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International Journal of Computational Fluid Dynamics

ISSN: 1061-8562 (Print) 1029-0257 (Online) Journal homepage: http://www.tandfonline.com/loi/gcfd20

Hydrodynamic modelling and CFD simulationof ferrofluids flow in magnetic targeting drugdelivery

Shigang Wang , Handan Liu & Wei Xu

To cite this article: Shigang Wang , Handan Liu & Wei Xu (2008) Hydrodynamic modelling andCFD simulation of ferrofluids flow in magnetic targeting drug delivery, International Journal ofComputational Fluid Dynamics, 22:10, 659-667, DOI: 10.1080/10618560802452009

To link to this article: http://dx.doi.org/10.1080/10618560802452009

Published online: 19 Nov 2008.

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Hydrodynamic modelling and CFD simulation of ferrofluids flow in magnetic targeting drug

delivery

Shigang Wang, Handan Liu* and Wei Xu

School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200240, China

(Received 25 January 2008; final version received 4 October 2008)

Magnetic targeting drug delivery is a method of carrying drug-loaded magnetic nanoparticles to a tissue target in thehuman body under the applied magnetic field. This method increases the drug concentration in the target andreduces the adverse side-effects. In this article, a mathematical model is presented to describe the hydrodynamics offerrofluids as drug carriers flowing in a blood vessel under the applied magnetic field. Numerical simulations areperformed to obtain better insight into the theoretical analysis with computational fluid dynamics. A 3D flow field ofmagnetic particles in an idealised blood vessel model containing an aneurysm is analysed to further understandclinical application of magnetic targeting drug delivery. Simulation samples demonstrate the important parametersleading to adequate drug delivery to the target site depending on the applied magnetic field in coincidence withreported results from animal experiments. Results of the analysis provide the important information and can suggeststrategies for improving delivery in favour of the clinical application.

Keywords: magnetic targeting drug delivery; ferrofluids; magnetic nanoparticles; hydrodynamic modelling; CFDsimulation

1. Introduction

In conventional drug delivery the drug is administeredby intravenous injection; it then travels to the heart fromwhere it is pumped to all regions of the body. For thesmall target region that the drug is aimed at, thismethodis extremely inefficient and leads to much larger doses(often of toxic drugs) than necessary. To overcome thisproblem, a number of targeted drug delivery methodshave been developed (Torchilin 2000, Lubbe et al. 2001,Vasir and Labhasetwar 2005). Among them, the mag-netic targeted drug delivery system is one of the mostattractive strategies because of its non-invasiveness,high targeting efficiency and its ability to minimise thetoxic side effects on healthy cells and tissues (Alexiouet al. 2000, 2005). Magnetic drug targeting therapy is apromising technique for the treatment of variousdiseases, especially cancer, arteriosclerosis, like stenosis,thrombosis and aneurysm, what is important is to keepthe therapeutic drug in the targeting site, which islocated along the inner wall of the blood vessel (Alksneet al. 1967, Alexiou et al. 2000, 2005, Chen et al. 2005,Udrea et al. 2006). Some in vitro (Chen et al. 2005,Udreaet al. 2006) and in vivo (Alksne et al. 1967,Goodwin et al.1999, Alexiou et al. 2002, Jurgons et al. 2006) experi-ments have been performed in this direction.

Typically, this compound in which drugs are boundwith nanoparticles is injected through a blood vessel

supplying the targeting tissue in the presence of anexternal magnetic field with sufficient field strengthand gradient to retain the carrier at the target site.Recent development on carriers has largely focused onnew polymeric or inorganic coatings on magnetite/maghemite nanoparticles (Ruuge and Rusetski 1993),such as ferrofluids. Ferrofluids are colloidal solutionsof ferro or ferromagnetic nanoparticles in a carrierfluid, which are widely used in technical applications.Because of a high magnetic moment of nanoparticles inferrofluids, ferrofluids are gaining increasing interest tobe utilised as drug carriers in magnetic targeting forbiological and medical applications (Riviere et al.2006). When they are used in medicine, ferrofluidsmust be bio-compatible and bio-degradable (Jurgonset al. 2006).

