Fit 2 parameters with fminsearch and use ODE15s to solve single differential equation

Hello ! I am currently working on a project looking at tumor growth. I have to try different mathematical models from the litterature and fit them using some data.

For the momment I am trying to fit the Gompertz model of a tumor grotwh which is the following differential equation: (a-b*ln(v))*v)

In my code a is p(1) ; b is p(2) ; v is y(1).

I have made a first file to solve the differential equation.

Then I made a second file to fit the model with some data that I've got.

My code seems to work but I am getting estimated parameters very different from what I was expected which was around 1 or even less for a and b respectively. Also I shouldn't get a negative value.

I have tried to follow some previous examples of the MATLAB answers from Star Strider, so I am not at all sure about what I have done.

If someone could tell me if my code is correct It would be much appreciated. I have attached my 2 files.

Thanks in advance !!!

[Soomin Lee]

How did you get p0 values which are educated guess of parameter y(1) and y(2)?

Answer 1

[Star Strider]

Using that model of the Gompertz equation, your code is correct.

If you want to avoid negative rate constants, use lsqcurvefit instead, with constraints on the lower limits of the parameters:

B = lsqcurvefit(@fit_GompModel, p0, t, y, [0 0])

producing:

B =
          68.5834500164402
      6.38601980323603e-12

This give a straight line, however that is likely the best you can do. I doubt that ga (genetic algorithm) or other solvers will produce any different result, although I did not try them.

[hna]

Thanks for you answer.

Now that I can use this function to fit my different models. Do you know a way that I could use to find the goodness of fit for each fitting model, so after I can see which one would be the best fit ?

[Star Strider]

As always, my pleasure.

The easiest way to assess the model fit would be to use the data and the estimated parameters with the Statistics and Machine Learning Toolbox fitnlm function. It will provide all the statistics you will need.

[hna]

Thanks again for the suggestion.

I had a look through this function. The thing is that to set this up fitnlm, don't you need to again specify some guessed parameters ? Because I don't really need this as I already got mine with the fminsearch function. I'd like to use the fitnlm to provide me some statiscal answers for my fit ..

[Star Strider]

With a constrained optimisation, goodness-of-fit may be a bit difficult to assess, since the usual tools such as fitnlm do not allow parameter constraints, and with parameter constraints, those would likely not be reliable anyway. Probably the best in that regard is the Coefficient of Determination. It is relatively easy to code, allows use of the fitted curve using constrained parameters, and should provide a reliable estimate for comparing different models.

[hna]

Thanks again. Last question ..

I tried to code my model using fitnlm just to see the difference in terms of parameters with fminsearch but it dosn't seem to work... I have attached my file if you have time to get a quick look..

Also what is the most accurate in terms of paramters calculation between the 2 functions ?

[Star Strider]

I cannot get your model and data to work with fitnlm. I am not certain what the problem is, however it may be that there are too few data. It works with nlinfit, however the parameters are both negative, so the statistics would be irrelevant.