# Deconvolution of two gamma variates

Started by January 26, 2007
```Hi! I'm busy with a project to determine cerebral blood flow using MRI. At
two places in the brain the concentration of a contrast solution is
measured every 1 second (or so.) The points are then fitted to a gamma
variate of the form:

f(t) = a (t-to)^b exp[ -(t-t0)/c ]   for t=> t0
f(t) = 0  t<0

where a, b, c, and t0 are parameters found by curve fitting.

The fitting is done because the blood recirculates into the brain. These
recirculation points are not included in the function fit. Please see:
http://s153.photobucket.com/albums/s235/s1020099/ for some pictures.

The question: How can I deconvolve two gamma variates? It has been
suggested to me to use the Fourier method and do it analyticaly. But the
Fourier transformation of the above function is of the form: p^(-1-b)
Gamma(1+b) and is this not too complicated?

I have tried the discrete fourier transformation: first extrapolating the
two gamma variates to a power of 2 (also to limit the discontinuties as
much as possible) . Then dividing in the fourier domain. etc

The results depend so much on the exact parameters of my two gamma
variates. The typical deconvolved graph I get has a peak followed by a
little valley. Is this what I should expect? With the deconvolved graph I
am measuring blood flow = max of peak / area . The little valley makes a
big difference in this determination.

I have also tried singular value decomposition. With this I zero-ed 20% of
the maximum of the singular values (on the diagonals). But with this I get
large oscilations in the result.

I am glad with any help,
Paul

```
```
On Jan 26, 3:48 pm, "PaulPaul" <p.ba...@student.rug.nl> wrote:
>
> The fitting is done because the blood recirculates into the brain. These
> recirculation points are not included in the function fit. Please see:http://s153.photobucket.com/albums/s235/s1020099/for some pictures.
>
> The question: How can I deconvolve two gamma variates? It has been
> suggested to me to use the Fourier method and do it analyticaly. But the
> Fourier transformation of the above function is of the form: p^(-1-b)
> Gamma(1+b) and is this not too complicated?
>
>
> I am glad with any help,
> Paul

Hello Paul,

The convolution of gamma distributed vars has a known function in
terms of the parameters of the individual gamma distributions. So you
can curve fit it to your data, and then back out the params of the two
constituent gamma distributions.

To result of convolving of gamma functions may be found on page 384 of
"Continuous Univariate Distributions", vol 1, by Johnson, Kotz, and
Balakrishnan. John Wiley, 1994

IHTH,

Clay

```