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Research Papers

Lagrangian Magnetic Particle Tracking Through Stenosed Artery Under Pulsatile Flow Condition OPEN ACCESS

[+] Author and Article Information
Sayan Bose

Department of Mechanical Engineering,
Future Institute of Engineering and Management,
Sonarpur Station Road,
Kolkata 700150, India

Amitava Datta, Ranjan Ganguly

Department of Power Engineering,
Jadavpur University,
Salt Lake Campus,
Kolkata 700098, India

Moloy Banerjee

Department of Mechanical Engineering,
Future Institute of Engineering and Management,
Sonarpur Station Road,
Kolkata 700150, India
e-mail: moloy_kb@yahoo.com

1Corresponding author.

Manuscript received December 5, 2013; final manuscript received February 5, 2014; published online February 26, 2014. Assoc. Editor: Abraham Wang.

J. Nanotechnol. Eng. Med 4(3), 031006 (Feb 26, 2014) (10 pages) Paper No: NANO-13-1085; doi: 10.1115/1.4026839 History: Received December 05, 2013; Revised February 05, 2014

Drug delivery technologies are an important area within biomedicine. Targeted drug delivery aims to reduce the undesired side effects of drug usage by directing or capturing the active agents near a desired site within the body. This is particularly beneficial in, for instance, cancer chemotherapy, where the side effects of general (systemic) drug administration can be severe. Herein, a numerical investigation of unsteady magnetic drug targeting (MDT) using functionalized magnetic microspheres in partly occluded blood vessels is presented considering the effects of particle-fluid coupling on the transport and capture of the magnetic particles. 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 cylindrical permanent magnet insert 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 flow Re, magnetic insert diameter, and its radial and axial position on the “targeting efficiency” is reported. Analysis shows that there exists an optimum regime of operating parameters for which deposition of the drug-carrying magnetic particles in a predesignated 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.

FIGURES IN THIS ARTICLE
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MDT is one of the major drug delivery methods that are used for treatment in different parts of our circulatory system. Among these different drug delivery methods, the magnetic targeted drug delivery system is one of the most attractive strategies due to its noninvasiveness, high targeting efficiency (TE) and minimizing of the toxic side effects on healthy cells and tissues [1,2].

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. [3] among many others provides sufficient introductory and background information on MDT. MDCPs typically comprise of a biocompatible polymer or metal oxide support, a magnetic material, and a drug or medically active agent, which is either encapsulated or attached to the surface. Torchilin [4], and Dobson and co-workers [5,6] provide detailed information on the types and applications of MDCPs.

MDT has been proposed as a potentially efficient and beneficial method for the medical treatment of various diseases and cardiovascular episodes, such as stenosis and thrombosis [7]. Atherosclerotic lesions and thrombi are important targets in the cardiovascular system for delivering diagnostic (imaging) and therapeutic agents [8]. For example, anti-angiogenic agents have been shown to decrease both neovascular proliferation and plaque development in animal models of atherosclerosis when administered chronically at high dosages [9]. However, chronic high doses of such agents are not free from side effects [10,11]. Targeted deposition of lipophilic agents, e.g., doxorubicin or paclitaxel, into the cell membrane inhibits the proliferation of vascular smooth muscle cells in vitro [12]. Recently, newer stent-based drug delivery systems, incorporating hydrophobic antiproliferative agents, have been successful in the clinic [13]. Similarly, tissue plasminogen activator-loaded magnetic nano- and microcarriers, guided directly to the site of vascular occlusion by external magnetic fields can be successfully used to induce effective thrombolysis and ameliorate acute ischemic stroke [14]. All these promising results indicate that local deposition and prolonged release of appropriate antiproliferative agents can effectively ameliorate restenosis. Several other clinical applications of MDT are also proposed in the literature, ranging from locoregional cancer treatment [15], magnetic fluid hyperthermia [16,17] and magnetic resonance imaging contrast enhancement [18].

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 [3]. However, for blood vessels embedded deep into the body, magnetizable inserts, e.g., stents [19,20], wires [21], and mesh [22] 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 [23], 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. [24] 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.

Here we have numerically investigated the targeting of micron-size magnetic beads in the partly occluded region of a blood vessel using a magnetic insert as proposed by Furlani and Furlani [21]. 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. 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.

Figure 1(a) shows the geometric configuration used for simulation of particle trajectories in the occluded vessel. The artery wall is considered to be noncompliant, which is consistent with the observation that atherosclerotic lesions accompany with hardening of arterial wall. The artery is treated as a rigid tube of diameter d (radius R0), having an inlet segment of length lu = 4d; a stenosed segment of length ls = d, and a downstream segment of length ld = 35d (see Fig. 1(a)). The degree of occlusion (S) is expressed as S=(Ro-Rmin/Ro)×100%, where Rmin denotes the radius at the throat of the occlusion. The study has been performed with 50% occlusion. We have assumed an idealized shape [25] of the occlusion that is expressed as

Magnetic particles injected at far upstream of the occlusion enter the region of interest (ROI), i.e., the occluded region, in the form of a homogeneous suspension. A cylindrical magnetic insert of radius Rm is placed at a distance ymag from the axis of the blood vessel (Fig. 1(a)), and xmag from the inlet plane and is used to produce the requisite magnetic field [21]. In the coordinate reference frame of the blood vessel, the magnetic field (see Fig. 1(b)) is expressed asDisplay Formula

(2a)Hx=MSRm22(xmag-x)2[(ymag-y)2+(xmag-x)2]2
Display Formula
(2b)Hy=MSRm22(ymag-y)2[(ymag-y)2+(xmag-x)2]2,andHz=0

where, MS is the remnant magnetization ( = 1.2 × 106 A/m) of the materials of the magnetic insert.

