Computational Modeling of Nanoparticle Targeted ... - Lehigh University

Reviews in Nanoscience and Nanotechnology Vol. 1, pp. 66–83, 2012 (www.aspbs.com/rnn)

Copyright © 2012 by American Scientific Publishers All rights reserved. Printed in the United States of America

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 process and design nanoparticles for Delivered by Ingentadelivery to: efficient targeted delivery. The modeling approaches span from continuum vascular flow, particle Brownian Guest User adhesion dynamics, to molecular level ligand-receptor binding. Computer simulation is envisioned to be able to IP : 76.98.2.41 optimize drug carrier design and predictSun, drug29 delivery efficiency for patient specific vascular environment. Apr 2012 01:39:09 KEYWORDS: Nanomedicine, Nanoparticle, Cancer, Drug Delivery, Computational Modeling.

CONTENTS 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Continuum Approach: Drug Dissolution to Convection-Diffusion-Reaction Model of Drug Delivery . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Dissolution of Drug Particles . . . . . . . . . . . . . . . . . . 3.3. Convection-Diffusion-Reaction Model of Drug Delivery . 3.4. Nanoparticle Binding in a Channel . . . . . . . . . . . . . . 3.5. Nanoparticle Deposition and Distribution in a Blood Vessel Network . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Particulate Approach: Rational Design of Nanoparticles . . . . 4.1. Introduction to Nanoparticle Design . . . . . . . . . . . . . 4.2. Influence of Nanoparticles Size and Shape on Targeted Delivery . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Theoretical Model of Nanoparticle Adhesion Probability 4.4. Particulate Model of Nanoparticle Delivery in a Vascular Environment . . . . . . . . . . . . . . . . . . . . . . 4.5. Simulation Results of Nanoparticle Targeted Delivery Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Future Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . .

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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

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1 to 100 nm in at least one dimension.1 Nanotechnology 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 treatment by enabling the administration of new intricate drugs, improving the bioavailability of existing drugs, and providing spatial and temporal targeting of drugs in order to dramatically reduce side effects and increase effectiveness. Through accomplishment of these revolutionary advantages, patients and physicians could benefit from personalized prescriptions, alleviated administration, increased patient compliance, reduced dosage frequency and less pain. Over the past decade, we have witnessed an explosive development of nanoparticulate systems for diagnostic imaging and targeted therapeutic applications.3–10 Various nanoplatforms, including liposomes,111 12 polymeric micelles,13–16 quantum dots,171 18 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 consideration and comprehensive analysis. The targeted drug 2157-9369/2012/1/066/018

doi:10.1166/rnn.2012.1014

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Fig. 1. The targeted drug delivery process spans across multiple spatial scales.

techniques such as Molecular dynamics, Brownian motion, based drug delivery model is introduced, which covers and stochastic approaches such as Monte Carlo simuthe basic governing equations and a few examples of lation to capture the nanoparticle motion. For example, targeted drug delivery under vascular conditions. Second, Shipley et al.27 and Modok et al.28 modeled delivery of particulate modeling based on coupled Brownian adhesion 30 spherical NPs in tumor. Mahmoudi et al.29 and Li et al. dynamicsto: method is described, where the motion and bindDelivered by Ingenta performed computational fluid dynamics studies of Guest mag- User ing of individual nanoparticles in the blood stream are 32 et al. netic NPs in vascular flow. Liu et al.31 and Zhang IP modeled. Finally, the future trend in computational mod: 76.98.2.41 studied the deposition of NPs in lung airway. Figure 2 eling of targeted drug delivery is briefly discussed. Sun, 29 Apr 2012 01:39:09 shows a multiscale simulation framework for targeted drug delivery ranging from continuum model to partic3. CONTINUUM APPROACH: DRUG ular model. It’s the recent advancement in the compuDISSOLUTION TO tational science that made computational modeling very CONVECTION-DIFFUSION-REACTION promising for targeted drug delivery application. The ligMODEL OF DRUG DELIVERY and coated nanoparticles, loaded with drugs inside, transport in blood stream, and adhere to diseased cells via 3.1. Introduction specific adhesion. However, this process becomes intriIn vivo drug release, transportation and targeted binding cate due to simultaneous involvement of hydrodynamic have been recognized as important elements in targeted force, adhesion force and Brownian force. In particular, drug delivery field. In order to target the disease area, the the ligand-receptor interaction is a sophisticated chemical drug loaded carriers are first injected into the circulation process. The surface property of functionalized nanopartisystem, where they transport through, across and within cles would play a crucial role to dictate the efficiency of vessels, tissues and cells. Due to specific binding between the targeted drug delivery by providing targeted selectivligand coated drug particles and receptors, expressed at ity. Computational modeling tool will lead to insights of the disease cell membrane, drug loaded particles deposit the dynamic delivery process, thus facilitate better design on the targeted disease region. Afterward process is folof nanoparticles. lowed by cellular uptaking and drug releasing. From the This article focuses on multiscale computational continuum stand point of view, the drug delivery process approach to the targeted drug delivery. First, continuum 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 controlled 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 spaFig. 2. A framework for multiscale modeling of the entire drug delivery tial distribution. A three dimensional (3D) simulation of system. 68

