A comprehensive model of magnetic particle motion during magnetic drug targeting
Introduction
One of the greatest challenges in treating cancer and other localized diseases is getting medication to the affected tissue. Because every vessel in the circulatory system is linked, drugs injected into the blood stream or taken orally end up spread throughout the body. This has several undesirable consequences. First, the majority of the drug is wasted. Second, perfusion of healthy tissue by drugs meant to treat unhealthy cells can cause severe system toxicity. Finally, a large amount of medication must be used in order to hit a target drug concentration at the site of the disease.
For these reasons, drug targeting has become an active area of medical research in recent years, and magnetic drug targeting (MDT) is one of the most promising of the developing techniques. In an ideal magnetic drug targeting treatment, drugs would be chemically bound to magnetic particles, injected into the body, and steered to the site of the diseased tissue using external magnetic fields. This treatment is highly desirable because not only would it deliver medication to the correct location in the body, but it would also do so non-invasively. It is important to note that magnetic particle steering and trapping will never have a 100% success rate; some particles will always be lost between their injection site and target. The goal then is to increase the probability that the particles make it to the target and increase the time that they remain there. Design of an effective MDT system requires research in three main areas: synthesis of composite drug and magnetic particles, real-time imaging of magnetic particles as they move through the blood stream, and development of technology to steer the particles through the circulatory system and hold them at the correct location. The current work focuses on the third area of research.
Previous work on steering magnetic particles through the bloodstream for MDT is summarized in several review papers (Cao et al., 2011, Nacev et al., 2012, Polyak and Friedman, 2009). Previous research has focused on applying magnetic force with permanent magnets that can effectively control magnetic particles over short ranges. Permanent magnets produce both a high ambient magnetic field and a high field gradient, both of which are necessary to apply force to paramagnetic materials. However, they cannot control particles at long ranges because their fields decay rapidly in space. Also, their fields are inflexible; a permanent magnet must be physically moved to alter the field it generates at a given point. Nevertheless, many researchers have proposed techniques for MDT that use permanent magnets. Most of the published research has been in vitro, such as the work of Senyei and Czerlinski (1978), who were the first to contain nanometer-sized iron oxide particles in a well-localized region of flow using a permanent magnet. More recently, Ruuge and Rusetski (1993) used permanent magnets to trap drops of ferrofluid in a flow. Babincova et al. (2001) used a ferromagnetic wire and external magnetic field to control nanosized superparamagnetic particles, and Gitter and Odenbach (2011) used a permanent magnet to steer magnetic nanoparticles through a model of an arterial bifurcation.
Only a few researchers have published papers on in vivo studies of MDT. Most of these have used small animals so that short-range permanent magnets are sufficient to control particles deep within their bodies. Alexiou et al. (2005) injected mitroxantrone bound to magnetic nanoparticles into tumorous rabbits and attracted them to the tumors with a permanent magnet. Riviere et al. (2007) installed cranial windows in seven mice and video-monitored concentrations of fluorescent magnetic particles in the brain as they attracted them with a permanent magnet, and Oechtering et al. (2011) used a 13 T/m permanent magnet to attract magnetic nanoparticles to cerebral aneurysms in rabbits. We found only a single study that attempted MDT in humans: Lubbe et al. (1996) used permanent magnets to attract 100-nm magnetic particles bound to epirubicin to advanced tumors that had been previously unsuccessfully treated with chemotherapy.
To our knowledge, there are no published studies on in vivo magnetic particle steering via electromagnet, but a few in vitro studies have been published, such as the work of Mathieu and Martel (2007), who used a Maxwell coil pair to steer magnetic particles through a Y-shaped model of an arterial bifurcation. They found that particles were easier to steer when the flow was slow and the particles were large. Electromagnetic steering is promising because electromagnets can generate strong magnetic fields far from their surfaces and the magnitudes and directions of these fields can be changed easily. However, our previous work (Cherry et al., 2010) has shown that it is extremely difficult to control individual μm-sized magnetic particles in the bloodstream because the fluid drag force is too large to overcome with magnetic force. Since large particles cannot be introduced into the bloodstream, MDT must be accomplished by injecting a cluster of particles. The hope is that the close proximity of these particles will reduce the average drag force per particle, thus making the entire cluster easier to magnetically trap and steer.
