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Magnetophoresis
Laboratory Techniques in Biochemistry and Molecular Biology, Volume 32 Magnetic Cell Separation M. Zborowski and J. J. Chalmers (Editors)
CHAPTER 5
Magnetophoresis Maciej Zborowski Department of Biomedical Engineering, Lerner Research Institute, Cleveland Clinic, Cleveland, OH 44195, USA
5.1. Magnetophoretic mobility Magnetic cell separation is made possible by a magnetic force acting on magnetically susceptible cells. The nature of the force is that of a magnetic dipole interaction with the external applied field. The magnetic dipole moment of a eukaryotic cell in its natural environment, in a tissue, is typically nonexistent in the absence of a magnetic field, and typically small, and negative, even with the application of high magnetic fields. The reason for that is that the major components of cells, such as water, phospholipid bilayers, proteins, and DNA, are diamagnetic [1, 2]. There are notable exceptions to that rule, such as cells exhibiting paramagnetic behavior (deoxygenated erythrocytes [3], and erythrocytes modified by the presence of methemoglobin [4] or hemozoin [5]) and ferromagnetic behavior of certain prokaryotic cells (such as magnetotactic bacteria) [6]; however, the overwhelming majority of current cell separation applications concerns cells that are diamagnetic in nature [4, 7]. Therefore, the diamagnetism of the cell is important for current applications of magnetic cell separation in that the cells do not substantially interact with the magnetic field unless artificially modified (typically, by the attachment of paramagnetic or ferromagnetic particles) [8–12]. # 2008 Elsevier B.V. All rights reserved DOI: 10.1016/S0075-7535(06)32005-0
106
MAGNETIC CELL SEPARATION
It is worth noting that the cell diamagnetic moment does not saturate even in the presence of very high magnetic fields, such as in excess of 50 T. Therefore, in principle, for very high magnetic fields and gradients, diamagnetic forces may eventually approach the typical paramagnetic or ferromagnetic forces acting on a cell labeled with current, commercial paramagnetic labels. The potential of these very large‐magnitude diamagnetic forces on tissues in high fields has been dramatically demonstrated by the levitation of tissues or entire organisms over superconducting magnets [13]. This potential has created increasing interest in diamagnetic separations of biological materials in fields exceeding 5 T, with the reported field magnitudes of up to 40 T. The magnetic force acting on a diamagnetic particle suspended in a diamagnetic, fluid medium is described by (Chapter 3) 2 B0 ð5:1Þ F ¼ DwV r 2m0 where Dw ¼ wp wf , V is the volume of the particle, and B0 is the applied magnetic field. For a micrometer‐sized particle in a viscous, aqueous solution, the drag force, Fd, of the continuous medium counteracts the eVect of the magnetic field to such an extent that the particle reaches its terminal velocity in a matter of microseconds, in which case it is possible to assume Fd ¼ F
ð5:2Þ
In the limit of a creeping flow, applicable to the motion of a diamagnetic particle in a viscous, diamagnetic media, the Stokes formula for the drag force applies Fd ¼ 6pRv
ð5:3Þ
From a combination of Eqs. (5.1)–(5.3) one obtains the terminal velocity of the particle: 2 DwV B0 r ð5:4Þ v¼ 6pR 2m0
Ch. 5
MAGNETOPHORESIS
107
An interesting property of the above expression is that it is in the form of a product of two quantities that are independent of each other, one being a combination of material properties of the particle and the suspending fluid medium, DwV=6pR, the other of the applied magnetic field, rðB0 2 =2m0 Þ. This suggests a definition of the magnetophoretic mobility as m
DwV 6pR
ð5:5Þ
because it has the desirable property of representing an intrinsic property of the particle in fluid suspension, independent of the applied field. Consequently, this fixes the definition of the force strength, associated with the magnetophoresis as 2 B0 Sm r ð5:6Þ 2m0 which has a familiar form of the magnetic pressure gradient (the Maxwell stress) in free space (in the absence of the magnetic particle and the fluid) [14]. The term ‘‘magnetophoresis’’ (a magnetic field‐induced motion) has been proposed in analogy to the term ‘‘electrophoresis’’ (an electric field‐induced motion) [15, 16]. The phenomenon has been measured for various field and particle combinations [17–24]. A combination of Eqs. (5.4) and (5.5) leads to the expression for the particle field‐induced velocity 2 B0 ð5:7Þ v ¼ mSm ¼ mr 2m0 Therefore, for a given particle and fluid system (described by m), the particle velocity follows the vector rðB0 2=2m0 Þ. In particular, for a field for which rðB0 2=2m0 Þ const, then v const, and the particles move in a nearly uniform motion, that is, in nearly parallel directions and nearly constant velocities. This type of a field has been named an isodynamic field and it has played an important role in separation and particle motion analysis [25–27].