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human brain tumor of primitive neuroectodermal tumor was performed by Wang et al.34 The simulation is conducted on CFD tools to solve simultaneously continuity, momentum and drug concentration equations. Using their model, the contribution of convective transport of macromolecular and micromolecular drugs in the vicinity of tumor were studied. In this section, the governing equations 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

Higuchi38 formulated following relation to model low soluble drug release problem: p (5) ft = D42C − Cs 5Cs t

Where C is the drug initial concentration, Cs is the drug solubility in the matrix media and D is the diffusion coefficient 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 system can be described by the convection-diffusion equation: ¡c

(1)

+− →U · ïc = ï · 4Dïc5

(6)

Delivered by Ingenta to: ¡t Guest User Where Q0 is the initial amount of drug in the pharmaceuWhere c is the concentration of nanoparticles, D is the IP : 76.98.2.41 tical dosage form, Qt is the amount of drug in the pharmadiffusion coefficient of nanoparticle and U is the flow Sun, 29 Apr 2012 01:39:09 ceutical dosage form at time t and K is a proportionality velocity. It is solved by using following Einstein-Stokes constant. Dividing the above equation by Q0 simplifying it to: (2) f0 = kt

Where ft = 1 − Qt /Q0 is often referred as fraction of drug. This relation can be used to describe the drug dissolution 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 dissolution rate of the drug is described by the Noyes-Whitney equation as shown below: dC = K4Cs − C5 dt

(3)

Where C is the concentration of solid in bulk dissolution 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 complex, 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:   4t − T i5b (4) m = 1 − exp − a Where a is time related constants, Ti represents time lag before onset of the dissolution, b is a curve characterized parameter. Rev. Nanosci. Nanotechnol., 1, 66–83, 2012

equation,

kB T (7) 6Œr Where kB is the Boltzmann constant, T is the temperature, Œ 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 complex biochemical reaction. To depict the effect of adsorption 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 continous 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: D=

¡cs + ï · 4−Ds ïcs 5 = ka cw ˆ − kd cs ¡t

(8)

Where Ds is the surface diffusivity (m2 /s), cw is the bulk concentration of the species at solid wall (unit mol/m3 5, ˆ is the surface concentration on the active site (mole/m2 5 Table I. Mathematical models used to describe drug dissolution curves. Zero order First order Hixson-crowell Weibull Higuchi Baker-lonsdale Korsmeyer-peppas Quadratic Logistic Gompertz Hopfenberg

Qt = Q0 + K0 t ln Qt = ln Q0 + K1 t Q01/3 − Qt1/3 = Ks t ln6− ln41 − 4Qt /Qˆ 557 √ = b ln4t5 − ln4a5 Qt = KH t 3/261 − 41 − Qt /Qˆ 52/3 7 − Qt /Qˆ = Kt Qt /Qˆ = Kk t n Qt = 1004K1 t 2 + K2 t5 Qt = A/61 + e−K4t−y5 7 Qt = Ae−e−K4t−y5 7 Qt /Qˆ = 1 − 61 − k0 t/C0 a0 7n

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and cs surface concentration of adsorbed species (mol/m2 5. Note that cs is different from c which is reflected in their units. ka (m3 /mol/s) and kd (s−1 5 are adhesion and detachment rates, respectively. However, the concentration of active sites is equal to the difference between the total concentration of active sites and the number of sites occupied by the adsorbed species. This gives the equation for the reaction rate as: ¡cs + ï · 4−Ds ïcs 5 = ka c4ˆ0 − cs 5 − kd cs ¡t

(9)

(A)

10 µm

Receptor coated reaction surface (B)

In above equation, ˆ0 represents the total number of active sites available on the active surface. Convection-diffusion equation and nanoparticle reaction equation are not independent, instead, they are coupled through Fick’s law: ¡cs = −D · ïc—w ¡t

Flow rate 0.1mm/s

Flow rate 1mm/s 10 µm

Receptor coated reaction

(10) Fig. 3.

