nanomaterials
Article
Characterization of Nanoparticle Dispersion in Red
Blood Cell Suspension by the Lattice
Boltzmann-Immersed Boundary Method
Jifu Tan 1 , Wesley Keller 2 , Salman Sohrabi 1 , Jie Yang 3, * and Yaling Liu 1,4, *
1
2
3
4
*
Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA;
[email protected] (J.T.); [email protected] (S.S.)
Department of Civil and Environmental Engineering, Lehigh University, Bethlehem, PA 18015, USA;
[email protected]
School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China
Bioengineering Program, Lehigh University, Bethlehem, PA 18015, USA
Correspondence: [email protected] (J.Y.); [email protected] (Y.L.);
Tel.: +86-28-8760-0797 (J.Y.); +1-610-758-5839 (Y.L.)
Academic Editor: Ming Su
Received: 16 November 2015; Accepted: 25 January 2016; Published: 5 February 2016
Abstract: Nanodrug-carrier delivery in the blood stream is strongly influenced by nanoparticle (NP)
dispersion. This paper presents a numerical study on NP transport and dispersion in red blood
cell (RBC) suspensions under shear and channel flow conditions, utilizing an immersed boundary
fluid-structure interaction model with a lattice Boltzmann fluid solver, an elastic cell membrane
model and a particle motion model driven by both hydrodynamic loading and Brownian dynamics.
The model can capture the multiphase features of the blood flow. Simulations were performed
to obtain an empirical formula to predict NP dispersion rate for a range of shear rates and cell
concentrations. NP dispersion rate predictions from the formula were then compared to observations
from previous experimental and numerical studies. The proposed formula is shown to accurately
predict the NP dispersion rate. The simulation results also confirm previous findings that the NP
dispersion rate is strongly influenced by local disturbances in the flow due to RBC motion and
deformation. The proposed formula provides an efficient method for estimating the NP dispersion
rate in modeling NP transport in large-scale vascular networks without explicit RBC and NP models.
Keywords: lattice Boltzmann method; immersed boundary method; cell suspension; nanoparticle
delivery; dispersion rate
1. Introduction
Accurately predicting drug delivery is a critical task in drug development research and clinical
trials [1,2]. It requires careful consideration of physiological conditions, such as hematocrit level
(the volume ratio between red blood cell and the whole blood) [3,4], vessel geometry and flow
conditions [5–7], drug carrier size and shape [3,8], dissolution rate [9] and external stimuli [10,11].
For small particles in red blood cell (RBC) suspensions, such as nanoparticles (NP) and platelets,
recent studies have demonstrated that local flow field disturbances caused by RBC translation and
deformation can enhance particle dispersion [3,12–16]. The migration of particles in RBC suspensions
under shear has been shown to behave like a random walk process [17,18], with a dispersion rate much
larger than thermal diffusion. Therefore, accurate predictions of NP dispersion in RBC suspensions
must consider fluid-structure interaction between the immersed solid bodies (particles and cells)
and the surrounding fluid. Previously-developed models for predicting NP dispersion in RBC
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www.mdpi.com/journal/nanomaterials
Nanomaterials 2016, 6, 30
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suspensions have relied primarily on empirical data fitting. Aarts et al. experimentally studied
shear-induced platelet diffusivity (D), which was fitted with shear rate (η) as a power law D “ kη n ,
where k is a constant and n is a function of hematocrit [19]. However, the model parameters are
obtained empirically rather than predicted from the underlying physics. Decuzzi et al. extended the
Taylor–Aris theory to calculate an effective NP diffusion rate that considers wall permeability and
blood rheology [20,21]. They also reported about a three-fold increase in dispersion rate of 1 µm
compared to thermal diffusion [16]. Recently, Fedosov’s group systematically studied micro- and
nano-particles in drug delivery, including particle size, shape effect and RBC influence on particle
margination and adhesion probabilities [3]. However, there is no analytical formula or quantitative
rule to directly predict the NP dispersion rate so far, which is much needed in large-scale drug delivery
simulations [20–22].
In order to address the deficiencies in previously-developed models for predicting NP dispersion,
this paper presents a numerical study on NP dispersion in RBC suspensions that considers the effects
of local flow field disturbances due to RBC motion. This study provides insight into the underlying
physics driving NP dispersion in these systems and develops simple, yet effective, formulae for
predicting dispersion rate as a function of characteristic physiological parameters. These simple
predictive formulae will provide an efficient approach for assessing NP dispersion under different
flow conditions and hematocrit level, thereby facilitating practical modeling of NP transport and
distribution in large-scale vascular systems [22].
The remainder of this paper is organized as follows. The fluid-structure interaction model is
introduced in Section 2, including a description of the modeling approach for the fluid environment,
RBCs and NPs, as well as the fluid-structure coupling scheme. Section 3 outlines the model setup and
test parameters for a parametric study on NP dispersion. Simulation results from the parametric study
are presented in Section 4, along with formulae derived from a regression analysis for predicting the
NP dispersion rate. Predictions from the formulae are compared to data reported in the literature.
Finally, conclusions and recommendations from the study are presented in Section 5.
2. Fluid-Structure Interaction Model
The transport of particles in RBC suspensions is governed by hydrodynamic forces and
fluid-structure interaction effects. In order to simulate this two-way coupling between the fluid
environment and the immersed soft matter, a numerical fluid-structure interaction (FSI) code was
developed utilizing an immersed boundary coupling scheme. Previous research has shown that the
immersed boundary method (IBM) is an efficient approach to simulate soft matter and biological
cells [23–30], flapping insect wings [31–33], harmonic oscillation of thin lamina in fluid [34] and other
FSI problems, such as particle settling [35]. The advantage of this approach is two-fold. First, the lattice
Boltzmann method (LBM) is very efficient at modeling fluid flow and ideal for parallel computing [36].
Second, the IBM uses an independent solid mesh moving on top of a fixed fluid mesh, thus removing
the burden of re-meshing in conventional arbitrary Lagrangian–Eulerian methods [37]. In this study,
the fluid was modeled using a lattice Boltzmann scheme, while the RBCs were modeled as elastic
membranes. The fluids inside and outside the cell membrane are modeled with the same viscosity,
following that in other works [38,39]. This approach greatly simplified the computation and without
losing too much generality. It also can capture the multiphase feature of the blood flow. NPs were
treated as rigid point elements with motions governed by both hydrodynamic loading and Brownian
dynamics. Additional details regarding the fluid-structure interaction model are provided in the
following sections.
2.1. Lattice Boltzmann Fluid Model
The lattice Boltzmann method (LBM) is a mesoscale approach to modeling fluid dynamics that
has been used extensively in blood flow modeling [24,38,40]. Reviews of the underlying theory for
the LBM can be found in the literature [41–44]. LBM is usually considered as a second order accurate
method in space and time [45]. The key concept of the LBM is the transport of particles, characterized
Nanomaterials 2016, 6, 30
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Ñ
Ñ
by a local density distribution function f i (x,t) in phase space (x, c i ), where t denotes the time and c i
denotes the lattice velocity. The dynamics of the LBM involves streaming and collision processes.
