A crucial part of the assay is the antibody-to-antigen binding. In this section we try to analyse its behaviour and try to determine if the system is going to have the appropriate amount of sensitivity.

Consider and antibody 'A' and its complimentary antigen 'a'. We have looked at a simple model, which can be illustrated by the following equations^[1]:

$$\frac{d[A-a]}{dt}=k_a[A][a] \\ \frac{d[A]}{dt}=-k_a[A][a] \\ \frac{d[a]}{dt}=-k_a[A][a]$$

• where

  [A]	- concentration of antibody 'A'
  [a]	- concentration of antigen 'a'
  ka	- affinity of antibody and antigen
  [A-a]	- concentration of antibody 'A'

In our case, as the antibody and antigen concentrations do not change in time individually, we can simplify this system to the following:

$$ \frac{[A-a]}{dt}=k_{a}(A_{0}-[A-a])(a_{0}-[A-a])$$

where A₀ and a₀ are the initial concentrations of the antibody and antigen.

We can clearly see that this system is not linear, but we can simulate it and analyse it.

Fig.1 Antibody-to-Antigen binding time evolution curves, according to different affinity values

References

[1] McFadden, R., Kwok, C. S., "Mathematical Model of Simultaneous Diffusion and Binding of Antitumor Antibodies in Multicellular Human Tumor Spheroids, CancerRes 1988; 48:4032-4037