Biomedical Engineering Reference
In-Depth Information
8.2.2 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
ð
Ab
Ag
Þ
t
p
2
,
ð
8
:
1c
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 or for some other reasons. Also, in some cases, a dual-fractal
analysis may be required to describe the dissociation kinetics.
8.3 Results
The fractal analysis will be applied to different analyte-receptor reactions occurring on bio-
sensor chip surfaces with the specific medical applications. Attempts will be made to relate
particularly changes in the fractal dimension on the biosensor chip surface with the changes
in the binding and the dissociation rate coefficients.
At the outset it should be pointed out that alternative expressions for fitting the binding and
dissociation data are available that include saturation, first-order reaction, and no diffusional
limitations, but these expressions are deficient in describing the heterogeneity that inherently
exists on the surface. It is this heterogeneity on the biosensor surface that one is attempting to
relate to the different biosensor performance parameters. More specifically the question we
wish to answer is how may one change the heterogeneity or the fractal dimension,
D
f
,on
the biosensor chip surface in order that one may be able to enhance the different biosensor
performance parameters.
