# Microfluidic immunomagnetic multi-target sorting - a model for controlling deflection of paramagnetic beads

*Written by Kevin Tian, AP 225, Fall 2011*

--Ktian 06:24, 12 September 2011 (UTC)

**Title:** Microfluidic immunomagnetic multi-target sorting – a model for controlling deflection of paramagnetic beads

**Authors:** Scott S. H. Tsai, Ian M. Griffiths and Howard A. Stone

**Journal:** Lab on a Chip, 2011, 11, 2577-2582

**Keywords:** microfluidics, electromagnetism, scaling, fluid mechanics, sorting, PDMS

## Paper Summary

The concept of immunomagnetic separation involves using functionalized paramagnetic beads that are capable of specific binding to desired targets, the result of which can be separated from its surrounding components via application of a magnetic field. These targets are often cells, proteins and various other biological components of interest. Such methods can be prove useful for cell capture, disease detection, cleansing of the blood. The basic technique of immunomagnetic separation has been shown to adequately perform the separation.

However one area that hasn't been aptly explored is the introduction of multi-target sorting, where microfluidic systems and paramagnetic beads of varying properties can be utilized to sort out several targets simultaneously. This paper investigates the dependence of bead deflection on bead size and susceptibility, magnet strength, fluid speed and viscosity as well as microfluidic device geometry, and develops a figure of merit that is potentially useful for immunomagnetic multi-target sorting.

## Experimental Design

**Channels**

The schematic design of the microfluidic channels is provided in figure 1(a). The channels were made using Polydimethylsiloxane (PDMS, Sylgard 184 Silicone Elastomer Kit from Dow Corning) via standard soft lithography processes. A 1.5mm thick layer of patterned PDMS was bonded to a 1.5mm non-patterned PDMS layer to enclose the channels. A rectangular slot was made to contain the permanent magnet. The entire system was then bonded to a glass microscope slide via plasma treatment.

All channels are <math>50 \mu m</math> in height, with the separation channel dimensions being <math>615 \mu m</math> by 20mm.

The three inlets pumped fluid in at constant rate, with outer inlets being pure fluid, the inner inlet a suspension of beads at a density of 3x<math>10^7</math> beads <math>mL^{-1}</math>

**Magnetic Components**

The paramagnetic beads are formed by polystyrene embedded with <math>Fe_2 O_3</math> nanoparticles and are suspended in DI water. Susceptibility is unknown but calculated after data is analyzed by use of models. It is assumed constant for each bead size. Bead size is one varied parameter.

The applied magnetic field is supplied with an <math>Nd_2 Fe_14 B</math> permanent magnet with magnetization M=<math>10^6 A m^{-1}</math>. The magnets are of cubes of dimension 3.175mm. These are stacked pole to pole in order to alter the length of the magnet in the system.

**Parameters**

- Average fluid speeds in the separation channel, <math>u_0</math>, varied from 1-15 cm <math>s^{-1}</math>
- Bead radii, <math>a</math>, is either 0.5 or 1.4 <math>\mu m</math>
- Magnet stack size, <math>l_m</math>, of either 6.4mm or 9.5mm (2-3 magnets stacked)
- The distance of the separation channel from the magnet center, <math>y_m</math>
- Distance from center-line of separation channel to channel wall, <math>y_c</math> of either 3mm or 4.5mm

## Results

**Observations**

- With no magnetic field applied (control) all the beads simply flow down the center-line of the separation channel (the spread was minimal, ~8<math>\mu m</math>)
- When magnetic fields were applied, as the fluid speed was decreased, the beads were deflected further away from the center line
- The 1.4<math>\mu m</math> beads have larger deflections than the 0.5<math>\mu m</math> beads
- Increasing the magnet stack length, <math>l_m</math> increases deflection (as it increases the magnetic moment of the magnet)
- Increasing magnet distance from the center-line of the channel, <math>y_m</math>, drastically decreases deflection.

Several frames that are representative of the observations above are present in Figure 2. Figure 3 depicts the quantitative measurements made from the experiment for the variations on the specified parameters, plotting the deflection against fluid flow rate.

