Biomedical Engineering Reference
In-Depth Information
t
ð
3
D
f
,
diss
Þ=
2
t
p
,
ð
Ab
Ag
Þ
¼
t
>
t
diss
ð
7
:
2
Þ
Here
D
f,diss
is the fractal dimension of the surface for the dissociation step. This corresponds
to the highest concentration of the analyte-receptor complex on the surface. Henceforth,
its concentration only decreases. The dissociation kinetics may be analyzed in a manner
“similar” to the binding kinetics.
7.2.2 Dual-Fractal Analysis
Binding Rate Coefficient
Sometimes, the binding curve exhibits complexities and two parameters (
k
,
D
f
) are not suffi-
cient to adequately describe the binding kinetics. This is further corroborated by low values
of
r
2
factor (goodness-of-fit). In that case, one resorts to a dual-fractal analysis (four parameters;
k
1
,
k
2
,
D
f1
, and
D
f2
) to adequately describe the binding kinetics. The single-fractal analysis pres-
ented above is thus extended to include two fractal dimensions. The time (
t
t
1
) at which the
“first” fractal dimension “changes” to the “second” fractal dimension is arbitrary and empirical.
For the most part, it is dictated by the data analyzed and experience gained by handling a single-
fractal analysis. A smoother curve is obtained in the “transition” region, if care is taken to select
the correct number of points for the two regions. In this case, the product (antibody-antigen; or
analyte-receptor complex, Ab
¼
Ag or analyte
receptor) is given by:
8
<
:
t
ð
3
D
f1
,
bind
Þ=
2
t
p
1
,
¼
t
<
t
1
t
ð
3
D
f2
,
bind
Þ=
2
ð
Ab
Ag
Þ
t
p
2
,
ð
7
:
3
Þ
¼
t
1
<
t
<
t
2
¼
t
c
t
1
=
2
,
t
>
t
c
In some cases, as mentioned above, a triple-fractal analysis with six parameters (
k
1
,
k
2
,
k
3
,
D
f1
,
D
f2
, and
D
f3
) may be required to adequately model the binding kinetics. This is when
the binding curve exhibits convolutions and complexities in its shape due perhaps to the very
dilute nature of the analyte (in some cases to be presented) or for some other reasons. Also, in
some cases, a dual-fractal analysis may be required to describe the dissociation kinetics.
7.3 Results
In this chapter we use fractal analysis to analyze the binding and dissociation (if applicable)
kinetics of (a) binding of 1 mM glucose in solution (using CV, cyclic voltametry) to Nf
(nafion)-CNTs-Cu (Cu-CNT-GCE) (Kang et al., 2007), (b) Influence of repeat runs on the
binding and dissociation of 0.1 mM glucose to DMG (dimethylglycoxime)-CuNP (copper
nanoparticles) CME (copper-based chemically modified electrodes) (
Xu et al., 2006
),
(c) binding of different concentrations (in mM) of glucose in 0.1 M PBS solution to Pt-Pb
