Correlation of simulation/finite element analysis to the separation of intrinsically magnetic spores and red blood cells using a microfluidic magnetic deposition system

Abstract

Magnetic separation of cells has been, and continues to be, widely used in a variety of applications, ranging from healthcare diagnostics to detection of food contamination. Typical, these technologies require cells labeled with antibody magnetic particle conjugate and a high magnetic energy gradient created in the flow containing the labeled cells (i.e. a column packed with magnetically inducible material), or dense packing of magnetic particles next to the flow cell. Such designs while creating high magnetic energy gradients, are not amenable to easy, highly detailed, mathematic characterization.

Our laboratories have been characterizing and developing analysis and separation technology that can be used on intrinsically magnetic cells or spores which are typically orders of magnitude weaker than typically immunomagnetically labeled cells. One such separation system is magnetic deposition microscopy (MDM) which not only separates cells, but which deposits cells in specific locations on slides for further microscopic analysis. In this study, the MDM system has been further characterized, using finite element and computational fluid mechanics software, and separation performance predicted, using a model which combines: 1) the distribution of the intrinsic magnetophoretic mobility of the cells (spores), 2) the fluid flow within the separation device, 3) accurate maps of the values of the magnetic field (max 2.27 T), and magnetic energy gradient (max of 4.41 T²/mm) within the system. Guided by this model, experimental studies indicated that greater than 95 percent of the intrinsically magnetic bacllus spores can be separated with the MDM system. Further, this model allows analysis of cell trajectories which can assist in the design of higher throughput systems.

Introduction

The traditional magnetic cell separation approach is to label the targeted cell(s) with a magnetic particle, typically facilitated by an antibody conjugated with a magnetic particle. The relative simplicity of magnetic cell separation has facilitated a significant commercial industry supporting magnetic separation systems for a variety of uses, ranging from human diagnostics to “sample prep” of cell suspensions for further analysis/studies.

In contrast to the significant investment in equipment and time required to perform a targeted magnetic separation, or subsequent analysis of the magnetically labeled cells using well-known systems in detection technology, such as the CellSearch™ system (Riethdorf et al., 2007), we have been developing over a number of years, a system referred to as magnetic deposition microscopy (MDM), which deposits magnetic cells on specific locations on a slide; the location of this deposition controlled by the design of a high, well characterized, magnetic energy gradient (Fang et al., 1999; Karl et al., 2008; Melnik et al., 2007; Zbrowski et al., 1995). The diagram of MDM system is presented in Figure 1A; Figure 1B presents an enlargement of the magnetic deposition zone. In contrast to most other high field and high gradient magnetic cell separation systems, HGMS (i.e. Miltenyi columns; Earhart et al., 2014; Jin et al., 2012), in this design an open flow field is used in which the magnetic field and gradient is created external to the fluid flow thereby reducing the potential for clumping and trapping of non-targeted cells. (Jin et al., 2012)

Figure 1.

Schematic diagram of magnetic deposition system.

In addition to demonstrating the potential to separate, through deposition, magnetically labeled lymphocytes and a magnetically labeled breast cancer cell line spiked into human blood, (Fang et al., 1999; Zborowski et al., 1995) we have demonstrated it is possible to deposit deoxygenated red blood cells (RBC), and several strains of Bacillus spores. (Karl et al., 2008; Melnik et al., 2007) This intrinsic magnetism of deoxygated RBCs was originally reported by Linus Pauling in 1936, (Pauling and Coryell, 1936; Pauling and Coryell, 1936) and subsequently quantified, in terms of both the mean and distribution, by Zborowski et al. (2003). Besides the magnetic properties of iron in hemoglobin, the element manganese, in several of its oxidation states, has significant magnetic susceptibility and has been shown to be present in a number of bacteria (Hastings and Emerson, 1986; Sprio et al., 2010). When in the sporulated state, several forms of Bacillus concentrate this manganese in and around the spores, thereby imparting a significant magnetic moment (Sun et al., 2011; Sun, 2010).

The ability to separate cells based on an intrinsic magnetic moment presents interesting possibilities. For example, the ability to remove RBCs magnetically, without traditional density separation or RBC lysis, is appealing when one is interested in further analysis of the nucleated blood cell population (Moore et al., 2013). Alternatively, RBC can be the focus of the separation and analysis, such in the case of malaria infection (Moore et al., 2006).

Other than applications of separation and detection of cells in blood, the detection of bacteria and spores in food also has important implications. Most bacteria can be killed during sterilization process; however, spores, resistant to heat and other preservation treatments in comparison to vegetative cells, require high temperatures and long heating times for inactivation. Such enhanced sterilization processes are costly and detrimental to the nutritional and organoleptic quality of most food products (Kort et al., 2005). Unfortunately, many food poisoning cases were caused by canned food, among which spore forming bacteria, i.e. Clostridium botulinum, commonly contribute (Devers et al., 2010). Bacillus cereus, a spore-forming Gram-positive strain, is another pathogenic bacterium that can cause food poisoning and produces gastrointestinal diseases.

In this study, we chose to further characterize the performance of the MDM system with a combination of finite element mathematical models that take into consideration not only the non-linear nature of the magnetic energy gradient, but the significant distribution of the intrinsic magnetic susceptibility of RBCs, and Bacillus spores, both prior to and after sterilization. Further, using this model, predictions of the performance capability of the system is presented as well as the potential to scale up such a system will be discussed. Specifically, the current way to remove RBCs from clinical samples (i.e typical 5 ml blood draws) is to centrifuge the sample; we suggest that it is reasonable to scale the results from the study presented here to remove the RBC from a 5 ml blood draw in 5 minutes if the RBCs are first deoxygenated.

