Particle-Based Simulations of Red Blood Cells - A Review

Abstract

Particle-based methods have been increasingly attractive for solving biofluid flow problems, because of the ease and flexibility in modeling complex structure fluids afforded by the methods. In this review, we focus on popular particle-based methods widely used in red blood cell (RBC) simulations, including dissipative particle dynamics (DPD), smoothed particle hydrodynamics (SPH), and lattice Boltzmann method (LBM). We introduce their basic ideas and formulations, and present their applications in RBC simulations which are divided into three classes according to the number of RBCs in the simulation: a single RBC, two or multiple RBCs, and RBC suspension. Furthermore, we analyze their advantages and disadvantages. On weighing the pros and cons of the methods, a combination of the immersed boundary (IB) method and some forms of smoothed dissipative particle hydrodynamics (SDPD) methods may be required to deal effectively with RBC simulations.

Full text

Review
Particle-based simulations of red blood cells—A review
Ting Ye
a
, Nhan Phan-Thien
b,
n
, Chwee Teck Lim
b,c,d,
nn
a
Department of Computational Mathematics, Jilin University, China
b
Department of Mechanical Engineering, National University of Singapore, Singapore
c
Department of Biomedical Engineering, National University of Singapore, Singapore
d
Mechanobiology Institute, National University of Singapore, Singapore
article info
Article history:
Accepted 7 November 2015
Keywords:
Dissipative particle dynamics
Smoothed particle hydrodynamics
Lattice Boltzmann method
RBC dynamics
RBC rheology
abstract
Particle-based methods have been increasingly attractive for solving biofluid flow problems, because of
the ease and flexibility in modeling complex structure fluids afforded by the methods. In this review, we
focus on popular particle-based methods widely used in red blood cell (RBC) simulations, including
dissipative particle dynamics (DPD), smoothed particle hydrodynamics (SPH), and lattice Boltzmann
method (LBM). We introduce their basic ideas and formulations, and present their applications in RBC
simulations which are divided into three classes according to the number of RBCs in the simulation: a
single RBC, two or multiple RBCs, and RBC suspension. Furthermore, we analyze their advantages and
disadvantages. On weighing the pros and cons of the methods, a combination of the immersed boundary
(IB) method and some forms of smoothed dissipative particle hydrodynamics (SDPD) methods may be
required to deal effectively with RBC simulations.
& 2015 Published by Elsevier Ltd.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Dissipative particle dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1. Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2. Applications in RBC simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3. Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3. Smoothed particle hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1. Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2. Applications in RBC simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3. Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4. Lattice Boltzmann method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.1. Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2. Applications in RBC simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3. Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5. Other particle-based methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6. Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Conflict of interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1. Introduction
Numerical simulation has increasingly become a very impor-
tant tool in solving red blood cells (RBCs) in microchannel flows,
leading to technological innovations in filtering, separation and
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0021-9290/& 2015 Published by Elsevier Ltd.
n
Corresponding author.
nn
Corresponding author at: Department of Mechanical Engineering, National
University of Singapore, Singapore.
E-mail addresses: nhan@nus.edu.sg (N. Phan-Thien),
ctlim@nus.edu.sg (C.T. Lim).
Please cite this article as: Ye, T., et al., Particle-based simulations of red blood cells—A review. Journal of Biomechanics (2015), http://dx.
doi.org/10.1016/j.jbiomech.2015.11.050i
Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

