Double integral parameter fit to dataset

Hello,

I am trying to fit an equation to my dataset by finding the optimal parameters. The equation (please find also attached is):

ds/dt = Prefactor * integral (from zero to "t") (((t - t')/ t0) ^ b ) * Delta T(t) dt'

where t0 is set to 1 and I have lower and upper boundaries for "Prefactor" and "b". My datasets consist of the evolution of "s" over "t" and "T" over "t" (no mathematical representation of T(t)).

I thought about integrating numerically using the "integral2" function in a "for" loop and then finding the optimal values for the parameters "Prefactor" and "b" by using "lsqnonlin" to minimise the difference between the simulated "s" in the "for" loop and the "s" from the dataset but I am not sure.

Any pointers on how to tackle this would be greatly appreciated.

Many thanks in advance!