Theoretical Analysis for trajectory simulation

The capability of modern finite element software, and computer power, allows highly detailed magnetic field maps to be combined with laminar flow conditions to theoretically predict the movement (trajectory) of magnetic cells and particles in some magnetic cell separation systems. The advantages of such models present the possibility to predict system performance as well as provide guidance in scaling up systems as well as predicting performance with cells of varying magnetic susceptibility. Unfortunately, many magnetic cell separation systems utilize the general high gradient magnetic separation (HGMS) approach which requires magnetically inducible material (i.e. steel wire or beads) packed within the flow path of the cell suspension (i.e. MACS columns). While relatively simpler to manufacture and operate, the combination of complex flow patterns and magnetic energy gradients within the HGMS are still beyond first principle, theoretical solutions; consequently only correlations can be used.

In contrast, for a number of years we have been designing, theoretically analyzing, and simulating, “open channel” (no magnetic elements directly in flow path) magnetic cell separation systems (Zborowski et al., 1995; Moore et al., 1998; Williams et al., 1999; Schneider et al., 2010). For example, while the magnetic force is non-linear, our MDM system is based on specific designs which are amenable to numerical solutions of the basic governing equations.

For the analysis and simulation presented in this work, the region of interest, ROI, is the inside of the flow channel within which the cells flow. For a portion of this region, a non-linear magnetic field is applied. Figure 1 presents a schematic diagram of the overall system, and Figure 2 presents a series of 2-D images which present an enlargement of this flow channel in the region upon which the magnetic field is applied. These images include a heat map of the magnetic field, B, as a function of position as well as the root mean square of B, ∇B², as a function of position

Figure 2.

Contour plots of magnetic field, B, (lower), and magnetic energy gradient, ∇B², (upper) to which cells in the flow channel of the MDM system are subjected. All dimension and location are to scale, in mm, expect for the upper contour plot which is offset to allow comparison with the lower contour plot. The arrows in the upper magnetic energy plot indicated the direction a cell, with a magnetic susceptibility greater than the suspending buffer, will experience a magnetic force.

The cells flowing through this channel are acted upon by magnetic, net buoyant, and drag forces. Cell trajectories result from the balance of these forces superimposed upon the steady, fully-developed, laminar flow in the rectangular duct. Cells with a net magnetic susceptibility greater than the suspending buffer, (χ_cell – χ_buffer) > 0, are attracted to increasing values of ∇B².

Capture rates are estimated from particle trajectories seeded at various y positions in the channel at the entrance to the field. The magnetic force, F_mag, is given by:

$$\stackrel{\rightharpoonup }{F_{mag}}=\stackrel{\rightharpoonup }{M}\stackrel{\rightharpoonup }{H}V\nabla \stackrel{\rightharpoonup }{B}$$ (1)

$$\stackrel{\rightharpoonup }{M}(\stackrel{\rightharpoonup }{B})=\chi (\stackrel{\rightharpoonup }{H})\stackrel{\rightharpoonup }{H}=\frac{\chi (\stackrel{\rightharpoonup }{H})\stackrel{\rightharpoonup }{B}}{{\mu }_0}$$ (2)

where $\stackrel{\rightharpoonup }{M}$ is the magnetization of the entity, $\stackrel{\rightharpoonup }{B}$ is the magnetic field induction (T), $\stackrel{\rightharpoonup }{H}$ is the strength of the applied magnetic field (A/m), μ₀ is the magnetic permeability of a vacuum (4π × 10⁻⁷ T m A⁻¹), and $\stackrel{\rightharpoonup }{B}={\mu }_0\stackrel{\rightharpoonup }{H}$ (S.I units used in this report). Since the magnetization of the spores and the red blood cells does not saturate in the range of magnetic field induction used, the linear representation of $\stackrel{\rightharpoonup }{M}$ presented in Equation 2 (with a constant value of χ) can be used unlike iron oxides (typically used in labeling of cells with antibody bead conjugates) that saturate at magnetic fields above 0.1T (Jin et al., 2008). We are further assuming that magnetic susceptibility is an average value of the molecular contributions of Mn atoms within the spores, or the hemoglobin molecules in the RBC’s. This is in contrast to recent models for magnetic separation of nanoparticles which takes into consideration Neil relaxation mechanism or a Brownian mechanism which are proposed to reduce the effective magnetic moment (Zhao et al. 2017). With these assumptions, Equations 1 and 2 can combined to give:

$$\stackrel{\rightharpoonup }{F_{mag}}=\chi \mathrm{V}\frac{\nabla \stackrel{\rightharpoonup }{B}}{2{\mu }_0}=\chi V\stackrel{\rightharpoonup }{S_m}$$ (3)

This magnetic force can be considered as a body force in a traditional force balance equation for the motion of a spherical particle in a fluid, such as the Basset-Boussienseq-Oseen, BBO, equation:

$$\begin{gathered} \frac{\pi D^3}{6}\left({\rho }_p+{\rho }_f\right)\frac{d\stackrel{\rightharpoonup }{u_p}}{dt}= \\ \left({\chi }_p-{\chi }_f\right)\frac{\pi D^3}{6}\nabla \stackrel{\rightharpoonup }{B}-3\pi D{\mu }_f\stackrel{\rightharpoonup }{u_p}-\frac{\pi D^3}{6}g\left({\rho }_p-{\rho }_f\right)-\frac{3}{2}\pi {\mu }_f{D}^2\left(\frac{1}{\sqrt{\pi v_f}}\right){\int }_{-\infty }^t\frac{\frac{d\stackrel{\rightharpoonup }{u_p}}{dt}(t_1)}{\sqrt{t-t_1}}d{t}_1 \end{gathered}$$ (4)

where u_p is the velocity of the particle (cell), ρ_f and ρ_p are the density of the fluid and particle (cell), g is acceleration of gravity, and μ_f and v_f are the dynamic and kinematic viscosity, respectively. The term on the left hand side corresponds to the “acceleration term”, corrected for the presence of “virtual mass”, the first term on the RHS is the magnetic body force term, the second term is the stokes drag term, the third term is the buoyancy term, and the fourth term is the Basset history effect term. Beyond these “corrections” to the traditional force balance (virtual mass and Basset history), other have suggested that this balance equation should account for hydrodynamic lift forces, and pressure gradient effects (Tchen 1947; Kim et al 2012).

