Background:
The simple linear pharmakokinetic model could describe for
example the time-dependent concentrations of plasma and urine based on a
simple bolus application.
The Mathematical Model:
The underlying ordinary differential equation is
y_{1}(t)_{t} = -k_{i}y_{1}(t)
y_{2}(t)_{t} = k_{i}y_{1}(t) - k_{2}y_{2}(t)
with initial values y_{1}(0)=D, the initial dose, and y_{2}(0)=0. Parameters to be estimated, are the rate constants k_{i} and k_{2}, and the dose D. Experimental data are available for y_{1}(t) and y_{2}(t) for 13 time values between 1 min and 60 min. The lower index t denotes the time derivative of the concentrations y_{1}(t) and y_{2}(t).
The solution of the differential equation can be given analytically in the form
y_{1}(t) = D exp(-k_{1} t)
y_{2}(t) = k_{1} D
(exp(-k_{2} t) - exp(-k_{1} t)) /(k_{1} - k_{2})
Literature:
1. Heinzel G., Woloszczak R., Thomann P. (1993): TOPFIT 2.0:
Pharmacokinetic and Pharmacodynamic Data Analysis System, G. Fischer,
Stuttgart, Jena, New York
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):