Simulations of MDT, such as the work of Kenjeres and Stuart (2009), have used Lagrangian particle tracking to predict the trajectories of magnetic particles subject to magnetic forcing in flows. Shaw and Murthy, 2010, Yue et al., 2012 performed simulations of magnetic drug targeting treatments in both Newtonian and shear-thinning flows. Kayal et al. (2011) published a 2D simulation of magnetic particles in a channel flow with magnetic force applied opposite the streamwise direction and reported significant slowing of particles in the flow near one wall. David et al. (2011) developed a 2D model for the motion of magnetic particles exposed to hydrodynamic drag, gravity, and buoyancy, and Banerjee et al. (2010) simulated magnetic particles flowing past an arterial occlusion with a permanent magnet implanted in the arterial wall. Haverkort et al. (2009) simulated magnetic particle-laden blood flow in coronary and carotid arteries and found that it was possible to trap a significant fraction of 4-micron particles in a section of artery using a small superconducting magnet placed a few centimeters away. Finally, Kenjeres and Righolt (2012) simulated the dynamics of drug-coated magnetic particles moving through the brain’s vascular system and found that they could use an external magnetic field to increase the particle deposition near a small diseased area. However, none of these simulations have included all of the effects needed to model the motion of a cluster of very small particles. In particular, none of the previous studies have included interparticle magnetic force and few have included the effect of blood cells in increasing particle diffusion.
The goal of the present work is to develop a model for the motion of magnetic particle-laden blood flow that accounts for all non-trivial forces. This model is applied to evaluate the feasibility of trapping and steering clusters of magnetic particles in two simplified configurations. Finally, we explore the parameter space of MDT, determining how parameters such as particle cluster concentration and applied magnetic field gradient affect the success with which magnetic particles can be controlled. This model is easily adaptable to different particle and flow inlet conditions, flow geometries, and periodic flowrate variations. Further development would be required to handle blood vessel distortion under the applied hydrodynamic loads.
Section snippets
Theory
The model follows the homogeneous flow or dusty gas approach (Carrier, 1958), which models the particles as a variable concentration continuum that moves at the local fluid velocity. This requires continuity and momentum equations for the mixture and a scalar transport equation for the particle concentration. The equations are derived for the specific case of a dilute suspensions of μm scale particles suspended in blood moving through human arteries that are large relative to blood cells. The
Shear thinning blood model
Blood is a shear thinning fluid because it contains deformable cells, and previous work has shown that this can significantly affect velocity profiles in large arteries if the flow remains laminar (Cherry and Eaton, 2013). Several different methods have been used to describe blood’s non-Newtonian behavior in previous simulations. One technique is to model arterial blood flow as a two-phase fluid which contains a thin layer of Newtonian plasma near the vessel walls and a non-Newtonian region of
Simulation setup
The simulation was performed on two 3D grids: a straight, round tube and an asymmetrically bifurcating tube. The straight tube was 8 mm in diameter and 80 mm long and was meant to model a section of an adult femoral artery. The bifurcating tube was designed to mimic the branching of the superior mesenteric artery into the ileocolic artery, and its geometry and grid are shown in Fig. 3. It consists of a straight, 60 mm long, 6 mm diameter tube (henceforth referred to as the “main branch”) with a 3 mm
Straight tube cases
The straight tube simulation cases were grouped into four sets. Each set varied one particle cluster or magnetic field parameter while holding everything else constant, and each set was designed to answer a question about the effect of one parameter on the success of trapping magnetic particles in a tube flow. The list below summarizes the cases. For all cases with a constant magnetic field gradient, the gradient was not applied to the first diameter-length of the tube. This was done to prevent
Bifurcating tube cases
All bifurcating tube simulation cases had the same injected particle cluster: a cylinder with a radius of 0.0015 m, a length of 0.05 m, and a volume fraction of 0.005. The bifurcation was oriented such that the force of gravity was in the −y direction. The goal of these simulations was to determine the optimal magnetic field magnitude and direction for steering a particle cluster down the ileocolic branch, so the magnetic field gradient was varied. Its values in the various cases are listed in
Conclusions
A comprehensive model for the motion of a cluster of micron scale magnetic particles in the human bloodstream was developed. It accounts for realistic forces on the particles, such as interparticle magnetic force and dispersion caused by collisions between particles and blood cells. A new numerical formulation for interparticle magnetic force in a cluster of many particles allows efficient and accurate implementation of this force in a two-phase CFD code. The dispersion coefficient due to
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