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 [26]. 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) asDisplay Formula

(3)dupdt=FD(u-up)+gx(ρp-ρ)ρp+Fx

where Fx is an additional acceleration (Magnetic force/unit particle mass) term, FD is the drag force per unit particle massDisplay Formula

(4)FD=18μCDReρpdp224

The drag coefficient, CD, for smooth particles using the spherical drag law can be taken fromDisplay Formula

(5)CD=a1+a2Re+a3Re2

where a1, a2, and a3 are constants that apply over several ranges of Re given by Morsi and Alexander [27].

The magnetic force term due to the presence of a finite gradient of the magnetic field, is expressed asDisplay Formula

(6)Fm=μ0(43πa3)χeff12(H·H)

where the effective susceptibility of the spherical particles is related to their intrinsic susceptibility χi as χeff=[χi/(1+(1/3)χi)] [28]. Dipole–dipole interaction between the magnetized microspheres is neglected in Eq. (6), since it is a very short-range force, and is not important for dilute suspension of the particles (which is the case here). Brownian motion can influence particle motion when the particle diameter is sufficiently small [29]. Gerber et al. [29] has suggested that the critical value of nanoparticles diameter is 40 nm below which the Brownian force cannot be neglected since it will sufficiently reduce the fluidic drag force. Hence, in our simulation, we ignore the effect of the Brownian force.

We have assumed an inelastic boundary for the particles on the blood vessel wall, signifying that the particle velocity vanishes after they impinge on the walls. Since there is a suspension of particles is considered, particle-fluid interaction is treated as two-way: i.e., the fluid drag influences the particle trajectories, the reaction of such drag on the fluid is also considered to alter the fluid flow. Thus, the fluid phase is kinetically coupled from the dispersed phase.

Three-dimensional continuity and Navier–Stokes equations are used as the governing equations of blood flowDisplay Formula

(7).u=0
Display Formula
(8)ρ(ut+u.u)=-p+μ2u

where u is three-dimensional velocity vector, ρ is density of the blood, t is time, p is pressure, and μ is viscosity of blood (flow).

Blood has been treated as an incompressible single phase fluid. It may be considered as a Newtonian fluid particularly for flow through large arteries [30]. 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 [31]. 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 the Newtonian model of blood and the density is taken as constant throughout the domain and time. Viscosity (μ) and density (ρ) are taken as 0.00345 kg/ms and1056 kg/m3, respectively.

In the unsteady flow, the physiological waveform was taken as the inlet velocity boundary condition (Eq. (9)) and the results of steady flow simulation are taken as initial conditions for unsteady flow simulationDisplay Formula

(9)u(y,t)=2u¯[1-(yR)2]{1+Asin(ωt)-Bcos(2ωt)}

where A and B are two constants (= 0.75), ω is the circular frequency of unsteady blood flow = 7.5, and R is the radius of the artery = 2 mm.

Wall of the stenosed blood vessel has been considered as rigid wall, implementing a no slip boundary condition at the wall, while a zero gauge output pressure has been considered at the output of the vessel due to the presence of the whole system (stenosed blood vessel) in the submerged body fluid.

The Lagrangian discrete phase model (DPM) in ansys fluent [32] 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 differencing scheme of second order accuracy is used to discretize the diffusion terms, whereas the convective terms are discretized by using Power law scheme and first order Euler scheme in time. The coupling of pressure velocity is done with semi-implicit method for pressure-linked equations technique, where the under relaxation parameter for pressure is taken to be as 0.3. A time step size of 0.003351 s is used which corresponds to 250 time step per cardiac cycle, while the dispersed phase is solved by tracking a large number of particles through the calculated flow field. The dispersed phase can exchange momentum, mass, and energy with the fluid phase.

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.

A fundamental assumption made in this model is that the dispersed second phase occupies a low volume fraction, even though high mass loading (m·particlem·fluid) is acceptable. It is 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 equation (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 Unsteady Particle Tracking is enabled, using particle time step size (0.001 s) to inject the particles. Sixty numbers of particles are allowed to enter inside the domain at each time step from the inlet plane. Particles are assumed to be inert with a density of 1800 kg/m3. Different diameters of particles ranging from 100 nm to 4000 nm are used for the entire simulation.

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 continuously 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 collection efficiency 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 ROI.

The physical flow domain was discretized into a large number of hexahedral computational cells. Model was tested for three different grid densities, i.e., 284, 568, and 852 cells in the cross-sectional flow area, and for different numbers of time steps, including 50, 100, 250, and 500 per cycle. The time-averaged 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 122.5 mm/s. Refinement in time steps, however, resulted little change in predicted velocities. Therefore, we adopted the scheme that contained 568 cells per cross-section, i.e., 230,608 for the whole mesh as shown in Fig. 2 and 250 time steps per cycle, the combination that gave the best grid independency and stability in solution within a reasonable CPU time.

