Background:
The relationship between one antibody, the receptor, and two antigens, the
ligands, can be described by equations based on the mass equilibrium under
certain assumptions. The resulting system of nonlinear equations does not
depend on the time, i.e., is a steady-state-system.
The Mathematical Model:
After some transformations, we get
z_{1}(c) (1 + p_{1}
z_{2}(c)
+ p_{2} z_{3}(c))
- p_{3} = 0
z_{2}(c)
(1 + p_{1} z_{1}(c))
- p_{4} = 0
z_{3}(c)
(1 + p_{2} z_{1}(c))
- c = 0
z_{1,} z_{2}, and z_{3} are the state variables, i.e., the solution variables of the system of equations, depending on a concentration c for which measurements subject to the fitting criterion p_{4} - z_{2}(c) are retrieved. Parameters to be estimated, are p_{1}, p_{2}, p_{3} and p_{4}.
Literature:
1. Rominger K.L., Albert H.J. (1985): Radioimmunological determination
of Fenoterol. Part I: Theoretical fundamentals, Arzneimittel-Forschung/Drug
Research, Vol. 35, No. 1a, 415-420
2. Schittkowski (2002):
Numerical Data Fitting in Dynamical Systems - A Practical Introduction with
Applications and Software,
Kluwer
Academic Publishers
Implementation:
The complete solution of a data fitting problem is described
in six
steps:
Results:
Then you would like to take a look at reports and graphs:
- parameter values
- experimental data versus fitting criterion
Model equations (or use your own favorite editor):
Measurement data (or use import function for text file or Excel):
Parameters, tolerances and start of a data fitting run:
Numerical results (computed by the least squares code DFNLP):
Report on parameter values, residuals, performance, etc. (or export to Word):
Experimental data versus fitting criterion (also available for Gnuplot):