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Copyright © 2012 by American Scientific Publishers
All rights reserved.
Printed in the United States of America
Reviews in Nanoscience and Nanotechnology
Vol. 1, pp. 66–83, 2012
(www.aspbs.com/rnn)
Computational Modeling of Nanoparticle
Targeted Drug Delivery
Yaling Liu∗, Samar Shah, and Jifu Tan
Department of Mechanical Engineering and Mechanics, Bioengineering Program, Lehigh University,
19 Memorial DR. W, Bethlehem, PA, 18015, USA
Nanomedicine is a promising application of nanotechnology in medicine, which can drastically improve drug
delivery efficiency through targeted delivery. However, characterization of the nanoparticle targeted delivery
process under vascular environment is very challenging due to the small scale of nanoparticles and the complex
in vivo vascular system. To understand such complicated system, various computational models are developed
to help reveal nanoparticle targeted delivery process and design nanoparticles for optimal delivery. This article
discusses a few computational tools to model the nanoparticle delivery process and design nanoparticles for
efficient targeted delivery. The modeling approaches span from continuum vascular flow, particle Brownian
adhesion dynamics, to molecular level ligand-receptor binding. Computer simulation is envisioned to be able to
optimize drug carrier design and predict drug delivery efficiency for patient specific vascular environment.
KEYWORDS: Nanomedicine, Nanoparticle, Cancer, Drug Delivery, Computational Modeling.
CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3. Continuum Approach: Drug Dissolution to
Convection-Diffusion-Reaction Model of Drug Delivery . . . . . 68
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2. Dissolution of Drug Particles . . . . . . . . . . . . . . . . . . . 69
3.3. Convection-Diffusion-Reaction Model of Drug Delivery . . 69
3.4. Nanoparticle Binding in a Channel . . . . . . . . . . . . . . . 70
3.5. Nanoparticle Deposition and Distribution in a Blood
Vessel Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4. Particulate Approach: Rational Design of Nanoparticles . . . . . 71
4.1. Introduction to Nanoparticle Design . . . . . . . . . . . . . . 71
4.2. Influence of Nanoparticles Size and Shape on
Targeted Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3. Theoretical Model of Nanoparticle Adhesion Probability . 73
4.4. Particulate Model of Nanoparticle Delivery in a
Vascular Environment . . . . . . . . . . . . . . . . . . . . . . . 73
4.5. Simulation Results of Nanoparticle Targeted Delivery
Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5. Future Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
1. INTRODUCTION
Nanotechnology refers to the study of matter on nanoscale,
in general dealing with structure size in between
∗Author to whom correspondence should be addressed.
Email: [email protected]
Received: 30 September 2011
Accepted: 12 March 2012
1 to 100 nm in at least one dimension.1 Nanotechnol-
ogy represents a broad range of applications; the medical
application of nanotechnology refers to “nanomedicine”.
Nanomedicine based drug delivery system hold great
promise in the next generation of medicine to improve
human health. Among all different research branches,
drug delivery contributes over 70% of scientific papers in
nanomedicine research field.2
The aim of drug delivery is to improve patient treat-
ment by enabling the administration of new intricate drugs,
improving the bioavailability of existing drugs, and pro-
viding spatial and temporal targeting of drugs in order to
dramatically reduce side effects and increase effectiveness.
Through accomplishment of these revolutionary advan-
tages, patients and physicians could benefit from person-
alized prescriptions, alleviated administration, increased
patient compliance, reduced dosage frequency and less
pain. Over the past decade, we have witnessed an explo-
sive development of nanoparticulate systems for diag-
nostic imaging and targeted therapeutic applications.3–10
Various nanoplatforms, including liposomes,11112 poly-
meric micelles,13–16 quantum dots,17118 Au/Si/polymer
shells,19–21 and dentrimers22–24 etc. have been developed.
Although recent data on in vivo nanoparticle (NP) drug
delivery has showed remarkably improved efficacy over
traditional drug, yet the challenges in nanomedicine field
are many.
For example, the study of drug delivery system is not
straightforward process, which requires further consid-
eration and comprehensive analysis. The targeted drug
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Computational Modeling of Nanoparticle Targeted Drug Delivery Liu et al.
Fig. 1. The targeted drug delivery process spans across multiple spatial scales.
techniques such as Molecular dynamics, Brownian motion,
and stochastic approaches such as Monte Carlo simu-
lation to capture the nanoparticle motion. For example,
Shipley et al.27 and Modok et al.28 modeled delivery of
spherical NPs in tumor. Mahmoudi et al.29 and Li et al.30
performed computational fluid dynamics studies of mag-
netic NPs in vascular flow. Liu et al.31 and Zhang et al.32
studied the deposition of NPs in lung airway. Figure 2
shows a multiscale simulation framework for targeted
drug delivery ranging from continuum model to partic-
ular model. It’s the recent advancement in the compu-
tational science that made computational modeling very
promising for targeted drug delivery application. The lig-
and coated nanoparticles, loaded with drugs inside, trans-
port in blood stream, and adhere to diseased cells via
specific adhesion. However, this process becomes intri-
cate due to simultaneous involvement of hydrodynamic
force, adhesion force and Brownian force. In particular,
the ligand-receptor interaction is a sophisticated chemical
process. The surface property of functionalized nanoparti-
cles would play a crucial role to dictate the efficiency of
the targeted drug delivery by providing targeted selectiv-
ity. Computational modeling tool will lead to insights of
the dynamic delivery process, thus facilitate better design
of nanoparticles.
This article focuses on multiscale computational
approach to the targeted drug delivery. First, continuum
Fig. 2. A framework for multiscale modeling of the entire drug delivery
system.
based drug delivery model is introduced, which covers
the basic governing equations and a few examples of
targeted drug delivery under vascular conditions. Second,
particulate modeling based on coupled Brownian adhesion
dynamics method is described, where the motion and bind-
ing of individual nanoparticles in the blood stream are
modeled. Finally, the future trend in computational mod-
eling of targeted drug delivery is briefly discussed.
3. CONTINUUM APPROACH: DRUGDISSOLUTION TOCONVECTION-DIFFUSION-REACTIONMODEL OF DRUG DELIVERY
3.1. Introduction
In vivo drug release, transportation and targeted binding
have been recognized as important elements in targeted
drug delivery field. In order to target the disease area, the
drug loaded carriers are first injected into the circulation
system, where they transport through, across and within
vessels, tissues and cells. Due to specific binding between
ligand coated drug particles and receptors, expressed at
the disease cell membrane, drug loaded particles deposit
on the targeted disease region. Afterward process is fol-
lowed by cellular uptaking and drug releasing. From the
continuum stand point of view, the drug delivery process
consists of drug dissolution, transport and binding, which
can be described by mass conservation law and chemical
kinetic reaction.
In recent years, in vitro release profile of drug from con-
trolled release platform has been combined with the state
of art Computational fluid dynamics (CFD) simulation to
predict the the spatial and temporal variation of the drug
transport in the living tissues. For example, Saltzman and
Radomsky33 developed a diffusion kinetics model for the
drug release in the brain tissue. The transport mechanism
was assumed to be mainly governed by diffusion due to
the selective permeability of the blood capillaries known
as blood-brain-barrier. This simplified model’s prediction
has been validated by the experimental data of drug spa-
tial distribution. A three dimensional (3D) simulation of
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Liu et al. Computational Modeling of Nanoparticle Targeted Drug Delivery
human brain tumor of primitive neuroectodermal tumor
was performed by Wang et al.34 The simulation is con-
ducted on CFD tools to solve simultaneously continuity,
momentum and drug concentration equations. Using their
model, the contribution of convective transport of macro-
molecular and micromolecular drugs in the vicinity of
tumor were studied. In this section, the governing equa-
tions of the continuum drug delivery model are described
and a few demonstration examples are presented.
3.2. Dissolution of Drug Particles
A number of mathematical models have been proposed
and effectively applied to describe the drug release and
dissolution in literature.35–38 The simplest form of drug
dissolution profile is zero order kinetics that assumes slow
drug releasing process,
Q0−Qt = Kt (1)
Where Q0 is the initial amount of drug in the pharmaceu-
tical dosage form, Qt is the amount of drug in the pharma-
ceutical dosage form at time t and K is a proportionality
constant. Dividing the above equation by Q0 simplifying
it to:
f0 = kt (2)
Where ft = 1−Qt/Q0 is often referred as fraction of drug.
This relation can be used to describe the drug dissolu-
tion of several types of modified pharmaceutical dosage
release forms, particularly with low soluble drugs.
3.2.1. First Order Kinetics
The application of this model was first proposed by
Gibaldi and Feldman,39 and later by Wagner.40 The disso-
lution rate of the drug is described by the Noyes-Whitney
equation as shown below:
dC
dt= K4Cs−C5 (3)
Where C is the concentration of solid in bulk dissolu-
tion medium, Cs is the concentration of solid in diffusion
layer surrounding solid, K is a first order constant and it
is associated with surface area of the solid drug, diffusion
coefficient and diffusion layer thickness.
Since the dissolution mechanism of drug is very com-
plex, various empirical equations are proposed to describe
this process. For example, the popular Weibull equation
expressed the fraction of drug, m at time t, in the simple
exponential form:
m= 1− exp
(
− 4t−T i5b
a
)
(4)
Where a is time related constants, Ti represents time lag
before onset of the dissolution, b is a curve characterized
parameter.
