ComputationalModelingofNanoparticle TargetedDrugDelivery
Reviews in Nanoscience and NanotechnologyVol. 1, pp. 66–83, 2012(www.aspbs.com/rnn)Copyright 2012 by American Scientific PublishersAll rights reserved.Printed in the United States of AmericaComputational Modeling of NanoparticleTargeted Drug DeliveryYaling Liu , Samar Shah, and Jifu TanDepartment of Mechanical Engineering and Mechanics, Bioengineering Program, Lehigh University,19 Memorial DR. W, Bethlehem, PA, 18015, USANanomedicine is a promising application of nanotechnology in medicine, which can drastically improve drugdelivery efficiency through targeted delivery. However, characterization of the nanoparticle targeted deliveryprocess under vascular environment is very challenging due to the small scale of nanoparticles and the complexin vivo vascular system. To understand such complicated system, various computational models are developedto help reveal nanoparticle targeted delivery process and design nanoparticles for optimal delivery. This articlediscusses a few computational tools to modelthe nanoparticleprocess and design nanoparticles forDeliveredby Ingentadeliveryto:efficient targeted delivery. The modeling approachesspanfromcontinuumvascular flow, particle BrownianGuest Useradhesion dynamics, to molecular level ligand-receptorbinding.Computersimulationis envisioned to be able toIP : 76.98.2.41optimize drug carrier design and ficvascularenvironment.Apr 2012 01:39:09KEYWORDS: Nanomedicine, Nanoparticle, Cancer, Drug Delivery, Computational Modeling.CONTENTS1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3. Continuum Approach: Drug Dissolution toConvection-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 BloodVessel Network . . . . . . . . . . . . . . . . . . . . . . . . . .4. Particulate Approach: Rational Design of Nanoparticles . . . .4.1. Introduction to Nanoparticle Design . . . . . . . . . . . . .4.2. Influence of Nanoparticles Size and Shape onTargeted Delivery . . . . . . . . . . . . . . . . . . . . . . . . .4.3. Theoretical Model of Nanoparticle Adhesion Probability4.4. Particulate Model of Nanoparticle Delivery in aVascular Environment . . . . . . . . . . . . . . . . . . . . . .4.5. Simulation Results of Nanoparticle Targeted DeliveryProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5. Future Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . .References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . .6667.6868696970.707171.7273.73.76798080801. INTRODUCTIONNanotechnology refers to the study of matter on nanoscale,in general dealing with structure size in between Author to whom correspondence should be addressed.Email: yal310@lehigh.eduReceived: 30 September 2011Accepted: 12 March 201266Rev. Nanosci. Nanotechnol. 2012, Vol. 1, No. 11 to 100 nm in at least one dimension.1 Nanotechnology represents a broad range of applications; the medicalapplication of nanotechnology refers to “nanomedicine”.Nanomedicine based drug delivery system hold greatpromise in the next generation of medicine to improvehuman health. Among all different research branches,drug delivery contributes over 70% of scientific papers innanomedicine research field.2The 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 todramatically reduce side effects and increase effectiveness.Through accomplishment of these revolutionary advantages, patients and physicians could benefit from personalized prescriptions, alleviated administration, increasedpatient compliance, reduced dosage frequency and lesspain. Over the past decade, we have witnessed an explosive development of nanoparticulate systems for diagnostic imaging and targeted therapeutic applications.3–10Various nanoplatforms, including liposomes,111 12 polymeric micelles,13–16 quantum dots,171 18 Au/Si/polymershells,19–21 and dentrimers22–24 etc. have been developed.Although recent data on in vivo nanoparticle (NP) drugdelivery has showed remarkably improved efficacy overtraditional drug, yet the challenges in nanomedicine fieldare many.For example, the study of drug delivery system is notstraightforward process, which requires further consideration and comprehensive analysis. The targeted 014
