Magnetic nanoparticles offer numerous promising biomedical applications, e.g. magnetic drug targeting. Here, magnetic drug carriers inside the human body are directed towards tumorous tissue by an external magnetic field. However, the success of the treatment strongly depends on the amount of drug carriers, reaching the desired tumor region. This steering process is still an open research topic.
In this paper, the previous study of a linear Halbach array is extended by an additional Halbach array with different magnetization angles between two adjacent magnets and investigated numerically using COMSOL Multiphysics.
The Halbach arrays are arranged with permanent magnets and generate a relatively large region of a moderately homogeneous, high magnetic field while having a strong gradient.
This results in a strong magnetic force, trapping many particles at the magnets. Afterwards, to avoid particle agglomeration, the Halbach array is flipped to its weak side. Therefore, the magnetic flux density, its gradient and the resulting magnetic force are computed for the different Halbach arrays with different constellations of magnetization directions.
Since the calculation of the gradient can lead to high errors due to the used mesh in Comsol, the gradient was derived analytically by investigating two different fitting functions.
Overall, the array with a 90

In the last decades, the interest in steering superparamagnetic iron-oxide nanoparticles (SPIONs) to a desired location, particularly in the context of biomedical engineering, has grown rapidly (

Nevertheless, the success of a MDT treatment strongly depends on the accumulation and therefore, on the ability to steer the SPIONs to the tumorous tissue.
However, the required guiding is a complex problem, since it depends on various multiphysical parameters like the velocity profile of the blood flow, the gradient of the applied magnetic field as well as the properties of the nanoparticles themselves.
The SPIONs should have a diameter of

The main parameter for guiding a particle swarm through the cardiovascular system is the applied magnetic field.
Here, the distance from the magnet between the SPIONs plays a crucial role, since the generated magnetic field decays approximately with

In practice, however, it is not always possible to place the magnets surrounding the vessel.
Thus,

In this work, the results of the previous conference paper (^{®}.

To evaluate the precise steering of the SPIONs, two different arrangements of Halbach arrays are proposed in this section. Furthermore, the magnetic flux density of a Halbach array and the resulting forces acting on the particles in a MDT scenario, namely magnetic and hydrodynamic drag force, will be introduced.

In the literature, Halbach arrays are primarily utilized for electromotors (

Magnetization pattern and field distribution of the adjustable Halbach array. The magnetic field can be shifted from below to above the magnets by changing

In extension to our previous published work (

Magnetization pattern and field distribution of the extended Halbach array. Here, the two sides cannot be changed by a simultaneous rotation of all single magnets.

In the proposed extended array, the magnetic field of the extended Halbach array is augmented and attenuated on one side, respectively, like in the normal Halbach array in Fig.

Most approaches in the literature use numerical methods or programs such as COMSOL Multiphysics^{®} (

The resulting magnetic force

From Eq.

The (hydrodynamic) drag force on one spherical SPION is

For a predominant movement of the SPIONs towards the magnets, the magnitude of the magnetic force

Proposed geometry model used in the 2D Comsol simulations. The magnetic field is analyzed at three positions, marked as red numbers. Whereas the measuring positions “1” and “3” are located underneath the fourth and the last magnet, respectively, “2” is located symmetrical in the center of the array at the gap between two adjacent magnets.

To investigate the performance of the particle steering, the linear Halbach array and the extended Halbach array were built up as a 2D model in COMSOL Multiphysics^{®} 5.6. Each array consists of 8 circular magnets and a Y-shaped geometry with a 30

Fixed simulation parameters.

In this section the simulation results will be presented and discussed. At first, the magnetic flux density of the different Halbach arrays at the three aforementioned evaluation positions will be investigated in detail. Afterwards, their performance on deflecting the SPIONs in the given scenario will be evaluated. For further investigating the magnetic force

In Fig.

Isolines of the magnetic flux density

In Fig.

Isolines of the magnetic flux density

The variation of

The behavior of the magnetic flux density

Magnetic flux density

For the adjustable Halbach array, the propagation of the SPIONs for a fixed rotation angle

Distribution of the SPIONs for different rotation angles

Without the magnet (labeled “none” in Fig.

For the extended Halbach array (not plotted in Fig.

Since

Comparison of the mean fitting errors of the the exponential Eq. (

Overall, the fitting performs quite well for both functions over the whole evaluation domain. By considering only the results inside the vessel, the errors even decrease, since the greatest errors occur directly at the boundary of the magnets. The fitting errors are smaller for the analytic fitting function Eq. (

Applying the fitting functions, the gradient of

Gradient of the magnetic flux density

To steer the particles, the main idea is to attract the SPIONs by a strong magnetic field and then rotate the array to get the particles washed out by the (hydrodynamic) drag force

Additionally,

Comparison of the results for different rotation angles

Besides, the main moving direction is given by

Ratio of

As expected,

It is worth mentioning, that the proposed study has some limitations: Within the simulations, the particles were considered separately. This means, that the interaction between the particles among each other were neglected. Also the change of the magnetic field due to the particles propagating through it, is not taken into account. Furthermore, the magnetization within the SPIONs is assumed to be constant.
However, it can be observed from measurements that the particles form chains parallel to the magnetic field lines (

Moreover, for the magnetic force

Another limitation of this study is caused by the permanent magnets. They cannot be switched off like an electromagnet. Therefore, there will be always a remaining magnetic field and, thus, a magnetic force towards the magnets, even for the weak side. In consequence, the magnetic array has to have a certain distance to the vessel, which, however, results in the decrease of the magnetic force for the strong side.

In this extended paper, an adjustable linear Halbach array and an extended version for steering magnetic nanoparticles were investigated numerically by using COMSOL Multiphysics^{®}.
Both Halbach arrays provide a strong and a weak magnetic side, which can be switched by rotating the single magnets.
In case of the adjustable Halbach array, this can be done by a simultaneous rotation of all single magnets around 90

The results reveal that overall the adjustable Halbach array performs better than the extended one in both, changing the magnetic force easily and deflecting the particles. In general it can be concluded that the magnetic force is the strongest directly underneath the magnets and by rotating the single magnets, the region where the drag force is able to wash the particles out, can be adjusted. In future research, the simulation model will be expanded and every second magnet of the adjustable Halbach array will be replaced by an electromagnet. Therefore, the configuration has to be optimized to design a proper electromagnet needing less space and power. By that, the strong and weak magnetic side can be switched by switching the direction of the current. With this, an electromagnetic field of travelling waves similar to a linear motor drive will be designed for steering the particles to a desired region.

The errors for the fitting of the magnetic flux density with the exponential Eq. (

Overview of the fitting results for the exponential Eq. (

Overview of the fitting results for the exponential Eq. (

The simulation data and MATLAB-code are available from the corresponding author upon request.

AST initiated the research project and developed the method. AST, SZ and ML conducted the simulation and carried out the visualization. AST interpreted the results. GF was supervisor. AST wrote the original draft. All authors were involved in editing and reviewing the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Kleinheubacher Berichte 2021”.

This paper was edited by Lars Ole Fichte and reviewed by two anonymous referees.