Biomedical Engineering Reference
In-Depth Information
Here
D
f,bind
or
D
f
(used later on in the topic) is the fractal dimension of the surface during the
binding step.
t
c
is the cross-over value.
Havlin (1989)
reports that the cross-over value may
be determined by
r
c
t
c
. Above the characteristic length,
r
c
, the self-similarity of the surface
is lost and the surface may be considered homogeneous. Above time,
t
c,
the surface may be
considered homogeneous, as the self-similarity property disappears, and “regular” diffusion
is now present. For a homogeneous surface where
D
f
is equal to 2, and when only diffusional
limitations are present,
p
½ case
(where
D
f,bind
is equal to 2) is that the analyte in solution views the fractal object, in our case,
the receptor-coated biosensor surface, from a “large distance.” In essence, in the association
process, the diffusion of the analyte from the solution to the receptor surface creates a deple-
tion layer of width (
Ðt
)
½
where
Ð
is the diffusion constant. This gives rise to the fractal
power law, (Analyte
¼
½ as it should be. Another way of looking at the
p
¼
t
(3
D
f,bind
)/2
. For the present analysis,
t
c
is arbitrarily chosen
and we assume that the value of the
t
c
is not reached. One may consider the approach as an
intermediate “heuristic” approach that may be used in future to develop an autonomous (and
not time-dependent) model for diffusion-controlled kinetics.
Receptor)
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 the solution may be
given, as a first approximation by:
t
ð
3
D
f
,
bind
Þ=
2
t
p
,
ð
Ab
Ag
Þ
¼
t
>
t
diss
ð
15
:
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.
15.2.2 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
¼