Nanoparticle binding in a channel at a flow rate of 0.1 and

Delivered by Ingenta to: concentration drops close to the receptor coated surface 1 mm/s. Particle 3.4. Nanoparticle Binding in a Channel due to adhesion, forming a depletion layer. Red color indicates highest Guest User concentration, while blue color indicates lowest concentration. IP : 76.98.2.41 To demonstrate application of continuum model in targeted Sun, Apr 2012 01:39:09 drug delivery, finite element modeling is used to 29 evaluate Figure 3 shows the depletion layer at shear rates 0.1 mm/s the nanoparticle transportation diffusion and biochemical reaction dynamics in a channel. In this model, the convection diffusion in 2D fluid domain is coupled with the adhesion reaction occurring on the reaction surface (disease 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 process of nanoparticle in 2D fluid domain is coupled with the adhesion reaction occurring only on the reaction surface 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 influenced 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. Symbol c0 ka kd ˆ0 Ds D kB T U ‹

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and 1 mm/s respectively. As the flow rate increases the depletion layer thickness decreases due to greater nanoparticle 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 delivery 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

Physical parameters used in nanoparticle binding in a channel. Value

Definition

1000 [mol/m3 ] 10−6 [m3 /(mol*s)] 10−3 -10−6 [1/s] 1000 [mol/m2 ] 10−11 [m2 /s] 10−9 [m2 /s] 1038 × 10−23 [m2 kg s−2 K−1 ] 300 [K] 0–25 [dyne/cm2 ] 10−10 [m]

Initial concentration Adhesion rate constant Detachment rate constant Active site concentration Surface diffusivity Particle diffusivity in the fluid Boltzmann constant Absolute temperature Maximum shear rate Equilibrium bond length

Fig. 4. (A) Drug injected at the top inlet of an idealized vascular network with three generations; (B) Receptors coated vessel section in the left branch of vascular network. Rev. Nanosci. Nanotechnol., 1, 66–83, 2012

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dedicated to comprehend their biological behaviors in vitro and in vivo. For example, it is known that spherical particles bigger than 200 nm are efficiently filtered by the Symbol Value Definition spleen, while particles smaller than 10 nm can be quickly c0 1000 [mol/m3 ] Initial concentration cleared by the kidney, thus making 10–200 nm as an ideal ka 10−6 [m3 /(mol*s)] Adhesion rate constant −9 size range for the spherical carriers. 10 [ 1/s] Detachment rate constant kd ˆ0 1000 [mol/m2 ] Active site concentration Similar to size, shape is a fundamental property of Ds 10−11 [m2 /s] Surface diffusivity micro/nanoparticles that may be critically important for Particle diffusivity in the fluid D 10−6 [m2 /s] their intended biological functions.44–50 Recent data begin V 1 [mm/s] Maximum velocity 3 Blood density  1063 [kg/m ] to reveal that particle shape may have a profound effect ‡ 0.003 [Pa.s] Blood dynamics viscosity on their biological properties. For example, cylindrically shaped filomicelles can effectively evade the non-specific uptake by the reticuloendothelial systems and persisted in nanoparticle surface. The particle depletion layer is clearly the circulation up to one week after intravenous injection. visible in the target region. The density of deposited drug From drug delivery stand point of view, non-spherical parparticles on the wall surface is plotted in Figure 5, which ticles will allow larger payload delivery than the spherindicates that most drug particles are deposited at the ical counterpart with same binding probability. Recently, entrance of the target region, while the rest of the target Mitragotri and coworkers have shown that the local shape region has low density of deposited drug particles. There of the particle Delivered to: at the point where a macrophage is attached, are no particles deposited in the healthy branch due to anby Ingenta not the overall shape, dictated whether the cell began Guest User assumption of zero non-specific adhesion at that particular internalization.51 These results indicate the importance of IP : 76.98.2.41 location. Such non-uniform distribution pattern indicates controlling particle shape for nanomedicine application. Apr 2012 01:39:09 possible impaired delivery dosage within theSun, target29 region, Theoretical studies of nanoparticle deposition are typwhich is important for delivery efficacy prediction and ically focused on simple spherical or oblate shape.52–54 dosage planning. Ideally, there should be a tool that can handle variety of shapes and sizes of nanoparticles, which enables endless 4. PARTICULATE APPROACH: RATIONAL possibilities of finding most suitable design of the nanoparDESIGN OF NANOPARTICLES ticle for a given application. Decuzzi and Ferrari.52–54 have studied the margination of nanoparticles in blood stream, 4.1. Introduction to Nanoparticle Design where nanoparticles diffusion in Newtonian fluid has been Most of the nanoparticles employed in the experimental analyzed. The same authors have also examined the adhestudies are spherical in shape. Extensive studies have been sion probability of nanoparticles under an equilibrium Table III. Physical parameters used for blood vessel network simulation.

