Biomedical Engineering Reference
In-Depth Information
14.2 Theory
14.2.1 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
¼ t
p
,
t < t
c
ð
Ab
Ag
Þ
ð
14
:
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, 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 ¼
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
)
1/2
where
Ð
is the diffusion constant. This gives rise to the fractal
power law, (Analyte
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 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
ð
3
D
f
,
bind
Þ=
2
t
p
,
ð
Ab
Ag
Þ
¼
t
>
t
diss
ð
14
:
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.