Like most models/simulations, simplifications are made to facilitate easier solving of the given problem. Our laboratories have published a significant number of studies which not only characterize the magnetic susceptibility, and subsequent movement, of labeled and unlabeled cells, but also compared these experimental studies to purely theoretical calculations. Most relevant to this current study, we have publish studies, which among other aspects, compares the experimentally measured magnetic velocity of RBCs to theoretically predicted magnetic velocities based on independently determined, fundamental properties of human RBCs. In one study (Zborowski et al.) this mean, measured, magnetic velocity of deoxygenated red blood cells was 95 percent of the theoretically predicted value.

These magnetic velocities measurements were made in an instrument referred to as a Cell Tracking velocimetry, CTV, in which the value of $\nabla \stackrel{\rightharpoonup }{S_m}$ is nearly constant (Nakamura et al. 2001) in the primary direction of movement, allowing the vector considerations to be reduced to a scalar. Consequently, this F_mag is balanced by an opposing Stokes^’ drag force (all other effects in Equation 3 are neglected). In this case, Equation 3 reduces to the magnetically induced velocity, u_m, given by:

$$u_m=m{S}_m=\frac{\mathrm{\Delta }\chi D^2}{18\eta }\frac{\nabla B^2}{2{\mu }_0}$$ (5)

where m is the magnetophoretic mobility of the cell or spore, S_m is the magnetostatic energy gradient, Δ𝜒 is the magnetic susceptibility difference between the cell and suspending fluid, D is the diameter of the cell or spore, and η is the fluid viscosity.

In this current presentation, a non-constant value of $\stackrel{\rightharpoonup }{S_m}$ is encountered as the cell or spore flow through the MDM. Prior theoretical and experimental analysis has demonstrated that any change in a magnetic force results in a nearly instantaneous, corresponding response in velocity; hence mass times acceleration term in the force balance is assumed to be negligible. Appendix A explores this assumption further. While not necessary, in our current application the flow channels are oriented vertically with respect to gravity.

As the aspect ratio of a channel is 25:1 (z-dimension: y-dimension), the infinite parallel plate model of the fluid velocity profile, may be used:

$$u_f(y)=-\frac{Q}{w^{3}h}\left(y^2-yh\right)$$ (6)

where Q is the flow rate and y is the displacement from the deposition slide. Unless the cells are neutrally buoyant, they will sediment at constant velocity, superimposed on the local fluid velocity. This sedimentation velocity is given by:

$$u_{sed}=\frac{\mathrm{\Delta }\rho D^{2}g}{18\eta }$$ (7)

(Notice similarities between equations 1 and 3). In the direction of flow (along x), magnetic velocity, flow rate and sedimentation velocity components are present. Thus the net velocity is:

$$\frac{dx}{dt}=\frac{m}{2{\mu }_0}\frac{d\left(B^2\right)}{dx}(x,y)+u_{sed}+u_f(y)$$ (8)

Toward the magnet (along y), a magnetic velocity component exists:

$$\frac{dy}{dt}=\frac{m}{2{\mu }_0}\frac{d\left(B^2\right)}{dy}(x,y)$$ (9)

Equations 4 and 5 are ordinary differential equations with independent variable t (time). Because of the complex nature of the magnetic field gradient, they must be solved numerically to predict the particle trajectories. The trajectory end points are the positions ending on the deposition slides or flowing out of the simulated ROI. There are three scenarios for particles passing through the MDM: 1) particles with strong magnetic susceptibility can be caught regardless the initial position in the flow stream, 2) particles of weak magnetic susceptibility have a high chance to pass through, especially those entering farthest from the deposition slide, and 3) particles with moderate magnetic susceptibility whose deposition depends on the initial distance from the deposition slide. These situations can be demonstrated using the probability associated with each magnetophoretic mobility and a specific predefined flow rate; both of which are also captured in this model.

The simulations are conducted in either a program written in Maple (Waterloo Maple, Canada) or MATLAB. Assumptions in the simulation include: 1) the flow is fully developed in the channel, 2) cells (spores) are uniformly distributed before acted upon by the magnetic field, 3) cells are point masses, 4) interparticle effects are negligible, 5) when cells reach the slide surface there is no further motion, 6) a “z” component of the magnetic field is negligible, enabling it to be modeled as 2D, and 7) the fluid velocity profile depends only on the variable y, as discussed above. The program reads in two external data files, 1) a 2D map of flux density, B, generated by the boundary element method, field modeling software, Magneto (IES, Winnipeg, Manitoba, Canada), and 2) magnetophoretic mobility distribution generated experimentally with the Cell Tracking Velocimetry (CTV) instrument. CTV is an instrument developed in-house that enables the measurement of magnetophoretic mobilities of single cell over a large cell population (hundreds to thousands; Jin et al., 2008; Jin et al., 2011; Xu et al., 2012).