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. Rev. Nanosci. Nanotechnol., 1, 66–83, 2012

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configuration. Mody et al.551 56 studied platelet motion near vessel wall surface under shear flow and concluded that hydrodynamic force influences platelet adhesion to the wall surface. The same authors551 56 also investigated the influence of Brownian motion on platelet movement and found that Brownian motion does not influence plateletshaped cells at physiological shear rates. However, size (∼2 Œm) and shape (oblate) of the platelet is not comparable to that of nanoparticles and the behavior observed for platelet might not be applicable for nanoparticles.

nanoparticle was found to become ellipsoidal with increasing 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 carrier 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. Further, 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. 4.2. Influence of Nanoparticles Size and Shape on Muro et al.59 studied controlled endothelial targeting and Targeted Delivery intracellular delivery by modulating size and shape of the drug carrier. Their study found that carrier geometry influThe targeted drug delivery process in general can be ences endothelial targeting efficiency. The non-spherical considered as a seamless combination of three stages: carrier had longer circulation time and higher targeting transport through the vessel network; adhesion process; specificity than regular spherical carrier. Shah and Liu and cellular update. Each stage is effectively governed et al.60 has by nanoparticle shape, size, and surface property. Spe-by Ingenta Delivered to:compared the transport phenomenon and binding probability of nanospheres and nanorods under shear cific combination of each parameter can accomplishGuest effi- User flow and revealed significantly higher binding efficiency cient targeted delivery. There have been a large number of IP : 76.98.2.41 of 01:39:09 nanorods due to their tumbling motion and larger constudies devoted on characterization of nanoparticle Sun, 29physiApr 2012 tact area.461 601 61 Winter et al.62 and Liu et al.631 64 have cal property. Djohari and Dormidontova57 studied kinetics performed numerical simulations of dielectrophoresis of of spherical nanoparticle for targeting cell surface using non-spherical particles. Theoretical models to determine 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 Œm; (d) Distribution of inert and degradable filomicelles in clearance organs for Lo = 4 or 8 Œm 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.

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adhesion probability of different shaped nanoparticles is discussed in the next section.

spherical particles of the same volume. Furthermore, the disc-shaped nanoparticles have the highest adhesion probability and the largest volume to the mass ratio, resulting in 300 times higher efficacy for cell targeting and 40 times 4.3. Theoretical Model of Nanoparticle Adhesion higher drug-loading capability than their spherical counterProbability part. The adhesion probability of the rod-shaped particle, A numerical model is built based on the previous work with the aspect ratio of 5, is about 20 times higher than the by Decuzzi and Ferrari521 65 to describe the cell targetspherical particle. Similarly, the oblate shaped nanopartiing process of nanorods and nanodisks under flow concles exhibit almost 10 times higher adhesion probability ditions. The adhesion probability (Pa 5 is characterized by compared to the spherical particles over the entire range the probabilistic kinetic formulation of McQuarrie66 and of volume considered. Decuzzi:67 This theoretical model is based the adhesion probability for a particle to adhere on receptor coated surface, ‹f Pa ' mr ml Kao Ac exp6− 7 (11) but lacks in revealing dynamic process of nanoparticle kB T transport and delivery. A coupled model that integrates Where mr is the receptor density on the substrate surmargination with adhesion kinetics, and applicable to face, ml is the ligand density on particle surface, Ac is nanoparticle of various shapes, is yet to be developed. the contact area of particle, f is force acting per unit Thus, analysis of this process for an arbitrarily shaped ligand-receptor pair, kB T is thermal energy ofDelivered system, ‹ isby Ingenta to: through a multiscale model is crucial to pronanoparticle o a characteristic length of ligand-receptor bond, and KGuest User a is vide biological insights on the transportation and adhethe affinity constant of ligand-receptor pair at zeroIPload. : 76.98.2.41 sion kinetics. In what follows, we will first introduces Normalized adhesion probability of oblate-, Sun, rod- and 29 discApr 2012 the01:39:09 nanoparticle adhesion kinetics theory and modelshaped nanoparticles for a wall shear stress of 1 (Pa) is ing method. Then, adhesion process and trajectories for plotted as a function of particle volume in Figure 7. The nanoparticles of different shapes and ligand densities are aspect ratio of disc (diameter over height) and rod (length presented. Next, the binding probability of nanoparticles over diameter) are chosen to be 5. is determined for a range of channel sizes. 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 4.4. Particulate Model of Nanoparticle Delivery in a bond formation, and then decreases due to the large volVascular Environment ume. Particles of larger volume faces larger dislodging Nanoparticles are usually introduced into the vascular force, which overwhelms bond forces and get the particle circulation stream through intravenous injection.68–70 The washed away, thus reduces the adhesion probability. Due targeted delivery efficiency is directly related to the the shape effect, the critical volume of the non-spherical nanoparticle selectively and ability to bind at the targeted particles is comparatively large and shifts to the right site. Though highly selective nanoparticles have reduced side of the plot (out of the plot range). For the range binding probability in non-target regions, the majority of of volume considered, the oblate, rod and disc particles nanoparticles are still lost in the vascular network due to show significantly higher adhesion probabilities than the non-specific adhesion. It is thus important to predict vanished 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