´
¯
Ñ
f i x ` ∆t c i , t ` ∆t ´ f i px, tq “ looΩ
mooin ` Fi
looooooooooooooooooomooooooooooooooooooon
(1)
Collision
Streaming
where Ωi is the collision operator and Fi is the external force term. The simplest and most
computationally-efficient collision scheme is the Bhatnagar–Gross–Krook (BGK) scheme [46],
Ωi “
1
eq
p f px, tq ´ f i px, tqq
τ i
(2)
eq
where τ is a relaxation constant and f i px, tq is the population distribution at equilibrium, which is
related to the local macroscale fluid velocity and the speed of sound cs .
¨
˛
´Ñ ѯ2
Ñ Ñ
Ñ Ñ
´Ñ ¯
c
¨
u
i
c ¨u
u¨u‹
eq
˚
fi
u , ρ “ wi ρ ˝1 ` i 2 `
´
‚
cs
2cs 2
2cs 4
(3)
where wi is the weight coefficients. An effective fluid viscosity is related to the relaxation time τ and
the speed of sound cs ,
pτ ´ 0.5q
(4)
ν “ c2s pτ ´ 0.5q “
3
and local macroscale density and velocity are calculated as:
ρ px, tq “
ÿ
Ñ
f i px, tq, ρ px, tq u px, tq “
ÿ
Ñ
i f i px, tq c i
`
i
∆t Ñ
f
2
(5)
Disturbances in the flow are introduced through a force density term Fi , which can be expressed
Ñ
Ñ
in terms of an external body force density f and fluid macroscale velocity u ,
1
Fi “ p1 ´
qw
2τ i
˜Ñ Ñ Ñ Ñ ¸
Ñ
ci´ u
c ¨uÑ
` i 4 ci ¨ f
2
cs
cs
(6)
For simplicity, the simulations presented in this paper consider a regular 2D domain with a D2Q9
(2D 9 velocities) fluid lattice, the details of which can be found in [45]. However, it is noted that the
modeling approach is readily adaptable to the 3D case (see Figure S9 in the Supplementary Materials)
and with more complex geometric configurations. Velocity and non-slip wall boundaries were used.
There are several ways to implement velocity boundary conditions, such as Zou/He boundaries [47]
and regularized boundary conditions [48]. The basic idea behind Zou/He boundaries is assuming that
the bounce back rule applies to the non-equilibrium part of the density population. The regularized
boundary condition is to use the moment flux tensor to reconstruct the non-equilibrium part of the
density distribution, thus it is more stable, but requires extra calculation. Interested readers are referred
to [49] for a summary of different boundary conditions in LBM. In this paper, treatment of velocity
boundary conditions followed the recommendations in [47] and is discussed in Section 3.
2.2. Spring Connected Network Cell Membrane Model
Cell models are commonly based on a continuum approach utilizing a strain energy function
to define membrane response [50–52]. Recently, however, a particle-based cell model governed by
molecular dynamics has emerged as a popular alternative due to its simplicity in mathematical
description [53–55]. These two models have been shown to provide consistent predictions [53]. For the
present study, a particle-based RBC model is adopted. The model consists of a network of vertices,
Nanomaterials 2016, 6, 30
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Xi , i P t1...NV u, connected with elastic springs, which are effective in axial extension/contraction and
flexure,
as shown
in6,Figure
1.
Nanomaterials
2016,
30
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8 µm
A
B
C
θ
2 µm
Cell membrane
Stretching
Bending
Figure 1. (A) Spring connected network cell membrane model. Kinematics for local stretching (B) and
Figure
1. (A) Spring connected network cell membrane model. Kinematics for local stretching (B) and
bending (C) modes of response.
bending (C) modes of response.
The potential energy of the membrane is defined as:
The potential energy of the membrane
as:
V { X i } =isVdefined
stretch + V ben
ding
+ V area
(7)
The stretch energy mimics V
the
spectrin`network
and
u“V
tXelastic
V
` Vis given by:
(7)
area
bending
2
1
Vstretchspectrin
= ks network
Lj − Land
j0
(8)
The stretch energy mimics the elastic
is given by:
2 j∈1...N
s
ÿ `
˘2
1
where k s is the stretching constant,
of L
the
L j is
L j0 and L j 0 is the equilibrium spring (8)
Vstretch
“theklength
s
j ´spring
2
i
stretch
(
)
jP1...Ns stretching, Fedosov investigated a wormlike
length. To consider the nonlinear effect of membrane
chain potential [56]. In a separate study, Wida developed an exponential relationship for mechanical
wherestiffness
ks is the
stretching constant, L is the length of the spring and L is the equilibrium spring
related to the bond stretch j ratio ( λ ) [57]. In this paper, the samej0exponential form of the
length.
To
consider
nonlinear effect of membrane stretching, Fedosov investigated a wormlike
spring constant, the
k s = k s 0 exp [ 2( λ − 1) ] , where λ is the bond stretch ratio, has been adopted.
chain potential [56]. In a separate study, Wida developed an exponential relationship for mechanical
The bending energy represents the bending resistance of the lipid bilayer and is defined as:
stiffness related to the bond stretch ratio (λ) [57]. In this paper, the same exponential form of the spring
2
1
constant, k s “ k s0 exp r2pλ ´ 1qs, where
λ is the
stretch
has been adopted.
Vbending
= bond
kb
θ j −θ ratio,
j0
(9)
2
The bending energy represents the bending resistance
of the lipid bilayer and is defined as:
j∈1...Nb
(
)
˘2
where k b is the bending constant and θ 1
andÿθ j 0 `are instantaneous
and equilibrium angles
j
Vbending “
kb
θ j ´ θ j0
between adjacent spring, respectively. In this2model,
we used the angles when cells are at rest as the
jP1...Nb
equilibrium angles θ j 0 ; each angle is also different from the others.
(9)
where k b is
the the
bending
constant
θ j andarea
θ j0 are
and equilibrium
between
When
cell deforms,
the and
cell surface
and instantaneous
volume remain relatively
constant. angles
To maintain
2
adjacent
spring,
respectively.
In
this
model,
we
used
the
angles
when
cells
are
at
rest
the
equilibrium
kas
(
A
−
A
)
0
is
a constant area in the 2D model, an artificial restoring force potential V area = a
2 A0
angles θ j0 ; each angle is also different from the others.
A area
When
the cell
deforms,
cell
surface
volume remain
constant.area
To maintain
introduced
to conserve
thethe
area
where
is theand
instantaneous
area, Arelatively
and
0 is the equilibrium
k a pA ´ A0 q2
k a is a constant.
a constant area in the 2D model, an artificial restoring force potential Varea “
is introduced
The equilibrium shape of cells is followed by the analytical curve from Fung2A
[58],
0 as shown in
to conserve
theThe
area
A is the
area,
A0 and
is the
equilibrium
area andbond
k a is length
a constant.
Figure 1.
cellwhere
membrane
wasinstantaneous
discretized to 52
nodes
angles.
The equilibrium
The
equilibrium
shape
of
cells
is
followed
by
the
analytical
curve
from
Fung
[58],
as
shown in
was set to be equal, but the equilibrium angle varies among all of the angles.
Figure 1. The cell membrane was discretized to 52 nodes and angles. The equilibrium bond length was
Immersed
Boundary
Coupling Scheme
set to2.3.
be equal,
but
the equilibrium
angle varies among all of the angles.
