Control of Magnetophoretic Mobility by Susceptibility-Modified

With the analytical expression for the magnetophoretic mobility of an ideal, ... Application of magnetic particle tracking velocimetry to quadrupole m...
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Anal. Chem. 2004, 76, 3899-3907

Control of Magnetophoretic Mobility by Susceptibility-Modified Solutions As Evaluated by Cell Tracking Velocimetry and Continuous Magnetic Sorting Lee R. Moore,† Sarah Milliron,‡ P. Stephen Williams,† Jeffrey J. Chalmers,§ Shlomo Margel,| and Maciej Zborowski*†

Department of Biomedical Engineering, The Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, Ohio 44195, Department of Bioengineering, University of Toledo, Toledo, Ohio 43606, Department of Chemical and Biomolecular Engineering, The Ohio State University, 140 West 19th Avenue, Columbus, Ohio 43210, and Department of Chemistry, Bar-Ilan University, Ramat Gan, Israel

With the analytical expression for the magnetophoretic mobility of an ideal, linearly polarizable sphere undergoing creeping motion in viscous medium, we have shown that both attractive and repulsive motions are possible in the magnetic field. We have validated theoretical predictions using magnetic monodisperse microspheres of 5.2-µm diameter and nonmagnetic polystyrene microspheres of 6.99-µm diameter suspended in solutions of paramagnetic ions. The microsphere magnetophoretic mobility was measured using a modified particle tracking velocimetry system, developed in-house and called a cell tracking velocimeter. The product of measured mobility and viscosity agrees well with the theoretical prediction, differing only by ∼11%. Further, a 26% increase in resolution between magnetic and nonmagnetic particle distributions was evaluated when paramagnetic ion carrier was used instead of water. Continuous particle sorting based on differences in magnetophoretic mobility was performed with another device developed by us, the quadrupole magnetic flow sorter (QMS). In the QMS, the introduction of paramagnetic ions into the carrier was effective in suppressing nonspecific crossover (i.e., the transport of low-mobility particles into the magnetic particle fraction) in particles and in biologically relevant red blood cells and thus showed promise as a means of increasing the purity of the magnetic separation. The term “magnetophoretic mobility” has been proposed to describe the behavior of a magnetic particle moving through a viscous medium under the influence of an external magnetic field.1-10 Its origin can be traced to a similar term used to describe particle motion in a viscous medium under the influence of an * Corresponding author. Tel.: 1-216-445-9330. Fax: 1-216-444-9198. E-mail: [email protected]. † The Cleveland Clinic Foundation. ‡ University of Toledo. § The Ohio State University. | Bar-Ilan University. (1) Hartig, R.; Hausmann, M.; Schmitt, J.; Herrmann, D. B. J.; Riedmiller, M.; Cremer, C. Electrophoresis 1992, 13, 674-676. 10.1021/ac049910f CCC: $27.50 Published on Web 06/17/2004

© 2004 American Chemical Society

electric field, the “electrophoretic mobility”.11-14 The concept of magnetophoretic mobility has not been as widely applied due to peculiarities of the magnetic dipole-field interaction, such as its strong dependence on position,15 and due to applications relying on equilibrium rather than transport separations. The former are exemplified by magnetic particle capture inside a high-gradient magnetic separator column and the latter by a continuous separation in a laminar flow exposed to an open field gradient.6,16 An examination of the expression for the magnetophoretic mobility reveals the potential for a wider range of magnetic separation applications than currently practiced in biological and clinical laboratories.10,17,18 In particular, it suggests the possibility of not only attractive, but repulsive particle motion in a magnetic field, by modifying the magnetic susceptibility of the solution. This could (2) Winoto-Morbach, S.; Tchikov, V.; Mueller-Ruchholtz, W. J. Clin. Lab. Anal. 1994, 8, 400-406. (3) Jones, T. B. Electromechanics of Particles; Cambridge University Press: Cambridge, U.K., 1995. (4) Hausmann, M.; Cremer, C.; Hartig, R.; Liebich, H.-G.; Luers, G. H.; Saalmueller, A.; Teichmann, R. In Cell Separation Methods and Applications; Radbruch, A., Ed.; Marcel Dekker: New York, 1998; pp 209-235. (5) Tchikov, V.; Schuetze, S.; Kroenke, M. J. Magn. Magn. Mater. 1999, 194, 242-247. (6) Williams, P. S.; Zborowski, M.; Chalmers, J. J. Anal. Chem. 1999, 71, 37993807. (7) Suwa, M.; Watarai, H. Anal. Chem. 2001, 73, 5214-5219. (8) Watarai, H.; Namba, M. J. Chromatogr., A 2002, 961, 3-8. (9) Wilhelm, C.; Gazeau, F.; Bacri, J.-C. Eur. Biophys. J. 2002, 31, 118-125. (10) Zborowski, M.; Moore, L. R.; Williams, P. S.; Chalmers, J. J. Sep. Sci. Technol. 2002, 37, 3611-3633. (11) Overbeek, J. T. G.; Bijsterbosch, B. H. In Electrokinetic Separation Methods; Righetti, P. G., Van Oss, C. J., Vanderhoff, J. W., Eds.; Elsevier/NorthHolland Biomedical Press: Amsterdam, The Netherlands, 1979; pp 1-32. (12) Melcher, J. R. Continuum Electromechanics; The MIT Press: Cambridge, MA, 1981. (13) Todd, P.; Plank, L. D.; Kunze, M. E.; Lewis, M. L.; Morrison, D. R.; Barlow, G. H.; Lanham, J. W.; Cleveland, C. J. Chromatogr. 1986, 364, 11-24. (14) Schu ¨ tt, W.; Hashimoto, N.; Shimizu, M. In Cell Electrophoresis; Bauer, J., Ed.; CRC Press: Boca Raton, FL, 1994; pp 255-266. (15) Becker, R. Electromagnetic Fields and Interactions; Dover Publications: New York, 1982. (16) Watson, J. H. P. J. Appl. Phys. 1973, 44, 4209-4213. (17) Helgesen, G.; Pieranski, P.; Skjeltorp, A. T. Phys. Rev. A 1990, 42, 72717280. (18) Rosensweig, R. E. Ferrohydrodynamics; Cambridge University Press: Cambridge, MA, 1985.

