Comput Mech (2014) 53:403–412DOI 10.1007/s00466-013-0968-y

ORIGINAL PAPER

Computational modeling of magnetic nanoparticle targetingto stent surface under high gradient field

Shunqiang Wang · Yihua Zhou · Jifu Tan ·Jiang Xu · Jie Yang · Yaling Liu

Received: 19 July 2013 / Accepted: 16 December 2013 / Published online: 8 January 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract A multi-physics model was developed to studythe delivery of magnetic nanoparticles (MNPs) to the stent-implanted region under an external magnetic field. The modelis firstly validated by experimental work in literature. Then,effects of external magnetic field strength, magnetic parti-cle size, and flow velocity on MNPs’ targeting and bindinghave been analyzed through a parametric study. Two newdimensionless numbers were introduced to characterize rel-ative effects of Brownian motion, magnetic force inducedparticle motion, and convective blood flow on MNPs motion.It was found that larger magnetic field strength, bigger MNPsize, and slower flow velocity increase the capture efficiencyof MNPs. The distribution of captured MNPs on the ves-sel along axial and azimuthal directions was also discussed.Results showed that the MNPs density decreased exponen-tially along axial direction after one-dose injection while itwas uniform along azimuthal direction in the whole stentedregion (averaged over all sections). For the beginning sectionof the stented region, the density ratio distribution of capturedMNPs along azimuthal direction is center-symmetrical, cor-responding to the center-symmetrical distribution of mag-netic force in that section. Two different generation mecha-

S. Wang and Y. Zhou contributed equally to this work.

S. Wang · Y. Zhou · J. Tan · Y. Liu (B)Department of Mechanical Engineering and Mechanics, LehighUniversity, Bethlehem, PA 18015, USAe-mail: yal310@lehigh.edu

Y. LiuBioengineering Program, Lehigh University, Bethlehem,PA 18015, USA

J. Xu · J. Yang (B)School of Mechanics and Engineering, Southwest Jiaotong University,Chengdu, Chinae-mail: yangchenjie@home.swjtu.edu.cn

nisms are revealed to form four main attraction regions. Theseresults could serve as guidelines to design a better magneticdrug delivery system.

Keywords Targeted delivery · Magnetic nano-particles ·Magnetic force · Particle size · Magnetic stent

1 Introduction

Magnetic nanoparticles (MNPs) have been widely used inmany bio-medicine applications such as drug delivery, drugreleasing, and cancer diagnosis [1–5]. It is a promising tech-nique for curing diseases like cancer [2], local injury [6,7], orlocal cell proliferation [8]. By applying an external magneticfield, the drug-carried MNPs can be targeted to the regionof interest to avoid full circulation in human body, reducethe curing time, and minimize dosages and side effects. Thetechnique of MNP targeting mainly consists of two parts:loading drugs to MNPs and capturing them via an externalmagnetic field to a targeted region. The latter is the focus ofthis paper.

Experimental work on MNPs targeting has been carriedout in research and clinic applications. Various MNPs havebeen synthesized [9]. MNPs’ fabrication process, physicalprinciples of magnetic targeting, and obstacles to the clinicalapplications have been summarized by Yigit et al. [2]. A fewsimulation works have also been carried out. Finite elementmethods (FEMs) have been widely used to investigate themotion of NPs under different physical conditions [10–14].Wong et al. [15] applied FEM simulations of magnetic parti-cle inspection to analyze the magnetic field around a defect.Furlani et al. [16] developed a FEM model to predict thecapture of magnetic micro/nano-particles in a bioseparationmicrosystem. Furlani [17] pointed out that FEM was typi-

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404 Comput Mech (2014) 53:403–412

cally used to determine the magnetic field and force whenstudying particles transport.

Based on studies of previous researchers, the targetingmethod of MNPs still needs to be improved due to its lim-ited capture efficiency. Forbes et al. [18] proposed a novelapproach that used a magnetizable stent to achieve effi-cient targeted drug delivery. Two independent sources of themagnetic field are exerted on MNPs to make them bettercaptured on regions of interest and also allow deep penetra-tion within the subject: one is external high gradient mag-netic field to attract the magnetic drug carriers to the stent,the other one is the magnetic field induced by the magne-tized stent. This approach can not only improve the captureefficiency of MNPs in the injury region of interest but alsosolve one of major problems caused by stent-restenosis [19],because MNPs can constantly and quantitatively provideanti-proliferative agents. It offers a new approach for resteno-sis treatment and MNPs accumulation. Later, Polyak andcoworkers [20], Chorny et al. [8,21–23] and other researchers[24] carried out a series of studies to verify and improve thismethod. However, their work only proved the feasibility ofthis approach. Quantitative analysis of magnetic drug deliv-ery system design combined with stents is still needed toobtain better capture efficiency of MNPs. The goal of ourwork is to characterize the effects of external magnetic field,MNP size, and flow velocity on the capturing of MNPs.Meanwhile, unveiling the mechanism of how the magneticforce influences the capturing of MNPs can provide a betterunderstanding of targeted MNP delivery.

