# Continuous-Flow Separation of Magnetic Particles from Biofluids: How Does the Microdevice Geometry Determine the Separation Performance?

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Methods

#### 2.1. CFD Model

_{p}and d

**v**

_{p}/dt are the bead mass and acceleration, and

**F**

_{ext}is the resultant force vector exerted on each bead. For this analysis, we take into account only the dominant magnetic and hydrodynamic forces acting on the beads.

**F**

_{m}is predicted with the “effective” dipole moment method as discussed in our previous work [1], and can be expressed as:

_{0}is the permeability of the free space (4π·10

^{−7}H·m

^{−1}), Vp is the particle volume,

**H**

_{a}is the applied magnetic field at the bead center, χ

_{p}and χ

_{f}represent the magnetic susceptibilities of the bead and the fluid, respectively. χ

_{p,e}and M

_{s,p}are the effective susceptibility of the bead (that can be related to the intrinsic susceptibility, χ

_{p}) [40] and the particle saturation magnetization. The particles are assumed spherical with a diameter of 4.9 µm and a density equal to 2000 kg·m

^{−3}. The saturation magnetization and susceptibility values employed are M

_{s,p}= 1.5 × 10

^{4}A·m

^{−1}and χ

_{p,e}= 0.25, which are within the range of magnetization and susceptibilities reported for commercial beads [27,41]. The surrounding fluids are considered non-magnetic due to their low susceptibilities in comparison to the beads. In order to calculate the magnetic field distribution inside the microdevice, we solved the analytical model developed by Furlani [42]. A permanent magnet of dimensions 10 × 5 × 3 mm

^{3}is modeled as the magnetic field source, which is commercially available [27]. This magnet generates a magnetic field of approximately 500 mT in its pole surface.

**F**

_{hd}is numerically predicted in this work using the following expression:

_{added}is the added mass, equal to 0.5ρV

_{p}, being ρ the fluid density [22].

**v**is the fluid velocity,

**F**

_{drag}is the drag force and A

_{P}is the bead cross sectional area, which can be written as A

_{P}= πr

_{p}

^{2}, being r

_{p}the particle radius. C

_{D}is the drag coefficient for steady-state flow around a sphere [43] and Re

_{p}is the particle Reynolds number [27,44].

**F**

_{hd}, the properties of the fluids as well as the velocity profile are needed. One of the fluids in our analysis, blood, presents a non-Newtonian rheology. However, we modeled it as a Newtonian fluid and its viscosity was quantified with an analytical, empirical expression, and has a value of 3.5 cP [45,46]. It should be noted that it has been demonstrated in previous work [47] that blood follows a Newtonian rheology when the shear rate exceeds about 100 s

^{−1}, and we have also experimentally validated this assumption [27]. On the other hand, the aqueous buffer solution was modeled as water with a constant viscosity equal to 1 cP.

**τ**) term denotes the contribution of shear stress on the fluid velocity.

**F**

_{P}can be written as:

#### 2.2. Geometric Configurations and Simulation Setup

_{min}) determines H, according to the following expression [50,51]:

_{h}) and the chip volume. In order to suitably compare rectangular and U-shaped cross sections, we also calculated the D

_{h}as well as the chip volume by using the appropriate equations for each type of device [56,57], which are detailed in Table 1. Rectangular and U-shaped devices were further characterized and compared by estimating the fluidic resistance (R

_{f}) of the channels according to Equation (12) [58,59,60]:

_{h}the hydraulic radius of the channel.

_{f}value for each of the studied geometries is listed in Table 1. By taking all of the previously described parameters into account, we can perform a realistic analysis of the geometrical features of the chip.

_{z}) is 190 µm. Correspondingly, Sep

_{y}, is defined as the separation between the magnet and the surface of the device (i.e., surface of the channel slide) in y-direction. This dimension was set to 1 mm for both types of devices. However, the total magnet-channel separation in y-direction should also account for the thickness of the cover slide, which was different for each configuration, as previously mentioned. In this case, the cover dimension in y is greater for the U device, as presented in Figure 2a. Although Sep

_{z}and Sep

_{y}take the same values in this work for the two cross section shapes studied, the cover thickness affects the magnetic field generated inside the device, and thus, the magnetic force acting on the beads. Therefore, in Figure 2b–d, we have also presented the magnetic force distribution in x and y directions inside both channels for the three tested lengths. To plot this distribution, we simulated the force acting on the particles that are located at the channel midplane, z = 0. Inspection of this plot indicated that, although the chip-magnet separation remains the same for both devices (same Sep

_{z}and Sep

_{y}), the thickness of the cover materials relative to, and combined with the depth of the channel, affects the magnitude of the magnetic force developed across the channel, impacting the performance of the separation. Thus, thick cover slides are not desirable when they are placed between the magnet and the particles to be separated.