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lead to improvements in the quality of the separands, such as increased purity and recovery. We set out to test the feasibility of such an approach. The use of ferrofluids and paramagnetic ions to modify solution susceptibility for the study of particle and cell motion in suspensions exposed to a magnetic field, has been described before.17,19-22 For instance, Hwang et al.21 proposed magnetic separation in solutions with a magnetic susceptibility gradient. Russell et al.22 used paramagnetic salts of various concentrations to immobilize magnetized cells in the vicinity of a rectangular slot magnet for the purpose of estimating mean cell susceptibility. Helgesen et al.17 studied the motion of nonmagnetic microparticles in colloidal suspensions of ferromagnetic submicrometer particles (ferrofluids) and dubbed the combination as a “magnetic hole system”. They observed that such a system offers the theoretical advantage of the “magnetic hole” dipole moment being collinear with an external field, which simplifies analysis of the dynamic phenomena and collective processes. Those studies are made relevant to biological separation applications by the introduction of paramagnetic lanthanide ion solutions normally used as contrast agents for NMR imaging (MRI) and demonstration of their low systemic toxicity.23,24 The availability of these MRI contrast agents makes possible a systematic study of cell motion in susceptibility-modified solutions and enables their application to cell sorting by magnetophoresis. We have developed an instrument for direct observation and measurement of particle motion in a highly regular magnetic field, the cell tracking velocimeter (CTV).25-28 It employs a constant field energy density gradient in the cell tracking volume and cell motion analysis software capable of measuring velocities of hundreds of cells per image frame, providing highly reliable cell motion statistics (such as mean and variance of velocity distribution).29 The instrument was designed to measure the velocities of cells tagged by nanometer-scale magnetic particles and suspended in aqueous electrolyte solutions. The accuracy of the CTV tracking algorithm has been validated by measurement of the sedimentation rate of particles of known density and size.28 The accuracy of magnetophoresis measurements has been validated by measuring the mobilities of magnetite-coated polystyrene microspheres, calculating their magnetizations, and comparing with magnetometer measurements.27 (19) Gill, S. J.; Malone, C. P.; Downing, M. Rev. Sci. Instrum. 1960, 31, 12991303. (20) Zimmels, Y.; Yaniv, I. IEEE Trans. Magn. 1976, MAG-12, 359-368. (21) Hwang, J. Y.; Takayasu, M.; Friedlaender, F. J.; Kullerud, G. J. Appl. Phys. 1984, 55, 2592-2594. (22) Russell, A. P.; Evans, C. H.; Westcott, V. C. Anal. Biochem. 1987, 164, 181-189. (23) Weinmann, H. J.; Laniado, M.; Mutzel, W. Physiol. Chem. Phys. Med. NMR 1984, 16, 167-172. (24) Evans, C. H. Biochemistry of the Lanthanides; Plenum Press: New York, 1990. (25) Reddy, S.; Moore, L. R.; Sun, L.; Zborowski, M.; Chalmers, J. J. Chem. Eng. Sci. 1996, 51, 947-956. (26) Chalmers, J. J.; Haam, S.; Zhao, Y.; McCloskey, K.; Moore, L.; Zborowski, M.; Williams, P. S. Biotechnol. Bioeng. 1999, 64, 519-526. (27) Moore, L. R.; Zborowski, M.; Nakamura, M.; McCloskey, K. E.; Gura, S.; Zuberi, M.; Margel, S.; Chalmers, J. J. J. Biochem. Biophys. Methods 2000, 44, 115-130. (28) Nakamura, M.; Zborowski, M.; Lasky, L.; Margel, S.; Chalmers, J. J. Exp. Fluids 2001, 30, 371-380. (29) Guezennec, Y. G.; Brodkey, R. S.; Trigui, N.; Kent, J. C. Exp. Fluids 1994, 17, 209-219.

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In the present study, uniform polystyrene microspheres were tracked in paramagnetic solutions of chelated gadolinium of known concentration. As mobility and viscosity depend on concentration, we set these as dependent variables and evaluated their product, by measurement and by calculation. This expected product is predicted from particle radius, solution properties, and magnetic field measurements. A comparison of means enables the validation of the assumptions underlying the magnetophoretic mobility expression. Error analysis of experimental variables enables an assessment of uncertainty attributable to the CTV method, providing a valuable tool for challenging the precision limits of the device. To test the effect of solution susceptibility modification on cell separation, preliminary experiments were performed to demonstrate the feasibility of using susceptibility-modified carrier solution to reduce nonspecific crossover in a continuous magnetophoretic separation device, the QMS.6,30 The nonspecific crossover of nonmobile cells into the enriched fraction of high-mobility cells reduces the purity of separands in the sorted fractions.31 This is undesirable, particularly for rare cell isolation. Numerous factors have been implicated in nonspecific crossover, making it difficult to predict. The typical method of controlling crossoversby changing the flow distribution inside the QMS flow channels results in a reduction of the recoverable mobility range.32,33 It is one of the goals of this paper to demonstrate the feasibility of using magnetic fluid to reduce crossover in continuous magnetic separation. THEORY Particle Magnetostatic Force. The magnetic force on a spherical paramagnetic (linearly polarizable) particle in a paramagnetic solution is3

µp - µs Fm ) 2πµsR3 ∇H 2 µp + 2µs 0

(1)

R is the particle radius, µs and µp are the permeabilities of the solution and particle, and H0 is the local field intensity in the absence of matter. The permeabilities can be expressed as susceptibilities through µ ) µ0(χ + 1), and the gradient term is rewritten as ∇H02 ) ∇(B0/µ0)2, through the relationship between field strength H0 and flux density B0 in the absence of matter. Substituting these into eq 1 leads to

