ANALYTICAL BIOCHEMISTRY 71, 6 1 5 - 6 2 2 (1976)
Design Principles for a Counterflow Centrifugation Cell Separation Chamber Counterflow centrifugation has been available as a technique for cell separation since 1948 when Lindahl first published an account of the method in Nature (1,2). It has been used comparatively little, however, since available systems have been complex or lacking sensitivity. We have overcome these difficulties using simple principles to design a separation chamber. The design analysis a n d results of the fractionation of populations of human erythrocytes and leukocytes are presented. In a counterflow centrifuge, fluid is pumped into the center of the centrifuge rotor through a rotating seal. This fluid moves through internal tubing to the outside of the rotor and is then turned toward the center through a divergent separation chamber. At the end of this chamber, the flow reconverges into a tube of small diameter through which it moves back to the center of the rotor. It then leaves the system through a second rotating seal. The cell suspension to be fractionated is introduced into the inlet tube at some convenient point. If the pump rate is correctly chosen, cells move through the system until they reach the separation chamber, where they will come to rest relative to the rotor at a point where the outwardly directed inertial forces are balanced by the inwardly directed fluid dynamic and buoyancy forces. Forces in the tangential plane, namely the Coriolis force and the opposing pressure gradients, are not important under the operating conditions of this system. The small circulation resulting from these forces is, in any case, in the tangential plane. The cell population distributes itself along the separation chamber in accordance with certain physical properties and the velocity gradient within the chamber. To elute a fraction of cells from the centrifuge, the pump rate is increased until the innermost group of cells reaches the end of the divergent chamber. At this point the fluid velocity is increased by the converging walls and the cells are swept downstream to escape from the rotor. The velocity gradient in the chamber is a function of chamber shape. Consequently, it is possible to design a chamber which separates cells having different physical properties. We may write for a cell at equilibrium pVoJ2r = pFVoJ~r + kx~)Vd
[1]
where/9 is the cell density, V its volume, d its diameter, and ks a shape factor; pr is the fluid density, ~/its viscosity, and V local velocity; while o~ is the centrifuge velocity and r the distance from the center of rotation to the cell. Equation [1] may be derived more rigorously from the general 615 Copyright © 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.
616
SHORT COMMUNICATIONS
equation of motion for a particle in combined centrifugal and hydrodynamic fields. This has been done in the Appendix, where justification of the inherent approximations is also made. Rearranging, and writing p' for p - Pr, and k2d3 for V, we have r = k~V/oJ2p'd 2
[2]
where k = kl/k2. Thus the value of r for equilibrium and for a given rotational speed is a function of p'd 2 and k, which are cell parameters, and V, which is a controllable function of radius. For example, we could make the chamber shape such that (d/dr)(p'd2/k) is constant, leading to a linear distribution of cells according to the magnitude of p'd2/k. This concept leads from Eq. [2] to the differential equation d p'd ~ -
dr
-
k
-
~1 d V co2 dr r
- A, a constant.
[3]
Integration of [3] gives V = (At 2 + Br)(co2/r/).