To ensure that the Magneto outputs provide accurate data of magnetic field, the program is first “calibrated” by measurement of the Mk I MDM magnet. The measurement was taken with a F.W. Bell Gaussmeter and Hall probe (Orlando, FL). The x-component of B is measured along y, where the probe is centered over the interpolar gap. Within Magneto the permanent magnet specification, B_r, the remnant field, was varied until predicted B closely matched the measured data, as determined by the method of least squares. Figures 2A presents a two dimensional, heat map plot of the magnetic field, while 2B presents a heat map plots of the calculated d(B²)/dy and d(B²)/dx values.

The Maple (Matlab) program reads in the B data, and calculates d(B²)/dy and d(B²)/dx with a 5-point Lagrange formula, at the same locations, and places these into arrays. Subroutines employing 2D linear interpolation allow these gradient components to be evaluated anywhere over the domain. Linear interpolation was found to be sufficiently accurate for closely-spaced points, totaling up to 50,000. The program uses a triple loop. In the first, outermost loop, the program reads in mobility-frequency pairs sourced from CTV analysis, usually under 100. In the second loop, trajectories originate at the nodes of typically 50-equally spaced divisions of the channel height, at the lower axial limit. The third and innermost loop is the trajectory solver.

The equations are solved numerically internal to Maple (MATLAB) using a Runga-Kutta method. The loop increases t and calculates x and y positions; these are compared with those of physical boundaries: the slide surface and the upper axial limit. Deceleration equations to estimate subsequent trial t values prevent overshoot. When a boundary is reached passage through the loop ceases. If the cell escapes the axial boundary, that trial is not added to the capture ratio; otherwise, the axial position of capture on the slide is recorded in a bin. Each trajectory terminus is weighted by the product of the mobility increment’s frequency in the CTV data, and by the mass flux (proportional to the fluid velocity) at its origin. Weighting factors assigned to a bin are summed to give the total capture ratio in that bin. From this it is possible to generate deposition density plots. Summing these capture ratio increments over all bins in the positive direction of flow results in a cumulative sum plot. The value of the cumulative sum at the axial limit yields the overall capture ratio.

Materials and methods

Red blood cells and Bacillus spores, and spore treatment.

Human red blood cells were obtained, and processed, as outlined in our previous publications (Zborowski et al., 2003, Moore et al., 2013). B. atrophaeus from ATCC (#9732) were cultured, and processed into spores, as described previously (Melnik et al., 2007). The freeze-dried powder of B. atrophaeus spores were resuspended either in water or chicken broth to make a final concentration of spores to 10⁷, 10⁶, 10⁵ and 10⁴ cfu/ml. Vitamin water, chicken broth and juices (grape juice and apple juice) were purchased from nearby grocery store. The clumps in chicken broth, which may possibly block the magnetic deposition system, were filtered out before suspending the spores using standard grade filter papers. The spore suspensions were subjected to heat treatment of 121° C for 30 min in an autoclave. To mimic acidic juices, HCl was added to buffer to create solutions of pH 1.8, 2.7, or 5.0 and the spores were suspended in these solutions overnight. The spore suspensions were vortexed for each trial.

Cell Tracking Velocimetry

The Cell Tracking Velocimetry, CTV, instrument has been extensive described previously (Jin et al., 2008; Jin et al., 2011; Xu et al., 2012; McCloskey et al., 2003). Basically, the instrument allows the magnetic susceptibility of individual cells, or particles, to be determined through microscopic observation, and subsequent computer analysis, of the magnetically induced velocity in a well-defined magnetic energy gradient. This magnetic energy gradient is constant in the viewing region with a magnitude of 142 ×10⁶ TA/m². Prior to analysis, a spore suspension was subjected to a 5000 ×g to discard the supernatant, and subsequently resuspended in distilled water to a final concentration of 10⁷ cfu/ml. A sample of approximately 1ml was pumped through the syringe into the channel and analyzed as previously described.

Separation of spores using magnetic deposition system

The magnetic deposition device (Figure 1) creates a fringing magnetic field at the interpolar gap, and combined with a thin flow channel pressed against this gap creates very magnetic high forces as indicated in Figure 2. We have previously reported that the maximum, predicted magnetic energy density gradient created in this system is 4,410 T²/m and the maximum flux density is 2.27 T (Karl et al., 2008). 500–600 μl of spores were delivered in a continuous manner into the flow channel (s) by syringes connected to inlet tubing, and evacuated from flow channels by outlet tubing leading to waste containers. The flow channel had an inner cross-section of 6.4 mm × 0.25 mm. After each use, the flow channel was disassembled with caution to avoid air intrusion which could change the deposition of the cells/spores on the slide. The plastic sheet and the septum were washed and stored in 70% ethanol.

Counting of B. atrophaeus spores

B. atrophaeus spore suspensions, before and after the magnetic deposition separation, were serially diluted with sterile distilled water, and 100 μl of 10⁻², 10⁻³ and 10⁻⁴ dilutions were plated on TSA plates. Caution was taken to spread the inoculums evenly but to the edge of agar plates. The plates were inverted in a 30°C incubator overnight to form countable colonies. The colonies formed on the plates were counted, and the concentration of spores in the suspension was calculated based on the colony number reading and dilution factor.

Result

Histograms of magnetically induced velocity.