The immersed boundary method (IBM) was selected to model the interaction between the fluid
2.3. Immersed
Boundarysolids
Coupling
Scheme
and the immersed
due to
the algorithm’s efficiency. The IBM was first proposed by Peskin to
study
blood flow
in the heart
[59]. The
fluidwas
is solved
on to
a spatially-fixed
Eulerian grid,
whilethe
thefluid
The
immersed
boundary
method
(IBM)
selected
model the interaction
between
immersed solids are modeled using a moving Lagrangian mesh, which is not constrained to the
and the immersed solids due to the algorithm’s efficiency. The IBM was first proposed by Peskin
geometric layout of the Eulerian fluid grid. Data are exchanged between the two domains through
to study blood flow in the heart [59]. The fluid is solved on a spatially-fixed Eulerian grid, while
nodal interpolation. The coupling scheme enforces velocity continuity at the fluid-structure
the immersed
solids are modeled using a moving Lagrangian mesh, which is not constrained to
boundary and transfers forces from the structure back into the fluid through an effective force
the geometric
layout
of the
Eulerian
fluid grid.
Dataimmersed
are exchanged
between
the two solid
domains
density. This
two-way
coupling
automatically
handles
body contact
and prevents
through
nodal interpolation.
The coupling
schemeforces
enforces
velocity
continuity
fluid-structure
penetration
through the development
of restoring
in the
fluid. The
approach at
hasthe
been
used for
boundary
andoftransfers
forcesinteraction
from theproblems,
structureincluding
back into
fluid through
an [60],
effective
a variety
fluid-structure
thethe
simulation
of jelly fish
bloodforce
flowThis
[4,24,26–30,38]
plateletautomatically
migration [14]. Comprehensive
reviews
of thecontact
IBM andand
its applications
density.
two-wayand
coupling
handles immersed
body
prevents solid
can be through
found in the
[25,61].
It is noticed
the current
approach
works great
softused
penetration
development
of that
restoring
forcescoupling
in the fluid.
The approach
hasfor
been
for a variety of fluid-structure interaction problems, including the simulation of jelly fish [60],
Nanomaterials 2016, 6, 30
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blood flow [4,24,26–30,38] and platelet migration [14]. Comprehensive reviews of the IBM and its
applications can be found in [25,61]. It is noticed that the current coupling approach works great for
soft deformable solids, an alternative coupling approach, e.g., treating a solid as a moving boundary
for the fluid and transferring the force back to the solid, should be used if the solid is relatively rigid,
such as a particulate settling study [44,62–64].
The immersed structure can be viewed as a parametric surface X(s,t), where the force exerted by
the structure on the fluid is interpolated as a source term in the momentum equation using:
ż
fpx, tq “
Fps, tqδpx ´ Xps, tqqds
(10)
where Fps, tq is the structural force at location s at time of t and ds is the discretized length of the
immersed structure. The force is spread to the local fluid nodes through a delta function δprq,
$
&
´ πr ¯¯
1´
1 ` cos
´2 ď r ď 2
δprq “
4
2
% 0
otherwise
(11)
Other delta functions can also be used. Peskin listed a few delta functions in [25].
The structure velocity is interpreted from the local fluid field through:
ż
upX, tq “
upx, tqδpx ´ Xps, tqqdx
(12)
The same delta function Equation (11) is used to interpolate cell node velocities from the local
fluid flow field.
2.4. Nanoparticle Model
NPs were modeled as rigid bodies with motion governed by hydrodynamic forces and Brownian
dynamics [65,66]. Langevin dynamics was used to simulate the motion of particles.
m
du
“ ´ςu ` Fc ` Fr
dt
(13)
where ς is the friction coefficient, Fc is the conservative force and Fr is the random force that satisfies
the fluctuation dissipation theorem.
xFr ptqy “ 0
(14)
@
D
Fr ptqFr pt1 q “ 2 k B Tς δpt ´ t1 q I
(15)
where kB T is the thermal energy, δ(t ´ t1 ) is the Dirac delta function and I is the unit-second order
tensor.
F c ` Fr
ςt
The solution of the above equation gives uptq “
` C expp´ q, where C is a constant.
ς
m
When the time step dt (4.2 ˆ 10´8 s in the present study) is much larger than the relaxation
m
time τR “ (2.2 ˆ 10´9 s), the particle position can be updated with the terminal velocity as
ς
F c ` Fr
uptq “
` u f . A position check was performed to avoid the penetration of particles into
ς
the RBCs, e.g., the random displacement induced by thermal fluctuation will be reversed if it will
jump inside cells at the next time step. In the diffusion coefficient calculation, periodic boundary
conditions were applied to particles when they crossed the channel wall to remove the effect of
geometric confinement.
The flow chart of the algorithm for the program is shown in Figure 2, which is a second
order accurate Runge–Kutta method based primarily on the midpoint rule [25]. The fluid-structure
interaction model was validated for a sphere settling process [67] and a lateral migration process [68]
in a viscous fluid (Figures S1–S5). Validation studies for fluid-structure interaction, cell tumbling and
Nanomaterials 2016, 6, 30
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tank treading motions (Figure S6, parameters listed in Table S1), as well as NP diffusion (Figures S7–S8)
are shown in the Supplementary Materials, which is provided in the supplementary document.
Nanomaterials 2016, 6, 30
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Figure 2. The algorithm flow chart of immersed boundary method.
Figure 2. The algorithm flow chart of immersed boundary method.
3. Model Setup and Parametric Study
3. Model Setup
and Parametric Study
Research has shown that particles in the core region of the vessel migrate toward the cell-free
layer regions,
wherethat
the migration
process
can region
be modeled
diffusion
[17,18].toward
This migration
is layer
Research
has shown
particles in
the core
of theasvessel
migrate
the cell-free
influenced by physical conditions, such as hematocrit level (Ht), cell membrane stiffness (ks), particle
regions, where the migration process can be modeled as diffusion [17,18]. This migration is influenced
size (r), shear rate ( η), fluid viscosity (ν) and cell size (dc). In this study, two parameters (hematocrit
by physical conditions, such as hematocrit level (Ht ), cell membrane stiffness (ks ), particle size (r), shear
level and shear rate) are considered, while the other parameters are kept constant [13]. NP dispersion
rate (η),isfluid
viscosity
(ν)pure
and shear
cell size
two
parameters
(hematocrit
level
and shear
c ). In this
first studied
under
flow(d
conditions
atstudy,
different
shear
rates for a given
hematocrit
level.
rate) areThis
considered,
the was
other
parameters
areNP
kept
constant
NP
dispersion
is first studied
pure shear while
flow setup
used
to explore the
dispersion
rate[13].
when
cells
undergo tumbling
and tank
treading
[69–71].