χp - χs ∇B02 Fm ) 3V(χs + 1) χp + 2χs + 3 2µ0

(2)

If χs and χp are much smaller than unityswhich is the case of the (30) Sun, L.; Zborowski, M.; Moore, L. R.; Chalmers, J. J. Cytometry 1998, 33, 469-475. (31) Williams, P. S.; Moore, L. R.; Chalmers, J. J.; Zborowski, M. Anal. Chem. 2003, 75, 1365-1373. (32) Hoyos, M.; Moore, L. R.; McCloskey, K. E.; Margel, S.; Zuberi, M.; Chalmers, J. J.; Zborowski, M. J. Chromatogr., A 2000, 903, 99-116. (33) Hoyos, M.; McCloskey, K. E.; Moore, L. R.; Nakamura, M.; Bolwell, B. J.; Chalmers, J. J.; Zborowski, M. Sep. Sci. Technol. 2002, 37 (4), 745-767.

experiments described belowsthen χs + 1 ≈ 1, χp + 2χs + 3 ≈ 3, and eq 2 reduces to the frequently reported expression16,18,34

solving for the velocity u, leads to the frequently cited expression for the particle magnetic migration velocity in one dimension:26,35

∇B02 ) V(χp - χs)(H0‚∇)B0 2µ0

u)

Fm ) V(χp - χs)

Fm ) V(χp - χs)

( )

2 dB0 d B0 ) V(χp - χs)H0 dx 2µ0 dx

where ∆χ ) χp - χs, and η is the solution viscosity. We can separate terms attributable to the particle and solution properties from those attributable to the external field, as6,10

m≡ Particle Susceptibility Relative to Solution. The volumetric susceptibility of a solution of paramagnetic ion (gadolinium) in the SI system of units χs is found from the concentrations and volumetric susceptibilities of the constituents, water w and gadolinium Gd:

4π([H2O]χ′M,w + [Gd]χ′M,Gd) 1000 cm3/L χw +

)

4π [Gd]χ′M,Gd (4) 1000 cm3/L

where [H2O] and [Gd] are molarities of water and gadolinium and χ′M,w and χ′M,Gd are the molar susceptibilities of water and gadolinium in the CGS system, still common in the literature tables. The conversion factors 1000 cm3/L and 4π are included to convert from CGS to SI units. The susceptibility difference between a paramagnetic particle and the displaced gadolinium solution becomes

∆χ ) χp - χw -

4π [Gd]χ′M,Gd 1000 cm3/L

(5)

The particles in this report are polymeric spheres or blood cells. When these are suspended in aqueous solution where [Gd] ) 0, no discernible velocity is detected by CTV (Table 3). This indicates that χp - χw ) 0, within experimental error, and thus justifies the simplification

∆χ ) -

(7)

(in three dimensions)

(in one dimension) (3)

χs )

( )

2 2 ∆χR2 d B0 9 η dx 2µ0

4π [Gd]χ′M,Gd 1000 cm3/L

(6)

In the final analysis, the net magnetic susceptibility of polystyrene particles in a paramagnetic ion solution (a “magnetic hole”) decreases linearly with increasing ion concentration. The above equation applies only to particles whose susceptibility equals that of the continuous medium (water). Note also that polystyrene particles and water are both diamagnetic. Magnetophoretic Mobility. Setting the magnetic force on a spherical particle (eq 3) equal to Stokes drag, Fd ) 6πRηu, and (34) Zborowski, M. In Scientific and Clinical Applications of Magnetic Microcarriers: An Overview; Ha¨feli, U., Schu ¨ tt, W., Teller, J., Zborowski, M., Eds.; Plenum Press: New York, 1997; pp 205-231.

( )

2 dB0 2 ∆χR2 d B0 ) H0 Sm ≡ 9 η dx 2µ0 dx

(8)

so that

u ) mSm

(9)

We designate m as magnetophoretic mobility and Sm as magnetic force field strength. The quantity reported by CTV is mobility, and it is found from the quotient of measured velocity and known force field strength. In many typical applications, such as those having solutions without paramagnetic ion, the mobility is constant between devices, so that particle velocities in different systems are easily predicted by comparing force field strengths.10 The magnetophoretic mobility of a particle in aqueous solution having no paramagnetic solute, from eq 8, is

mw )

2 2 R (χp - χw) 9 ηw

(10)

Similarly, the magnetophoretic mobility of this particle suspended in a solution containing paramagnetic ion (gadolinium) is

ms )

2 2 R (χp - χs) 9 ηs

(11)

Solving eq 10 for χp and substituting this into eq 11 gives

ms )

2 ηw 2 R (χw - χs) mw + ) ηs 9 ηs

ηw πR2 8 mw [Gd]χ′M,Gd (12) ηs 9 (1000 cm3/L)η s

The right-most equality derives from the substitution of eq 4 for the solution susceptibility. A magnetic particle of unknown mobility is measured by CTV in aqueous solution to evaluate mw. With the aid of eq 12, we may predict the (reduced) mobility of this particle when placed in a solution of gadolinium, ms. Resolution of Particle Distributions. We may evaluate the mobility resolutionsa measure of separation between Gaussian (35) Takayasu, M.; Gerber, R.; Friedlaender, F. J. IEEE Trans. Magn. 1983, MAG-19, 2112-2114.

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mobility distributions of particles 1 and 2sas36

EXPERIMENT Monodisperse Magnetic Microspheres. Uniform-size magnetic microspheres MM are synthesized by one of us (S.M.). As several batches have been supplied, the microspheres used in this paper, internally designated as MMC-3 and MMB-2, will subsequently be referred to as MM1 and MM2. Both are 5.2 µm in diameter, but MM1 has the higher magnetite/maghemite content. A description of their synthesis has been given elsewhere.37 As mentioned above, an earlier set of magnetic microspheres (MMB series) was used as standards for CTV evaluation.27 They have also been used to evaluate QMS performance.32 Nonmagnetic Microspheres. Monodisperse polystyrene microspheres (PSM) were chosen to minimize the impact of diameter dispersion on mobility, eq 8. The reported diameter was 6.992 ( 0.050 µm, and composition was 95% polystyrene crosslinked with 5% divinylbenzene (Duke Scientific, Palo Alto, CA). These monodisperse PSM will be henceforth referred to as PSM1.