In this paper, a finite element model of MNP binding onstent is firstly developed and verified by experimental resultsin Forbe’s work [18]. Then, effects of external magnetic field,MNP size and flow velocity on capturing of MNPs are dis-cussed by using the presented model. Two dimensionlessnumbers are introduced to characterize effects of these threefactors on MNPs transport. Lastly, a general case is built tostudy the specific distribution of captured MNPs along thestented region. The mechanism of magnetic force in local-ized regions is unveiled and it reveals that magnetic force caneither attract MNPs towards or repel MNPs away from thestented surface.

2 Methods

2.1 Model description

The channel with a diameter of 3 mm [25] and a length of20 mm is built to represent the blood vessel. The Palmaz-Schatz type of stent [26–29] with a length of 15 mm isimplanted in the middle of channel, embedded into the chan-nel wall tightly. The inner diameter of the stent is 3 mm,same as the channel diameter; the outer diameter of stent

is 3.2 mm. Incompressible fluid flow and magnetic field areapplied to the system. The whole problem is solved usingFEM approach. The mesh elements used in the work aretetrahedral with a minimum element quality of 0.3278 forthe whole domain. Finer meshes are applied in the stent andother regions close to the stent. Moreover, four times of meshelements have been applied and differences within 2 % areobserved for both pressure and magnetic field. Thus, thismesh setup can guarantee the correctness of our simulationresults.

2.2 Flow field

The blood flow is considered to have a density of 1,060 kg/m3

[30] and a dynamic viscosity of 4×10−3 Pa s [31]. The flow isassumed to be fully developed in the blood. For the pressureat outlet, from Navier–stokes equations, only the pressuregradient influences the velocity of fluid flow. Thus, as longas the pressure gradient is fixed, the outlet pressure can bechosen as a reference pressure, which is expressed as 0 Pain the model. Based on dp

dy = − 32μuaved2 for a pipe flow, the

analytical pressure gradient is specified as −355.6, −711.1,−2133.3, −3555.6 and −4977.8 Pa/m, which correspond topeak flow velocity of 0.05, 0.1, 0.3, 0.5 and 0.7 m/s [intro-duced in Sect. (2.7)], respectively. However, in our numer-ical model, the pressure gradient is not imposed and it is aresult of simulation which relies on the simulation accuracy.In order to verify that our numerical model has a reliablesimulation accuracy, the typical case of peak flow velocityof 0.1 m/s is considered. The exact analytical pressure gra-dient is −711.1 Pa/m, while the computed pressure gradientis −705.0 Pa/m. With the difference of 0.86 %, it is demon-strated that the simulation results can well mimic the actualfluid field based on analytical solution. The stent is tightlyembedded into the artery wall so it doesn’t affect the flowvelocity. The flow in the channel is treated as Newtonian,incompressible, fully developed and steady. The flow field issimulated by solving the Navier–Stokes equation:

ρ

(∂u∂t

+ u · ∇u)

= −∇ p + μ∇2u (1)

∇ · u = 0 (2)

where u is the fluid velocity in the fluid domain, ρ and μ

are density and viscosity of the fluid respectively, p is thehydraulic pressure in the fluid domain, and t stands for time.

2.3 Magnetic field

A uniform external magnetic field is applied to the model inZ direction. A 6×20×6 mm3 box is created to represent theexternal environment of the vessel including human tissuesand air. It is challenging to take all components of human tis-

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sues, such as bone, fat, muscle, etc., into account in magneticfield modeling. Thus, following others’ approaches [32–34],the intra-tissue permeability differences are not consideredand all tissues are treated as water. The relative permeabilityof water is 0.999992, which is close to 1.0000004 for air. Thematerial of stent is 304 grade stainless steel with a relativepermeability of 1.8 [22]. Uniform magnetic flux density inZ direction is applied to the system. The four faces of thebox in X and Y direction are set as magnetic insulation foran ideal uniform magnetic field. The magnetic field distrib-ution is simulated by solving the magnetostatics equation, aspecial case of Maxwell’s equations:

−∇ · μ0μr∇Vm = 0 (3)

where μ0 is the permeability constant with a value of −4π ×10−7H m−1, μr is the relative permeability of medium, andVm is the magnetic potential.

2.4 Mathematical model for MNPs

A multi-physics model is developed to study the transport ofMNPs to the targeted stented region in the magnetic field.The motion of MNP is dominated by the magnetic force,fluid drag force and thermal kinetic force induced by Brown-ian motion (BM). These forces are typically in the orderof 10−1 or a few pN. However, inertia force, gravitationalforce and buoyancy force are neglected due to their smallmagnitude (typically in the order of 10−4 pN) comparedto dominant forces listed above [35–39]. Other factors suchas cell–particle and particle–particle interaction forces willmake our model complex and they are also computationallytoo expensive to be included in our large scale stent model[40]. These factors are studied in our previous work [40,41]about nanoparticle delivery in microcirculation, but are notconsidered in this paper. A uniform external magnetic fieldis applied in 3D model, while both MNPs and the implantedstent are paramagnetic. One dose of MNPs is injected intothe artery and trajectories of MNPs are described by the fol-lowing governing equation:

m M N P u̇M N P = FM N P + F f + FB = 0 (4)

where FM N P is the magnetic force applied on the MNP,F f the fluid drag force and FB the force due to BM.mM N P and u̇M N P are the mass and velocity of MNPs,respectively. Details about how to consider these three forcesin Eq. (4) are introduced as follows.

