# Modeling Superparamagnetic Particles in Blood Flow for Applications in Magnetic Drug Targeting

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## Abstract

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## 1. Introduction

## 2. Governing Equations

## 3. Results and Discussion

#### 3.1. Viscosity Models

#### 3.2. Magnetic Field

#### 3.3. Release Point

#### 3.4. Red Blood Cell Collisions

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic for the visualization of the implemented model of magnetic drug targeting. The velocity profile, as shown, is entirely in the axial direction with no turbulence. It should be noted that red blood cells are not pictured and that the magnet is represented by a cylinder.

**Figure 2.**Comparison of (

**a**) shear rate at different positions in the blood vessel and (

**b**) the relationship between shear rate and viscosity over different viscosity models. All parameters were kept constant among the models and only shear rate, as a function of position, changed. These included: $\frac{2}{\pi}\times {10}^{-2}$ m/s for the maximum velocity, $5.60\times {10}^{-2}$ Pa s and $3.5\times {10}^{-3}$ Pa s for ${\eta}_{0}$ and ${\eta}_{\infty}$, respectively, $3.31$ s for relaxation time, $0.035$ Pa s${}^{n}$ for the flow consistency index [23], and ${\eta}_{\infty}$ as the constant viscosity based on the asymptotic behavior. Specifically, the flow behavior index for all models is the same, 0.708.

**Figure 3.**Comparison of magnetic moment and capture rate over different viscosity models. Each curve was generated by running a simulation of 100 trajectories for particles released at the center of vessel for that viscosity model, where a captured particle is one whose path terminates within the tumor. All parameters were kept constant among the models, consistent with the values in Figure 2, and the paths of every particles are independent and identically distributed by definition.

**Figure 4.**Graphs comparing simulations with single release points versus simulations with randomly chosen release points for magnetic particles. The simulations show 100 particle trajectories with

**m**= [0, 0, 700] Am

^{2}and use the Cherry viscosity model with n = 0.708.

**Figure 5.**Graphs comparing simulations with and without red blood cell collisions. The simulations show 100 particle trajectories with

**m**= [0, 0, 700] Am

^{2}and use the Cherry viscosity model with n = 0.708. Without red blood cell collisions, the capture rate was 100%; with red blood cell collisions, the capture rate was 91%.

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**MDPI and ACS Style**

Rukshin, I.; Mohrenweiser, J.; Yue, P.; Afkhami, S.
Modeling Superparamagnetic Particles in Blood Flow for Applications in Magnetic Drug Targeting. *Fluids* **2017**, *2*, 29.
https://doi.org/10.3390/fluids2020029

**AMA Style**

Rukshin I, Mohrenweiser J, Yue P, Afkhami S.
Modeling Superparamagnetic Particles in Blood Flow for Applications in Magnetic Drug Targeting. *Fluids*. 2017; 2(2):29.
https://doi.org/10.3390/fluids2020029

**Chicago/Turabian Style**

Rukshin, Iris, Josef Mohrenweiser, Pengtao Yue, and Shahriar Afkhami.
2017. "Modeling Superparamagnetic Particles in Blood Flow for Applications in Magnetic Drug Targeting" *Fluids* 2, no. 2: 29.
https://doi.org/10.3390/fluids2020029