Biomedical Engineering Reference
In-Depth Information
14.2.2 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 the
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,
¼
the product
(antibody-antigen; or analyte-receptor complex, Ab
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
t
p
2
,
ð
Ab
Ag
Þ
¼
t
1
<
t
<
t
2
¼
t
c
ð
14
:
3
Þ
:
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 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.
14.3 Results
A fractal analysis is applied to the binding and dissociation (if applicable) kinetics of dif-
ferent analyte-receptor reactions occurring on different biosensor surfaces. Alternative
expressions for fitting the data are available that include saturation, first-order reaction,
and no diffusion limitations, but these expressions are apparently deficient in describing
the heterogeneity that inherently exists on the surface. One might justifiably argue that the
appropriate modeling may be achieved by using a Langmuirian or other approach. The
Langmuirian approach may be used to model the data presented if one assumes the presence
of discrete classes of sites (e.g., double exponential analysis as compared with a single-fractal
analysis).
Lee and Lee (1995)
report that the fractal approach has been applied to surface
science, for example, adsorption and reaction processes. These authors point out that the frac-
tal approach provides a convenient means to represent the different structures and the
morphology at the reaction surface. These authors also draw attention to the use of the fractal