Computational Modeling of Nanoparticle Targeted Drug DeliveryLiu et al.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 coversand stochastic approaches such as Monte Carlo simuthe basic governing equations and a few examples oflation 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 ofparticulate modeling based on coupled Brownian adhesion30spherical NPs in tumor. Mahmoudi et al.29 andLietal.dynamicsto:method is described, where the motion and bindDelivered by Ingentaperformed computational fluid dynamics studies of Guestmag- Usering of individual nanoparticles in the blood stream are32et al.netic NPs in vascular flow. Liu et al.31 and Zhang IPmodeled. Finally, the future trend in computational mod: 76.98.2.41studied the deposition of NPs in lung airway.Figure2elingof targeted drug delivery is briefly discussed.Sun, 29 Apr 201201:39:09shows a multiscale simulation framework for targeteddrug delivery ranging from continuum model to partic3. CONTINUUM APPROACH: DRUGular model. It’s the recent advancement in the compuDISSOLUTION TOtational science that made computational modeling veryCONVECTION-DIFFUSION-REACTIONpromising for targeted drug delivery application. The ligMODEL OF DRUG DELIVERYand coated nanoparticles, loaded with drugs inside, transport in blood stream, and adhere to diseased cells via3.1. Introductionspecific adhesion. However, this process becomes intriIn vivo drug release, transportation and targeted bindingcate due to simultaneous involvement of hydrodynamichave been recognized as important elements in targetedforce, adhesion force and Brownian force. In particular,drug delivery field. In order to target the disease area, thethe ligand-receptor interaction is a sophisticated chemicaldrug loaded carriers are first injected into the circulationprocess. The surface property of functionalized nanopartisystem, where they transport through, across and withincles would play a crucial role to dictate the efficiency ofvessels, tissues and cells. Due to specific binding betweenthe targeted drug delivery by providing targeted selectivligand coated drug particles and receptors, expressed atity. Computational modeling tool will lead to insights ofthe disease cell membrane, drug loaded particles depositthe dynamic delivery process, thus facilitate better designon the targeted disease region. Afterward process is folof nanoparticles.lowed by cellular uptaking and drug releasing. From theThis article focuses on multiscale computationalcontinuum stand point of view, the drug delivery processapproach to the targeted drug delivery. First, continuumconsists of drug dissolution, transport and binding, whichcan be described by mass conservation law and chemicalkinetic reaction.In recent years, in vitro release profile of drug from controlled release platform has been combined with the stateof art Computational fluid dynamics (CFD) simulation topredict the the spatial and temporal variation of the drugtransport in the living tissues. For example, Saltzman andRadomsky33 developed a diffusion kinetics model for thedrug release in the brain tissue. The transport mechanismwas assumed to be mainly governed by diffusion due tothe selective permeability of the blood capillaries knownas blood-brain-barrier. This simplified model’s predictionhas been validated by the experimental data of drug spaFig. 2. A framework for multiscale modeling of the entire drug deliverytial distribution. A three dimensional (3D) simulation ofsystem.68Rev. Nanosci. Nanotechnol., 1, 66–83, 2012
Computational Modeling of Nanoparticle Targeted Drug DeliveryLiu et al.human brain tumor of primitive neuroectodermal tumorwas performed by Wang et al.34 The simulation is conducted on CFD tools to solve simultaneously continuity,momentum and drug concentration equations. Using theirmodel, the contribution of convective transport of macromolecular and micromolecular drugs in the vicinity oftumor were studied. In this section, the governing equations of the continuum drug delivery model are describedand a few demonstration examples are presented.3.2. Dissolution of Drug ParticlesA number of mathematical models have been proposedand effectively applied to describe the drug release anddissolution in literature.35–38 The simplest form of drugdissolution profile is zero order kinetics that assumes slowdrug releasing process,Q0 Qt KtHiguchi38 formulated following relation to model lowsoluble drug release problem:p(5)ft D42C Cs 5Cs tWhere C is the drug initial concentration, Cs is the drugsolubility 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 paperby Paulo Casta et al.41 The commonly used mathematicalmodels are listed in Table I.3.3. Convection-Diffusion-Reaction Model of DrugDeliveryThe 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: ¡tGuest UserWhere Q0 is the initial amount of drug in the pharmaceuWhere c is the concentration of nanoparticles, D is theIP : 76.98.2.41tical dosage form, Qt