Medical Applications of Biosensors - Biosensors and Biosensor Kinetics - page 198

8.2.1 Single-Fractal Analysis

Binding Rate Coefficient

Havlin (1989) points out 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 Analyte Receptor Þ ð 8 : 1a Þ

t 1 = 2 ,

t > t c

Here D f,bind or D f (used later on in the chapter) 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 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 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 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
receptor surface creates a depletion layer of width ( Ðt ) ½ where Ð is the diffusion constant.
This gives rise to the fractal power law,

t ð 3 D f , bind Þ= 2 . For 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.

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 , ð Analyte Receptor Þ t > t diss

ð : Þ

8 1b

¼k diss t ð 3 D f , diss Þ= 2

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.