Biomedical Engineering Reference
In-Depth Information
value may be determined by
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 sur-
face may be considered 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 diffusional limitations are present,
p
¼
½ as it should be. Another way of looking
at the
p
½ case (where
D
f,bind
is equal to two) is that the analyte in solution views the frac-
tal 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 recep-
tor surface creates a depletion layer of width (
Ðt
)
½
where
Ð
is the diffusion constant. This
gives rise to the fractal power law,
¼
t
ð
3
D
f
,
bind
Þ=
2
. For the present anal-
ysis,
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 the
future to develop an autonomous (and not time-dependent) model for diffusion-controlled
kinetics.
ð
Analyte
Receptor
Þ
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
12
:
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.
12.2.2 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.
¼