Fig. 7. Adhesion probabilities of nanoparticles of various shapes as a function of particle volume, ƒ is the aspect ratio. Rev. Nanosci. Nanotechnol., 1, 66–83, 2012

To achieve targeted drug delivery, nanoparticles are usually coated with ligands that bind specifically to a particular 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 interaction results in a bond formation between nanoparticle and 73

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vascular surface. The adhesive strength of bond is medikf0 are the reverse and forward reaction rates at zero load ated by the specific binding of ligand-receptor. Other facof ligand-receptor pair, respectively; L is the difference tors such as steric interactions, electrostatic, van der Waals between bond length y and equilibrium length ‹. During (vdW) or hydrodynamic forces also influence interactions dynamic interaction process, the bond length of a ligandbetween nanoparticle and vessel surface. However, the receptor pair may vary based on particle location. The vdW force is usually several orders of magnitude smaller ligand-receptor bonds are modeled as springs with spring compared to specific adhesive force, whilst the effect of constant ‘ and equilibrium length ‹, thus the bond forces steric interaction and electrostatic is limited to very short are described as a function of bond length y. Then, the distance (less than nanometer scale).53 Thus, these factors ligand-receptor interaction forces can be summed on finite are neglected for this model. The ligand-receptor bindelement surface through integration over the nanoparticle ing process is integrated with Brownian dynamics and this surface. Equations of bond forces fL and integrated adhecombined model is embedded into the Immersed Finite sion forces ‘ s on particle surface â are given as: Element (IFEM) platform.72–75 IFEM can be used for fully fL = ‘4y − ‹5 (15) coupled fluid-structure interaction problems, i.e., solving Z particle motion in a fluid while capturing the influence of ‘ S · n = Nb fL 4X c 5dâ (16) particle on fluid flow. However, due to Brownian motion, it is computationally expensive to calculate the change of Such adhesion force is coupled with the fluid-structure fluid flow caused by particle motion at every time step. interaction (FSI) force in the IFEM formulation. Similar Also, the effect of nanoparticle motion is limited to theby Ingenta to: Delivered adhesion model has been used by Chang et al.77 and Dong local surrounding region. Thus, the influence of partiGuest User et al.78 in the study of white blood cell rolling. The physcle motion on the fluid flow is neglected and theIP model : 76.98.2.41 ical parameters used in the model are listed in Table IV. is only solved for the particle motion andSun, the adhesion 29 Apr 2012 Besides 01:39:09 adhesion forces, the Brownian force acting on process. to the nanoparticles is also important and is integrated into When a particle approaches the vascular wall, ligands the IFEM formulation by adding a Brownian force term, on the particle surface form bonds with receptors on the which is described in the next section. vascular wall, as demonstrated in Figure 8. An adhesion kinetic equation is used to calculate the bond density Nb :76 4.4.2. Brownian Dynamics at Nanoscale ¡Nb (12) = kf 4Nl − Nb 54Nr − Nb 5 − kr Nb Fundamental theories of Brownian dynamics have indi¡t cated that random collisions from surrounding liqWhere Nl and Nr are the ligand and receptor densities; kr uid molecules impacts motion of an immersed small and kf are the reverse and forward reaction rates, respecparticle.79–81 The influence of Brownian motion on tively. This interaction model represents a conservation behavior of nanoparticles in microfluidic channel and equation of the different species (ligands, receptors, and platelets and blood cells in blood flow has been studied bonds). The kr and kf are function of bond length: extensively.82–85 Patankar et al.86 have proposed an algorithm for direct numerical simulation of Brownian motion kr = kr0 exp4−4ks − kts 5L2 /2Bz 5 (13) by adding random disturbance in fluid. At microscale, the 0 2 drag force acting on particles such as blood cells is sigkf = kf exp4−kts L /2Bz 5 (14) nificantly large (>50 pN for particle size >1 Œm), thus Where ks is the bond elastic constant; kts is the bond elasBrownian motion is neglectable.82 At nanoscale, Brown0 tic constant at transient state; Bz is thermal energy; kr and 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 dynamShear Flow ics at nanoscale and governing equations are described as Receptor following. Ligand The random forces R4t5 and torque T4t5 acting on a nanoparticle is responsible for Brownian motion and rotation and satisfy the fluctuation-dissipation theorem:87 “Ri 4t5” = 01