sorting out different cells based solely on their hydrodynamic
behavior in a flow process. There are three broad types of widely-
used numerical methods: mesh-based, particle-based and hybrid
mesh-particle methods. We pay our attention to the particle-based
methods in this review.
A particle-based method in general refers to the class of mesh-
free methods that employ a set of finite number of discrete par-
ticles to represent the state of a flow system and to record the
evolution of the system (i.e., their positions and velocities) (Liu
and Liu, 2003). Compared with mesh-based methods, it is based
on a set of arbitrarily distributed particles rather than a system of
pre-defined meshes, and hence it is attractive in dealing with
problems with complex structures. However, its computational
cost is often higher than that of the mesh-based method due to the
requirement of a large number of particles to guarantee the
simulation accuracy and the resolution required. Considering that
biological systems are generally complex and personal/cluster
computing power has been continually improving, it can be fore-
seen that the particle-based method has a bright prospect in
simulations of biological systems. Particle-based methods can be
divided into microscopic, mesoscopic and macroscopic particle-
based methods according to their length scales. For example,
molecular dynamics (MD), dissipative particle dynamics (DPD) and
smoothed particle hydrodynamics (SPH) are the representative
methods of each class, respectively.
In RBC simulations, the length scale is usually about several to
several hundreds of micrometers. Hence, both the mesoscopic and
macroscopic particle-based methods are applicable to RBC simu-
lations. Past numerical studies on RBCs may be roughly classified
into three groups according to the number of RBCs in the com-
putational domain. The first group focuses on a single RBC, dealing
with a RBC modeling including its deformation, relaxation ( Dao et
al., 2003; Fedosov et al., 2010; Ye et al., 2013), and its dynamic
behaviors in simple flows, such as a simple shear or a tube flow
(Lac et al., 2004; Sui et al., 2008a; Hosseini and Feng, 2009; Ye et
al., 2010 ; Yazdani and Bagchi, 2011; Ye et al., 2014c). In the second
group, two or multiple RBCs are considered to mainly probe their
interactions, including the simulations of their aggregation and
disaggregation (Liu et al., 2004; Bagchi et al., 2005; Zhang et al.,
2008; Li et al., 2014a). In the last group, a large number of RBCs are
considered to study their rheology, including the simulations of
their motion in a shear or tube flow (Pries et al., 1992; Bagchi,
2007; Doddi and Bagchi, 2009; Zhang, 2011a; Xu et al., 2013; Ye et
al., 2014b).
In this review, we mainly focus on the applications of three
particle-based methods used in RBC simulations: DPD (meso-
scopic), SPH (macroscopic) and LBM (mesoscopic).
2. Dissipative particle dynamics
2.1. Theoretical background
Dissipative particle dynamics (DPD) is originally designed as a
mesoscopic stochastic particle-based simulation technique by
Hoogerbrugge and Koelman (1992). It is quickly adopted as a
promising approach for simulating dynamic and rheological
properties of simple and complex fluids, such as multiphase fluids
and cell suspensions. In its framework, the whole computational
domain is discretized into a set of particles. Each particle, called
DPD particle, may be considered as a cluster of physical particles,
and moves according to its Newton's second law. Later on, as
physical quantities (density, linear momentum) constructed from
the state space of DPD particles (their positions and velocities)
satisfy conservation principles (Marsh, 1998), these DPD particles
may be regarded as a fictitious construct (hence a particle-based
method) to solve these conservative equations.
In RBC modeling, different DPD particles are employed to dis-
tinguish different components in the computational domain,
enabling the fluid-structure interactions easier to deal with. Taking
the modeling task of a RBC flowing in a vessel for example, there
are four types of DPD particles, the membrane, the internal fluid,
the suspending fluid and the wall particles for representing the
RBC membrane, its cytoplasm, the suspending fluid and the vessel
wall, respectively, as shown in Fig. 1. All these particles move in
accordance with their Newton's second law of motion (Ye et al.,
2014c),
m
i
d
2
r
i
dt
2
¼
X
j a i;j A P
A
f
DPD
ij
; iA P
F
[ P
I
; ð1Þ
m
i
d
2
r
i
dt
2
¼
X
j a i;j A P
A
f
DPD
ij
þf
Mem
i
þf
Int
i
; iA P
M
; ð2Þ
where m
i
and r
i
are the mass and location of the ith particle; t is
the time; all the particles are collected in a set P
A
, and P
A
¼ P
M
[
P
F
[ P
I
[ P
W
in which P
M
, P
F
, P
I
and P
W
are the collections of
membrane, fl uid, cytoplasm and wall particles; f
DPD
ij
, f
Mem
i
and f
Int
i
are the DPD, membrane deformation and intercellular forces,
respectively. Note that the wall particles are frozen, only providing
interaction to other particles. DPD forces are used to describe the
fluid properties, consisting of conservative, dissipative and random
forces. The conservative force is related to the fluid compressi-
bility; the dissipative force mainly determines the fluid viscosity;
the random force keeps the system temperature (specific kinetic
energy) constant via a fluctuation-dissipation theorem. The
detailed formulations can be found in the work of Español (1988),
or in Marsh (1998). The membrane deformation force is used to
describe the stretching and bending deformation of the RBC.
Currently, there are two types of widely-used RBC models, shell-
Fig. 1. Four types of DPD particles are used to model a RBC flowing a vessel: the
membrane particles (circles), the internal particles (diamonds), the fluid particles
(triangles), and the wall particles (squares).
T. Ye et al. / Journal of Biomechanics ∎ (∎∎∎ ∎) ∎∎∎–∎∎∎2
Please cite this article as: Ye, T., et al., Particle-based simulations of red blood cells—A review. Journal of Biomechanics (2015), http://dx.
doi.org/10.1016/j.jbiomech.2015.11.050i