Higuchi38 formulated following relation to model low
soluble drug release problem:
ft =√
D42C−Cs5Cst (5)
Where C is the drug initial concentration, Cs is the drug
solubility in the matrix media and D is the diffusion coef-
ficient of the drug molecules in the matrix substance.
A few other models are summarized in a review paper
by Paulo Casta et al.41 The commonly used mathematical
models are listed in Table I.
3.3. Convection-Diffusion-Reaction Model of Drug
Delivery
The concentration of nanoparticle c inside a vascular sys-
tem can be described by the convection-diffusion equation:
¡c
¡t+−→U ·ïc = ï · 4Dïc5 (6)
Where c is the concentration of nanoparticles, D is the
diffusion coefficient of nanoparticle and U is the flow
velocity. It is solved by using following Einstein-Stokes
equation,
D = kBT
6��r(7)
Where kB is the Boltzmann constant, T is the temperature,
� is the viscosity of fluid medium and r is the NP radius.
The biorecognition of the targeted drug delivery site is
similar to a key lock mechanism which is in reality a com-
plex biochemical reaction. To depict the effect of adsorp-
tion of nanoparticles on a functionalized surface, Langmuir
reaction model is employed.42 The ligand-receptor binding
process is a weak reversible process, which leads to con-
tinous attachment and detachment of nanoparticles.43 The
material balance for the active surface including surface
diffusion and the reaction rate expression for the formation
of the adsorbed species cs is defined by:
¡cs¡t
+ï · 4−Dsïcs5= kacw�−kdcs (8)
Where Ds is the surface diffusivity (m2/s), cw is the bulk
concentration of the species at solid wall (unit mol/m35,
� is the surface concentration on the active site (mole/m25
Table I. Mathematical models used to describe drug dissolution curves.
Zero order Qt =Q0 +K0tFirst order lnQt = lnQ0 +K1t
Hixson-crowell Q1/30 −Q
1/3t = Kst
Weibull ln6− ln41− 4Qt/Q�557= b ln4t5− ln4a5
Higuchi Qt = KH
√t
Baker-lonsdale 3/261− 41−Qt/Q�52/37−Qt/Q� = Kt
Korsmeyer-peppas Qt/Q� = Kktn
Quadratic Qt = 1004K1t2 +K2t5
Logistic Qt = A/61+ e−K4t−y57Gompertz Qt = Ae−e−K4t−y57Hopfenberg Qt/Q� = 1− 61−k0t/C0a07
n
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Computational Modeling of Nanoparticle Targeted Drug Delivery Liu et al.
and cs surface concentration of adsorbed species (mol/m25.
Note that cs is different from c which is reflected in their
units. ka (m3/mol/s) and kd (s
−15 are adhesion and detach-
ment rates, respectively. However, the concentration of
active sites is equal to the difference between the total con-
centration of active sites and the number of sites occupied
by the adsorbed species. This gives the equation for the
reaction rate as:
¡cs¡t
+ï · 4−Dsïcs5= kac4�0− cs5−kdcs (9)
In above equation, �0 represents the total number of active
sites available on the active surface. Convection-diffusion
equation and nanoparticle reaction equation are not inde-
pendent, instead, they are coupled through Fick’s law:
¡cs¡t
=−D ·ïc�w (10)
3.4. Nanoparticle Binding in a Channel
To demonstrate application of continuum model in targeted
drug delivery, finite element modeling is used to evaluate
the nanoparticle transportation diffusion and biochemical
reaction dynamics in a channel. In this model, the con-
vection diffusion in 2D fluid domain is coupled with the
adhesion reaction occurring on the reaction surface (dis-
ease site). When a portion of the blood vessel is injured,
significant P-selectin is expressed on damaged endothelial
cells, which can be targeted by nanoparticles coated with
GPIb ligand. In this model, the convection-diffusion pro-
cess of nanoparticle in 2D fluid domain is coupled with
the adhesion reaction occurring only on the reaction sur-
face which mimics the target site for drug delivery. The
physical parameters used to create this model are listed in
Table II.
To initiate adhesion, nanoparticles must stay close to
the vessel wall, inside the so called depletion layer also
known as a near-wall layer where adhesion process take
place. The thickness of the depletion layer is largely influ-
enced by the flow rate, evident from the simulation result
shown in Figure 3. When drug particles bind with the
receptors coated surface, drug concentration drops near the
surface, effectively forms a “depletion layer” near the wall.
Table II. Physical parameters used in nanoparticle binding in a channel.
Symbol Value Definition
c0 1000 [mol/m3] Initial concentrationka 10−6 [m3/(mol*s)] Adhesion rate constantkd 10−3 -10−6 [1/s] Detachment rate constant�0 1000 [mol/m2] Active site concentrationDs 10−11 [m2/s] Surface diffusivityD 10−9 [m2/s] Particle diffusivity in the fluidkB 1038×10−23 [m2kg s−2 K−1] Boltzmann constantT 300 [K] Absolute temperatureU 0–25 [dyne/cm2] Maximum shear rate� 10−10 [m] Equilibrium bond length
Receptor coated reaction
Flow rate 1mm/s
10 µm
Flow rate 0.1mm/s
10 µm
(A)
Receptor coated reaction surface
(B)
Fig. 3. Nanoparticle binding in a channel at a flow rate of 0.1 and
1 mm/s. Particle concentration drops close to the receptor coated surface
due to adhesion, forming a depletion layer. Red color indicates highest
concentration, while blue color indicates lowest concentration.
Figure 3 shows the depletion layer at shear rates 0.1 mm/s
and 1 mm/s respectively. As the flow rate increases the
depletion layer thickness decreases due to greater nanopar-
ticle flux and shorter retention time of the nanoparticles.
3.5. Nanoparticle Deposition and Distribution in a
Blood Vessel Network
Another example application of continuum model is to
determine nanoparticle deposition and distribution in a
complex vascular geometry. Figure 4 shows the drug deliv-
ery process in an idealized vascular network with three
generations. The physical parameters used to create this
model in listed in Table III.
Drug loaded nanoparticles of a given concentration are
injected at the top inlet and are transported through the
vascular network along with fluid flow. The left branch
of the network is assumed to be a receptor coated target
surface that can form bonds with ligands on drug loaded
Fig. 4. (A) Drug injected at the top inlet of an idealized vascular net-
work with three generations; (B) Receptors coated vessel section in the
left branch of vascular network.
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Liu et al. Computational Modeling of Nanoparticle Targeted Drug Delivery
Table III. Physical parameters used for blood vessel network
simulation.
Symbol Value Definition
c0 1000 [mol/m3] Initial concentrationka 10−6 [m3/(mol*s)] Adhesion rate constantkd 10−9[ 1/s] Detachment rate constant�0 1000 [mol/m2] Active site concentrationDs 10−11 [m2/s] Surface diffusivityD 10−6 [m2/s] Particle diffusivity in the fluidV 1 [mm/s] Maximum velocity� 1063 [kg/m3] Blood density� 0.003 [Pa.s] Blood dynamics viscosity
nanoparticle surface. The particle depletion layer is clearly
visible in the target region. The density of deposited drug
particles on the wall surface is plotted in Figure 5, which
indicates that most drug particles are deposited at the
entrance of the target region, while the rest of the target
region has low density of deposited drug particles. There
are no particles deposited in the healthy branch due to an
assumption of zero non-specific adhesion at that particular
location. Such non-uniform distribution pattern indicates
possible impaired delivery dosage within the target region,
which is important for delivery efficacy prediction and
dosage planning.
4. PARTICULATE APPROACH: RATIONALDESIGN OF NANOPARTICLES
4.1. Introduction to Nanoparticle Design
Most of the nanoparticles employed in the experimental
studies are spherical in shape. Extensive studies have been
Fig. 5. Drug concentration as it flows from parent vessel through the vascular network with the receptor coated target region marked by the black
circle. Red color indicates highest concentration, while blue color indicates lowest concentration.
dedicated to comprehend their biological behaviors in vitro
and in vivo. For example, it is known that spherical par-
ticles bigger than 200 nm are efficiently filtered by the
spleen, while particles smaller than 10 nm can be quickly
cleared by the kidney, thus making 10–200 nm as an ideal
size range for the spherical carriers.
Similar to size, shape is a fundamental property of
micro/nanoparticles that may be critically important for
their intended biological functions.44–50 Recent data begin
to reveal that particle shape may have a profound effect
on their biological properties. For example, cylindrically
shaped filomicelles can effectively evade the non-specific
uptake by the reticuloendothelial systems and persisted in
the circulation up to one week after intravenous injection.
From drug delivery stand point of view, non-spherical par-
ticles will allow larger payload delivery than the spher-
ical counterpart with same binding probability. Recently,
Mitragotri and coworkers have shown that the local shape
of the particle at the point where a macrophage is attached,
not the overall shape, dictated whether the cell began
internalization.51 These results indicate the importance of
controlling particle shape for nanomedicine application.
