Effect of non-Newtonian characteristics of blood on magnetic particle capture in occluded blood vessel

Journal of Magnetism and Magnetic Materials 374 (2015) 611–623

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Effect of non-Newtonian characteristics of blood on magnetic particle capture in occluded blood vessel Sayan Bose, Moloy Banerjee n Department of Mechanical Engineering, Future Institute of Engineering and Management, Sonarpur Station Road, Kolkata-700150, India

art ic l e i nf o

a b s t r a c t

Article history: Received 16 June 2014 Received in revised form 12 August 2014 Available online 10 September 2014

Magnetic nanoparticles drug carriers continue to attract considerable interest for drug targeting in the treatment of cancer and other pathological conditions. Magnetic carrier particles with surface-bound drug molecules are injected into the vascular system upstream from the desired target site, and are captured at the target site via a local applied magnetic field. Herein, a numerical investigation of steady magnetic drug targeting (MDT) using functionalized magnetic micro-spheres in partly occluded blood vessel having a 901 bent is presented considering the effects of non-Newtonian characteristics of blood. An Eulerian–Lagrangian technique is adopted to resolve the hemodynamic flow and the motion of the magnetic particles in the flow using ANSYS FLUENT. An implantable infinitely long cylindrical current carrying conductor is used to create the requisite magnetic field. Targeted transport of the magnetic particles in a partly occluded vessel differs distinctly from the same in a regular unblocked vessel. Parametric investigation is conducted and the influence of the insert configuration and its position from the central plane of the artery ðzoffset Þ, particle size ðdp Þ and its magnetic property ðχ Þ and the magnitude of current (I) on the “capture efficiency” (CE) is reported. Analysis shows that there exists an optimum regime of operating parameters for which deposition of the drug carrying magnetic particles in a target zone on the partly occluded vessel wall can be maximized. The results provide useful design bases for in vitro set up for the investigation of MDT in stenosed blood vessels. & 2014 Elsevier B.V. All rights reserved.

Keywords: Magnetic drug targeting Stenosed blood vessel Rheology Computational fluid dynamics

1. Introduction In conventional drug delivery the drug is administered by intravenous injection; it then travels to the heart from where it is pumped to all regions of the body. For the small target region that the drug is aimed at, this method is extremely inefficient and leads to much larger doses (often of toxic drugs) than necessary. To overcome this problem, a number of targeted drug delivery methods have been developed [1–3]. Among them, the magnetic targeted drug delivery system (MDT) is one of the most attractive strategies because of its non-invasiveness, high targeting efficiency and its ability to minimize the toxic side effects on healthy cells and tissues [4,5]. MDT is a therapeutic technique that uses an external magnetic field to retain magnetic drug carrier particles (MDCPs) at a specific site in the body. Häfeli et al. [6], Jurgons [7] among many others provides sufficient introductory and background information on MDT. MDT therapy is a promising technique for the treatment of various diseases, especially cancer, arteriosclerosis, like stenosis, thrombosis and aneurysm, what is important is to keep the therapeutic drug in the targeting site,

n

Corresponding author. E-mail address: [email protected] (M. Banerjee).

http://dx.doi.org/10.1016/j.jmmm.2014.09.012 0304-8853/& 2014 Elsevier B.V. All rights reserved.

which is located along the inner wall of the blood vessel [8,4,5,9,10]. Some in vitro [9,10] and in vivo [8,11,12,7] experiments have been performed in this direction. The newer magnetic particles are composed of a magnetic core material (usually iron or iron oxide) that has been treated to retain drugs on its surface or internally. The particles are small enough ð o 5 μmÞ to navigate the artery and capillary systems of the body, but larger particles can lodge into certain tissues. By configuring magnets externally (outside the body), the particles can be stopped in the bloodstream or pulled in a direction of flow until the particles are positioned at the disease site. For tumors, the particles are drawn into the blood supply arteries of the tumor and lodged in the tissue. Radioisotopes or chemotherapeutic drugs attached to the particles can irradiate the tumor cells or desorb from the particles and attack the tumor cells. Indeed, given the intricate network of blood vessels and capillaries in the human body, precise guidance of particles to capillary beds is prohibitive. A sophisticated and powerful magnet system capable of operating in parallel with an imaging scheme (e.g., magnetic resonance imaging or computed tomography) may be needed to target or concentrate the magnetic particles in all but the most accessible (e.g., a superficial target or an organ that displays other conditions such as enhanced blood flow, natural filters, or sheer mass that naturally accumulates the particles) sites in the body. Crude bar