[4]
The desired variation in cross-sectional area S of the chamber then follows. For any pump rate, P (measured in volume per unit time), we have S = P/V = P~/o~2(Ar2 + Br)
[5]
A convenient value of A may be obtained from the range of p'd2/k of the cell population to be fractionated. B may be obtained from an arbitrary inlet cross-sectional area and pump rate P. A chamber designed according to these principles will have a volume which is small if the inlet dimensions are small. Large inlet dimensions lead to pump rates which are unacceptably high. However, we may look for a chamber shape in which cells having high values of p'd2/k and those having low values are not separated significantly in the outer portion of the chamber. This condition is fulfilled by making (d/dr)(p'd2/k) large, which necessarily, from Eq. [3], leads to the condition that rdV/dr must be large compared with V, because (d/dr)(V/r) = (1/r2)[(rdV/dr) - V]. Physically, this is achieved by making the outer portion of the chamber cross-sectional area diverge rapidly. In the chamber we have designed, the zone of rapid expansion occupies the outer third of its length. This is run smoothly into an inverse square contour, as given by Eq. [5] with B = 0, which occupies the remaining two-thirds of the chamber length. The exact contours are not critical, so long as the outer portion of the chamber diverges rapidly and the inner position slowly. The two photographs, shown in Fig. 1, demonstrate the functional characteristics of the two regions during a red cell separation. In Fig. la, cells occupy only the outer region where the divergence is large. The cells furthest downstream form a clearly defined front. As the pump
SHORT COMMUNICATIONS
617
FIG. 1. Photographs showing the location of red blood cells in the separation chamber (a) at the completion of loading and (b) prior to elution of a fraction,
rate is increased, the cells move into the slowly divergent portion of the chamber, dispersing according to their values of the separation parameter and the front disappears (see Fig. lb). There is a physical limit to the slowness of divergence of the inner portion dictated by the necessity that the hydrodynamic forces acting on a cell as it moves inward should not decrease tess rapidly than the sum of buoyant and inertial forces. This would lead to unstable equilibrium. The limiting value of convergence is that which leads to a velocity proportional to the radius. The chamber described has been used with the Beckman Elutriator Rotor system to fractionate human red cells into different age groups. As red cells age, they become more dense (3), smaller (4), and lose activity of certain enzymes (5,6). Shape and membrane flexibility also change significantly (1). The fluid medium used was Hank's balanced salt solution (Ca 2+, Mg 2+ free) containing 0.2% bovine serum albumin. Centrifuge speed was 2000 RPM. The assay systems used to establish the effectiveness of the separation were the decrease in activity of GOT (aspartate aminotransferase, EC 2.6.1.1), expressed as a ratio of activity to hemoglobin (5,6), and the measurement of cell volume distribution for each fraction using the Coulter ZBI sizing apparatus. Results of these assays for a number of separations are combined in Fig. 2.
618
SHORT COMMUNICATIONS
,o
;o
do
~
,;o
¢,o -~o
Cell Volume (p~)
FIG. 2. Erythrocyte separation. The ratio of GOT to hemoglobin is plotted against mean cell volume for each fraction taken from the centrifuge. Each curve represents a different separation and different donor.
Old cells constitute early fractions, while younger cells make up later fractions. Although p'd 2 is lower for young than old cells, the young cell has a lower drag, either because of its basic shape or because its flexibility allows it to conform to a low drag configuration under the system of prevailing forces. This results in a value of p'd2/k which is higher in young cells. Consequently, higher fluid velocities are required to elute them. In addition to erythrocyte fractionations, this chamber has proven useful for other cell systems. Two of us (NP, JB) have used it to separate human peripheral leukocytes into their component species. Some results of this fractionation are shown in Fig. 3. Summary. A concise approximation of the equilibrium condition for particles in a counterflow centrifuge has been used as the basis for the design of a cell separation chamber. Used with the Beckman Elutriator rotor system, the chamber has proven to be an effective means of separating populations of human erythrocytes into different age groups, and of separating leukocyte preparations into their various cell types.
FIG. 3. Leucocyte fractionation. (a) The photograph shows a sample of the cells loaded into the separation chamber. A preliminary separation of leukocytes from erythrocytes was made using a ficoll-hypaque gradient. The cell types are seen to be diverse. (b) The lymphocytes emerge from the chamber in the first four fractions. The photograph is of the second fraction and the cells contain somewhat more cytoplasm than those of the first fraction. (c) The fifth fraction consists almost entirely of polymorphs. (Magnification 87.5x).
