How to fit a model that has a convolution

Hi, I am trying to fit some data with a model that includes a convolution (it's an input function convolved with an exponential). It has 3 parameters, ktrans, kep, and a shift term (Since the data does not start at 0). I have attached a script that includes some data (CtPix and the time it was acquired over), and the steps I have been trying to fit it using the conv function in matlab. The output is completely wrong (when I plot kFit). One thing to note is that both the exponential (expConv) and Cpt should have the same shift factor but due to something I don't understand, the convolution should only have 1 of the functions shifted (see my other topic here). I have 2 ideas of what it could be but not sure:

1. Not enough "samples"

2. The vectors are incorrectly defined. Since the Conv function doesn't have any idea of the time input, I may have to 0 pad the data to 0 but not sure that is the correct way to do it for fitting...

Any suggestions would be really helpful. Thanks