Recently, some theoretical studies of magneticallytargeted drug delivery considered tracking individualparticles under the influence of Stokes drag and amagnetic force alone (Grief and Richardson 2005), andformulated a two-dimensional (2D) model, suitable forstudying the deposition of magnetic particles within anetwork of blood vessels (Richardson et al. 2000).Other theoretical studies investigated the basic inter-action between magnetic and fluid shear forces in ablood vessel (Voltairas et al. 2002) or utilised highgradient magnetic separation principles to study a

*Corresponding author. Email: helenlew818@hotmail.com

International Journal of Computational Fluid Dynamics

Vol. 22, No. 10, December 2008, 659–667

ISSN 1061-8562 print/ISSN 1029-0257 online

� 2008 Taylor & Francis

DOI: 10.1080/10618560802452009

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magnetic drug targeting system (Ritter et al. 2004).Although there have been a number of theoreticalstudies for magnetic drug targeting, very few research-ers have addressed the hydrodynamic models ofmagnetic fluids in magnetic drug targeting delivery.Thus, the transport issues related to magnetic drugtargeting delivery are yet poorly understood and itretards the extensive application of the magnetic drugtargeting delivery. Therefore, it is very necessary tostudy the flow of the ferrofluids in the blood vesselunder the action of the external magnetic field.

In this article, we focused on investigating the flowrules of ferrofluids as drug carriers under the appliedmagnetic field. A mathematical model presented in thisarticle describes the hydrodynamics of ferrofluids in ablood vessel under the action of the magnetic field.Numerical simulations are performed to obtain betterinsight into the theoretical analysis. Furthermore, theferrofluids flow is analysed numerically with com-putational fluid dynamics (CFD) in a model of anidealised 3D blood vessel containing an aneurysm tounderstand the clinical application of ferrofluids.Magnetic fluids have been used in medicine since 1960for, e.g. the magnetically controlled metallic thrombosisof intracranial aneurysms (Alksne et al. 1967). Thebiokinetic behaviour of ferrofluids in vivo was investi-gated and showed that the retention of ferrofluids intarget region is dependent on the magnetic fieldstrength (Goodwin et al. 1999, Alexiou et al. 2002,Jurgons et al. 2006). Results of the analysis providedimportant information leading to adequate drug deliv-ery to the target site and can suggest strategies forimproving delivery in favour of the clinical application.The simulation results coincide with these animalexperiments.

2. Mathematical model

Ferrofluids is a nano-scale colloid mixture. Because ofthe magnitude of nanometre scale of the magneticparticle, the hysteresis of its flow velocity and tempera-ture can be neglected. The body forces are the gravity aswell as the magnetic force. In the absence of a magneticfield the stress is symmetric, whereas the momentumequation is the Navier-Stokes equation. In the presenceof a magnetic field, the magnetic field brings anasymmetric stress and produces a magnetic force.Hereby, except for the conventional Navier-Stokesequation, the magnetic force and the asymmetric stressare included in the momentum equation. The fluid flowsfrom macroscopically to microscopically through theartery to arterioles and to capillary vessels. If stillconsidered the microfluid flow with continuum theory,the viscous force is more dominant than the gravityforce, in addition to magnetic force as a new body force.

So, the equations of motion for incompressible ferro-fluids dynamics are (LiuHan-dan et al. 2008) as follows:

Continuity equation

r � u ¼ 0 ð1Þ

Momentum equation

rDu

Dt¼ �rpþ Zr2uþ m0 M � rð ÞH ð2Þ

When the magnetisation M is aligned with theapplied magnetic field H or H is large enough, themagnetisation equation is

M ¼ wH ð3Þ

Maxwell equation

r�H ¼ 0 ð4Þ

This set of equations represents 10 equations for 10variables (p (1), u (3), M(3) and H (3)). The viscosity Z,density r, and magnetic susceptibility w are consideredas known and constants. m0 is the magnetic perme-ability in free space and is equal to 4p 6 1077H/m.The numerical solution for these equations is obtainedfrom the CFD.

