Formula of normalization when using GATE simulation data

General Discussions
1

Dear CASToR users,
I am trying to apply normalization in CASToR recon based on the method in the article: “Normalization of Monte Carlo PET data using GATE”, Normalization of Monte Carlo PET data using GATE | IEEE Conference Publication | IEEE Xplore. I am confused about 3 details in this article.

  1. About auivj in formula (5). How to understand “the inverse of the analytical projection of the source” ? On what method should I used to get the projection?
  2. About Φ(i,j) mod D=d in formula (7). How to understand “mod” in this formula? Because it does not seem to mean the remainder operation in division.
  3. How to define an annular source as in the article. I found out that in GATE Official Guide website, only 2-D annular source are supported and I can not find any 3-D annular source definition guidance on that.

Any help would be very appreciated.

Regards,
Zhao Xin

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2

Hi Zhao,

  1. The normalization factor depends on the projector used for the reconstruction. Ideally, you shall use the same projector for a_{uivj} and for your CASToR reconstruction.
  2. It is indeed \Phi modulo D (azimuthalBin % D).
  3. In 2011, the Forbid command was used (not sure if it still works) :

/gate/source/addSource OuterCylinder

….

/gate/source/OuterCylinder/gps/shape Cylinder

….

/gate/source/OuterCylinder/gps/Forbid internalCylinder

/gate/world/daughters/name internalCylinder

/gate/world/daughters/insert cylinder

…

Hope this help,

Claude

3

Dear Claude,

Thanks for clearing all my confusion. It is really helpful for my current work. I really appreciate it.

Many thanks,
Zhao

4

Hi!

I came across this conversation and I am still unsure to understand how to construct the factor “a” of equation (5).

I have the following questions:

  • What is the “source” referring to in the sentence “analytical projection of the source”? Is it the annular source?
  • Given that the projection is said to be “analytical”, then why does it seem to be projector-dependent, as suggested by @COMTAT_Claude? I tried to implement it by computing the analytical line integral of the source along the LOR, but I am now unsure whether I implemented it correctly.

More generally, I would like to have a deeper understanding of the factor a. Do you have any references introducing it in more details? I tried to look it up into the literature but haven’t found anything so far.

Thank you very much!

Aurélien.

5

Yes, the « source » refers to the annular source.

Ignoring attenuation, random and scattered coincidences, the principle of direct normalization is to compute the ratio N=y/Px, where y are the high statistics measured data (after correction for attenuation, random and scattered coincidences), P the projection used for the reconstruction, and x the pixelized image of the known source. In other words, the ratio between the measured data and the expected data, for a given P.

For a component-based normalization, the same idea applies, in particular for the computation of a=1/Px, needed for the estimation of the transverse geometric factors.

In practice, it is common to compute the analytical line integral of the annular source along the LoRs (what was done for the 2011 proceeding) instead of an algebraic projection Px.

All this work was based on a book chapter: Quantitative Techniques in PET by S. Meikle and R. Badawy, in Positron Emission Tomography, Basic Sciences and Clinical Practice. Springer, 2003.

Hope this helps,

Claude

image001.png

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6

Thank you very much for your answer! I think it helped greatly.

One last thing remains unclear to me. I am trying, without success so far, to find in the literature more examples and more details about that factor “a”. I especially went through the reference you provided but cannot find where that factor is defined. Could you please pinpoint to the exact page where it appears? Perhaps I am missing something quite obvious…

Thank you once again!

Aurélien.

7

In the paragraph entitled “The Transaxial Geometric Factors g_{uivj}^{tr}”, it’s just written “Once the data have been collected, an analytic correction is applied to compensate for non-uniform illumination of the LORs by the source”, without further explanation on how to compute the analytic correction. ‘a’ refers to this analytic correction.

Best,

Claude

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