based and spring-based membrane models (Pozrikidis, 2001; Li et
al., 2005; Fedosov et al., 2010). The former shell-based model
assumes the RBC membrane as a highly deformable shell without
thickness, while the latter spring-based model treats the RBC
membrane as a triangular network connected by elastic or vis-
coelastic springs. The intercellular force is used to describe the
cell–cell interaction. Currently, there are two well-known theore-
tical descriptions: the bridging model (Chien and Jan, 1973; Bagchi
et al., 2005) and the depletion model (Neu and Meiselman, 2002).
Both models suggest that the cells tend to aggregate when the
attractive force between them is larger than the repulsive force,
and the repulsive force is generated by electrostatic repulsion,
membrane shearing and membrane bending. However, the main
difference between them is that the former assumes that the
attractive force results from the bridging of macromolecules, but
the latter believes that it is due to the existence of the polymer
depletion layer between the cells. Mathematically, these two
models are so complex that they are not widely used in numerical
simulations to describe the cell interactions (Chung et al., 2007).
Liu and Liu (2006) proposed a simple model, where a Morse
potential function is employed to fi t the total interaction energy, as
a representation of the depletion model. A detailed review about
deformation and interaction models of RBCs can be found in a
recent paper by Ju et al. (2015).
2.2. Applications in RBC simulations
Most of studies on DPD simulations of RBCs have been done in
Karniadakis's group, including Fedosov (2010), Li et al. (2014b),
Pivkin and Karniadakis (2008), Pan et al. (2010) Recently, Ye et al.
(2013; 2014b; 2014a) have also done some related RBCs
modeling work.
Fig. 2 shows some DPD applications on the simulations of a
single RBC. Most of these studies focus on the RBC modeling
including the stretching and relaxation of RBC, and its dynamic
behaviors in a simple shear or a tube flow. It is well known that a
RBC exhibits viscoelastic behavior (Hochmuth et al., 1979). It
deforms quickly when being pulled diametrically, showing its
elastic characteristics. After being released, it gradually recovers its
original shape, showing its viscoelastic characteristics. Therefore,
many studies have started with simulations of RBC stretching and
relaxation to examine or validate the RBC model, as shown in
Fig. 2(a)–(c). Fedosov et al. (2010) studied the stretching defor-
mation of a healthy or malaria-infected RBC under a coarse-
graining and spectrin levels. Pan et al. (2010) adopted a low-
dimensional model to simulate the stretching deformation of a
healthy RBC. Ye et al. (2013) also did similar studies, and further
simulated the shape relaxation of healthy and malaria-infected
RBC, where the malaria-infected RBC is modeled with an embed-
ded rigid parasite. Once the RBC model is developed and validated,
simulations of a RBC in a shear or tube flow are performed next.
Generally, a RBC in a shear flow has three typical motions: tum-
bling, trembling and tank-treading motions, as shown in Fig. 2(d).
The tumbling motion is an unsteady state, where the RBC flips or
tumbles continuously in the original shape as a rigid body. The
trembling motion is a transitional state, characterized by a shape
variation and an angular oscillation about the RBC orientation. The
tank-treading motion is a steady state, where the RBC has a sta-
tionary confi
guration but its membrane rotates around its internal
fluid like a tank treading. Which of the three motions occurs
mainly depends on the fl ow shear rate and the fluid viscosity (Sui
Fig. 2. DPD simulations of a single RBC, (a) the stretching of RBC using spectrin level model (Fedosov et al., 2010); (b) the stretching of RBC based on a low-dimensional
model (Pan et al., 2010); (c) the shape relaxation to initial biconcave shape of RBC (Ye et al., 2013); (d) the three different modes (tumbling, trembling and tank-treading
motions from top to bottom) of RBC in a shear flow (Ye et al., 2014c); (e) the asymmetric and axisymmetric motions of RBC at different shear rates in a tube flow (Fedosov et
al., 2014c); (f) the axisymmetric and asymmetric motions of RBC at the different time in a tube flow (Ye et al., 2014a); (g) the RBC undergoing a symmetric motion by driving
a body force, and then recovering its initial shape by releasing the body force (Pivkin and Karniadakis, 2008); (h) the RBC flowing a stenotic channel (Li et al., 2014b); (i) the
RBC rolling on a plane subject to the adhesion (Fedosov et al., 2011).
T. Ye et al. / Journal of Biomechanics ∎ (∎∎∎ ∎) ∎∎∎–∎∎∎ 3
Please cite this article as: Ye, T., et al., Particle-based simulations of red blood cells—A review. Journal of Biomechanics (2015), http://dx.
doi.org/10.1016/j.jbiomech.2015.11.050i

