Biomedical Engineering Reference
In-Depth Information
The concentration
c
is obtained by the mass balance taking into account the deple-
tion due to immobilization of the targets [18]
d
G
= -
dc
S
V
(7.64)
dt
d t
where
S
is the functionalized surface and
V
the volume of the microchamber. Taking
into account the initial conditions, integration of (7.64) leads to
S
c
=
c
-
G
(7.65)
0
V
where
c
0
= c
(
t =
0) is the initial (uniform) concentration
.
Upon substituting (7.65) in
(7.63), one obtains the differential equation for the surface concentration G
d
G
æ
S
ö
S
2
=
k c
G -
k c
+
k
+
k
G
G +
k
G
(7.66)
ç
÷
on
0
0
on
0
off
on
0
on
è
ø
dt
V
V
and, for the concentration
d c
æ
S
ö
2
=
k c
+
k c
-
k
-
k
G
c k c
-
(7.67)
ç
÷
off
0
on
0
off
on
0
on
è
ø
dt
V
By considering an infinite volume
V
=
¥, (7.66) collapses to the usual Langmuir equa-
tion. The two differential equations (7.66) and (7.67) are of the mathematically well-
known Ricatti type [19]. Ricatti equations can be solved if a particular solution is
known. In such a case, a change of variable using the particular solution transforms the
Ricatti equation in a Bernoulli equation, which has a closed form solution.
Usually, in order to promote hybridization, the temperature of reaction is set to
a value well beneath the “fusion” temperature (i.e., the temperature where dissocia-
tion dominates), so that
k
off
is usually kept small. However,
k
off
is not necessarily
vanishing in front of
k
on
c
, especially in our case where the initial concentration
c
0
is small. Thus, all the terms in (7.67) have their importance. Let us mention that
values of
k
on
and
k
off
for immunoassays have been investigated thoroughly in the
literature [20, 21].
The algebraic manipulations to obtain to the solution are somewhat long. We
will only briefly indicate the approach: First, we search the solution of the second-
order characteristic polynomial in
c
:
æ
S
ö
2
k c
-
k c
-
k
-
k
G
c k c
-
=
0
(7.68)
ç
÷
on
on
0
off
on
0
off
0
è
ø
V
The two roots of the polynomial are given by
2
k
k
k
1
æ
S
ö
1
æ
S
ö
off
off
off
-
c
=
c
-
-
G
-
c
-
-
G
+
4
c
ç
÷
ç
÷
0
0
0
0
0
2
è
k
V
ø
2
è
k
V
ø
k
on
on
on
(7.69)
2
k
k
k
1
æ
S
ö
1
æ
S
ö
off
off
off
+
c
=
c
-
-
G
+
c
-
-
G
+
4
c
ç
÷
ç
÷
0
0
0
0
0
è
ø
è
ø
2
k
V
2
k
V
k
on
on
on