The fluid drag force is calculated using Stokes’ approxi-mation for a spherical particle in laminar flow [42]:

F f = Cd(u − uM N P ) (5)

where Cd = 6πμrM N P is the drag coefficient, where rM N P

is the radius of the MNPs.Displacement induced by BM is directly calculated using

traditional diffusion theory expressed as:

⟨|r|2

⟩= 6Dt (6)

where 〈 〉 means the time average, r is the position vector ofMNPs, D for the diffusion coefficient which is formulated asD = kT

6πμrM N P, where k and T are Boltzmann constant and

temperature, respectively.A mathematic model is developed to describe the mag-

netic characteristics of MNPs and the magnetic field whichis the superposition of applied external uniform magneticfield and non-uniform field induced by the magnetized stent.Here MNP is treated as an equivalent magnetic point dipolethrough an effective dipole moment approach [43]. For aparamagnetic sphere, the magnetic moment m of the pointdipole is aligned with the external magnetic field:

m = χM N P

1 + DM N PχM N PV H (7)

where H is the total magnetic field strength applied on thecenter of MNP, V is the volume of the MNP,DM N P is thedemagnetization coefficient (DM N P = 1/3 for a sphere),χM N P and μM N P are the susceptibility and permeability ofMNPs respectively with a relationship of χM N P =μM N P−1.

The magnetic force is calculated as: FM N P = (m·∇) B =μ0 (m·∇) H [44]. The formula of the magnetic force isexpanded to calculate the component of magnetic force ini direction, where i could be x, y or z:

FM N Pi = μ0

(mx

∂ Hi

∂x+ my

∂ Hi

∂y+ mz

∂ Hi

∂z

)(8)

where mx , my and mz are the components of magneticmoments in x, y and z direction respectively. Hi is the com-ponent of magnetic field in i direction. Accordingly, thevelocity component in i direction induced by magnetic forceis calculated based on Stokes’ law as:

uM N Pi = FM N Pi

6πμrM N P(9)

2.5 Model validation with experimental results

To further verify this model, simulated results are comparedwith experimental data reported in Ref. [18]. In Forbe’sexperimental work [18], capture efficiency on stents, which isdefined as the ratio between the amount of captured MNPs inthe stented region and overall MNPs released,with different

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406 Comput Mech (2014) 53:403–412

Fig. 1 Setup of the system: aPalmaz-Schatz type of stent isembedded in a blood vessel. Auniform magnetic field along Zdirection is applied in thesystem. MNPs are released fromthe inlet. The size unit ismillimeter

relative permeability is presented. In our numerical model,MNPs with a distance to the stent smaller than their radiusare treated to be captured. Validation of our current numeri-cal model is performed by comparing our numerical resultswith the experimental results. Details of the numerical modelare provided as follows. The whole setup of the system is thesame as the one shown in Fig. 1. A few minor changes aremade to make numerical simulation consistent with experi-mental work. Specifically, the stent used in [18] is 5 mm indiameter and 20 mm in length. The blood vessel embeddedwith stent is also extended correspondingly. The diameter ofthe wire is 150 µm. The external uniform magnetic field isset as 0.05 T. The peak flow velocity at inlet is set as 0.2 m/s,which is a typical value for artery [10]. Diameter and sus-ceptibility for MNPs are set as 350 nm and 1 (in the order of∼1–2), respectively. Material of the stent is 304 grade stain-less steel with relative permeability of 1.80, 6.94 and 10.44,which correspond to different thicknesses of plated magneticmaterial.

Based on our numerical model setup proposed above,transport of MNPs is determined and results on capture effi-ciency in different cases are obtained. Comparison betweennumerical and experimental results is shown in Fig. 2. Resultsshow that the capture efficiency on stents of various rela-tive permeability is consistent between numerical and exper-imental cases, although difference in absolute values exists.These differences could be due to several reasons. Firstly, theexperimental work considered an in vivo test, where MNPscan also be captured on non-targeted regions, thus decreasingthe amount of MNPs entering the stented blood vessel. Theresulting capture efficiency in experiments becomes smaller.Secondly, other factors such as blood cells, non-uniform ves-sel geometry, and pulsatile blood flow may also contribute tothis difference. With that, the rate of change for capture effi-ciency in terms of relative permeability is also calculated forboth numerical and experimental models to demonstrate thevalidation of our model. Cases of µr equal to 6.94 and 10.44are considered with the case of µr = 1.8 as the reference.As shown in the inset of Fig. 2, the rate of change for cap-

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

Relative permeability: r

Cap

ture

effi

cien

cy:

u=0.2m/s, r=350nm

num

exp

6 7 8 9 10 110.01

0.011

0.012

0.013

0.014

Relative permeability: r

Cha

nge

rate

of

u=0.2m/s, r=350nm

numexp

α

α

μ

μ

Fig. 2 Capture efficiency for stents of various relative permeability,with u = 0.2 m/s and r = 350 nm. The inset shows the normalizedcapture efficiency, which is defined as ratio of capture efficiency to thatin the reference case of µr = 1.8. Solid line represents experimentalresults in Forbe’s work and dashed line represents numerical results.Error bars stand for standard error

ture efficiency agrees very well between the modeling resultsand experimental results. Thus, the model provides insightson how relative permeability of stent influences the captureefficiency qualitatively.