108
MAGNETIC CELL SEPARATION
The above formula is reminiscent of the expression for the particle velocity in dielectrophoresis, due to the force exerted by an inhomogeneous electric field, E, on the induced electric dipole v ¼ mDEP rE 2
ð5:8Þ
where mDEP is the dielectrophoretic mobility [28, 29]. As mentioned above, the intrinsic cell magnetophoresis in the currently available magnetic separators is of limited practical value, such as high‐gradient magnetic separator (HGMS) enrichment of malaria‐infected erythrocytes and magnetic field‐oriented motility of the magnetotactic bacteria [30, 31]. In addition, direction of a diamagnetic cell in an aqueous electrolyte solution, approximating physiological composition of interstitial fluids, is away from the magnetic field source, rather than toward it. However, this creates an opportunity for a highly selective separation process whereby the targeted binding of paramagnetic or ferromagnetic particles makes the cells strongly attracted by the source of the magnetic field, and therefore makes them easily separated from the rest of the cell population [12]. To the lowest‐order approximation, the magnetic properties of a cell‐label ensemble are a weighed average of the properties of the cell and the magnetic label. Typically, because of the weak diamagnetic properties of the cell, the magnetization of the paramagnetic or ferromagnetic label dominates the magnetic forces. This, in turn, reaches saturation in the magnetic fields of the cell separators. Therefore, for most practical applications, the motion of the cell‐ label ensemble is described as induced by the force acting on a saturated dipole (Eq. 3.23) F ¼ mrB0
ð5:9Þ
where m ¼ const is the magnitude of the magnetic dipole moment (note that the above formula applies only to a freely suspended magnetic dipole). Because the saturated dipole‐field interactions are much stronger than the diamagnetic forces acting on the cell itself and the aqueous electrolyte suspension, the magnetic properties of the cell and the fluid can be omitted from the above equation.
Ch. 5
MAGNETOPHORESIS
109
Following the same line of reasoning as that used for the derivation of the diamagnetic particle velocity induced by the magnetic field, the velocity of the cell‐label ensemble is v¼
m rB0 6pR
ð5:10Þ
Here the motion of the cell‐label ensemble follows the lines of rB0 , which, interestingly, are identical with the lines of rðB0 2=2m0 Þ, which govern the motion of the diamagnetic cells. This identity exists by virtue of the relationship rB0 2 ¼ 2B0 rB0 and that B0 is a scalar quantity, and therefore does not aVect the direction of the vector B0 rB0 . The magnitude of the cell‐label ensemble velocity is, however, much greater than that of the unlabeled cells, typically by several orders of magnitude, which is more than suYcient to achieve separation. The diVerence between the motion of an induced magnetic dipole and a saturated magnetic dipole in a magnetic field gradient is illustrated in Figs. 5.1–5.4. Although the motion of the induced and the saturated dipoles follows the same path, the velocity changes diVerently with position for the two cases, one being proportional to jrðB0 2=2m0 Þj and the other to jrB0 j, as discussed above. This poses a problem of a proper definition of the magnetophoretic mobility that would be applicable to both cases. Clearly, the magnetophoretic mobility as defined for a linearly inducible dipole, Eq. (5.5), becomes a function of position, or, more generally, of the magnetic field intensity, for a saturated dipole because in that case wp ¼ wp(H). Conversely, if one defines the magnetophoretic mobility as m=6pR on the basis of Eq. (5.10), then this quantity becomes the function of particle coordinates, or generally, of the magnetic field intensity for an induced dipole because in that case m ¼ MV ¼ wVH. An ad hoc definition that changes depending of the type of the application (diamagnetic separation in strong fields or paramagnetic or ferromagnetic separations in weak fields) and dictated by a particular circumstance does not appear satisfactory. The review of the published literature does not settle the issue either [16–20].
110
MAGNETIC CELL SEPARATION
B
A
M
m
H
H
H at reference point
cp =
H at reference point
M = const H
m≈
cpV f
= const
Fig. 5.1. Magnetophoretic mobility is independent of H for paramagnetic and diamagnetic particles. Here f ¼ 6pR is particle friction coefficient (Eq. 5.5). (A) Paramagnetic particles, magnetic susceptibility. (B) Paramagnetic particles, magnetophoretic mobility.
B
A M
m
Ms H
H H at reference point
cp ≈ Ms ∝ 1 H H
H at reference point
m ≈
cp (H ) V f
∝
1 H
Fig. 5.2. Magnetophoretic mobility depends on H for ferromagnetic and superparamagnetic particles. (A) Ferromagnetic particle, magnetic susceptibility. (B) Ferromagnetic particle, magnetophoretic mobility. Note that the magnetophoretic mobility of ferromagnetic particles greatly exceeds that of the paramagnetic particles in the low H field region (compare with Fig. 5.1).
