# Magnetic Forces by Permanent Magnets to Manipulate Magnetoresponsive Particles in Drug-Targeting Applications

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Governing Equations

#### 2.2. Hydrodynamic Force

_{x}is the velocity x-component, R is the tube radius, and y is the distance from the tube center line. The dimensionless Reynolds number controls the relationship between a flow’s inertial and viscous forces.

_{p}is the particle radius, η is the fluid viscosity, ${\overrightarrow{V}}_{f}$ is the fluid velocity, and ${\overrightarrow{V}}_{p}$ is the particle velocity [19].

#### 2.3. Magnetic Force

_{mag}—magnetophoretic force;

_{0}—is the magnetic permeability of vacuum (μ

_{0}= 4π × 10

^{−7}N/A

^{2});

_{p}—is the volume of the magnetic particle (V

_{p}= $\frac{4}{3}\pi {R}_{p}^{3}$), R

_{p}—magnetic particle radius.

## 3. Materials

#### 3.1. Problem Description

#### 3.2. Magnetic Particle

^{3}and size of 4–6 μm were utilized in this investigation to mimic the magnetic carrier (same as in our previous study [5]). As a result, the magnetic field needs of the experimental setup are significantly reduced by their magnetic moments, which are approximately 10

^{6}÷ 10

^{7}times larger than those of typical drug-carrying iron oxide nanocomposites.

#### 3.3. B-Field Experimental Measurement

#### 3.4. Magnets and Magnetic Field Generation

_{r}—is the magnet residual flux density, and z—distance from the magnet surface (where z ≥ 0) on the magnet’s centerline.

- (1)
- To have the certainty of the correctness of the numerical simulation results obtained (by comparing the results generated by the two programs);
- (2)
- To compare the accuracy of the numerical simulation solutions with the analytical and experimental results, respectively;
- (3)
- To identify the effectiveness (in terms of the computation time and computations cost) of the two numerical simulation programs for solving problems related to the properties of the magnetic field associated with different objects or magnetic equipment (in our case, two different permanent magnets).

#### 3.5. Experimental Test Rig

^{−1}(assuming laminar flow):

#### 3.6. The Carrier Fluid

^{3}) as the working fluid (carrier fluid—CF). The glycerol-water solution in use guarantees that the rheological behavior of blood is reproduced [5,6,39]. The aqueous glycerol solutions employed as the carrier fluid were combined with iron (Fe) particles at a mass concentration of 5% (corresponding to 1 g Fe dispersed in 20 mL carrier fluid) to create the model suspension of magnetic carriers utilized in the studies.

## 4. Results

_{mag}) and the viscous force caused by the flow were considered (F

_{d}). The magnetic particles in our experiment have a diameter of around 6 μm. The difference between the velocity of the particles and the flow surrounding them is minimal because the particles are inertia-free. As a result, there is little hydrodynamic contact between the particles. To represent how the MSMPs transport with the flow, three independent steps can be taken:

- (1)
- Calculating the permanent magnets’ magnetic fields.
- (2)
- Calculating the magnetic force of a magnetic particle.
- (3)
- An experimental investigation of the particle deposition for both mentioned magnets.

#### 4.1. Calculation of the Permanent Magnets’ Magnetic Fields

_{mag}= h + R, h—distance from magnet surface to the vessel wall, R—vessel radius, according to Figure 7.

#### 4.2. Calculation of the Magnetic Force Acting on the Micro-Sized Magnetoresposive Particle

_{r,mag}is the relative permeability of the permanent magnet, H is the magnetic field strength (A/m), and M

_{b}(H) is the magnetization vector of the model suspension stream (A/m) (is a function of H). As mentioned in the previous paragraph, we used Fe particles as the magnetic particles in this study. The magnetic properties of the Fe particle and the used carrier fluid are shown in Table 7. Also, all the parameters used in this paper for calculation are defined in Table 8.

_{eff}—is the particle’s mass, including the added mass; ${v}_{p}$—is the particle total velocity; F

_{mag}—magnetophoretic force; F

_{d}—fluid drag force; F

_{g}—gravitational force; and F

_{bouy}—Buoyancy force.

_{mag}) in the model suspension solution, v

_{mag}= F

_{mag}/6πμ R

_{p}, where R

_{p}denotes the radius of an MSMP.

_{mag}, where n is the number of MPs in the model suspension, was used as the magnetic force per unit volume assumption. Equation (4) provides the magnetic force operating on each particle.

