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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