The company reports that the PSM are nonporous and have a specific gravity of 1.05. As no reported susceptibility value of this copolymer has been found, the volumetric susceptibility of polystyrene, in the SI system of units, is used as an approximation, χp ) -8.21 × 10-6, which is similar to that of water, χw ) -9.04 × 10-6. These PSM are similar in size and density to peripheral blood cells, which makes them good candidates for mimicking the behavior of nonmagnetic cells in studies with our continuous separation device, the QMS. The concentration used in the CTV experiments was (0.25-0.5) × 106/mL. For evaluating QMS crossover, with and without magnetic solution, larger, more disperse polystyrene microspheres were used: 15.8 µm, 14.9% CV, at a concentration of 2 × 106/mL. These will be referred to as PSM2 and were also obtained from Duke Scientific. Oxygenated Red Blood Cells. Blood was collected at the NIH (Bethesda, MD) under an IRB-approved protocol, from normal volunteers and for research purposes. The samples were drawn into EDTA Vacutainer tubes (Becton-Dickinson, Rutherford, NJ). The blood sample was washed three times with phosphatebuffered saline (PBS) [7.4 pH, without calcium chloride and without magnesium chloride (Life Technologies, Inc. Gaithersburg, MD)] and resuspended in 4 mL of PBS. It was then shipped to the Cleveland Clinic Foundation on ice pack. Upon arrival, the sample concentration was evaluated with a particle counter (model Z-1, Coulter Corp., Hialeah, FL), following dilution with PBS to a concentration of ∼2 × 106/mL. Paramagnetic Solutions. Lanthanides are a class of elements that are paramagnetic due to unpaired inner 4f electrons.24 These are sufficiently screened so that their magnetic properties are not lost upon ionization or binding. Lanthanides almost always form ionic rather than covalent bonds, and the predominant valence number is 3+. Lanthanides chelated to diethylenetriaminepentaacetic acid (DTPA) have very high stability constants, ensuring there is no interaction with physiological ligands when administered parenterally for MRI studies.38 Their presence is detected by a decrease in T1 and T2 relaxation times compared to surrounding structures. Gadolinium has the fourth highest magnetic moment (7.95 Bohr magnetons) of the 15 lanthanides. The tabulated molar susceptibility of Gd3+ in CGS units is 0.027.39 The high magnetic moment, plus the extreme stability of the GdDTPA complex, makes it an ideal choice for MRI. Commercial preparations of chelated gadolinium, such as Magnevist (Berlex Labs, Richmond, CA) and Optimark (Mallinckrodt Inc., St. Louis, MO), have undergone clinical trials verifying their lack of toxicity and rapid renal clearance rates. Their physical properties are listed in Table 1. Both have trivalent gadolinium, Gd3+, at a concentration of 0.5 M, allowing significant dilution in most of our applications, which is fortuitous in bringing the osmolality nearer the plasma value of 285 mOsm/kg. In CTV experiments, Magnevist is added in such quantity to PBS to achieve the desired concentration (serial dilutions beginning with 1:5) and 50 mL total volume. Then 200 µL of stock PSM1 microspheres is added. The viscosity of Magnevist solutions is measured by a plate-and-cone viscometer (model LVTDV-IICP, Brookfield Engineering Laboratories, Stoughton, MA).

(36) Schimpf, M.; Caldwell, K.; Giddings, J. C., Eds. Field-Flow Fractionation Handbook; John Wiley & Sons: New York, 2000. (37) Bamnolker, H.; Nitzan, B.; Gura, S.; Margel, S. J. Mater. Sci. Lett. 1997, 16, 1412.

(38) Weinmann, H. J.; Brasch, R. C.; Press: W. R.; Wesbey, G. E. AJR, Am. J. Roentgenol. 1984, 142, 619-624. (39) Weast, R. C., Ed. CRC Handbook of Chemistry and Physics, 62nd ed.; CRC Press: Boca Raton, FL, 1981.

Rs )

∆m 2(σ1 + σ2)

(13)

where ∆m is the difference in mobility means and σ1 and σ2 are the standard deviations, which in practice, are estimated by the standard error. In eq 13, the particle mobility means are chosen so that ∆m is positive: m2 - m1 > 0. In some cases, we may increase the resolution between two particle distributions by suspending them in gadolinium solution compared with water. In such a case Rs,s/Rs,w will be greater than 1.0, where the subscripts s and w refer to gadolinium solution and aqueous solution, as above. Looking at eq 13, the resolution ratio of particles suspended in gadolinium solution and water is just

(

)

Rs,s ∆ms σ1,w + σ2,w ) Rs,w ∆mw σ1,s + σ2,s

(14)

With the aid of eq 12 and some algebra, it is easy to show that

[

]

π[Gd]χ′M,Gd Rs,s ηw 8 ) + (R12 - R22) Rs,w ηs 9 (1000 cm3/L)η ∆m s w

(

)

σ1,w + σ2,w (15) σ1,s + σ2,s

If we suppose that small changes in mean mobility will have a negligible impact on the peak dispersion ratio, this ratio becomes 1.0 in eqs 14 and 15 and Rs,s/Rs,w ≈ ∆ms/∆mw. At low gadolinium concentration where the change in viscosity is slight, the particle resolution in gadolinium solution compared with aqueous solution will be increased, provided that the more mobile particle (particle 2) is the smaller one. Conversely, the use of gadolinium in solution may be expected to reduce the resolution when the most mobile particles are also the largest.