et al., 2008a). Fedosov (2010) reproduced the tumbling and tank-
treading modes at the different shear rates. Ye et al. (2014c)
reproduced all the three modes of healthy RBC by varying the fl ow
shear rate and fluid viscosity, and further studied the dynamic
motion of malaria-infected RBC. In a tube flow, however, a RBC has
two typical motions: axisymmetric and asymmetric motions, as
shown in Fig. 2(e)–(g). The axisymmetric motion is characterized
by the RBC deforming into an axisymmetric parachute shape, but
the asymmetric motion by the RBC deforming into a non-
axisymmetric slipper shape. Several factors have been found to
influence the RBC to undergo either an axisymmetric or an
asymmetric motion, and these can include the tube diameter and
flow velocity (Kaoui et al., 2009). In general, the RBC prefers to
experience an axisymmetric motion in a narrow tube than in a
wide tube (with respect to the cell largest dimension). It also
prefers axisymmetric configuration if its initial center is close to
the central axis of tube, or if the flow velocity is large. Pivkin, Pan,
Fedosov, Ye et al. (2008; 2011; 2014c; 2014a) reproduced both the
axisymmetric and asymmetric motions by changing the flow
velocity. In addition, Li et al. (2014b) simulated a single RBC
flowing a stenotic tube, as shown in Fig. 2(h); Fedosov et al. (2011)
simulated a RBC rolling on a plane due to the adhesion, as shown
in Fig. 2(i).
Fig. 3 shows some DPD applications on simulations of two or
multiple RBCs. Most of these studies focus on the aggregation and
disaggregation of cells under no flow condition, or in a shear or
tube flow. Under no flow condition, the RBCs move towards each
other if their distance is larger than an equilibrium separation
called the zero-force separation. Otherwise, they repel each other.
As a result, they finally achieve a steady state with a rouleau
structure, as shown in Fig. 3(a). Ye et al. (2014a) studied the steady
state configuration of two RBCs subject to the different inter-
cellular interaction strengths, and compared their results with
those obtained by a mesh-based method. In a shear or tube flow,
the RBCs may also aggregate to form rouleau, but may dis-
aggregate with increasing shear rate or increasing flow velocity, as
shown in Fig. 3(b) and (c). Fedosov et al. (2014b) studied the
rouleau formation and its disaggregation in a shear flow, and Ye
et al. (2014a)
studied the aggregation and disaggregation of a
healthy and a malaria-infected RBCs in a tube flow. There are not
too many DPD studies on the aggregation and disaggregation of
two or multiple RBCs to date, which may be due to heavy com-
putational cost. Physically, the RBCs are in an equilibrium state at
the zero-force separation of about 13 nm. In order to identify such
small distance, a huge amount of DPD particles are required,
quickly increasing computational cost. However, if the cell–cell
interaction is not considered in the DPD simulations, the RBCs will
collide each other, leading to nonphysical results. Hence, a com-
mon way is to enlarge the zero-force separation to avoid cells
collision, and at the same time roughly describe the phenomenon
of cell aggregation.
Fig. 4 shows some DPD simulations involving a large number of
RBCs. Experimental studies showed that RBCs often arrange
themselves into a single file, when passing through a narrow tube
with a diameter about 10
μm (comparable to a RBC diameter)
(Pries and Secomb, 2003), or a slightly wide tube with a diameter
of about 25
μm but at a high fluid viscosity (Sakai et al., 2009).
Otherwise, the RBCs develop into dual or multiple files and then
transit to a suspension flow (Nagayama and Honda, 2012). Most of
studies involving a large number of RBCs focused on the following
phenomena: (i) Fåhraeus–Lindqvist effect, the dependence of
apparent viscosity on the tube diameter; (ii) Fåhraeus effect, the
dependence of hematocrit on the tube diameter; (iii) existence of a
cell-free layer near the tube wall; and (iv) plug velocity profile of
tube flow. Fedosov et al. (2014b) examined the apparent viscosity
of RBC suspension in a shear flow under different shear rates and
hematocrits, as shown in Fig. 4(a). Fedosov et al. (2014a) and Pan
et al. (2010; 2011) also studied the apparent viscosity of RBC
suspension in a tube flow, analyzed the existence of a cell-free
layer near the tube wall, and demonstrated a plug velocity profile
in tube flow, as shown in Fig. 4(b) and (c). Ye et al. (2014b) also
discussed the rheology of a file of RBCs in a tube flow, and ana-
lyzed the effect of cell spacing on the apparent viscosity of RBC
suspension, as shown in Fig. 4(d). Fedosov (2010) also examined
the dynamic and rheological behaviors of RBCs flowing a channel
with a sudden constriction, and the dynamic and rheological
behaviors of mixed healthy and malaria-infected RBCs (Fedosov et
al., 2014b,a) or mixed RBCs and white blood cells (WBCs) (Fedosov
and Gompper, 2014) in a tube flow, as shown in Fig. 4(e)–(g).
2.3. Advantages and disadvantages
The main advantage of DPD as a particle-based method is the
ease and flexibility in modeling complex structure fluids. The
method satisfies conservation laws, and the relevant constitutive
equation for the complex fl uid can be back-tracked from the state
of the system. These features make the method especially suitable
to study the rheology of complex fluids.
However, DPD has some obvious disadvantages. First, adopting
a set of DPD parameters means adopting the whole rheology of the
fluid – a physical parameter like viscosity cannot be independently
varied keeping other properties constant. Specifying a physical
parameter requires manually tuning in the conservative and dis-
sipative forces. (For a standard DPD fluid (Marsh, 1998), there are
excellent estimates for its viscosity, compressibility, etc.) Second,
there are no clear physical scales (mass, length and time), because
the DPD forces are not derived by a physical model such as Navier–
Stokes equations (Kulkarni et al., 2013). The Navier–Stokes
Fig. 3. DPD simulations of two or multiple RBCs, (a) the steady state of two RBCs under no flow condition ( Ye et al., 2014a); (b) the schematic diagram of RBC interaction in a
shear flow (Fedosov et al., 2014b); (c) the interaction of a healthy and a malaria-infected RBCs in a tube flow (Ye et al., 2014a).
T. Ye et al. / Journal of Biomechanics ∎ (∎∎∎ ∎) ∎∎∎–∎∎∎4
Please cite this article as: Ye, T., et al., Particle-based simulations of red blood cells—A review. Journal of Biomechanics (2015), http://dx.
doi.org/10.1016/j.jbiomech.2015.11.050i

