A model for predicting magnetic targeting of multifunctional particles in the microvasculature

A mathematical model is presented for predicting magnetic targeting of multifunctional carrier particles that are designed to deliver therapeutic agents to malignant tissue in vivo. These particles consist of a nonmagnetic core material that contains embedded magnetic nanoparticles and therapeutic agents such as photodynamic sensitizers. For in vivo therapy, the particles are injected into the vascular system upstream from malignant tissue, and captured at the tumor using an applied magnetic field. The applied field couples to the magnetic nanoparticles inside the carrier particle and produces a force that attracts the particle to the tumor. In noninvasive therapy, the applied field is produced by a permanent magnet positioned outside the body. In this paper, a mathematical model is developed for predicting noninvasive magnetic targeting of therapeutic carrier particles in the microvasculature. The model takes into account the dominant magnetic and fluidic forces on the particles and leads to an analytical expression for predicting their trajectory. An analytical expression is also derived for predicting the volume fraction of embedded magnetic nanoparticles required to ensure capture of the carrier particle at the tumor. The model enables rapid parametric analysis of magnetic targeting as a function of key variables including the size of the carrier particle, the properties and volume fraction of the embedded magnetic nanoparticles, the properties of the magnet, the microvessel, the hematocrit of the blood and its flow rate.     

Ferromagnetic seeding for the magnetic targeting of drugs and radiation in capillary beds

A new implant assisted-magnetic drug targeting approach is introduced and theoretically analyzed to demonstrate its feasibility. This approach uses ferromagnetic particles as seeds for collecting magnetic drug carrier particles at the desired site in the body, such as in a capillary bed near a tumor. Based on the capture cross section (λ_c) approach, a parametric study was carried out using a 2-D mathematical model to reveal the effects of the magnetic field strength (μ₀H₀=0.01–1.0 T), magnetic drug carrier particle radius (R_p=20–500 nm), magnetic drug carrier particle ferromagnetic material content (x_fm,p=20–80 wt%), average blood velocity (u_B=0.05–1.0 cm/s), seed radius (R_s=100–2000 nm), number of seeds (N_s=1–8), seed separation (h=0–8R_s), and magnetic drug carrier particle and seed ferromagnetic material saturation magnetizations (iron, SS 409, magnetite, and SS 304) on the performance of the system. Increasing the magnetic field strength, magnetic drug carrier particle size, seed size, magnetic drug carrier particle ferromagnetic material content, or magnetic drug carrier particle or seed saturation magnetization, all positively and significantly affected λ_c, while increasing the average blood velocity adversely affected it. Increasing the number of seeds or decreasing the seed separation, with both causing less significant increases in λ_c, verified that cooperative magnetic effects exist between the seeds that enhance the performance. Overall, these theoretical results were encouraging as they showed the viability of this minimally invasive, implant assisted-magnetic drug targeting approach for targeting drugs or radiation in capillary beds.     

In vitro study of ferromagnetic stents for implant assisted-magnetic drug targeting

Implant-assisted-magnetic drug targeting (IA-MDT) was studied in vitro using a coiled ferromagnetic wire stent made from stainless steel 430 or 304, and magnetic drug carrier particle (MDCP) surrogates composed of poly(styrene/divinylbenzene) embedded with 20 wt% magnetite. The fluid velocity, particle concentration, magnetic field strength, and stent material all proved to be important for capturing the MDCP surrogates. Overall, this in vitro study further confirmed the important role of the ferromagnetic implant for attracting and retaining MDCPs at the target zone.     

In vitro study of magnetic particle seeding for implant assisted-magnetic drug targeting