Theoretical studies of nanoparticle deposition are typ-
ically focused on simple spherical or oblate shape.52–54
Ideally, there should be a tool that can handle variety of
shapes and sizes of nanoparticles, which enables endless
possibilities of finding most suitable design of the nanopar-
ticle for a given application. Decuzzi and Ferrari.52–54 have
studied the margination of nanoparticles in blood stream,
where nanoparticles diffusion in Newtonian fluid has been
analyzed. The same authors have also examined the adhe-
sion probability of nanoparticles under an equilibrium
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Computational Modeling of Nanoparticle Targeted Drug Delivery Liu et al.
configuration. Mody et al.55156 studied platelet motion near
vessel wall surface under shear flow and concluded that
hydrodynamic force influences platelet adhesion to the
wall surface. The same authors55156 also investigated the
influence of Brownian motion on platelet movement and
found that Brownian motion does not influence platelet-
shaped cells at physiological shear rates. However, size
(∼2 �m) and shape (oblate) of the platelet is not compa-
rable to that of nanoparticles and the behavior observed
for platelet might not be applicable for nanoparticles.
4.2. Influence of Nanoparticles Size and Shape on
Targeted Delivery
The targeted drug delivery process in general can be
considered as a seamless combination of three stages:
transport through the vessel network; adhesion process;
and cellular update. Each stage is effectively governed
by nanoparticle shape, size, and surface property. Spe-
cific combination of each parameter can accomplish effi-
cient targeted delivery. There have been a large number of
studies devoted on characterization of nanoparticle physi-
cal property. Djohari and Dormidontova57 studied kinetics
of spherical nanoparticle for targeting cell surface using
dissipative particle dynamics. The shape of the adsorbed
Fig. 6. Kinetics of filomicelle length reduction in vivo. (a) Inert filomicelles shorten, with the rate of shortening decreasing as they shorten. The grey
region represents the optical limit of L measurements; (b) Degradable filomicelles (OCL3) shorten at a rate that depends on initial length. The inset
plots the length dependent shrinkage rate; (c) Filomicelles show a saturable increase in half-life of circulating mass, fitting a cooperative clearance
model with �max = 502 days, m= 201 and L� = 205 �m; (d) Distribution of inert and degradable filomicelles in clearance organs for Lo = 4 or 8 �m
after four days in the circulation of rats. All error bars show the standard deviation for three or more animals. Reprinted with permission from [58],
Y. Geng, et al., Shape effects of filaments versus spherical particles in flow and drug delivery. Nat. Nanotechnol. 2, 249 (2007). © 2007, Macmillan
Publishers Ltd: Nature.
nanoparticle was found to become ellipsoidal with increas-
ing binding energy. Janus-like nanoparticles with ligands
coated on one side of the nanoparticle was observed to
bind faster than that with uniformly coated ligands. Geng
et al.58 revealed the potential of non-spherical shaped car-
rier for drug delivery application. Figure 6 shows the
results of their experimental study on the circulation time
of filomicelles of different lengths in a mice model. Fur-
ther, the same group performed in vivo study to investigate
the shape effect and discovered that non-spherical shape
carrier has 10 times longer circulation time compared to
its spherical counterpart.
Muro et al.59 studied controlled endothelial targeting and
intracellular delivery by modulating size and shape of the
drug carrier. Their study found that carrier geometry influ-
ences endothelial targeting efficiency. The non-spherical
carrier had longer circulation time and higher targeting
specificity than regular spherical carrier. Shah and Liu
et al.60 has compared the transport phenomenon and bind-
ing probability of nanospheres and nanorods under shear
flow and revealed significantly higher binding efficiency
of nanorods due to their tumbling motion and larger con-
tact area.46160161 Winter et al.62 and Liu et al.63164 have
performed numerical simulations of dielectrophoresis of
non-spherical particles. Theoretical models to determine
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Liu et al. Computational Modeling of Nanoparticle Targeted Drug Delivery
adhesion probability of different shaped nanoparticles is
discussed in the next section.
4.3. Theoretical Model of Nanoparticle Adhesion
Probability
A numerical model is built based on the previous work
by Decuzzi and Ferrari52165 to describe the cell target-
ing process of nanorods and nanodisks under flow con-
ditions. The adhesion probability (Pa5 is characterized by
the probabilistic kinetic formulation of McQuarrie66 and
Decuzzi:67
Pa ' mrmlKoaAc exp6−
�f
kBT7 (11)
Where mr is the receptor density on the substrate sur-
face, ml is the ligand density on particle surface, Ac is
the contact area of particle, f is force acting per unit
ligand-receptor pair, kBT is thermal energy of system, � is
a characteristic length of ligand-receptor bond, and Koa is
the affinity constant of ligand-receptor pair at zero load.
Normalized adhesion probability of oblate-, rod- and disc-
shaped nanoparticles for a wall shear stress of 1 (Pa) is
plotted as a function of particle volume in Figure 7. The
aspect ratio of disc (diameter over height) and rod (length
over diameter) are chosen to be 5.
As shown in Figure 7, for the spherical particle
with increasing particle volume, the adhesion probability
increases first, due to larger available surface area for the
bond formation, and then decreases due to the large vol-
ume. Particles of larger volume faces larger dislodging
force, which overwhelms bond forces and get the particle
washed away, thus reduces the adhesion probability. Due
the shape effect, the critical volume of the non-spherical
particles is comparatively large and shifts to the right
side of the plot (out of the plot range). For the range
of volume considered, the oblate, rod and disc particles
show significantly higher adhesion probabilities than the
Fig. 7. Adhesion probabilities of nanoparticles of various shapes as a
function of particle volume, � is the aspect ratio.
spherical particles of the same volume. Furthermore, the
disc-shaped nanoparticles have the highest adhesion prob-
ability and the largest volume to the mass ratio, resulting
in 300 times higher efficacy for cell targeting and 40 times
higher drug-loading capability than their spherical counter-
part. The adhesion probability of the rod-shaped particle,
with the aspect ratio of 5, is about 20 times higher than the
spherical particle. Similarly, the oblate shaped nanoparti-
cles exhibit almost 10 times higher adhesion probability
compared to the spherical particles over the entire range
of volume considered.
This theoretical model is based the adhesion probabil-
ity for a particle to adhere on receptor coated surface,
but lacks in revealing dynamic process of nanoparticle
transport and delivery. A coupled model that integrates
margination with adhesion kinetics, and applicable to
nanoparticle of various shapes, is yet to be developed.
Thus, analysis of this process for an arbitrarily shaped
nanoparticle through a multiscale model is crucial to pro-
vide biological insights on the transportation and adhe-
sion kinetics. In what follows, we will first introduces
the nanoparticle adhesion kinetics theory and model-
ing method. Then, adhesion process and trajectories for
nanoparticles of different shapes and ligand densities are
presented. Next, the binding probability of nanoparticles
is determined for a range of channel sizes.
4.4. Particulate Model of Nanoparticle Delivery in a
Vascular Environment
Nanoparticles are usually introduced into the vascular
circulation stream through intravenous injection.68–70 The
targeted delivery efficiency is directly related to the
nanoparticle selectively and ability to bind at the targeted
site. Though highly selective nanoparticles have reduced
binding probability in non-target regions, the majority of
nanoparticles are still lost in the vascular network due to
non-specific adhesion. It is thus important to predict van-
ished concentration of nanoparticles in the upstream and
nanoparticle concentration when it reaches the targeted
region. The focus of a particulate model is to explore an
optimum design of nanoparticle to achieve high binding
probability in the diseased region and high overall delivery
efficiency under given vascular environment.
4.4.1. Nanoparticle Adhesion Kinetics
To achieve targeted drug delivery, nanoparticles are usu-
ally coated with ligands that bind specifically to a par-
ticular type of receptors expressed on the diseased vessel
cell surface.71 Once nanoparticles marginate to vascular
surface, the ligand coated nanoparticles interact with the
specific receptors expressed at target surface. Such interac-
tion results in a bond formation between nanoparticle and
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Computational Modeling of Nanoparticle Targeted Drug Delivery Liu et al.
vascular surface. The adhesive strength of bond is medi-
ated by the specific binding of ligand-receptor. Other fac-
tors such as steric interactions, electrostatic, van der Waals
(vdW) or hydrodynamic forces also influence interactions
between nanoparticle and vessel surface. However, the
vdW force is usually several orders of magnitude smaller
compared to specific adhesive force, whilst the effect of
steric interaction and electrostatic is limited to very short
distance (less than nanometer scale).53 Thus, these factors
are neglected for this model. The ligand-receptor bind-
ing process is integrated with Brownian dynamics and this
combined model is embedded into the Immersed Finite
Element (IFEM) platform.72–75 IFEM can be used for fully
coupled fluid-structure interaction problems, i.e., solving
particle motion in a fluid while capturing the influence of
particle on fluid flow. However, due to Brownian motion,
it is computationally expensive to calculate the change of
fluid flow caused by particle motion at every time step.
Also, the effect of nanoparticle motion is limited to the
local surrounding region. Thus, the influence of parti-
cle motion on the fluid flow is neglected and the model
is only solved for the particle motion and the adhesion
process.