## Discussion

**Scaling Analysis**

If we were to consider the forces that are acting upon the beads there are many possible considerations. However we note the following aspects of the experiment:

- The Reynolds number of the beads is of order <math>10^{-2}</math>, thus inertial forces can be safely neglected
- The diffusion coefficient of the beads due to thermal fluctuations is of order <math>10^{-12}m^2s^{-1}</math>
- This yields effective diffusion due to thermal fluctuations of order <math>10^{-1} \mu m</math> at room temperature
- This is significantly smaller than the average deflection of the beads, thus one can ignore thermal diffusion

- The authors choose to ignore bead-wall and bead-bead hydrodynamic interactions due to the assumption that the bead suspension is dilute

This leaves the dominant forces on the bead being Stokes drag and magnetic force (<math>F_d, F_m</math> respectively), which in equilibrium must satisfy: <math>F_d+F_m=0\,\!</math>

By noting that Stokes drag has the form, <math>F_d=-6 \pi \eta a(\nu-u)\,\!</math>, and that the magnetic force is of the form, <math> F_m = 4 \pi a^3 \mu_{0} \left ({\chi \over {\chi + 3}} \right) \nabla H^2</math>, we can rearrange the equilibrium equation to solve for velocity of the bead to obtain the following:

<math>v = {2 \over 3} {{\mu_0 a^2} \over \eta} {\chi \over {\chi +3}} \nabla H^2 + u</math>

If we now consider the y-component of the above equation and divide by the channel width, <math>y_c</math>, we obtain a dimensionless figure of the exit position of the beads. Essentially...

<math>y'_e = {\bar{y_e} \over y_c} = \kappa \Omega</math>, where, <math>\Omega = {\chi \over {\chi+3}} {{\mu_0 a^2 l_{m}^3 w_{m}^2 h_{m}^2 M^2} \over {y_c \eta u_0 y_{m}^7}}</math>

and <math>\kappa</math> is a dimensionless scaling constant.

We note that the predicted exit position of the bead is linear with respect to <math>\Omega</math> and thus we can observe the dependencies of the deflection of the bead with respect to the various parameters in the expression of <math>\Omega</math>. For example we note that there is an inverse relation to the 7th power between distance of the magnet from the center line (<math>y_m</math>) and final deflection (<math>y'_e</math>). Thus slight changes to <math>y_m</math> will drastically affect the ultimate deflection. Similar observations can be made for other parameters.

Regarding the scaling factor, <math>\kappa</math>, we note that detailed calculations are possible but are detailed more closely in the paper. The essential result is that numerical calculations can be made to determine <math>\kappa</math> by observing the partial derivative of magnetic field H and rewriting the velocity equation above in a dimensionless form.

**Comparisons**

The dimensionless parameter <math>\Omega</math> can be considered to be measure of the expected deflection of a magnetic bead in the system. Observing Figure 4, where <math>y'_e</math> is plotted against <math>\Omega</math> for a computed value of <math>\kappa \approx 0.01</math>. Data points are essentially replotted from Figure 3 (the same key holds).

We observe that the data points essentially collapse onto the line with slope equal to <math>\kappa</math>, which quantitatively supports the expression, <math>y'_e = \kappa \Omega</math>.

## Conclusions

It has been demonstrated that the various parameters described above present in simple a microfluidic system made for immunomagnetic separation can be approximated to have a mathematical relation described in the definition of <math>\Omega</math>. Based on that analysis it was found that

- Bead size and susceptibility
- Magnet size and magnetization
- Fluid speed and viscosity
- Channel geometry

were the most important parameters with regards to changing the exit position of the beads following the sorter. Since the parameter <math>\Omega</math> has been shown to agree well with the linear relationship provided above, it is possible to use this as a design parameter to develop a multi-target sorter, by designing specific targets to be deflected specific amounts. By varying the bead parameters it is very well possible to alter the deflection in this way, paving the way for multi-target sorting.