Ch. 5
111
MAGNETOPHORESIS
A
B
m
0.2 2.0
0.4
0.6
0.8
1
1.2
−12 HB0 = const 1.8 H
1.6
H at P(0.2,1)
1.4 1.2
m≈
cpV f
= const
1.0 P(0.2,1)
Fig. 5.3. Paramagnetic particles travel smaller distances in equal time intervals under the influence of the magnetic pressure gradient (Maxwell stress), shown in panel B, than the ferromagnetic particles (Fig. 5.4) because of their relatively smaller, field‐independent magnetophoretic mobility, m, shown in panel A.
A
B m
0.2 2.0
0.4
0.6
0.8
−12 HB0 = const 1.8 H H at P(0.2,1)
1.6 1.4 1.2
m≈
cp (H) V f
∝
1 H
1.0 P(0.2,1)
1
1.2
112
MAGNETIC CELL SEPARATION
Yet, it is highly desirable to arrive at a definition of the magnetophoretic mobility as a property that is independent of a particular magnetic system and provides means of calculating motion of a cell for any field geometry. The usefulness of such a definition can be compared to that of the electrophoretic mobility in electrophoresis, or sedimentation coeYcient in centrifugal separations. It is the opinion of these authors that the magnetophoretic mobility defined on the basis of Eq. (5.4), as discussed above, is more fundamental than the one suggested by Eq. (5.10). First of all, the diamagnetic properties are exhibited by all matter, whereas paramagnetism and ferromagnetism are limited to only a few, albeit very important, cases [32]. Second, the magnetophoretic mobility as defined in Eq. (5.5) combines particle radius with the material properties of the particle and the fluid media (magnetic susceptibilities and fluid viscosity), unlike the mobility suggested by Eq. (5.10) (magnetic moment is not a material property). Third, there is a precedent in dielectrophoresis in defining the mobility of moving dipoles in viscous media as the ratio of velocity to the field energy gradient, rather than field gradient, as expressed in Eq. (5.8) [28, 29]. Fourth, the extension of the magnetophoretic mobility definition from diamagnetic materials to para‐ and ferromagnetic materials follows a natural path of admitting the dependence of the magnetic susceptibility on the applied field (Fig. 5.2B), as it is routinely done in the studies of magnetic properties of materials [33]. In conclusion, we propose the following definition of the magnetophoretic mobility that is equally applicable to the linearly
Fig. 5.4. Ferromagnetic particle. The initial position of the particle was set the same as in the case of the paramagnetic particle, shown in Fig. 5.3. The saturation magnetization of the ferromagnetic particle was assumed the same as that of the induced magnetization of the paramagnetic particle at the point P(0.2,1) (such a case is more realistic for the superparamagnetic particle than for the paramagnetic particle). Note that the trajectory of the ferromagnetic particle has the same shape as that of the paramagnetic particle, and that it travels a comparatively greater distance than that shown in Fig. 5.3.
Ch. 5
113
MAGNETOPHORESIS
polarizable magnetic materials (diamagnetic and paramagnetic) and nonlinear materials, whose magnetic susceptibility is a function of the field (ferromagnetic and superparamagnetic): m
½wp ðHÞ wf ðHÞV 6pR
ð5:11Þ
where we indicate explicitly the dependence of the particle and the fluid susceptibilities on the applied field, wp ¼ wp ðHÞ and wf ¼ wf ðHÞ, respectively.
t
2t 3t
N
S
Fig. 5.5. Cross section for capture—infinite cylinder. Solid lines represent magnetic particle trajectories traveled in equal periods of time, ending at the surface of the ferromagnetic cylinder magnetized by an external magnetic field (as in Figs. 4.1 and 4.2). Broken lines represent boundary of the area cleared of the magnetic particles in a given time, t. Note diminishing increase in magnetic capture area with the increasing time, as indicated by broken lines marked 2t and 3t.
114
MAGNETIC CELL SEPARATION
t
2t
3t
N
S
Fig. 5.6. Cross‐section for capture—rectangular magnet. See caption for Fig. 5.5 for explanation.