SHORT COMMUNICATIONS
619
620
SHORT COMMUNICATIONS
APPENDIX
A Derivation of the Equation of Motion of a Particle under Combined Centrifugal and H y d r o d y n a m i c Fields The equation of motion of a particle relative to a set of moving axes may be written, using vector notation, as follows: F p = m { ~ + [to x ( t o x r ) + ~ x
r + 2 o ) x i-]}
[A1]
where Fp is the sum of the forces acting on the particle and m is its mass. In the system which concerns us, the only forces of significance are the hydrodynamic forces, Fo, and the buoyancy forces, FB. We will assume that these are independent of one another and that the buoyancy forces can be calculated from a knowledge of the pressure distribution in the absence of the particle. Then, Stokes' Law gives Fo = kl"OdV
[A2]
for the hydrodynamic forces, while the buoyancy forces are given by the pressures integrated over the particle surface, S, i.e., by FB = f pndS Js
[A3]
where fi is a unit vector normal to the surface. Expression of the buoyancy term as a surface integral is not convenient, as we do not have an explicit equation for the pressure, p. However, we may use Gauss' Theorem to replace the surface integral by a volume integral which uses the gradient of p (Vp) instead of p, that is
IsPfidS=-IvVpdV.
[14]
Vp follows from the equation of motion of the fluid, which in the absence of large shears may be written DV'
- -
Dt
Vp
--
[15]
p
Here, V' has to be the velocity relative to inertial axes and D/Dt is the substantial derivative D/Dt --- d/dt + V.V. Thus, V' = V+to
x r
[A6]
Expansion of [A5] and [A6] gives
{[dV
Vp = o F -
~
+ (V'V)V+ 2to × V + to × (to × r)
[A7]
SHORT
621
COMMUNICATIONS
Equations [A1-A4] and [A7] may then be combined to yield the equation of motion of the particle.
kl dV+_m Iv fl ~dv + ( V ' V ) V +
j) dV
2to x V + to x (to x r) - t o × (to x r ) -
2 0 z i" [A8]
As the particle under consideration is small, the lengths, velocities, velocity gradients, and angular velocities within the integral sign may be taken to be constants. Equation [A8] then simplifies to
, _ k ~ d V + pV [ -OV J m - ~ + ( V ' V ) V + 2to × V + to x (to x r)
-toz(toxr)-2toxi-
[A9]
If we consider rotation only about the z axis, equation [A9] may be expanded and resolved into components as follows: i'=
[ klr/d
f0v
V . + pV V .
m
m
+ V~ Ox
"
Design Principles for a Counterflow Centrifugation Cell Separation Chamber Counterflow centrifugation has been available as a technique for cell separation since 1948 when Lindahl first published an account of the method in Nature (1,2). It has been used comparatively little, however, since available systems have been complex or lacking sensitivity. We have overcome these difficulties using simple principles to design a separation chamber. The design analysis a n d results of the fractionation of populations of human erythrocytes and leukocytes are presented. In a counterflow centrifuge, fluid is pumped into the center of the centrifuge rotor through a rotating seal. This fluid moves through internal tubing to the outside of the rotor and is then turned toward the center through a divergent separation chamber. At the end of this chamber, the flow reconverges into a tube of small diameter through which it moves back to the center of the rotor. It then leaves the system through a second rotating seal. The cell suspension to be fractionated is introduced into the inlet tube at some convenient point. If the pump rate is correctly chosen, cells move through the system until they reach the separation chamber, where they will come to rest relative to the rotor at a point where the outwardly directed inertial forces are balanced by the inwardly directed fluid dynamic and buoyancy forces. Forces in the tangential plane, namely the Coriolis force and the opposing pressure gradients, are not important under the operating conditions of this system. The small circulation resulting from these forces is, in any case, in the tangential plane. The cell population distributes itself along the separation chamber in accordance with certain physical properties and the velocity gradient within the chamber. To elute a fraction of cells from the centrifuge, the pump rate is increased until the innermost group of cells reaches the end of the divergent chamber. At this point the fluid velocity is increased by the converging walls and the cells are swept downstream to escape from the rotor. The velocity gradient in the chamber is a function of chamber shape. Consequently, it is possible to design a chamber which separates cells having different physical properties. We may write for a cell at equilibrium pVoJ2r = pFVoJ~r + kx~)Vd
[1]
where/9 is the cell density, V its volume, d its diameter, and ks a shape factor; pr is the fluid density, ~/its viscosity, and V local velocity; while o~ is the centrifuge velocity and r the distance from the center of rotation to the cell. Equation [1] may be derived more rigorously from the general 615 Copyright © 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.