equations only come in at the averaging stage for the mean density
and mean momentum; any scale associated with these equations
may not be directly employed in scaling the equations of motion
for the DPD particles. Finally, in modeling of RBC, its membrane is
modeled as a set of DPD particles with some physical properties to
describe the membrane deformation. This membrane has the
negligible mass compared with the cytoplasm. If a large number of
particles are used to model the membrane, their masses must be
far less than the mass of cytoplasm particle, and thus a very small
time step is required in simulations for stability, leading to an
unaffordable computational cost. If the mass of membrane particle
is set to be that of a cytoplasm particle for computational effi-
ciency, only a few particles are required to model the membrane,
not enough describing the membrane deformation accurately.
3. Smoothed particle hydrodynamics
3.1. Theoretical background
Smoothed particle hydrodynamics (SPH) is a macroscopic
particle-based method proposed by Gingold and Monaghan (1977)
and Lucy (1977), initially for astrophysical problems. Until 1990s, it
was extended to laboratory-scale situations like viscous flow
(Takeda et al., 1994; Watkins et al., 1996) and thermal conduction
(Kum et al., 1995; Cleary and Monaghan, 1999). It has been widely
used in many areas, such as astrophysics, oceanography, and
biology. Different to DPD, SPH starts with the Navier–Stokes
equations, employs a continuous Lagrangian interpolation using a
kernel function (delta sequence) to discretize the whole compu-
tational domain into a set of particles (Violeau, 2012). Each particle
has a spatial distance, called the smoothing length, over which its
physical properties are given by the kernel function.
SPH particle discretization for a RBC also leads to different
types of particles to distinguish different components in compu-
tational domain, in the same manner as in the DPD method, as
shown in Fig. 1. All particles move according to Newton's second
law (Hosseini and Feng, 2009),
m
i
d
2
r
i
dt
2
¼
X
j a i;j A P
A
f
SPH
ij
; iA P
F
[ P
I
; ð3Þ
m
i
d
2
r
i
dt
2
¼
X
j a i;j A P
A
f
SPH
ij
þf
Mem
i
þf
Int
i
; iA P
M
; ð4Þ
where f
SPH
ij
are the SPH forces, and their detailed expressions can
be found in the work of Hosseini and Feng (2009). Note that SPH
formulations (Eqs. (3) and (4)) are similar to DPD formulations
(Eqs. (1) and (2)) in appearance. SPH forces results directly from
discretizing the Navier–Stokes equations (assuming a Newtonian
fluid), and hence it has specific physical parameters like viscosity
with clear physical scales. DPD forces are arbitrarily chosen con-
servative, dissipative and random forces. In SPH, the physical field
quantities (like density, and velocity) are obtained directly after
each solution time step; in DPD, a further average need to be
processed to obtain filed quantities.
3.2. Applications in RBC simulations
There have not been too many SPH simulations on RBCs to date,
most of which have focused on a single RBC deformation, as
shown in Fig. 5. Hosseini and Feng (2012) studied the stretching
deformation of a healthy or malaria-infected RBC, and analyzed
the effect of a parasite on the RBC deformation in detail, as shown
in Fig. 5(a). They pointed out that the presence of a sizeable
parasite greatly reduces the deformability of RBC under stretching.
They also simulated the motion and deformation of a single
Fig. 4. DPD simulations of a large number of RBCs, (a) RBCs suspending in a simple shear fl ow (Fedosov et al., 2014b); (b) RBCs suspending in a tube flow based on a coarse-
graining model (Fedosov et al., 2014a), (c) RBCs suspending in a tube flow based on a low-dimensional model (Pan et al., 2011); (d) a file of RBCs flowing in a tube flow (Ye et
al., 2014b); (e) RBCs flowing a channel with a sudden constriction (Fedosov, 2010); (f) mixed healthy and malaria-infected RBCs in a tube flow (Fedosov et al., 2014b);
(g) mixed RBCs and WBCs in a tube flow (Fedosov and Gompper, 2014).
T. Ye et al. / Journal of Biomechanics ∎ (∎∎∎ ∎) ∎∎∎–∎∎∎ 5
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healthy RBC in a shear and Poiseuille flow, and reproduced the
tank-treading and axisymmetric motions (Hosseini and Feng,
2009), as shown in Fig. 5(b) and (c). They also simulated the
behavior of a single healthy RBC fl owing through a sudden con-
striction (Hosseini and Feng, 2009), as shown in Fig. 5(d). Fol-
lowing the above work, Wu and Feng (2013) conducted 3D
simulations of a healthy and malaria-infected RBC flowing through
a sudden constriction, to find out whether the RBC can go through
this constriction, as shown in Fig. 5(e). Nayanajith et al. (2012,
2013) also performed 2D simulations of a single RBC flowing a
uniform and stenotic microchannel, as shown in Fig. 5(f) and (g).
3.3. Advantages and disadvantages
Compared to DPD, the main advantage of SPH in RBC simula-
tions is that SPH has specific physical parameters (e.g., viscosity)
and clear physical scales embedded in its method, because it is
directly derived from Navier–Stokes equations. Moreover, SPH is
also easy and flexible to model complex structure fluid as a
particle-based method. Implicit in the starting Navier–Stokes
equations are the Newtonian fluid assumption for the fluids
(exterior and interior to the RBC). Extending the method to
accommodate viscoelastic behavior may be a challenge.
In addition, it has been noted that thermal fluctuations may be
important at length scale of the order about 200 nm or smaller
(Freund, 2014). SPH is truly a particle-based discretization of the
Navier–Stokes equations, and the concept of thermal fluctuations
is irrelevant here. In modeling the cell membrane by particles of
the same physical properties to the fluids, SPH suffers from the
same problems to DPD.
4. Lattice Boltzmann method
4.1. Theoretical background
Lattice Boltzmann method (LBM) is proposed by McNamara
and Zanetti (1998) for the purpose of removing statistical noise in
the lattice gas automata (LGA), which can be considered as a
simplified, fictitious version of MD, by replacing the Boolean par-
ticle number with its ensemble average, the so-called density
distribution function. Due to its origination from LGA, it can be
regarded as a mesoscopic method like DPD. In the LBM framework,
Fig. 5. Applications of SPH in RBC simulations, (a) 3 D simulation of RBC stretching (Hosseini and Feng, 2012); (b-c) 2 D simulation of a RBC in a shear and a tube flows
(d) 2 D simulation of (d) 2 D simulation of a RBC flowing the channel with a sudden constriction (Hosseini and Feng, 2009); (e) 3 D simulation of a RBC flowing the channel
with a sudden constriction (Wu and Feng, 2013); simulation of a RBC in a Poiseuille flow (Nayanajith et al., 2012); (g) 2 D simulation of a RBC flowing a stenotic channel
(Nayanajith et al., 2013).
T. Ye et al. / Journal of Biomechanics ∎ (∎∎∎ ∎) ∎∎∎–∎∎∎6
Please cite this article as: Ye, T., et al., Particle-based simulations of red blood cells—A review. Journal of Biomechanics (2015), http://dx.
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the fluid is modeled as a set of fictitious particles, and such par-
ticles undergo consecutive propagations and collision processes
over a discrete lattice mesh. Hence, it is in some sense a hybrid
mesh-particle method as it is not only mesh-based, but also
inherits some aspects of a particle-based method.
Different to DPD and SPH, LBM usually uses the immersed
boundary method (IBM) to handle the fluid-RBC interaction in RBC
simulations, instead of the different types of particles, as shown in
Fig. 6. The RBC membrane is modeled as an immersed boundary
discretized into a set of computational nodes without physical
meaning. The particle motion is governed by the discrete lattice
Boltzmann equation (LBE), and a common LBE with Bhatnagar–
Gross–Krook (BGK) is given by Sui et al. (2008b)
f
i
ðxþ e
i
Δt; tþΔtÞf
i
ðx; tÞ¼
1
τ
f
i
ðx; tÞf
eq
i
ðx; tÞ