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Table 1. Physical Properties of MRI Contrast Agents, Magnevist and Optimark (Company Literature) property

Magnevist

Optimark

Gd concentration molecular weight empirical formula osmolality, 37 °C

0.5 M 938 C28H54GdN5O20 1960 mOsmol/ kg of water 4.9 cP 1.195 g/cm3

0.5 M 661.77 C20H34GdN5O10 1110 mOsmol/ kg of water 3.1 cP 1.16 g/cm3

viscosity, 20 °C density, 25 °C

Figure 2. Midaxial view (A) and cross-sectional view (B) of the quadrupole magnetic sorter (QMS).

Figure 1. Cell tracking velocimeter (CTV) showing coordinate orientation.

Cell Tracking Velocimetry for Measuring Cell Magnetophoresis. The principles and practice of CTV have been described.26-28 The central part of the apparatus is the permanent magnet assembly, with specially designed pole pieces that produce a nearly constant value for the force field strength, Sm, over an area of observation of 1.71 × 1.28 mm (width × height), and a nominal depth of field of 20 µm, Figure 1. Spaced 2.5 mm apart, the pole pieces conduct the magnetic flux into an air gap, into which a 0.6 mm × 1.7 mm i.d., 0.4-mm wall rectangular channel is placed. The orientation of the magnet and flow channel ensures that the field is essentially, two-dimensional: no z component. The major component of the magnetic force is orthogonal to gravity, to avoid a sedimentation contribution to the magnetophoretic mobility. In computing mobilities, only the horizontal component of the gradient is used. In the viewing area, the magnitude of average force field strength Sm is 146 TA/mm2 ( 0.7%, the mean flux density B is 1.41 T, and the mean field gradient dB/dx is 0.131 T/mm. The cell motion in the viewing area is observed with a 5× microscope objective and 2.5× photo eyepiece (Olympus, Tokyo, Japan). A Cohu (San Diego, CA) CCD 4915 camera operating at a frame speed of 30 Hz and a µ-Tech Vision 1000 PCI Bus Frame Grabber (MuTech Corp., Billerica, MA) is used to convert the image into a 640 × 480 pixel array, where each pixel contains eight bits of gray-level information ranging from 0 (black) to 255 (white). The cell tracking velocimetry algorithm is a modification of a 3-D version called particle tracking velocimetry.29 The code uses five successive images to establish the most probable path for a specific particle. From this most probable path, the algorithm determines and reports the 2-D location. A linear fit of the location-time data gives the magnetic migration velocity of each particle, u. Mobilities are found using the known value of

Sm and eq 9. Macros compute statistics for the entire set of particle velocities, including mean, standard deviation, and confidence limits. A wide range of particle mobilities may be measured by varying the acquisition rate, with the highest rate (each frame is acquired) used for the fast particles, and lower rates (every nth frame is acquired, n ) 2, ..., 100 or higher) used for the very slow particles. We have determined that the mobility discrimination power of CTV is below the particle Brownian motion noise. Quadrupole Magnetic Flow Sorter (QMS). The effect of solution susceptibility modification on magnetic field-induced particle motion is further studied using the QMS system. The reader is referred to previous publications on this device, which has also been designated as “magnetic annular SPLITT”.6,40,41 QMS continuously sorts cells (or particles) into two fractions from a feed of typically two populations, magnetic and nonmagnetic, while the magnetic population is disperse in magnetophoretic mobility. Figure 2 is an illustration of the device with panel A showing a midaxial view, and panel B showing a cross-sectional view. The device comprises the A3 quadrupole permanent magnet and the Mark III column (in-house designations). The magnet (1) produces a maximum field of 1.42 T, and a force whose component is primarily radial, with minimal circumferential or axial components. The bore radius of the magnet is 4.82 mm. The channel annulus is bounded by a concentric rod (2) and an outer cylinder (3). The rod radius ri is 2.38 mm and the inner radius of the cylinder ro is 4.53 mm, giving an annulus of 2.15 mm. The top splitter (4) separates feed a′ and carrier b′ streams entering the separation zone, and the bottom splitter (5) separates magnetically depleted a from enriched b streams. The top splitter has an outer radius of 3.20 mm and wall thickness of 0.152 mm, while the bottom splitter has an outer radius of 3.62 mm and wall thickness of 0.152 mm. The inlet streams come together below the top splitter, and the virtual boundary separating them is called the inner splitting cylinder (ISC). An analogous outer splitting cylinder (OSC) separates the outlet flows (dashed lines in panel A). The (40) Zborowski, M.; Williams, P. S.; Sun, L.; Moore, L. R.; Chalmers, J. J. J. Liq. Chromatogr. Relat. Tech. 1997, 20, 2887-2905. (41) Moore, L. R.; Rodriguez, A. R.; Williams, P. S.; McCloskey, K.; Bolwell, B. J.; Nakamura, M.; Chalmers, J. J.; Zborowski, M. J. Magn. Magn. Mater. 2001, 225, 277-284.

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cylindrical volume between them is the transport lamina, and its thickness, d, is the difference in the radii of the OSC and the ISC, rOSC - rISC. The transport lamina acts as a resistive element to the radial migration of magnetic and nonmagnetic particles. Inlet flows at a′ and b′ are provided by a dual-syringe pump with independent drives (Harvard 33, Harvard Apparatus, Inc., Holliston, MA). Another Harvard 33 pump is used in withdraw mode for outlets a and b. Samples are introduced to the column with a six-way valve typically used in liquid chromatography (Rheodyne 7725i, Alltech Associates, Deerfield, IL). The sample loop of the Rheodyne valve is filled by the method of partial loop filling described by the manufacturer. In this case, if the sample volume is less than or equal to half the loop volume, the sample will be quantitatively transferred to the loop. The loop volume is 1.06 mL, and the sample volume is 0.45 mL. Evaluating QMS Magnetophoretic Sorting with UV Detectors. To improve the speed of data processing, we adopted an on-line particle detection method, typical of analytical chromatography.32 Two 254-nm UV detectors with 32-µL flow cells (VUV10, Hyperquan Inc., Colorado Springs, CO) are connected to outlets a and b of the QMS separator. Both detector signals ((4 V) are fed to a two-channel, 12-bit analog-to-digital converter (DI190, Dataq Instruments, Akron, OH), whose output is displayed and saved to PC via Dataq’s Windaq Lite software. Peak area integration is performed with the aid of other software (PeakFit, SPSS, Chicago, IL). The recovery F is defined as the ratio of the number of particles recovered in a particular outlet (a or b) to the number introduced to inlet a′.32 Component flow rates, peak areas, and detector gains are used to calculate recoveries in outlets a and b. For practical purposes, the inlet response is evaluated by direct injection of the sample into the same detector for which recovery is evaluated. For instance, if we wish to evaluate the recovery in outlet a, F(a), then we need a reference response of the a detector, denoted below as ar. The equation is