Fig. 8. Model of ligand-receptor adhesion kinetics between ligand coated nanoparticle surface and receptor coated vascular wall surface.

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“Ti 4t5” = 0

(17)

“Ri 4t5Rj 4t 0 5” = 2kB T ‚t „ij „4t − t 0 5„1 “Ti 4t5Tj 4t 0 5” = 2kB T ‚r „ij „4t − t 0 5„

(18)

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Table IV. List of physical parameters used in the nanoparticle adhesion kinetics model. Definition

Symbol

Ligand density Receptor density Reverse reaction rate Forward reaction rate Equilibrium bond length Static bond spring constant Transient bond elastic constant Thermal energy Fluid viscosity

Value 10

Reference 2

Lawrence and Springer (1991)107 Bell et al. (1984)108 Bell (1978)109 Bell (1978)109 Bell (1978)109 Dembo et al. (1988)76 Dembo et al. (1988)76 Dembo et al. (1988)76 —

200 × 10 (sites/cm 5 2.0–5.0×1010 (sites/cm2 5 0.5 (1/s) 100 × 10−9 (cm2 /s) 20 nm 0.5 (dyne/cm) 0.48 (dyne/cm) 400 × 10−14 (erg) 0.01 (g/cm-s)

Nl Nr kr0 kf0 ‹ ‘ kts Bz Œ

Where „ is the unit-second order tensor, „ij is the The friction coefficient of a rod-shaped particle for an arbitrary orientation is given by Ref. [92]: Kronecker delta, „4t − t 0 5 is the Dirac delta function, kB T is thermal energy of system, ‚t and ‚r are the trans(19) ‚t = 3Œdeqv × 4f˜ · —cos ˆ— + f⊥ · —sin ˆ—5 lational and rotational friction coefficient of nanoparticle, respectively. 3 (20) ‚r = Œdeqv The friction coefficient depends on several physical Delivered by Ingenta to: Where Œ is the fluid viscosity, deqv is the diameter of partiparameters, such as fluid viscosity, size and shape of the Guest User cle volume equivalent sphere, ˆ is the angle between flow nanoparticle. The friction coefficient for spherical-shaped IP : 76.98.2.41 direction and the long axis of the particle, f— and f− are particles can be easily derived from Stokes’ law. However, Sun, 29 Apr 2012 01:39:09 Stokes correction factors for a spheroid particle moving there is no empirical formula available for determining the parallel and perpendicular to the flow, respectively. These friction coefficient of particles with complex shapes. In correction factors are expressed as Ref. [92]: literature, there are empirical formulas for friction coeffi  cients for particles, but limited to simple shapes and ori4 ƒ −1/3 entations such as oblate or rod-shaped particles.88–91 In a f˜ = + ƒ (21) 5 5 recent work by Loth,92 new empirical formula is proposed   to compute friction coefficient for a non-spherical particle. 3 2ƒ −1/3 + ƒ = (22) f ⊥ Friction coefficient of rod shaped particles in this work 5 5 92 is derived based on Loth and extended with an angle Where ƒ is the aspect ratio of the spheroid particle. The factor to incorporate arbitrary orientations. When a partivelocity of a particle moving under a deterministic force cle travels along the fluid flow, the relative velocity of the in a fluid with velocity Vf is given by: particle can be divided into components in two directions:   parallel to flow and perpendicular to flow, as shown in Fdet Vs = +V f 41 − e−4‚t /m5t 5 (23) Figure 9. ‚t

V cosθ θ V

r

V sinθ

Fig. 9. Illustration of friction coefficient measurement of arbitrarily orientated nanorod. Rev. Nanosci. Nanotechnol., 1, 66–83, 2012

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 vectors, respectively. For a time step (typically ∼1 Œs) much greater than characteristic time constant m/‚t (∼10 ns), the nanoparticle moves with a terminal velocity, thus Eq. (23) reduces to: F Vs = det + Vf (24) ‚t 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 nanoparticle position in translational direction. Similarly, the angular velocity of a nanoparticle can be obtained through: —s =