Particle Trajectories and Capture on the Wall.

Simulations are performed in a partly occluded blood vessel having a diameter d of 4 mm, and a stenosed length ls of 4 mm and a degree of occlusion S = 50%. The density of the blood is taken as ρ = 1056 kg/m3. The magnetic beads are assumed to have the following properties: a = 500 nm, χeff = 3.0. The base case simulation is performed for a steady flow of blood at Re = 150. The magnetic insert, for the base case, is assumed to have a radius of Rm = 2 mm and its center placed at rmag = 4 mm, and zmag = 24 mm (= 6 d). The saturation of magnetization is assumed to be MS = 1.2 × 106 A/m. The time period (T) for the physiological pulsatile profile has been selected as 0.838 s. Sixty numbers of magnetic nanoparticles are continuously injected from the inlet with a step size of 0.001 s. The total time for the injection is considered to be one full cardiac cycle, once the injection is stopped then another 3 or 4 cycle is studied until and unless all the particles injected from the inlet plane either been trapped or escaped. From the previous work of the researcher [33], it is quite evident that the pulsatile flow boundary condition at the inlet has a strong effect on different hemodynamic parameters under consideration compared to the case of simple parabolic velocity at the inlet. Figure 3 shows the streamline describing the flow through the partially occluded blood vessel contours under the steady fully developed velocity boundary condition at the. At far upstream from the occlusion, the streamlines are parallel, indicating a fully developed flow. The streamlines converge toward the centerline as the occluded region is approached, attaining high velocity at the throat region. For the flow conditions described in Fig. 3, the streamlines cannot negotiate the curvature of the restriction profile in the vessel, thereby creating a separated zone and then spread again at the postocclusion region. Further downstream, the streamlines become parallel again, indicating that the fully developed profile is re-established. The figure also shows the distribution of total pressure, which clearly shows that there is a large drop in pressure is observed at the throat of the stenosis, but there is also a sign of pressure recovery just after the throat. When a homogeneous suspension of magnetic particles is allowed to enter the section with the flow, the particles are dragged along with the flow to the downstream. In absence of the magnetic insert, the particles follow the streamlines as their inertia is negligible compared to the viscous drag on them. Thus, no particle is found to attach to the vessel walls anywhere in the domain. When a magnetic field is established using the magnetic insert, the particles are found to be influenced appreciably by the magnetic force. It is to be noted that although the flow is axisymmetric in nature, the magnetic arrangement does not show angular symmetry. Therefore, the particle trajectories show three-dimensional motion with respect to the coordinate systems in which the flow is described. Figure 4 shows the particle capture contour for the base case. The result clearly shows that no particle has been captured in the recirculation zone. Most of the particles are captured around the angular position of 90 deg, which is nothing but the location of the cylindrical magnetic insert. Out of 50,280 numbers of particle injected from the inlet, a total of 17,372 are captured on the wall. To get more insight into the particle capture several histograms are plotted. Figure 5 shows the particle capture histogram along the axial direction. From the result, it is quite evident that major portion of the particles (almost 43% and 39%) is captured in the range of 5.5d–6.0d and 6.0d–6.5d. This observation is quite evident since the magnetic insert is placed at 6d which ensures a large magnetic force around that location. Since the effect of magnetic force diminishes as we passed the insert and hence no particles are captured beyond 7d. In order to check the distribution of the maximum particle capture (43%) as reported in the Fig. 4, next we plot the angular distribution of all those particles that are captured in the range of 5.5d–6.0d (Fig. 6). From the diagram it has been observed that out of total 7442 particles captured, most of the particles (50.17% and 48.5%) are being captured in the angular position of 60–90 deg and 90–120 deg. A very few number of particles are also captured in the range of 30–60 deg as well as 120–150 deg maintaining almost symmetrical particle distribution along the angular direction. Next, we put our attention (Fig. 7) to plot the histogram for all the particles that are captured on the wall of the artery for different angular position. This plotting is quite similar to the plot that we observed in Fig. 6, only difference is that now the diagram clearly shows that some of the particles are also captured on the stenosis wall.

When magnetophoretic separation of particles from a pressure-driven forced flow are achieved in straight channels [34] or tubes [35], or in unstenosed microvascularature [21], the maximum capture of magnetic particles are observed on the wall in the region of the largest magnetic field gradient, i.e., at z = zmag. However, the same is not true for the flow through the occluded veins or arteries.

Clinical efficacy of the MDT warrants that at least a certain fraction of the particle laden drug is deposited on the wall of the affected or occluded zone of the blood vessel. Defining such a zone (both in terms of its axial and radial span) on the vessel wall would stem purely from a clinical experience, and vary from case to case. As a representative case, here we have considered the entire vessel wall periphery within the axial span of 3.5 ≤ z/d ≤ 6.0 as the “target zone of capture” (ZC), which is marked by the vertical pair of lines in Fig. 3.

Parametric Study.

Trajectories of the magnetic microspheres in the occluded vessel depend on several factors, viz., the hemodynamic flow, size of the magnetic inserts, size of the particles, property of the magnetic material and its position with respect to the occluded vessel.

