Computational Modeling of Nanoparticle Targeted Drug Delivery

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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

66 Rev. Nanosci. Nanotechnol. 2012, Vol. 1, No. 1 2157-9369/2012/1/066/018 doi:10.1166/rnn.2012.1014


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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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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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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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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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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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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)



IP : 76.98.2.41Sun, 29 Apr 2012 01:39:09


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)



IP : 76.98.2.41Sun, 29 Apr 2012 01:39:09


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,



IP : 76.98.2.41Sun, 29 Apr 2012 01:39:09


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



IP : 76.98.2.41Sun, 29 Apr 2012 01:39:09


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



IP : 76.98.2.41Sun, 29 Apr 2012 01:39:09


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.



IP : 76.98.2.41Sun, 29 Apr 2012 01:39:09


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.

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