Biomedical Engineering Reference
In-Depth Information
Table 10.7: Fractal dimensions for the binding and the dissociation phase for NH
3
in air to a
sol-gel derived thin film. Influence of film thickness in mm (
Roy et al., 2005
).
Film thickness (mm)
D
f
D
f1
D
f2
D
fd
150
0.7844 0.1715
0.2200 0.3098
1.6212 0.1155
0.2126 0.1728
320
1.0898 0.08686
0.7546 0.1252
1.7064 0.0416
0.8576 0.03100
480
1.7638
0.09774
1.7268
0.09774
2.4048
0.1414
1.3568
0.2092
factor of 6.89 from a value of
k
1
equal to 0.2384 to
k
2
equal to 1.6449. In this case, the affin-
ity,
K
1
(
¼
k
1
/
k
d
) is equal to 0.424, and
K
2
(
¼
k
2
/
k
d
) is equal to 2.93.
Figure 10.8c
shows the binding and the dissociation of NH
3
in ppm in air to the sol-gel
derived 480 nM thin film biosensor. A dual-fractal analysis is required to adequately describe
the binding kinetics. A single-fractal analysis is adequate to describe the dissociation kinetics.
The values of (a) the binding rate coefficient,
k
, and the fractal dimension,
D
f
, for a single-
fractal analysis, (b) the dissociation rate coefficient,
k
d
, and the fractal dimension,
D
fd
, for
a single-fractal analysis, and (c) the binding rate coefficients,
k
1
and
k
2
, and the fractal
dimensions,
D
f1
and
D
f2
, for a dual-fractal analysis, are given in
Tables 10.6
and
10.7
.
It is of interest to note that as the fractal dimension increases by a factor of 1.392 from a
value of
D
f1
equal to 1.7268 to
D
f2
equal to 2.4048, the binding rate coefficient increases
by a factor of 4.90 from a value of
k
1
equal to 2.7898 to
k
2
equal to 13.692. In this case,
the affinity,
K
1
(
¼
k
1
/
k
d
) is equal to 1.306, and
K
2
(
¼
k
2
/
k
d
) is equal to 6.41.
Figure 10.9a
and
Tables 10.6
and
10.7
show for a dual-fractal analysis the increase in the
binding rate coefficient,
k
1
, with an increase in the film thickness, in mm. For the data shown
in
Figure 10.9a
, the binding rate coefficient,
k
1
, for a dual-fractal analysis is given by:
3
:
12
1
:
08
10
09
10
09
k
1
¼ð
7
:
5
þ
10
:
5
Þ½
film thickness, mm
ð
10
:
8a
Þ
The fit is poor. Only three data points are available. The availability of more data points
would lead to a better and more reliable fit. The poor fit is also reflected in the error in
the binding rate coefficient,
k
1
, value presented. Only the positive value is presented since
the binding rate coefficient cannot have a negative value. The binding rate coefficient,
k
1
,
is sensitive to the film thickness, in mm since it exhibits a greater than third (equal to
3.12) order of dependence on the film thickness. The non-integer order of dependence
exhibited by
k
1
on the film thickness lends support to the fractal nature of the system.
Figure 10.9b
and
Tables 10.6
and
10.7
show the increase in the binding rate coefficient,
k
2
,
with an increase in the film thickness, in mm for a dual-fractal analysis. For the data shown
in Figure 10.b, the binding rate coefficient,
k
2
, for a dual-fractal analysis is given by:
2
:
065
1
:
156
k
2
¼ð
2
:
4
þ
3
:
9
Þ
10
05
½
film thickness, mm
ð
10
:
8b
Þ