When a particle approaches the vascular wall, ligands
on the particle surface form bonds with receptors on the
vascular wall, as demonstrated in Figure 8. An adhesion
kinetic equation is used to calculate the bond density Nb:76
¡Nb
¡t= kf4Nl−Nb54Nr −Nb5−krNb (12)
Where Nl and Nr are the ligand and receptor densities; krand kf are the reverse and forward reaction rates, respec-
tively. This interaction model represents a conservation
equation of the different species (ligands, receptors, and
bonds). The kr and kf are function of bond length:
kr = k0r exp4−4ks−kts5L2/2Bz5 (13)
kf = k0f exp4−ktsL2/2Bz5 (14)
Where ks is the bond elastic constant; kts is the bond elas-
tic constant at transient state; Bz is thermal energy; k0r and
ReceptorLigand
Shear
Flow
Fig. 8. Model of ligand-receptor adhesion kinetics between ligand
coated nanoparticle surface and receptor coated vascular wall surface.
k0f are the reverse and forward reaction rates at zero load
of ligand-receptor pair, respectively; L is the difference
between bond length y and equilibrium length �. During
dynamic interaction process, the bond length of a ligand-
receptor pair may vary based on particle location. The
ligand-receptor bonds are modeled as springs with spring
constant � and equilibrium length �, thus the bond forces
are described as a function of bond length y. Then, the
ligand-receptor interaction forces can be summed on finite
element surface through integration over the nanoparticle
surface. Equations of bond forces fL and integrated adhe-
sion forces � s on particle surface â are given as:
fL = �4y−�5 (15)
�S ·n=∫
NbfL4Xc5dâ (16)
Such adhesion force is coupled with the fluid-structure
interaction (FSI) force in the IFEM formulation. Similar
adhesion model has been used by Chang et al.77 and Dong
et al.78 in the study of white blood cell rolling. The phys-
ical parameters used in the model are listed in Table IV.
Besides adhesion forces, the Brownian force acting on
to the nanoparticles is also important and is integrated into
the IFEM formulation by adding a Brownian force term,
which is described in the next section.
4.4.2. Brownian Dynamics at Nanoscale
Fundamental theories of Brownian dynamics have indi-
cated that random collisions from surrounding liq-
uid molecules impacts motion of an immersed small
particle.79–81 The influence of Brownian motion on
behavior of nanoparticles in microfluidic channel and
platelets and blood cells in blood flow has been studied
extensively.82–85 Patankar et al.86 have proposed an algo-
rithm for direct numerical simulation of Brownian motion
by adding random disturbance in fluid. At microscale, the
drag force acting on particles such as blood cells is sig-
nificantly large (>50 pN for particle size >1 �m), thus
Brownian motion is neglectable.82 At nanoscale, Brown-
ian force becomes a dominant force to drive nanoparticle
near vascular wall surface, while the drag force acting on
a nanoparticle is relatively small. Shah and Liu et al.60
developed novel hybrid model to study Brownian dynam-
ics at nanoscale and governing equations are described as
following.
The random forces R4t5 and torque T4t5 acting on a
nanoparticle is responsible for Brownian motion and rota-
tion and satisfy the fluctuation-dissipation theorem:87
�Ri4t5� = 01 �Ti4t5� = 0 (17)
�Ri4t5Rj4t′5� = 2kBT�t�ij�4t− t′5�1
�Ti4t5Tj4t′5� = 2kBT�r�ij�4t− t′5� (18)
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Table IV. List of physical parameters used in the nanoparticle adhesion kinetics model.
Definition Symbol Value Reference
Ligand density Nl 200×1010 (sites/cm25 Lawrence and Springer (1991)107
Receptor density Nr 2.0–5.0×1010 (sites/cm25 Bell et al. (1984)108
Reverse reaction rate k0r 0.5 (1/s) Bell (1978)109
Forward reaction rate k0f 100×10−9 (cm2/s) Bell (1978)109
Equilibrium bond length � 20 nm Bell (1978)109
Static bond spring constant � 0.5 (dyne/cm) Dembo et al. (1988)76
Transient bond elastic constant kts 0.48 (dyne/cm) Dembo et al. (1988)76
Thermal energy Bz 400×10−14 (erg) Dembo et al. (1988)76
Fluid viscosity � 0.01 (g/cm-s) —
Where � is the unit-second order tensor, �ij is the
Kronecker delta, �4t − t′5 is the Dirac delta function,
kBT is thermal energy of system, �t and �r are the trans-
lational and rotational friction coefficient of nanoparticle,
respectively.
The friction coefficient depends on several physical
parameters, such as fluid viscosity, size and shape of the
nanoparticle. The friction coefficient for spherical-shaped
particles can be easily derived from Stokes’ law. However,
there is no empirical formula available for determining the
friction coefficient of particles with complex shapes. In
literature, there are empirical formulas for friction coeffi-
cients for particles, but limited to simple shapes and ori-
entations such as oblate or rod-shaped particles.88–91 In a
recent work by Loth,92 new empirical formula is proposed
to compute friction coefficient for a non-spherical particle.
Friction coefficient of rod shaped particles in this work
is derived based on Loth92 and extended with an angle
factor to incorporate arbitrary orientations. When a parti-
cle travels along the fluid flow, the relative velocity of the
particle can be divided into components in two directions:
parallel to flow and perpendicular to flow, as shown in
Figure 9.
r
θ
V cosθ
V sinθ
V
Fig. 9. Illustration of friction coefficient measurement of arbitrarily ori-
entated nanorod.
The friction coefficient of a rod-shaped particle for an
arbitrary orientation is given by Ref. [92]:
�t = 3��deqv× 4f� · �cos��+ f⊥ · �sin ��5 (19)
�r = ��d3eqv (20)
Where � is the fluid viscosity, deqv is the diameter of parti-
cle volume equivalent sphere, � is the angle between flow
direction and the long axis of the particle, f� and f− are
Stokes correction factors for a spheroid particle moving
parallel and perpendicular to the flow, respectively. These
correction factors are expressed as Ref. [92]:
f� =(
4
5+ �
5
)
�−1/3 (21)
f⊥ =(
3
5+ 2�
5
)
�−1/3 (22)
Where � is the aspect ratio of the spheroid particle. The
velocity of a particle moving under a deterministic force
in a fluid with velocity Vf is given by:
Vs =(
Fdet
�t
+V f
)
41− e−4�t/m5t5 (23)
Where Fdet is the total deterministic force acting on the
nanoparticle (including Brownian force, adhesion force,
etc.), Vs and Vf are the solid and fluid velocity vec-
tors, respectively. For a time step (typically ∼1 �s) much
greater than characteristic time constant m/�t (∼10 ns), the
nanoparticle moves with a terminal velocity, thus Eq. (23)
reduces to:
Vs =Fdet
�t
+Vf (24)
Equation (24) actually describes that the deterministic
force acting on a particle is balanced by the drag force
from the fluid. This is reasonable since the mass of a
nanoparticle is so small that inertia effect can be neglected.
This terminal velocity is then use to update the nanoparti-
cle position in translational direction. Similarly, the angu-
lar velocity of a nanoparticle can be obtained through:
�s =Tdet�r
+�f (25)
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Computational Modeling of Nanoparticle Targeted Drug Delivery Liu et al.
Where �f is the angular velocity due to fluid flow. Com-
bining the translational and angular velocities, particle
nodal positions are updated based on its distance from the
particle center as:
vi = Vs+�s× ri (26)
The fluid flow in our simulation is assumed to
be an incompressible viscous fluid governed by the
Navier-Stokes equations:
�
(
¡vf¡t
+vf ·ïvf)
=−ïp+�ï 2vf (27)
ï ·vf = 0 (28)
It should be noticed that vf is the fluid velocity in the
fluid main, while Vf is the fluid velocity interpolated onto
the solid domain. The Navier-Stokes equations are solved
through finite element method. To reduce numerical oscil-
lations, the velocity test function is employed along with
stabilization parameters. Using integration by parts and
the divergence theorem, the Patrov-Galarkin weak form
is obtained. Then, the nonlinear system is solved using
the Newton-Raphson method. Moreover, Generalized Min-
imum Residual (GMRES) iterative algorithm is employed
to improve computation efficiency and to compute residu-
als based on matrix-free techniques.93 Details of the imple-
mentation can also be referred to Zhang et al. and Liu
et al.72–74194
4.5. Simulation Results of Nanoparticle Targeted
Delivery Process
Mathematical modeling of targeted drug delivery system
provides quantitative description of the drug transporta-
tion in biological systems. Therefore, it can be utilized to
evaluate efficiency of drug delivery and to estimate dose
response.
4.5.1. Effect of Nanoparticle Shape on Adhesion
Kinetics
The following section discusses about influence of
nanoparticle geometry on adhesion kinetics. Two separate
sets of simulation studies have been performed to evaluate
near wall behavior of spherical particle and non-spherical
particle.
Comparing Deposition Process of Nanoparticles. To
investigate the influence of nanoparticle shape on adhesion
kinetics, two nanoparticles of different shapes, spherical
and non–spherical, but of the same volume are consid-
ered in this study. The length of the rod shaped particle
considered is 1000 nm with an aspect ratio of 5. The
diameter of spherical particle is 380 nm. Such constant
volume comparison helps to understand whether nanorod
or nanosphere bind easily to wall surface for a given drug
load capacity. The simulations are carried over a channel
of 5 �m long and 2 �m high. In the simulation, a spherical
particle and a rod-shaped particle are initially positioned
with their centers 600 nm above a receptor-coated surface,
as shown in Figure 10.