þΔtF
i
; ð5Þ
where f
i
ðx; tÞ is the distribution function for particles with velocity
e
i
at position x and time t, Δt is the lattice time interval, f
i
eq
is the
equilibrium distribution function,
τ
is the relaxation time, and F
i
is
the force term represented by the IBM method. Once the particle
density distribution is known, the fl uid density and momentum
are calculated by
ρ ¼
X
i
f
i
; ρu ¼
X
i
e
i
f
i
þ
1
2
f
Δt; ð6Þ
Fig. 6. Schematic illustration of IBM to deal with the fluid-RBC interaction (Dadvand et al., 2014). The velocity U
k
of RBC membrane is calculated by interpolating the velocity
u
i;j
of fluid, and the membrane force F
k
is spread into the fluid domain as a body force f
i;j
.
Fig. 7. LBM simulations of a single RBC, (a) the stretching deformation of RBC (Krüger et al., 2014); (b) the tumbling motion of RBC in a shear flow (Shi et al., 2013a); (c) the
trembling motion of RBC in a shear flow (Sui et al., 2008a); (d) the tank-treading motion of RBC in a shear flow (Sui et al., 20 08b); (e) the axisymmetric motion of RBC in a
Poiseuille flow (Xiong and Zhang, 2010); (f) the lateral motion of RBC in a Poiseuille flow (Shi et al., 2013b); (g) the RBC in a curved channel (Krüger et al., 2014); (h) the RBC
in a T-type channel (Hyakutake and Nagai, 2015); (i-j) the RBC in a stenotic channel (Vahidkhah and Fatouraee, 2012; Xu et al., 2014).
T. Ye et al. / Journal of Biomechanics ∎ (∎∎∎ ∎) ∎∎∎–∎∎∎ 7
Please cite this article as: Ye, T., et al., Particle-based simulations of red blood cells—A review. Journal of Biomechanics (2015), http://dx.
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where f is the body force from the immersed boundary to the flow
field. The detailed formulations of LBM can be found in the work of
Zhang (2011b) and Sui et al. (2008b).
4.2. Applications in RBC simulations
There have been numerous applications of LBM on RBC simu-
lations, far more than the applications of DPD and SPH. However,
they have the similar research scopes. Some typical work among
them has been done by Zhang et al. (2008), Zhang et al. (2009),
Zhang (2011b), Sui et al. (2007), Sui et al. (2008a), Sui et al.
(2008b), Sui et al. (2008c), Shi et al. (2013a), Shi et al. (2013b),
and so on.
Fig. 7 shows some applications of LBM on a single RBC. Some
researchers simulated the stretching deformation of a RBC to
validate their models to numerical methods (Daniel et al., 2012;
Shi et al., 2013a; Krüger et al., 2014), as shown in Fig. 7(a). They
also simulated a RBC in a simple shear flow to reproduce the
tumbling, trembling and tank-treading motions (Sui et al., 2007,
2008a, 2008b, 2008c; Shi et al., 2013a), as shown in Fig. 7 (b)–(d),
and a RBC in a tube flow to undergo its axisymmetric and asym-
metric motions (Xiong and Zhang, 2010; Daniel et al., 2012;
Vahidkhah and Fatouraee, 2012; Shi et al., 2013a; Dadvand et al.,
2014), as shown in Fig. 7(e). Moreover, Shi et al. (2013b) studied
the lateral motion of a RBC in a tube flow, where a RBC is placed
laterally instead of facing to the fluid flow at the initial state, as
shown in Fig. 7(f). Researchers also paid attentions to the simu-
lation of a RBC in complex channels, such as the curved channel
(Krüger et al., 2014), T-type channel (Hyakutake and Nagai, 2015)
and stenotic channel (Vahidkhah and Fatouraee, 2012
; Xu et al.,
2014), as shown in Fig. 7(g)–(j).
Fig. 8 shows some applications of LBM on the simulations of
two or multiple RBCs. Most concentrated on the RBCs aggregation
under no flow condition, the RBCs aggregation and disaggregation
in a shear flow, and the aggregated RBCs flowing some complex
channels. Zhang et al. (2008) studied the RBCs aggregation under
no flow condition and the RBCs disaggregation in a shear flow
based on the Morse potential model, as shown in Fig. 8(a) and (b).
Ju et al. (2013) studied a doublet consisted of two RBCs in a shear
flow, and reported the three different aggregation modes: flat-
contact, sigmoid-contact and relaxed sigmoid-contact modes, as
shown in Fig. 8(c). They also examined the effect of RBC deform-
ability on its aggregation in details, and pointed out that the
deformability difference between two RBCs could significantly
reduce their aggregating tendency, as shown in Fig. 8(d). The
dynamics behaviors of aggregated RBCs flowing complex channels
were also studied, including stenotic (Vahidkhah and Fatouraee,
2012), T-type (Hyakutake and Nagai, 2015) and bifurcating (Shen
and He, 2012) channels, as shown in Fig. 8(e)–(g). There are some
studies on the motion and deformation of RBCs in channels
without considering their interactions (Xiong and Zhang, 2010; Shi
et al., 2013a, 2014). For example, Fig. 8(h) and (i) shows the motion
and deformation of four RBCs in a Poiseuille flow, where the initial
spacing of RBC should be large enough to avoid the RBC collision
during the whole simulation.
Fig. 9 shows applications of LBM on RBC suspensions; most
focused on the rheology of RBC suspension in a Poiseuille flow.
Zhang et al. (2009); Yin and Zhang (2012) conducted 2D
Fig. 8. LBM simulations of two or multiple RBCs, (a) the aggregation of four RBCs under no flow condition (Zhang et al., 2008 ); (b) the disaggregation of a RBC rouleau in a
shear flow (Zhang et al., 2008); (c-d) the aggregation and disaggregation of a RBC doublet in a shear flow (Ju et al., 2013); (e) the aggregated RBCs flowing a stenotic channel
(Vahidkhah and Fatouraee, 2012); (f) the four RBCs flowing a T-type channel (Hyakutake and Nagai, 2015); (g) the several RBCs flowing a bifurcating channel (Shen and He,
2012); (h-i) the four RBCs in a Poiseuille flow without considering RBC interaction (Xiong and Zhang, 2010; Shi et al., 2014).
T. Ye et al. / Journal of Biomechanics ∎ (∎∎∎ ∎) ∎∎∎–∎∎∎8
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simulations of RBCs suspending in a Poiseuille flow, to examine
the cell-free layer development process and the effects of cell
deformability and aggregation on hemodynamic and hemorheo-
logical behaviors, as shown in Fig. 9(a). In order to simulate the
real whole blood, some researchers took platelets (Crow and