5.1.1. Cross‐section for magnetic capture Cell magnetophoretic mobility, m, is the primary determinant of the magnetic separation outcome. There are other parameters, however, that have to be taken into account if the separation is to be accomplished within the constraints imposed by cell biology; such constraints include a maximum separation time and minimum mechanical stresses. A convenient way of summarizing the magnetic
Ch. 5
115
MAGNETOPHORESIS
t 2t N 3t
S
Fig. 5.7. Cross‐section for capture—interpolar gap. See caption for Fig. 5.5 for explanation.
separator characteristics is to determine its cross‐section for magnetic capture. In simple terms, it describes the area that is cleared of the magnetically mobile cells down to a certain, desirably small fraction of the initial purity. The actual percentage may depend on the application, and it can be perhaps as high as 5% or as low as 0.1%. In geometrical terms, the cross section for capture is an area bound by a curve made of all those points that are the initial positions of the farthest cells captured in a given time t. This applies to a suitable projection of the three‐dimensional magnet assembly on a two‐dimensional surface that is used to calculate the magnetic capture cross section. Typically, such a surface is normal to the direction of the bulk flow of the cell mixture. For a single wire
116
MAGNETIC CELL SEPARATION
HGMS and the bulk flow parallel to the long axis of the wire, the cross section for capture is calculated for a surface normal to the wire. On that surface, the capture cross‐section is determined by a family of cell trajectories parametrized by their lengths, l, that are traveled by the equal amounts of time between the point on the curve bounding the cross‐section for capture to the surface of the magnet: * 1 + l¼m
d HB0 2
dl
Dt,
Dt ¼ const
ð5:12Þ
l
In other words, the linear extension of the capture cross section is directly proportional to cell magnetophoretic mobility, m, the average magnitude of the Maxwell stress gradient along the cell trajectory, hd12 HB0 =dlil , and the cell residence time in the magnetic field, Dt. The cross sections are illustrated in Figs. 5.5–5.7 for selected magnet geometries.
References [1] Torbet, J. (1987). Using magnetic field orientation to study structure and assembly. Trends Biochem. Sci. 12, 327–30. [2] Torbet, J. and Dickens, M. J. (1984). Orientation of skeletal muscle actin in strong magnetic fields. FEBS Lett. 173, 403–6. [3] Fabry, M. E. and San George, R. C. (1983). EVect of magnetic susceptibility on nuclear magnetic resonance signals arising from red cells: A warning. Biochemistry 22, 4119–25. [4] Zborowski, M., Ostera, G. R., Moore, L. R., Milliron, S., Chalmers, J. J. and Schechter, A. N. (2003). Red blood cell magnetophoresis. Biophys. J. 84, 2638–45. [5] Bohle, D. S., Debrunner, P., Jordan, P. A., Madsen, S. K. and Schulz, C. E. (1998). Aggregated heme detoxification byproducts in malarial trophozoites: Beta‐hematin and malaria pigment have a single S ¼ 5/2 iron environment in the bulk phase as determined by EPR and magnetic Moessbauer spectroscopy. J. Am. Chem. Soc. 120, 8255–6.
Ch. 5
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117
[6] Bazylinski, D. A. and Frankel, R. B. (2004). Magnetosome formation in prokaryotes. Nat. Rev, Microbiol, 2, 217–30. [7] Rosenblatt, C., Torres de Araujo, F. F. and Frankel, R. B. (1982). Birefringence determination of magnetic moments of magnetotactic bacteria. Biophys. J. 40, 83–5. [8] Ugelstad, J., Stenstad, P., Kilaas, L., Prestvik, W. S., Herje, R., Berge, A. and Hornes, E. (1993). Monodisperse magnetic polymer particles. New biochemical and biomedical applications. Blood Purif. 11, 347–69. [9] Miltenyi, S., Mu¨ller, W., Weichel, W. and Radbruch, A. (1990). High gradient magnetic cell separation with MACS. Cytometry 11, 231–8. [10] Liberti, P. A. and Feeley, B. P. (1991). Analytical‐ and process‐scale cell separation with bioreceptor ferrofluids and high gradient magnetic separation. In: Cell Separation Science and Technology (Compala, D. S. and Todd, P., eds.), Vol. 464. ACS Symposium Series, Washington, pp. 268–88. [11] Thomas, T. E., Richards, A. J., Roath, O. S., Watson, J. H. P., Smith, R. J. S. and Lansdorp, P. M. (1993). Positive selection of human blood cells using improved high gradient magnetic separation filters. J. Hematother. 2, 297–303. [12] Pankhurst, Q. A., Connolly, J., Jones, S. K. and Dobson, J. (2003). Applications of magnetic nanoparticles in biomedicine. DOI:10.1088/ 0022–3727/36/13/201. J. Phys. D. Appl. Phys. 36, R167–81. [13] Simon, M. D. and Geim, A. K. (2000). Diamagnetic levitation: Flying frogs and floating magnets (invited). Appl. Phys. A. 87, 6200–4. [14] Sutton, G. W. and Sherman, A. (2006). Engineering magnetohydrodynamics. Dover Publications Inc., Mineola, NY. [15] Winoto‐Morbach, S., Tchikov, V. and Mueller‐Ruchholtz, W. (1994). Magnetophoresis: I. Detection of magnetically labeled cells. J. Clin. Lab. Anal. 8, 400–6. [16] Leenov, D. and Kolin, A. (1954). Theory of electromagnetophoresis. I. Magnetohydrodynamic forces experienced by spherical and symmetrically oriented cylindrical particles. J. Chem. Phys. 22, 683–8. [17] Gill, S. J., Malone, C. P. and Downing, M. (1960). Magnetic susceptibility measurements of single small particles. Rev. Sci. Instrum. 31, 1299–303. [18] Wilhelm, C., Gazeau, F. and Bacri, J.‐C. (2002). Magnetophoresis and ferromagnetic resonance of magnetically labeled cells. Eur. Biophys. J. 31, 118–25. [19] Suwa, M. and Watarai, H. (2001). Magnetophoretic velocimetry of manganese(II) in a single emulsion droplet at the femtomole level. Anal. Chem. 73, 5214–9. [20] Watarai, H. and Namba, M. (2002). Capillary magnetophoresis of human blood cells and their magnetophoretic trapping in a flow system. J. Chromatogr. A. 961, 3–8.