The Gauss law gives the magnetic induction B as

r � B ¼ 0 ð5Þ

B ¼ m0 HþMð Þ ¼ m0 1þ wð ÞH ¼mH ð6Þ

3. Computational modelling and numerical simulation

In this research, the finite volume method is used toobtain the numerical simulations of ferrofluids flowingin a blood vessel. In the finite volume method thegeneral formation of the governing equations is(Versteeg and Malalasekera 1995):

@ rfð Þ@t

¼ div � grad fð Þ þ S

where f is the general variable, � is the generaliseddiffusive coefficient and S is the generalised sourceterm. Its physical meaning is the conservation princi-ples of dependent variable f in the finite controlvolume. In Equation (2) f is the velocity and Sindicates the pressure gradient and the magnetic force.

When studying the flow in a blood vessel, Nettiet al. (1996) presented the ratio of resistance to flow

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along the vessels to that through the vessel walls todenote the relative size of extravasations from a bloodvessel.

b ¼ 128Z0Lpl2=d3

where Z0 is the blood viscosity, Lp is the vascularpermeability and the vessel is of length l anddiameter d. The value of the parameter b for a singlevessel is within the order of magnitude of 1073, andhence the amount of fluid filtration across the vascularwall of a blood vessel will always be negligiblecompared to the amount of perfusion flow (the rangeof b is from 10 to 100). So the flow in a blood vessel isprimarily an axial flow (Netti et al. 1996). As arepresentative application the laminar, incompressible,three-dimensional, fully developed viscous flow of aNewtonian fluid in a straight cylindrical duct forsimplified aneurysm geometry is numerically studied(Castro et al. 2006). The ferrofluids as drug carriersflowing in the blood vessel is studied under the actionof the applied magnetic field, and the change of thepressure and velocity at the target site is analysed asthe change of the magnetic intensity.

3.1. Simulation model

To further understand the clinical application offerrofluids, the ferrofluids flow is analysed in anidealised 3D model of a blood vessel containing ananeurysm. Aneurysms are common vascular abnor-malities that represent a disruption in arterial wallcontinuity. Except for the traditional surgical option,the drug injection option is used for therapy, such asthrombin (Gorge et al. 2003). Under the action of theapplied magnetic field, ferrofluids are concentrated onthe aneurysm wall and then release therapeutic agents.An aneurysm is a bulge that is formed in a bloodvessel. A schematic drawing of an aneurysm is shownin Figure 1.

Simplifying the computational domain of theaneurysm, we chose the diameter of the blood vessel ofthe aneurysm d ¼ 0.8 mm, and the length L ¼ 10 mm.In the Cartesian coordinates, the axial direction is inthe X direction with X ¼ 0 set at the inlet and the originis set at the centre of the circle at the cross-section ofthe inlet. There is a bulge at X ¼ 5 mm from the originof theX direction and down fromY ¼ 70.4 mm,whichis the spherical centre of the aneurysm blood vessel.The aneurysm is modelled as a sphere whose diameteris 1 mm. The applied magnetic field acts on the wholesegment of the aneurysm blood vessel.

The grid distribution in the above computationaldomain is generated by GAMBIT. Illustrations of thecomputational mesh are given in Figure 2.

3.2. Boundary conditions

For 3D computation, the computational domain is thewhole segment, as shown in Figure 2. The appliedmagnetic field acts on the whole segment of theaneurysm blood vessel.

(1) Boundary of the inlet. A parabolic inlet velocityprofile is applied at the inlet, where the axialvelocity u is given by White (1974) as

u y; zð Þ ¼ 16d2

Zp3� dp

dx

� � X1i¼1;3;5;...

� �1ð Þi�1ð Þ2 1�

cosh ipz2d

cosh ip2

" #�cos ipy

2d

i3

where d is the diameter of the blood vessel, Zis the viscosity and p ¼ pþ rgz. The othercomponents of the velocity v and w are 0.