_{mag}is [44]:

_{mag}

_{_x}is most significant on the left and right sides of the magnet (Figure 11A). The magnetic particle is drawn to the magnet surface by the horizontal magnetic force. As a result, the magnetic nanoparticles move toward the magnet more quickly and away from it more slowly. On the other hand, the magnet’s center is where the vertical component of the magnetic force achieves its highest magnitude (Figure 11B). The magnetic micro-sized particles are caught and collected by this magnetic force as they travel within the vessel.

## 5. Discussion

_{mag_x}). In addition, the magnetic force’s vertical component (F

_{mag_z}) strength is inversely proportional to the distance from the magnets’ centers to their edges and are responsible for particle deposition.

#### 5.1. The Effect of Flow Rate on Particle Accumulation Evolution

#### 5.2. The Effect of Magnet Type on Particle Accumulation

_{Fe}was determined experimentally from the weight difference between the injected amount of Fe particles in the suspension (m

_{Fe_total}= 1 g) and the amount of the accumulated Fe particles in the targeted area. The Fe particles that had built up in the target area were collected after stopping the flow system and taking the permanent magnet out of the target area.

_{f}= 0.12 m/s, flow rate Q = 362 mL/min, Reynolds number of Re = 281, and an injection period of T = 30 s.

_{Fe}) and the total amount of the injected Fe particles m

_{Fe_total}during an injection period of 30 s (Table 11). This definition highlights the differences between the investigated magnets from the perspective of the particle targeting technique (Equation (16)).

#### 5.3. Correlation between Magnet Distance and Particle Deposition

## 6. Conclusions

- (1)
- The deposition on the vessel wall is greatly influenced by the intensity of the magnetic field, the magnet type, the magnet size, and the magnetic characteristics of the ferromagnetic particles.
- (2)
- How well particles may be targeted depends on how the magnetic and drag forces are balanced. This is because the magnetic and the drag forces are proportional to the particle’s size in terms of its cube.
- (3)
- The results from the CFD models are qualitatively comparable with the measured magnetic field induction, magnetic field strength, and their fluctuation with the distance from the magnet surface.

- (1)
- The used magnets are clearly defined and, more importantly, investigated from the magnetic field point of view.
- (2)
- As described in Chapter 3.6, we used a glycerol-water solution during our experimental investigation to ensure that the rheological behavior of blood was replicated. There is a fundamental difference, to quantify the magnetic particle deposition in the in vitro test section, which tries to mimic the real working environment.
- (3)
- Our investigation consists of a comparison between two different software and analytical solutions, respectively, which is a novelty in terms of results validation.
- (4)
- In our research, we used ordinary commercially available magnets rather than special magnets or a magnet configuration designed for specific applications. Using ordinary magnets, we highlighted the possibility of creating an efficient magnetic system for an optimized drug-targeting technique.

## 7. Limitation

- (1)
- The use of relatively large Fe particles (4–6 μm).
- (2)
- The use of two different magnets (one neodymium and one ferrite) without being able to quantify the impact of each magnet’s geometry variation and level of magnetization on the effectiveness of capturing magnetizable particles.
- (3)
- The large diameter of the test section (8 mm tubes).
- (4)
- The vessel wall rugosity.
- (5)
- The use of a single-flow regime.
- (6)
- A single value for the artery bifurcation angle.

## 8. Outlooks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Permanent magnets used in this study. (

**A**) Dimensions, axis association, and polarization direction. (

**B**) Magnet M1—Neodymium-type magnet with grade N52. (

**C**) Magnet M2—ferrite-type magnet with grade Y35.

**Figure 5.**The investigated artery bifurcation model: (

**A**) numerical model; (

**B**) experimental model. The constant internal diameter of 8 mm was used for numerical and experimental model generation (both, for the main vessel and the branch).

**Figure 6.**Theoretical and numerical investigations of the magnetic field generated by the magnet M1 (40×20×10 mm), type NdFeB grade N52. Numerical results: (

**A**) using FEMM 4.2, (

**B**) using Ansys Maxwell. (

**C**) B

_{z}evolution function of the magnet surface distance. Comparison between theoretical, experimental, and numerical results.

**Figure 7.**Theoretical and numerical investigations of the magnetic field generated by the magnet M2 (29×26×5 mm), type ferrite grade Y35. Numerical results: (

**A**) using FEMM 4.2, (

**B**) using Ansys Maxwell. (

**C**) B

_{z}evolution function of the magnet surface distance. Comparison between theoretical, experimental, and numerical results.