616
SHORT COMMUNICATIONS
equation of motion for a particle in combined centrifugal and hydrodynamic fields. This has been done in the Appendix, where justification of the inherent approximations is also made. Rearranging, and writing p' for p - Pr, and k2d3 for V, we have r = k~V/oJ2p'd 2
[2]
where k = kl/k2. Thus the value of r for equilibrium and for a given rotational speed is a function of p'd 2 and k, which are cell parameters, and V, which is a controllable function of radius. For example, we could make the chamber shape such that (d/dr)(p'd2/k) is constant, leading to a linear distribution of cells according to the magnitude of p'd2/k. This concept leads from Eq. [2] to the differential equation d p'd ~ -
dr
-
k
-
~1 d V co2 dr r
- A, a constant.
[3]
Integration of [3] gives V = (At 2 + Br)(co2/r/).
[4]
The desired variation in cross-sectional area S of the chamber then follows. For any pump rate, P (measured in volume per unit time), we have S = P/V = P~/o~2(Ar2 + Br)
[5]
A convenient value of A may be obtained from the range of p'd2/k of the cell population to be fractionated. B may be obtained from an arbitrary inlet cross-sectional area and pump rate P. A chamber designed according to these principles will have a volume which is small if the inlet dimensions are small. Large inlet dimensions lead to pump rates which are unacceptably high. However, we may look for a chamber shape in which cells having high values of p'd2/k and those having low values are not separated significantly in the outer portion of the chamber. This condition is fulfilled by making (d/dr)(p'd2/k) large, which necessarily, from Eq. [3], leads to the condition that rdV/dr must be large compared with V, because (d/dr)(V/r) = (1/r2)[(rdV/dr) - V]. Physically, this is achieved by making the outer portion of the chamber cross-sectional area diverge rapidly. In the chamber we have designed, the zone of rapid expansion occupies the outer third of its length. This is run smoothly into an inverse square contour, as given by Eq. [5] with B = 0, which occupies the remaining two-thirds of the chamber length. The exact contours are not critical, so long as the outer portion of the chamber diverges rapidly and the inner position slowly. The two photographs, shown in Fig. 1, demonstrate the functional characteristics of the two regions during a red cell separation. In Fig. la, cells occupy only the outer region where the divergence is large. The cells furthest downstream form a clearly defined front. As the pump
SHORT COMMUNICATIONS
617
FIG. 1. Photographs showing the location of red blood cells in the separation chamber (a) at the completion of loading and (b) prior to elution of a fraction,
rate is increased, the cells move into the slowly divergent portion of the chamber, dispersing according to their values of the separation parameter and the front disappears (see Fig. lb). There is a physical limit to the slowness of divergence of the inner portion dictated by the necessity that the hydrodynamic forces acting on a cell as it moves inward should not decrease tess rapidly than the sum of buoyant and inertial forces. This would lead to unstable equilibrium. The limiting value of convergence is that which leads to a velocity proportional to the radius. The chamber described has been used with the Beckman Elutriator Rotor system to fractionate human red cells into different age groups. As red cells age, they become more dense (3), smaller (4), and lose activity of certain enzymes (5,6). Shape and membrane flexibility also change significantly (1). The fluid medium used was Hank's balanced salt solution (Ca 2+, Mg 2+ free) containing 0.2% bovine serum albumin. Centrifuge speed was 2000 RPM. The assay systems used to establish the effectiveness of the separation were the decrease in activity of GOT (aspartate aminotransferase, EC 2.6.1.1), expressed as a ratio of activity to hemoglobin (5,6), and the measurement of cell volume distribution for each fraction using the Coulter ZBI sizing apparatus. Results of these assays for a number of separations are combined in Fig. 2.
618
SHORT COMMUNICATIONS
,o
;o
do
~
,;o
¢,o -~o
Cell Volume (p~)
FIG. 2. Erythrocyte separation. The ratio of GOT to hemoglobin is plotted against mean cell volume for each fraction taken from the centrifuge. Each curve represents a different separation and different donor.
