Biomedical Engineering Reference
In-Depth Information
r
c
2
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 consid-
ered homogeneous, since 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 diffu-
sional limitations are present,
p
½
case (where
D
f,bind
is equal to two) is that the analyte in solution views the fractal object,
in this 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 depletion layer of width (
Ðt
)
½
where
Ð
is the diffusion constant. This
gives rise to the fractal power law,
¼
½ as it should be. Another way of looking at the
p
¼
t
ð
3
D
f
,
bind
Þ=
2
.Forthepresent
analysis,
t
c
is arbitrarily chosen and we assume that the value of
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
ð
Analyte
Receptor
Þ
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
Ab
Ag
Þ
¼
t
>
t
diss
ð
13
:
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.
13.2.2 Dual-Fractal Analysis
Binding Rate Coefficient
Sometimes, the binding curve exhibits complexities and two parameters (
k
,
D
f
) are not
sufficient 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.
¼
