Biomedical Engineering Reference
In-Depth Information
6.2 Theory
6.2.1 Single-Fractal Analysis
Binding Rate Coefficient
Havlin (1989)
reports that the diffusion of a particle (analyte [Ag]) from a homogeneous
solution to a solid surface (e.g., receptor [Ab]-coated surface) on which it reacts to form a
product (analyte-receptor complex; (Ab
Ag)) is given by:
t
ð
3
D
f
,
bind
Þ=
2
t
p
,
¼
t
<
t
c
ð
Ab
Ag
Þ
ð
6
:
1
Þ
t
1
=
2
,
t
>
t
c
Here
D
f,bind
or
D
f
is the fractal dimension of the surface during the binding step.
t
c
is the
cross-over value.
Havlin (1989)
points out 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 pres-
ent. 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 depletion 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
. In the present 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 the 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 solution may be given,
as a first approximation by:
t
3
D
f
,
diss
Þ=
2
t
p
,
ð
Ab
Ag
Þ
¼
t
>
t
diss
ð
6
:
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.