Then, the study
extended
investigate
NP dispersion
channel
under pure
shear
flow motion
conditions
at different
shearis rates
for to
a given
hematocrit
level.inThis
pure shear
flow
at
different
hematocrit
levels,
which
was
used
to
model
the
blood
flow
in
capillaries
with
flow setup was used to explore the NP dispersion rate when cells undergo tumbling and tankatreading
shear rate between 0 and 500 s−1 [72].
motion physiologically-relevant
[69–71]. Then, the study
is extended to investigate NP dispersion in channel flow at different
In the study, RBCs were assumed to be healthy with typical physical parameters and with a size
hematocrit
levels,
which
was
used
to model the blood flow in capillaries with a physiologically-relevant
of 8 µm. NPs were assumed to be spherical with a typical size of 100 nm. For simplicity, surface
´
1
shear rate
between
0 and 500sos that[72].
charges
were neglected,
NPs did not adhere to other NPs or to the RBCs. It is noted that the
In NP
theconcentration
study, RBCswas
were
assumed
beinhealthy
typicalascertain
physical
parameters
with a size
kept
relativelyto
low
order to with
more readily
the
effect of RBCand
motion
−3 Pa·s. Through dimensional
on
NP
dispersion.
Dynamic
viscosity
of
the
fluid
was
fixed
at
1
×
10
of 8 µm. NPs were assumed to be spherical with a typical size of 100 nm. For simplicity, surface
an empiricalsofunction
between
diffusion
coefficient,
cellthe
sizeRBCs.
and hematocrit
wasthat the
chargesanalysis,
were neglected,
that NPs
did not
adhere
to othershear
NPsrate,
or to
It is noted
D
defined as: was
, where D low
is thein
dispersion
is the cellascertain
size, η isthe
theeffect
shear rate
and motion
=kept
f ( H t )relatively
NP concentration
order torate,
moredcreadily
of RBC
d c2η
´
3
on NP dispersion. Dynamic viscosity of the fluid was fixed at 1 ˆ 10 Pa¨ s. Through dimensional
H is hematocrit. The same dimensionless dispersion rate was also suggested in [15]. This formula
analysis, tan empirical function between diffusion coefficient, shear rate, cell size and hematocrit was
was validated
through simulations presented in later sections.
D
defined as: The
“ case
f pHconsisted
is the dispersion
rate,with
dc isathe
celllength,
size, aη 25-µm
is the width
shearand
ratea and Ht
test
ofD
a rectangular
fluid domain
50-µm
t q, where
2
dc η
lattice grid
size of 0.5 µm, as shown in Figure 3A. Simulation with a finer grid of 0.25 µm showed a
is hematocrit. The same dimensionless dispersion rate was also suggested in [15]. This formula was
similar fluid velocity field. In the shear flow case, the top and bottom surfaces were defined as
validated
through
simulations
in right
later sections.
velocity
boundaries,
whilepresented
the left and
edges of the domain were modeled as periodic
The
test case In
consisted
of aflow
rectangular
fluid domain
a 50-µm
length,
25-µm
width and
boundaries.
the channel
case, a parabolic
velocitywith
profile
was applied
at athe
left inlet
the µm,
right as
outlet
boundary
was modeled
as an openwith
condition.
a latticeboundary,
grid sizeand
of 0.5
shown
in Figure
3A. Simulation
a finerNon-slip
grid ofboundaries
0.25 µm showed
were
defined
along the
upper
Noticethe
thattop
the and
physical
systemsurfaces
has to be were
converted
a similar
fluid
velocity
field.
In and
thelower
shearsurfaces.
flow case,
bottom
defined as
into LB units through reference dimensions, such as length (dx), time (dt) and mass (dm). Here, we
velocity boundaries, while the left and right edges of the domain were modeled as periodic boundaries.
In the channel flow case, a parabolic velocity profile was applied at the left inlet boundary, and the right
outlet boundary was modeled as an open condition. Non-slip boundaries were defined along the upper
and lower surfaces. Notice that the physical system has to be converted into LB units through reference
Nanomaterials 2016, 6, 30
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dimensions, such as length (dx), time (dt) and mass (dm). Here, we select dx = 0.5 µm. The reference
mass dm = ρdx3 = 1.28 ˆ 10´16 kg assuming the blood plasma density is 1025 kg/m3 . Depending on
the Reynolds number, a relaxation time τ is typically recommended between 0.8 and 1.0; see [73,74].
As suggested in [44], a relaxation time τ of 1.0 was used for all simulations for computational efficiency
and larger time steps. The time step dt can be determined through normalizing the fluid viscosity
τ ´ 0.5 Nanomaterials
dt 2016, 6, 30
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“ ν 2 , where ν is the kinematic viscosity of the fluid. Thus, the time step can be calculated
3
dx
8 s.
= 0.5
µm. The
reference mass
dm normalized,
= ρdx3 = 1.28 × 10
kg assuming
the blood
density
as 4.2 ˆselect
10´dx
Other
parameters
can be
as−16well,
since the
basic plasma
reference
values for
3
is
1025
kg/m
.
Depending
on
the
Reynolds
number,
a
relaxation
time
τ
is
typically
recommended
length, time and mass have been selected. As all of the simulations performed here have low Reynolds
between 0.8 and 1.0; see [73,74]. As suggested in [44], a relaxation time τ of 1.0 was used for all
numbers, the compressibility effect of the Mach number can be safely ignored.
simulations for computational efficiency and larger time steps. The time step dt can be determined
RBC
membranes
were
as bi-concave
τ − 0.5
dt curves with the dimensions shown in Figure 1A.
through
normalizing
themodeled
fluid viscosity
= ν 2 , where ν is the kinematic viscosity of the fluid.
3
dx
A single RBC was composed of 52 nodes. The cell parameters were selected based on recommended
parameters
canartificial
be normalized,
well, since kthe= 1 was
Thus, the time
step literature
can be calculated
as 4.2
10−8 s. Other
values reported
in the
[53,75],
as× listed
in Table
1. The
area as
constraint
a
basic reference values for length, time and mass have been selected. As all of the simulations performed
selected so that the area change was within 1%. Periodic boundary conditions (PBC) were applied to
here have low Reynolds numbers, the compressibility effect of the Mach number can be safely ignored.
the left and RBC
rightmembranes
boundaries
of modeled
the fluidasdomain
for curves
both RBCs
and
NPs. Theshown
area ratio
between
were
bi-concave
with the
dimensions
in Figure
1A. RBCs
and theAfluid
domain
defined
the hematocrit
level. were selected based on recommended
single
RBC waswas
composed
of as
52 nodes.
The cell parameters
values reported in the literature [53,75], as listed in Table 1. The artificial area constraint ka = 1 was
Table
1. Cell
membrane
model
parameters.
selected so that the area change
was
within
1%. Periodic
boundary
conditions (PBC) were applied to
the left and right boundaries of the fluid domain for both RBCs and NPs. The area ratio between
RBCs andParameters
the fluid domain was defined
as the
hematocritRecommended
level.
Specified
Value
Range
References
Stretching coefficient ks0
5 µN/m
5~12 µN/m
Table 1. Cell membrane
model parameters.
Bending coefficient kb
8 ˆ 10´19 J
2 ˆ 10´19 to 1 ˆ 10´17 J
Parameters
Stretching coefficient ks0
4. Results and Discussion
Bending coefficient kb
Specified Value
5 µN/m
8 × 10−19 J
Recommended Range
5~12 µN/m
2 × 10−19 to 1 × 10−17 J
[53,75]
[53,75]
References
[53,75]
[53,75]
4.1. NP 4.Dispersion
under Pure Shear Flow
Results and Discussion
The
NP
ratePure
wasShear
studied
4.1.
NPdispersion
Dispersion under
Flow over a range of shear rates for a single layer of three cells.