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magnets have been determined to work well for superficial tumor sites and for targeting liver tumors, and offer a relatively inexpensive capital investment compared with superconducting magnet designs. The most advanced superconducting magnet system has been developed by Stereotaxis, Inc. [13], for guiding catheters within the human vasculature. Typically, this compound in which drugs are bound with nanoparticles is injected through a blood vessel supplying the targeting tissue in the presence of an external magnetic field with sufficient field strength and gradient to retain the carrier at the target site. Recent development on carriers has largely focused on new polymeric or inorganic coatings on magnetite/maghemite nanoparticles [14], such as ferrofluids. Ferrofluids are colloidal solutions of ferro or ferromagnetic nanoparticles in a carrier fluid, which are widely used in technical applications. Because of a high magnetic moment of nanoparticles in ferrofluids, ferrofluids are gaining increasing interest to be utilized as drug carriers in magnetic targeting for biological and medical applications [15]. When they are used in medicine, ferrofluids must be biocompatible and bio-degradable [7]. Recently, some theoretical studies of magnetically targeted drug delivery considered tracking individual particles under the influence of Stokes drag and a magnetic force alone [16], and formulated a two-dimensional (2D) model, suitable for studying the deposition of magnetic particles within a network of blood vessels [17]. Other theoretical studies investigated the basic interaction between magnetic and fluid shear forces in a blood vessel [18] or utilized high gradient magnetic separation principles to study a magnetic drug targeting system [19]. The hydrodynamic drag on the magnetic particles in the arteries is very large. Therefore, establishing sufficiently strong magnetic field and gradients (by permanent magnets or electromagnets) for the guided transport of MDCPs and their localized aggregation in an arterial system remains to be major challenge. For superficial arteries magnetic bandages employing buttons or Halbach array arrangements can provide strong holding force [6]. However, for blood vessels embedded deep into the body, magnetizable inserts, e.g., stents [20,21], wires [22], and mesh [23] have been proposed. Although these studies have successfully demonstrated the feasibility of targeting MDCPs in regular blood vessels, to the knowledge of the authors, such studies in occluded vessels have not been well-reported. Since the hydrodynamics of occluded blood vessels differ considerably from that of regular ones [24], targeted localization of magnetic drug careers in a stenosed arterial geometry can provide useful information for magnetically targeted anti-angiogenic drug therapy, or for thrombolytic treatment in a partly occluded blood vessel. A recent numerical study by Haverkort et al. [25] has demonstrated feasibility of using superconducting magnets and magnetic particles of 250 nm–4 μm diameters for targeted delivery in mildly stenosed coronary and carotid arteries. The magnetic force on a particle is strongly dependent on the 3 size of the particle (i.e., the force is proportional to d , where d is the diameter of the particle) and many studies have consequently focused on micron-sized magnetic particles. In a recent simulation study on the retention of magnetic particles in small arteries exposed to a magnetic field generated by a Maxwell coil pair electromagnet it was concluded that particles should be significantly larger than 10 μm to be captured efficiently. Such large particles are not suitable for in vivo applications since they will obstruct the blood capillaries [26]. Although there have been a number of theoretical studies for magnetic drug targeting, very few researchers have addressed the hydrodynamic models of magnetic fluids in magnetic drug targeting delivery. Thus, the transport issues related to magnetic drug targeting delivery are yet poorly understood and it retards the

extensive application of the magnetic drug targeting delivery. Therefore, it is very necessary to study the flow of the ferrofluids in the blood vessel under the action of the external magnetic field. The Blood being a suspension of cells in plasma, it exhibits a non-Newtonian behavior at the low shear stress mainly when it flows through small arteries and micro-vessels [27]. The biophysical aspects of blood flow in the micro-vessel has been analyzed by Prier et al. [28], gives intricate interaction between the mechanical behavior of blood and its cellular constituents, and also its effect in the complex and irregular geometries of microvessels. Bugliarello and Sevilla [29] and Cokelet [30] have shown experimentally that when blood flows through a narrow blood vessel there is a peripheral layer of plasma and a core region of suspension of all the erythrocytes. They also suggested that Casson model and Herschel–Bulkley model are more suitable for the study of blood flow through narrow arteries. It has been established by Merrill et al. [31] that Casson model holds satisfactory for blood flowing in tubes of diameter 130–1300 mm, where as Herschel–Bulkley fluid model could be used in micro-tubes of diameter 20–100 mm. Recently Shaw and coworkers [32–38] has been doing a considerable work in the field of MDT, both theoretical as well as experimental to find out the trajectories of the drug dosed magnetic carrier particle in a micro vessel, which is subjected to the external magnetic field using different rheological model of blood. Here we have numerically investigated the targeting of micronsize magnetic beads in the partly occluded region of a blood vessel having a 901 bend using a magnetic insert as proposed by Haverkort et al. [25] considering the non-Newtonian characteristics of blood. This technique can offer strong local magnetic field gradient, causing the desired magnetic capture of the drug carrying magnetic beads even for vessels that are located deep inside the body. A mathematical model presented in this article describes the hydrodynamics of ferrofluids in a blood vessel under the action of the magnetic field. Numerical simulations are performed to obtain better insight into the theoretical analysis. Furthermore, the ferrofluids flow is analyzed numerically with computational fluid dynamics (CFD) in a model of an idealized 3D blood vessel containing an occlusion to understand the clinical application of ferrofluids. Distribution of particle capture along the endothelial wall of the occluded region of the vessel, and the particle targeting efficiencies are investigated as functions of the hemodynamic, geometric and magnetic parameters for four different rheological models. Results of the analysis provided important information leading to adequate drug delivery to the target site and can suggest strategies for improving delivery in favour of the clinical application.

2. Model equations 2.1. Blood flow Three dimensional continuity and Navier–Stokes (NS) equations are used as the governing equations of blood flow. ∇u¼0


ρ

 ∂u þu  ∇u ¼  ∇p þ μ∇2 u ∂t

ð1Þ ð2Þ

where, u is three dimensional velocity vector, ρ is density of the blood, t is time, p is pressure, μ is viscosity of blood (flow). The fully developed velocity profile is used for the inlet, while the outlet is set to be of zero gauge pressure and the no slip velocity boundary condition is used for the artery wall.