F(a) )

N(a) Q(a)h(a)/G(a) ) N(a′) Q(ar)h(ar)/G(ar)

(16)

Here N is the number of particles, Q is component flow rate, h is peak area, and G is the detector output voltage gain. The direct injection process is repeated with regard to the b detector, and eq 16 is usedsexcept with b and br replacing a and arsto find F(b). Equation 16 strictly holds true only for a homogeneous particle population. RESULTS AND DISCUSSION Attractive- and Repulsive-Mode Particle Magnetophoresis in Susceptibility-Modified Solutions. To demonstrate the effect of magnetic fluids on particle mobility in the magnetic field, MM1 and PSM1 were combined in approximately equal microsphere number concentrations (0.25 × 106/mL) in a 1:10 Optimark dilution, yielding a gadolinium concentration of 0.05 M. This suspension was then processed using CTV with a frame rate of 3/s and an acquisition of 30 frames. Figure 3 shows sample trajectories plotted from position-time data obtained by CTV. Arrowheads indicate the direction of migration. The rightward 3904 Analytical Chemistry, Vol. 76, No. 14, July 15, 2004

Figure 3. Trajectories of polystyrene microspheres (PSM1) and magnetic microspheres (MM1) in 0.05 M gadolinium solution, as recorded by CTV. The time interval between points was 1/3 s. Table 2. CTV Results of PSM1 in Varying Concentrations of Gadolinium Solution sample number

Gd3+ concn, M

N

m, × 1015 m3/TAs

95% CI of mean, × 1015, m3/TAs

1 2 3 4 5 6 7

0 0.00311 0.00622 0.0124 0.0249 0.0498 0.0995

750 750 511 1000 1522 1325 997

-0.1 ( 1.3 -2.7 ( 1.4 -5.7 ( 1.3 -10.6 ( 1.1 -20.6 ( 3.1 -38.1 ( 4.9 -67.9 ( 7.4

(-0.16, 0.03) (-2.8, -2.6) (-5.8, -5.5) (-10.6, -10.5) (-20.7, -20.4) (-38.3, -37.8) (-68.4, -67.5)

trajectories are those of PSM1, whose lower-than-solution magnetic susceptibility causes them to move against the field gradient and, thus, behave in a manner analogous to a “magnetic hole” or a bubble.17 The leftward trajectories belong to the MM1 that are of higher-than-solution magnetic susceptibility and thus, conventionally, follow the field gradient. As the trajectories encompass the same total time, their lengths are proportional to particle velocities. While it may appear that some trajectories overlap and result in collision, this is usually not the case as CTV tracks particles at different depths. Trajectories derive from horizontal (uniform) and vertical (nonuniform, but relatively weaker) force components. In evaluating mobilities, only the uniform horizontal velocity component is considered. Particle Magnetophoretic Mobility as a Linear Function of the Product of Solution Susceptibility and Viscosity. Manipulation of particle mobility by altering solution susceptibility offers the most tractable experimental model of particle magnetophoresis.17 Suspensions of PSM1 were analyzed by CTV in gadolinium solutions from 0 to 99.5 mM, produced by approximately serial 1:2 dilution of Magnevist in PBS, Table 2. The large number of particles analyzed (N, third column) results in a narrow confidence interval of the mean. A paired t-test between adjacent sample pairs, e.g., sample numbers 1 and 2, 2 and 3, etc., yielded a one-sided P < 0.0005 in all cases (not shown), indicating a high probability that the mean mobilities are distinct. As sample 2 is statistically distinct from sample 1, the negative control, the lower detection limit of PSM1 lies below 2.7 × 10-15

Figure 4. Moblity-viscosity histograms of PSM1 in serial dilutions of gadolinium.

Figure 5. Relationship between measured and expected mobilityviscosity product of PSM1 in gadolinium solution.

m3/TAs. By a more conventional definitionsthe negative control mean + 2 SDsthe detection limit is 2.5 × 10-15 m3/TAs. The motion of PSM1 in gadolinium solution is described by eq 8. However, the solution viscosity depends on the chelated gadolinium preparation concentration, resulting in a nonlinear effect of concentration on mobility. The empirically determined concentration-viscosity relationship from plate-and-cone viscometer data is ηs ) 0.9332 + 1.622 [Gd] + 11.45 [Gd]2, where ηs is in centipoises and [Gd] is the molar concentration of gadolinium from Magnevist in PBS (data not shown). We elected to use the product of viscosity and mobility, mηs, rather than the mobility alone, as the experimental parameter, because of its predicted linear dependence on concentration, eq 12, and thus its suitability for a linear regression analysis. The mobility-viscosity product distributions appear to be normal with high resolution between adjacent histograms except at the lowest concentrations, Figure 4. Moreover, the mηs dispersion increases with Gd concentration, except again, at the lowest values where they appear to be independent of concentration. The mean mηs data are plotted as a function of molar concentration of gadolinium, [Gd], with error bars showing the limits of one standard deviation about the mean, Figure 5. A regression line through the data indicates a high degree of linearity, r2 ) 0.9997, and the expression of the regression line is mηs ) (0.0012 - 8.2037 [Gd]) × 10-16. The software SigmaPlot (SPSS, Inc., Chicago, IL) gives the standard errors of the intercept and slope as 0.0028 × 10-16 and 0.064 × 10-16, respectively. This gives rise to a P value for the null hypothesissthat the intercept