This setup was designed to eliminate the cell-cell interaction between different layers, so that we can
The NP dispersion rate was studied over a range of shear rates for a single layer of three cells.
focus onThis
thesetup
shear
rate
effect to
oneliminate
NP dispersions.
ratesbetween
rangingdifferent
from 20
to 500
s´1 we
were
was
designed
the cell-cellShear
interaction
layers,
so that
canselected
in orderfocus
to cover
and RBC
tank
treading
of500
thes−1flow
on theboth
shearthe
rateRBC
effecttumbling
on NP dispersions.
Shear
rates
ranging regions
from 20 to
wereregime.
selected For all
in order
to cover both
tumbling
and RBC tank treading
regions
of the
flowat
regime.
For all
shear rates
investigated
inthe
theRBC
study,
the dimensionless
number
ηt was
held
25. Therefore,
the
shear
rates
investigated
in
the
study,
the
dimensionless
number
was
held
at
25.
Therefore,
the
η
t
simulation time is longer for lower shear rate case. Three RBCs and 792 NPs were considered for each
1 and
simulation
time is of
longer
lower shear
rate case.NPs
Threeand
RBCs
and 792
NPs were
considered
each200 s´1 ,
simulation.
Snapshots
the for
interaction
between
RBCs
at shear
rates
of 40 s´for
simulation. Snapshots of the interaction between NPs and RBCs at shear rates of 40 s−1 and 200 s−1,
representative of RBC tumbling and RBC tank treading regions of the flow regime, respectively, are
representative of RBC tumbling and RBC tank treading regions of the flow regime, respectively, are
shown in
Figure
3 (see3 the
theSupplementary
Supplementary
Materials).
shown
in Figure
(see corresponding
the correspondingmovies
movies in
in the
Materials).
B
25 µm
A
50 µm
Figure 3. Interaction between nanoparticle (NP) and red blood cell (RBC) at shear rates of (A) 40 s−1
Figure 3. Interaction between nanoparticle
(NP) and red blood cell (RBC) at shear rates of (A) 40 s´1
(RBC tumbling) and (B) 200 s−1 (RBC tank treading). The bold red lines outline the RBC membranes,
´1
(RBC tumbling)
and (B)
200 sdenote
(RBC
bold red
lines
outline the RBC membranes,
while the green
markers
NPs.tank
Flowtreading).
streamlinesThe
are shown
in the
background.
while the green markers denote NPs. Flow streamlines are shown in the background.
The mean square displacement over the y direction at different shear rates was calculated to
obtain the NP dispersion rates, as shown in Figure 4. It shows that the dispersion rate is strongly
The mean square displacement over the y direction −1at different shear rates was calculated
to
influenced by cell motion. In the RBC tumbling ( η < 40 s ) and RBC tank treading ( η > 200 s−1)
obtain the NP dispersion rates, as shown in Figure 4. It shows that the dispersion rate is strongly
regions of the flow regime, the NP dispersion rate is approximately
linear with the shear rate.
1 ) and RBC tank treading (η > 200 s´1 )
influenced
by cell
the isRBC
tumbling
(η <motion
40 s´transits
Between
40 s−1motion.
and 200 s−1In
, there
a region
where RBC
from tumbling to tank treading
regions of the flow regime, the NP dispersion rate is approximately linear with the shear rate. Between
Nanomaterials 2016, 6, 30
8 of 14
40 s´1 and 200 s´1 , there is a region where RBC motion transits from tumbling to tank treading motion.
In this transition region, there is a drop in NP dispersion with increased shear rate. This shows that
the cell tumbles first and then it deforms to a shape that cannot sustain tumbling motion, then evolves
into the tank treading shape. Thus, the overall average dispersion rate is decreased compared to the
pure tumbling
regime,
which leads to a peak around a shear rate of 40 s´1 . For the range of8 ofshear
rates
Nanomaterials
2016, 6, 30
14
investigated in the study, the dispersion rate initially increases in the tumbling region, then decreases
motion. In this transition region, there is a drop in NP dispersion with increased shear rate. This
in the transition region and increases again with the shear rate in the tank treading region. A linear
shows that the cell tumbles first and then it deforms to a shape that cannot sustain tumbling motion,
regression
was
to fittreading
both theshape.
tumbling
three average
data points
at a low
rate) and tank
thenmodel
evolves
intoused
the tank
Thus, (first
the overall
dispersion
rateshear
is decreased
treadingcompared
data (last
data
pointsregime,
at a high
shear
rate),
to three
the pure
tumbling
which
leads
to a peak around a shear rate of 40 s−1. For the
range of shear rates investigated
in the study, the dispersion rate initially increases in the tumbling
#
12
region, then decreases in 7.8
the ˆ
transition
and
increases
again with the shear rate in the tank
10´14 η region
` 4.7 ˆ
10´
tumbling
´
15
´
12
treading region. A linear 8.5
regression
model
was
used
to
fit
both
tumbling (first three data points
ˆ 10 η ` 4.0 ˆ 10
tankthe
treading
at a low shear rate) and tank treading data (last three data points at a high shear rate),
−12 effect of shear rate on NP dispersion in the
where η is the shear rate. The formulae
the
 7.8indicate
× 10−14η + that
4.7 ×10
tumbling
−15
−12
tumbling region is roughly an order of8.5
magnitude
larger
thantank
thattreading
in the tank treading region. This can
×
+
×
10
η
4.0
10

be attributed to larger RBC motions in the tumbling region, where RBCs undergo rigid-like rotations
where η is the shear rate. The formulae indicate that the effect of shear rate on NP dispersion in the
that trigger larger local flow disturbances that promote the dispersion of adjacent NP away from
tumbling region is roughly an order of magnitude larger than that in the tank treading region. This
the cell. It is also worth noting that the constant terms in the formulae are close to the NP thermal
can be attributed to larger RBC motions in the tumbling region, where RBCs undergo rigid-like
diffusion
coefficient.
The larger
theoretical
diffusion
rate that
for promote
100-nmthe
particles
at aoftemperature
of 300 K is
rotations
that trigger
local flow
disturbances
dispersion
adjacent NP away
´ 12 m2 /s. This observation agrees with the physical requirement that the dispersion
about 4.4
ˆ
10
from the cell. It is also worth noting that the constant terms in the formulae are close to the NP thermal
diffusion
coefficient.
The theoretical
rate for 100-nm
at a temperature
300 K
is
rate should
be close
to thermal
diffusiondiffusion
in the absence
of shearparticles
flow. Therefore,
for a of
given
hematocrit
−12 ´21
4.4η×10
Thisηobservation
with the physical
requirement
that theas:
dispersion rate
about
level, and
with
< 40
sm /s.and
> 200 s´1 ,agrees
the dispersion
rate D
can be written
should be close to thermal diffusion in the absence of shear flow. Therefore, for a given hematocrit
level, and with η < 40 s−1 and η > 200 s−1,Dthe
“dispersion
kη ` D rate D can be written as:
0
D = kη + D0
(16)
(16)
where D0 is the thermal diffusion coefficient and k is a constant that depends on the hematocrit level.
where D0 is the thermal diffusion coefficient and k is a constant that depends on the hematocrit
It is noted
that this formula is readily adaptable to different particle sizes, because the constant term D0
level. It is noted that this formula is readily adaptable to different particle sizes, because the constant
already contains the particle size effect. The contribution of RBC motion is represented in the constant
term D0 already contains the particle size effect. The contribution of RBC motion is represented in the
k. It is noted
that the influence of particle concentration on dispersion rate has been neglected.
constant k . It is noted that the influence of particle concentration on dispersion rate has been neglected.