As presented previously in the Theoretical Analysis for trajectory simulation section, the simulations of cell trajectories in the MDM system requires prior knowledge of the magnetophoretic mobility of the cells. Figure 3A presents histograms of the distribution of the magnetically induced velocity of oxygenated, deoxygenated, and met- hemoglobin form of human red blood cells. For clarity, the magnetophoretic mobility is also presented as a second x-axis (magnetic velocity divided by the magnetic energy gradient used to make the measurement). Figure 3B presents a similar histogram for the B. atrophaeus spores, prior to and after heat treatment, in the same CTV instrument. Note the approximate two orders of magnitude higher value of mean, magnetically induced velocity in the spores compare to the RBC, the larger distribution of magnetic velocity, and the significant decrease in the number of spores after heat-treatment, the highest decrease occurring with the more magnetic spores. This decrease in total number of spores is consistent with only 0.73 % of the original number of spores making colonies. It is also worth noting that the average of magnetophoretic mobility of the spores after sterilization did not change significantly compared to before sterilization (Table I); however, the distribution of the magnetism (as shown in induced magnetic velocities) in Figure 3B changed. Before sterilization, a peak in the distribution of magnetic velocities occurred at 0.04 mm/s; after sterilization, the peak shifted to approximately 0 mm/s.

Figure 3.

Histograms of the distribution of magnetically induced velocity of oxygenated, met-hemoglobin, and deoxygenated RBCs, 3A, and pre and post sterilization of bacillus spores, 3B. Note the 100X higher range of the x-axis in 3B. This data was measured in a cell tracking velocity instrument.

Table 1.

Mean, magnetophoretic mobility (×10⁴) (mm³/T•A•s) of B. atrophaeus before and after sterilization

	Before Sterilization	Chicken Broth after sterilization	DI water after Sterilization
Run 1	2.23	2.49	1.94
Run 2	2.11	1.83	1.74

In addition to magnetically induced velocity, the CTV system can also measure settling velocity, and thereby assuming a specific density, determine an effective diameter. For both the prior and after heat treatment, the average settling velocity did not statistically change, and the mean diameter of 4.7 microns (before and after sterilization) suggests that the spores are clumped, which we previously reported (Sun et al., 2011).

Magnetically Deposited RBCs.

High-spin methemoglobin RBCs were used as a model of magnetically susceptible cells. For normal RBC suspensions in equilibrium with ambient air the intracellular hemoglobin exists in the form of a diamagnetic oxyhemoglobin which makes the RBC slightly less susceptible than the diamagnetic, aqueous suspension media and therefore unresponsive to the applied magnetic field (for the stated purpose of this study). However, when intracellular oxyhemoglobin was converted to the paramagnetic methemoglobin by incubation with sodium nitrite, as described previously (Zborowski et al., 2003), the RBC become less diamagnetic than the aqueous media and thus results in the RBCs in suspension to migrate along the magnetic field gradient in the fringing field of the interpolar gap, towards the magnetic deposition zone.

Figure 4 is a collection of images demonstrating this deposition of RBC in which the hemoglobin was converted to methemoglobin. 0.5 ml of the methemoglobin form at a concentration of 5×10 RBCs/mL was pumped through the MDM twice; a flow rate of 2 ml/hr going up and 1.2 mL/hr going down (relative to gravity) The suspension was aspirated and then dispensed through the flow channel using programmable syringe pump (PHD 2000 Programmable Syringe Pump, Harvard Apparatus Inc.).

Figure 4.

Actual and predicted fraction of RBC deposited using the MDM instrument. 4A is an enlargement of the deposition region of the slide, 4B, upon which RBC were deposited. 4C is a 2-D plot of fraction of cells deposited at a function of x location, LHS side y-axis corresponds to the simulated and experimental studies, and the RHS y-axis corresponds to the theoretical, cumulative deposition of cells.

Figure 4B is an image of a slide upon which the cells were deposited, while Figure 4A is an enlargement of the deposition zone. The imposed dotted lines correspond to the location of the two magnet pole pieces. Figure 4C presents a 2-D plot of fraction of cells deposited as a function of x location; LHS side y-axis corresponds to simulated and experimental studies, and the RHS side y-axis corresponds to the theoretical, cumulative deposition of cells. Given the assumption that there is no variability of magnetic forces in the z-axis, the fraction and cumulative deposited cells in Figure 4C is a sum of all the cells in the “z-direction” for a given x location. The experimental cell deposition distribution shown in Figure 4C was obtained by first using a high-resolution cell deposition image acquisition with a flat-bed scanner (Epson Perfection 164OSU Scanner, Epson Corp.). After image acquisition, the image was segmentation into 8×383 = 3,064 pixel rectangular segments covering the cell deposition image and a portion of the adjacent areas devoid of cells, and the mean 8-bit pixel luminosity for each segment (Adobe Photoshop, Adobe Inc.) as a function of distance across the cell deposition area was recorded. From this, the distribution was determined. Given the significant number of experimental variables used in the simulations, we suggest that the general agreement of the experimental data and theoretical predictions are acceptable.

Figure 5A demonstrates the predicted (simulated) total capture rate of met RBC as a function of flow rate and, as expected, higher capture rate is associated with lower flow rate Figure 5B presents the simulated distribution of met RBC deposition (capture) as a function of location and flow rate. The location of −2.45mm (x-axis) corresponds to the entrance of the flow channel. Note the change in the second peak (RHS) as the flow rate changes; it is suggested that this represents the capture of the lower magnetic susceptibility cells that started out the furthest from the deposition slide. It takes a second region of attraction to move them all the way to the slide, and subsequent deposition (capture).

Figure 5.

Predicted capture rate of met hemoglobin RBC as a function of flow rate (5A) and predicted deposited position and capture rate, 5B, as a function of different flow rates.

Magnetic deposited Spores.

Three general types of experiments were conducted to test the effectiveness and or performance of the MDM system using spores: 1) the ability of the system to detect the presence of spores in a suspension, including spores spiked into liquid foods, 2) the effectiveness of removing the spores from the suspension, and 3) comparison of the separation before and after sterilization. Starting with spore concentrations of 10⁷ cfu/ml in buffer, clear deposition bands could be observed down to a concentration of 10⁴ cfu/ml. Despite having a mild pigment in the spores, their visual contrast is less than RBC^’s, and visual detection of deposited spores at initial concentrations below 10⁴ cfu/ml is challenging to detect under microscopic evaluation. Similar performance was obtained when the spores were spiked into milk and chicken broth; however, with juices, to our surprise, no obvious deposition line formed.