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[21] Watarai, H. and Suwa, M. (2004). Magnetophoresis and electromagnetophoresis of microparticles in liquids. Anal. Bioanal. Chem. 378, 1693–9. [22] Furlani, E. P. (2007). Magnetophoretic separation of blood cells at the microscale. J. Phys. D: Appl. Phys. 40, 1313–9. [23] Tchikov, V., Schuetze, S. and Kroenke, M. (1999). Comparison between immunofluorescence and immunomagnetic techniques of cytometry. J. Magn. Magn. Mater. 194, 242–7. [24] McCloskey, K. E., Moore, L. R., Hoyos, M., Rodriguez, A., Chalmers, J. J. and Zborowski, M. (2003). Magnetophoretic cell sorting is a function of antibody binding capacity. Biotechnol. Prog. 19, 899–907. [25] Frantz, S. G. (1936). Magnetic separation methods and means. Patent No. 2,056,426 (US). [26] Smolkin, M. R. and Smolkin, R. D. (2006). Calculation and analysis of the magnetic force acting on a particle in the magnetic field of separator. Analysis of the equations used in the magnetic methods of separation. IEEE Trans. Magn. 42, 3682–93. [27] Moore, L. R., Zborowski, M., Nakamura, M., McCloskey, K., Gura, S., Zuberi, M., Margel, S. and Chalmers, J. J. (2000). The use of magnetite‐ doped polymeric microspheres in calibrating cell tracking velocimetry. J. Biochem. Biophys. Methods. 44, 115–30. [28] Pohl, H. A. (1978). Dielectrophoresis: The behavior of neutral matter in nonuniform electric fields. Published in Cambridge Monographs on Physics, edited by Woolfson, M. M. and Ziman, J. M. Cambridge University Press, Cambridge. [29] Markx, G. H. and Pethig, R. (1995). Dielectrophoretic separation of cells: Continuous separation. Biotechnol. Bioeng. 45, 337–43. [30] Carter, V., Cable, H. C., Underhill, B. A., Williams, J. and Hurd, H. (2003). Isolation of Plasmodium berghei ookinetes in culture using Nycodenz density gradient columns and magnetic isolation. Malar. J. 2, 35. [31] Dunin‐Borkowski, R. E., McCartney, M. R., Frankel, R. B., Bazylinski, D. A., Posfai, M. and Buseck, P. R. (1998). Magnetic microstructure of magnetotactic bacteria by electron holography. Science 282, 1868–70. [32] Purcell, E. M. (1985). Electricity and Magnetism. Berkeley Physics Course‐ Volume 2. McGraw‐Hill Book Co., New York. [33] Melcher, J. R. (1981). Continuum electromechanics. The MIT Press, Cambridge, MA.