-12
9
x 10
D (m2/s)
8
7
6
5
4
0
100
200
300
η (s-1)
400
500
Figure 4. NP dispersion rate as a function of shear rate. Error bars indicate the standard variance for
Figure 4.
NP dispersion rate as a function of shear rate. Error bars indicate the standard variance for
three simulations. RBCs undergo tumbling motion at a low shear rate ( η < 40 s−1) and tank treading
three simulations.
RBCs undergo tumbling motion at a low shear rate (η < 40 s´1 ) and tank treading
motion at a high shear rate ( η > 200 s−1). In between, there is a transition region. Linear regression
motion lines
at a for
high
shear rate (η > 200 s´1 ). In between, there is a transition region. Linear regression
the tumbling and tank treading regions are shown, as well.
lines for the tumbling and tank treading regions are shown, as well.
4.2. NP Dispersion under Channel Flow
4.2. NP Dispersion
under
Channel
Flow
It should
be noted
that the
pure shear flow case does not exist in physiological flow. While the
shear case provides an understanding of NP dispersion enhancement under different shear flow
It should be noted that the pure shear flow case does not exist in physiological flow. While the
rates, it is important to simulate NP dispersion in realistic channel flow cases. Figure 5A,B presents
shear case
provides
andispersion
understanding
of NP
dispersion
under
different
shearrate
flow rates,
snapshots
of NP
in a channel
flow
simulationenhancement
with a hematocrit
of 23.5%
and a shear
it is important
toatsimulate
NP
dispersion
in realistic
channel
flow
5A,B
presentsthe
snapshots
of 200 s−1
0.26 s and
0.46
s, respectively.
The channel
width
is cases.
25 µm.Figure
For these
simulations,
specified shear
was measured
as the shear
rateaathematocrit
the wall, unless
noted and
otherwise.
Due
to the
of NP dispersion
in arate
channel
flow simulation
with
of 23.5%
a shear
rate
of 200 s´1
at 0.26 s and 0.46 s, respectively. The channel width is 25 µm. For these simulations, the specified
Nanomaterials 2016, 6, 30
9 of 14
shear rate was measured as the shear rate at the wall, unless noted otherwise. Due to the increased cell
2016,
30 pure shear flow simulations, the number of NPs was reduced to 378.
9 of 14The NPs
volume,Nanomaterials
compared
to6,the
were initially positioned in the core region of the channel. Since the shear rate is linearly changing
increased cell volume, compared to the pure shear flow simulations, the number of NPs was reduced
across the
channel,
RBCs
not positioned
exhibit distinctive
such
as tumbling
tankrate
treading,
as
to 378.
The NPs
weredid
initially
in the coremotions,
region of the
channel.
Since theorshear
is
shown linearly
in
the
previous
pure
shear
flow.
The
higher
hematocrit
and
cell-cell
interaction
also
confined
Nanomaterials
2016, 6, across
30
of 14
changing
the channel, RBCs did not exhibit distinctive motions, such as tumbling or9tank
in theThe
previous
pure of
shear
higher hematocrit
andtreading
cell-cell interaction
also some
the cell treading,
motion as
inshown
the flow.
majority
theflow.
cellsThe
behaved
like a tank
motion, while
increasedthe
cellcell
volume,
to the
pure
shear flow
simulations,
the like
number
of NPs
was reduced
motion
in the flow.
The
majority
the
cells due
behaved
a tank
treading
motion,
cells in confined
the core region
werecompared
bended
or folded
in theofchannel
to
the
symmetry
of
the
velocity near
to
378.
The
NPs
were
initially
positioned
in
the
core
region
of
the
channel.
Since
the
shear rate
is
while
some
cells
in
the
core
region
were
bended
or
folded
in
the
channel
due
to
the
symmetry
of the
the center
line of the channel.
The RBC
motion
under
other hematocrit
levels and shear
rates was
linearly changing
across line
the channel,
RBCs did
not
exhibit
distinctive
motions,
such as tumbling
tank
velocity
near the center
of the channel.
The
RBC
motion
under other
hematocrit
levels andorshear
similar treading,
to Figure
However,
they are
notshear
shown
as 5.
shown
in the previous
pure
flow.here.
The higher hematocrit and cell-cell interaction also
rates was similar to Figure 5. However, they are not shown here.
confined the cell motion in the flow. The majority of the cells behaved like a tank treading motion,
while some cells in the core region were bended or folded in the channel due to the symmetry of the
velocity near the center line of the channel. The RBC motion under other hematocrit levels and shear
rates was similar to Figure 5. However, they are not shown here.
(A)
(B)
Figure 5. Snapshots from a channel flow simulation for a cell-particle mixture with a hematocrit level
Figure of
5. 23.5%
Snapshots
channel
simulation
a cell-particle
mixture with
a hematocrit
level
−1 at 0.26
and a from
shear arate
of 200 sflow
s (A) andfor
0.46
s (B). Fluid streamlines
are shown
in the
of 23.5%
and a shear
rate
200 s´1
at 0.26
s (A) and
0.46nanoparticles.
s (B). FluidFor
streamlines
shown in the
background,
while
theofyellow
markers
represent
100-nm
illustrationare
purposes,
(A)
(B)
background,
while
the yellow
markers
represent
100-nm
illustration
purposes,
cells and
particles
are not shown
to scale.
A short
movie nanoparticles.
of the simulationFor
is provided
in the
Figure
5. Snapshots
fromshown
a channel
simulation
for amovie
cell-particle
mixture
with a hematocrit
level in the
Supplementary
Materials.
cells and
particles
are
not
toflow
scale.
A short
of the
simulation
is provided
of 23.5% and a shear rate of 200 s−1 at 0.26 s (A) and 0.46 s (B). Fluid streamlines are shown in the
Supplementary
Materials.
As
illustrated
in Figure
5A,B,markers
the NPsrepresent
tend to 100-nm
migratenanoparticles.
toward the wall.
In order topurposes,
characterize
background,
while
the yellow
For illustration
cells
and particles
are the
not channel,
shown tothe
scale.
A short
movie
the simulation
in number
the
the NP
distribution
across
channel
width
wasofdivided
into binsisofprovided
1 µm. The
Supplementary
Materials.
Asofillustrated
Figure
5A,B,
theand
NPs
tend to
toward
In order
characterize
NPs
within in
each
bin
was
counted
divided
by migrate
the total number
ofthe
NPswall.
to obtain
the NP to
fraction
each bin.
The NP
fraction
across
channel width
height is
plotted
in Figure
at time
of 0, number
the NP within
distribution
across
the
channel,
thethechannel
was
divided
into6A
bins
of 1points
µm. The
As illustrated
Figure 5A,B, shear
the NPs
to migrate
toward
the wall.