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Blood has been treated as an incompressible single phase fluid. It may be considered as a Newtonian fluid particularly for flow through large arteries [31]. However, blood exhibits non-Newtonian behavior at shear rate less than 100 s  1, which is often typical in the recirculating regions formed distal to the stenoses [39]. The Newtonian assumption is valid only for the case when the blood flows inside larger arteries [40]. Since blood flow in small arteries often exhibits non-Newtonian characteristics, how these affect flow behavior has attracted a considerable research interest. In order to determine the most suitable model for simulating the changes of viscosity in blood steam, Cho and Kensey [41] investigated several non-Newtonian models, including Power Law and Carreau models, and compared their results with actual blood samples. Perktold et al. [42] found that the Power Law model demonstrated more significant non-Newtonian influence. Most recently, Johnston and coworkers [43] found that the Carreau model seemed to be far more suitable for blood flow as the results from Carreau model agreed best with most of their experimental data. They also found that the Carreau model did not over predict the fluid behavior near the vessel wall for cases of high velocities with a significant non-Newtonian impact on the flow. The effect of gravity is neglected in comparison with the inertia and viscous force. Energy equation is not included in the governing equations since the effect of temperature change is assumed relatively small. For the present study, we have considered four different non-Newtonian models for blood namely Carreau, Quemada, generalized power law and modified Casson model in order to get the insights of non-Newtonian behavior of blood on the MDT. The relative importance of inertial forces compared to viscous forces is given by the dimensionless Reynolds number Re ¼ ρu0 d=μ with u0 and d characteristic velocity and length scales of the flow under consideration. For the left main coronary artery for example u0  0:1 m/s and d  7:0 mm such that with μ  0:00345 Pa s the Reynolds number is Re  200, which is well within the laminar flow regime. The rheology of blood can be described by four different model.

2.1.1. Carreau model Carreau [44] proposed a four-parameter non-Newtonian viscosity model which is given by for this model the shear stress is given by   μðγ_ Þ ¼ μ1 þ μ0  μ1 ½1 þ ðλ_γ Þ2 ðn  1=2Þ

ð3Þ

where μ0 ¼ 0:056 Pa s is the blood viscosity at zero shear rate, λ ¼ 3:131 is the time constant associated with the viscosity that changes with the shear rate, and n ¼ 0:3568.

613

2.1.3. Generalized power law model The generalized power law model [45] is given by   μ ¼ λγ_ n  1 "

ð5Þ

# b   exp   γ_

ð6Þ

"
#   γ_    d exp   n γ_  ¼ n1 þ Δnexp  1 þ c γ_

ð7Þ




  λ γ_  ¼ μ1 þ Δηexp  1 þ

  γ_  a

  where γ_  is the strain rate, the rest values are μ1 ¼ 0:00345 Pa s, n1 ¼ 1, Δη ¼ 0:025 Pa s, Δn ¼ 0:45, a¼ 50 s  1, b¼ 3 s  1, c ¼50 s  1, d¼ 4 s  1. 2.1.4. Modified Casson model For large diameter vessels, like arteries, a modified and more general Casson model was formulated by González and Moraga [46], which is given by pffiffiffiffiffi !2    pffiffiffiffiffi τ0   _ ffiffiffi p þ μ γ ¼ ηc ð8Þ pffiffiffi λ þ γ_ where ηc ¼ 0:3:45  10  3 Pa s is the blood viscosity at zero shear rate, τ0 ¼ 2:1  10  2 s  1 is the yield stress and λ ¼ 11:5 s  1 is a constant when the shear rate tends to zero. 2.2. Magnetic force calculation The magnetization force Fm , also known as the Kelvin force or magneto-phoretic force, on a magnetized particle of volume V is given by Fm ¼ ∭V μ0 M  ∇HdV

ð9Þ 2

NA is the magnetic permeability of where μ0 ¼ 4π  10 vacuum, M is the materials magnetization and H is the magnetic field. Above a certain magnetic field strength the magnetization saturates to a constant value M sat , resulting in the following model: 8 < χ ef f H H o M sat =χ ef f ð10Þ M¼ ^ H Z M sat =χ : M sat H ef f 7

where, the effective susceptibility of the spherical  particles is related to their intrinsic susceptibility χi as χ eff ¼ χ i = 1 þ 1=3χ i . ^ A hat is used to indicate the unit vector, i.e., H ¼ H=H, with H ¼ jHj. For a curl free magnetic field, the above equation can be reduced to 8 3 > < π6dp μ0 χ2ef f ∇H 2 H oM sat =χ ef f Fm ¼ ð11Þ 3 > : π dp μ0 M sat ∇H H ZM sat =χ ef f 6

2.3. Particle trajectory 2.1.2. Quemada model This model was developed by Quemada to predict the viscosity of concentrated systems based on the shear rate and haematocrit. This viscosity model is given by [44] 0

12 qffiffiffiffiffiffiffiffiffiffiffiffi  γ_ =_γ c K þ K 1 0 1 B C qffiffiffiffiffiffiffiffiffiffiffiffi ϕA μðγ_ Þ ¼ μp @1  2 1 þ γ_ =_γ

ð4Þ

c

where μp ¼ 1:2  10  3 Pa s is the viscosity of plasma (suspending medium) and for haematocrit ϕ ¼ 0:45 the values of the parameters are γ_ c ¼ 1:88 s  1 , K 1 ¼ 2:07 and K 0 ¼ 4:33.

When the magnetic career particles are transported in the vasculature, they simultaneously experience magnetic force, viscous drag force, gravitational (including buoyancy) forces, particle inertial effect, and thermal Brownian effects. The trajectory of a discrete phase particle is obtained by integrating the force balance on the particle, which is written in a Lagrangian reference frame [47]. This force balance equates the particle inertia with the forces acting on the particle, and can be written (for the x direction in Cartesian coordinates) as

   g x ρp  ρ dup ¼ F D u  up þ þF x ð12Þ dt ρp

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where F x is an additional acceleration (Magnetic force/unit particle mass) term, F D is the drag force per unit particle mass. FD ¼

18μC D Re

ð13Þ

ρp d2p 24

The drag coefficient, C D , for smooth particles using the spherical drag law can be taken from a2 a3 C D ¼ a1 þ þ 2 Re Re