A velocity is applied at the top of channel to generate
a shear rate of 8.0 s−1. Nanoparticles are allowed to move
freely through the channel under the influence of shear
flow and Brownian forces. For a typical simulation demon-
strated in Figure 10, the spherical particle fails to make
any contact with the vessel wall while it travels through the
channel. Under given velocity and channel length, Brow-
nian diffusion is not large enough to make the spherical
particle to reach close enough to the wall surface to ini-
tiate binding process. Compared to nanospheres, nanorods
make contact and adhere to vessel wall quickly and fre-
quently. The rod-shaped particle exhibits tumbling motion
by virtue of non-spherical shape while flowing through the
channel. Due to the tumbling motion, a nanorod usually
contacts with the receptor coated wall with bonds formed
at the long axis end first. Such initial contact is followed
by nanoparticle rotation along the contact end and steadily
growing adhesion force, which ensure firm adhesion to the
vessel wall and at the end settle down at equilibrium state
with full contact. The simulation results reveal typical tra-
jectories of a nanosphere and a nanorod, which illustrate
different dynamic adhesion processes. A more quantitative
description of the adhesion process will be presented in
later sections.
One question that might arise at this point is the exis-
tence of such near wall particle tumbling motion. In lit-
erature, tumbling of non-spherical particles near a wall
surface has been reported.56195196 The combined effects
of shear flow and Brownian rotation have been found to
enhance rotation of nanorods.97198
Comparing Trajectories of Nanospheres and Nanorods.
Nanorods are expected to have higher probability to con-
tact with the wall surface than their spherical counter parts
because of tumbling motion. To test this theory, trajecto-
ries of spherical and non-spherical nanoparticles under the
same flow condition are compared. A shear rate of 8.0 s−1
is employed for the both cases. The simulations are carried
over the channel with the length equal to 15 �m and the
height equal to 5 �m.
To illustrate the fluctuations of nanoparticle-wall dis-
tance, minimum distance between the nanoparticle surface
and the wall surface is recorded over the time, as shown
in Figure 11(A). Such nanoparticle trajectory indicates the
path of nanoparticle during its motion through the channel.
In a series of simulation runs, a nanosphere and a nanorod
are placed initially 650 nm above the wall surface. The
trajectories of nanorod and nanosphere of 20 independent
simulations are plotted in Figure 11(B).
The simulation result elucidates that a nanorod has
larger fluctuations in trajectories due to tumbling motion,
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Liu et al. Computational Modeling of Nanoparticle Targeted Drug Delivery
Fig. 10. Shape dependent particle adhesion kinetics. The left column shows a spherical particle washed away without contact with surface; the right
column shows a nanorod tumbles and gets deposited. A, B, C, D are at times t = 0 s, 0.25 s, 0.5 s, and 0.75 s, respectively. The line labeled on
the spherical particle indicates its rotation. The vectors in fluid domain indicate flow field and arrows indicate magnitude and direction of bonding
forces.
thus it has more contact/adhesion events compared to that
of nanosphere, as shown in Figure 11(C). Moreover, in a
fixed number of trials, ten nanorods are deposited while
only three nanospheres are deposited. Probability of spher-
ical particle to contact with wall surface solely depends on
Brownian diffusion; while in case of non-spherical parti-
cle, probability of contact is enhanced by tumbling motion.
Thus, this result indicates that nanorod has higher contact
probability than the nanosphere for given physiological
flow condition.
4.5.2. Nanoparticle Binding Probability
The simulation method developed in previous sections is a
rigorous way to model the full transportation and adhesion
dynamics of arbitrarily-shaped nanoparticles. However, to
model the adhesion process of large number of nanopar-
ticles, it is computationally cost-effective and more con-
venient to derive a binding probability for nanoparticles
under various configurations. The binding probability is
the probability of a nanoparticle located within a certain
distance from the wall surface to bind with the vascular
wall. Binding probability directly determines how many
nanoparticles will actually bind to the wall surface among
total number of nanoparticles present within the fluid chan-
nel considered? This is an important parameter to deter-
mine drug concentration for desired application.
It should be noted that only nanoparticles are considered
in this particular section. Blood cells have been observed
to influence the dispersion rate of nanoparticles. However,
the focus of this section is to characterize the influ-
ence of particle shape on its binding property. Although,
multi-scale model that can handle blood cells along with
nanoparticle would certainly be covered in future requiring
further study and development. Now, it is known that to
initiate bond formation, nanoparticles must stay very close
to the wall surface, inside a cell free layer (CFL) or deple-
tion layer,99 as shown in Figure 12. The red blood cells
flow with relatively higher velocity in the core region of
vessel, leaving a pure plasma region with lower velocity
close to vessel wall. The existence of CFL makes it rea-
sonable to only consider nanoparticles in the deposition
process. The thickness of the cell free layer is found to be
varying from 2–5 �m, independent of vessel size for ves-
sels with diameter above 20 �m.100–102 This suggests that
binding probabilities of nanoparticles should be studied for
a range of depletion layer or CFL thicknesses.
This particular section focuses on studying the effect
of two parameters; shear rate and depletion layer thick-
ness, on nanoparticle binding probability. To ensure con-
sistency and study sole effect of mentioned parameters
among all the cases, the rest of the parameters are kept
constant. For example, the value of ligand density is
assumed to be sufficiently high to guarantee firm adhesion
of nanoparticles (adhesion force typically varies between
1 pN–100 pN, while dislodging forces are limited around
0.01 pN). Moreover, it has been shown recently that once
a nanoparticle tethers to the receptor coated surface, it
is unlikely to get detached under hydrodynamics force103
due large adhesion force which overwhelms other forces
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t1 t3
t4Nanoparticle trajectory
Minimum
distance
t2
0 40002000 6000 8000 10000 120000
200
400
600
800
1000
(A)
(B)
Length in X dir (nm) →
Heig
ht in
Y d
ir (
nm
) →
Trajectories of particles (~ 20 trials)
non-sph
sph
(C)
Fig. 11. Comparing trajectories of nanorod and nanosphere to study
shape effect on particle adhesion kinetics. (A) Illustration of measurement
method of minimum distance between nanoparticle and wall surface at
different times. (B) Trajectories of 20 independent trials of nanorod and
nanosphere, where red spot indicates adhesion of nanorod and blue spot
indicates adhesion of nanosphere at that location. (C) Mean trajectory of
20 trials of nanorod and nanosphere with standard deviation shown as
vertical bar.
present that scale. As a result, this section focuses on deter-
mining binding probability of nanoparticles rather than dis-
sociation probability. The simulation parameters are listed
in Table IV, unless otherwise noted. The diameter and
length of nanorod is 200 nm and 1000 nm, respectively.
The diameter of nanosphere is 380 nm.
The simulation begins with randomly assigned initial
positions of nanoparticle at the channel inlet. Range of
shear velocities is applied at the top of the channel to
generate different shear rates. The nanoparticle transporta-
tion is simulated by the Brownian adhesion dynamics
model as discussed in the previous section. To ensure sta-
tistical accuracy, binding probability is evaluated based
on the results of 200 independent trials. The number of
bonded nanoparticles is counted and normalized by the
total number of nanoparticles to obtain the binding prob-
ability for a given depletion layer thickness under a given
flow condition.
Fig. 12. Multiscale model of the targeted drug delivery.
Binding probability of nanoparticles as a function ofdepletion layer or CFL thickness is plotted in Figure 13for two different shear rates, 10 s−1 and 2 s−1, respectively.The nanorods show significantly higher adhesion probabil-ity than the nanospheres at both shear rates. Figure 13(A)shows the binding probability of nanoparticles under ashear rate of 10 s−1. As the CFL thickness increases, bind-ing probability of nanoparticle decreases. Due to limiteddiffusion length, the binding probability of a nanospheredecreases almost linearly with CFL thickness, except forlow CFL thickness of 1.5 �m. At 1.5 �m CFL thickness,the size of nanoparticle becomes comparable to the CFLthickness, thus results in higher deposition probability. Incomparison, the binding probability of nanorod decreasesalmost quadratically with CFL thickness, mainly due to thetumbling motion. In particular, a nanorod has significantlyhigher binding probability than nanosphere at smaller CFLthicknesses. As shear rate decreases, binding probabilitiesfor both particles increase. At a shear rate of 10 s−1 andCFL thickness of 1.5 �m, the binding probability of thenanorod is around 2.5 times of that for the nanosphere.At a shear rate of 2 s−1, the difference in the bindingprobability between nanorod and nanosphere is reduced,as shown in Figure 13(B). At lower shear rates, Brownianmotion becomes a dominant factor, thus it overwhelms thecontribution of tumbling motion.Besides shape, the effect of nanoparticle aspect ratio is
also investigated. Nanorods of two aspect ratios (5 and 10)are considered in the study and compared with nanosphere.The binding probability of nanoparticles under different
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2 3 4 5 60
0.2
0.4
0.6
0.8
1
Depletion layer thickness (µm) →
Binding Probability vs. depletion layer thickness(10s–1)
nanorod
nanosphere
2 3 4 5 60
0.2
0.4
0.6
0.8
1
Depletion layer thickness (µm) →
Bin
din
g P
rob
ab
ility
→B
ind
ing
Pro
ba
bili
ty →
Binding Probability vs. Depletion layer thickness(2s–1)
nanorod
nanosphere
(b)
(a)
Fig. 13. Binding probabilities of a nanorod and a nanosphere for a
range of depletion layer thicknesses. Binding probability of nanorod and
nanosphere at shear rates of (A) 10 s−1 and (B) 2 s−1, respectively.
shear rates is plotted in Figure 14. A depletion layer thick-
ness of 5 �m is considered for the study. It is found that
nanoparticle with higher aspect ratio has higher binding
probability than that of lower aspect ratio or spherical
nanoparticles. The binding probabilities for nanorods are
proportional to the aspect ratio with a scaling factor of
around 1.6 in a range of shear rates. The simulation result
also elucidates that increase in shear rates reduces bind-
ing probability of nanoparticles, but the degree of reduc-
tion of binding probability varies with different aspect
ratio of nanoparticles. Binding probability of nanosphere
drops largely with increase in shear rate. While that of
nanorods drops only marginally with increase in shear rate.
