Here, y = c/co, 2 = x/L (where L is a characteristic length dimension—for example, the diameter of a fiber-optic biosensor), and 6 = t/(L2/D).
The appropriate initial condition for Eq. (8.1a) in dimensionless form is yfcO) = l for z>0,0 = 0 >(0,0) =0 for z = 0,0 = 0.
This initial condition is equivalent to the rapid immersion of a sensor into a solution with antigen.
A boundary condition in dimensionless form for Eq. (8.1a) is
This boundary condition has not been mentioned in similar previous analyses (Stenberg et al, 1986; Sadana and Sii, 1991, 1992; Sadana and Madagula, 1993, 1994; Sadana and Beelaram, 1994). We now feel that this more correctly represents the actual situation.
Another boundary condition for Eq. (8.1a) is dc drAp dn"
Eq. (8.2c) arises because of mass conservation, wherein the flow of antigens to the surface must be equal to the rate of the antigen reacting with the antibody on the surface (specific binding) as well as the binding of the antigen to the surface itself (nonspecific binding). Here, dTsAg/dt and dQF^Jdt represent the rates of specific and nonspecific binding, respectively. The right-hand side is different for different reaction orders. We will present the details for obtaining this boundary condition for first- and second-order reactions. The analysis is then easily extended for one-and-one-half and general nth-order reactions. The analysis for first- and second-order reactions where only specific binding is present is available in the literature (Sadana and Sii, 1991, 1992).
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