Optimizing Magnetic Levitation:

Transcription

1 Optimizing Magnetic Levitation: How to Tune Parameters for Best Results Overview Magnetic levitation is dependent on many aspects of gravitational force. Because the residual gravitational system and assay design. The LeviCell system has been force depends on particle density, the levitation height designed to provide maximum flexibility to customers is different for particles of different densities. The exploring new applications. Therefore, understanding LeviCell system provides a unique way to differentiate the parameters that provide such flexibility is important cells or particles (e.g. beads) based on unique physical to experimental design and outcome. The underlying properties their density and magnetic permeability physics that determines how particles (cells for and ultimately separates or enriches cells with example) behave in the LeviCell system can be varying levitation profiles for further study. The system described by a few key equations and parameters. can operate in label-free mode or in the presence of Some parameters are fixed and inherent to the design of the system, while others are controlled by the user. fluorescently stained or tagged objects to aid in analysis. In combination, these parameters determine the ability to identify and separate particles with different magnetic and density signatures. System Introduction 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 The core of the LeviCell system is a long channel sandwiched between two magnets of fixed strength behavior of objects subjected to such a system. As described above, the system differentiates cells of which create a magnetic field within the channel. Cells different levitation profiles by taking advantage of enter on one end, suspended in a paramagnetic gravity and the small forces cells experience in the solution. The field and the paramagnetic properties of presence of a magnetic field. The modeling and the solution cause the cells to levitate. As they flow equations below assume that magnetic properties of through the channel, they approach an equilibrium the cell are negligible for simplicity and broad application. The buoyant force is proportional to the from the magnetic field balances the residual weight of the fluid that is displaced, and the residual position or levitation height, where the force applied 1

2 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). 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. 2

3 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 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. equilibration time (fig2) almost linearly as the magnetic force is increased (eq2). 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. 3

4 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. 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. 4

5 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 Xaxis. 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. 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. 5

6 Equations and References (eq1) Residual gravitational force on particle From this, we find the relationship of levitation height to particle density, and levitation agent concentration: V = volume of particle (m 3 ) g = 9.81 gravitational acceleration (m/s 2 ) ρ p = particle density (kg/m 3 ) ρ f = paramagnetic fluid density (kg/m 3 ) (eq2) Magnetophoretic force on particle (eq4) Determining particle density based on levitation height Using COMSOL, we find that H 2 y 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 y eq. r = particle radius μ 0 = permeability of free space χ f = susceptibility of paramagnetic fluid = constant * concentration of levitation agent H = magnetic field (eq3) Levitation Height Estimation (eq5) Drag Force The drag force experienced by a particle of radius rp and velocity v in a fluid of viscosity μ. At equilibrium, the forces in equations 1 and 2 equate to form the equation: (eq6) Flow rate estimate based on equilibration time We can approximate H y 2 as a linear function of y, the distance along the gravitational axis, for our magnet and channel geometry: 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 t eq, the equilibrium time. where S H is the slope or rate of change of H 2 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 y eq, the equilibrium or levitation height. 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. 6