This result clearly demonstrates advantage of nanorod over
nanosphere in terms of binding probability over a range of
shear rates.
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
Shear Rate γ (s–1) →
Bin
din
g P
rob
ab
ility
→
Binding Probability vs. Shear rate for D.L.T. (5 µm)
nanorod(A.R.=10)
nanorod(A.R.=5)
nanosphere
Fig. 14. Effect of shape on binding probability. Binding probabilities
of nanosphere and nanorods of two different aspect ratios for depletion
layer thickness (DLT) of 5 �m.
5. FUTURE TREND
The future of nanoparticle based targeted drug delivery is
very promising. We have witnessed exponential growth of
research related to nanoparticle based drug delivery in the
past decade. Engineering design of drug carrier is playing
an important role in nanomedicine field. To improve effi-
ciency, magnetic particles have also been proposed to offer
better imaging property and targeting efficiency under
localized magnetic field compared to polymer particles.104
The large variety of material selection (metallic or non-
metallic particles), sizes (10 nm to 200 nm), shape (spher-
ical or non-spherical), and complex vascular conditions
(healthy or tumor vasculature) have raised needs on faster
and efficient nanocarrier design. It is very time consuming
and challenging task for researchers to predict behavior
of various nanocarriers under physiological environment.
Owing to the limitation of experiments, computational
0
5
10
15
20
25
30
35
Nu
mb
er
of
pu
blic
atio
ns
2000 – 2003 2004 – 2007 2008 – present
Years
Fig. 15. Evolution of paper published on “modeling nanoparticle drug
delivery” over the last decade. Source: PubMed search engine.
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Fig. 16. Nanoparticle deposition and distribution in vascular geometry
reconstructed from MRI scanned images.61
work would be crucial tool for engineering shape and
size of these nanocarriers. As a result, there have been a
significant growth in the number of papers on modeling
nanoparticle targeted delivery published lately4616016111051106
as shown in Figure 15.
6. CONCLUSION
Multi-scale modeling of targeted drug delivery system pro-
vides quantitative description and in-depth analysis of the
drug transportation and delivery process in dynamic bio-
logical system. Such detailed results are very useful to
determine delivery efficiency of particular nanocarrier for
a given vascular condition and indeed to estimate dose
quantity and toxicity. An accurate computational model
needs to represent the actual physiological condition of
drug delivery. Most current studies focus on modeling
nanoparticle transport and binding under idealized vascular
environment such as simple straight and branched chan-
nels. To mimic the real vascular environment, the vascular
geometry could be reconstructed from CT/MRI scanned
images. The simulated nanoparticle deposition in a branch
vessel reconstructed from MRI scanned images is shown in
Figure 16. Such virtual tool can be used to predict nanocar-
rier bio-distribution and the delivery efficiency under a
given patient vascular geometry and hemodynamic condi-
tions, and help design nanoparticles for maximum target-
ing efficiency and minimum drug dosage.
Acknowledgments: The authors acknowledge the sup-
ports of this work from National Science Foundation
(NSF) CAREER grant CBET-1113040, NSF CBET-1067502, and National Institute of Health (NIH) grantEB009786.
References and Notes
1. Wikipedia, Nanotechnolgy (2011).
2. T. J. Webster, Welcome to the international journey to improving
human health. Int. J. Nanomedicine 1, 1 (2006).
3. C. Chauvierre, D. Labarre, P. Couvreur, and C. Vauthier, Novel
polysaccharide-decorated poly(isobutyl cyanoacrylate) nanoparti-
cles. Pharmaceutical Research 20, 1786 (2003).
4. O. C. Farokhzad and R. Langer, Nanomedicine: Developing smarter
therapeutic and diagnostic modalities. Advanced Drug Delivery
Reviews 58, 1456 (2006).
5. E. Mathiowitz, J. S. Jacob, Y. S. Jong, G. P. Carino,
D. E. Chickering, P. Chaturvedi, C. A. Santos, K. Vijayaraghavan,
S. Montgomery, M. Bassett, and C. Morrell, Biologically erodable
microsphere as potential oral drug delivery system. Nature 386, 410
(1997).
6. N. Nasongkla, E. Bey, J. M. Ren, H. Ai, C. Khemtong,
J. S. Guthi, S. F. Chin, A. D. Sherry, D. A. Boothman, and
J. M. Gao, Multifunctional polymeric micelles as cancer-targeted,
MRI-ultrasensitive drug delivery systems. Nano Letters 6, 2427
(2006).
7. N. A. Peppas, Intelligent biomaterials as pharmaceutical carriers
in microfabricated and nanoscale devices. Mrs Bulletin 31, 888
(2006).
8. C. Roney, P. Kulkarni, V. Arora, P. Antich, F. Bonte, A. M. Wu,
N. N. Mallikarjuana, S. Manohar, H. F. Liang, A. R. Kulkarni,
H. W. Sung, M. Sairam, and T. M. Aminabhavi, Targeted
nanoparticles for drug delivery through the blood-brain barrier for
Alzheimer’s disease. J. Controlled Release 108, 193 (2005).
9. P. Shah, Use of nanotechnologies for drug delivery. Mrs Bulletin
31, 894 (2006).
10. G. B. Sukhorukov and H. Mohwald, Multifunctional cargo systems
for biotechnology. Trends in Biotechnology 25, 93 (2007).
11. W. T. Al-Jamal and K. Kostarelos, Liposomes: From a clinically
established drug delivery system to a nanoparticle platform for ther-
anostic nanomedicine. Accounts of Chemical Research 44, 1094
(2011).
12. G. Sharma, S. Anabousi, C. Ehrhardt, and M. N. Ravi Kumar, Lipo-
somes as targeted drug delivery systems in the treatment of breast
cancer. J. Drug Target 14, 301 (2006).
13. R. P. Choudhury, V. Fuster, and Z. A. Fayad, Molecular, cellular
and functional imaging of atherothrombosis. Nature Reviews Drug
Discovery 3, 913 (2004).
14. D. Maysinger, J. Lovric, A. Eisenberg, and R. Savic, Fate of
micelles and quantum dots in cells. Eur. J. Pharm. Biopharm.
65, 270 (2007).
15. D. Sutton, N. Nasongkla, E. Blanco, and J. Gao, Functionalized
micellar systems for cancer targeted drug delivery. Pharm. Res.
24, 1029 (2007).
16. V. P. Torchilin, Targeted polymeric micelles for delivery of poorly
soluble drugs. Cell Mol. Life Sci. 61, 2549 (2004).
17. X. Gao, L. Yang, J. A. Petros, F. F. Marshall, J. W. Simons, and
S. Nie, In vivo molecular and cellular imaging with quantum dots.
Curr. Opin. Biotechnol. 16, 63 (2005).
18. A. M. Smith, G. Ruan, M. N. Rhyner, and S. Nie, Engineering
luminescent quantum dots for in vivo molecular and cellular imag-
ing. Ann. Biomed. Eng. 34, 3 (2006).
19. S. Koenig and V. Chechik, Shell cross-linked Au nanoparticles.
Langmuir 22, 5168 (2006).
20. X. Lou, C. Wang, and L. He, Core–shell Au nanoparticle for-
mation with DNA-polymer hybrid coatings using aqueous ATRP.
Biomacromolecules 8, 1385 (2007).
80 Rev. Nanosci. Nanotechnol., 1, 66–83, 2012
Delivered by Ingenta to:Guest User
IP : 76.98.2.41Sun, 29 Apr 2012 01:39:09
Liu et al. Computational Modeling of Nanoparticle Targeted Drug Delivery
21. Y. Yang, M. Nogami, J. Shi, and M. Ruan, Template guided
self-assembling two-dimensional array of Au@SiO2 core–shell
nanoparticles for room-temperature single electron transistors.
J. Nanosci. Nanotechnol. 5, 179 (2005).
22. Y. Cheng, Y. Gao, T. Rao, Y. Li, and T. Xu, Dendrimer-based
prodrugs: Design, synthesis, screening and biological evaluation.
Comb. Chem. High Throughput Screen 10, 336 (2007).