We have used a term “TE” to quantify the fraction of the inlet particles that are ultimately captured in the target region ZC in Fig. 3Display Formula

(10)TE=NumberofparticlescapturedinZC(i.e.,in3.5z/d6.0)Numberofparticlesreleasedattheinlet×100%

For evaluation of TE, particle counts from all the time steps are added.

Variation With Particle Size.

Parametric investigation has been carried out to see the variation of TE with the particle diameter. Five different sets of particle diameter have been considered over here varying from 4000 nm to 100 nm apart from the base case. The result (Fig. 8(a)) shows that almost 95% TE is observed for a very high particle diameter, since more the diameter of the particle means more amount of magnetic force on the particle ensures better amount of capture. From the result it is quite interesting to observe that a reduction in particle diameter from 500 nm to 100 nm reduces the TE by almost 1.5 times, where as this reduction is more pronounced for the case of 1000 nm to 500 nm (almost 2.6 times). Larger diameter particle shows better capture against the smaller diameter particle, but in most of the MDT system the selection of the 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 diameters arterioles and hence they should be avoided.

Particle capture histograms are also plotted considering the variation of particle diameter (see Figs. 8(b) and 8(c)). It is quite interesting to observe from the figure that a good percentage of the particle (approximately 5.2%) is trapped by the stenosis wall and the remaining particles are allowed to escape in the downstream location. From Fig. 8(b), it has been observed that almost 14.5% particles are captured after the ZC for 1000 nm diameter particles, while this is almost 5% for 500 nm particles. This information provides valuable information about the drug loading at the target site, since more the amount of drug deposited at the downstream to the ZC, more the chance the drug may convected further downstream with the flow of blood. The results of Fig. 8(c) shows the same distribution of particle capture for the higher size of the particle. It is quite interesting to note that for a lower sized particle (e.g., 2000 nm), the percentage of particle captured after the ZC is quite large (approx 24%) compared to the case of the maximum sized particle (e.g., 4000 nm), where only 6% of the total number of the particles are captured after the ZC.

Variation With Flow Re.

It can be seen from Fig. 9(a) that, both the number of captured particles and the axial distribution of the captured particle number density vary considerably with Re. For example, at Re = 50, out of 50,280 total number of particles almost 28,000 particles are captured almost at the last zone of capture (i.e., 5.75d–6d), but this is interestingly almost 2.33 times less for the case of high flow Re (= 200). This gives strong information about the efficacy of the MDT system. At larger Re, (e.g., Re = 100, 150, and 200) particle capture occurs more on the postocclusion region, and also many particles evade capture. At Re = 200, almost 35,000 particles escaped from the outlet indicating a total loss of drug.

The above particle distribution is further justified the TE versus Re plot (Fig. 9(b)), which shows a large drop in TE value as we increase flow Re from 50 to 200. Figure 10 provides evidence that there exist a very narrow flow regime when the targeting efficiency is high, and beyond which the capture efficiency shows a near hyperbolic relationship with the flow Re.

Variation With Magnetic Insert Size.

For a given flow condition and occlusion geometry, the TE is also influenced by the magnetic field and its gradient. The local magnetic condition in the occluded vessel depends on the size of the insert (Rm), and its radial and axial locations with respect to the occluded vessel (i.e., δ and zmag, respectively). Figures 10(a) and 10(b) shows the variation of the targeting efficiency as a function of Rm/d. It is apparent from Eqs. (2) and (6) that the magnetic force scales with the fourth power of Rm. Therefore, for very small value of Rm, the magnetic influence on the particles is too weak to overcome the viscous drag, and most of the particles exit the domain without being captured. With increased ratio of Rm/d, the TE value is expected to increase. As can be seen from Fig. 10(a), the TE increases from 5.17% to 19.88% as Rm/d is increased from 0.125 to 0.5. Beyond this value, the magnetic field becomes so strong in comparison to the fluid drag that a larger fraction of particles are captured much ahead of the ZC. This is further justified by Fig. 10(b), which shows that maximum amount of particles (almost 15%) are captured after the ZC region for Rm/d = 0.5.

Variation With Axial Location of the Insert.

Location of the magnetic insert is another decisive parameter in achieving the desired target efficiency. Since the ZC is defined within a range of 3.5d–6d, so the insert should be placed within that range in order to obtain the maximum TE. The result of Fig. 11(a) also shows the same, when the insert is placed at a location of 4d, the TE is observed to be maximum (almost 41%), but this will sharply decreases by almost 20% when the insert is placed at 6d. From the result of Fig. 11(b), it is observed that the point of maximum capture corresponds to the same point where the insert is placed and this point will shift toward the downstream direction as the insert is shifted. When the insert is placed at 6d location a very few number of particle (103) are captured in the range of 5–5.5d, but this is not the case for the insert when it is placed at 5.5d, where almost 7000 number of particles are captured at the same region, ensuring better distribution of the drug at the low shear region, which is essentially be the site more prone to the atherosclerosis.

Variation With Radial Location of the Insert.