The previous study by Sun et al. indicated that the magnetic susceptibility of these spores is a strong function of pH; low pH can disrupt the spore clusters and lower the magnetic susceptibility of remaining spore, or spore clusters by two orders of magnitude. Analysis of the juices used in this current studied gave values of pH 3.3. While not a primary focus of this study, we did observe that if these acid treated spores are suspended back in neutral pH solutions with Mn solutions of 220 uM overnight the magnetically induced velocities, as measured by CTV, can be restored.

Given the difficulty in visually confirming the presence of spores, plating was performed using the spore solution before deposition through the MDM system, and on the effluent collected after flowing over the magnetic deposition system. As is generally practiced with such assays, the targeted number of colonies per plate is on the order of 20 to 100. Consequently, a range of dilutions were used on the spore suspension prior to, and after flow through the MDM, and 100 μl aliquots of the diluted spore suspensions were plated. Table I presents the results for two separate trials, which both resulted in a 98–99% depletion of the spores.

Figure 6 is a collection of images and plots of an actual deposition of Bacillus spores using the MDM system, before and after heat treatment. Figure 6B is an image of a slide upon which the spores were deposited and Figure 6A is an enlargement of the deposition zone. Figure 6C presents a 2-D plot of the simulated fraction of spores deposited at a function of x location, presented in a similar manner to the RBC deposition, the LHS side y-axis corresponds to simulated and experimental studies, and the RHS y-axis corresponds to the theoretical, cumulative deposition of spores. The cumulative capture rate of spores before sterilization can be as high as 95% from simulation, which is consistent with the experiment results. It should be noted that the capture rate from the experiments is slightly higher than the prediction; it is possible that not all of the spores escaping the magnetic deposition can germinate on the plates.

Figure 6.

Actual and predicted fraction of spores deposited using the MDM instrument. 6A is an enlargement of the deposition region of the slide, 6B, upon which RBC were deposited. 6C is a 2-D plot of fraction of cells deposited at a function of x location, LHS side y-axis for simulated studies, and the theoretical, cumulative deposition of cells on the RHS y-axis (note: all cells in the “z-direction are summed). Unlike the RBC study presented in Figure 5, due to the low visual contrast of deposited spores, no image processing of the spores on the slides was conducted.

Simulated cell trajectories.

Figure 7 is an example of trajectories of deoxygenated RBC in the MDM system with a flow rate of 0.5ml/hr. The borders of the plot correspond to the boundaries of the simulation, including the top wall and slide surface defining the channel, and the field extent of +/− 2.5 mm. For clarity, 15 rather than the usual 50 seed positions are shown. Each seed position has 10 trajectories, corresponding to a coarser-than-usual division of the mobility distribution. Visual inspection shows that a preponderance of trajectories terminates on the slide surface opposite the corners of the magnet pole pieces, at +/− 0.64 mm. These regions correspond to the highest magnetic gradient. As described elsewhere, weighting of the trajectory termini on the slide allows calculation of deposition density and overall capture ratio. At each seed position, except the lowest, some of the trajectories, corresponding to lower mobilities, deflect toward the magnet but not sufficiently to be captured. This becomes more common at higher elevations.

Figure 7.

Simulated trajectories of deoxygenated RBC in the deposition region of the MDM system. The two regions on the y=0, x = ±0.6 mm surface with the high number of trajectory terminations correspond to the location of the two magnet poles.

Each trajectory terminus is weighted by the product of the mobility increment’s frequency in the CTV data, and by the mass flux (proportional to the fluid velocity) at its origin. The fluid velocity is evaluated for a rectangular duct of finite dimension, rather than from the assumption of infinite parallel plate flow. Weighting factors assigned to a bin are summed to give the total capture ratio in that bin from deposition density calculations and plots. Summing these capture ratio increments over all bins in the positive direction of flow results in a cumulative sum plot. Visual inspection of particle trajectories in Figure 7 clearly shows the location of the two magnetic poles located approximately 0.6 mm on either side of the x origin.

Relations between Capture rate, Magnetophoretic mobility and flow rate.

In this report, we demonstrated with two cases, RBC and Bacillus spores, experimentally and through simulation, the relationship between the distribution of magnetic susceptibility, flow rate, and capture rate in the MDM system. For spores and metRBC, which both have constant magnetization over the magnetic fields used in this study, capture rate is only correlated with flow rate at a specific magnetic mobility. The relationship can be generalized for all particles or cells that have constant magnetization (Figure 8). Figure 8A presents the theoretical capture rate (y-axis) of hypothetical cells with specific magnetic mobilities (x-axis) at different flow rates (A negligible sedimentation velocity was assumed). A linear increase of capture rate with the magnetic mobility, from 0 to almost 100% capture can be observed for all flow rates presented. Alternatively, Figure 8B, the theoretical magnetic mobility cut-off is presented. This cut-off is defined as the minimum magnetic mobility allowing 100% capture rate as a function of flow rate. Again, a linear increase is observed.

Figure 8.

8A, the simulated capture rate of cells vs magnetic mobility and, 8B magnetic mobility cut-off vs flow rate.

