Dear CASToR developers and users :
I am trying to apply normalization in our 2layer small animal PET system. The crystal size is 0.8mm for the 1st layer and 1.2mm for 2nd layer.
I applied normalization, the method learned from the article: " Pepin, Audrey et al. ‘Normalization of Monte Carlo PET data using GATE.’ 2011 IEEE Nuclear Science Symposium Conference Record (2011): 41964200".
The result was good for conventional PET systems in that all the crystal sizes are the same， but when applied normalization in our system with 2 different crystal sizes, severe artifacts appear in the image after normalization. （Normalized cylinder imaging: A. conventional PET with the same crystal size of 0.8mm for 2 layers; B. our PET with 2 different crystal sizes for 2 layers）
All data was generated by GATE simulation and transferred into CASToR listmode Cdf/Cdh file.
Can anyone give me some guidance on the normalization of PET systems with different crystal sizes? I will be grateful for any reply. Any help will be appreciated!
Best
Xin
I don’t have much experience in this, so I can’t help much. However, I looked quickly at the paper you cited and it doesn’t seems to take into account mutlilayers détector. How did you adapt the decomposition to take this into account?
Note that I assume that size refer to width and not length.
If that is the case, I think that two samesized detector could be similar enough to single detector that the decomposition shown in the paper could work. Even tough it surprise me a little since deeper crystal should still be a little different than the others.
Dear Maxime,
Thank you for your reply!
(1) The paper I cited is designed for a singlelayer PET system, and we do some adjustments to fit the condition of the multilayers détector. The interference functions f_n^tr are no more introduced and the transverse geometric factors g_tr are designed for LORs in 1 single crystal ring of each layer. The transverse geometric factors g_tr are calculated by the same approach for each LOR independently.
(2) As you say, two samesized detectors could be similar enough to single detectors so we get acceptable normalization, no matter the crystals are staggered or not. (A.crystal staggered detector; B.Conventional DOI detector)
Now I am confused when trying to address the normalization for 2layer detectors with different crystal sizes (width size not length). I would be grateful if someone could provide ideas on this issue.
Best,
Xin
Greetings Xin,
Did you make any progress on this problem? I do not have new insights to provide at the moment, but I was curious to see if you found a solution.
Bests,
Maxime Toussaint
H Maxime，
Thank you for replying. I am trying to introduce oversampling into the normalization of our 2Layer PET scanner. Now I am focusing on it and do not get satisfactory results yet.
If I found a solution I will answer you still on this topic.
Best regards,
Xin
Here are some naive questions:
 Have you tried to index every components with DOI layer? The beauty of component based normalization, if my understanding is correct, is to reduce the statistics burden by reducing the number of variables. Thus, indexing all components with DOI might require much more statistics, but it might work? One could do this analysis on a smaller example to see if a simple bruteforce approach give something adequate. In your previous message, you mention that you extend the g^{tr} indexing. Was that extension only per DOIlayer or did it includes LORs with different DOI layer ID?
 Why did you exclude the f^{tr} from the decomposition? Your image show that your scanner is defined by block, so theses variables should be of interest.
 Was the u,v,i,j indexing extended to include detectors of both DOI layer? In other words, is the possible values of u,v different between a staggered scanner vs a nonstaggered scanner in the example provided previously.
Do remember that I am in no way an expert on the question. It just picked my interest as something I assumed to be solved and discovering that some, seemingly unanswered, questions remained.
Oh, Have you taken a look at “Foudray AMK, Chinn G, Levin CS. Component based normalization for PET systems with depth of interaction measurement capability. IEEE Nucl Sci Symp 2005;2108–2111”? The title seems very promising.
Dear Maxime,
Thank you for your response! I sincerely apologize for the delay in getting back to you. I have been occupied with some matters over the past few weeks.

Regarding your question about the indexation with DOI layer, I did apply a certain form of indexation. It involved combining components from all DOI layers into a single matrix. This was achieved by assigning larger layerID factors after smaller layer ID factors. For example, in our small animal PET, there are 120 crystal rings in the inner layer and 80 crystal rings in the outer layer. When calculating the axial block profile factors <b_ax>, a 200×1 array is generated. The top 120 values correspond to the <b_ax> factor of 120 inner rings, while the last 80 values correspond to the <b_ax> factor of 80 outer single rings. 3 of normalization factors(the axial block profile factors b_ax, the axial geometric factors g_ax and the intrinsic detector efficiencies ε) were calculated using the same method as described in the reference paper for singlelayer PET.
Now, regarding the transverse factor, the reference article employed <g_tr> and <f_tr> to accurately and simply module the transverse factor due to the high symmetry of the crystals in the transverse direction of the singlelayer DOI PET. However, in the case of multilayer DOI PET, the combinations of line of response (LOR) are not limited within the single layer, so they cannot be modeled using the previous method. To address this, I separately considered the LOR from transaxial crystals of each Rsector to transaxial crystals of other Rsectors. I accounted for their projection with annular source and obtained a unique transverse factor, which I denoted as <g_tr>.

The method described in the reference literature for singlelayer PET utilizes two modular factors, g_tr and f_tr, where each value corresponds to multiple LORs. However, in the case of multilayer PET, these LORs cannot be treated as equivalent. Consequently, we will calculate separate transverse normalization factors for each LOR. Given the nonmodular nature of multilayer PET, there is no necessity to calculate two coefficients. Instead, one transverse factor, denoted as g_tr, will suffice for the correction.

① We utilize the indices u and v to identify the crystal ring IDs. In our 2layer PET system, the final crystal ring ID in the inner layer is represented by u = 119, while the subsequent crystal ring corresponds to the first ring in the outer layer, denoted as u = 110+1=120. Due to the difference in crystal size between the two layers, the scanner of our PET design is staggered. However, to assess the applicability of the normalization method in a nonstaggered scanner, I conducted a test using a similar PET system with identical crystal size and quantity in each layer(120 crystal rings in the inner layer & 120 crystal rings in the outer layer). It is different from our design so with different u,v.
② The indices i and j are employed to indicate the crystals within a ring.

I have read this article “Foudray AMK, Chinn G, Levin CS. Component based normalization for PET systems with depth of interaction measurement capability. IEEE Nucl Sci Symp 2005;2108–2111”. They addressed the multilayer PET normalization problem by considering all lines of response (LORs) passing through the same crystal surface of two crystals as a LOR bin. Within each LOR bin, the LORs share the same normalization factor. This clever approach effectively transformed the challenge of normalizing a multilayer PET system into the normalization of a Monolithic system. But considering the significant number of geometric position calculations required and the associated computational complexity, we have decided not to pursue this implementation method at present.
In the meantime, I am exploring oversampling method as a potential solution to this problem. If I achieve promising results, I will gladly share them with you here.
Thank you once again for your response. I am eager to continue our discussion on this matter.
Best,
Xin