The variation of TE has also been studied with the radial position of the insert (see Fig. 12). It is quite clear from the result that almost full capture is achieved when the insert is placed very close to the wall, but a slight change in the position, the TE drops sharply and becomes almost independent of the location beyond a critical value (≈1.5d).

Variation With Magnetic Susceptibility.

Figures 13(a) and 13(b) gives the information about the TE with respect to particle magnetic susceptibility. As the susceptibility increases, the magnetic force, which is directly proportional to the susceptibility will also increase, results in larger TE. Since from the physiological condition, higher the value of magnetic susceptibility may damage the cell, hence maximum value of susceptibility used in the simulation is 3. The histogram results (Fig. 13(b)) also shows that more the number of particles will deposit at the same locations as the susceptibility value increases.

3D Numerical simulation of the transport of magnetically guided drug-carrying magnetic microspheres in a partly occluded vessel is conducted under pulsatile flow boundary condition at inlet. A cylindrical magnetic insert is used to separate the magnetic microspheres and target them on the vessel wall near the occlusion region. Flow near the occluded region deviate noticeably from that in a regular blood vessel. Therefore, the particle trajectories and the spatial distribution of their accumulation on the vessel wall also are found to differ considerably from that observed in straight channels or tubes. Several parametric variations have been studied on the targeting efficiency of the beads (in a designated “zone of capture” on the vessel wall). Except the regime of extremely low particle diameter the TE value is found to increase with increase in diameter in a nearly exponential fashion. It is quite interesting to note that for a lower sized particle, the percentage of particle captured after the ZC is quite large compared to the case of the maximum sized particle. The influence of flow Re on TE shows that there exists a very narrow flow regime when the TE is high, and beyond which the TE shows a near hyperbolic relationship with the flow Re. The TE increases with the increase in insert radius, but more the insert size more is the amount of drug deposited much ahead of ZC. The TE has been found to be strongly dependent on the position of the insert as well as the magnetic property. From the result, it can be concluded that there exists an optimum value of the parameters for which the TE will be maximum.

The results provide useful design bases for in vitro set up for the investigation of MDT in occluded blood vessels.

The authors gratefully acknowledge the financial support from the DST under SERC FAST TRACK project (Ref. No. SERC/ET-0173/2011).