2.6 Initial settings of MNPs

According to Ref. [8], if the distance between the wall andMNP is bigger than 50 um, the influence of the local highgradient magnetic field induced by the stent on MNPs can beignored. Thus, all MNPs are released in the vicinity of stentsurface with an initial distance of smaller than 50 um. Ineach case, 824 MNPs, with a same surface concentration of1.8 × 1010/m2 reported in Ref. [45], are released at the inletand transported within the blood flow under effects of mag-

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Comput Mech (2014) 53:403–412 407

netic force, drag force and Brownian motion. The magneticsusceptibility of MNPs is 0.27 [8].

2.7 Selection of parameters

The magnetic field strength is selected as 0, 0.01, 0.5, 2, 6 and8 T within the FDA suggested range [46]. Based on the organfiltration mechanism, particles bigger than 200 nm are readilyfiltered by liver and spleen while those smaller than 10 nmare easily cleared by the kidney or through extravasationsduring the blood circulation [47,48]. So the radius of MNPsis chosen as 25, 50, 100 and 150 nm. A case of 500 nm is alsoadded for the comparison purpose. Peak velocity of the inletparabolic flow is chosen as 0.05, 0.1, 0.3, 0.5 and 0.7 m/s[10], with corresponding shear rate of 67, 133, 400, 667 and933 /s, respectively (based on the formula γ̇ = 8uave

d , whereuave is the average blood flow and d is the diameter of theblood vessel).

Two dimensionless numbers βm and Pem are created toevaluate relative effects of BM, magnetic force induced par-ticle motion and convective blood flow. βm measures the ratioof the time for particles to reach the wall by diffusion to thatby magnetic force.

βm = d2c

/6D

dc/

um= dcum

6D(10)

where dc is the capture distance, which is chosen to be 50 umas mentioned above, D is the diffusion coefficient, and um isthe magnetic field induced velocity in the normal direction ofthe stent surface. βm >> 1 indicates that the capture processof MNPs is magnetic force dominated. βm << 1 indicatesthat the capture process of MNPs is diffusion dominated.Meanwhile, Pem is a modified Peclet number characterizingthe ratio between MNP radial traveling time toward the walland the convection time in the channel. It evaluates whetherthe flushing effect or capturing effect is dominant in the par-ticle delivery.

Pem = dc/(6D/dc + um)

L/

U= d2

c U

L (6D + umdc)(11)

where U is the characteristic convection velocity in the regionaway from the wall with a distance of dc, and L is the charac-teristic length and chosen as the length of stent in this case.

Pem can also be expressed by another dimensionless num-ber Reynolds number Re as:

Pem = d2c μRe

ρd L (6D + umdc)(12)

where Re = ρuavedμ

and d is the diameter of the vessel.Among the three factors (magnetic field strength, diameter

of the MNPs and fluid velocity) studied in this work, Re isonly influenced by flow velocity. Since fluid velocity effectis already considered in Pem , a separate discussion on Re isnot provided.

After the release, particles are transported in the chan-nel under effects of external magnetic field, high gradientmagnetic field induced by the stent, flow drag force, andBM. When distances between MNPs and the artery wall aresmaller than MNPs’ radius, they are treated to be capturedby the stented region.

3 Results and discussion

In consideration of the case of the flow rate as 0.7 m/s andthe magnetic flux density as 2 T, the simulation results ofvelocity in the channel and the system’s magnetic flux densityare shown in Fig. 3.

The capture efficiency of MNPs α is defined as the frac-tion of captured MNPs over those injected from the inlet. Inthe following session, the effects of magnetic field, MNPsize and flow velocity on capture efficiency are studied.βm and Pem are analyzed to evaluate the relative dominanteffect of MNPs motion in each case. The distribution ofcaptured MNPs is studied along axial and azimuthal direc-tions.

Figure 4a shows the capture efficiency under various mag-netic fields. The flow velocity is set as 0.1 m/s and theradius of particles is 100 nm. The magnetic field strengthis set as 0 T at first to prove the feasibility and high effi-ciency of capture of MNPs under magnetic field. The cap-ture efficiency without magnetic field is 6 % while it reaches79 % under the magnetic field strength of 6 T. Furthermore,in the range of weak magnetic fields, namely from 0.01to 2 T, the capture efficiency increases rapidly with mag-netic field, reaching 58 % under magnetic field strength of2 T. Capture efficiency reaches a saturation value of 80 %when magnetic field reaches 8 T. The saturation is alsoobserved by other researchers [45]. The enhancement of cap-ture efficiency induced by magnetic fields can be explainedthrough Fig. 4b. βm increases with larger magnetic fieldstrength, indicating a more significant impact of magneticforce on MNPs motion with larger magnetic field strength.More MNPs are attracted to the wall and the capture effi-ciency increases correspondingly under a higher magneticfield strength. As for Pem , it decreases with increasing mag-netic fields, leading to a faster capturing process because theenhanced magnetic force pulls MNPs towards the stentedregion faster.