(2) Boundary of the outlet. A fully developedassumption is applied at the outlet. The normalderivatives of the physical variables along theoutlet section are 0.

(3) Boundary of the blood vessel wall. It is definedas solid, no slip conditions. And the wallboundary is assumed as an insulating stateunder the applied magnetic field.

(4) Initial pressure. As the blood pressure of anadult arteriole is about 30 mm Hg, i.e. 4 kPa,the initial pressure value is set to 4,000 Pascals.The below-mentioned pressure drop values inFigures 7 and 8 are based on this initialpressure value to fluctuate.

3.3. Parameters

When the water is the carrier fluid at a temperatureof 298 K, the related physical properties of Fe3O4

Figure 1. Schematic drawing of an aneurysm.

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Figure 2. (a) The grid distribution of Computational domain. (b) The various directions of views after being meshed. Availablein colour online.

ferrofluids (Rosensweig 1997) are the density r ¼ 1460kg/m3, the viscosity Z ¼ 0.035 N s/m2 (measured inthe absence of a magnetic field), the magnetic satura-tion Ms ¼ 15.9 kA/m. In addition, the diameter ofmagnetic particles is 20 nm and the volume fraction is0.046.

After confirming the governing equations and theboundary conditions of the model, the CFD softwareFLUENT is used to perform steady state numericalsimulation by means of the finite volume method.User-defined functions are written in the C program-ming language to describe the applied magnetic fieldand the velocity. The pressure term of the momentumequation is dealt with by SIMPLEC algorithm. Thesolution was obtained under four different magneticfields, namely B ¼ 0, 0.1, 0.5, 1.0T. When the residualsof the source term of mass in the continuity equationand each component of the velocity are less than1.0 6 1073, the iteration is considered as theconvergence.

4. Computation results and discussion

4.1. Distribution of induced current density andelectromagnetic force

The vector diagrams of induced current density andelectromagnetic force are given in Figure 3. Ferrofluidsmove through the static magnetic field and the inducedcurrent is formed in the blood vessel as shown inFigure 3(a). An electromagnetic force is formed by theinteraction between the current and magnetic field,whose direction is opposite to the flow direction offerrofluids as shown in Figure 3(b). This force reducesthe flow of ferrofluids.

4.2. The effect of magnetic induction intensity onvelocity field

At differentmagnetic induction intensityB ¼ 0, 0.1, 0.5,1.0T, the velocity contours near the bulge in x–y plane(z ¼ 0) are as shown in Figure 4(a–d), respectively.

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Figure 3. (a) The vector diagram of induced current density in the blood vessel. (b) The vector diagram of electromagnetic forcein the blood vessel.

Figure 4. Velocity contours in x-y plane at different magnetic fields. (a) B ¼ 0T, (b) B ¼ 0.1T, (c) B ¼ 0.5T,(d) B ¼ 1.0T. Available in colour online.

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Figure 5 gives the velocity field of the cross-sectionof the bulge of the aneurysm when x ¼ 5 mm, i.e. themost outstanding part in the bulge of the aneurysm, atdifferent magnetic fields B ¼ 0, 0.1, 0.5, 1.0T.

As Figures 4 and 5 shows, without magnetic fieldthe ferrofluids flow slowly and flux is low, nothing ischanged after the ferrofluids flow near the bulgeregion of the aneurysm. At a weaker magnetic fieldof B ¼ 0.1T there is no obvious change near thebulge region, as shown in Figure 5(b). With an

increase of the magnetic field, the flux into the bulgeincreases.

Figure 6 shows the distribution curves of velocityalong y ¼ 0 in x–y plane (z ¼ 0) at different magneticfields B ¼ 0, 0.1, 0.5, 1.0T.