is the amount of drug in the pharmadiffusion coefficient of nanoparticle and U is the flowSun, 29 Apr 201201:39:09ceutical dosage form at time t and K is a proportionalityvelocity.It is solved by using following Einstein-Stokesconstant. Dividing the above equation by Q0 simplifyingit to:(2)f0 ktWhere 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 dosagerelease forms, particularly with low soluble drugs.3.2.1. First Order KineticsThe application of this model was first proposed byGibaldi and Feldman,39 and later by Wagner.40 The dissolution rate of the drug is described by the Noyes-Whitneyequation as shown below:dC K4Cs C5dt(3)Where C is the concentration of solid in bulk dissolution medium, Cs is the concentration of solid in diffusionlayer surrounding solid, K is a first order constant and itis associated with surface area of the solid drug, diffusioncoefficient and diffusion layer thickness.Since the dissolution mechanism of drug is very complex, various empirical equations are proposed to describethis process. For example, the popular Weibull equationexpressed the fraction of drug, m at time t, in the simpleexponential form: 4t T i5b(4)m 1 exp aWhere a is time related constants, Ti represents time lagbefore onset of the dissolution, b is a curve characterizedparameter.Rev. Nanosci. Nanotechnol., 1, 66–83, 2012equation,kB T(7)6 rWhere 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 issimilar 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, Langmuirreaction model is employed.42 The ligand-receptor bindingprocess is a weak reversible process, which leads to continous attachment and detachment of nanoparticles.43 Thematerial balance for the active surface including surfacediffusion and the reaction rate expression for the formationof the adsorbed species cs is defined by:D ¡cs ï · 4 Ds ïcs 5 ka cw kd cs¡t(8)Where Ds is the surface diffusivity (m2 /s), cw is the bulkconcentration of the species at solid wall (unit mol/m3 5, is the surface concentration on the active site (mole/m2 5Table I. Mathematical models used to describe drug dissolution curves.Zero orderFirst smeyer-peppasQuadraticLogisticGompertzHopfenbergQt Q0 K0 tln Qt ln Q0 K1 tQ01/3 Qt1/3 Ks tln6 ln41 4Qt /Q 557 b ln4t5 ln4a5Qt KH t3/261 41 Qt /Q 52/3 7 Qt /Q KtQt /Q Kk t nQt 1004K1 t 2 K2 t5Qt A/61 e K4t y5 7Qt Ae e K4t y5 7Qt /Q 1 61 k0 t/C0 a0 7n69
Computational Modeling of Nanoparticle Targeted Drug DeliveryLiu et al.and cs surface concentration of adsorbed species (mol/m2 5.Note that cs is different from c which is reflected in theirunits. ka (m3 /mol/s) and kd (s 1 5 are adhesion and detachment rates, respectively. However, the concentration ofactive sites is equal to the difference between the total concentration of active sites and the number of sites occupiedby the adsorbed species. This gives the equation for thereaction rate as:¡cs ï · 4 Ds ïcs 5 ka c4 0 cs 5 kd cs¡t(9)(A)10 µmReceptor coated reaction surface(B)In above equation, 0 represents the total number of activesites available on the active surface. Convection-diffusionequation and nanoparticle reaction equation are not independent, instead, they are coupled through Fick’s law:¡cs D · ïc w¡tFlow rate 0.1mm/sFlow rate 1mm/s10 µmReceptor coated reaction(10)Fig. 3.Nanoparticle binding in a channel at a flow rate of 0.1 andDelivered by Ingentato: concentration drops close to the receptor coated surface1 mm/s. Particle3.4. Nanoparticle Binding in a Channeldue to adhesion, forming a depletion layer. Red color indicates highestGuest Userconcentration, while blue color indicates lowest concentration.IP : 76.98.2.41To demonstrate application of continuum model in targetedSun,Apr 201201:39:09drug delivery, finite element modeling is usedto 29evaluateFigure 3 shows the depletion layer at shear rates 0.1 mm/sthe nanoparticle transportation diffusion and biochemicalreaction dynamics in a channel. In this model, the convection diffusion in 2D fluid domain is coupled with theadhesion reaction occurring on the reaction surface (disease site). When a portion of the blood vessel is injured,significant P-selectin is expressed on damaged endothelialcells, which can be targeted by nanoparticles coated withGPIb ligand. In this model, the convection-diffusion process of nanoparticle in 2D fluid domain is coupled withthe adhesion reaction occurring only on the reaction surface which mimics the target site for drug delivery. Thephysical parameters used to create this model are listed inTable II.To initiate adhesion, nanoparticles must stay close tothe vessel