^{−1}depending on the geometry. These hydrodynamic conditions ensure medium shear rates in the channels, under which blood can be treated as a Newtonian fluid. Nevertheless, shear stresses are still too low to observe hemolysis [64,65]. The calculated average velocity for every flow rate was used as an initial condition. As for the boundary conditions, a no slip condition (zero velocity) was applied along the microchannel walls while at the outlet, we used a calculated outflow boundary. Beads were introduced into the domain at a constant concentration of 0.1 g·L

^{−1}, which corresponds to a flow of 10–2500 particles·s

^{−1}, depending on the fluid flow rate; they were randomly injected through the cross section of the blood inlet as seen in Figure 1a, with a velocity equal to the blood stream. The simulation time was kept below 15 s for all the cases, which allowed us to track the path of 100–400 beads.

#### 2.3. Dimensionless Analysis

_{res}) and the time they require to travel from the blood to the buffer solution considering that they move completely perpendicular to the flow direction (t

_{m}) [48,67]. Apart from the variables and parameters included in J, the channel length (L)/width-depth (D

_{h}) ratio, defined as the aspect ratio, is included in θ. Hence, this design parameter can be written as:

## 3. Results and Discussion

#### 3.1. Influence of Microchannel Cross Section Shape

^{−1}) is applied at the inlets.

**v**

_{p}(flow into the figures) has a greater magnitude for the R channel, implying that, higher recoveries can be achieved within the R device. Thus, in rectangular channels the operating conditions acting on the beads seem to be more favorable for particle magnetophoresis than in U-shaped devices. In other words, for obtaining the same particle capture, the inlet flow rate in U-shaped channels must be reduced.

#### 3.2. Influence of Microchannel Length

**F**

_{m}for the three tested lengths is approximately 0.013 nN in the U-shaped and 0.038 nN in rectangular devices.

_{f}values, which are one or two orders of magnitude lower than those observed for U-shaped devices with the same length. Comparing channels with R

_{f}values of the same order of magnitude, that is 10 mm long rectangular and 2 mm long U-shaped chips, it can be noticed that the R-10mm device allows the management of flow rates 100 times higher than U-2 mm while attaining the entire bead recovery, which implies a significant enhancement in the system throughput.

#### 3.3. Dimensionless Channel Design

**F**

_{m}is fixed for each cross section shape. Furthermore, complete particle capture is achieved working in hydrodynamic conditions (i.e., drag forces larger than magnetic ones) for all the systems (J

_{2}

^{R}= 0.054; J

_{5}

^{R}= 0.022; J

_{10}

^{R}= 0.011; J

_{2}

^{U}= 0.084; J

_{5}

^{U}= 0.017; J

_{10}

^{U}= 0.008). In addition, we found that the J value that corresponds to complete particle recovery is similar for both devices; as seen in Figure 6, the recovery as a function of J is dependent on L but independent on the cross sectional shape. Since the J parameter scales directly proportional to the magnetic and inversely to the drag force (Equation (13)), similar J values for each channel length stem from the compensation of the higher values of

**F**

_{m}and

**F**

_{drag}for rectangular channels in comparison to the U-shaped. Additionally, due to the similarity of the magnetic force exerted on the particles regardless the channel length for each cross section shape, the J parameter necessary to achieve complete recovery decreases as the channel length is increased. This is due to the fact that, by using longer devices, the applied flow rate can be also increased, and thus the fluid velocity inside the channels, which leads to higher drag forces (Figure S2). Therefore, optimum systems will be those that allow complete bead recovery at the smallest J, since the throughput could be enhanced up to 5 times by working with 10 mm R (J

_{10}

^{R}= 0.011) compared to 2 mm R (J

_{2}

^{R}= 0.054) channels, or 10 times when 10 mm U (J

_{10}

^{U}= 0.008) devices are used instead of 2 mm U (J

_{2}

^{U}= 0.084).

