Biomedical Engineering Reference
In-Depth Information
surface irregularities show scale invariance they can be characterized by a single number, the
fractal dimension. Later on in the topic we will characterize the surfaces of the biosensors
used in the different examples by a fractal dimension. More specifically, we will characterize
the heterogeneity present on these biosensor surfaces by a fractal dimension.
The fractal dimension is a global property, and it is insensitive to structural or morphological
details ( Pajkossy and Nyikos, 1989 ). Markel et al. (1991) point out that fractals are scale self-
similar mathematical objects that possess nontrivial geometrical properties. Furthermore,
these authors state that rough surfaces, disordered layers on surfaces, and porous objects
all possess fractal structure. A consequence of the fractal nature is a power-law dependence
of a correlation function (in our case the analyte-receptor on the biosensor surface) on a coor-
dinate (e.g., time).
Pfeifer (1987) shows that fractals may be used to track topographical features of a surface
at different levels of scale. Lee and Lee (1995) point out that the fractal approach permits
a predictive approach for transport (diffusion-related) and reaction processes occurring on
catalytic surfaces. This approach may presumably be extended to diffusion-limited analyte-
receptor reactions occurring on biosensor surfaces.
The binding of an analyte in solution to a receptor attached to a solid (albeit flow cell or
biosensor surface) is a good example of a low dimension reaction system in which the distri-
bution tends to be “less random” ( Kopelman, 1988 ), and a fractal analysis would provide
novel physical insights into the diffusion-controlled reactions occurring at the surface. Also,
when too many parameters are involved in a reaction, which is the case for these analyte-
receptor reactions on a solid (e.g., biosensor surface), a fractal analysis provides a useful
lumped parameter. It is appropriate to pay particular care to the design of such systems
and to explore new avenues by which further insight or knowledge may be obtained on these
biosensor systems. The fractal approach is not new and has been used previously in analyzing
different phenomena on lipid membranes.
Fatin-Rouge et al. (2004) have recently presented a summary of cases where the analysis of
diffusion properties in random media has provoked significant theoretical and experimental
interest. These cases include soils ( Sahimi, 1993 ), gels ( Starchev et al., 1997; Pluen et al.,
1999 ), bacteria cytoplasm ( Berland et al., 1995; Schwille et al., 1999 ), membranes ( Saffman
and Delbruck, 1975; Peters and Cherry, 1982; Ghosh and Webb, 1988 ), and channels ( Wei
et al., 2000 ). Coppens and Froment (1995) have analyzed the geometrical aspects of diffusion
and the reaction occurring in a fractal catalyst pore. In this chapter, and in this topic as
a whole, we are extending the analysis to analyte-receptor binding (and dissociation) on
biosensor surfaces.
Fatin-Rouge et al. (2004) show that in most real systems disorder may exist over a finite
range of distances. Harder et al. (1987) and Havlin (1989) point out that in this range the
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