ð15Þ

ð14Þ 2.7. Numerical solution methods

where, a1 , a2 and a3 are constants that apply over several ranges of Re given by Morsi and Alexander [48]. The trajectory Eq. (12) is solved by stepwise integration over discrete time steps using the trapezoidal discretization scheme. 2.4. Capture efficiency A major challenge for Magnetic Drug Targeting is to create a large enough magnetic force to capture particles of reasonably small size. The material properties and magnetic field are therefore chosen such as to maximize the capture efficiency (or collection efficiency). The capture efficiency (CE) for a piece of artery is defined as the fraction of inserted particles that is attracted by the magnetic field towards the vessel wall and are captured or collected there [47,49]. 2.5. Materials used MDCP of varying diameters ranging from 250, 500 nm, 1, 2 and 4 μm were inserted into the flow, homogenously distributed over the inlet. Particles are assumed to be inert with a density of 1800 kg/m3 and a saturation magnetization of M sat ¼ 106 A/m. For the magnetic field we used an infinitely long straight wire carrying a current of I. In this paper, we consider four different location of the magnet (see Fig. 1) and for the infinitely long current carrying conductor, the magnitude of the field at a distance s (measured normal to the axial direction) is given by H ¼ I=2π s. A current of I ¼ 105 A was chosen, yielding a magnetic field of B ¼ μ0 H ¼ 2T on the artery centerline halfway the bend. 2.6. Geometry Fig. 1 shows the computational domain considered for the simulation study. A tetrahedral mesh was used, containing approximately 535,000 nodes and 380,000 elements. The artery model is considered to be 901 bend with circular cross section of

10R

radius R¼3.5 mm and the upstream and downstream side of the stenosis zone are considered to be of 10R length. The occlusion geometry is assumed to be of Cosine shaped, which is expressed as: (
 2 ) 2π ðx  lu Þ Rz ¼ R 1  0:5S 1  cos ; for lu r x r lu þls ; ls

Configuration 4

R

Configuration 3 Stenosed artery

Configuration 1

Configuration 2

10R

The Lagrangian discrete phase model (DPM) in ANSYS FLUENT [50] follows the Euler–Lagrange approach. The fluid phase is treated as a continuum by solving the Navier–Stokes equations. A finite volume solver is used for simulation. Central differncing scheme of second order accuracy is used to discretize the diffusion terms whereas the convective terms are discretized by using Power law scheme. The coupling of pressure velocity is done with SemiImplicit Method for Pressure-Linked Equations (SIMPLE) technique, where the under relaxation parameter for pressure is taken to be as 0.3. The standard Lagrangian part of the DPM calculates the trajectory based on the translational force balance that is formulated for a representative particle as in Eq. (3). In the standard DPM, each particle represents a parcel of particles. In our case, a DPM parcel is subjected to a fluidic drag force, a magnetic force and gravity. The magnetic force is programmed based on the particle position (not the cell position) and then compiled into FLUENT using a user defined function (UDF). A fundamental assumption made in this model is that the dispersed second phase occupies a low volume fraction, even _ particle Z m _ f luid Þ is acceptable. It is though high mass loading ðm essential to monitor this volume loading in each cell to ensure that the value of this parameter should not increase beyond 12%, which is the upper limit for the validity of the DPM. The trajectory Eq. (3) is solved by stepwise integration over discrete time steps using the trapezoidal discretization scheme. The maximum number of time steps used to compute a single particle trajectory via integration of Eq. (3) used in the simulation is 5000, when the maximum number of steps is exceeded; ANSYS FLUENT abandons the trajectory calculation for the current particle injection and reports the trajectory fate as “incomplete”. Step Length Factor is inversely proportional to the integration time step and is roughly equivalent to the number of time steps required to traverse the current continuous phase control volume. In this simulation the step length factor is chosen as 5. In this problem the Steady Particle Tracking is enabled, using Four Seventy Eight (478) numbers of particles are allowed to enter inside the domain at through each cell centroid from the inlet plane. A “two-way coupling” method is used to predict the effect of discrete phase on the continuum. This two-way coupling is accomplished by alternately solving the discrete and continuous phase equations until the solutions in both phases have stopped changing. In our analysis particle streams were released from the inlet plane. The outlet was set to allow particles to escape while the wall of the artery is set to trap particles. Thus, the CE could be determined by calculating the percentage of particle streams that were trapped. In our numerical studies, we imposed a “trap” boundary condition to investigate the amount of particles that are captured at the wall.

3. Grid independence

Fig. 1. Computational domain.

The physical flow domain was discretized into a large number of hexahedral computational cells. Model was tested for three

S. Bose, M. Banerjee / Journal of Magnetism and Magnetic Materials 374 (2015) 611–623

different grid densities i.e., 239, 478 and 717 cells in the crosssectional flow area. The absolute difference in centerline axial velocity between the coarse and fine cross-sectional mesh was 2 mm/s, and that between fine and finer one was only 1 mm/s, which was very small compared to the mean centerline axial velocity value of 200 mm/s. Therefore, we adopted the scheme that contained 478 cells per cross-section, i.e., 380,000 for the whole mesh as shown in Fig. 2 that gave the best grid independency and stability in solution within a reasonable CPU time. Computations were carried out on a Microsoft Windows7 Ultimate workstation with Intels Core(TM) i7–3770, 3.40 GHz processor with 16 GB random access memory (RAM) and 2 TB hard drive available at the DST sponsored project laboratory. The average CPU time for one set of calculations was 50 min.

