# Two-Phase Biofluid Flow Model for Magnetic Drug Targeting

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Governing Equations

**u**-p) formulation are as follows:

**M**as:

#### 2.2. Blood Viscosity—Power Law Model

**τ**and the shear rate $\dot{\gamma}$ is given by $\tau =\mu \dot{\left(\gamma \right)}$. The shear rate is as follow $\dot{\gamma}=\left(\nabla \mathit{u}+\nabla {\mathit{u}}^{T}\right)$. The tensor of shear is $\mathit{\tau}=\mu \left(\nabla \mathit{u}+\nabla {\mathit{u}}^{T}\right)$. The viscosity $\left(\mu \right)$ for the Power Law model is given by the relation $\mu =m{\left(\dot{\gamma}\right)}^{n-1}$. Considering all the above, we conclude:

#### 2.3. Dipole-Dipole Interaction of MNPs

## 3. Algorithm Verification

^{®}(ANSYS Inc., Canonsburg, PA, USA) [44]. The flow domain is a rectangular channel of length $L=0.02\mathrm{m}$ and width $H=0.001\mathrm{m}$. The fluid flow is incompressible, laminar, non-Newtonian (power-law model with $n=0.8,\mathrm{m}=0.012$ (Equation (16)). The density is set to $\rho 1010{\mathrm{kg}\text{}\mathrm{m}}^{-3}$ and the viscosity to $0.00309\mathrm{Pa}\text{}\mathrm{s}\le \mu \le 0.02\mathrm{Pa}\text{}\mathrm{s}$. We apply a sinusoidal velocity $u=0.0281sin\left(2.82485t\right)+0.0281{\mathrm{ms}}^{-1}$ at the inlet. We compare the centerline velocity of the two methods for different time instances and the difference (for all three velocity vector components) is less than $1\%$. Figure 3 shows a comparison of the two methods at time $t=0.22243\mathrm{s}$.

## 4. Results and Discussion

#### 4.1. MDT in a Planar Microchannel

#### 4.1.1. Effect of the MNPs Volume Fraction

#### 4.1.2. Effect of the Magnetic Field Location

#### 4.1.3. Effect of the Magnetic Field Strength

#### 4.2. MDT When MNPs Are Injected from the Upper Wall of the Microchannel

#### 4.3. MDT in a Microchannel with Stenosis

#### 4.4. MDT in a Microchannel with Bifurcation

## 5. Conclusions

- MNPs loaded with medicinal drugs can be carried to a tissue target in the human body under the applied magnetic field.
- Recirculation zones are formed upstream and downstream of the targeting area (which is near to the location of the magnetic field), as the volume fraction of MNPs increases.
- The capture of MNPs on the tissue target increases with the volume fraction of MNPs and high values of the TAWSS are presented.
- When the magnetic source is placed close to the tissue target, recirculation zones are formed upstream and downstream of the targeting area.
- The increase of the strength of magnetic field causes the increase of TAWSS close to the tissue target.
- The MDT flow case in a microchannel with bifurcation shows that the decrease of the bifurcation angle results to the formation of one recirculation zone at the lower branch of the bifurcation, while at the upper branch, MNPs capture is larger.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Non dimensional concentration for 2000 nm MNPS’ diameter and average velocity 0.0281 ms

^{−1}at time $t=0.595\mathrm{s}$ with MPC method.

**Figure 6.**Stream function contours for (

**a**) φ$=0.03$, (

**b**) φ $=0.06$, and (

**c**) φ $=0.09$ at $t=0.8\mathrm{s}.$

**Figure 7.**(

**a**) Time average volume concentration (TAVC) and (

**b**) Time average wall shear stress (TAWSS), along the upper wall for one period of cardiac cycle and various MNPs volume fractions φ.

**Figure 8.**Stream function contours for (

**a**) b = 0.003 m, (

**b**) b = 0.002 m, and (

**c**) b = 0.001 m at t = 0.8 s.

**Figure 9.**(

**a**) Time average volume concentration (TAVC) and (

**b**) Time average wall shear stress (TAWSS), along the upper wall for one period of cardiac cycle and different locations of the magnetic source b.

