Background:
An unstructured growth model of trichosporon cutaneum is to be developed in
form of a system of two ordinary differential equations.
The Mathematical Model:
The underlying ordinary differential equations are
c_{x}(t)_{t} = m(t) c_{x}(t)
- D(t) c_{x}(t)
c_{s}(t)_{t} = (-m(t) /y_{xs}
- m_{s}) c_{x}(t) + D(t) (c_{sf} - c_{s}(t))
with m(t)
= m_{max}
c_{s}(t)
/ (K_{s} + c_{s}(t))
and D(t) = u(t)/V_{0}.
We have u(t)
= 0 if t < t_{f} and u(t)
= a (t - t_{f}) otherwise. Initial
values are c_{x}(0)=0.5 and c_{s}(0)=6.
We have to guarantee that there is a continuous transition at t=t_{f},
where the model changes. Parameters to be estimated, are K_{s},
m_{max}, a, and an intermediate
time value t_{f},_{ }which is unknown in
advance and where the model changes. The lower index t denotes the
time derivative of the time-dependent state variables c_{x}(t)
and c_{s}(t).
Literature:
Baltes M., Schneider R., Sturm C., Reuss M. (1994): Optimal experimental
design for parameter estimation in unstructured growth models, Biotechnical
Progress, Vol. 10, 480-488
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):