4. Code validation study The present numerical result has been validated taking the result of Cohen et al. [51], where the particles are allowed to inject inside the artery which is straight and an infinitely long current carrying conductor is placed parallel to the axis of the artery,

Mn

y (m )

p

Mn

0.2250

p

5.1. Particle trajectories and capture on the wall (base case): Casson model

0.1687

0.006

Mn

p

0.1125

0.004

0.002

0 0.03

Mn p 0.0563

0.04

0.05

0.06

0.07

0.08

x (m ) 1.6

Shaw 2010c Shawetetal., al. 2010c PresentNumerical Numerical Present

1.4

Volume fraction of particle

1.2 1 0.8 0.6 0.4 0.2 0 300

which ensures that particles are being trapped near the wall. The wire is located at hwire ¼ 2 cm above the centerline of the pipe. The particle diameter is dp ¼ 2 μm, its density is ρp ¼ 6450 kg=m3 and its susceptibility is χ ef f ¼ 3. The trajectories of individual particles are computed for different particle magnetization numbers Mnp ¼ Fm0 =3πμdp u0 by adjusting the current through the wire. Fm0 and u0 are the magnetic force and the fluid velocity at the centerline of the pipe flow. The result of the present numerical simulation is compared with the solution given by Cohen et al. [51] and is shown in Fig. 2a. From the figure it is quite clear that the solution matches quite well and hence validates the numerical scheme used over here. As a further step towards the validation of the commercial code used herein, the present numerical code is also compared with the analytical results of Shaw et al. [38], for the flow of carrier particle in a straight tube under the action of external magnetic field keeping the same input parameters as mentioned in the literature. The corresponding variation of particle volume fraction with the size of magnetic carrier particles have been plotted (see Fig. 2b), which shows that the present numerical method matches quite well with the result, which will certainly validate the mathematical model presented herein.

5. Results and discussion

0.01

0.008

615

400

500

600

700

800

900

1000

Particle radius (nm) Fig. 2. (a) Trajectories of magnetic particles released at the centerline of the pipe for different values of the magnetic number. The symbols are the analytical solution, whereas the lines are the present numerical solution. (b) Comparison of the present study with Shaw et al. [38] for Newtonian fluid.

Simulations are performed in a 901 bend partly occluded blood vessel having a diameter d of 7 mm, and a stenosed curved length ls of 7 mm and a degree of occlusion S¼ 50%. The density of the blood is taken as ρ ¼1056 kg/m3. The base case simulation is performed for a steady flow of blood at Re¼200 and corresponding particle diameter of 1 μm,  effective susceptibility χ of 3 (these values are considered to be same for the four different non-Newtonian rheological model considered herein). The magnetic insert, for the base case, is assumed to carry a current of I ¼ 105 A and it’s placed at 1 cm from the axis of the artery, which results in yielding a magnetic field of B ¼ μ0 H ¼ 2 T on the tube centerline. The saturation of magnetization is assumed to be M s ¼ 1:2  106 A=m. The fluid velocity, resulting from the simulations, is shown in Fig. 3 considering the modified Casson model for the blood rheology. The presence of two characteristic counter-rotating Dean-vortices can clearly be seen. The centrifugal force acting on the fluid particle further forces the fluid particles towards the outer arterial wall. The presence of occlusion will further increase this velocity compared to the case where there is no occlusion. At the throat of the stenosis, since there is a large increase in velocity magnitude ðuthroat =u0 ¼ 5Þ, the presence of two counter-rotating vortex are very much prominent, shown in the inset of Fig. 3. Due to presence of the strong centrifugal force the location of maximum velocity is also shifted from the center of the artery. After the bend portion is over all the fluid particles tries to adheres with the outer wall, which in turn results into the formation of a strong recirculation length occupying major portion of the inner arterial wall. It has also been observed that the numerical value of the maximum velocity is not very much affected by the different rheological model considered (see Table 1). From the result shown in Fig. 3, the maximum velocity is almost 0.545 m/s, which is almost same as the Quemada model, but both the Carreau and generalized power law model predicts almost 5% higher estimated value corresponding to the other two rheological models. Fig. 4 shows the variation of wall shear stress (WSS). From the figure it is quite clear that maximum WSS will occur at the throat

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100

y m

CE %

80

Config 1 Config 2 Config 3 Config 4

60

40

20

0

1

1.5

2

2.5

3

3.5

4

d p µm

x m Fig. 3. Velocity magnitude contour for the base case simulation (inset shows the velocity vectors also for the section plane as shown above).

Table 1 Maximum velocity observed for different rheological model. Rheology

Velocity (m/s)

0.5

Carreau

Quemada

Generalized power law

0.570

0.54

0.574

Fig. 4. WSS (Pa) contour for the base case simulation.

of the stenosis and it is almost 6.5 Pa. A strong recirculation zone is observed at the downstream side of the stenosis, which will certainly affect the chance of further deposition. Hence, from the point of view of targeting the drug, it should reach the zone where the chance of deposition is high. In order to find the effect of different insert configuration on the CE, next we present the variation of CE with the particle diameter. Since, the size of the magnetic drug carrier particle has a strong influence on the efficacy of the MDT system; hence it is quite important to find the influence of it. Fig. 5 shows the variation of CE with the particle diameter, for the different inset configuration. From the diagram it is quite clear that 100% CE value

Fig. 5. Variation of CE with the particle size for different inset configuration (other parameters remains same as that of the base case considered using modified Casson Model).