CHAPTER 5
Magnetophoresis Maciej Zborowski Department of Biomedical Engineering, Lerner Research Institute, Cleveland Clinic, Cleveland, OH 44195, USA
5.1. Magnetophoretic mobility Magnetic cell separation is made possible by a magnetic force acting on magnetically susceptible cells. The nature of the force is that of a magnetic dipole interaction with the external applied field. The magnetic dipole moment of a eukaryotic cell in its natural environment, in a tissue, is typically nonexistent in the absence of a magnetic field, and typically small, and negative, even with the application of high magnetic fields. The reason for that is that the major components of cells, such as water, phospholipid bilayers, proteins, and DNA, are diamagnetic [1, 2]. There are notable exceptions to that rule, such as cells exhibiting paramagnetic behavior (deoxygenated erythrocytes [3], and erythrocytes modified by the presence of methemoglobin [4] or hemozoin [5]) and ferromagnetic behavior of certain prokaryotic cells (such as magnetotactic bacteria) [6]; however, the overwhelming majority of current cell separation applications concerns cells that are diamagnetic in nature [4, 7]. Therefore, the diamagnetism of the cell is important for current applications of magnetic cell separation in that the cells do not substantially interact with the magnetic field unless artificially modified (typically, by the attachment of paramagnetic or ferromagnetic particles) [8–12]. # 2008 Elsevier B.V. All rights reserved DOI: 10.1016/S0075-7535(06)32005-0
106
MAGNETIC CELL SEPARATION
It is worth noting that the cell diamagnetic moment does not saturate even in the presence of very high magnetic fields, such as in excess of 50 T. Therefore, in principle, for very high magnetic fields and gradients, diamagnetic forces may eventually approach the typical paramagnetic or ferromagnetic forces acting on a cell labeled with current, commercial paramagnetic labels. The potential of these very large‐magnitude diamagnetic forces on tissues in high fields has been dramatically demonstrated by the levitation of tissues or entire organisms over superconducting magnets [13]. This potential has created increasing interest in diamagnetic separations of biological materials in fields exceeding 5 T, with the reported field magnitudes of up to 40 T. The magnetic force acting on a diamagnetic particle suspended in a diamagnetic, fluid medium is described by (Chapter 3) 2 B0 ð5:1Þ F ¼ DwV r 2m0 where Dw ¼ wp wf , V is the volume of the particle, and B0 is the applied magnetic field. For a micrometer‐sized particle in a viscous, aqueous solution, the drag force, Fd, of the continuous medium counteracts the eVect of the magnetic field to such an extent that the particle reaches its terminal velocity in a matter of microseconds, in which case it is possible to assume Fd ¼ F
ð5:2Þ
In the limit of a creeping flow, applicable to the motion of a diamagnetic particle in a viscous, diamagnetic media, the Stokes formula for the drag force applies Fd ¼ 6pRv
ð5:3Þ
From a combination of Eqs. (5.1)–(5.3) one obtains the terminal velocity of the particle: 2 DwV B0 r ð5:4Þ v¼ 6pR 2m0
Ch. 5
MAGNETOPHORESIS
107
An interesting property of the above expression is that it is in the form of a product of two quantities that are independent of each other, one being a combination of material properties of the particle and the suspending fluid medium, DwV=6pR, the other of the applied magnetic field, rðB0 2 =2m0 Þ. This suggests a definition of the magnetophoretic mobility as m
DwV 6pR
ð5:5Þ
because it has the desirable property of representing an intrinsic property of the particle in fluid suspension, independent of the applied field. Consequently, this fixes the definition of the force strength, associated with the magnetophoresis as 2 B0 Sm r ð5:6Þ 2m0 which has a familiar form of the magnetic pressure gradient (the Maxwell stress) in free space (in the absence of the magnetic particle and the fluid) [14]. The term ‘‘magnetophoresis’’ (a magnetic field‐induced motion) has been proposed in analogy to the term ‘‘electrophoresis’’ (an electric field‐induced motion) [15, 16]. The phenomenon has been measured for various field and particle combinations [17–24]. A combination of Eqs. (5.4) and (5.5) leads to the expression for the particle field‐induced velocity 2 B0 ð5:7Þ v ¼ mSm ¼ mr 2m0 Therefore, for a given particle and fluid system (described by m), the particle velocity follows the vector rðB0 2=2m0 Þ. In particular, for a field for which rðB0 2=2m0 Þ const, then v const, and the particles move in a nearly uniform motion, that is, in nearly parallel directions and nearly constant velocities. This type of a field has been named an isodynamic field and it has played an important role in separation and particle motion analysis [25–27].
108
MAGNETIC CELL SEPARATION
The above formula is reminiscent of the expression for the particle velocity in dielectrophoresis, due to the force exerted by an inhomogeneous electric field, E, on the induced electric dipole v ¼ mDEP rE 2
ð5:8Þ
where mDEP is the dielectrophoretic mobility [28, 29]. As mentioned above, the intrinsic cell magnetophoresis in the currently available magnetic separators is of limited practical value, such as high‐gradient magnetic separator (HGMS) enrichment of malaria‐infected erythrocytes and magnetic field‐oriented motility of the magnetotactic bacteria [30, 31]. In addition, direction of a diamagnetic cell in an aqueous electrolyte solution, approximating physiological composition of interstitial fluids, is away from the magnetic field source, rather than toward it. However, this creates an opportunity for a highly selective separation process whereby the targeted binding of paramagnetic or ferromagnetic particles makes the cells strongly attracted by the source of the magnetic field, and therefore makes them easily separated from the rest of the cell population [12]. To the lowest‐order approximation, the magnetic properties of a cell‐label ensemble are a weighed average of the properties of the cell and the magnetic label. Typically, because of the weak diamagnetic properties of the cell, the magnetization of the paramagnetic or ferromagnetic label dominates the magnetic forces. This, in turn, reaches saturation in the magnetic fields of the cell separators. Therefore, for most practical applications, the motion of the cell‐ label ensemble is described as induced by the force acting on a saturated dipole (Eq. 3.23) F ¼ mrB0
ð5:9Þ
where m ¼ const is the magnitude of the magnetic dipole moment (note that the above formula applies only to a freely suspended magnetic dipole). Because the saturated dipole‐field interactions are much stronger than the diamagnetic forces acting on the cell itself and the aqueous electrolyte suspension, the magnetic properties of the cell and the fluid can be omitted from the above equation.