23. R. Duncan and L. Izzo, Dendrimer biocompatibility and toxicity.
Adv. Drug Deliv. Rev. 57, 2215 (2005).
24. M. Najlah and A. D’Emanuele, Crossing cellular barriers using
dendrimer nanotechnologies. Curr. Opin. Pharmacol. 6, 522
(2006).
25. C. G. Galbraith and M. P. Sheetz, Forces on adhesive contacts affect
cell function. Curr. Opin. Cell Biol. 10, 566 (1998).
26. W. R. Sanhai, J. H. Sakamoto, R. Canady, and M. Ferrari, Seven
challenges for nanomedicine. Nat. Nano 3, 242 (2008).
27. R. J. Shipley and S. J. Chapman, Multiscale modelling of fluid
and drug transport in vascular tumours. B Math Biol. 72, 1464
(2010).
28. S. Modok, R. Scott, R. A. Alderden, M. D. Hall, H. R. Mellor,
S. Bohic, T. Roose, T. W. Hambley, and R. Callaghan, Trans-
port kinetics of four-coordinate and six-coordinate platinum com-
pounds in the multicell layer tumour model. Brit. J. Cancer 97, 194
(2007).
29. M. Mahmoudi, M. A. Shokrgozar, A. Simchi, M. Imani,
A. S. Milani, P. Stroeve, H. Vali, U. O. Häfeli, and S. Bonakdar,
Multiphysics flow modeling and in vitro toxicity of iron oxide
nanoparticles coated with poly(vinyl alcohol). J. Phys. Chem. C
113, 2322 (2009).
30. X. L. Li, K. L. Yao, and Z. L. Liu, CFD study on the magnetic fluid
delivering in the vessel in high-gradient magnetic field. J. Magn.
Magn. Mater. 320, 1753 (2008).
31. Y. Liu, R. M. C. So, and C. H. Zhang, Modeling the bifurcating
flow in a human lung airway. J. Biomech. 35, 465 (2002).
32. Z. Zhang and C. Kleinstreuer, Airflow structures and nano-particle
deposition in a human upper airway model. Journal of Computa-
tional Physics 198, 178 (2004).
33. M. L. R. W. Mark Saltzman, Drugs released from polymers: Dif-
fusion and elimination in brain tissue. Chem. Eng. Sci. 46, 2429
(1991).
34. C.-H. Wang and J. Li, Three-dimensional simulation of IgG deliv-
ery to tumors. Chem. Eng. Sci. 53, 3579 (1998).
35. J. M. S. L. Paulo Costa, Modeling and comparison of dissolution
profiles. European Journal of Pharmaceutical Sciences 13, 123
(2001).
36. C. G. Varelas, D. G. Dixon, and C. A. Steiner, Zero-order release
from biphasic polymer hydrogels. J. Controlled Release 34, 185
(1995).
37. R. W. Korsmeyer, R. Gurny, E. Doelker, P. Buri, and N. A. Peppas,
Mechanisms of solute release from porous hydrophilic polymers.
Int. J. Pharm. 15, 25 (1983).
38. Higuchi, Rate of release of medicaments from ointment bases con-
taining drugs in suspension. J. pharm. Sci. 50, 874 (1961).
39. M. F. S. Gibaldi, Establishment of sink conditions in dissolution
rate determinations. Theoretical considerations and application to
nondisintegrating dosage forms. J. Pharm. Sci. 56, 1238 (1967).
40. J. Wagner, Interpretation of percent dissolved-time plots derived
from in vitro testing of conventional tablets and capsules. J. Pharm.
Sci. 58, 1253 (1969).
41. P. Costa and J. M. Sousa Lobo, Modeling and comparison of dis-
solution profiles. European Journal of Pharmaceutical Sciences
13, 123 (2001).
42. J. Berthier and P. Silberzan, Microfluidics for biotechnology. Artech
House Microelectromechanical System (MEMS) Series. Artech
House, Inc., Norwood, MA (2006).
43. B. Goldstein and M. Dembo, Approximating the effects of diffusion
on reversible reactions at the cell surface: Ligand-receptor kinetics.
Biophysical Journal 68, 1222 (1995).
44. B. D. Chithrani, A. A. Ghazani, and W. C. W. Chan, Determining
the size and shape dependence of gold nanoparticle uptake into
mammalian cells. Nano Letters 6, 662 (2006).
45. H. Lee, B. Hoang, R. M. Reilly, and C. Allen, The effects of parti-
cle size and molecular targeting on the intratumoral and subcellular
distribution of polymeric nanoparticles. Molecular Pharmaceutics
7, 1195 (2010).
46. J. B. Haun and D. A. Hammer, Quantifying nanoparticle adhesion
mediated by specific molecular interactions. Langmuir 24, 8821
(2008).
47. N. Nishiyama, Nanomedicine: Nanocarriers shape up for long life.
Nat. Nanotechnol. 2, 203 (2007).
48. M. Ferrari, Nanogeometry: Beyond drug delivery. Nat. Nanotech-
nol. 3, 131 (2008).
49. S. Cai, D. Cheng, E. M. Lima, and D. E. Discher, Micelles of
different morphologies—Advantages of worm-like filomicelles of
PEO-PCL in paclitaxel delivery. Pharmaceutical Research 24, 2099
(2007).
50. V. R. S. Patil, Y. H. Yun, S. M. Slack, and D. J. Goetz, Parti-
cle diameter influences adhesion under flow. Biophysical Journal
80, 1733 (2001).
51. J. A. Champion, S. Mitragotri, Role of target geometry in phago-
cytosis. Proc. Natl. Acad. Sci. USA 103, 4930 (2006).
52. P. Decuzzi and M. Ferrari, The adhesive strength of non-spherical
particles mediated by specific interactions. Biomaterials 27, 5307
(2006).
53. P. Decuzzi, S. Lee, B. Bhushan, and M. Ferrari, A theoretical model
for the margination of particles within blood vessels. Annals of
Biomedical Engineering 33, 179 (2005).
54. P. Decuzzi, S. Lee, M. Decuzzi, and M. Ferrari, Adhesion of micro-
fabricated particles on vascular endothelium: A parametric analysis.
Annals of Biomedical Engineering 32, 793 (2004).
55. N. A. Mody, O. Lomakin, T. A. Doggett, T. G. Diacovo,
and M. R. King, Mechanics of transient platelet adhesion to
von willebrand factor under flow. Biophysical Journal 88, 1432
(2005).
56. N. A. Mody and M. R. King, Three-dimensional simulations of
a platelet-shaped spheroid near a wall in shear flow. Phys Fluids
17 (2005).
57. H. Djohari and E. E. Dormidontova, Kinetics of nanoparticle tar-
geting by dissipative particle dynamics simulations. Biomacro-
molecules 10, 3089 (2009).
58. Y. Geng, P. Dalhaimer, S. S. Cai, R. Tsai, M. Tewari, T. Minko,
and D. E. Discher, Shape effects of filaments versus spherical
particles in flow and drug delivery. Nat. Nanotechnol. 2, 249
(2007).
59. S. Muro, J. A Champion, J. Leferovich, C. Gajewski, E. H. Schuch-
man, S. Mitragotri, and V. R. Muzykantov, Control of endothelial
targeting and intracellular delivery of therapeutic enzymes by mod-
ulating the size and shape of ICAM-1-targeted carriers. Molecu-
lar Therapy the Journal of the American Society of Gene Therapy
16, 1450 (2008).
60. Y. L. S. Shah, W. Hu, and J. Gao. Modeling particle shape-
dependent dynamics in nanomedicine. J. Nanosci. Nanotechnol.
11, 919 (2011).
61. S. Shah, Numerical Simulation of Particle Adhesion Dynamics for
Applications in Nanomedicine and Biosensing, University of Texas
at Arlington, Arlington (2009).
62. W. T. Winter and M. E. Welland, Dielectrophoresis of non-spherical
particles. Journal of Physics D: Applied Physics 42, 045501
(2009).
63. Y. L. Liu, J. H. Chung, W. K. Liu, and R. S. Ruoff, Dielec-
trophoretic assembly of nanowires. J. Phys. Chem. B 110, 14098
(2006).
Rev. Nanosci. Nanotechnol., 1, 66–83, 2012 81
Delivered by Ingenta to:Guest User
IP : 76.98.2.41Sun, 29 Apr 2012 01:39:09
Computational Modeling of Nanoparticle Targeted Drug Delivery Liu et al.
64. Y. L. Liu, K. Oh, J. G. Bai, C. L. Chang, W. Yeo, J. H. Chung,
K. H. Lee, and W. K. Liu, Manipulation of nanoparticles and
biomolecules by electric field and surface tension. Comput. Method
Appl. M 197, 2156 (2008).
65. P. Decuzzi and M. Ferrari, Design maps for nanoparticles targeting
the diseased microvasculature. Biomaterials 29, 377 (2008).
66. D. A. McQuarrie, Kinetics of small systems I. J. Chem. Phys.
38, 433 (1963).
67. P. Decuzzi and M. Ferrari, The adhesive strength of non-spherical
particles mediated by specific interactions. Biomaterials 27, 5307
(2006).
