## 3 Results and Discussions  
### Model of Competitive Immunoussay  
In the course of a competitive immunoassay with a modified antigen, three compounds interact simultaneously: the antigen-binding sites of antibodies (Ab), an unlabeled antigen in the tested sample (Ag), and a modified competing antigen (Ag*). This leads to the formation of two types of complexes having the compositions AbAg and AbAg*. The registered signal reflects the content of the second complex. Therefore, the calibration dependence of the assay accords with the dependence of the concentration of [AbAg*] on the initial concentration of the added unlabeled antigen [Ag]\({}_{0}\). (The subscript \"\({}_{0}\)\" here and below indicates the starting time \(t\) = 0.) Various formats of competitive immunoassay differ only in specifications for the considered reactants while demonstrating the same calibration dependence. Thus, for ELISA, Ag* is an antigen immobilized on the solid phase, and the registered signals are determined by the formation of the AbAg* complexes. The introduction of a detectable label directly during the competition (as in the considered FPIA) or at a subsequent stage (as in the experimentally studied variant of ELISA) affects the signal intensity, but not its concentration dependence. Therefore, the following chemical processes should be considered when describing a competitive immunoassay, taking into account the notations introduced above:  
\[Ab+Ag\leftrightarrow AbAg \tag{1}\]  
\[Ab+Ag^{*}\leftrightarrow AbAg^{*} \tag{2}\]The rates of change in the concentrations of all reagents are described by a system of differential equations:  
\[\frac{\partial[AbAg]}{\partial t}=\frac{-\partial[Ag]}{\partial t}=\mathrm{k_{a1} [Ab][Ag]-k_{d1}[AbAg]} \tag{3}\]  
\[\frac{\partial[AbAg^{*}]}{\partial t}=\frac{-\partial[Ag^{*}]}{\partial t}= \mathrm{k_{a2}[Ab][Ag^{*}]-k_{d2}[AbAg^{*}]} \tag{4}\]  
\[\frac{\partial[Ab]}{\partial t}=\frac{\partial[Ag^{*}]}{\partial t}+\frac{ \partial[Ag]}{\partial t} \tag{5}\]  
where \(\mathrm{k_{ai}}\) and \(\mathrm{k_{di}}\) denote the kinetic constants of association and dissociation, respectively. When carrying out the analysis in equilibrium mode (the rates of change in concentrations reach zero values), the system of differential Equations (3)-(5) turns into a system of algebraic equations:  
\[\mathrm{k_{a1}[Ab][Ag]-k_{d1}[AbAg]=0} \tag{6}\]  
\[\mathrm{k_{a2}[Ab][Ag\,^{*}]-k_{d2}[AbAg\,^{*}]=0} \tag{7}\]  
The concentrations of the reactants are also related by the equations of the mass conservation law:  
\[[Ag]_{0}=[Ag]+[AbAg] \tag{8}\]  
\[[Ag\,^{*}]_{0}=[Ag^{*}]+[AbAg^{*}] \tag{9}\]  
\[[Ab]_{0}=[Ab]+[AbAg]+[AbAg^{*}] \tag{10}\]  
The equilibrium dissociation constants of the \(AbAg\) and \(AbAg^{*}\) complexes are denoted below as \(\mathrm{K_{d1}}\) and \(\mathrm{K_{d2}}\), respectively:  
\[Ab+Ag^{*}\leftrightarrow AbAg^{*} \tag{2}\]0  
\[Ab+Ag^{*}\leftrightarrow AbAg^{*} \tag{2}\]1  
The complex of Equations (8)-(12) completely describes the equilibrium competitive system and allows a cubic equation to be derived for the equilibrium concentrations of each of the components, as was demonstrated in [40; 41; 42]. For example, for the concentration of free antibodies, the cubic equation is as follows:  
\[Ab+Ag^{*}\leftrightarrow AbAg^{*} \tag{2}\]2  
where \(\mathrm{a}=\mathrm{K_{d1}}+\mathrm{K_{d2}}+[Ag]_{0}+[Ag^{*}]_{0}-[Ab]_{0}\); \(\mathrm{b}=\mathrm{K_{d2}}\) (\([Ag]_{0}-[Ab]_{0}\)) \(+\mathrm{K_{d1}}\) (\([Ag^{*}]_{0}-[Ab]_{0}\)) \(+\mathrm{K_{d2}}\mathrm{K_{d1}};\mathrm{c}=-\mathrm{K_{d2}}\mathrm{K_{d1}}[Ab]_{0}\). A trigonometric transformation provides a solution for this equation:  
\[Ab+Ag^{*}\leftrightarrow AbAg^{*} \tag{2}\]3  
where \(\mathrm{\theta}=\mathrm{arccos}\frac{-2\mathrm{a}^{3}+9\mathrm{ab}-27\mathrm{c} }{2\sqrt{(\mathrm{a}^{2}-3\mathrm{b})}^{3}}\). Using this solution, we can calculate the concentration of the detected complex [\(AbAg^{*}\)]:  
\[Ab+Ag^{*}\leftrightarrow AbAg^{*} \tag{2}\]4  
Equation (15) describes the calibration curve of the assay.