Biomedical Engineering Reference
In-Depth Information
æ
2
ö
y
d x
3
V
P
-
c t
0
-
c t
P
=
V
e
+
1
-
[1
-
e
]
1
1
ç
÷
p x
, 0
2
d t
2
d
è
ø
(9.36)
d y
c
P
-
c t
2
1
-
c t
=
V
e
+
[1
-
e
]
1
1
p y
, 0
d t
c
where
x
P
and
y
P
are the coordinates of the particle at time
t
. This time the system
(9.36) is coupled because the
y
-coordinate appears in the first
x
-equation—which
means that the trajectory of the particle will not be linear.
9.10.1.1 A Case of Analytical Solution to the Trajectory
If we examine in detail the system, it can be seen that, if the velocity of the particle
is zero at the inlet
V
p,x
=V
p,y
=0
—this is the case if the particles start at the top, along
the upper (depletion) wall—time can be eliminated from (9.36) and the differential
equation has the following particular form
æ
2
ö
d x
3
V
y
c
P
0
1
P
=
1
-
(9.37)
ç
÷
2
d y
2
c
d
è
ø
P
2
Equation (9.37) is easily solved (taking into account with
x
0
=
0,
y
0
= -d
)
and we find
V c
0
1
2
2
3
x
= -
[
y y
(
-
3 ) 2
d
-
d
]
(9.38)
P
P
P
2
c
2
d
2
This is the equation of a cubic; the corresponding trajectory has been plotted in
Figure 9.25.; one verifies that
x
P
= 0 for
y
P
= -
d
. By setting
y
P
=
d
in (9.38) one finds
the distance from inlet at which a particle meets the accumulation wall.
c
1
x
=
2
dV
c
d
0
(9.39)
It can be readily checked that the larger the magnetic force, the smaller the distance
x
d
and the larger the hydrodynamic drag, the larger the distance
x
d
.
Equation (9.39) can be used to define the separation efficiency of the MFFF. If
the particles are injected at the top of the channel through a nozzle, the separation
distance between the accumulation sites on the accumulation plate for two differ-
ent types of particles (1 and 2) is given by (9.39). Provided that the particles are
sufficiently small to neglect the gravity force before the magnetic force, one obtains
a very simple relation [21]
2
2
L
L
D
=
D
χ
χ
r
2
1 1
2
(9.40)
r
1
2
2
where
L
is the distance from the channel entrance. Usually, smaller paramagnetic
particles have lower magnetic susceptibility, so that we can expect an efficient sepa-
ration between two populations of magnetic beads (i.e., the two types of magnetic
beads are gathering in two distinct packets on the accumulation wall).
