An equation for fitting distance-based measurements with analyte concentrations: From discrete segments simulation to closed-form solution

In this study, an equation for fitting the band lengths in µPADs to the concentrations/amount of analyte added to the µPAD sample area is derived. A simulation of the band formation is carried out using a discrete segment model. The detector channel is divided into equal segments with the same amount of reagent R in each segment. The sample moves into the channel in steps corresponding to segments of the same size as the detector segment. Each sample segment contains analyte A at C mole ratio to reagent R. Assuming a stoichiometric ratio of 1:1 for reaction between A and R, there will be formation of only one product band in each detector segment. By examining the number of bands (n) formed after N steps, a set of linear algebraic equations is derived to determine the number of bands (n) for any integer values of N and C. By extrapolating this result to real positive numbers, we obtain the equation L=a.C_A/(b + C_A), where L represents the band length, and C_A represents the concentration/amount of analyte. The equation represents a rectangular hyperbola.