Overview

Magnetic levitation is dependent on many aspects of

system and assay design. The LeviCell system has been

designed to provide maximum flexibility to customers

exploring new applications. Therefore, understanding

the parameters that provide such flexibility is important

to experimental design and outcome. The underlying

physics that determines how particles (cells for

example) behave in the LeviCell system can be

described by a few key equations and parameters.

Some parameters are fixed and inherent to the design

of the system, while others are controlled by the user.

In combination, these parameters determine the ability

to identify and separate particles with different

magnetic and density signatures.

System Introduction

The core of the LeviCell system is a long channel

sandwiched between two magnets of fixed strength

which create a magnetic field within the channel. Cells

enter on one end, suspended in a paramagnetic

solution. The field and the paramagnetic properties of

the solution cause the cells to levitate. As they flow

through the channel, they approach an equilibrium

position or levitation height, where the force applied

from the magnetic field balances the residual

gravitational force. Because the residual gravitational

force depends on particle density, the levitation height

is different for particles of different densities. The

LeviCell system provides a unique way to differentiate

cells or particles (e.g. beads) based on unique physical

properties — their density and magnetic permeability

— and ultimately separates or enriches cells with

varying levitation profiles for further study. The system

can operate in label-free mode or in the presence of

fluorescently stained or tagged objects to aid in

analysis.

Physics of Levitation

To help with experimental design and interpretation, it

is useful to describe the physics of levitation in the

LeviCell system and the key parameters that impact

behavior of objects subjected to such a system. As

described above, the system differentiates cells of

different levitation profiles by taking advantage of

gravity and the small forces cells experience in the

presence of a magnetic field. The modeling and

equations below assume that magnetic properties of

the cell are negligible for simplicity and broad

application. The buoyant force is proportional to the

weight of the fluid that is displaced, and the residual

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Optimizing Magnetic Levitation:How to Tune Parameters for Best Results

gravitational force is shown (eq1). Similarly, in the

presence of the magnetic field, cells experience a force

proportional to the volume of the paramagnetic fluid

they displace (eq2). At equilibrium, these forces cancel

each other, and the equilibrium height depends on

how the magnetic field varies across the channel and

the susceptibility of the paramagnetic fluid, as well as

the density of the cell (eq3). (One can use the analogy

density:residual gravitational force as fluid

permeability:magnetic force.)

Levitation Height, Dynamic Range, and Resolution

The strength of the magnets and their position relative

to the channel determine the magnetic field and,

ultimately, the levitation height and time taken to reach

equilibrium. The magnet locations are fixed within the

design and set the dynamic range of cell density that

can be separated by the system, with one caveat. The

concentration of paramagnetic fluid also changes the

magnetic force applied, and thus influences the

equilibrium levitation height and time to reach it.

Increasing the paramagnetic fluid concentration

increases the dynamic range within the channel,

allowing the detection of a broader range of densities

(fig1). However, this also results in reduced resolution,

so densities that are closer in value are harder to

separate. Adjusting the concentration gives the user

flexibility to adjust the dynamic range and resolution of

the system depending on the application needs. A

titration of the concentration of levitation agent is

recommended when characterizing the density

distribution of a new cell population to determine ideal

conditions (ref user guide).

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Figure 1 - Dynamic range and resolution as a function of levitation buffer concentration. Using COMSOL Multiphysics Simulation

software, the magnetic field is simulated for the LeviCell system geometry. With this information, cell density is calculated for a

given levitation height and concentration of levitation agent in the levitation buffer (eq 4). The levitation buffer density varies

little with concentration changes of the levitation agent, 1.02g/cc for 30mM levitation agent) to 1.04g/cc for 100mM levitation

agent. Note that for particles with density less than that of the levitation buffer, decreasing the concentration of levitation agent

raises the levitation height, whereas for particles with density greater than that of the levitation buffer, decreasing the

concentration lowers the levitation height.

Equilibration Time

It is important to understand the parameters that affect

the time needed for a cell to reach its equilibrium

position in order to: (1) understand the flow rate best

suited to an application; and (2) avoid mis-interpreting

experimental results, for example, interpreting a wide

range in levitation heights as density variation rather

than incomplete equilibration.

The magnet strength and position relative to the

channel affect equilibration time by changing the field

and force, but as these are fixed parameters in the

system they will not be discussed. The levitation agent

concentration also affects equilibration time and

provides the user with more flexibility in designing the

experiment; increasing the concentration reduces the

equilibration time (fig2) almost linearly as the magnetic

force is increased (eq2).

The equilibration time also depends on the radius of

the particle, as it is subject not just to the gravitational

and magnetic forces, but also to drag force as it

traverses the channel. The equilibration time is

determined by the particle’s terminal velocity, which is

achieved when the drag force is equal to the

gravitational and magnetic forces combined. The

equilibrium time is therefore also dependent on the

viscosity of the solution (eq5), and this changes

negligibly with varying levitation agent concentration.

Increasing particle diameter reduces the time required

to reach equilibrium by nearly the square of the radius

(fig3). Note that the size of the particle does not affect

its final equilibrium position, which depends only on

density; it affects only how long it takes to reach

equilibrium.

