Systematic over-correction with scatter correction in TOF list-mode reconstruction

General Discussions
1

Dear CASToR developers and users,

I am experiencing a systematic apparent over-correction when applying scatter correction in TOF list-mode PET reconstruction with CASToR. I have already checked several common causes discussed in previous CASToR forum posts, but the problem remains.

I am using simulated PET data with only the true and scatter events. Normalization, attenuation, and scatter corrections are applied.

For non-TOF reconstruction, the correction works as expected: the contrast recovery coefficients (CRCs) of 37mm ROIs obtained from true+scatter events with scatter correction (85%) are very close to those obtained from true-only reconstruction (90%).

However, when TOF information is enabled, the scatter-corrected reconstruction shows substantially higher CRCs, typically around 115–134%.

For comparison, the corresponding TOF true-only reconstruction gives a CRC of approximately 90%. Therefore, the CRC values above 100% appear to be specifically associated with the TOF-dependent scatter correction rather than with TOF reconstruction itself.

For clarity, I am using TOF list-mode data. The discrete TOF bins described below are used only during scatter estimation and are not reconstruction TOF bins.

Scatter estimation procedure

I first estimate the scatter event rate between detector-module pairs rather than directly estimating the scatter rate for every individual crystal-to-crystal LOR.

The estimated scatter rate for each module pair is then distributed uniformly among all crystal-pair LORs connecting the corresponding pair of modules.

Therefore, the crystal-level scatter estimate is based on a module-level estimate and is spatially smoothed by construction, rather than directly using noisy Monte Carlo scatter counts for each individual crystal-pair LOR.

For the TOF dependence, I divide the TOF range into a number of bins and sort the simulated scatter events according to their TOF values.

For each LOR and TOF bin, I calculate the scatter rate density as

scatter density = N_scatter / (acquisition duration × spatial TOF-bin width),

where

spatial TOF-bin width = c × Δt / 2.

The resulting quantity therefore has units of counts/s/mm.

This scatter correction is then written into the CASToR list-mode CDF for reconstruction.

Checks already performed

I have already investigated the following possible explanations:

  1. Data mode

    The reconstruction uses TOF list-mode data, not TOF histogram data.

    The discrete TOF bins are used only for estimating the scatter-rate density and are not used as reconstruction TOF bins.

  2. Number of TOF bins used for scatter estimation

    I initially used 4 TOF bins.

    I then repeated the scatter estimation using 10 TOF bins, while keeping the same simulated dataset and reconstruction settings.

    Unexpectedly, the apparent over-correction became even stronger when increasing the number of scatter-estimation TOF bins from 4 to 10 (CRCs increased from 115% to 134%).

    Since the TOF-bin width becomes smaller as the number of bins increases, this observation makes me suspect that the problem may be related to the normalization or dimensional definition of the scatter term.

  3. Normalization and attenuation

    Both normalization and attenuation correction factors are explicitly written into the CDF as multiplicative correction factors according to the CASToR requirements.

    They are not included only through the sensitivity image.

    I have checked these factors and they appear to be consistent.

  4. Acquisition duration

    The acquisition duration in the CDF header is correct.

    Scatter counts are divided by the acquisition duration when calculating the scatter rate.

    I have also checked that the acquisition duration is not unintentionally applied more than once.

  5. TOF sign convention and detector ordering

    The detector ID ordering is kept consistent throughout the processing.

    I do not exchange detector ID1 and ID2, and the TOF sign remains negative for all event.

  6. True-only TOF reconstruction

    I reconstructed the true coincidences only using exactly the same TOF reconstruction parameters.

    The 37mm-CRC of the true-only reconstruction is approximately 90% for both TOF and non-TOF reconstruction.

    In contrast, the true+scatter data reconstructed with scatter correction give CRC values of approximately 115–134%.

    Therefore, the high CRC does not appear to be caused simply by TOF convergence behavior.

Main questions

At this stage, I would particularly appreciate clarification regarding the expected normalization and units of the scatter term for continuous TOF list-mode reconstruction in CASToR.

  1. For TOF list-mode reconstruction, is the scatter value stored for each event expected to represent a continuous scatter-rate density in counts/s/mm?

  2. If a scatter rate is estimated within a finite TOF interval, is dividing the scatter rate by the spatial TOF-bin width

    c × Δt / 2

    the correct way to convert it into the quantity expected by CASToR?

  3. Is there any dependence on the TOF resolution or TOF projector normalization that needs to be considered when defining the scatter correction term?

  4. Could the stronger over-correction observed when increasing the number of scatter-estimation TOF bins from 4 to 10 indicate that an additional TOF-bin-width-dependent scaling is being applied somewhere?

  5. Is there a recommended numerical consistency test for verifying the normalization of the TOF scatter term?

    For example, if the true number of simulated scatter coincidences is known, is there a way to verify that the scatter correction values written into the CDF represent the same total scatter contribution after integration over the TOF coordinate?

The most puzzling observation is that the same general normalization, attenuation, and scatter correction framework works well for non-TOF reconstruction, while the large quantitative bias appears only after making the scatter correction TOF-dependent.

In addition, the fact that the apparent over-correction becomes stronger when increasing the number of scatter-estimation TOF bins from 4 to 10 makes me particularly suspicious of a TOF-width-dependent normalization issue.

If useful, I can provide the CDF header, reconstruction command, TOF resolution, TOF range, scatter-estimation bin widths, scatter fraction, representative scatter correction values.

Any clarification regarding the expected physical units and normalization of the TOF list-mode scatter correction would be greatly appreciated.

Kinds regards,

Da