Biomedical Engineering Reference
In-Depth Information
intermediate “heuristic” approach that may be used in the future to develop an autonomous
(and not time-dependent) model for diffusion-controlled kinetics.
Dissociation Rate Coefficient
The diffusion of the dissociated particle (receptor [Ab] or analyte [Ag]) from the solid sur-
face (e.g., analyte [Ag]-receptor [Ab] complex coated surface) into solution may be given,
as a first approximation by:
t
ð
3
D
f
,
diss
Þ=
2
t
p
,
ð
Þ
¼
t
>
t
diss
ð
:
Þ
Ab
Ag
9
2
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.
9.2.2 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 present,
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:
<
t
ð
3
D
f1
,
bind
Þ=
2
¼
t
p
1
,
t
<
t
1
t
ð
3
D
f2
,
bind
Þ=
2
ð
Ab
Ag
Þ
t
p
2
,
ð
9
:
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 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.
