Simulation studies of a novel electromagnetic actuation scheme for focusing magnetic micro/nano-carriers into a deep target region

Magnetic drug targeting (MDT) is a method for steering magnetic nanoparticles (MNPs) to, and concentrating them in, desired locations, such as tumors,^1–3 infections⁴ and blood clots.⁵ MDT can reduce the levels of drugs in the rest of the body, thereby increasing the effectiveness of treatment.

Due to the physical properties of magnetic fields and magnetic forces, the static field of any single coil or magnet always attracts magnetic micro/nano-particles.^6–8 Hence, the majority of previous systems have been designed to attract therapeutic particles to target regions.^9–15 Because two or more magnets can be utilized to push particles,¹⁶ the method can be applied to therapies directed toward the back of the eye^6,7 and the inner ear.^17–19 However, there is a difficulty associated with magnetic focusing in central targets, which is explained in part by Samuel Earnshaw’s 1842 theorem.²⁰ The theorem proves that spherical particles (SPPs) cannot be focused in a static magnetic field. To overcome this issue, a new scheme using ferromagnetic rods and fast magnetic pulses was introduced in.Ref. 21 However, the intensive use of spherical magnetic nanoparticles in the drug delivery encouraged us to develop a new scheme for concentrating SPPs in the desired location in body.

In this paper, we propose a new actuation scheme for pushing and focusing SPPs, based on the magnetophoresis (MAP) force and a combination of two coaxial coils with opposite current directions. COMSOL multi-physics software simulation results show that the new scheme can focus SPPs onto a target region of a block filled with an environment that has the characteristics of blood. The effects of the focusing scheme were then investigated in a realistic blood vessel channel in the brain, with a maximum length of about 10–12 cm. The results also show that SPPs can be concentrated onto a target tumor region after they have been steered to the target vessel branch. The proposed scheme will promote MDT research concerned with targeting drugs to deep tissues, to maximize MDT efficiency while minimizing its side effects.

An actuation coil (ACC) for pushing can be designed using two coaxial coils with opposite current directions, as shown in Fig. 1a. The total magnetic field on the x-axis generated by the two coils can be represented by:²² 

where I_inner and R_inner are the current and radius of the inner coil, and I_outer and R_outer are the current and radius of the outer coil, respectively; x is distance from the center of coil to position on x–axis (Assume the inner coil and the outer coil have the same center). This coil combination can generate two field-free points (FFP) on the x-axis, denoted FFP₁ and FFP₂, respectively, in Fig. 1(a), (b). Setting H_x(x_FFP) to zero yields the following equation:

FFP₁, located in front of the ACC coil, can be regarded as a reference point for marking the pushing region. The magnetic gradient along the x-axis is shown in Fig. 1(c) and we can observe the pushing region in this figure (note that FFP₁ and the zero magnetic gradient point are not the same).

If the position of FFP₁ is changed, the position of the pushing region may also change. So, the distance between the pushing region and the ACC can be controlled by moving FFP₁. From Eq. (2), if the distance between the two coils is kept constant, the location of the FFP depends only on the ratio of the currents I_outer/I_inner. Thus, changing the current in one or both of the coils generates an oscillation in the FFP along a line on the x-axis. Theoretically, the FFP can be moved to any point on the x-axis if the currents are chosen appropriately. If we keep the current ratio unchanged, the FFP will not move, but the gradient strength at the FFP or the pushing region can be changed.

Based on the same coil structure as used in the pushing scheme, two independent ACCs can be oppositely placed, as shown in Fig. 1(d). By adjusting the distance between the two ACCs and/or changing the current of the coils, two FFP₁ locations can be generated on the x-axis. The space between these FFP₁ locations is called the focusing region in Fig. 1(d), (e). Figure 1(f) shows the magnetic gradient along the x-axis, the gradient strength in x-direction with a different sign is twice higher than in y-direction and z-direction (Gauss’s law); we can observe that the working range of the focusing region is about 5.4 cm. The proposed scheme can focus the SPPs in the x-axis, but after focusing the SPPs tend to emerge toward both sides of the y and z-axis. Therefore, another scheme is needed for focusing the SPPs.

For focusing a surface region, we need to combine two of the focusing coil configurations for the line segment. If four ACCs are in operation simultaneously, as shown in Fig. 2(a), the directions of the resultant magnetic gradients as shown in Fig. 2(b) cannot concentrate the SPPs, which is the main reason why SPPs cannot be focused using a static magnetic field. Thus, we have to consider another scheme, as shown in Fig. 2(c). The x- and y-axis coils should operate alternately with the time function (TF) of the currents shown in Fig. 2(d). The TF of the currents can be represented by:

where T_x and T_y are the times at which the currents are applied for the x-axis and y-axis coils, respectively. By applying Eq. (3), the SPPs can be pushed from the four sides of the surface region and concentrated into a target surface region. Since the direction of the magnetic gradient in the xz-plane is similar to that in the xy-plane (Fig. 1e), and the SPPs are focused in the x-axis and y-axis; the SPPs tend to emerge toward both sides of the z-axis. Thus, to focus on one side of the surface region, the particle sample should be placed on one side (positive or negative) of the z-axis. This is the main reason why it is only possible to focus on surfaces. If the currents in the pushing scheme are maintained, the workspace of the focusing scheme to the surface region is 5.4 cm × 5.4 cm.

