## 3 Results and Discussions
### Theoretical Dependence of CR on the Analysis Format  
When determining several structurally similar compounds under the same concentrations of analytical reagents, the cross-reactivity will be determined by the constants of the interactions of these compounds with antibodies. Figure 3 shows theoretical calibration curves for three model analytes (I, II, and III) with the equilibrium dissociation constants of their immune complexes equal to 0.1, 1, and 10 nM. The curves were calculated for three chosen concentrations of analytical reactants: assay (a), \(\left[Ab\right]_{0}\) = 1 nM, \(\left[Ag^{*}\right]_{0}\) = 1 nM; assay (b), \(\left[Ab\right]_{0}\) = 10 nM, \(\left[Ag^{*}\right]_{0}\) = 1 nM; and assay (c), \(\left[Ab\right]_{0}\) = 10 nM, \(\left[Ag^{*}\right]_{0}\) = 10 nM. Figure 3: \(Cont.\)  
Figure 2: Theoretical calibration dependences of equilibrium competitive immunoassay at different values of the initial concentrations of the labeled antigen and antibodies. The IC\({}_{50}\) values calculated using these curves are integrated in Table 1. Cross-reactivity can be quantified in terms of the ratio of the IC\({}_{50}\) values for compounds I, II, and III. The presented data confirm the change in IC\({}_{50}\) with varying concentrations of analytical reactants. Thereby, cross-reactivity is not an intrinsic fixed characteristic of either the antibody itself or its combination with a competitor used in the analysis, but rather reflects the properties of a particular assay and its conditions. Therefore, the same antigen-antibody pair used in different methods of analysis will give different spectra of cross-reactivity values for structurally similar compounds. The data obtained show that cross-reactivity depends both on the absolute value of the concentrations of the reactants and on their ratio. In the case of equal concentrations of antibodies and labeled/immobilized antigens, the higher their concentration is, the higher the cross-reactivity will be. Thus, for \([Ab]_{0}=[Ag^{*}]_{0}=1\) nM, the IC\({}_{50}\) values for antigens with K\({}_{\rm d}\) = 0.1 and 10 nM differ by a factor of almost eight (Figure 3a, Table 1), while with \([Ab]_{0}\) and \([Ag^{*}]_{0}\) being 10 times higher, the IC\({}_{50}\) values for the same antigens will differ by a factor of 2.7 (Figure 3c, Table 1). The ratio of reagents affects the cross-reactivity even more strongly. For example, at concentrations of \([Ag^{*}]_{0}\) and \([Ab]_{0}\) equal to 1 and 10 nM, respectively, the IC\({}_{50}\) values for antigens with K\({}_{\rm d}\) = 0.1 and 10 nM differ by only 1.5 times (Figure 3b, Table 1). Thus, an approximately five-fold variation in CR values can be achieved (as follows from the comparison of Figure 3a, Table 1 and Figure 3b, Table 1). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Analytes with Different Affinities**} & \multicolumn{2}{c}{**Assay (a)**} & \multicolumn{2}{c}{**Assay (b)**} & \multicolumn{2}{c}{**Assay (c)**} \\ \cline{2-7}  & **IC\({}_{50}\)**, nM** & **CR, \%** & **IC\({}_{50}\)**, nM** & **CR, \%** & **IC\({}_{50}\)**, nM** & **CR, \%** \\ \hline \hline Analyte I (K\({}_{\rm d}\) = 0.1 nM) & 1.6 & 100 & 17.5 & 100 & 11.1 & 100 \\ Analyte II (K\({}_{\rm d}\) = 1 nM) & 3.0 & 53 & 17.9 & 98 & 15.1 & 74 \\ Analyte III (K\({}_{\rm d}\) = 10 nM) & 12.4 & 13 & 26.0 & 67 & 29.5 & 38 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: IC\({}_{50}\) values (nM) for the calibration dependences given in Figure 3.  
Figure 3: Theoretical calibration dependences of equilibrium competitive immunoassay for three model structurally similar compounds (I, II, and III; black, red, and blue curves in each sub-figure, respectively) and three assay formats differing in the used concentrations of analytical reactants: (**a**) \([Ab]_{0}=1\) nM, \([Ag^{*}]_{0}=1\) nM; (**b**) \([Ab]_{0}=10\) nM, \([Ag^{*}]_{0}=1\) nM; (**c**) \([Ab]_{0}=10\) nM, \([Ag^{*}]_{0}=10\) nM, \([Ag^{*}]_{0}=10\) nM.