68. P. Calvo, B. Gouritin, H. Chacun, D. Desmaele, J. D’Angelo,
J. P. Noel, D. Georgin, E. Fattal, J. P. Andreux, and P. Couvreur,
Long-circulating PEGylated polycyanoacrylate nanoparticles as
new drug carrier for brain delivery. Pharmaceut. Res. 18, 1157
(2001).
69. J. F. Kukowska-Latallo, K. A. Candido, Z. Y. Cao, S. S. Nigavekar,
I. J. Majoros, T. P. Thomas, L. P. Balogh, M. K. Khan, and J. R.
Baker, Nanoparticle targeting of anticancer drug improves thera-
peutic response in animal model of human epithelial cancer. Cancer
Res. 65, 5317 (2005).
70. J. D. Heidel, Z. P. Yu, J. Y. C. Liu, S. M. Rele, Y. C. Liang,
R. K. Zeidan, D. J. Kornbrust, and M. E. Davis, Administra-
tion in non-human primates of escalating intravenous doses of
targeted nanoparticles containing ribonucleotide reductase sub-
unit M2 siRNA, Proceedings of the National Academy of
Sciences of the United States of America (2007), Vol. 104,
pp. 5715–5721.
71. B. C. H. Lutters, M. A. Leeuwenburgh, C. C. M. Appeldoorn, T. J.
M. Molenaar, T. J. C. Van Berkel, and E. A. L. Biessen, Block-
ing endothelial adhesion molecules: A potential therapeutic strategy
to combat atherogenesis. Current Opinion in Lipidology 15, 545
(2004).
72. L. Zhang, A. Gerstenberger, X. D. Wang, and W. K. Liu, Immersed
finite element method. Computer Methods in Applied Mechanics
and Engineering 193, 2051 (2004).
73. W. K. Liu, Y. L. Liu, D. Farrell, L. Zhang, X. S. Wang, Y. Fukui,
N. Patankar, Y. J. Zhang, C. Bajaj, J. Lee, J. H. Hong, X. Y. Chen,
and H. Y. Hsu, Immersed finite element method and its applications
to biological systems. Computer Methods in Applied Mechanics
and Engineering 195, 1722 (2006).
74. W. K. Liu, D. W. Kim, and S. Q. Tang, Mathematical foundations
of the immersed finite element method. Comput. Mech. 39, 211
(2007).
75. T. R. Lee, Y. S. Chang, J. B. Choi, D. W. Kim, W. K. Liu, and Y. J.
Kim, Immersed finite element method for rigid body motions in
the incompressible Navier-Stokes flow. Comput. Method Appl. M
197, 2305 (2008).
76. M. Dembo, D. C. Torney, K. Saxman, and D. Hammer, The
reaction-limited kinetics of membrane-to-surface adhesion and
detachment. P Roy Soc. Lond B Bio. 234, 55 (1988).
77. K. C. Chang, D. F. J. Tees, and D. A. Hammer, The state diagram
for cell adhesion under flow: Leukocyte rolling and firm adhesion,
Proceedings of the National Academy of Sciences of the United
States of America (2000), Vol. 97, pp. 11262–11267.
78. C. Dong, J. Cao, E. J. Struble, and H. W. Lipowsky, Mechanics of
leukocyte deformation and adhesion to endothelium in shear flow.
Annals of Biomedical Engineering 27, 298 (1999).
79. A. Einstein, Investigations on the Theory of Brownian Movement,
New York: Dover (1956).
80. D. L. Ermak and J. A. Mccammon, Brownian dynamics with hydro-
dynamic interactions. J. Chem. Phys. 69, 1352 (1978).
81. A. Li and G. Ahmadi, Dispersion and deposition of spherical-
particles from point sources in a turbulent channel flow.
Aerosol. Sci. Tech. 16, 209 (1992).
82. N. A. Mody and M. R. King, Influence of Brownian motion on
blood platelet flow behavior and adhesive dynamics near a planar
wall. Langmuir 23, 6321 (2007).
83. P. W. Longest and C. Kleinstreuer, Comparison of blood parti-
cle deposition models for non-parallel flow domains. J. Biomech.
36, 421 (2003).
84. F. Gentile, M. Ferrari, and P. Decuzzi, The transport of nanoparti-
cles in blood vessels: The effect of vessel permeability and blood
rheology. Ann. Biomed. Eng. 36, 254 (2008).
85. T. R. Lee, Y. S. Chang, J. B. Choi, W. K. Liu, and Y. J. Kim,
Numerical simulation of a nanoparticle focusing lens in a microflu-
idic channel by using immersed finite element method. J. Nanosci.
Nanotechnol. 9, 7407 (2009).
86. N. Sharma and N. A. Patankar, Direct numerical simulation of the
Brownian motion of particles by using fluctuating hydrodynamic
equations. J. Comput. Phys. 201, 466 (2004).
87. N. Mori, M. Kumagae, and K. Nakamura, Brownian dynamics sim-
ulation for suspensions of oblong-particles under shear flow. Rheol
Acta 37, 151 (1998).
88. K. Inthavong, H. Wen, Z. F. Tian, and J. Y. Tu, Numerical study
of fibre deposition in a human nasal cavity. J. Aerosol. Sci. 39, 253
(2008).
89. A. Haider and O. Levenspiel, Drag coefficient and terminal velocity
of spherical and nonspherical particles. Powder Technol. 58, 63
(1989).
90. S. Tran-Cong, M. Gay, and E. E. Michaelides, Drag coeffi-
cients of irregularly shaped particles. Powder Technol. 139, 21
(2004).
91. M. Dimaki and P. Boggild, Dielectrophoresis of carbon nano-
tubes using microelectrodes: A numerical study. Nanotechnology
15, 1095 (2004).
92. E. Loth, Drag of non-spherical solid particles of regular and irreg-
ular shape. Powder Technol. 182, 342 (2008).
93. Y. Saad and M. H. Schultz, Gmres—A generalized minimal resid-
ual algorithm for solving nonsymmetric linear-systems. Siam. J.
Sci. Stat. Comp. 7, 856 (1986).
94. Y. Liu, W. K. Liu, T. Belytschko, N. Patankar, A. C. To, A. Kopacz,
J. H. Chung, Immersed electrokinetic finite element method. Int. J.
Numer. Meth. Eng. 71, 379 (2007).
95. C. Pozrikidis, Flipping of an adherent blood platelet over a sub-
strate. J. Fluid Mech. 568, 161 (2006).
96. E. Gavze and M. Shapiro, Particles in a shear flow near a solid wall:
Effect of nonsphericity on forces and velocities. Int. J. Multiphas
Flow 23, 155 (1997).
97. S. N. Naess and A. Elgsaeter, Transport properties of non-spherical
nanoparticles studied by Brownian dynamics: Theory and numeri-
cal simulations. Energy 30, 831 (2005).
98. M. G. Forest, S. Heidenreich, S. Hess, X. F. Yang, and R. H. Zhou,
Robustness of pulsating jet-like layers in sheared nano-rod disper-
sions. J. Non-Newton Fluid 155, 130 (2008).
99. J. F. Zhang, P. C. Johnson, and A. S. Popel, Effects of erythrocyte
deformability and aggregation on the cell free layer and appar-
ent viscosity of microscopic blood flows. Microvasc. Res. 77, 265
(2009).
100. J. P. Woodcock, Physical-properties of blood and their influ-
ence on blood-flow measurement. Rep. Prog. Phys. 39, 65
(1976).
101. L. E. Bayliss, The axial drift of the red cells when blood
flows in a narrow tube. The Journal of physiology 149, 593
(1959).
102. M. Taylor, The flow of blood in narrow tubes. II. The axial stream
and its formation, as determined by changes in optical density. The
Australian Journal Of Experimental Biology and Medical Science
33, 1 (1955).
103. J. B. Haun and D. A. Hammer, Quantifying nanoparticle adhesion
mediated by specific molecular interactions. Langmuir 24, 8821
(2008).
82 Rev. Nanosci. Nanotechnol., 1, 66–83, 2012
Delivered by Ingenta to:Guest User
IP : 76.98.2.41Sun, 29 Apr 2012 01:39:09
Liu et al. Computational Modeling of Nanoparticle Targeted Drug Delivery
104. M. Babincova and P. Babinec, Magnetic drug delivery and tar-
geting: Principles and applications. Biomed. Pap. Med. Fac. Univ.
Palacky Olomouc Czech Repub. 153, 243 (2009).
105. K.T.A.-J. Mingguang Li, Kostas kostarelos and joshua reineke.
Physiologically based pharmacokinetic modeling of nanoparticles.
ACS Nano 4, 6303 (2010).
106. Chae-Kyu Kim, Multimodal drug delivery using gold nanoparticles.
Nanoscale 1 (2009).
107. M. B. Lawrence and T. A. Springer, Leukocytes roll on a selectin
at physiological flow-rates—Distinction from and prerequisite for
adhesion through integrins. Cell 65, 859 (1991).
108. G. I. Bell, M. Dembo, and P. Bongrand, Cell-adhesion - competition
between nonspecific repulsion and specific bonding. Biophys. J.
45, 1051 (1984).
109. G. I. Bell, Models for specific adhesion of cells to cells. Science
200, 618 (1978).
Rev. Nanosci. Nanotechnol., 1, 66–83, 2012 83