Biomedical Engineering Reference
In-Depth Information
Equation (9.40) for separation efficiency of a MFFF device shows the advan-
tage of analytical methods when possible. They can produce a relation between
parameters that is not always obvious when using numerical solutions. It is strongly
recommended to always search first for analytical solutions—even at the price of
simplifications—before turning to numerical methods.
9.10.1.2 General Case: Numerical Approach
The system (9.34) determines the velocity field of the particles in the carrier fluid.
The question is now to solve the system (9.36) for particle trajectory. More or less
sophisticated methods can be used. But if we take advantage of the very slow veloc-
ity of the carrier fluid, a very simple predictor-corrector method can be set up.
Suppose that the particle has the coordinates
x
i
and
y
i
at time
t
i
.
The first step of the numerical scheme is to find a predictor point at time
t
i+1
.
This predictor point is obtained by making use of the velocity
�
V
=
(
V
,
V
)
p i
,
p x i
,
,
p y i
,
,
at time
t
i
x
�
=
x
+ D
t V
i
+
1
i
p x i
,
,
(9.41)
�
y
=
y
+ D
t V
i
+
1
i
p y i
,
,
�
In reality, the particle velocity is not constantly
V
during the time interval [
t
i
, t
i+1
].
p i
Because we have found the predictor point
(
x
� �
, we now know the velocity at
,
y
)
i
+
1
i
+
1
�
�
�
this point
V
=
(
V
,
V
)
and a more accurate velocity in the time interval
p i
, 1
+
p x i
,
, 1
+
p y i
,
, 1
+
æ
�
�
ö
V
+
V
V
+
V
p x i
,
, 1
+
p x i
,
,
p y i
,
, 1
+
p y i
,
,
[
t
i
, t
i+1
] is
.
,
ç
÷
2
2
è
ø
The second step is then the following correction
V
�
+
V
p x i
,
, 1
+
p x i
,
,
x
=
x
+ D
t
i
+
1
i
2
(9.42)
�
V
+
V
p y i
,
, 1
+
p y i
,
,
y
=
y
+ D
t
i
+
1
i
2
This two-step predictor-corrector method can be schematized graphically (Figure
9.27). The particle is located at the point
M
i
at time
t
i
,
the predictor point is
P
i+1
and the corrected point
M
i+1
. The distance between these two points is the first
order error. The larger the carrier fluid velocity, the larger the distance [
P
i+1,
M
i+1
].
A second-order method using the same principle can be easily set up to verify that
the precision is satisfactory.
9.10.2 Concentration of Magnetic Beads
The starting point is (9.28) where the velocity field is given by (9.32) and the mag-
netic force by (9.21)
�
æ
2
ö
∂
c
3
V
y
æ
1
ö
0
2
=
D c
D -
1
-
Ñ -
c uv
D Ñ
χ
H
Ñ
c
(9.43)
ç
÷
ç
÷
p
2
è
ø
∂
t
2
2
d
è
ø
