Corrections for Physical Factors - Correction Techniques in Emission Tomography - page 72

5.3.1 Singles-based correction
This approach requires the knowledge of the singles rate on each detector.
A statistical consideration then leads to the following relation:

R ij = 2 coin s i s j ; (5.6)

where s i , s j are the single counting rates measured on detector i and j, re-
spectively, R ij is the expected randoms rate between the detectors, and coin
is the length of the coincidence window [48]. This rate can be subtracted from
the measured coincidence rate P ij between the two detectors, resulting in a
good estimate of the true coincidence rate T ij (in addition to scattered events
S ij ):

T ij + S ij P ij R ij : (5.7)

An advantage of this method is the low variance of the calculated randoms
rate as single rates in PET are much higher than coincidence rates. However,
a proper knowledge of system dead times, variances in coin between differ-
ent detector elements and detector eciencies is crucial for correct randoms
estimation, as otherwise the calculated rate may be biased [14].

5.3.2 Delayed window correction
Another method, the delayed window approach, is based on a more direct
measurement of random events. For this, a measured count on a detector
not only opens the usual prompt coincidence window of width coin , but also
another window after some time delay > coin (see Figure 5.3). This ensures
that delayed coincidences cannot be caused by a single annihilation event and
must therefore be random coincidences. Subtracting the delayed coincidence
rate D ij from a line of response between detectors i and j from the prompt

FIGURE 5.3: Estimation of random coincidences using the delayed window
approach. Counts on a detector open two windows, the coincidence window
and the delayed window. Measured coincidences during the coincidence win-
dow can be either true, scattered or random coincidences; coincidences in the
delayed window can only be random coincidences.