The effects of particle size on the capture efficiency areshown in Fig. 5a. The capture efficiency increases withincreased particle size. In this case βm is always larger than 1as shown in Fig. 5b, which means this process is always

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408 Comput Mech (2014) 53:403–412

Fig. 3 a Flow velocity distribution across the channel. The flow veloc-ity ranges from 0 to 0.7 m/s as indicated by the color bar; b Cross sectionview of flow velocity; c Magnetic flux density distribution across the

system. The magnetic flux density ranges from 1.4 to 3.04 T as shownin the color bar; d Cross section view of magnetic field. (Color figureonline)

(a) (b)

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Magnetic field: B [T]

Cap

ture

effi

cien

cy: α

u=0.1m/s, r=100nm

0 5 10 15 20

x 10-3

0.05

0.06

0.07

0.08

Magnetic field: B [T]

Cap

ture

effi

cien

cy: α

u=0.1m/s, r=100nm

0 2 4 6 8

100

105

Magnetic field: B [T]

u=0.1m/s, r=100nm

βm

Pem

Fig. 4 a Capture efficiency under various magnetic fields, with u = 0.1 m/s and r = 100 nm. The inset magnifies the results under the magneticfield of 0 and 0.01 T; b βm and Pem in terms of magnetic fields under the same situation as (a)

(a) (b)

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

Particle Size: r [nm]

Cap

ture

effi

cien

cy: α

u=0.1m/s, B=1T

0 100 200 300 400 50010

-2

10 0

10 2

104

106

Particle Size: r [nm]

u=0.1m/s, B=1T

βm

Pem

Fig. 5 a Capture efficiency of particle of various sizes, with u = 0.1 m/s and B = 1 T; b βm and Pem in terms of particle sizes under the samesituation as (a)

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Comput Mech (2014) 53:403–412 409

(b)(a)

0 0.2 0.4 0.6 0.80.2

0.3

0.4

0.5

0.6

0.7

Flow velocity: u [m/s]

Cap

ture

effi

cien

cy: α

r=100nm, B=1T

0 0.2 0.4 0.6 0.810-2

100

102

104

Flow velocity: u [m/s]

r=100nm, B=1T

βm

Pem

Fig. 6 a Capture efficiency under various flow velocities, with r = 100 nm and B = 1 T; b βm and Pem in terms of flow velocities under the samesituation as (a)

magnetic force dominated. From Eqs. (7)–(10), magneticforce and βm are proportional to r3 which verifies that βm

increases with the particle size. Referring to Eq. (9), um

is proportional to r2. As a result, with r increasing, thetransport time to the wall decreases as the velocity compo-nent induced by magnetic force increases. Meanwhile, theincrease of um with particle size leads to a decrease of Pem .The decreasing trend of Pem indicates that MNPs tend to bepulled to the stent surface, resulting in an increased captureefficiency.

Figure 6a shows the trend of capture efficiency under vari-ous flow velocities. It demonstrates that the capture efficiencydecreases as the flow velocity increases, since higher flowvelocity gives MNPs less time to enter the captured regionand a higher chance to be flushed away. Figure 6b indicatesthat βm is not a function of flow velocity based on Eq. (10). Asfor Pem , it increases because the convection time of MNPsdecreases due to the increased flow velocity. In other words,more MNPs are flushed away, resulting in a decreased cap-ture efficiency.

To study the distribution of captured MNPs in axial andazimuthal direction, a general case is studied with the para-meters of B = 1 T, r = 100 nm, u = 0.1 m/s. The distrib-ution of captured MNPs in the whole stented region is shownin Fig. 7, while the distribution of MNPs and the magneticforce distribution in the beginning section of the stent sectionare displayed in Fig. 8.

As shown in Fig. 7a, the number of captured MNPs alongthe axial direction decreases exponentially because onlyone dose of MNPs are injected into the model, resulting indecreased free particle concentration along the vessel. How-ever, the distribution of MNPs along the azimuthal directionin the whole stented region is uniform (averaged over allsections), which can be explained by the uniform distribu-

tion of the stent structure and average magnetic force alongthe azimuthal direction.

As shown in Fig. 8b, the distribution of captured MNPsin azimuthal direction is center-symmetrical in the beginningsection of the stented region, while they are uniformly distrib-uted for the whole stented region, as shown in Fig. 7d. Thiscenter-symmetrical distribution of captured MNPs is corre-sponding to the center-symmetrical distribution of magneticforce, as shown in Fig. 8a. In this case, βm = 2.12 × 103

implies that this is a magnetic force dominated case. Mag-netic force can act as either attraction or repulsion for MNPs,which depends on whether its radial direction componentpoints to the stent surface or points to the channel center. Ifthe radial direction component of magnetic force points tothe stent surface, the magnetic force acts as an attraction; ifit points to the channel center, the magnetic force acts as therepulsion respectively. From Fig. 8b, 60 % of captured MNPsare concentrated in four main attraction regions named as 1©–4©, and much less MNPs are captured in other regions. The

generation mechanisms of these attraction regions are dif-ferent. For region 1© and 3© which are located between twostent struts, the local magnetic force is small, but the mag-netic forces in the nearby vicinity are large and they act asrepulsive forces. When particles are in the vicinity of thesetwo regions, they are pushed into and captured in these tworegions. In another word, the attraction region 1© and 3©are formed indirectly by the repulsion of MNPs from nearbystent struts. On the other hand, the attraction region 2© and 4©which are located on stent struts, are directly formed by theattraction of stent struts there. For region 2© and 4©, the accu-mulative local magnetic force acts as attractions even thoughboth attractive and repulsive forces exist in each region. Thus,particles are attracted to these two attraction regions directlyby magnetic force in these two regions.