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Figure 2 - Equilibration time dependence on levitation agent concentration. Using COMSOL Multiphysics Simulation software,

the levitation height for two particle sets with different levitation agent concentration is simulated for 20um diameter particles

and a total flow rate of 10ul/min. Twenty particles under each condition are randomly distributed across the height of the channel

at time zero. As they flow through the channel, they approach their equilibrium height at different rates depending on the

concentration of levitation agent. The final levitation height at equilibrium also differs between the two sets as a result of the

levitation agent concentration. A comparison of the distribution of each particle set at specific timepoints highlights the

difference in equilibration time, with the 50mM particle set having a standard deviation of levitation height of ~2% or ~65um

after 90s, compared to the 100mM particle set with a near equivalent levitation height range after 40s.

Sample Flow Rate

The total flow rate of the sample is another important

consideration. The sample needs sufficient residence

time within the magnetic field to achieve equilibrium; if

the flow rate is too fast, a wider distribution of

levitation heights will be observed, which may be

misinterpreted as density variation (fig 4). The

maximum flow rate for a given particle radius and

levitation buffer can be estimated based on the

channel geometry (eq6). To confirm that the total flow

rate chosen is appropriate for the sample, a user can

compare static levitation height with in-flow levitation

height. If the range in levitation height is unacceptably

high while flowing for the application, the user can

reduce the total flow rate and/or increase the levitation

buffer concentration for improved performance (ref the

user guide).

Another consideration regarding sample flow rate is

the differential flow required to separate cells of

different densities and levitation heights. The end of

the flow cell has a separating feature that divides

particles based on their levitation heights and the

differential flow ratio (upper outlet flow compared to

lower outlet flow, the sum is the total flow rate). In

other words, by making the upper outlet flow rate

higher than that of the lower outlet, cells of higher

density can be pulled into the upper outlet (fig5). The

upper and lower outlets of the flow cell have a smaller

cross-sectional area compared with the main

separation chamber and extend beyond the magnetic

field. Because they are smaller, the linear velocity

increases as particles approach these outlets, and their

trajectories change as users vary the ratio of upper to

lower flow rate. It is important to maintain the total

flow rate that allows for equilibration.

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Figure 3 - Equilibration time dependence on cell radius. Using COMSOL Multiphysics Simulation software, the levitation height

for two particle sets with either a 10 or 20 micron diameter, both with the same density (1.03g/cc), is simulated with 50mM

levitation reagent concentration and a total flow rate of 10ul/min. Twenty particles of each diameter are randomly distributed

across the height of the channel at time zero. As they flow through the channel, they approach their equilibrium height at

different rates depending on their diameter. A comparison of the distribution of each particle set at specific timepoints highlights

the difference in equilibration time, with the 10um diameter particle set having a standard deviation in levitation height of ~2% or

~65um after 350s, compared to a 20um diameter particle set with a near equivalent levitation height distribution after 90s.

Experimental Results

Using commercially available beads of known density,

levitation heights for four different bead densities were

measured on two systems as shown, compared with

simulation.

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Figure 4 - Particle trajectories along the channel X axis with varied flow rates. Using COMSOL Multiphysics Simulation software,

levitation height and associated particle density estimates were determined for a 10um diameter particle set (n=20) with 50mM

levitation agent concentration and various flow rates. The total flow rate determines the linear velocity of particles along the X-

axis. If a particle’s velocity through the channel separation region (ending near x=40mm) results in a shorter residence time than

needed to equilibrate, a larger distribution of levitation heights is observed. Note that for some applications a larger distribution

in levitation heights is an acceptable tradeoff for increased throughput.

Figure 5 - Particle trajectories through the channel as observed from the side (perpendicular to flow), with symmetric flow

rate (left image, top = 20ul/min = bottom) compared to asymmetric flow (right image, top = 30ul/min, bottom = 10ul/min).

Particle densities are 1.014g/cc (blue) and 1.053g/cc (burgundy), both densities with a 20um diameter. There are 20

particles simulated for each density, and a levitation agent concentration of 50mM.

Equations and References (eq1) Residual gravitational force on particle

V = volume of particle (m3)

g = 9.81 gravitational acceleration (m/s2)

ρp = particle density (kg/m3)

ρf = paramagnetic fluid density (kg/m3)

(eq2) Magnetophoretic force on particle

r = particle radius

μ0 = permeability of free space

χf = susceptibility of paramagnetic fluid =

constant * concentration of levitation agent

H = magnetic field

(eq3) Levitation Height Estimation

At equilibrium, the forces in equations 1 and 2 equate

to form the equation:

We can approximate ∇Hy2 as a linear function of y, the

distance along the gravitational axis, for our magnet

and channel geometry:

where SH is the “slope” or rate of change of ∇H2 across

the channel along y and is a constant for a given

hardware design – it scales with magnetic field strength

and also distance of the channel from magnet. At

equilibrium, y is yeq, the equilibrium or levitation

height.

From this, we find the relationship of levitation height

to particle density, and levitation agent concentration:

(eq4) Determining particle density based on levitation

height

Using COMSOL, we find that ∇Hy2 is not perfectly

linear across the channel and so apply a polynomial fit

to data generated from the model. The fit parameters

(constants a, b, c and d) combined with eq3 provide

the equation to determine particle density at a given

yeq.

(eq5) Drag Force

The drag force experienced by a particle of radius rp

and velocity v in a fluid of viscosity μ.

(eq6) Flow rate estimate based on equilibration time

The separation volume is the region in the channel that

is subject to the magnetic field before being split.

During the PrimeSample script, you may estimate the

time required to adequately levitate your sample, and

call this teq, the equilibrium time.

For example, for a 100ul separation volume, if you hold

your sample for 10min prior to flow to achieve a

minimum distribution acceptable for you application,

you can assume you need a total flow rate of

approximately 10ul/min.

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