As Figure 6 shows, the ferrofluids velocitydecreases when flowing through the bulge region.When under no magnetic field (B ¼ 0T) or a weakermagnetic field (B ¼ 0.1T), the velocity is slow or hasno obvious change. As the magnetic induction

Figure 5. Velocity Fields in the cross-section of the aneurysm when x ¼ 5mm at different magnetic fields. (a) B ¼ 0T,(b) B ¼ 0.1T, (c) B ¼ 0.5T, (d) B ¼ 1.0T. Available in colour online.

Figure 6. Velocity curves in x-y plane when y ¼ 0 at different magnetic induction intensity. (a) B ¼ 0T, (b) B ¼ 0.1T,(c) B ¼ 0.5T, (d) B ¼ 1.0T. Available in colour online.

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intensity increases, the velocity decreases, especiallyinto the bulge zone. This will make more stagnancy offerrofluids in the target site. The concentration offerrofluids in the target increases and the carriers havemore chance to relieve drugs.

4.3. The effect of magnetic induction intensity onpressure field

At different magnetic induction intensities, Figure 7shows the distribution curves of pressure dropnear the bulge region along y ¼ 70.4 mm in x–y

plane, i.e. along the line of the vessel wall growing thebulge.

Figure 8 shows the comparison of the above fourcurves in the same coordinate.

The blood pressure of an adult arteriole is about30 mmHg, i.e. 4 kPa, as an initial pressure. Asshown in Figures 7 and 8, the pressure falls as theflow field from the inlet to the outlet in accordancewith the specialty of human haemodynamics (Fung1984). In simulation cases, if the length of a vessel is20 times more than its diameter, the effect of importand export can be neglected (Fung 1984). In this

Figure 7. Pressure drop near the bulge along y ¼ 70.4 mm in x-y plane at the different magnetic fields. (a) B ¼ 0T,(b) B ¼ 0.1T, (c) B ¼ 0.5T, (d) B ¼ 1.0T. Available in colour online.

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case, as the length of the vessel is not enough long,the pressure drop at the inlet and outlet is affectedmore or less as shown in Figures 7 and 8. So thepressure changes more near the inlet and outlet.In the presence of the magnetic field, the pressuredrop increases. The reason is that the formation ofmagnetisation pressure of ferrofluids results in theincrease of pressure drop with the increasing ofmagnetic field intensity (Rosensweig 1997). Thepressure drop of the fluid increases to stagnateferrofluids at the targeting region.

5. Conclusion

For magnetic targeting drug delivery, we haveformulated a dynamic model of ferrofluids as drugcarriers flowing in a blood vessel, which introduce themagnetic force. This model allows better understand-ing the flow of ferrofluids under the applied magneticfield. All of the required physical parameters such asmagnetic induced intensity, viscosity, velocity, etc. areincorporated in the model. A 3D numerical simulationand analysis of ferrofluids flow are carried out in ablood vessel to understand the clinical application formagnetic targeting drug delivery. From the simulationresults and analysis, after imposing a magnetic fieldthe velocity decreases and the pressure drop increasesat the target position as the magnetic field intensityincreases. It makes more stagnancy of ferrofluids toget the required concentration of drug delivery. Thus,this model better simulates the flow state in the bloodvessel for magnetic targeting drug delivery. Simulationresults are provided in accordance with those animal

experiments (Alksne et al. 1967, Goodwin et al. 1999,Alexiou et al. 2002, Jurgons et al. 2006).

In summary, the numerical study of the mathema-tical model characterises the ferrofluid accumulationand dispersion in a simplified case for magnetictargeting drug delivery. The ferrofluid pressure profile,mass flux vectors and velocity nature provide importantinformation about the specialty of the ferrofluidaggregate at the target region. This is vital for thebiomedical transport that will allow a therapeutic agentto be used for various magnetic drug targetingapplications.

Acknowledgements

This research work was supported by the National BasicResearch Program of China (973 Program, 2007CB936004)and the National Nature Science Foundation of China (No.50875169) for fundamental research. The authors would liketo express their gratitude to Prof. Zunji Ke (ShanghaiInstitutes for Biological Sciences, Chinese Academy ofSciences) for helpful discussions.

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