InNP
order
to characterize
−1. Figure
and
0.52 s,bin
forwas
aincharacteristic
ratetend
of 200
presents
the fraction
of NPs 0.26
within
each
counted and divided
bysthe
total 6B
number
ofthe
NPs
tofraction
obtainacross
the NP
the NP distribution
channel,
channel
was
into bins
µm.NP
The
number
−1. 1
channel
height at t =across
0.52 sthe
under
shearthe
rates
of 100width
s−1, 200
s−1divided
, 300 s−1 and
500 sof
The
fractional
within each
The
NPbinfraction
across
channel
is plotted
intoFigure
6ANP
at time points of
of NPsbin.
within
each
wasare
counted
andthe
divided
bysample
theheight
total
number
NPs
obtain
the
values
shown
in
the figure
the average
of three
usingofdifferent
random
seedsfraction
for the
´1runs
0, 0.26 and
0.52
s,
for
a
characteristic
shear
rate
of
200
s
.
Figure
6B
presents
the
NP
fraction
within
each bin.
The NP
fraction
across
channel
height is plotted
in Figure
6A at time
points
of 0, across
NP
Brownian
motion
model.
Figure
6Bthe
clearly
demonstrates
that particle
migration
speed
toward
´
1
´
1
´
1
´
1 . The NP
−1
0.26channel
and
0.52walls
s,atfortincreases
a characteristic
shear
raterates
of 200of
s 100
. Figure
presents
across
the channel
height
=
0.52 swith
under
shear
s 6B
, 200
s , the
300NP
s fraction
and 500
s the
the
shear
rate.
−1, 200 s−1, 300 s−1 and 500 s−1. The NP fractional
channel
height
at
t
=
0.52
s
under
shear
rates
of
100
s
fractional values shown in the figure are the average of three sample runs using different random
0.1 in the figure are the average of three sample runs using different random seeds for the
values shown
-1
seeds for
the
Figure 6B
demonstrates
particle migration
A NP Brownian motion
B clearly
t = 0.52model.
s
0.12
500 sthat
NP Brownian motion model.t =Figure
speed toward
-1
0.26 s 6B clearly demonstrates that particle migration
300 s
speed toward the
0.08channel walls increases with shear rate.
0.1
the channel
walls increases with
rate.
t = 0 shear
s
-1
t = 0.52 s
t = 0.26 s
t=0s
0.04
0.08
0
0.04
5
10
15
20
Channel height(μm)
25
100 s-1
0.12
0.06
500 s-1
NP fraction
0.02
0.06
B
0.08
0.1
0.04
300 s-1
200 s-1
NP fraction
NP fraction
A
NP fraction
200 s
0.06
0.1
0.08
0.02
100 s-1
0.06
0
0.04
5
10
15
20
Channel height(μm)
25
0.02
0.02
Figure 6. The NP fraction across the channel height for a hematocrit
level of 23.5%. (A) NP fraction at
−1
t = 0, 0.26
0 s and 0.52 s for a shear rate of 200 s ; and (B) NP0fraction at t = 0.52 s for shear rates of
5 s−1 and
10 500 15
25
5
10
15
20
25
s−1. μ 20
100 s−1, 200 s−1, 300
Channel height( m)
Channel height(μm)
In
order
to characterize
NP migration
thea dispersion
rate of
was
calculated
from the
Figure
6. The
NP fraction across
the channelspeed,
height for
hematocrit level
23.5%.
(A) NP fraction
at NP
Figure 6. The NP fraction across the channel height
for a hematocrit level of 23.5%. (A) NP fraction
and
(B) NP fraction
at t =levels,
0.52 s and
for shear
rates ofshear
= 0, 0.26displacement.
s and 0.52 s forThe
a shear
rate of 200
s−1;for
meant square
dispersion
rates
different
hematocrit
at various
´1 ; and (B) NP fraction at t = 0.52 s for shear rates of
at t = 0,100
0.26
s
and
0.52
s
for
a
shear
rate
of
200
s
s−1, 200 s−1, 300 s−1 and 500 s−1.
100 s´1 , 200 s´1 , 300 s´1 and 500 s´1 .
In order to characterize NP migration speed, the dispersion rate was calculated from the NP
mean square displacement. The dispersion rates for different hematocrit levels, and at various shear
In order to characterize NP migration speed, the dispersion rate was calculated from the NP mean
square displacement. The dispersion rates for different hematocrit levels, and at various shear rates,
Nanomaterials 2016, 6, 30
10 of 14
Nanomaterials 2016, 6, 30
10 of 14
are shown in Figure 7A. From the pure shear simulation results shown in Equation (14), a modified
rates, are
shown in Figure
7A. developed,
From the pure shear simulation results shown in Equation (14), a
dimensionless
dispersion
rate was
modified dimensionless dispersion rate was developed,
D ´ D0
f pH
q
Dr D“r = D −2 D0 =“f ( H
t) t
ddcc2ηη
(17)
(17)
The dimensionless
dispersion
rate rate
Dr isDplotted
in Figure 7B, where the error bars show standard
The dimensionless
dispersion
r is plotted in Figure 7B, where the error bars show
variancestandard
for threevariance
sample
forruns.
three sample runs.
-11
6
Ht 23.5%
Ht 20.2%
Ht 13.4%
Ht 10.1%
B
x 10
2
1.5
4
2
0
0
2.5
Dr
Dispersion rate (m2/s)
A
-3
x 10
Dimensionless
8
100
200
300
400
Shear (s-1)
500
600
1
0.5
0.1
0.15
0.2
0.25
Hematocrit (Ht)
Figure 7. Dispersion rate of NPs in blood flow. (A) NP dispersion rate at different hematocrit (Ht) and
Figure 7.shear
Dispersion
rate of NPs in blood flow. (A) NP dispersion rate at different hematocrit (Ht) and
rates; (B) relationship between dimensionless dispersion rate (Dr) and hematocrit (Ht). Error
shear rates;
(B)
relationship
dimensionless
dispersion rate (Dr ) and hematocrit (Ht ). Error bars
bars show the standardbetween
variance from
three samples.
show the standard variance from three samples.
Figure 7 shows that the lateral dispersion of NPs (i.e., migration of NPs toward the vessel walls)
is much larger than what is predicted by thermal diffusion alone. This migration is influenced by
Figure
7 shows that the lateral dispersion of NPs (i.e., migration of NPs toward the vessel walls) is
both the hematocrit level and the shear rate. While the relationship between dispersion rate and shear
much larger
what is predicted
by 7A),
thermal
diffusion between
alone. This
migration
is influenced
rate isthan
approximately
linear (Figure
the relationship
dispersion
and hematocrit
is not by both
the hematocrit
level
and 7B).
the Nevertheless,
shear rate. While
the
between
dispersion
andas:
shear rate
fully linear
(Figure
a best fit
linerelationship
with reasonable
approximation
can berate
written
is approximately linear (Figure 7A), the
between dispersion and hematocrit is not fully
D −relationship
D0
Dr =
= 4.643 × 10 −3 H t +5.834 × 10 −4
(18)
2
linear (Figure 7B). Nevertheless, a best fit
d c ηline with reasonable approximation can be written as:
Particle dispersion rate predictions
(18) were also compared to dispersion rates
D ´ D0 from Equation
´4
“ 4.643
ˆ1-µm
10´3particles
Ht ` 5.834
ˆis10
(18)
Dr “
published in the literature
for platelets
[15,76]
and
[13]. It
noted that the comparison
2
dc η
of Equation (18) with platelets and microparticles was considered due to the lack of the NP dispersion
rate in the literature. The predictions and measured dispersion rates are summarized in Table 2. The
Particle
dispersion rate predictions from Equation (18) were also compared to dispersion rates
dimensionless dispersion rate is also plotted in Figure 8. As shown in both the table and the figure,
published
in the literature for platelets [15,76] and 1-µm particles [13]. It is noted that the comparison
the dispersion rate predictions using Equation (18) are in good agreement with measured rates
of Equation
(18) in
with
microparticles
considered
to thedispersion
lack of the
NP
dispersion
reported
the platelets
literature. and
Discrepancies
betweenwas
the predicted
and due
measured
rates
may
be due
to the linear
correlation
assumption
for hematocrit
and/or the effect
particle
concentration,
rate in the
literature.