As a general example of MP focusing, a block with dimensions of 0.7 × 0.7 × 0.5 cm was initially used for the simulation, as shown in Fig 3(a). The block was filled with an environment that has the characteristics of blood (the density is 1050 kg/m³ and the viscocity is 0.004 Pa.s²⁵). Then, the proposed focusing scheme was applied to a 3-D model of a realistic vessel, as shown in Fig 3b. Steadily-creeping blood flow was considered, flowing into the channel from the inlet and exiting the channel from the outlets. Fluid modeling parameters were selected to be similar to the behavior of blood,²³ and the surrounding environment was assumed to be air. The particles were assumed to be spherical magnetite particles with a diameter (d_p) of 500 nm.²⁵ The blood flow inside the channels was simulated as a steady state laminar flow, and its velocity profile was calculated using the computational fluid dynamics (CFD) module in COMSOL.

The block model has been designed to mainly demonstrate that the proposed actuation scheme can deliver magnetic particles to a target surface. In the simulation, 1000 particles of diameter 500 nm and distributed uniformly in the block, were released; their trajectories according to the magnetic force were captured after 920 s with f = 0.025 Hz (T_x = T_y). The SPPs were concentrated at the center of the desired surface region (0.05 × 0.05 cm) as shown in Fig. 3(a). To reduce the concentration time of the SPPs, we can increase the current or particle diameter. The frequency has an effect on the particle movement limits; at low frequency, particles have limited movement, which affects the concentration of particles onto a target region. Based on the simulation results, the proposed focusing scheme was applied to a tumor region inside a realistic blood vessel channel.

A simulation was performed to investigate the effects of focusing on a surface region in a realistic blood vessel channel. A specific vessel in the human brain was considered and the geometry of the vessel was extracted using magentic resonance imaging (MRI).²⁴ Based on the extracted information, a 3-D model of the vessel was created using CAD software and imported to the COMSOL software to simulate particle trajectories.²⁵ The channel consisted of one inlet for blood flow and SPPs and six outlets. The blood flows into the channel from the inlet and exits the channel from the outlets. The diameters of the channels in all paths were varied, resulting in varying velocity profiles in each section, as shown in Fig. 3(b). Since particles could path through the endothelial cell under a gradient magnetic field²⁶ and the main objective of this simulation is to concentrate the particles on the surface of a tumor, the studies for penetration of particles into the tumor surface are not considered in this simulation.

Because the realistic blood vessel channels had a maximum length of about 10–12 cm, the workspace was extended. It was necessary to increase I_outer from 29 A to 40 A to move FFP₁ near to the ACCs (current amplitude ratio I_outer/I_inner = 1); then, the workspace had dimensions of 14 × 14 cm. To focus the SPPs onto a target tumor region, we first considered a target tumor region at the center of the workspace. Then, we applied the focusing scheme along the vessel branch, which contained a tumor with T_x = 50 s (step 1). From that, the SPPs can be focused close to the target tumor region. Then, we continued to apply the focusing scheme perpendicular to the tumor vessels to concentrate the SPPs onto the tumor (step 2) with T_y = 5 s. Note that T_y should be smaller than T_x because a larger T_y will direct the SPPs to the outlet.

In the simulation, 1000 particles with diameters of 500 nm and distributed uniformly on the inlet surface, were released into the inlet channel and their trajectories captured over 275 s. When there were no magnetic fields, the SPPs were scattered around blood vessels and almost no SPPs arrived at the tumor. However, applying the focusing scheme for a surface region (applied after the SPPs move for 55 s) concentrated up to 97.9% of the SPPs to the target tumor region; the number of SPPs going to the outlet channel was also reduced. The bar chart in Fig. 3e demonstrates the percentages of the particles reaching to the tumor with respect to particle size and current of coils (keeping amplitude ratio I_outer/I_inner = 1). The simulation results show that the particles in all the cases can reach to the tumor region through the proposed scheme with high reaching rates. We can clearly observe that the focusing scheme for the surface region can greatly improve the concentration efficiency, and also that the simulation results are very promising. For real implementtions, it should be noted that a large power supply is required to generate sufficient magnetic gradients for nanoparticles.

Addressing the limitations associated with focusing SPPs into deep tissues inside the body is important for improving the concentration efficiencies of MDTs. SPP distributions subjected to a focusing force toward a surface region were investigated using numerical methods and commercially available software for a simple block and a realistic vessel. Theory, modeling descriptions, and simulation results were presented for the proposed concept. The results showed that we can focus on any target surface. The results in the realistic vessel also showed that the SPPs of 500 nm are successfully concentrated to a target tumor region with a performance of 97.9%. The applicability of this scheme is very high in practice, due to the simple coil structure and easy guidance scheme. For realistic MP guidance in vivo experiments, an exact 3-D blood map of each patient should be extracted using an imaging modality such as MRI. Subsequently, we need to determine the target position exactly; then we can apply the focusing scheme for targeting. Our future work will include practical experiments on targeted drug delivery in a mouse brain, as well as development of the actuation device with optimal parameters for the best performance.

This work was supported by the National Research Foundation Korea (NRF) (2012-0009524 and 2014R1A2A1A11053989).