Fitting coupled differential equations to experimental data
- To: mathgroup at smc.vnet.net
- Subject: [mg83812] Fitting coupled differential equations to experimental data
- From: "Ausman, Kevin" <ausman at okstate.edu>
- Date: Sat, 1 Dec 2007 05:43:26 -0500 (EST)
I have been beating my head against a problem for several weeks now, so
I'm at the point of begging for help. Please help!
I have a set of coupled differential rate equations with several
parameters (this is a chemical system, and these variable parameters are
rate constants and initial concentrations):
{DHR'[t] == -kadD*DHR[t]*C60[t] + kdeD*DHR1[t],
DHR1'[t] == kadD*DHR[t]*C60[t] - kdeD*DHR1[t] - kox*DHR1[t],
RH1'[t] == kox*DHR1[t] - kdeR*RH1[t] + kadR*RH[t]*C60[t],
RH'[t] == kdeR*RH1[t] - kadR*RH[t]*C60[t],
C60'[t] == -kadD*DHR[t]*C60[t] + kdeD*DHR1[t] + kdeR*RH1[t] -
kadR*RH[t]*C60[t], Fluorescence[t] == ScaleFactor*(RH[t] + =
RH1[t]),
DHR[0] == initDHR, DHR1[0] == 0, RH[0] == initRH, RH1[0] =
== 0,
C60[0] == initC60}
My variables in this case are:
variables={DHR, DHR1, RH, RH1, C60, Fluorescence}
And my parameters are:
{initC60,initRH,initDHR,ScaleFactor,kadD,kdeD,kox,kadR,kdeR}
I can successfully create a model function at a given set of values for
these parameters using NDSolve:
model[kadD_?NumberQ, kdeD_?NumberQ, kox_?NumberQ, kadR_?NumberQ,
kdeR_?NumberQ, initDHR_?NumberQ, initRH_?NumberQ, initC60_?NumberQ,
ScaleFactor_?
NumberQ] := (model[kadD, kdeD, kox, kadR, kdeR, initDHR, initRH,
initC60, ScaleFactor] =
First[NDSolve[{DHR'[t] == -kadD*DHR[t]*C60[t] + kdeD*DHR1[t],
DHR1'[t] == kadD*DHR[t]*C60[t] - kdeD*DHR1[t] - kox*DHR1[t],
RH1'[t] == kox*DHR1[t] - kdeR*RH1[t] + kadR*RH[t]*C60[t],
RH'[t] == kdeR*RH1[t] - kadR*RH[t]*C60[t],
C60'[t] == -kadD*DHR[t]*C60[t] + kdeD*DHR1[t] + kdeR*RH1[t] -
kadR*RH[t]*C60[t],
Fluorescence[t] == ScaleFactor*(RH[t] + RH1[t]),
DHR[0] == initDHR, DHR1[0] == 0, RH[0] == initRH, =
RH1[0] == 0,
C60[0] == initC60}, variables, {t, 0, 7200}]])
solution =
model[0.018, 0.024, 0.394, 0.024, 0.018, 0.78, 0, 0.018, 90700];
Plot[Fluorescence[t] /. solution, {t, 0, 7200}]
<Graphic>
I can even treat the parameters as adjustible using the Manipulate
operation, plotting it with the experimental data that I want to fit:
XPData :=
Import["C:\Documents and Settings\xxx\My \
Documents\Briefcase\Research\Kinetic Fitting with \
Mathematica\DHR123_7.PRN", "Table"]
XPTime = XPData[[All, 1]];
XPFluor = XPData[[All, 2]];
Manipulate[(solution =
model[kadD, kdeD, kox, kadR, kdeR, initDHR, initRH, initC60,
ScaleFactor];
Show[ListPlot[XPData],
Plot[Fluorescence[t] /. solution, {t, 0, 7200}]]), {{initC60, 0.1},
0, 1}, {{initDHR, 0.1}, 0, 1}, {{initRH, 0}, 0,
1}, {{ScaleFactor, 10000}, 0, 1000000}, {{kadD, 0.1}, 0,
1}, {{kdeD, 0.1}, 0, 1}, {{kox, 0.1}, 0, 1}, {{kadR, 0.1}, 0,
1}, {{kdeR, 0.1}, 0, 1}]
What I cannot seem to do is fit this model to experimental data using
FindFit, or by generating a least-squares error function and using
Minimize, NMinimize, or FindMinimum.
For example:
FindFit[XPFluor, (Model2 =
model[kadD, kdeD, kox, kadR, kdeR, initDHR, initRH, initC60,
ScaleFactor];
Fluorescence[t] /. Model2), {{initC60, 0.018}, {initDHR,
1}, {initRH, 0}, {ScaleFactor, 134000}, {kadD, 0.012}, {kdeD,
0.118}, {kox, 1}, {kadR, 0.012}, {kdeR, 0.134}}, t]
generates ReplaceAll::reps errors and InterpolatingFunction::dmval
errors.
LSE[kadD_?NumberQ, kdeD_?NumberQ, kox_?NumberQ, kadR_?NumberQ,
kdeR_?NumberQ, initDHR_?NumberQ, initRH_?NumberQ, initC60_?NumberQ,
ScaleFactor_?NumberQ, XPT_?VectorQ,
XPF_?VectorQ] := (LSE[kadD, kdeD, kox, kadR, kdeR, initDHR, initRH,
initC60, ScaleFactor, XPT,
XPF] = (solution =
model[kadD, kdeD, kox, kadR, kdeR, initDHR, initRH, initC60,
ScaleFactor]; M = Fluorescence[XPT] /. solution; Norm[XPF - M]))
FindMinimum[
LSE[kadD, kdeD, kox, kadR, kdeR, initDHR, initRH, initC60,
ScaleFactor, XPTime,
XPFluor], {{initC60, 0.018}, {initDHR, 1}, {initRH,
0}, {ScaleFactor, 134000}, {kadD, 0.012}, {kdeD, 0.118}, {kox,
1}, {kadR, 0.012}, {kdeR, 0.134}}]
generates NDSolve::icfail, First::first, ReplaceAll::reps, and
FindMinimum::nrnum errors.
I have tried dozens of other formulations, and can evaluate fits
one-at-a-time or by using Manipulate, and then evaluate LSE (or other
formulations) one-at-a-time or by using Manipulate, but I cannot seem to
get Mathematica to perform an optimization.
Does anyone have an idea of how to help here? Thanks!
Kevin Ausman
-------------------------
Prof. Kevin Ausman
Oklahoma State University
Department of Chemistry
342 Physical Science
Stillwater, OK 74078
Email: ausman at okstate.edu
Office: 405-744-4330
Fax: 405-744-6007
--------------------------