is equal to 0sof 0.696, from which we conclude that, at [Gd] ) 0, mηs and therefore m are likely to equal zero. This supports the assumption, stated above, that the magnetic susceptibilities of PSM1 particles and water are indistinguishable by CTV (χp - χw ) 0). Applying eq 12, the theoretical expression is mηs ) -9.215 [Gd3+] × 10-16. Comparing the two expressions, we see that the measurements are ∼11% lower in magnitude than predicted, due to some indeterminate cause. CTV Instrument Error Contribution to the Measured Particle Magnetophoretic Mobility. To determine the CTV instrument contribution to the overall experimental error, we revert to the particle magnetic migration velocity, u, because it is the experimental parameter that is directly measurable by CTV. The variables whose dispersions contribute to the measured dispersion in the mean of velocity are particle radius R, force field strength Sm, susceptibility difference ∆χ, and solution viscosity ηs, eq 7. These are listed with their uncertainties at 1 SD in the heading and in the third and fourth columns of Table 3. The horizontal component of the magnetic energy density gradient is very uniform, as discussed above, so the relative error in Sm is therefore low (less than 1%). Likewise, PSM1 are uniform particle standards with a low CV in diameter. The uncertainty in susceptibility difference is attributed to uncertainty in preparing solutions to desired concentration, and a 3% relative error is ascribed based on experimental technique. Notice that no uncertainty is given where the concentration of gadolinium is zero, as the susceptibility of pure water is regarded as being known exactly. Solution preparation uncertainty also results in viscosity uncertainty and is evaluated by inserting the concentration error limits into the viscosity-concentration expression. The relative error in viscosity is seen to rise disproportionately at increasing concentration. Comparing the mean of measured velocity u (Table 3, second column) with predicted u from eq 7 (fifth column) leads to a nearconstant percent difference with concentration (sixth column). One possibility is that the actual gadolinium concentration is less than that stated by the manufacturer. At present, we have no means to verify this. Reexamining eq 7, we see that within a given experiment the only variables that contribute to the expected dispersion in the measured velocity are R and Sm, as ∆χ and η are constant for a well-mixed solution. The theoretical variance in u, ∆2u, is found from the relative variances in R and Sm:42

[ (∆RR) + ( S

∆2u ) u2 4

2

)]

∆Sm m

2

(17)

and the method error attributed to CTV is just the square root of the measured variance minus the theoretical variance (∆2um ∆2u)1/2. This is listed in the final column of Table 3. The CTV error is seen to be a near-linear function of concentration when the concentration is equal to or greater than 0.0124 M but seems to reach a lower limit below this concentration. Effect of Solution Susceptibility on CTV Mobility Resolution. The ability to separate two distributions in a device such as the QMS depends on the resolution, eq 13. Distributions are (42) Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry, 5th ed.; McGraw-Hill: New York, 1989.

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Table 3. Error Propagation of PSM1 in Gadolinium Solution from Magnevista [Gd]

meas u × 107, m/s

∆χ × 106, SI

ηs × 104, kg/m‚s

pred u × 107, m/s

mean u, % diff

CTV error in u × 107

0 0.00311 0.00622 0.0124 0.0249 0.0498 0.0995

-0.1 ( 2.0 -4.1 ( 2.1 -8.5 ( 1.9 -15.8 ( 1.6 -30.8 ( 4.7 -57.1 ( 7.3 -102 ( 11

0.0 ( 0.0 -1.055 ( 0.032 -2.110 ( 0.063 -4.21 ( 0.13 -8.45 ( 0.25 -16.90 ( 0.51 -33.8 ( 1.0

9.332 ( 0.000 9.384 ( 0.022 9.437 ( 0.023 9.551 ( 0.027 9.807 ( 0.036 10.424 ( 0.062 12.08 ( 0.14

0.0 -4.58 ( 0.17 -9.10 ( 0.33 -17.94 ( 0.66 -35.1 ( 1.3 -66.0 ( 2.5 -113.8 ( 4.4

10.5 7.03 11.8 12.1 13.5 10.5

2.0 2.1 1.9 1.6 4.7 7.3 11

a

Properties common to all concentrations are radius R ) 3.496 ( 0.035 µm and Sm ) (1.499 ( 0.010) × 10-8 TA/m2.

Figure 6. Improved resolution between MM2 and PSM2 particles due to the addition of gadolinium to the carrier.

regarded as well separated when the resolution is greater than 1.0 and completely separated with resolution of at least 1.5. To verify eq 15, we tested whether a gadolinium solution could increase the resolution between two particle populations of different sizes and mobilities, compared to aqueous solution. In gadolinium-free PBS, the PSM2 and MM2 mobilities were found to be (-0.08 ( 0.54) × 10-13 and (4.9 ( 1.8) × 10-13 m3/ TAs, Figure 6. Applying eq 13, the resolution, Rs, between the two microsphere populations is 1.09. The microsphere mobilities decrease with the addition of the paramagnetic ion to the solution, as discussed above. In 0.025 M gadolinium, the PSM2 and MM2 mobilities were measured as (-1.13 ( 0.53) × 10-13 and (4.37 ( 1.46) × 10-13 m3/TAs, giving a resolution of 1.38. Thus, gadolinium solution increased the resolution by a factor of 1.26. With the aqueous mobility difference, ∆mw, and other parameter values inserted into eq 15, and with the term in standard deviations ignored, it is predicted that the resolution in gadolinium solution, Rs,s, will be 1.17 times as large as the aqueous resolution Rs,w. The experimental resolution increase being more than expected may be due to ignoring the standard deviation ratio in eq 15. Evaluation of the SD ratio retroactively from experimental data yields 1.15, so now the resolution ratio of eq 15 is 1.34, close to the experimental result. We view the increase in resolution as a hopeful finding, applicable, for instance, to the QMS separation of magnetically labeled lymphocytes from larger nonmagnetic tumor cells, a project underway in the laboratory of J.J.C. Attractive- and Repulsive-Mode Magnetophoresis in Application to Magnetic Flow Cell Sorting. We postulated that 3906