The concept of using magnetic particles (seeds) as the implant for implant assisted-magnetic drug targeting (IA-MDT) was analyzed in vitro. Since this MDT system is being explored for use in capillaries, a highly porous (ε∼70%), highly tortuous, cylindrical, polyethylene polymer was prepared to mimic capillary tissue, and the seeds (magnetite nanoparticles) were already fixed within. The well-dispersed seeds were used to enhance the capture of 0.87 μm diameter magnetic drug carrier particles (MDCPs) (polydivinylbenzene embedded with 24.8 wt% magnetite) under flow conditions typically found in capillary networks. The effects of the fluid velocity (0.015–0.15 cm/s), magnetic field strength (0.0–250 mT), porous polymer magnetite content (0–7 wt%) and MDCP concentration (C=5 and 50 mg/L) on the capture efficiency (CE) of the MDCPs were studied. In all cases, when the magnetic field was applied, compared to when it was not, large increases in CE resulted; the CE increased even further when the magnetite seeds were present. The CE increased with increases in the magnetic field strength, porous polymer magnetite content and MDCP concentration. It decreased only with increases in the fluid velocity. Large magnetic field strengths were not necessary to induce MDCP capture by the seeds. A few hundred mT was sufficient. Overall, this first in vitro study of the magnetic seeding concept for IA-MDT was very encouraging, because it proved that magnetic particle seeds could serve as an effective implant for MDT systems, especially under conditions found in capillaries.     

Implant assisted-magnetic drug targeting: Comparison of in vitro experiments with theory

Implant assisted-magnetic drug targeting (IA-MDT) was studied both in vitro and theoretically, with extensive comparisons made between model and experiment. Magnetic drug carrier particles (MDCPs) comprised of magnetite encased in a polymer were collected magnetically using a ferromagnetic, coiled, wire stent as the implant and a NdFeB permanent magnet for the applied magnetic field. A 2-D mathematical model with no adjustable parameters was developed and compared to the 3-D experimental results. The effects of the fluid velocity, stent and MDCP properties, and magnetic field strength on the performance of the system were evaluated in terms of the capture efficiency (CE) of the MDCPs. In nearly all cases, the parametric trends predicted by the model were in good agreement with the experimental results: the CE always increased with decreasing velocity, increasing magnetic field strength, increasing MDCP size or magnetite content, or increasing wire size. The only exception was when experiments showed an increase in the CE with an increase in the number of loops in the wire, while the model showed no dependence. The discrepancies between experiment and theory were attributed to phenomena not accounted for by the model, such as 3-D to 2-D geometric and magnetic field orientation differences, and interparticle interactions between the MDCPs that lead to magnetic agglomeration and shearing force effects. Overall, this work showed the effectiveness of a stent-based IA-MDT system through both in vitro experimentation and corroborated theory, with the designs of the ferromagnetic wire and the MDCPs both being paramount to the CE.     

Biodegradable nanocomposite magnetite stent for implant-assisted magnetic drug targeting

This study shows, for the first time, the fabrication of a biodegradable polymer nanocomposite magnetic stent and the feasibility of its use in implant-assisted-magnetic drug targeting (IA-MDT). The nanocomposite magnetic stent was made from PLGA, a biodegradable copolymer, and iron oxide nanopowder via melt mixing and extrusion into fibers. Degradation and dynamic mechanical thermal analyses showed that the addition of the iron oxide nanopowder increased the polymer’s glass transition temperature (T_g) and its modulus but had no notable effect on its degradation rate in PBS buffer solution. IA-MDT in vitro experiments were carried out with the nanocomposite magnetic fiber molded into a stent coil. These stent prototypes were used in the presence of a homogeneous magnetic field of 0.3 T to capture 100 nm magnetic drug carrier particles (MDCPs) from an aqueous solution. Increasing the amount of magnetite in the stent nanocomposite (0, 10 and 40 w/w%) resulted in an increase in the MDCP capture efficiency (CE). Reducing the MDCP concentrations (0.75 and 1.5 mg/mL) in the flowing fluid and increasing the fluid velocities (20 and 40 mL/min) both resulted in decrease in the MDCP CE. These results show that the particle capture performance of PLGA-based, magnetic nanocomposite stents are similar to those exhibited by a variety of different non-polymeric magnetic stent materials studied previously.     

