# deconvolution and fitting of gamma variates

Started by May 24, 2007
```Hi Everyone,

I'm looking for help on deconvolving and the fitting of two gamma
variates. I'm using MATLAB. My two gamma variates have the form:

modelFun =  @(p,x) ( p(1) .* ( (x-t0)./(p(2)-t0)).^ p(3).*
exp(p(3).*(1-(x-t01)./(p(2)-t01))) .* heaviside2nd(x-t01));

Where P has to be given initial values to perform the fit. Here the
function heaviside2nd is the modified version of the heaviside function
where Y=1 at X=0 rather than NaN.

In my project I have to deconvolve the gamma variate graph Cm with the
gamma variate graph AIF.  invF [ F(Cm)/F(AIF) ]

The thoery says that the AIF (arterial input function) are the average of
those brain pixel chosen which have a peak that arrives earlier than
average and which have a breadth (FWHM) that is smaller than average, and
also the maximum is as large as possible. These are the properties of a
concentration bolus that passes through the brain arteries.

I use the AIF to determine blood flow in the other regions. The blood flow
in the other regions can only be determined if there is an instant
injection of blood: hence this is the residue function.

http://s153.photobucket.com/albums/s235/s1020099/?action=view&current=Afb1.jpg

This is a case in which it has gone right. The deconvolved curve shoots up
almost instantly and gradualy decreases as the blood leaves the region.

But it doesn't always look like this and I don't know why:

http://s153.photobucket.com/albums/s235/s1020099/?action=view&current=Afb2.jpg

http://s153.photobucket.com/albums/s235/s1020099/?action=view&current=Afb3.jpg

In the last photo the deconvolved curves oscilates. Is this a sign of
noise? But how can there be noise in my sampled analytical gamma varite
function?

Maybe I'm not fitting the gamma variate curve right. The curve is fitted
to force the concentration to go back to zero. You see that after the main
peak the points remain higher than before the peak started.

Please can you help? I don't understand why the deconvolution sometimes
works great and sometimes gives strange results.

I would very much appreciate email contact with some expert in this field
too if possible.

Thanks.

_____________________________________
Do you know a company who employs DSP engineers?
Is it already listed at http://dsprelated.com/employers.php ?
```