Biomedical Engineering Reference
In-Depth Information
dimension,
D
f
, for a single-fractal analysis, (b) the dissociation rate coefficient,
k
d
, and the
fractal dimension for the dissociation phase,
D
fd
for a single-fractal analysis, (c) the binding
rate coefficients,
k
1
and
k
2
, and the fractal dimensions,
D
f1
and
D
f2
, for a dual-fractal
analysis, and (d) the dissociation rate coefficients,
k
d1
and
k
d2
, and the fractal dimensions,
D
fd1
and
D
fd2
, for the dissociation phase for a dual-fractal analysis.
An increase in the fractal dimension for a dual-fractal analysis by a factor of 3.18 from a
value of
D
f1
equal to 0.8628 to
D
f2
equal to 2.7406 leads to an increase in the binding rate
coefficient by a factor of 80.77 from a value of
k
1
equal to 0.06332 to
k
2
equal to 5.1146.
Note that changes in the fractal dimension or the degree of heterogeneity on the ZnO-glass
substrate and in the binding rate coefficient are, once again, in the same direction.
In this case for the dissociation phase and for a dual-fractal analysis an increase in the fractal
dimension for dissociation by a factor of 1.57 from a value of
D
fd1
equal to 1.7272 to
D
fd2
equal to 2.7066, the dissociation rate coefficient increases by a factor of 7.78 from a value
of
k
d1
equal to 0.6243 to
k
d2
equal to 4.859. Once again, an increase in the degree of hetero-
geneity on the biosensor surface in the dissociation phase leads to an increase in the dissoci-
ation rate coefficient.
Figure 10.2a
and
Tables 10.1
and
10.2
show the increase in the binding rate coefficient,
k
2
,
with an increase in the fractal dimension,
D
f2
, for a dual-fractal analysis. For the data shown
in
Figure 10.2a
, the binding rate coefficient,
k
2
, is given by:
10
13
10
13
D
30
:
4
k
2
¼ð
2
:
8
1
:
5
Þ
ð
10
:
4a
Þ
f
2
The fit is reasonable. Only three data points are available. The availability of more data
points would lead to a more reliable fit. The binding rate coefficient,
k
2
, is extremely sensi-
tive to the fractal dimension,
D
f2
, that exists on the biosensor surface as noted by the 30.4
order of dependence exhibited.
Figure 10.2b
and
Tables 10.1
and
10.2
show the increase in the ratio of the binding rate
coefficients,
k
2
/
k
1
, with an increase in the fractal dimension ratio,
D
f2
/
D
f1
, for a dual-fractal
analysis. For the data shown in
Figure 10.2b
, the ratio of the binding rate coefficients,
k
2
/
k
1
,
is given by:
1
:
438
0
:
122
k
2
=
k
1
¼ð
16
:
43
1
:
78
Þð
D
f2
=
D
f1
Þ
ð
10
:
4b
Þ
The fit is very good. Once again, only three data points are available. The availability of more data
points would lead to a more reliable fit. The ratio of the binding rate coefficients,
k
2
/
k
1
, exhibits
close to a one and a half order of dependence on the ratio of fractal dimensions,
D
f2
/
D
f1
.
Figure 10.2c
and
Tables 10.1
and
10.2
show the increase in the ratio,
k
/
k
d
,
k
1
/
k
d1
,and
k
2
/
k
d2
with an increase in the fractal dimension ratio,
D
f
/
D
fd
,
D
f1
/
D
fd1
,and
D
f2
/
D
fd2
.