Numerical study of magnetic nanoparticles concentration in biofluid (blood) under influence of high gradient magnetic field

Ferrofluids are widely used in pharmaceutical industries as magnetic separation tools, anti-cancer drug carriers and micro-valve applications. The purpose of the current study is to investigate the effect of a magnetic field on the volume concentration of magnetic nanoparticles of a non-Newtonian biofluid (blood) as a drug carrier. The effect of particles on the flow field is considered. The governing non-linear differential equations, concentration and Naviar-stokes are coupled with the magnetic field. To solve these equations, a finite volume based code is developed and utilized. The results show accumulation of magnetic nanoparticles near the magnetic source until it looks like a solid object. The accumulation of nanoparticles is due to the magnetic force that overcomes the fluid drag force. As the magnetic strength and size of the magnetic particles increase, the accumulation of nanoparticles increases, as well. The magnetic susceptibility of particles also affects the flow field and the contour of the concentration considerably.     

Application of high gradient magnetic separation principles to magnetic drug targeting

A hypothetical magnetic drug targeting system, utilizing high gradient magnetic separation (HGMS) principles, was studied theoretically using FEMLAB simulations. This new approach uses a ferromagnetic wire placed at a bifurcation point inside a blood vessel and an externally applied magnetic field, to magnetically guide magnetic drug carrier particles (MDCP) through the circulatory system and then to magnetically retain them at a target site. Wire collection (CE) and diversion (DE) efficiencies were defined and used to evaluate the system performance. CE and DE both increase as the strength of the applied magnetic field (0.3–2.0 T), the amount of ferromagnetic material (iron) in the MDCP (20–100%) and the size of the MDCP (1–10 μm radius) increase, and as the average inlet velocity (0.1–0.8 m s⁻¹), the size of the wire (50–250 μm radius) and the ratio (4–10) of the parent vessel radius (0.25–1.25 mm radius) to wire radius decrease. The effect of the applied magnetic field direction (0° and 90°) on CE and DE was minimal. Under these plausible conditions, CEs as high as 70% were obtained, with DEs reaching only 30%; however, when the MDCPs were allowed to agglomerate (4–10 μm radius), CEs and DEs of 100% were indeed achieved. These results reveal that this new magnetic drug targeting approach for magnetically collecting MDCPs at a target site, even in arteries with very high velocities, is feasible and very promising; this new approach for magnetically guiding MDCPs through the circulatory system is also feasible but more limited. Overall, this study shows that magnetic drug targeting, based on HGMS principles, has considerable promise as an effective drug targeting tool with many potential applications.     

In vitro study of magnetic nanoparticles as the implant for implant assisted magnetic drug targeting

Magnetic nanoparticle (MNP) seeds were studied in vitro for use as an implant in implant assisted-magnetic drug targeting (IA-MDT). The magnetite seeds were captured in a porous polymer, mimicking capillary tissue, with an external magnetic field (70 mT) and then used subsequently to capture magnetic drug carrier particles (MDCPs) (0.87 μm diameter) with the same magnetic field. The effects of the MNP seed diameter (10, 50 and 100 nm), MNP seed concentration (0.25-2.0 mg/mL), and fluid velocity (0.03-0.15 cm/s) on the capture efficiency (CE) of both the MNP seeds and the MDCPs were studied. The CE of the 10 nm MNP seeds was never more than 30%, while those of the 50 and 100 nm MNP seeds was always greater than 80% and in many cases exceeded 90%. Only the MNP seed concentration affected its CE. The 10 nm MNP seeds did not increase the MDCP CE over that obtained in the absence of the MNP seeds, while the 50 and 100 nm MNP seeds increased significantly, typically by more than a factor of two. The 50 and 100 nm MNP seeds also exhibited similar abilities to capture the MDCPs, with the MDCP CE always increasing with decreasing fluid velocity and generally increasing with increasing MNP seed concentration. The MNP seed size, magnetic properties, and capacity to self-agglomerate and form clusters were key properties that make them a viable implant in IA-MDT.     

In vitro study of magnetic particle seeding for implant-assisted-magnetic drug targeting: Seed and magnetic drug carrier particle capture

An implant-assisted-magnetic drug targeting system using seed particles as the implant to increase the capture of magnetic drug carrier particles (MDCPs) in capillary tissue was studied in vitro. Dextran-coated magnetite particles were used as seeds, polydivinylbenzene magnetite particles were used as MDCPs, and a polyethylene porous cylinder was used as surrogate capillary tissue. The results showed that seeds could be magnetically captured first and then used to magnetically capture the MDCPs, causing a significant increase in their collection compared to when the seeds were absent.     