**Figure 8.**Definition of the investigation sections during numerical analysis. L1, L2, and L3 (dashed lines) depict the site of investigation of the magnetic field strength H. Figure shows the magnitude of the magnetic field generated produced by the ferrite-type magnet. The magnet is placed at h mm from the vessel’s bottom wall. This study employs three distinct domains: (

**D1**)-permanent magnet; (

**D2**)-carrier fluid; and (

**D3**)-air.

**Figure 9.**Comparison of the magnetic field components ${\mathrm{H}}_{\mathrm{x}}\text{}\mathrm{and}\text{}{\mathrm{H}}_{\mathrm{y}}$ obtained analytically, experimentally, and using FEA (FEMM and Maxwell software) along the x direction of the bottom wall of the vessel. B−field magnitude distribution for neodymium permanent magnet M1 grade N52 bottom. The distance between the vessel’s bottom wall and the magnet surface is h = 5 mm.

**Figure 10.**Comparison of the magnetic field components ${\mathrm{H}}_{\mathrm{x}}\text{}\mathrm{and}\text{}{\mathrm{H}}_{\mathrm{y}}$ obtained analytically, experimentally, and using FEA (FEMM and Maxwell software) along the x direction of the bottom wall of the vessel. B−field magnitude distribution for the permanent ferrite magnet (magnet M2) grade Y35 bottom. The distance between the vessel’s bottom wall and the magnet surface is h = 5 mm.

**Figure 11.**Comparison of the magnetic force components for magnet M2 along the vessel axial coordinate. (

**A**) Evolution of magnetic force x and z components at various distances from the magnet surface (lines L4, L3, and L2 in Figure 8). (

**B**) A comparison of the vertical magnetic force component F

_{mag_z}along the lines L4, L3, and L2.

**Figure 12.**(

**A**) Comparison of the magnetic force component F

_{mag_z}generated by magnets M1 and M2 along the bottom wall of the vessel. The centers of both magnets are placed in the same position as the vessel bifurcation center. For both magnets that were looked at, the distance between the bottom of the vessel and the magnet surface was h = 2 mm. (

**B**) The magnetic field induction B

_{z}(T) produced by magnets M1 and M2 are compared along the z−axis as a function of the distance to the magnets’ surfaces. Due to the different magnet types (ferrite (M2) and neodymium (M1)) and magnet grades (Y35 and N52, respectively), there are significant variances between the magnetic field inductions generated by the investigated magnets.

**Figure 13.**Flow field evolution in the artery bifurcation. (

**A**) For a flow rate of Q1 = 362 mL/min and (

**B**) for a flow rate of Q2 = 754 mL/min. The bifurcation induces the generation of two recirculation areas, one in the main artery (Vortex V1) and the other in the side branch (Vortex V2). The shape and extension of both vortices are directly related to the flow rate and the bifurcation angle.

**Figure 14.**Fe particles deposit under the magnetic force generated by the permanent ferrite magnet having a grade of Y35 (magnet M2). Fe particles were injected for a period of 30 s. (

**A**) Particle depositions over 5 s. (

**B**) Depositions of particles at the end of the 30 s injection period. The magnet’s center in the z direction is in line with the vertical axis of the vessel bifurcation. The deposition shape shows how the drag force balances the magnetic force. Both investigations were conducted for identical working conditions, fluid velocity V

_{f}= 0.12 m/s, flow rate Q = 362 mL/min, Reynolds number of Re = 281, and distance between the vessel bottom wall and the magnet surface h = 5 mm.

**Figure 15.**Particle retention dependency of the used magnet types (

**A**,

**B**). Particle accumulation evolution at different time steps during the injection period

**A1**,

**A2**,

**B1**,

**B2**. Magnetic field vertical component values for both magnets at h = 5 mm from the bottom wall of the artery (

**C**). Magnetic force distribution for magnets M1 and M2 along the main artery wall (

**D**). Comparison of the accumulated particle quantities and percentage differences between the used magnets, corresponding to the different time steps during injection (

**E**). Investigations were conducted in the same conditions; distance from the wall of h = 5 mm, injection period of 30 s, flow rate of 362 mL/min, and Reynolds number Re = 281.

**Figure 16.**Particle accumulation dependency of the magnet distance h = 10 mm for both investigated magnets. (

**A**) Magnet M2 is a ferrite magnet, and (

**B**) Magnet M1 is a neodymium magnet. (

**C**) Vertical magnetic force component for magnets M1 and M2 at a distance of h = 10 mm from the artery wall. (

**D**) Accumulated Fe particle mass and mass percentage differences between magnets M1 and M2 at the end of the injection period. Working conditions were identical for both investigations; injection period of 30 s, the flow rate of 362 mL/min, and Reynolds number of Re = 281.