is observed at a particle diameter of 1.5 μm for configuration 2 and 4, where as CE value is obtained at 2 μm and 1.75 μm for configuration1 and 3 respectively. At any particular diameter, the fourth configuration always predicts better CE value compared to all other configuration. It is also interesting to note from the result that up to 7.5 μm, there is not so much variation in CE values predicted by configuration 2 and 3, but increase in particle size beyond this limit clearly separates the two curve. Fig. 6a–d shows the contour for magnetic force per unit particle volume considering four different configuration of the insert (see Fig. 1) under the base case simulation. For all configurations the distance of closest approach from the wire to the centerline of the artery is chosen to be 1 cm. From the figure it is interesting to note that though there is little variation in the maximum value of the force, but for the fourth configuration the force distributes through a larger section of the artery and hence produces better capture. In the next part of the simulation, we focus our attention to find the effect of different insert configuration on CE under the base case simulation. As a first step towards this we present the particle capture histogram plotted against the axial distance (x/d), since the geometry under consideration is a curved one (see Fig. 1), the histograms are shown up to a non-dimensional length of 8 and beyond this point the artery becomes stretched along the y-direction. In order to visualize the amount of particles that are trapped on the wall in the downward part of the artery, we also present the histograms along the y-direction (shown in the insets of Fig. 7a–d). Out of the 478 number of particle injected from the inlet plane, the particle capture histograms shows that 230, 326, 296 and 433 numbers of particles are captured on the wall for configuration1, 2, 3 and 4 respectively. A closure look into the histogram diagrams shows that for configuration 1 (Fig. 7a), very few number of particle (almost 2.5%) has been captured in the pre stenosis zone i.e., 0 r x=d r 5d. Most of the particles  (almost 31%)of the particles are captured in the stenosis zone 5d r x=d r7d and very few number of particles (almost 15%) are captured in the downstream  side of the stenosis 7d r x=d r 8d , which is more prone to further deposition of plaque. Remaining 51.5% particles are escaped out of the domain, which is certainly total loss for the MDT system. The inset of Fig. 7a shows the y-distribution of all the particles (i.e., 28 number of particles) that are captured at the last part of the artery.

S. Bose, M. Banerjee / Journal of Magnetism and Magnetic Materials 374 (2015) 611–623

617

Fig. 6. Contour of Magnetic force per unit particle volume for four different configuration of the insert, (a) configuration 1, (b) configuration 2, (c) configuration 3 and (d) configuration 4.

The histogram diagram for configuration 2 (Fig. 7b), shows that almost 28% of particles are captured both in the stenosis and downstream side of the stenosis zone, which certainly improves the efficacy of the MDT system. The percentage of loss of drug is reduced significantly (almost 31%) compared to the earlier configuration. The prediction of the result for the third configuration (Fig. 7c) is almost same as that of the first one, with only improvement in the loss of drug (38%). Fig. 7d shows the histogram plotting for the fourth configuration; it is quite clear from the diagram that this configuration produces better distribution of particles over other. Almost 32% and 28% of particles are captured in the stenosis and downstream side of the stenosis zone, which is certainly produces best result among the different configuration of the insert we have taken into consideration. The amount of drug loss (almost 9%) also reduces significantly. The y-distribution of the particles (inset of Fig. 7d) shows that out

of the 28% of particles that are captured at the downstream zone almost 34% of particles are captured at the zone of minimum WSS ðy=d  2dÞ. From the above result we can certainly conclude that from the efficacy of MDT system is considered the configuration 2 and 4 produces the best result and hence in the next part, we restrict our discussions only on these two configurations.

5.2. Parametric investigation The above results presented so far clearly justifies the fact the configuration 4 will predict the more value of CE and hence we limit our discussion with this configuration only and we focus our attention to find the effect of different rheological model considered herein.

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90 80

25

80

20

60

20

15

15

No. of particles captured

No. of particles captured

70

10

50

5

40

0

25

1

2

3

4

5

6

7

8

30 20

60

10 5

40

0

1

2

3

4

5

6

7

8

20

10 0

2

4

x

6

0

8

x

80

15 10

50

5

40

0

1

2

3

4

5

6

7

8

30

70 60 50 40

6

8

d

50

20

10

10

2

4

x

6

8

30 20 10 0

1

2

3

4

5

0

2

4

x d

6

30

20

0

8

40

No. of particles captured

No. of particles captured

60

6

90 25 20

70

4

d

90 80

2

d

Fig. 7. Particle capture histogram for different inset configuration (all other parameters remains same as that of the base case for modified Casson model), insets shows the distribution of particles along the y-direction, (a) configuration 1, (b) configuration 2, (c) configuration 3 and (d) configuration 4.

5.2.1. Variation of particle diameter Simulations are performed considering different particle diameter ranging from 250 nm to 4000 nm for two different non-Newtonian models, keeping the other parameters same as that of the base case. Fig. 8 shows the parametric variation of CE with the particle diameter for four different rheological model of blood considered here. As the particle diameter increases the magnetic force on the MDCPs also increases and results in higher value of CE. From the figure it is quite clear that the CE value increases exponentially as the diameter increases and reaches the 100% capture limit. Apart from the generalized power law model all the other three rheological model predicts almost same value of CE and 100% capture is achieved for all the three models for the particle diameter of 1.45 μm (approx), which is almost half the value (3.01 μm) at which the saturation occurs for the generalized power law model. For a particular diameter (say 1000 nm),

modified Casson model predicts a CE value of 90%, whereas Carreau and Quemada model predicts almost 3.3% and 5.5% lower value, but it is almost 33.33% for the case of generalized power law model. It is also observed that this difference between the CE values decreases as we increase the particle size. One method to determine the effect of the non-Newtonian model, proposed by Johnston et al. [43], is the importance factor I L which is derived from the concept introduced by Ballyk et al. [52] who defined this as I L ¼ μef f =μ1 , μef f is the effective viscosity characteristic of a particular flow and μ1 is the reference Newtonian viscosity (¼0.00345 Pa s). Clearly, I L ¼ 1 indicates Newtonian flow and deviations from unity indicate regions of non-Newtonian flow. Johnston et al. [43] improved on this concept by calculating an average of these importance values that would be more representative of the actual flow in the artery. Instead of simply averaging I L , they averaged the relative difference of each value of viscosity from

S. Bose, M. Banerjee / Journal of Magnetism and Magnetic Materials 374 (2015) 611–623

Table 3 Capture efficiency value for different rheology and insert configuration for particle diameter of 1 μm.