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410 Comput Mech (2014) 53:403–412

(c)

(a) (b)

(d)

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

MNP distribution in axial direction

Nor

mal

ized

cap

ture

rat

io

1 2 3 10

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

MNP distribution in azimuthal direction

Nor

mal

ized

cap

ture

rat

io

12

10

outlet

inlet

Fig. 7 a Distribution of captured MNPs (dots in the figure) in the ves-sel (only MNPs and stents are shown). The capture ratio decreasesfrom inlet (left) to outlet (right). b Cross section view of the distribu-tion of captured MNPs. c Normalized capture ratio of MNPs along theaxial direction. The inset shows the stented region is divided into 10

equal regions in the axial direction, numbering from 1 to 10. d Normal-ized capture ratio of MNPs along azimuthal direction. The inset showsthe cross-section region is also divided into 10 equal sections alongazimuthal direction

(b)(a)

15%

16%

13%

Fig. 8 a Magnetic force distribution in the beginning section of thestented region. Arrows show the magnitude and direction of the mag-netic force near the inner channel surface. b Ratio distribution of cap-tured MNPs in the azimuthal direction. The blue bold dashed line illus-trates the normalized value of capture ratio of MNPs in the section of

stented regions. The circular dashed line marks zero ratio of capturedMNPs. The stent struts are corresponded in two views using dot-dashlines. In order to make the figure more concise, the stent is plottedpartially. (Color figure online)

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Comput Mech (2014) 53:403–412 411

4 Conclusion

A multi-physics computational model is developed and val-idated by experimental work in literature [18]. Then, effectsof magnetic field, particle size and flow velocity on the cap-ture efficiency of MNPs in the stented region are studied. Theimplanted paramagnetic stent generates large local magneticforce on the MNPs under the external magnetic field. Thefeasibility of this approach is validated by comparing cap-ture efficiency of MNPs without magnetic field and that withmagnetic field strength of 6 T. A parametric study was per-formed on the capture efficiency of MNPs. It is concludedthat that larger magnetic field strength, bigger MNPs, andslower flow velocity enhance the MNPs capture efficiency.The magnetic force might either attract MNPs toward or repelMNPs away from the stent surface, depending on whether itsforce component in radial direction points to the stent surfaceor to the channel center. In addition, MNPs density decreasesexponentially along the axial direction after one-dose injec-tion while the distribution of captured MNPs along azimuthaldirection is uniform in the whole stented region. A center-symmetrical distribution of captured MNPs is observed atbeginning cross-section of the stent, with MNPs concentratedin four main attraction regions. Two generation mechanismsof attraction regions are revealed and explained by the dis-tribution profile of magnetic force. Such non-uniform MNPsdistribution might be improved through the design of stentand the way magnetic field is applied, i.e., the application ofa rotation magnetic field.

In the current model, an ideal velocity field in a circularblood vessel is used, while the influence of the deformedvessel caused by the expansion of stent is neglected. In thefuture, blood flow velocity in a more realistic stented vesselwill be considered to provide more accurate predictions fornanoparticle delivery in stented region under clinical setup.

Acknowledgments The authors acknowledge the supports of thiswork from National Institute of Health (Grant EB015105) and NationalScience Foundation (Grant CBET-1113040, CBET-1067502).

References

1. Chen GJ, Wang LF (2011) Design of magnetic nanoparticles-assisted drug delivery system. Curr Pharm Des 17(22):2331–2351

2. Yigit MV, Moore A, Medarova Z (2012) Magnetic nanoparticles forcancer diagnosis and therapy. Pharm Res-Dordr 29(5):1180–1188.doi:10.1007/s11095-012-0679-7

3. Azhar SL, Lotfipour F (2012) Magnetic nanoparticles for antimi-crobial drug delivery. Die Pharmazie 67(10):817–821

4. Park H, Yang J, Seo S, Kim K, Suh J, Kim D, Haam S, YooKH (2008) Multifunctional nanoparticles for photothermally con-trolled drug delivery and magnetic resonance imaging enhance-ment. Small 4(2):192–196. doi:10.1002/smll.200700807

5. Dave SR, Gao X (2009) Monodisperse magnetic nanoparticles forbiodetection, imaging, and drug delivery: a versatile and evolv-

ing technology. Wiley Interdiscip Rev Nanomed Nanobiotechnol1(6):583–609. doi:10.1002/wnan.51

6. McGill SL, Cuylear CL, Adolphi NL, Osinski M, Smyth HD (2009)Magnetically responsive nanoparticles for drug delivery applica-tions using low magnetic field strengths. IEEE Trans Nanobiosci8(1):33–42. doi:10.1109/TNB.2009.2017292