Tdet + —f ‚r

(25) 75

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load capacity. The simulations are carried over a channel of 5 Œm long and 2 Œm 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. vi = Vs + —s × ri (26) A velocity is applied at the top of channel to generate a shear rate of 8.0 s−1 . Nanoparticles are allowed to move The fluid flow in our simulation is assumed to freely through the channel under the influence of shear be an incompressible viscous fluid governed by the flow and Brownian forces. For a typical simulation demonNavier-Stokes equations: strated in Figure 10, the spherical particle fails to make   any contact with the vessel wall while it travels through the ¡vf  + vf · ï vf = −ïp + Œï 2 vf (27) channel. Under given velocity and channel length, Brow¡t nian diffusion is not large enough to make the spherical particle to reach close enough to the wall surface to iniï · vf = 0 (28) tiate binding process. Compared to nanospheres, nanorods It should be noticed that vf is the fluid velocity in the make contact and adhere to vessel wall quickly and frefluid main, while Vf is the fluid velocity interpolated onto quently. The rod-shaped particle exhibits tumbling motion the solid domain. The Navier-Stokes equations are solved by virtue of non-spherical shape while flowing through the through finite element method. To reduce numerical oscil-by Ingenta to: Delivered channel. Due to the tumbling motion, a nanorod usually lations, the velocity test function is employed along with Guest User contacts with the receptor coated wall with bonds formed stabilization parameters. Using integration by parts and IP : 76.98.2.41 at the long axis end first. Such initial contact is followed the divergence theorem, the Patrov-Galarkin weak form Sun, 29 Apr 2012 by 01:39:09 nanoparticle rotation along the contact end and steadily is obtained. Then, the nonlinear system is solved using growing adhesion force, which ensure firm adhesion to the the Newton-Raphson method. Moreover, Generalized Minvessel wall and at the end settle down at equilibrium state imum Residual (GMRES) iterative algorithm is employed with full contact. The simulation results reveal typical trato improve computation efficiency and to compute residujectories of a nanosphere and a nanorod, which illustrate als based on matrix-free techniques.93 Details of the impledifferent dynamic adhesion processes. A more quantitative mentation can also be referred to Zhang et al. and Liu description of the adhesion process will be presented in et al.72–741 94 later sections. One question that might arise at this point is the exis4.5. Simulation Results of Nanoparticle Targeted tence of such near wall particle tumbling motion. In litDelivery Process erature, tumbling of non-spherical particles near a wall surface has been reported.561 951 96 The combined effects Mathematical modeling of targeted drug delivery system of shear flow and Brownian rotation have been found to provides quantitative description of the drug transportaenhance rotation of nanorods.971 98 tion in biological systems. Therefore, it can be utilized to Comparing Trajectories of Nanospheres and Nanorods. evaluate efficiency of drug delivery and to estimate dose Nanorods are expected to have higher probability to conresponse. tact with the wall surface than their spherical counter parts because of tumbling motion. To test this theory, trajecto4.5.1. Effect of Nanoparticle Shape on Adhesion ries of spherical and non-spherical nanoparticles under the Kinetics same flow condition are compared. A shear rate of 8.0 s−1 is employed for the both cases. The simulations are carried The following section discusses about influence of over the channel with the length equal to 15 Œm and the nanoparticle geometry on adhesion kinetics. Two separate height equal to 5 Œm. sets of simulation studies have been performed to evaluate To illustrate the fluctuations of nanoparticle-wall disnear wall behavior of spherical particle and non-spherical tance, minimum distance between the nanoparticle surface particle. and the wall surface is recorded over the time, as shown Comparing Deposition Process of Nanoparticles. To in Figure 11(A). Such nanoparticle trajectory indicates the investigate the influence of nanoparticle shape on adhesion path of nanoparticle during its motion through the channel. kinetics, two nanoparticles of different shapes, spherical In a series of simulation runs, a nanosphere and a nanorod and non–spherical, but of the same volume are considare placed initially 650 nm above the wall surface. The ered in this study. The length of the rod shaped particle trajectories of nanorod and nanosphere of 20 independent considered is 1000 nm with an aspect ratio of 5. The simulations are plotted in Figure 11(B). diameter of spherical particle is 380 nm. Such constant The simulation result elucidates that a nanorod has volume comparison helps to understand whether nanorod or nanosphere bind easily to wall surface for a given drug larger fluctuations in trajectories due to tumbling motion, Where —f is the angular velocity due to fluid flow. Combining the translational and angular velocities, particle nodal positions are updated based on its distance from the particle center as:

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Delivered by Ingenta to: Guest User Fig. 10. Shape dependent particle adhesion kinetics. The left column shows a spherical particle washed away without contact with surface; the right IP : 76.98.2.41 column shows a nanorod tumbles and gets deposited.Sun, A, B, 29 C, DApr are at2012 times t01:39:09 = 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 spherical particle to contact with wall surface solely depends on Brownian diffusion; while in case of non-spherical particle, 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 nanoparticles, it is computationally cost-effective and more convenient 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 channel considered? This is an important parameter to determine 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, Rev. Nanosci. Nanotechnol., 1, 66–83, 2012

the focus of this section is to characterize the influence 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 depletion 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 reasonable to only consider nanoparticles in the deposition process. The thickness of the cell free layer is found to be varying from 2–5 Œm, independent of vessel size for vessels with diameter above 20 Œm.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 thickness, on nanoparticle binding probability. To ensure consistency 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 77

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(A) t2 t1

t4

Nanoparticle trajectory t3

Minimum distance

Trajectories of particles (~ 20 trials) Height in Y dir (nm) →

(B)

(C)

1000 800

non-sph sph

600 400 200 0 0

2000

8000 4000 6000 Length in X dir (nm) →

10000

12000

Delivered by Ingenta to: Guest User IP : 76.98.2.41 Sun, 29 Apr 2012 01:39:09 Fig. 12.

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 determining binding probability of nanoparticles rather than dissociation 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 transportation is simulated by the Brownian adhesion dynamics model as discussed in the previous section. To ensure statistical 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 probability for a given depletion layer thickness under a given flow condition. 78

Multiscale model of the targeted drug delivery.

Binding probability of nanoparticles as a function of depletion layer or CFL thickness is plotted in Figure 13 for two different shear rates, 10 s−1 and 2 s−1 , respectively. The nanorods show significantly higher adhesion probability than the nanospheres at both shear rates. Figure 13(A) shows the binding probability of nanoparticles under a shear rate of 10 s−1 . As the CFL thickness increases, binding probability of nanoparticle decreases. Due to limited diffusion length, the binding probability of a nanosphere decreases almost linearly with CFL thickness, except for low CFL thickness of 1.5 Œm. At 1.5 Œm CFL thickness, the size of nanoparticle becomes comparable to the CFL thickness, thus results in higher deposition probability. In comparison, the binding probability of nanorod decreases almost quadratically with CFL thickness, mainly due to the tumbling motion. In particular, a nanorod has significantly higher binding probability than nanosphere at smaller CFL thicknesses. As shear rate decreases, binding probabilities for both particles increase. At a shear rate of 10 s−1 and CFL thickness of 1.5 Œm, the binding probability of the nanorod is around 2.5 times of that for the nanosphere. At a shear rate of 2 s−1 , the difference in the binding probability between nanorod and nanosphere is reduced, as shown in Figure 13(B). At lower shear rates, Brownian motion becomes a dominant factor, thus it overwhelms the contribution 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 Rev. Nanosci. Nanotechnol., 1, 66–83, 2012

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Binding Probability vs. Shear rate for D.L.T. (5 µm)

Binding Probability vs. depletion layer thickness(10s –1)

(a) 1

0.25

nanorod

nanorod(A.R.=10) nanorod(A.R.=5)

nanosphere

Binding Probability →

Binding Probability →

0.8

0.6

0.4

0.1

0 2

3 4 5 Depletion layer thickness (µm) →

6

0

5

15 10 Shear Rate γ (s–1) →

20

Fig. 14. Effect by Ingenta to: of shape on binding probability. Binding probabilities of nanosphere and nanorods of two different aspect ratios for depletion Guest User layer thickness (DLT) of 5 Œm. nanorod IP : 76.98.2.41 nanosphere Sun, 29 Apr 2012 01:39:09

–1 Delivered Binding Probability vs. Depletion layer thickness(2s )

1

5. FUTURE TREND

0.8

0.6

0.4

0.2

0

2

3

4

5

6

Depletion layer thickness (µm) →

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 thickness of 5 Œm 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 binding probability of nanoparticles, but the degree of reduction 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. Rev. Nanosci. Nanotechnol., 1, 66–83, 2012

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 efficiency, 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 nonmetallic particles), sizes (10 nm to 200 nm), shape (spherical 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

35

Number of publications

(b)

Binding Probability →

0.15

0.05

0.2

0

nanosphere

0.2

30 25 20 15 10 5 0

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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(NSF) CAREER grant CBET-1113040, NSF CBET1067502, and National Institute of Health (NIH) grant EB009786.

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