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magnetic solution would reduce the undesirable, nonspecific crossover of low-mobility particles into the enriched outlet (outlet b, Figure 2), thereby increasing the purity of positive cells in outlet b. We confirmed this postulate by performing magnetophoretic particle sorting in the QMS apparatus. Nonspecific crossover experiments are routinely performed in our laboratory to evaluate new channel designs and the effects of concentration and flow rate.31,32 The retrieval factor in outlet b, F(b), is defined as the ratio of the number of cells (or particles) appearing in outlet b to the number introduced to the channel at inlet a′. Nonspecific crossover, also referred to as simply “crossover”, occurs when F(b) * 0 for nonmagnetic particles. As these particles are unresponsive to the magnetic field and the transport lamina provides a resistive element to radial migration (Figure 2), ideally, crossover should not occur. It is almost always undesirable, as the migration of nonmagnetic cells into the enriched outlet reduces the magnetic cell purity. (Purity is defined as the ratio of cells of one type to the total number of cells.) This is especially problematic in rare cell isolation, where the crossover of even a few percentages of nonmagnetic cells dramatically reduces purity. It can be shown that the maximum purity in b, P(b), which can be obtained in the ideal case of all the magnetic cells recovered in b, is

P(b) )

P(f) P(f) + (1 - P(f))F(b)

(18)

where P(f) refers to the initial purity of magnetic particles fed into inlet a′ and F(b) is the recovery of nonmagnetic particles in outlet b. An example calculation with realistic conditions shows that, for a feed purity of 1% and a crossover of 1%, the maximum obtainable purity in outlet b is only 50.3%. Williams et al. demonstrated that nonspecific crossover is caused by various channel imperfections.31 Further, several hydrodynamic effects have been suggested to contribute, making it difficult to predict.32 It can often be reduced to an acceptably low level by altering the flow rate ratios between inlets a′ and b′, and outlets a and b, to increase the transport lamina thickness.33,41 The disadvantage of such an approach is that it reduces the magnetic cell mobility range, which can be recovered in outlet b,6 an undesirable feature as most cells of interest are disperse in mobility.43 Here, we postulate that the use of magnetic solutions can reduce crossover without the loss of recovery and mobility range (43) McCloskey, K. E.; Chalmers, J. J.; Zborowski, M. Cytometry 2000, 40, 307315. Erratum in: Cytometry 2000, 41, 150.

- rISC (so that it is positive when rOSC > rISC , as in most applications, and negative when rOSC < rISC). Ideal behavior in the absence of magnetic solution predicts no crossover, which means that F(b) ) 0 for a transport lamina thickness equal to or greater than 0, but for negative d, F(b) can be predicted directly from the flow rates:

F(b) ) 0 F(b) )

Figure 7. Reduced crossover of PSM2 particles in QMS in 0.005 M gadolinium solution as compared with aqueous carrier.

Figure 8. Reduced crossover of red blood cells in QMS in 0.005 M gadolinium solution as compared with aqueous carrier.

associated with thick transport laminae. This was tested on samples of PSM2 and dilute red blood cells in the QMS, with and without magnetic solution. A typical drop-off of the PSM2 and RBC recovery, Fb, with increasing transport lamina thickness is observed in Figures 7 and 8, respectively. The experiments were performed at constant inlet flow rate ratio and varying outlet flow rate ratio. The radial positions of the splitting cylinders, rISC and rOSC, were calculated from inlet and outlet flow rate ratios as described,6 with the transport lamina thickness d defined as rOSC (44) Zborowski, M.; Ostera, G. R.; Moore, L. R.; Milliron, S.; Chalmers, J. J.; Schecter, A. N. Biophys. J. 2003, 84, 2638-2645.

Q(a′) - Q(a) Q(a′)

Q(a) g Q(a′) 0 e Q(a) < Q(a′)

(19)

The continuous smooth line of Figures 7 and 8 shows the ideal theoretical plot using eq 19. Looking first at gadolinium-free results, both PSM2 and RBC exhibit approximately ideal behavior with increasing d in the negative range of dswhere the small influence of crossover is overwhelmed by that of the gross fluid flowsbut deviate from ideal in the positive range of d. When paramagnetic gadolinium is added to the solution at a concentration of 0.005 M, PSM2 and RBC crossover was lowered, consistent with the repulsive-mode magnetophoresis CTV results, discussed above. The effect was larger for PSM2 than for RBC. As eq 12 contains R2, the mobility reduction is expected to be greater for the larger PSM2 compared to RBC. Another interesting feature of Figures 7 and 8 is that the aqueous crossover of RBC is higher than PSM2. This is believed to be due to hydrodynamic causes: the nonspherical, flexible shape of RBC interacts more strongly with fluid shear stress giving rise to hydrodynamic lift force. This causes a horizontal displacement and contributes to crossover. Evidence of this is given by the increased mobility dispersion, measured by CTV, of sedimenting oxygenated RBC compared to PSM1.44 Current work focuses on the effect of magnetic solution in QMS separations with binary particle suspensions, a mixture of low- and high-mobility particles. The goal is to show that, in a practical separation, low-mobility cells or particles can be deterred from crossover by repulsive-mode magnetophoresis, without significantly lowering the recovery of the high-mobility (magnetic) particles. ACKNOWLEDGMENT This work is supported by Grants R01 CA62349 (M.Z.) and R33 CA81662 and R01 CA97391 (J.J.C.) from the National Institutes of Health, and Grants BES-9731059 and BES-0124897 (J.J.C. and M.Z.) and CTS-0125657 (P.S.W.) from the National Science Foundation. The authors also acknowledge expert technical help in RBC preparation by Dr. Graciela Ostera (The National Institutes of Health, Bethesda, MD). Received for review January 14, 2004. Accepted May 5, 2004. AC049910F

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