100

Rheology configuration

80

CE %

619

Carreau Quemada Gen. PL Mod. Casson

60

1 2 3

Carreau

Quemada

Generalized power law

44.77 58.79 55.02

40.17 52.30 50.21

37.45 42.47 42.68

40

20

0.5

1

1.5

2

dp

2.5

3

3.5

4

m

Fig. 8. Variation of CE with the particle size for different rheological model (other parameters remains same as that of the base case considered using modified Casson model).

No. of particles captured

0

80 1000 nm 500 nm 250 nm

60

40

20

Table 2 Area weighted and volume average of the importance factor value for different rheological model of blood. Rheology

Area weighted average

Volume average

Modified Casson Carreau Quemada Generalized power law

1.147 1.411 1.558 2.641

1.216 1.627 1.701 2.983

2 1=2 ½∑ μef f  μ1  N

μ1

2

4

x

6

8

d

Fig. 9. Particle capture histogram for three different particle diameters.

the Newtonian value that is then expressed as a percentage. This global non-Newtonian importance factor is defined as:

100 IG ¼ N

0

ð16Þ

This equation is evaluated at each of the N nodes on the entire area of the wall as well as the volume of fluid. For the nonNewtonian simulations, the global importance factor is tabulated in Table 2. From the result it seems that the generalized power law model has larger deviations from the Newtonian viscosity due to the nature of the equation that has a steeper strain-viscosity relation, while the modified Casson model deviates less from the Newtonian viscosity. This pattern clearly suggests the variation in particle trajectory because of different non-Newtonian model considered herein. Larger diameter particle shows better capture against the smaller diameter particle, but in most of the MDT system the selection of proper size of the particle is quite important in terms of the efficacy of the targeting. Since, the use of larger diameter particles can create further blockage to the smaller diameter arterioles and hence they should be avoided. The effect of all the rheological model of blood has also been studied on different magnetic insert configuration and the representative values of CE are tabulated in Table 3 for a particular particle diameter of 2 μm. From the table it is quite clear that configuration 1 will predict the least value of CE and it is true irrespective of the rheological model considered herein. Hence, as

far as the efficacy of the MDT system is considered, the first configuration is not at all suitable. Out of the three different configurations, the second one predicts the highest value of CE and hence it is more suitable compared to the other two. The Carreau model predicts the highest value of CE for the second configuration, whereas the other two predicts almost 10% and 28% lower estimated value. It is quite interesting to note that there is hardly any change in CE for the generalized power law model, with the change in configuration. The corresponding particle capture histograms are also plotted showing the variation along axial direction, considering three different particle diameters (see Fig. 9) only for the modified Casson model. For the lower particle size, the effect of magnetic force is less and hence less number of particles is captured in the pre stenosis zone. As the particle size increases more number of particle are being trapped and the axial distribution of all those particles are also uniform in nature.

5.2.2. Variation with current Fig. 10 shows the variation of CE with the current through the conductor. The nature of the diagram is almost similar for all the rheological model, apart from the generalized power law model. The diagram shows the gradual increase in CE values. It is also observed from the results that beyond a current value of 1:75  105 A, all of the three model predicts 100% CE value, where as the generalized power law model predicts almost 15% lower CE value at the same current. The same 100% CE value is achieved at 2:0  105 A for the generalized power law model. The effect of all the rheological model of blood has also been studied on different magnetic insert configuration and the

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100

90

90

80

CE %

No. of particles captured

Carreau Quemada Gen .PL Mod .Casson

80 70 60 50 40

70

1.25e5 amp 1.00e5 amp 0.50e5 amp

60 50 40 30 20

30

10

1 105

3 105

2 10 5

0

I Amp

2

4

x

Fig. 10. Variation of CE with current.

6

8

d

Fig. 11. Particle capture histogram for three different current values. Table 4 Capture efficiency value for different rheology and insert configuration for current intensity of 1:25  105 A.

1 2 3

90

Rheology Carreau

Quemada

Generalized power law

58.99 82.00 55.02

54.18 74.06 59.41

46.44 52.72 51.67

representative values of CE are tabulated in Table 4 for a particular magnitude of current flowing through the insert as 1:25  105 A. From the table it is interesting to note that there is very little change in CE value between configuration 1 and 3 for all the rheological model, whereas there is a large increase in the CE value for the second configuration. The Carreau model predicts almost 1.46 times value of CE corresponding to configuration 1 and 3, and this is almost 1.32 times for the Quemada model. But, it is quite interesting to note that there is hardly any change in the value of CE is observed for the generalized power law model for the configuration 2 and 3. Fig. 11 shows the histogram plot for three different current values. From the result it is quite interesting to note that though the increase in current value increase the CE, but at higher magnitude of current most of the particles are trapped either in the pre stenosis zone (42%) or in the stenosis zone (39%), resulting in a very few particle concentration in the downstream side (19%) of the stenosis. Hence it can be concluded that though 1:25  105 A produces maximum value of CE, but from the distribution of drug in the low shear zone is considered, it produces less effect compared to somewhat lower current magnitude ð1:0  105 AÞ. A further reduction in current magnitude reduces the CE value very significantly (almost 45%).