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Sousa, J. E., Costa, M. A., Abizaid, A., Abizaid, A. S., Feres, F., Pinto, I. M. F., Seixas, A. C., Staico, R., Mattos, L. A., Sousa, A. G. M. R., Falotico, R., Jaeger, J., Popma, J. J., and Serruys, P. W., 2001, “Lack of Neointimal Proliferation After Implantation of Sirolimus-Coated Stents in Human Coronary Arteries: A Quantitative Coronary Angiography and Three-Dimensional Intravascular Ultrasound Study,” Circulation, 103, pp. 192–204. [CrossRef] [PubMed]
Chen, H., Kaminski, M. D., Pytel, P., MacDonald, L., and Rosengart, A. J., 2008, “Capture of Magnetic Carriers Within Large Arteries Using External Magnetic Fields,” J. Drug Targeting, 16, pp. 262–271. [CrossRef]
Alexiou, C., Arnold, W., Klein, R. J., Parak, F. G., Hulin, P., Bergemann, C., Erhardt, W., Wagenpfeil, S., and Lübbe, A. S., 2000, “Locoregional Cancer Treatment With Magnetic Drug Targeting,” Cancer Res., 60, pp. 6641–6648. Available at: http://cancerres.aacrjournals.org/content/60/23/6641.long [PubMed]
Jordan, A., Scholz, R., Maier-Hauff, K., Johannsen, M., Wust, P., Nadobny, J., Schirra, H., Schmidt, H., Deger, S., Loening, S., Lanksch, W., and Felix, R., 2001, “Presentation of a New Magnetic Field Therapy System for the Treatment of Human Solid Tumors With Magnetic Fluid Hyperthermia,” J. Magn. Magn. Mater., 225(1-2), pp. 118–126. [CrossRef]
Johannsen, M., Gneveckow, U., Eckelt, L., Feussner, A., WaldöFner, N., Scholz, R., Deger, S., Wust, P., Loening, S. A., and Jordan, A., 2005, “Clinical Hyperthermia of Prostate Cancer Using Magnetic Nanoparticles: Presentation of a New Interstitial Technique,” Int. J. Hyperthermia, 21, pp. 637–652. [CrossRef] [PubMed]
Pankhurst, Q. A., Connolly, J., Jones, S. K., and Dobson, J., 2003, “Applications of Magnetic Nanoparticles in Biomedicine,” J. Phys. D: Appl. Phys., 36, pp. R167–R181. [CrossRef]
Avilés, M. O., Ebner, A. D., and Ritter, J. A., 2008, “Implant Assisted-Magnetic Drug Targeting: Comparison of in vitro Experiments With Theory,” J. Magn. Magn. Mater., 320, pp. 2704–2713. [CrossRef]
Forbes, Z. G., Yellen, B. B., Halverson, D. S., Fridman, G., Barbee, K. A., and Friedman, G., 2008, “Validation of High Gradient Magnetic Field Based Drug Delivery to Magnetizable Implants Under Flow,” IEEE Trans. Biomed. Eng., 55, pp. 643–649. [CrossRef] [PubMed]
FurlaniE. J., and Furlani, E. P., 2007, “A Model for Predicting Magnetic Targeting of Multifunctional Particles in the Microvasculature,” J. Magn. Magn. Mater., 312, pp. 187–201. [CrossRef]
Yellen, B. B., Forbes, Z. G., Halverson, D. S., Fridman, G., Barbee, K. A., Chorny, M., Levy, R., and Friedman, G., 2005, “Targeted Drug Delivery to Magnetic Implants for Therapeutic Applications,” J. Magn. Magn. Mater., 293, pp. 647–662. [CrossRef]
Ku, D. N., Giddens, D. P., Zarins, C. K., and Glagov, S., 1985, “Pulsatile Flow and Atherosclerosis in the Human Carotid Bifurcation,” Arteriosclerosis, 5, pp. 293–302. [CrossRef] [PubMed]
Haverkort, J. W., Kenjereš, S., and Kleijn, C. R., 2009, “Computational Simulations of Magnetic Particle Capture in Arterial Flows,” Ann. Biomed. Eng., 37, pp. 2436–2444. [CrossRef] [PubMed]
Neofytou, P., and Tsangaris, S., 2004, “Computational Haemodynamics and the Effects of Blood Rheological Models on the Flow Through an Arterial Stenosis,” European Congress on Computational Methods in Applied Sciences and Engineering.
Banerjee, M. K., Ganguly, R., and Datta, A., 2010, “Magnetic Drug Targeting in Partly Occluded Blood Vessels Using Magnetic Microspheres,” ASME J. Nanotechnol. Eng. Med., 1(4), p. 041005. [CrossRef]
Morsi, S. A., and Alexander, A. J., 1972, “An Investigation of Particle Trajectories in Two-Phase Flow Systems,” J. Fluid Mech., 55(2) pp. 193–208. [CrossRef]
Modak, N., Datta, A., and Ganguly, R., 2009, “Cell Separation in a Microfluidic Channel Using Magnetic Microspheres,” Microfluid. Nanofluid., 6, pp. 647–660. [CrossRef]
Gerber, R., Takayasu, M., and Frieslander, FJ., 1983, “Generalization of HGMS Theory: the Capture of Ultrafine Particles,” IEEE Trans. Magn., 19(5), pp. 2115–2117. [CrossRef]
Merrill, E. W., 1965, “Rheology of Human Blood and Some Speculations on its Role in Vascular Homeostasis,” Biomechanical Mechanisms in Vascular Homeostasis and Intravascular Thrombosis, P. N.Sawyer, ed., Appleton-Century-Crofts, New York, pp. 121–137.
BergerS. A., and JouL. D., 2000, “Flow in Stenotic Vessels,” Annu. Rev. Fluid Mech., 32, pp. 347–382. [CrossRef]
ansys® Academic CFD Research 14.5, Ansys, Inc.
Banerjee, M. K., Ganguly, R., and Datta, A., 2012, “Effect of Pulsatile Flow Waveform and Womersley Number on the Flow in Stenosed Arterial Geometry,” ISRN Biomath., 2012, pp. 1–17. [CrossRef]
Sinha, A., Ganguly, R., and Puri, I. K., 2009, “Magnetic Separation From Superparamagnetic Particle Suspensions,” J. Magn. Magn. Mater., 321, pp. 2251–2267. [CrossRef]
Ganguly, R., Gaind, A. P., and Puri, I. K., 2005, “A Strategy for the Assembly of Three-Dimensional Mesoscopic Structures Using a Ferrofluid,” Phys. Fluids, 17, pp. 1–9. Available at: http://www.researchgate.net/publication/228332503_A_strategy_for_the_assembly_of_3-D_mesoscopic_structures_using_a_ferrofluid
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References