According to the linear relationship shown in Figure 8, the capture rate, CR, for a particular cell population with uniform and positive magnetic mobility m (mm^{^}3 s/kg) and no sedimentation, under the flow rate of Q(ml/h) is:

$$CR=\{\begin{array}{ll}1& ifm>2x{10}^{-5}Q\\ \frac{m}{2x{10}^{-5}Q}& ifm<2x{10}^{-5}Q\end{array}$$

With this equation, for any cell we can estimate the maximum flow rate for the required capture rate. For example, for the spores before sterilization, if a 90% capture rate is targeted, the maximum flow rate is approximately 1.9 mL/min. Beyond spores, we have previously, theoretically defined, and experimentally measured, the range of magnetophoretic mobility one can obtain through labeling of various cell types with commercial antibody-magnetic particle conjugates. As can be imaged, given the range in diameter of commercial magnetic particles from on the order of 200 nanometers to 5 microns, the magnetophoretic mobility can range over at least 6 orders of magnitude (Jin et al., 2008; Jin et al., 2011; Xu et al., 2012).

Discussion and conclusion

Magnetic cell separations within small fluidic regions is not only popular commercially, (i.e the MACS™ family or products), but is also being used/described in a number of research publications on microfluidic systems. However, as we demonstrate in this current and previous work, the characterization and modeling of such separations is complex as a result of the non-linear nature of the magnetic energy gradient used in most systems and the distribution of magnetophoretic mobility of labeled and unlabeled cells (Figures 2 and 3). While complex, as we have demonstrated above, modern boundary and or finite element, field modeling software, such as Magneto allow highly accurate magnetic field maps to be combined with fluid dynamic modeling software to produce accurate predictions of particle trajectories if well-defined magnetic pole pieces are used. This is in contrast to the non-structured distribution of magnetic poles in typical HGMS systems.

The performance of any magnetic cell separation system is based on the relative magnitude of magnetic force being applied to the cells, which is this case is proportional to ∇B². When the magnetic separation technology requires that the cell needs to move, either to be deposited and held on a surface, as in this study, or to be deflected into a different fluid stream line, the higher the magnetically induced velocity, the faster the separation can be accomplished. Equation 1 reduces this velocity to two primary terms: the cells magnetophoretic mobility, m, and the magnetic energy gradient, S_m. When using the intrinsic magnetic susceptibility of the cell, as we are doing in this presentation, the primary variables left to design or manipulate are: 1) the magnitude and direction of S_m, 2) the geometry in which S_m is applied (including the total are in which Sm is applied), and 3) the residence time of the cell in zone of S_m.

While this current presentation demonstrates the ability to theoretically, and experimentally verify the magnetic separation of weakly magnetic cells (deoxygenated RBCs), the throughput is relatively slow, on the order of mls/hr, and as Figure 5A demonstrates, the capture rate non-linearly drops as the flow rate through the system is increased. However, the actual separation volume simulated within which significant magnetic energy is imparted was 5.6 × 10⁻³ mls. At a flow rate of 0.2 mls/hr (the theoretical flow rate to achieve over 90% removal of deoxygentated RBCs), this corresponds to a residence time of 1.7 min. The value of S_m in this separation volume ranges from 136 to 4,400 T/m², with an average value of 1390 T/m².

Added to this range in values of Sm is the distribution in the magnetophoretic mobility in a cell population. For example, studies in our laboratory have indicated that the distribution in RBC magnetophoretic mobility presented in Figure 3A is typical; in fact, we have recently demonstrated that both the mean, and distribution of this mobility changes as RBCs age (Chalmers et al. 2017). This distribution in mobility has a significant effect on the paths that RBCs take in the MDM. To further demonstrate this effect, Figure 7 was further modified to only have three cell injection locations, and within each of these injection locations, three RBCs were injected; one with the mean RBC magnetophoretic mobility, one with 1 standard deviation above the mean, and 1 with 1 standard deviation below the mean, Figure 9. The paths for these three types of cells are indicated with a dotted, solid, and dashed lines. Two flow rates were used: 0.5 and 0.1 mls/hr, 9A, 9B, respectively.

Figure 9.

Simulated paths of RBCs of three different magnetophoretic mobilites, mean, one with 1 standard deviation above the mean, and 1 with 1 standard deviation below the mean, at three different injection locations. Two flow rates were used: 0.5 and 0.1 mls/hr, 9A, 9B, respectively. 9C presents the particle velocity, the Vx, Vy and Vrms. The specific path from which this data is obtained is highlight with a star in 9B.

As expected, the distribution in the mobility and the non-linear nature of the value of Sm results in a large spreading of the trajectories, with this spreading increasing with distance from the deposition surface. As presented in Figure 5A, the percent of RBC captures increases rapidly between a flow rate 1 and 0.1 ml/hr. What is not necessarily obvious is the cell trajectory as a function of magnetophoretic mobility. For example, for cells injected in the middle position, at a flow rate of 0.1 ml/hr, the cells with the lowest value of manetophoretic mobility barely is captured; yet only the lowest mobility cell is not captured from the injection location furthest from the deposition surface.

Ideally, to simplify magnetic cell separator design, it is desirable to create a constant zone of very high S_m to scale up a practical separator for cells with intrinsic magnetization, such as RBCs. Since the current systems presented here and in our previous publications indicates that we are achieving near the maximum value of B that is practical, without resorting to superconducting magnets, to achieve the desired value of ∇B² we are only left with the option of creating separation systems with a large surface area of high ∇B², and as in the case of the MDM, a flowing cell suspension very near this large surface area. If one assumes a linear scale with surface area, a rough estimate indicates that a 103 fold increase in surface area would result in the potential to separate RBCs from blood at a rate of 1 ml/min. This would correspond to a surface area of approximately 222 cm². Given the size of a typical centrifuge used to separate RBCs from blood, a separation area of this size is potentially practical. Future research in our lab is focusing on attempting to demonstrate such a system.

Appendix A. Validity of assumption to ignore acceleration term in Equation 4.