Quantitative targeting maps based on experimental investigations for a branched tube model in magnetic drug targeting

Magnetic drug targeting (MDT), because of its high targeting efficiency, is a promising approach for tumour treatment. Unwanted side effects are considerably reduced, since the nanoparticles are concentrated within the target region due to the influence of a magnetic field. Nevertheless, understanding the transport phenomena of nanoparticles in an artery system is still challenging. This work presents experimental results for a branched tube model. Quantitative results describe, for example, the net amount of nanoparticles that are targeted towards the chosen region due to the influence of a magnetic field. As a result of measurements, novel drug targeting maps, combining, e.g. the magnetic volume force, the position of the magnet and the net amount of targeted nanoparticles, are presented. The targeting maps are valuable for evaluation and comparison of setups and are also helpful for the design and the optimisation of a magnet system with an appropriate strength and distribution of the field gradient. The maps indicate the danger of accretion within the tube and also show the promising result of magnetic drug targeting that up to 97% of the nanoparticles were successfully targeted.     

Experimental investigations on a branched tube model in magnetic drug targeting

Magnetic drug targeting (MDT) has been established as a promising technique for tumour treatment. Due to its high targeting efficiency unwanted side effects are considerably reduced, since drug-loaded nanoparticles are concentrated within a target region due to the influence of a magnetic field. This work presents experimental results that are based on systematic quantitative measurements on a branched tube model as a model system for a blood vessel supplying a tumour. The systematic measurements are summarized in novel drug targeting maps, combining e.g. the net amount of targeted nanoparticles, the magnetic volume force and also the position of the magnet. The model, the injection procedure and the ferrofluid are chosen close to the parameters of a medical application. This will allow transfer of the results to future medical investigations. This work will present a targeting map, where the concentration of the injected ferrofluid is in the range of experiments with an ex vivo bovine artery model.     

Inclusion of magnetic dipole-dipole and hydrodynamic interactions in implant-assisted magnetic drug targeting

Mathematical modelling of the implant-assisted magnetic drug targeting system of Avilés, Ebner and Ritter is performed. In order to model the agglomeration of particles known to occur in this system, the magnetic dipole-dipole and hydrodynamic interactions are included. Such interactions were calculated previously by Mikkelsen et al. under low magnetic fields (~0.05 T) in microfluidic systems. Here, a higher magnetic field (0.7 T) is considered and the effect of interactions on two nanoparticles with a seed implant is calculated. The calculations were performed with the open-source software OpenFOAM. Different initial positions are considered and the system performance is assessed in terms of capture cross section. Inclusion of both interactions was seen to alter the capture cross section of the system by up to 7% in absolute terms.     

Optimal Halbach permanent magnet designs for maximally pulling and pushing nanoparticles

Optimization methods are presented to design Halbach arrays to maximize the forces applied on magnetic nanoparticles at deep tissue locations. In magnetic drug targeting, where magnets are used to focus therapeutic nanoparticles to disease locations, the sharp fall off of magnetic fields and forces with distances from magnets has limited the depth of targeting. Creating stronger forces at a depth by optimally designed Halbach arrays would allow treatment of a wider class of patients, e.g. patients with deeper tumors. The presented optimization methods are based on semi-definite quadratic programming, yield provably globally optimal Halbach designs in 2 and 3-dimensions, for maximal pull or push magnetic forces (stronger pull forces can collect nanoparticles against blood forces in deeper vessels; push forces can be used to inject particles into precise locations, e.g. into the inner ear). These Halbach designs, here tested in simulations of Maxwell's equations, significantly outperform benchmark magnets of the same size and strength. For example, a 3-dimensional 36 element 2000 cm³ volume optimal Halbach design yields a 5× greater force at a 10 cm depth compared to a uniformly magnetized magnet of the same size and strength. The designed arrays should be feasible to construct, as they have a similar strength (≤1 T), size (≤2000 cm³), and number of elements (≤36) as previously demonstrated arrays, and retain good performance for reasonable manufacturing errors (element magnetization direction errors ≤5°), thus yielding practical designs to improve magnetic drug targeting treatment depths.     