Characteristics | Value |
---|---|

particle diameter | 4–6 μm |

density | 7.86 g/cm^{3} |

molar mass | 55.8 g/mol |

Saturation Magnetization | Saturation Field | Coercive Field | Remanent Magnetization |
---|---|---|---|

Ms (A·m^{2}/kg): 177 | Hs (kA/m): 600 | Hc (kA/m): 1.32 | Mr (A·m^{2}/kg): 0.891 |

Magnet | Shape | Material | Length—L (mm) | Width—W (mm) | Thickness—T (mm) |
---|---|---|---|---|---|

M1 | Rectangular | Neodymium—NdFeB | 40 | 20 | 10 |

M2 | Rectangular | Ferrite | 26 | 29 | 5 |

Magnet | Shape | Magnetization Direction | Material | Grade | Br (T) | Hcb (kA/m) | Hcj (kA/m) | BHmax (kJ/m ^{3}) |
---|---|---|---|---|---|---|---|---|

M1 | block | thickness | Neodymium | N52 | 1.42–1.47 | 860–995 | ≥955 | 380–422 |

M2 | block | thickness | Ferrite | Y35 | 0.43–0.45 | 215–239 | 217–241 | 33.1–38.3 |

**Table 5.**B-field values at a different position from the magnet surface and the percentage differences between analytical values and values obtained experimentally and by numerical analysis for neodymium N52-type permanent magnet.

Z-Position (mm) | Bz_Equation (T) | Bz_exp (T) | Bz_FEMM (T) | Bz_Maxwell (T) | (%) Diff. Equation—Exp | (%) Diff. Equation—FEMM | (%) Diff. Equation—Maxwell |
---|---|---|---|---|---|---|---|

0.00 | 0.4090 | 0.367 | 0.3532 | 0.3566 | 10.27 | 13.65 | 12.81 |

5.00 | 0.2727 | 0.247 | 0.2343 | 0.2371 | 9.42 | 14.08 | 13.05 |

10.00 | 0.1675 | 0.151 | 0.1456 | 0.148 | 9.85 | 13.09 | 11.64 |

15.00 | 0.1041 | 0.094 | 0.0923 | 0.0961 | 9.70 | 11.31 | 7.68 |

20.00 | 0.0671 | 0.061 | 0.0628 | 0.0663 | 9.09 | 6.43 | 1.19 |

25.00 | 0.0450 | 0.041 | 0.0443 | 0.0482 | 8.89 | 1.54 | 7.11 |

30.00 | 0.0312 | 0.026 | 0.0330 | 0.0367 | 16.67 | 5.80 | 17.63 |

**Table 6.**Ferrite Y35-type permanent magnet B-field values compare different distances from the magnet surface.

Z-Position (mm) | Bz_Equation (T) | Bz_Exp (T) | Bz_FEMM (T) | Bz_Maxwell (T) | (%) Diff. Equation—Exp | (%) Diff. Equation—FEMM | (%) Diff. Equation—Maxwell |
---|---|---|---|---|---|---|---|

0.00 | 0.0780 | 0.0732 | 0.0476 | 0.0504 | 6.15 | 39.02 | 35.38 |

5.00 | 0.0589 | 0.0551 | 0.0371 | 0.04 | 6.45 | 37.00 | 32.09 |

10.00 | 0.0379 | 0.0351 | 0.0251 | 0.0281 | 7.39 | 33.71 | 25.86 |

15.00 | 0.0234 | 0.0215 | 0.0165 | 0.0194 | 8.12 | 29.40 | 17.09 |

20.00 | 0.0146 | 0.0131 | 0.0110 | 0.0138 | 10.27 | 24.98 | 5.48 |

25.00 | 0.0095 | 0.0085 | 0.0074 | 0.0101 | 10.53 | 22.32 | 6.32 |

30.00 | 0.0064 | 0.0058 | 0.0050 | 0.0077 | 9.38 | 21.64 | 20.31 |

Materials | Properties | Value | Unit |
---|---|---|---|

Carrier fluid | ρ_{f}—fluid densityη _{f}—fluid dynamic viscosityχ _{f}—fluid magnetic susceptibility | 1055 0.0036 −6.6 × 10 ^{−7} | kg/m^{3}kg/(m.s) [-] |

Fe | ρ_{Fe}—Fe particle densityχ _{Fe}—Fe particle magnetic susceptibility | 7860 4 | kg/m^{3}[-] |