Ch. 5
MAGNETOPHORESIS
109
Following the same line of reasoning as that used for the derivation of the diamagnetic particle velocity induced by the magnetic field, the velocity of the cell‐label ensemble is v¼
m rB0 6pR
ð5:10Þ
Here the motion of the cell‐label ensemble follows the lines of rB0 , which, interestingly, are identical with the lines of rðB0 2=2m0 Þ, which govern the motion of the diamagnetic cells. This identity exists by virtue of the relationship rB0 2 ¼ 2B0 rB0 and that B0 is a scalar quantity, and therefore does not aVect the direction of the vector B0 rB0 . The magnitude of the cell‐label ensemble velocity is, however, much greater than that of the unlabeled cells, typically by several orders of magnitude, which is more than suYcient to achieve separation. The diVerence between the motion of an induced magnetic dipole and a saturated magnetic dipole in a magnetic field gradient is illustrated in Figs. 5.1–5.4. Although the motion of the induced and the saturated dipoles follows the same path, the velocity changes diVerently with position for the two cases, one being proportional to jrðB0 2=2m0 Þj and the other to jrB0 j, as discussed above. This poses a problem of a proper definition of the magnetophoretic mobility that would be applicable to both cases. Clearly, the magnetophoretic mobility as defined for a linearly inducible dipole, Eq. (5.5), becomes a function of position, or, more generally, of the magnetic field intensity, for a saturated dipole because in that case wp ¼ wp(H). Conversely, if one defines the magnetophoretic mobility as m=6pR on the basis of Eq. (5.10), then this quantity becomes the function of particle coordinates, or generally, of the magnetic field intensity for an induced dipole because in that case m ¼ MV ¼ wVH. An ad hoc definition that changes depending of the type of the application (diamagnetic separation in strong fields or paramagnetic or ferromagnetic separations in weak fields) and dictated by a particular circumstance does not appear satisfactory. The review of the published literature does not settle the issue either [16–20].
110
MAGNETIC CELL SEPARATION
B
A
M
m
H
H
H at reference point
cp =
H at reference point
M = const H
m≈
cpV f
= const
Fig. 5.1. Magnetophoretic mobility is independent of H for paramagnetic and diamagnetic particles. Here f ¼ 6pR is particle friction coefficient (Eq. 5.5). (A) Paramagnetic particles, magnetic susceptibility. (B) Paramagnetic particles, magnetophoretic mobility.
B
A M
m
Ms H
H H at reference point
cp ≈ Ms ∝ 1 H H
H at reference point
m ≈
cp (H ) V f
∝
1 H
Fig. 5.2. Magnetophoretic mobility depends on H for ferromagnetic and superparamagnetic particles. (A) Ferromagnetic particle, magnetic susceptibility. (B) Ferromagnetic particle, magnetophoretic mobility. Note that the magnetophoretic mobility of ferromagnetic particles greatly exceeds that of the paramagnetic particles in the low H field region (compare with Fig. 5.1).
Ch. 5
111
MAGNETOPHORESIS
A
B
m
0.2 2.0
0.4
0.6
0.8
1
1.2
−12 HB0 = const 1.8 H
1.6
H at P(0.2,1)
1.4 1.2
m≈
cpV f
= const
1.0 P(0.2,1)
Fig. 5.3. Paramagnetic particles travel smaller distances in equal time intervals under the influence of the magnetic pressure gradient (Maxwell stress), shown in panel B, than the ferromagnetic particles (Fig. 5.4) because of their relatively smaller, field‐independent magnetophoretic mobility, m, shown in panel A.