wall, inside the so called depletion layer alsoknown as a near-wall layer where adhesion process takeplace. The thickness of the depletion layer is largely influenced by the flow rate, evident from the simulation resultshown in Figure 3. When drug particles bind with thereceptors coated surface, drug concentration drops near thesurface, effectively forms a “depletion layer” near the wall.Table II.Symbolc0kakd 0DsDkBTU 70and 1 mm/s respectively. As the flow rate increases thedepletion layer thickness decreases due to greater nanoparticle flux and shorter retention time of the nanoparticles.3.5. Nanoparticle Deposition and Distribution in aBlood Vessel NetworkAnother example application of continuum model is todetermine nanoparticle deposition and distribution in acomplex vascular geometry. Figure 4 shows the drug delivery process in an idealized vascular network with threegenerations. The physical parameters used to create thismodel in listed in Table III.Drug loaded nanoparticles of a given concentration areinjected at the top inlet and are transported through thevascular network along with fluid flow. The left branchof the network is assumed to be a receptor coated targetsurface that can form bonds with ligands on drug loadedPhysical parameters used in nanoparticle binding in a channel.ValueDefinition1000 [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 concentrationAdhesion rate constantDetachment rate constantActive site concentrationSurface diffusivityParticle diffusivity in the fluidBoltzmann constantAbsolute temperatureMaximum shear rateEquilibrium bond lengthFig. 4. (A) Drug injected at the top inlet of an idealized vascular network with three generations; (B) Receptors coated vessel section in theleft branch of vascular network.Rev. Nanosci. Nanotechnol., 1, 66–83, 2012
Liu et al.Computational Modeling of Nanoparticle Targeted Drug Deliverydedicated to comprehend their biological behaviors in vitroand in vivo. For example, it is known that spherical particles bigger than 200 nm are efficiently filtered by theSymbolValueDefinitionspleen, while particles smaller than 10 nm can be quicklyc01000 [mol/m3 ]Initial concentrationcleared by the kidney, thus making 10–200 nm as an idealka10 6 [m3 /(mol*s)]Adhesion rate constant 9size range for the spherical carriers.10 [ 1/s]Detachment rate constantkd 01000 [mol/m2 ]Active site concentrationSimilar to size, shape is a fundamental property ofDs10 11 [m2 /s]Surface diffusivitymicro/nanoparticlesthat may be critically important forParticle diffusivity in the fluidD10 6 [m2 /s]theirintendedbiologicalfunctions.44–50 Recent data beginV1 [mm/s]Maximum velocity3Blood density 1063 [kg/m ]to reveal that particle shape may have a profound effect 0.003 [Pa.s]Blood dynamics viscosityon their biological properties. For example, cylindricallyshaped filomicelles can effectively evade the non-specificuptake by the reticuloendothelial systems and persisted innanoparticle surface. The particle depletion layer is clearlythe circulation up to one week after intravenous injection.visible in the target region. The density of deposited drugFrom drug delivery stand point of view, non-spherical parparticles on the wall surface is plotted in Figure 5, whichticles will allow larger payload delivery than the spherindicates that most drug particles are deposited at theical counterpart with same binding probability. Recently,entrance of the target region, while the rest of the targetMitragotri and coworkers have shown that the local shaperegion has low density of deposited drug particles. Thereof the particleDeliveredto: at the point where a macrophage is attached,are no particles deposited in the healthy branchdue to anby Ingentanot the overall shape, dictated whether the cell beganGuest Userassumption of zero non-specific adhesion at that particularinternalization.51 These results indicate the importance ofIP : 76.98.2.41location. Such non-uniform distribution pattern indicatescontrollingparticle shape for nanomedicine application.Apr 201201:39:09possible impaired delivery dosage within theSun,target29region,Theoretical studies of nanoparticle deposition are typwhich is important for delivery efficacy prediction andically focused on simple spherical or oblate shape.52–54dosage planning.Ideally, there should be a tool that can handle variety ofshapes and sizes of nanoparticles, which enables endless4. PARTICULATE APPROACH: RATIONALpossibilities of finding
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
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