5.2.3. Variation with magnetic susceptibility Fig. 12 shows the variation of CE with the susceptibility of the MDCPs. The general trend is the CE value increases as the susceptibility increases, which is true for all the rheological model considered herein. A closer look into the variation of CE reveals that for a particular value of magnetic susceptibility (say χ ¼ 2), the modified Casson model predicts a CE value of almost 82%,

80 70

CE %

Configuration

60 50 40

Carreau Quemada Gen .PL Mod .Casson

30 20 1

1.5

2

2.5

3

Fig. 12. Variation of CE with magnetic susceptibility.

which is almost 11%, 13% and 38% lower for Carreau, Quemada and generalized power law model respectively. In order to get the effect of different configuration on the rheological models, the CE are tabulated in Table 5, for a particular value of magnetic susceptibility of 2. It is very much interesting to note that the effect of magnetic susceptibility has very little effect on the different models with the change in insert configuration. Fig. 13 shows the particle distribution histogram for three different magnetic susceptibility value. From Fig. 12, it is quite clear that CE value increases as the susceptibility increases, but a closure look into the histogram distribution shows that for a smaller value of susceptibility ðχ ¼ 2Þ gives better concentration (32%) of the particle compared to the case of higher susceptibility (28% for χ ¼ 3) in the downstream side of the stenosis, which is certainly the zone where the chance of further deposition is very high.

5.2.4. Variation with the position of the insert Finally the variation in CE is studied considering different distance of the insert measured from the central axis of the artery.

S. Bose, M. Banerjee / Journal of Magnetism and Magnetic Materials 374 (2015) 611–623

Table 5 Capture efficiency value for different rheology and insert configuration for magnetic susceptibility of 2. Configuration

1 2 3

Rheology

Table 6 Capture Efficiency value for different rheology and insert configuration for insert position of 7.5d from the central axis. Configuration

Carreau

Quemada

Generalized power law

33.47 42.89 42.88

30.54 38.28 38.28

30.33 33.68 35.12

621

1 2 3

Rheology Carreau

Quemada

Generalized power law

65.06 84.31 80.54

58.79 74.01 71.96

50.84 54.60 52.93

120

120 100

1 2 3

80

No. of particles captured

No. of particles captured

100

60

40

20

5 mm 10 mm

80

12.5 mm

60 40 20

0

2

4

x

6

8

0

d

2

4

x

Fig. 13. Particle capture histogram for three different magnetic susceptibility values.

6

8

d

Fig. 15. Particle capture histogram for three insert distance.

100

90

CE %

80

70

60 Carreau Quemada Gen .PL Mod .Casson

50

6

8

10

12

z offset Fig. 14. Variation of CE with magnetic inset position from the central axis of the artery.

As the distance is increased, the magnetic force on the particles tends to reduce which is clearly reflected in Fig. 14. The reduction in CE is much more prominent for the first configuration compared to the second one. The drop in CE is maximum (almost 60%) between configuration 2 and 1 for the case of offset distance of 1 cm. It is also observed from the results that as the insert moves close to the axis, the CE tends to increase and 100% capture is observed for both the configuration for the case of offset distance of 5 mm. Further reduction in offset value does not alter the CE.

In order to realize the effect of different configuration on the rheological models, the CE is tabulated in Table 6, for a particular value of insert distance from the central axis of the artery as 7.5d. Similar to the earlier observations, here also there is a very little change in the value of CE for the generalized power law model, this can be justified by the fact of strong effect of hydrodynamic drag force observed. The Carreau model predicts almost 1.3 times value of CE corresponding to configuration 1, and this is almost 1.25 and 1.07 times for the Quemada and generalized power law model respectively. Fig. 15 shows the particle capture histogram for three different offset position of the insert from the central axis. The results from Fig. 14 suggests that whenever the magnetic inset is very close to the artery, the CE value will be the highest because of the strong magnetic force acting on the particle. But a closure look to the distribution of particles reveals that there is no deposition of the particles at the low shear zone. On the other hand though, the increase in distance reduces the CE value significantly, but it offers better percentage of particle in the low shear zone (almost 32% and 28% respectively for offset distance of 10 and 12.5 mm).

6. Conclusion 3D Numerical simulation of the transport of magnetically guided drug-carrying magnetic micro spheres in a partly occluded 901 bent artery is conducted under steady flow boundary condition. A current carrying conductor is used to create the required magnetic field and is used to separate the magnetic micro spheres and target them on the artery wall near the occlusion and post occlusion region. Four different insert configuration and rheological model are used in the

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numerical study. Flow near the occluded blood vessel differs considerably from the regular blood vessel. Therefore, the particle trajectories and the spatial distribution of their accumulation on the vessel wall also are found to differ considerable from that observed in straight channels or tubes. Several parametric variations have been studied on the CE of the beads. Based on the simulation results following are the important conclusions: 1. The flow patterns inside the bend portion suggest the existence of two counter rotating vortices, which in turn tries to force the fluid flow back through the more viscous region near the side walls and the maximum velocity is shifted towards the outer curvature of the artery due to the presence of centrifugal force. 2. The existence of strong recirculation zone just downstream side of the occlusion indicates the zone of low WSS and hence the region is more prone to restenosis. 3. The magnetic force distribution clearly suggests that configuration 4 produces the better efficient MDT system, since the force is distributed through a larger section of the artery. The same is further justified by the CE value. 4. For all the rheological model, the modified Casson model predicts the highest value of CE, whereas the generalized power law gives the least value of CE, under different parametric variations. This can be attributed by the presence of strong drag force in the case of generalized power law compared to the modified Casson model. 5. The CE value is found to increase exponentially with diameter of the particle, magnitude of current through the conductor and the susceptibility of the MDCPs and reaches a saturation value of 100% after a certain point. But a closure look into the histogram distribution it reveals that though the CE is maximum at those parameters but the amount of particle deposited in the low shear zone is less. Hence, from the design point of view the selection of those parameters is quite important. The results provide useful design basis for in vitro set up for the investigation of MDT in occluded blood vessel.

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