_{h}ratio is compensated with the decrease in the J parameter, due to the rise of the fluid drag force which results from the larger velocities that could be applied to obtain the same bead collection. For this reason, particle recovery as a function of θ does not change with the channel length. Thereby, regardless the aspect ratio, with rectangular section channels complete recoveries are accomplished approximately 2 times faster in comparison to U-shaped ones, as derived from the linear relationship (R

^{2}≈ 0.98) between the θ parameter and the percentage of particle capture for all the geometries except for the 2 mm long U device. This fact leads to lower θ values for rectangular channels (θ

^{R}= 0.45) than for U-shaped ones (θ

^{U}= 1.73) for attaining complete particle capture. Furthermore, for comparable aspect ratios, such as 5 mm long U (L/D

_{h}= 51) and 10 mm long R (L/D

_{h}= 42) or 2 mm long U (L/D

_{h}= 21) and 5 mm long R (L/D

_{h}= 21), the values of the J parameter that yield entire capture are lower for rectangular channels (J

_{5}

^{U}= 0.017; J

_{10}

^{R}= 0.011; J

_{2}

^{U}=0.084; J

_{5}

^{R}= 0.022), and hence lower θ. It should be noted that because of the limitations for producing deeper glass devices, the cross section area of U is lower than for rectangular channels, since the limitations regarding the channel depth in R chips are less restrictive. Thus, similar aspect ratios entail longer R than U channels. Consequently, the higher magnetic forces and residence time combined with the different velocity distribution of rectangular channels make possible the use of higher velocities (i.e., higher drag forces) than in U-shaped, thus reinforcing the evidence that rectangular sections promote the improvement of the system throughput. As such, systems that bear complete recoveries with the lowest θ are preferred.

^{−1}). It should be mentioned that the functional channel volume corresponds to the branch of the Y-Y shaped device whereby blood flows; since the microdevices considered in this study are symmetric in the z-direction (see Figure 1a), the total system volume is twice the channel volume. As presented in Figure 7a, by increasing the channel volume (microchannel lengthening), the particle residence time in the chip also increases, which promotes their collection at the buffer channel outlet (Figure S2). Furthermore, the analysis of the functional volume of the channels reveals that U devices are more efficient since they deliver higher recoveries than R devices when comparing the same volume. In this regard, for having comparable channel volumes, the length of U chips must be five times higher than for rectangular, due to the smaller cross section shape of U devices that arises from the weaknesses of the fabrication method as previously stated. Under these conditions, U-shaped chips provide higher residence time to the particles, which fosters their capture, thus leading to higher recoveries than in rectangular devices, although the magnetic force exerted in the rectangular channel is higher. This means that the effect of channel elongation has a greater impact than the application of higher magnetic forces. Similarly, for attaining comparable bead recoveries, the length of U channels should be approximately 2.5 times higher than the length of R channels. However, due to the lower deepness of U-shaped channels in comparison to rectangular, the volume of U is lower than of R devices, which demonstrates the more efficient performance of high aspect ratio (L/D

_{h}) channels in terms of bead capture, given that they render similar recoveries with lower channel volumes.

^{−1}, whereas U-shaped devices must be elongated to a greater extent (or several channels in series must be used) in order to increase the residence time of the beads in the device and thus promote their entire recovery (Figure S2). In other words, the particle residence time in U channels should be approximately twice than that in rectangular ones for achieving similar recoveries in both types of channels when applying the same inlet velocity; thus, rectangular devices allow attaining complete particle capture while treating blood flow rates 4.4 times higher than in U channels. Due to the higher residence time required by U-shaped chips, several devices have to be arranged in parallel for concurrently delivering comparable recoveries and throughputs to rectangular channels. Accordingly, for successfully treating a flow rate of 1 mL of blood per hour while capturing entirely the beads, one 10 mm long R device or six 10 mm long U devices, operating in parallel, are required. However, it should be taken into account that the use of several microseparators either in series or in parallel, which is needed so that the performance of U devices is similar to that of rectangular ones, may lead to higher fabrication and operational costs, thus negatively affecting the system efficiency.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Top view of the microfluidic-magnetophoretic device, (

**b**) Schematic representation of the channel cross-sections studied in this work, and (

**c**) the magnet position relative to the channel location (Sep

_{y}and Sep

_{z}are the magnet separation distances in y and z, respectively).