**Figure 10.**Stream function contours for different values of magnetic field intensity, (

**a**) $\gamma ={10}^{3}\text{}\mathrm{A}$, (

**b**) $\gamma =3\times {10}^{3}\text{}\mathrm{A}$, (

**c**) $\gamma =5\times {10}^{3}\mathrm{A}$, and (

**d**) $\gamma =7\times {10}^{3}\mathrm{A}$ at $t=0.8\mathrm{s}$.

**Figure 11.**(

**a**) Time average volume concentration (TAVC) and (

**b**) Time average wall shear stress (TAWSS), along the upper wall for one period of cardiac cycle and different values of magnetic field intensity γ.

**Figure 12.**Computational domain for planar microchannel, where the MNPs are injected from the upper wall.

**Figure 13.**Dimensionless volume concentration of MNPs at different times during a cardiac cycle (

**a**) $t=0.2\mathrm{s}$, (

**b**) $t=0.4\mathrm{s}$, (

**c**) $t=0.6\mathrm{s}$, and (

**d**) $t=0.8\mathrm{s}$ when they are injected from the upper wall.

**Figure 14.**Stream function contours when (

**a**) the MNPs are inserted in the microchannel from the inlet (

**b**) the MNPs are injected from the upper wall of the microchannel at $t=0.8\text{}\mathrm{s}$.

**Figure 15.**(

**a**) Time average volume concentration (TAVC) and (

**b**) Time average wall shear stress (TAWSS), along the upper wall for one period of cardiac cycle and two ways of MNPs approach to the targeted tissue.

**Figure 17.**Stream function contours for (

**a**) ${x}_{Mag,1}=0.009\mathrm{m}$, (

**b**) ${x}_{Mag,2}=0.01\mathrm{m}$, and (

**c**) ${x}_{Mag,3}=0.011\mathrm{m}$ at t = 0.8 s.

**Figure 18.**(

**a**) Time average volume concentration (TAVC) and (

**b**) Time average wall shear stress (TAWSS), along the upper wall for one period of cardiac cycle and various horizontal position of the magnetic source.

**Figure 20.**Stream functions contours for three bifurcation angles (

**a**) $30\xb0$, (

**b**) $45\xb0$ and (

**c**) $60\xb0$ at time $t=1.6\mathrm{s}$.

**Figure 21.**Dimensionless volume concentration of MNPs at various times during two cardiac cycle (

**a**) $t=0.4\mathrm{s}$, (

**b**) $t=0.8\mathrm{s}$, (

**c**) $t=1.2\mathrm{s}$, and (

**d**) $t=1.6\mathrm{s}$ when the bifurcation is $60\xb0$.

**Figure 22.**(

**a**) Time average volume concentration (TAVC) and (

**b**) Time average wall shear stress (TAWSS), along the upper wall of the upper branch of the bifurcation for one period of cardiac cycle and various bifurcation angles.

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## Share and Cite

**MDPI and ACS Style**

Boutopoulos, I.D.; Lampropoulos, D.S.; Bourantas, G.C.; Miller, K.; Loukopoulos, V.C.
Two-Phase Biofluid Flow Model for Magnetic Drug Targeting. *Symmetry* **2020**, *12*, 1083.
https://doi.org/10.3390/sym12071083

**AMA Style**

Boutopoulos ID, Lampropoulos DS, Bourantas GC, Miller K, Loukopoulos VC.
Two-Phase Biofluid Flow Model for Magnetic Drug Targeting. *Symmetry*. 2020; 12(7):1083.
https://doi.org/10.3390/sym12071083

**Chicago/Turabian Style**

Boutopoulos, Ioannis D., Dimitrios S. Lampropoulos, George C. Bourantas, Karol Miller, and Vassilios C. Loukopoulos.
2020. "Two-Phase Biofluid Flow Model for Magnetic Drug Targeting" *Symmetry* 12, no. 7: 1083.
https://doi.org/10.3390/sym12071083