The
predictions
and measured
dispersion
ratesofare
summarized
in Table 2.
which
was
very
low
for
this
study
and
was
assumed
to
have
a
negligible
effect
on
NP
dispersion.
The dimensionless dispersion rate is also plotted in Figure 8. As shown in both the table and the
Another possible reason is that platelets generally have on average an aspect ratio of 1.4, and this may be
figure, the dispersion rate predictions using Equation (18) are in good agreement with measured rates
a confounding factor and contribute to the differences [77,78]. Nevertheless, the order of magnitude of the
reporteddispersion
in the literature.
betweenare
thecorrectly
predicted
andby
measured
dispersion rates may
rate, as wellDiscrepancies
as its trend with hematocrit
predicted
Equation (18).
be due to the linear correlation assumption for hematocrit and/or the effect of particle concentration,
Tablelow
2. Comparison
particleand
dispersion
predictions
Equation
(18) with dispersion
which was very
for thisofstudy
was rate
assumed
to from
have
a negligible
effect onrates
NP dispersion.
reported in the literature.
Another possible reason is that platelets generally have on average an aspect ratio of 1.4, and this may
Htfactor and
Shear
(s−1) Dispersion
Rate (cm2/s)[77,78].
Prediction
(cm2/s) theReference
be a confounding
contribute
to the differences
Nevertheless,
order of magnitude
−6
[0.2, rate,
0.4] as well400
0.68]
× 10
[0.39, 0.63] predicted
× 10−6
of the dispersion
as its trend[0.5,
with
hematocrit
are correctly
by[76]
Equation (18).
[0.2, 0.4]
1100
[1.5, 2.1] × 10−6
[1.1, 1.7] × 10−6
[76]
−9
−9
[0.1,
0.15,
0.2]
44.8
[8.2,
11.9,
17.2]
×
10
[31.3,
37.9,
44.6]
×
10
[13]
Table 2. Comparison of particle dispersion rate predictions from Equation (18) with dispersion rates
[0.1, 0.2]
804
[0.9, 1.4] × 10−7
[5.4, 7.8] × 10−7
[15]
reported in the literature.
Ht
Shear (s´1 )
Dispersion Rate (cm2 /s)
Prediction (cm2 /s)
Reference
[0.2, 0.4]
[0.2, 0.4]
[0.1, 0.15, 0.2]
[0.1, 0.2]
400
1100
44.8
804
[0.5, 0.68] ˆ 10´6
[1.5, 2.1] ˆ 10´6
[8.2, 11.9, 17.2] ˆ 10´9
[0.9, 1.4] ˆ 10´7
[0.39, 0.63] ˆ 10´6
[1.1, 1.7] ˆ 10´6
[31.3, 37.9, 44.6] ˆ 10´9
[5.4, 7.8] ˆ 10´7
[76]
[76]
[13]
[15]
Nanomaterials 2016, 6, 30
11 of 14
Nanomaterials 2016, 6, 30
11 of 14
-3
Dimensionless
Dr
4
x 10
3
Prediction
Saadatmand,2011
Zhao,2012
Crowl,2011
Simulation
2
1
0
0
0.1
0.2
0.3
0.4
0.5
Hematocrit (Ht)
Figure 8. Comparison of the particle dispersion rate predicted from Equation (18) with the data
Figure 8. Comparison of the particle dispersion rate predicted from Equation (18) with the data
reported in the literature. The dashed line is the prediction from Equation (18).
reported in the literature. The dashed line is the prediction from Equation (18).
5. Conclusion and Future Work
5. Conclusion and Future Work
This paper presents a numerical study on NP dispersion in blood flow considering the influence
of
RBC
motion
and deformation.
which did
not exist
from
previous modeling
This paper
presents
a numericalAnalytical
study onformulae,
NP dispersion
in blood
flow
considering
the influence
results
[3,16],
were
proposed
to
characterize
NP
migration
to
the
vessel
walls
as
a function
of shear
of RBC motion and deformation. Analytical formulae, which did not exist from
previous
modeling
rate
at
different
hematocrit
levels.
The
formula’s
predictions
agree
well
with
data
reported
the rate
results [3,16], were proposed to characterize NP migration to the vessel walls as a function ofinshear
literature, thereby facilitating practical modeling of NP transport and distribution in large-scale
at different hematocrit levels. The formula’s predictions agree well with data reported in the literature,
vascular systems.
thereby facilitating practical modeling of NP transport and distribution in large-scale vascular systems.
In the future, the model will be extended to 3D to explore NP binding and distribution in
In
the future,
the The
model
will beformula
extended
3Ddispersion
to explorerate
NP will
binding
distribution
in capillary
capillary
vessels.
proposed
fortothe
also and
be used
to evaluate
NP
vessels.
The
proposed
formula
for
the
dispersion
rate
will
also
be
used
to
evaluate
NP
transport
and
transport and distribution in a large-scale vascular network.
distribution in a large-scale vascular network.
Supplementary Materials: Figures S1–S9, Table S1 and three short movies are available online at
Supplementary
Materials: Figures S1–S9, Table S1 and three short movies are available online at
http://www.mdpi.com/2079-4991/6/2/30/s1.
http://www.mdpi.com/2079-4991/6/2/30/s1.
Acknowledgments: The study presented in this paper was supported by the National Science Foundation (NSF)
Acknowledgments:
The study
presented
in this(CAREER)
paper was
supported
by the National
Science Foundation
Faculty Early Career
Development
Program
Grant
CBET-1113040,
NSF CBET-1067502,
NSF
(NSF)DMS-1516236
Faculty Early
Career
Development
Program
(CAREER)
Grant CBET-1113040, NSF CBET-1067502, NSF
and
the National
Institute of
Health (NIH)
Grant EB015105.
DMS-1516236 and the National Institute of Health (NIH) Grant EB015105.
Author Contributions: J.T. developed the code, performed the simulation and wrote the manuscript. W.K.
Author
Contributions:
J.T. developed
performed
the and
simulation
wrote
manuscript.
developed
the code, analyzed
the data.the
S.S.code,
contributed
to the code
analyzedand
the data.
J.Y.the
helped
design the W.K.
developed
the
code,
analyzed
the
data.
S.S.
contributed
to
the
code
and
analyzed
the
data.
J.Y.
helped
simulation. Y.L. conceived, designed the simulation and analyzed the data. All revised the manuscript. design the
simulation. Y.L. conceived, designed the simulation and analyzed the data. All revised the manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
Conflicts of Interest: The authors declare no conflict of interest.
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