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Lübbe, A. S., Alexiou, C., and Bergemann, C., 2001, “Clinical Applications of Magnetic Drug Targeting,” J. Surg. Res., 95, pp. 200–206. [CrossRef] [PubMed]
Häfeli, U. O., Gilmour, K., Zhou, A., Lee, S., and Hayden, M. E., 2007, “Modeling of Magnetic Bandages for Drug Targeting: Button vs. Halbach Arrays,” J. Magn. Magn. Mater., 311, pp. 323–329. [CrossRef]
Torchilin, V. P., 2000, “Drug Targeting,” Eur. J. Pharm. Sci., 11(2), pp. S81–S91. [CrossRef] [PubMed]
Dobson, J., 2006, “Magnetic Nanoparticles for Drug Delivery,” Drug Dev. Res., 67, pp. 55–60. [CrossRef]
Dobson, J., 2006, “Magnetic Nanoparticle-Based Targeting for Drug and Gene Delivery,” Nanomedicine, 1, pp. 31–37. [CrossRef] [PubMed]
Voltairas, P. A., Fotiades, D. I., and Michalis, L. K., 2002, “Hydrodynamics of Magnetic Drug Targeting,” J. Biomech., 35, pp. 813–829. [CrossRef] [PubMed]
Torchilin, V. P., 1995, “Targeting of Drugs and Drug Carriers Within the Cardiovascular System,” Adv. Drug Delivery Rev., 17, pp. 75–101. [CrossRef]
Moulton, K. S., Heller, E., Konerding, M. A., Flynn, E., Palinski, W., and Folkman, J., 1999, “Angiogenesis Inhibitors Endostatin or TNP-470 Reduce Intimal Neovascularization and Plaque Growth in Apolipoprotein E–Deficient Mice,” Circulation, 99, pp. 1726–1732. [CrossRef] [PubMed]
Herbst, R. S., Madden, T. L., Tran, H. T., Blumenschein, G. R., Jr., Meyers, C. A., Seabrooke, L. F., Khuri, F. R., Puduvalli, V. K., Allgood, V., Fritsche, H. A., Jr., Hinton, L., Newman, R. A., Crane, E. A., Fossella, F. V., Dordal, M., Goodin, T., and Hong, W. K., 2002, “Safety and Pharmacokinetic Effects of TNP-470, an Angiogenesis Inhibitor, Combined With Paclitaxel in Patients With Solid Tumors: Evidence for Activity in Non-Small-Cell Lung Cancer,” J. Clin. Oncol., 20, pp. 4440–4447. [CrossRef] [PubMed]
Liu, S., Widom, J., Kemp, C. W., Crews, C. M., and Clardy, J., 1998, “Structure of Human Methionine Aminopeptidase-2 Complexed With Fumagillin,” Science, 282, pp. 1324–1327. [CrossRef] [PubMed]
Lanza, G. M., Yu, X., Winter, P. M., Abendschein, D. R., Karukstis, K. K., Scott, M. J., Chinen, L. K., Fuhrhop, R. W., Scherrer, D. E., and Wickline.S. A., 2002, “A Novel Site-Targeted Ultrasonic Contrast Agent With Broad Biomedical Application,” Circulation, 106, pp. 2842–2867. [CrossRef] [PubMed]
Sousa, J. E., Costa, M. A., Abizaid, A., Abizaid, A. S., Feres, F., Pinto, I. M. F., Seixas, A. C., Staico, R., Mattos, L. A., Sousa, A. G. M. R., Falotico, R., Jaeger, J., Popma, J. J., and Serruys, P. W., 2001, “Lack of Neointimal Proliferation After Implantation of Sirolimus-Coated Stents in Human Coronary Arteries: A Quantitative Coronary Angiography and Three-Dimensional Intravascular Ultrasound Study,” Circulation, 103, pp. 192–204. [CrossRef] [PubMed]
Chen, H., Kaminski, M. D., Pytel, P., MacDonald, L., and Rosengart, A. J., 2008, “Capture of Magnetic Carriers Within Large Arteries Using External Magnetic Fields,” J. Drug Targeting, 16, pp. 262–271. [CrossRef]
Alexiou, C., Arnold, W., Klein, R. J., Parak, F. G., Hulin, P., Bergemann, C., Erhardt, W., Wagenpfeil, S., and Lübbe, A. S., 2000, “Locoregional Cancer Treatment With Magnetic Drug Targeting,” Cancer Res., 60, pp. 6641–6648. Available at: http://cancerres.aacrjournals.org/content/60/23/6641.long [PubMed]
Jordan, A., Scholz, R., Maier-Hauff, K., Johannsen, M., Wust, P., Nadobny, J., Schirra, H., Schmidt, H., Deger, S., Loening, S., Lanksch, W., and Felix, R., 2001, “Presentation of a New Magnetic Field Therapy System for the Treatment of Human Solid Tumors With Magnetic Fluid Hyperthermia,” J. Magn. Magn. Mater., 225(1-2), pp. 118–126. [CrossRef]
Johannsen, M., Gneveckow, U., Eckelt, L., Feussner, A., WaldöFner, N., Scholz, R., Deger, S., Wust, P., Loening, S. A., and Jordan, A., 2005, “Clinical Hyperthermia of Prostate Cancer Using Magnetic Nanoparticles: Presentation of a New Interstitial Technique,” Int. J. Hyperthermia, 21, pp. 637–652. [CrossRef] [PubMed]
Pankhurst, Q. A., Connolly, J., Jones, S. K., and Dobson, J., 2003, “Applications of Magnetic Nanoparticles in Biomedicine,” J. Phys. D: Appl. Phys., 36, pp. R167–R181. [CrossRef]
Avilés, M. O., Ebner, A. D., and Ritter, J. A., 2008, “Implant Assisted-Magnetic Drug Targeting: Comparison of in vitro Experiments With Theory,” J. Magn. Magn. Mater., 320, pp. 2704–2713. [CrossRef]
Forbes, Z. G., Yellen, B. B., Halverson, D. S., Fridman, G., Barbee, K. A., and Friedman, G., 2008, “Validation of High Gradient Magnetic Field Based Drug Delivery to Magnetizable Implants Under Flow,” IEEE Trans. Biomed. Eng., 55, pp. 643–649. [CrossRef] [PubMed]
FurlaniE. J., and Furlani, E. P., 2007, “A Model for Predicting Magnetic Targeting of Multifunctional Particles in the Microvasculature,” J. Magn. Magn. Mater., 312, pp. 187–201. [CrossRef]
Yellen, B. B., Forbes, Z. G., Halverson, D. S., Fridman, G., Barbee, K. A., Chorny, M., Levy, R., and Friedman, G., 2005, “Targeted Drug Delivery to Magnetic Implants for Therapeutic Applications,” J. Magn. Magn. Mater., 293, pp. 647–662. [CrossRef]
Ku, D. N., Giddens, D. P., Zarins, C. K., and Glagov, S., 1985, “Pulsatile Flow and Atherosclerosis in the Human Carotid Bifurcation,” Arteriosclerosis, 5, pp. 293–302. [CrossRef] [PubMed]
Haverkort, J. W., Kenjereš, S., and Kleijn, C. R., 2009, “Computational Simulations of Magnetic Particle Capture in Arterial Flows,” Ann. Biomed. Eng., 37, pp. 2436–2444. [CrossRef] [PubMed]
Neofytou, P., and Tsangaris, S., 2004, “Computational Haemodynamics and the Effects of Blood Rheological Models on the Flow Through an Arterial Stenosis,” European Congress on Computational Methods in Applied Sciences and Engineering.
Banerjee, M. K., Ganguly, R., and Datta, A., 2010, “Magnetic Drug Targeting in Partly Occluded Blood Vessels Using Magnetic Microspheres,” ASME J. Nanotechnol. Eng. Med., 1(4), p. 041005. [CrossRef]
Morsi, S. A., and Alexander, A. J., 1972, “An Investigation of Particle Trajectories in Two-Phase Flow Systems,” J. Fluid Mech., 55(2) pp. 193–208. [CrossRef]
Modak, N., Datta, A., and Ganguly, R., 2009, “Cell Separation in a Microfluidic Channel Using Magnetic Microspheres,” Microfluid. Nanofluid., 6, pp. 647–660. [CrossRef]
Gerber, R., Takayasu, M., and Frieslander, FJ., 1983, “Generalization of HGMS Theory: the Capture of Ultrafine Particles,” IEEE Trans. Magn., 19(5), pp. 2115–2117. [CrossRef]
Merrill, E. W., 1965, “Rheology of Human Blood and Some Speculations on its Role in Vascular Homeostasis,” Biomechanical Mechanisms in Vascular Homeostasis and Intravascular Thrombosis, P. N.Sawyer, ed., Appleton-Century-Crofts, New York, pp. 121–137.
BergerS. A., and JouL. D., 2000, “Flow in Stenotic Vessels,” Annu. Rev. Fluid Mech., 32, pp. 347–382. [CrossRef]
ansys® Academic CFD Research 14.5, Ansys, Inc.
Banerjee, M. K., Ganguly, R., and Datta, A., 2012, “Effect of Pulsatile Flow Waveform and Womersley Number on the Flow in Stenosed Arterial Geometry,” ISRN Biomath., 2012, pp. 1–17. [CrossRef]
Sinha, A., Ganguly, R., and Puri, I. K., 2009, “Magnetic Separation From Superparamagnetic Particle Suspensions,” J. Magn. Magn. Mater., 321, pp. 2251–2267. [CrossRef]
Ganguly, R., Gaind, A. P., and Puri, I. K., 2005, “A Strategy for the Assembly of Three-Dimensional Mesoscopic Structures Using a Ferrofluid,” Phys. Fluids, 17, pp. 1–9. Available at: http://www.researchgate.net/publication/228332503_A_strategy_for_the_assembly_of_3-D_mesoscopic_structures_using_a_ferrofluid