In vitro investigation of the behaviour of magnetic particles by a circulating artery model

Magnetic drug targeting is the use of coated magnetic nanoparticles as carriers for cytostatic drugs. After intraarterial application of these carriers, they are attracted with an external magnetic field to, for example, an experimental VX2 tumour. The biological compatibility of this system depends on several physiological and physical parameters. We established an in vitro model to simulate in vivo conditions in a circulating system consisting of a circuit with an intact bovine femoral artery close to an external magnetic field. Nanoparticle suspensions were applied by a side inlet. After the magnetisation procedure particle size, concentration and distribution was examined.     

Towards ferrofluids with enhanced magnetization

It is well known that the saturation magnetization of a sterically stabilized magnetic fluid (ferrofluid) is limited by the presence of a surfactant coating on the surface, and in some cases, by an effectively demagnetized surface layer in the solid magnetic particle. These surface layers take up a disproportionate volume in the colloidal dispersion thereby severely limiting the volume fraction of the core magnetic substance. This work proposes and analyzes Janus particles having the objective of increasing the magnetic loading beyond the present day constraints. Using numerical computation of the virial coefficient it is calculated that the magnetic volume fraction of magnetite ferrofluids might be increased by a factor approaching 2 and that of iron-based ferrofluids by a factor of 3.     

Many particle magnetic dipole-dipole and hydrodynamic interactions in magnetizable stent assisted magnetic drug targeting

The implant assisted magnetic targeted drug delivery system of Avilés, Ebner and Ritter is considered both experimentally (in vitro) and theoretically. The results of a 2D mathematical model are compared with 3D experimental results for a magnetizable wire stent. In this experiment a ferromagnetic, coiled wire stent is implanted to aid collection of particles which consist of single domain magnetic nanoparticles (radius ). In order to model the agglomeration of particles known to occur in this system, the magnetic dipole-dipole and hydrodynamic interactions for multiple particles are included. Simulations based on this mathematical model were performed using open source C++ code. Different initial positions are considered and the system performance is assessed in terms of collection efficiency. The results of this model show closer agreement with the measured in vitro experimental results and with the literature. The implications in nanotechnology and nanomedicine are based on the prediction of the particle efficiency, in conjunction with the magnetizable stent, for targeted drug delivery.     

Analysis of high gradient magnetic field effects on distribution of nanoparticles injected into pulsatile blood stream

Magnetic nanoparticles are widely used in a wide range of applications including data storage materials, pharmaceutical industries as magnetic separation tools, anti-cancer drug carriers and micro valve applications. The purpose of the current study is to investigate the effect of a non-uniform magnetic field on bio-fluid (blood) with magnetic nanoparticles. The effect of particles as well as mass fraction on flow field and volume concentration is investigated. The governing non-linear differential equations, concentration and Navier-stokes are coupled with the magnetic field. To solve these equations, a finite volume based code is developed and utilized. A real pulsatile velocity is utilized as inlet boundary condition. This velocity is extracted from an actual experimental data. Three percent nanoparticles volume concentration, as drug carrier, is steadily injected in an unsteady, pulsatile and non-Newtonian flow. A power law model is considered for the blood viscosity. The results show that during the systole section of the heartbeat when the blood velocity increases, the magnetic nanoparticles near the magnetic source are washed away. This is due to the sudden increase of the hydrodynamic force, which overcomes the magnetic force. The probability of vein blockage increases when the blood velocity reduces during the diastole time. As nanoparticles velocity injection decreases (longer injection time) the wall shear stress (especially near the injection area) decreases and the retention time of the magnetic nanoparticles in the blood flow increases.     

Modeling of magnetic bandages for drug targeting: Button vs. Halbach arrays

Magnetic targeting of drugs to diseased tissues, such as non-healing wounds or skin tumors, is a promising clinical use of magnetic microspheres. For successful magnetic targeting, a magnet must be placed in close proximity to the target tissue. In this work the forces exerted on magnetic microspheres by different arrangements of magnets including a simple square magnet, a number of button magnet arrays, and a Halbach array were simulated and compared. Magnetic bandages utilizing a Halbach array configuration were found to yield the best trapping characteristics (large and uniform force distributions) for magnetic targeting applications close to a surface.