Biomedical Engineering Reference
In-Depth Information
6.2.2 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 presented above is thus extended to include two fractal dimensions. At pres-
ent, 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
6
:
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 bind-
ing curve exhibits convolutions and complexities in its shape due perhaps to the very dilute
nature of the analyte (in some of the 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.
6.3 Results
A fractal analysis is applied to the binding and dissociation (if applicable) kinetics of differ-
ent analyte-receptor reactions occurring on different biosensor surfaces. Understandably,
alternative expressions for fitting the data that include saturation, first-order reaction, and
no diffusion limitations are available, but these expressions are apparently deficient in
describing the heterogeneity that inherently exists on the surface. Another advantage of this
technique is that the analyte-receptor binding (as well as the dissociation reaction) is a com-
plex reaction, and the fractal analysis via the fractal dimension and the rate coefficient pro-
vides a useful lumped parameter(s) analysis of the diffusion-limited reaction occurring on a
heterogeneous surface.
In the classical situation to demonstrate fractality, one should make a log-log plot, and one
should definitely have a large amount of data. It may be useful to compare the fit to some other
forms, such as an exponential form or one involving saturation. At present, we do not present