7. Liu Y, Shah S, Tan J (2012) Computational modeling of nanopar-ticle targeted drug delivery. Rev Nanosci Nanotechnol 1(1):66–83.doi:10.1166/rnn.2012.1014

8. Chorny M, Fishbein I, Yellen BB, Alferiev IS, Bakay M, GantaS, Adamo R, Amiji M, Friedman G, Levy RJ (2010) Targetingstents with local delivery of paclitaxel-loaded magnetic nanoparti-cles using uniform fields. Proc Natl Acad Sci USA 107(18):8346–8351. doi:10.1073/pnas.0909506107

9. Dobson J (2006) Magnetic nanoparticles for drug delivery. DrugDevelop Res 67(1):55–60. doi:10.1002/Ddr.20067

10. Grief AD, Richardson G (2005) Mathematical modelling of mag-netically targeted drug delivery. J Magn Magn Mater 293(1):455–463. doi:10.1016/j.jmmm.2005.02.040

11. Tzirtzilakis EE, Loukopoulos VC (2005) Biofluid flow in a channelunder the action of a uniform localized magnetic field. ComputMech 36(5):360–374. doi:10.1007/s00466-005-0659-4

12. Liu Y, Tan J, Thomas A, Ou-Yang D, Muzykantov VR (2012) Theshape of things to come: importance of design in nanotechnologyfor drug delivery. Ther Deliv 3(2):181–194

13. Tan JF, Shah S, Thomas A, Ou-Yang HD, Liu YL (2013) Theinfluence of size, shape and vessel geometry on nanoparticledistribution. Microfluid Nanofluid 14(1–2):77–87. doi:10.1007/s10404-012-1024-5

14. Shah S, Liu YL, Hu W, Gao JM (2011) Modeling particle shape-dependent dynamics in nanomedicine. J Nanosci Nanotechnol11(2):919–928. doi:10.1166/jnn.2011.3536

15. Wong BS, Low YG, Xin W, Jee-Hou H, ChingSeong T, Jong BoonO (2010) 3D finite element simulation of magnetic particle inspec-tion. In: IEEE Conference on sustainable utilization and develop-ment in engineering and technology (STUDENT) Nov 20–2, pp.50–55. doi:10.1109/student.2010.5687008

16. Furlani EP, Sahoo Y, Ng KC, Wortman JC, Monk TE (2007) Amodel for predicting magnetic particle capture in a microfluidicbioseparator. Biomed Microdevices 9(4):451–463. doi:10.1007/s10544-007-9050-x

17. Furlani EP (2006) Analysis of particle transport in a magne-tophoretic microsystem. J Appl Phys 99(2):Artn 024912. doi:10.1063/1.2164531

18. Forbes ZG, Yellen BB, Halverson DS, Fridman G, Barbee KA,Friedman G (2007) Validation of high gradient magnetic field baseddrug delivery to magnetizable implants under flow. IEEE TransBio-Med Eng 55(2):643–649. doi:10.1109/Tbme.899347

19. Gay M, Zhang LT (2009) Numerical studies of blood flow inhealthy, stenosed, and stented carotid arteries. Int J Numer MethodsFluids 61(4):453–472. doi:10.1002/Fld.1966

20. Sensenig R, Sapir Y, MacDonald C, Cohen S, Polyak B (2012)Magnetic nanoparticle-based approaches to locally target therapyand enhance tissue regeneration in vivo. Nanomedicine 7(9):1425–1442. doi:10.2217/Nnm.12.109

21. Chorny M, Fishbein I, Alferiev I, Levy RJ (2009) Magneticallyresponsive biodegradable nanoparticles enhance adenoviral genetransfer in cultured smooth muscle and endothelial cells. MolPharm 6(5):1380–1387. doi:10.1021/Mp900017m

22. Polyak B, Fishbein I, Chorny M, Alferiev I, Williams D, YellenB, Friedman G, Levy RJ (2008) High field gradient targeting ofmagnetic nanoparticle-loaded endothelial cells to the surfaces ofsteel stents. Proc Natl Acad Sci USA 105(2):698–703. doi:10.1073/pnas.0708338105

23. Chorny M, Polyak B, Alferiev IS, Walsh K, Friedman G, Levy RJ(2007) Magnetically driven plasmid DNA delivery with biodegrad-

123

412 Comput Mech (2014) 53:403–412

able polymeric nanoparticles. FASEB J 21(10):2510–2519. doi:10.1096/fj.06-8070com

24. Chorny M, Fishbein I, Forbes S, Alferiev I (2011) Mag-netic nanoparticles for targeted vascular delivery. IUBMB Life63(8):613–620. doi:10.1002/Iub.00479

25. Forbes ZG, Fridman G, Yellen BB, Chomy M, Friedman G, BarbeelKA (2004) Locally targeted drug delivery to magnetic stents fortherapeutic applications. IEEE/EMBS international summer schoolon medical devices and biosensors (ISSS-MD)

26. Liu Y, Yang J, Zhou Y, Hu J (2013) Structure design of vascu-lar stents. In: Multiscale simulations and mechanics of biologicalmaterials. Wiley, pp. 301–317. doi:10.1002/9781118402955.ch16