Old cells constitute early fractions, while younger cells make up later fractions. Although p'd 2 is lower for young than old cells, the young cell has a lower drag, either because of its basic shape or because its flexibility allows it to conform to a low drag configuration under the system of prevailing forces. This results in a value of p'd2/k which is higher in young cells. Consequently, higher fluid velocities are required to elute them. In addition to erythrocyte fractionations, this chamber has proven useful for other cell systems. Two of us (NP, JB) have used it to separate human peripheral leukocytes into their component species. Some results of this fractionation are shown in Fig. 3. Summary. A concise approximation of the equilibrium condition for particles in a counterflow centrifuge has been used as the basis for the design of a cell separation chamber. Used with the Beckman Elutriator rotor system, the chamber has proven to be an effective means of separating populations of human erythrocytes into different age groups, and of separating leukocyte preparations into their various cell types.
FIG. 3. Leucocyte fractionation. (a) The photograph shows a sample of the cells loaded into the separation chamber. A preliminary separation of leukocytes from erythrocytes was made using a ficoll-hypaque gradient. The cell types are seen to be diverse. (b) The lymphocytes emerge from the chamber in the first four fractions. The photograph is of the second fraction and the cells contain somewhat more cytoplasm than those of the first fraction. (c) The fifth fraction consists almost entirely of polymorphs. (Magnification 87.5x).
SHORT COMMUNICATIONS
619
620
SHORT COMMUNICATIONS
APPENDIX
A Derivation of the Equation of Motion of a Particle under Combined Centrifugal and H y d r o d y n a m i c Fields The equation of motion of a particle relative to a set of moving axes may be written, using vector notation, as follows: F p = m { ~ + [to x ( t o x r ) + ~ x
r + 2 o ) x i-]}
[A1]
where Fp is the sum of the forces acting on the particle and m is its mass. In the system which concerns us, the only forces of significance are the hydrodynamic forces, Fo, and the buoyancy forces, FB. We will assume that these are independent of one another and that the buoyancy forces can be calculated from a knowledge of the pressure distribution in the absence of the particle. Then, Stokes' Law gives Fo = kl"OdV
[A2]
for the hydrodynamic forces, while the buoyancy forces are given by the pressures integrated over the particle surface, S, i.e., by FB = f pndS Js
[A3]
where fi is a unit vector normal to the surface. Expression of the buoyancy term as a surface integral is not convenient, as we do not have an explicit equation for the pressure, p. However, we may use Gauss' Theorem to replace the surface integral by a volume integral which uses the gradient of p (Vp) instead of p, that is
IsPfidS=-IvVpdV.
[14]
Vp follows from the equation of motion of the fluid, which in the absence of large shears may be written DV'
- -
Dt
Vp
--
[15]
p
Here, V' has to be the velocity relative to inertial axes and D/Dt is the substantial derivative D/Dt --- d/dt + V.V. Thus, V' = V+to
x r
[A6]
Expansion of [A5] and [A6] gives
{[dV
Vp = o F -
~
+ (V'V)V+ 2to × V + to × (to × r)
[A7]
SHORT
621
COMMUNICATIONS
Equations [A1-A4] and [A7] may then be combined to yield the equation of motion of the particle.
kl dV+_m Iv fl ~dv + ( V ' V ) V +
j) dV
2to x V + to x (to x r) - t o × (to x r ) -
2 0 z i" [A8]
As the particle under consideration is small, the lengths, velocities, velocity gradients, and angular velocities within the integral sign may be taken to be constants. Equation [A8] then simplifies to
, _ k ~ d V + pV [ -OV J m - ~ + ( V ' V ) V + 2to × V + to x (to x r)
-toz(toxr)-2toxi-
[A9]
If we consider rotation only about the z axis, equation [A9] may be expanded and resolved into components as follows: i'=
[ klr/d
f0v
V . + pV V .
m
m
+ V~ Ox
"
