Neighbouring clock cycles
2.5 nsec single events on channel j, the number of single events acquired during a PET acquisition is typically 1 to 2 orders of magnitude greater than the number of coincidences. In such an environment, equation (1) provides a good estimate of the random coincidence rate.
The timing of commercial tomographs is usually governed by a system clock. A timing signal on channel i is thus assigned to a particular clock cycle. If there is a timing signal on channel j within a certain range of, say, n neighbouring clock cycles, a coincidence is recorded on Lj (Fig. 5.3). Therefore the randoms rate on Lij would be given by
where tc is the duration of a single clock cycle. A typical BGO tomograph might have a 2.5 nanosecond clock cycle, and n = 5 clock cycles. Thus, the total coincidence time window ntc (equivalent to 2rfor an analog system) would equal 12.5 nanoseconds.
Equations (1) and (2) indicate that the overall randoms rate for an acquisition will change at a rate proportional to the square of the overall singles rate. Provided dead time is small, this means that for a given source distribution the randoms will change roughly in proportion to the square of the activity concentration.
Random coincidences can form a significant fraction of all recorded coincidences in PET imaging, particularly if large amounts of activity are used or if scans are performed in 3D mode. The number of randoms detected may be reduced by shortening the coincidence window. However, the window must be large enough to prevent loss of true coincidences due to the difference in arrival times (which may be up to 2 ns for an annihilation pair originating 30 cm from the centre of the tomograph) or statistical variations in the triggering of the event timing circuitry. Thus, selection of the coincidence window is a trade-off between minimising acceptance of randoms and loss of sensitivity to true coincidences. The coincidence window is typically set to 3 to 4 times the full width half maximum (FWHM) timing resolution of the tomograph.
The use of fast scintillators such as LSO or GSO reduces timing uncertainty (compared to that obtainable with slower scintillators such as BGO or NaI), but the window width cannot be less than 3 nsec to 4 nsec without accounting for time-of-flight effects. Randoms may also be reduced by shielding the detectors from activity that lies outside the tomograph field of view-this reduces the singles rates without adversely affecting sensitivity to true coincidences [1,2].
Randoms tend to be fairly uniformly distributed across the field of view. This contrasts with true coincidences, which follow activity concentration and are reduced in regions of high attenuation. Thus, the fraction of random coincidences in regions of high attenuation can become very large and, if uncorrected, substantial quantitative errors can arise.
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