A
B m
0.2 2.0
0.4
0.6
0.8
−12 HB0 = const 1.8 H H at P(0.2,1)
1.6 1.4 1.2
m≈
cp (H) V f
∝
1 H
1.0 P(0.2,1)
1
1.2
112
MAGNETIC CELL SEPARATION
Yet, it is highly desirable to arrive at a definition of the magnetophoretic mobility as a property that is independent of a particular magnetic system and provides means of calculating motion of a cell for any field geometry. The usefulness of such a definition can be compared to that of the electrophoretic mobility in electrophoresis, or sedimentation coeYcient in centrifugal separations. It is the opinion of these authors that the magnetophoretic mobility defined on the basis of Eq. (5.4), as discussed above, is more fundamental than the one suggested by Eq. (5.10). First of all, the diamagnetic properties are exhibited by all matter, whereas paramagnetism and ferromagnetism are limited to only a few, albeit very important, cases [32]. Second, the magnetophoretic mobility as defined in Eq. (5.5) combines particle radius with the material properties of the particle and the fluid media (magnetic susceptibilities and fluid viscosity), unlike the mobility suggested by Eq. (5.10) (magnetic moment is not a material property). Third, there is a precedent in dielectrophoresis in defining the mobility of moving dipoles in viscous media as the ratio of velocity to the field energy gradient, rather than field gradient, as expressed in Eq. (5.8) [28, 29]. Fourth, the extension of the magnetophoretic mobility definition from diamagnetic materials to para‐ and ferromagnetic materials follows a natural path of admitting the dependence of the magnetic susceptibility on the applied field (Fig. 5.2B), as it is routinely done in the studies of magnetic properties of materials [33]. In conclusion, we propose the following definition of the magnetophoretic mobility that is equally applicable to the linearly
Fig. 5.4. Ferromagnetic particle. The initial position of the particle was set the same as in the case of the paramagnetic particle, shown in Fig. 5.3. The saturation magnetization of the ferromagnetic particle was assumed the same as that of the induced magnetization of the paramagnetic particle at the point P(0.2,1) (such a case is more realistic for the superparamagnetic particle than for the paramagnetic particle). Note that the trajectory of the ferromagnetic particle has the same shape as that of the paramagnetic particle, and that it travels a comparatively greater distance than that shown in Fig. 5.3.
Ch. 5
113
MAGNETOPHORESIS
polarizable magnetic materials (diamagnetic and paramagnetic) and nonlinear materials, whose magnetic susceptibility is a function of the field (ferromagnetic and superparamagnetic): m
½wp ðHÞ wf ðHÞV 6pR
ð5:11Þ
where we indicate explicitly the dependence of the particle and the fluid susceptibilities on the applied field, wp ¼ wp ðHÞ and wf ¼ wf ðHÞ, respectively.
t
2t 3t
N
S
Fig. 5.5. Cross section for capture—infinite cylinder. Solid lines represent magnetic particle trajectories traveled in equal periods of time, ending at the surface of the ferromagnetic cylinder magnetized by an external magnetic field (as in Figs. 4.1 and 4.2). Broken lines represent boundary of the area cleared of the magnetic particles in a given time, t. Note diminishing increase in magnetic capture area with the increasing time, as indicated by broken lines marked 2t and 3t.
114
MAGNETIC CELL SEPARATION
t
2t
3t
N
S
Fig. 5.6. Cross‐section for capture—rectangular magnet. See caption for Fig. 5.5 for explanation.
5.1.1. Cross‐section for magnetic capture Cell magnetophoretic mobility, m, is the primary determinant of the magnetic separation outcome. There are other parameters, however, that have to be taken into account if the separation is to be accomplished within the constraints imposed by cell biology; such constraints include a maximum separation time and minimum mechanical stresses. A convenient way of summarizing the magnetic
Ch. 5
115
MAGNETOPHORESIS
t 2t N 3t
S
Fig. 5.7. Cross‐section for capture—interpolar gap. See caption for Fig. 5.5 for explanation.
separator characteristics is to determine its cross‐section for magnetic capture. In simple terms, it describes the area that is cleared of the magnetically mobile cells down to a certain, desirably small fraction of the initial purity. The actual percentage may depend on the application, and it can be perhaps as high as 5% or as low as 0.1%. In geometrical terms, the cross section for capture is an area bound by a curve made of all those points that are the initial positions of the farthest cells captured in a given time t. This applies to a suitable projection of the three‐dimensional magnet assembly on a two‐dimensional surface that is used to calculate the magnetic capture cross section. Typically, such a surface is normal to the direction of the bulk flow of the cell mixture. For a single wire
116
MAGNETIC CELL SEPARATION
HGMS and the bulk flow parallel to the long axis of the wire, the cross section for capture is calculated for a surface normal to the wire. On that surface, the capture cross‐section is determined by a family of cell trajectories parametrized by their lengths, l, that are traveled by the equal amounts of time between the point on the curve bounding the cross‐section for capture to the surface of the magnet: * 1 + l¼m
d HB0 2
dl
Dt,
Dt ¼ const
ð5:12Þ
l
In other words, the linear extension of the capture cross section is directly proportional to cell magnetophoretic mobility, m, the average magnitude of the Maxwell stress gradient along the cell trajectory, hd12 HB0 =dlil , and the cell residence time in the magnetic field, Dt. The cross sections are illustrated in Figs. 5.5–5.7 for selected magnet geometries.
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