The following derivation follows the same methodology as presented by Denn (1979). Starting with Equation 4 from the manuscript:

$$\begin{gathered} \frac{\pi D_p^3}{6}\left({\rho }_p+\frac{{\rho }_f}{2}\right)\frac{d{u}_p}{dt}=\frac{\pi D_p^3}{6}\mathrm{\Delta }\chi \frac{\nabla B^2}{2{\mu }_0}-3\pi \mathrm{\eta }{D}_p{u}_p+\frac{\pi D_p^3}{6}g\left({\rho }_p-{\rho }_f\right) \\ -\frac{3}{2}\pi \mathrm{\eta }{D}_p^2\left(\frac{1}{\sqrt{\pi v_f}}\right){\int }_{-\infty }^t\frac{\frac{d{u}_p}{dt}\left(t_1\right)}{\sqrt{t-t_1}}d{t}_1 \end{gathered}$$ (A.1)

the Basset history term is set to zero. Further, it is assumed that the magnetic force and gravity force act in the same, one dimension. Rewriting A.1:

$$D_p^2\left({\rho }_p+\frac{{\rho }_f}{2}\right)\frac{d{u}_p}{dt}=D_p^2\mathrm{\Delta }\chi \frac{\nabla B^2}{2{\mu }_0}-18\mathrm{\eta }{u}_p+D_p^{2}g\left({\rho }_p-{\rho }_f\right)$$ (A.2)

After further rearrangement:

$$\frac{d{u}_p}{dt}=\frac{18\mu }{\left({\rho }_p+\frac{1}{2}{\rho }_f\right)D_p^2}\left[\frac{D_p^2}{18\mu }\mathrm{\Delta }\chi \frac{\nabla B^2}{2{\mu }_0}+\frac{D_p^2}{18\mu }\mathrm{\Delta }\rho g-u_p\right]$$ (A.3)

For simplicity, we will let: $\frac{D_p^2}{18\mathrm{\mu }}\mathrm{\Delta }\chi \frac{\nabla B^2}{2{\mu }_0}=M$ and $\frac{D_p^2}{18\mu }\mathrm{\Delta }\rho g=G$. At steady state, the LHS is zero, therefore the sum of M and G must equal u_p, which is the terminal velocity, u_p∞. Since we are already assuming Stokes flow, Re << 1, we can introduce a Re and a dimensionless term, θ = V_p,_∞t/D_p, which allows Equation A.3 to be rewritten:

$$\frac{d{u}_p}{d\theta }=\frac{18}{\left(\frac{{\rho }_p}{{\rho }_f}+\frac{1}{2}\right)R{e}_{\infty }}\left[M+G-u_p\right]$$ (A.4)

Solving for the most general case, in which the particle goes from no velocity to the final terminal velocity, the solution can be written as:

$$\frac{V_p}{V_{p,\infty }}=1-exp\left[-\frac{18\theta }{\left(\frac{{\rho }_p}{{\rho }_f}+\frac{1}{2}\right)R{e}_{\infty }}\right]$$ (A.5)

The particle velocity will reach 95 percent of its terminal velocity when the argument of the exponential is −3, therefore setting the term in the bracket equal to 3, we will call this θ_∞

$${\mathrm{\theta }}_{\infty }=R{e}_{\infty }\left(\frac{{\rho }_p}{6{\rho }_f}+\frac{1}{12}\right)$$ (A.6)

The density of human RBCs and spores range from 1.10 × 10³ to 1.165 × 10³ kg/m³ and the density of PBS is 1.063 × 10³ kg/m³; therefore, the value in the parentheses in Equation A.6 ranges from 0.26 to 0.27.

For the current study, the highest value of u_p∞, 1.2 × 10⁻⁵ m/s, occurs when the value of S_m, $\left(\frac{\nabla B^2}{2{\mu }_0}\right)$, is 1.8 × 10⁹ T-A/m², and a mobility of the RBC is assumed to be 6.8 × 10⁻¹⁵ m³-s/kg (1 standard deviation above the mean). At this value of u_p∞, the Re for a RBC is 1 × 10⁻⁴, satisfying the assumption of Re << 1. For comparison, the experimentally measured settling velocity of a RBC, without any magnetic forces, is on the order of 1.5 × 10⁻⁶ m/s.

With the value of θ_∞ calculated from Equation A.6, and the definition of θ, the time it takes for a cell to reach its terminal velocity is given by:

$$t_{\infty }=\frac{D_p}{V_{p,\infty }}{\theta }_{\infty }$$ (A.7)

For the extreme case presented here, it takes approximately 1.6 ×10⁻⁵ seconds to accelerate a RBC from 0 velocity to 1.2 × 10⁻⁵ m/s. Assuming a velocity 10% of the termination velocity, it only a very small fraction of a cell diameter to reach this terminal velocity.

It is also interesting to note that if A.6 is substituted into A.7, one obtains:

$${\theta }_{\infty }=\frac{D_p}{V_{p,\infty }}Re\left(\frac{{\rho }_p}{6{\rho }_f}+\frac{1}{12}\right)=\frac{D_p^2}{\mathrm{\eta }}\left(\frac{{\rho }_p}{6{\rho }_f}+\frac{1}{12}\right)$$ (A.8)

which indicates that the time it takes to reach terminal velocity is independent of the specific terminal velocity, as long as the Re <<1. For the simulation used in this study, on the order of 500 node points were used in the direction perpendicular to the deposition surface. Consequently, except for the first increment in nodes points in the direction of the deposition surface, the cells or spores will not have an initial zero velocity, assumed to solve Equation A.4, but will have a finite, non-zero initial magnetically induced velocity. This would reduce the time to reach terminal velocity as compared to starting with a zero initial magnetically induced velocity.