Symbol | Description | Default Value | Unit |
---|---|---|---|

m_{eff} | Mass of the Fe particle (one particle) | 8.88945 × 10^{−13} | (kg) |

${v}_{p}$ | Magnetic particle total velocity | ${v}_{p}={v}_{f}+{v}_{mag}$ | (m/s) |

${v}_{f}$ | Fluid velocity (carrier fluid) | 0.12 | (m/s) |

${v}_{mag}$ | Magnetic particle (Fe particle) velocity due to the magnetic force acting on the particle | ${F}_{mag}=\frac{1}{2}{\mu}_{0}\chi {V}_{p}\nabla {H}^{2}$ | (m/s) |

${v}_{m}$ | Fluid mean velocity (carrier fluid) | ${V}_{m}=\frac{Q}{S}=\frac{Q}{\pi {R}^{2}}$ | (m/s) |

${V}_{p}$ | The volume of magnetic particle | V_{p} = $\frac{4}{3}\pi {R}_{p}^{3}$ | (m^{3}) |

${R}_{p}$ | Magnetic particle radius (Fe particle) | 4 ÷ 6 × 10^{−6} | (m) |

R | Artery (vessel) radius | 8 × 10^{−3} | (m) |

z | Magnetic particle (Fe) coordinates along the z direction | (m) | |

μ_{0} | The magnetic permeability of air | μ_{0} = 4π × 10^{−7} | (N/A^{2}) |

ρ_{f} | Fluid density (carrier fluid) | 1055 | (kg/m^{3}) |

$Q$ | Fluid flow rate (carrier fluid) | 6.0288 × 10^{−6} | (m^{3}/s) |

g | Gravity acceleration | 9.81 | (m/s^{−2}) |

Item | Flow Rate Q (ml/min) | Flow Velocity v_{f} (m/s) | Reynolds Number Re (-) | Main Vortex Length L_V1 (mm) | Particle Deposition * (g) | Magnet Type | Magnet Distance h (mm) |
---|---|---|---|---|---|---|---|

Q1 | 362 | 0.12 | 281 | 17 | 0.285 ± 0.00616 | M2 | 5 |

Q2 | 754 | 0.25 | 586 | 22 | 0.214 ± 0.00734 | M2 | 5 |

**Table 10.**Characteristics of the particle accumulation shape along the artery lower wall at the end of the injection period.

Magnet Distance h (mm) | Magnet Type | Magnetic Flux Density Bz (T) | Particle Deposition Quantity m_{Fe} (g) | Particle Deposition Length L_Deposition (mm) | Particle Deposition AVERAGE Thickness H_Deposition (mm) |
---|---|---|---|---|---|

2 | M1 | 0.35 | 0.405 ± 0.00714 | 44 | 3 |

M2 | 0.065 | 0.261 ± 0.00831 | 36 | 1.85 | |

5 | M1 | 0.27 | 0.382 ± 0.00817 | 41.5 | 2.2 |

M2 | 0.051 | 0.285 ± 0.00616 | 34 | 1.11 | |

10 | M1 | 0.16 | 0.268 ± 0.00778 | 40 | 1.2 |

M2 | 0.032 | 0.161 ± 0.00857 | 32 | 0.75 |

Magnet Distance h (mm) | Magnet Type | Targeting Efficiency TE (%) | Targeting Efficiency Percentage Difference between M1 and M2 |
---|---|---|---|

2 | M1 | 40.5% | 14.4% |

M2 | 26.1% | ||

5 | M1 | 38.2% | 9.7% |

M2 | 28.5% | ||

10 | M1 | 26.8% | 10.7% |

M2 | 16.1% |

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**MDPI and ACS Style**

Bernad, S.I.; Bernad, E.
Magnetic Forces by Permanent Magnets to Manipulate Magnetoresponsive Particles in Drug-Targeting Applications. *Micromachines* **2022**, *13*, 1818.
https://doi.org/10.3390/mi13111818

**AMA Style**

Bernad SI, Bernad E.
Magnetic Forces by Permanent Magnets to Manipulate Magnetoresponsive Particles in Drug-Targeting Applications. *Micromachines*. 2022; 13(11):1818.
https://doi.org/10.3390/mi13111818

**Chicago/Turabian Style**

Bernad, Sandor I., and Elena Bernad.
2022. "Magnetic Forces by Permanent Magnets to Manipulate Magnetoresponsive Particles in Drug-Targeting Applications" *Micromachines* 13, no. 11: 1818.
https://doi.org/10.3390/mi13111818