27. Gay M, Zhang L, Liu WK (2006) Stent modeling using immersedfinite element method. Comput Method Appl Mech Eng 195(33–36):4358–4370. doi:10.1016/j.cma.2005.09.012

28. Liu WK, Liu YL, Farrell D, Zhang L, Wang XS, Fukui Y, PatankarN, Zhang YJ, Bajaj C, Lee J, Hong JH, Chen XY, Hsu HY (2006)Immersed finite element method and its applications to biologicalsystems. Comput Method Appl Mech Eng 195(13–16):1722–1749.doi:10.1016/j.cma.2005.05.049

29. Gay M, Zhang LT (2009) Numerical studies on fluid-structureinteractions of stent deployment and stented arteries. Eng Com-put 25(1):61–72. doi:10.1007/s00366-008-0105-2

30. Cutnell J, Kenneth J (1998) Density of blood. Physics. Wiley, NewYork

31. Erica M. Cherrya JKE (2013) Simulation of magnetic particles inthe bloodstream for magnetic drug targeting applications. Eighthinternational conference on multiphase flow

32. Li SZ, Williams GD, Frisk TA, Arnold BW, Smith MB (1995)A computer-simulation of the static magnetic-field distribution inthe human head. Magnet Reson Med 34(2):268–275. doi:10.1002/mrm.1910340219

33. Bhagwandien R (1994) Object Induced Geometry and Inten-sity Distortions in Magnetic Resonance Imaging: Met EenSamenvatting in Het Nederlands. Univerisiteit Utrecht, FaculteitGeneeskunde

34. Li SZ, Dardzinski BJ, Collins CM, Yang QX, Smith MB (1996)Three-dimensional mapping of the static magnetic field inside thehuman head. Magnet Reson Med 36(5):705–714. doi:10.1002/mrm.1910360509

35. Dames P, Gleich B, Flemmer A, Hajek K, Seidl N, Wiekhorst F,Eberbeck D, Bittmann I, Bergemann C, Weyh T, Trahms L, Rose-necker J, Rudolph C (2007) Targeted delivery of magnetic aerosoldroplets to the lung. Nat Nanotechnol 2(8):495–499. doi:10.1038/nnano.2007.217

36. Alexandra Heidsieck BG (2010) Analysis of particle trajectoriesfor magnetic drug targeting. COMSOL conference, Paris

37. Gleich B, Hellwig N, Bridell H, Jurgons R, Seliger C, Alexiou C,Wolf B, Weyh T (2007) Design and evaluation of magnetic fieldsfor nanoparticle drug targeting in cancer. IEEE Trans Nanotechnol6(2):164–170. doi:10.1109/Tnano.891829

38. Tan J, Wang S, Yang J, Liu Y (2013) Coupled particulate and con-tinuum model for nanoparticle targeted delivery. Comput Struct122:128–134. http://dx.doi.org/10.1016/j.compstruc.2012.12.019

39. Liu YL, Guo QJ, Wang SQ, Hu W (2012) Electrokinetic effects ondetection time of nanowire biosensor. Appl Phys Lett 100(15):Artn153502. doi:10.1063/1.3701721

40. Tan JF, Thomas A, Liu YL (2012) Influence of red blood cellson nanoparticle targeted delivery in microcirculation. Soft Matter8(6):1934–1946. doi:10.1039/C2sm06391c

41. Liu YL, Oh K, Bai JG, Chang CL, Yeo W, Chung JH, Lee KH,Liu WK (2008) Manipulation of nanoparticles and biomoleculesby electric field and surface tension. Comput Method Appl MechEng 197(25–28):2156–2172. doi:10.1016/j.cma.2007.08.012

42. Wasilewski J, Mirota K, Peryt-Stawiarska S, Nowakowski A,Polonski L, Zembala M (2012) An introduction to computationalfluid dynamics based on numerical simulation of pulsatile flowin the left coronary artery. Kardiochir Torakochi 9(3):366–374.doi:10.5114/kitp.2012.30851

43. Furlani EP, Ng KC (2006) Analytical model of magnetic nanopar-ticle transport and capture in the microvasculature. Phys Rev E73(6):Artn 061919. doi:10.1103/Physreve.73.061919

44. Shire ES (1960) Classical electricity and magnetism. CambridgeUniversity Press, Cambridge

45. Ravnik J, Hribersek M (2013) High gradient magnetic particle sep-aration in viscous flows by 3D BEM. Comput Mech 51(4):465–474.doi:10.1007/s00466-012-0729-3

46. Zaremba LA (2003) Guidance for industry and FDA staff: criteriafor significant risk investigations of magnetic resonance diagnosticdevices. Center for Devices and Radiological Health

47. Petros RA, DeSimone JM (2010) Strategies in the design ofnanoparticles for therapeutic applications. Nat Rev Drug Discov9(8):615–627. doi:10.1038/Nrd2591

48. Wang J, Byrne JD, Napier ME, DeSimone JM (2011) More effec-tive nanomedicines through particle design. Small 7(14):1919–1931. doi:10.1002/smll.201100442

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