**Figure 2.**(

**a**) Channel-magnet configuration and (

**b**–

**d**) magnetic force distribution in the channel midplane for 2 mm, 5 mm and 10 mm long rectangular (left) and U-shaped (right) devices.

**Figure 3.**(

**a**) Velocity distribution in a section perpendicular to the flow for rectangular (left) and U-shaped (right) cross section channels, and (

**b**) particle location in these cross sections.

**Figure 4.**Influence of fluid flow rate on particle recovery when the applied magnetic force is (

**a**) different and (

**b**) equal in U-shaped and rectangular cross section microdevices.

**Figure 6.**Influence of (

**a**) magnetic and fluidic forces (J parameter) and (

**b**) channel geometry (θ parameter) on particle recovery. Note that U-2mm does not accurately fit a line.

**Figure 7.**Dependence of bead capture on the (

**a**) functional channel volume and (

**b**) particle residence time (t

_{res}). Note that in the curve fitting expressions V represents the functional channel volume and that U-2mm does not accurately fit a line.

Rectangular Shape (R) | U Shape (U) | |||||
---|---|---|---|---|---|---|

L (mm) | 2 | 5 | 10 | 2 | 5 | 10 |

W (µm) | 300 | 300 | 300 | 280 | 280 | 280 |

H (µm) | 200 | 200 | 200 | 60 | 60 | 60 |

D_{h} (µm) | 240 | 240 | 240 | 97 | 97 | 97 |

L/D_{h} | 8 | 21 | 42 | 21 | 51 | 103 |

Volume (mm^{3}) | 0.12 | 0.3 | 0.6 | 0.03 | 0.08 | 0.15 |

R_{f} (10^{12} Pa·s·m^{−3}) | 0.33 | 0.83 | 1.65 | 6.46 | 16.14 | 32.29 |

**Table 2.**Comparative analysis of all channel configurations for the same inlet fluid velocity (1.92 cm·s

^{−1}).

Particle Recovery (%) | Throughput (µL·s^{−1}) | J (-) | Θ (-) | t_{res} (s) | |
---|---|---|---|---|---|

U-2 mm | 7.09 | 0.1 | 0.004 | 0.086 | 0.1 |

U-5 mm | 24.78 | 0.1 | 0.004 | 0.216 | 0.26 |

U-10 mm | 53.5 | 0.1 | 0.004 | 0.432 | 0.52 |

R-2 mm | 21.57 | 0.44 | 0.012 | 0.103 | 0.11 |

R-5 mm | 63.02 | 0.44 | 0.012 | 0.258 | 0.26 |

R-10 mm | 100 | 0.44 | 0.012 | 0.516 | 0.53 |

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## Share and Cite

**MDPI and ACS Style**

González Fernández, C.; Gómez Pastora, J.; Basauri, A.; Fallanza, M.; Bringas, E.; Chalmers, J.J.; Ortiz, I. Continuous-Flow Separation of Magnetic Particles from Biofluids: How Does the Microdevice Geometry Determine the Separation Performance? *Sensors* **2020**, *20*, 3030.
https://doi.org/10.3390/s20113030

**AMA Style**

González Fernández C, Gómez Pastora J, Basauri A, Fallanza M, Bringas E, Chalmers JJ, Ortiz I. Continuous-Flow Separation of Magnetic Particles from Biofluids: How Does the Microdevice Geometry Determine the Separation Performance? *Sensors*. 2020; 20(11):3030.
https://doi.org/10.3390/s20113030

**Chicago/Turabian Style**

González Fernández, Cristina, Jenifer Gómez Pastora, Arantza Basauri, Marcos Fallanza, Eugenio Bringas, Jeffrey J. Chalmers, and Inmaculada Ortiz. 2020. "Continuous-Flow Separation of Magnetic Particles from Biofluids: How Does the Microdevice Geometry Determine the Separation Performance?" *Sensors* 20, no. 11: 3030.
https://doi.org/10.3390/s20113030