2):708 –759.
Tsubota K, Wada S, Yamaguchi T. 2006a. Particle method for
computer simulation of red blood cell motion in blood flow.
Comput Meth Prog Bio. 83(2):139 –146.
Tsubota K, Wada S, Yamaguchi T. 2006b. Simulation study on
effects of hematocrit on blood flow properties using particle
method. J Biomech Sci Eng. 1(1):159 – 170.
Udaykumar HS, Kan HC, Shyy W, TranSonTay R. 1997.
Multiphase dynamics in arbitrary geometries on fixed
Cartesian grids. J Comput Phys. 137(2):366– 405.
Ulker P, Sati L, Celik-Ozenci C, Meiselman HJ, Baskurt OK.
2009. Mechanical stimulation of nitric oxide synthesizing
mechanisms in erythrocytes. Biorheology. 46(2):121– 132.
Zhang JF, Johnson PC, Popel AS. 2007. An immersed boundary
lattice Boltzmann approach to simulate deformable liquid
capsules and its application to microscopic blood flows. Phys
Biol. 4(4):285 – 295.
Zhang JF, Johnson PC, Popel AS. 2008. Red blood cell
aggregation and dissociation in shear flows simulated by
lattice Boltzmann method. J Biomech. 41(1):47 – 55.
Zhang JF, Johnson PC, Popel AS. 2009. Effects of erythrocyte
deformability and aggregation on the cell free layer and
apparent viscosity of microscopic blood flows. Microvasc
Res. 77(3):265– 272.
Zhongcan OY, Helfrich W. 1989. Bending energy of vesicle
membranes – General expressions for the 1st, 2nd, and 3rd
variation of the shape energy and applications to spheres and
cylinders. Phys Rev A. 39(10):5280– 5288.
Zhu C. 1991. A thermodynamic and biomechanical theory of
cell-adhesion .1. General formulism. J Theoret Biol.
150(1):27– 50.
Ziherl P, Svetina S. 2005. Nonaxisymmetric phospholipid
vesicles: rackets, boomerangs, and starfish. Europhys Lett.
70(5):690– 696.