Figures

Grahic Jump Location
Fig. 1

(a) Geometrical configuration of the occlusion with the cylindrical magnet for drug targeting and (b) magnetic field produced by the insert (flux lines superposed on the |H| contours)

Grahic Jump Location
Fig. 3

Contour plot of total pressure and the arrow lines indicating the stream lines for the base case

Grahic Jump Location
Fig. 4

Particle capture contour for the base case

Grahic Jump Location
Fig. 5

Particle capture histogram for the base case along axial direction

Grahic Jump Location
Fig. 6

Angular distribution of particles for the base case at an axial location of z = 5.5d

Grahic Jump Location
Fig. 7

Angular distribution of all the particles that are captured at the wall for the base case

Grahic Jump Location
Fig. 2

3D geometrical model of the stenosis

Grahic Jump Location
Fig. 8

(a) Variation of TE with particle diameter (all the other parameters are same as the base case), (b) particle capture histogram for different Re along axial direction (small sized particle), and (c) particle capture histogram for different Re along axial direction (large sized particle)

Grahic Jump Location
Fig. 9

(a) Particle capture histogram for different Re along axial direction and (b) variation of TE with flow Re (all the other parameters are same as the base case)

Grahic Jump Location
Fig. 10

(a) Variation of TE with radius of the insert (all the other parameters are same as the base case) and (b) particle capture histogram for different insert radius

Grahic Jump Location
Fig. 11

(a) Variation of TE with flow axial position of the insert (all the other parameters are same as the base case) and (b) particle capture histogram for different axial location of the magnet

Grahic Jump Location
Fig. 12

Variation of TE with radial position of the insert (all the other parameters are same as the base case).

Grahic Jump Location
Fig. 13

(a) Variation of TE with effective susceptibility of the magnetic material (all the other parameters are same as the base case) and (b) particle capture histogram for different susceptibility of the magnetic material

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