Fogelson, 2010; Skorczewski et al., 2013), or even white blood cells
into account (WBCs) (Sun and Munn, 2005), as shown in Fig. 9(b)–
(d). Flows in complex channels such as bifurcating channels were
also considered (Hyakutake et al., 2008; Sun and Munn, 2008;
Melchionna, 2011; Xiong and Zhang, 2012; Yin et al., 2013), as
shown in Fig. 9(e)–(h). However, the majority of past studies have
usually made some assumptions or idealizations to avoid huge
computational cost. For example, 2D simulations were conducted
instead of 3D simulations in Fig. 9(a)–(f) and (h), or RBCs were
modeled as rigid objects instead of deformable ones in Fig. 9
(d) and (f)–(h). Therefore, 3D LBM simulations of suspensions of
deformable RBCs are still lacking so far.
4.3. Advantages and disadvantages
LBM has several advantages over other conventional CFD
methods, and has attracted a lot of interest in computational
physics. Due to its particle-based nature, it is also easy and flexible
to model complex structure fluids, like DPD and SPH. As a meso-
scopic method, it is more straightforward to incorporate micro-
scopic interactions, like DPD. Besides, a main advantage may be
that it can be easily implemented in a massive parallel computing
environment because of its local dynamics. Finally, it deals with
the fluid-RBC interaction by coupling to the immersed boundary
method, instead of modeling RBC membrane as physically con-
nected particles, which is completely different with DPD and SPH.
Therefore, LBM can be considered as a powerful mesoscopic
method for complex fluid systems, especially in dealing with
incorporating microscopic interactions and parallelization to the
algorithm.
Perhaps we can point to two main issues in the LBM. One is that
LBM requires high computational cost, in that not only it models
the fluid as a set of fictitious particles but also because it has an
additional set of meshes. The other stems from its mesh-based
nature that the regular square or cubic mesh is required to con fi ne
the particle movement, and hence it is somewhat complex to treat
irregular boundaries. There are recent effort in addressing irre-
gular boundaries with LBM (Guo and Chang, 2013).
5. Other particle-based methods
Some other particle-based methods have also been applied to
modeling RBCs, such as moving-particle semi-implicit (MPS),
discrete-particle (DP) and multi-particle collision dynamics
(MPCD) methods. The first one is similar in spirit to SPH (Xiang
and Chen, 2015), where a kernel function is also defined to cal-
culate the particle physical properties including density, pressure,
etc. Moreover, the governing equation of particle motion is also
derived from discretizing the Navier
–Stokes equations, and hence
it is a macroscopic method. The second one is similar in to DPD
(Boryczko et al., 2003). It not only has the conservative, dissipative
and random forces like DPD, but also takes into the particle rota-
tion into account. It is derived on the basis of the microscopic
description, and therefore it is a mesoscopic approach with sto-
chastic excitations, like DPD. The last one is somewhat similar to
LBM (Peltomäki and Gompper, 2013), which also consists of two
Fig. 9. LBM simulations of RBC suspension, (a) RBCs in a Poiseuille flow (Zhang et al., 2009); (b-c) RBCs with platelets in a Poiseuille flow (Skorczewski et al., 2013; Crow and
Fogelson, 2010); (d) RBCs and WBCs in a Poiseuille flow (Sun and Munn, 2005); (e-g) RBCs in a bifurcating channel (Yin et al., 2013; Hyakutake et al., 2008; Melchionna,
2011); (h) RBCs and WBCs in a bifurcating channel (Sun and Munn, 2008).
T. Ye et al. / Journal of Biomechanics ∎ (∎∎∎ ∎) ∎∎∎–∎∎∎ 9
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alternating steps: streaming and collision. In the streaming step,
the particles move ballistically without any interactions. In the
collision step, the particles are binned in cells of a cubic lattice, and
undergo stochastic rotations of their relative velocities. The MPCD
method fully incorporates thermal fluctuations and hydrodynamic
interactions, and hence it is a mesoscopic method.
Fig. 10 shows the applications of MPS and DP methods. Tsubota
et al. (2006) applied MPS method to a 2D simulation of a RBC in a
Poiseuille flow, and reproduced the axisymmetric motion of RBC,
as shown in Fig. 10(a). Dzwinel et al. (2003; 2003; 2006) used DP
method to simulate RBCs suspension in different capillaries,
including straight, stenotic, curved and bifurcating capillaries, and
they pointed out that there is a strong tendency to produce RBCs
clusters in capillaries, and the irregularities in geometry influence
both the flow and RBC shapes, considerably increasing the clotting
effect. Noguchi, Gompper et al. (2005; 2005; 2005; 2013) used
MPCD method to simulate the dynamics of a single RBC in shear
and capillary flow, and also studied the sedimentation of a RBC
subject to a gravitation.
6. Summary and outlook
In this review, we mainly focus on three particle-based meth-
ods used in RBC simulations: DPD, SPH and LBM. Compared with
mesh-based methods, particle-based methods are easy to imple-
ment and are flexible in dealing with complex fluids. However, the
computational cost is often high due to the requirement of a large
number of particles to guarantee the simulation accuracy and
resolution. In RBC studies, the flow configurations are usually
complex, such as RBCs flowing in a complex capillary network, or
in bio-chips in a separation process. Therefore, particle-based
methods are attracting increasing attention. Particle-based meth-
ods can be divided into microscopic, mesoscopic and macroscopic
classes. Because the length scale in RBC simulations is about sev-
eral to several hundreds of micrometers, both the mesoscopic and
macroscopic methods may be suitable to RBC simulations.
Although microscopic methods can also be applied to RBCs pro-
blems in principle, their computational cost will be unaffordable
for today typical computer resources.
In order to emphasize the respective benefits and drawbacks of
different particle-based methods, we attempt to provide a com-
parison among DPD, SPH and LBM, as follows:

As a mesoscopic method, both DPD and LBM take into account
thermal fluctuations in RBC simulations. For some problems
where fluctuations are important (such as RBCs aggregation),
they may be more suitable than SPH (a macroscopic method).

As particle-based methods, both DPD and SPH consider a set of
particles only, and hence they involve a lower computational
cost than LBM, because LBM is in some sense a hybrid mesh-
particle method. However, LBM can be easily adapted to parallel
computation because of its local dynamics, which may easily tip
the scale to its favor if computational cost is the only issue.

SPH results from a particle-based discretization of the Navier–
Stokes equations, and LBM is based on the discrete lattice
Boltzmann equation. For these method, the choice of length/
time scales and the fluid physical properties are clear. For DPD, a
change in one of the DPD parameters imply a change in the fluid
rheology, and although there are good estimate for the fluid
physical parameters based on kinetic theory in the high damp-
ing limit, changing a parameter independently of the rest is a
challenge.

One of challenges in RBC simulations is the treatment of fluid-
RBC interactions. DPD and SPH model the RBC membrane as a
set of physical particles, like common fluid particles, and the
fluid-RBC interactions are directly treated as the fluid-fluid
interaction. This treatment is simple, effective but devoid of
Fig. 10. Applications of MPS, DP and MPCD methods in RBC simulations, (a) a RBC in a Poiseuille flow (Tsubota et al., 2006); (b-e) RBC suspension in a straight, stenotic,
curved and bifurcating capillaries (Boryczko et al., 2003), (f) a RBC in shear flow (Noguchi and Gompper, 2005), (g) a RBC in a capillary flow (Noguchi and Gompper, 2005),
(h) the sedimentation of a RBC (Peltomäki and Gompper, 2013).
T. Ye et al. / Journal of Biomechanics ∎ (∎∎∎ ∎) ∎∎∎–∎∎∎10
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doi.org/10.1016/j.jbiomech.2015.11.050i

physical foundation. As we know, the RBC membrane has a
negligible mass compared with that of the cytoplasm. If a large
number of particles are used to model the membrane, their
combined masses should be far less than the mass of cytoplasm,
and thus either a very small time step is required, or using only
a few particles and thus not able to capture the membrane
deformation. However, LBM treats the RBC membrane as an
immersed boundary, and then adopts IBM to deal with the
fluid-RBC interactions. This is more physical approach.

Another challenge in RBC simulations is the treatment of RBC-
RBC interaction. The distance at which these interactions are
important is about 13 nm, the so-called zero-force separation.
Currently, all these three methods have not been able to resolve
such small distance due to the limitation in computational
resources. A common trick is to enlarge the zero-force separa-
tion, in order to avoid cells collision and also roughly describe
the phenomenon of cells aggregation.
On balance, if a mesoscopic particle-based method is to be
recommended, then a combination of the immersed boundary
method (IBM) and a smoothed dissipative particle dynamics
(SDPD) method may be the way forward in particle-based simu-
lation of RBCs flows. SDPD is a hybrid method coupling the
advantages of both SPH and DPD, having specific physical meaning
like SPH and thermal fluctuations like DPD. IBM can handle fluid-
RBC interactions, like LBM. It remains to be seen if new phenom-
ena involving flows of RBCs can be brought to light using particle-
based methods, leading to new innovations and technologies in
health care diagnostics.
Conflict of interest
The authors declare that they have no proprietary, financial,
professional or other personal interest of any nature or kind in any
product, service and/or company that could be construed as
influencing the position presented in the paper.
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