Estimation from Singles Rates

The total number of randoms on a particular line of response Lij can, in principle, be determined directly from the singles rates t{ and Tj using equation (1) or (2). Consider an acquisition of duration T. The random coincidences Rj in the data element corresponding to the line of response Lj may be found by integrating equation (1) or (2) over time:

If r,(t) and Tj(t) change in the same way over time, we can factor out this variation to obtain

where k is a constant and s, and Sj are the single event rates at, say, the start of the acquisition. For an emission scan, f(t) is simply the square of the appropriate exponential decay expression, provided that tracer redistribution can be ignored. Rj can then be determined from the single events accumulated on channels i and j over the duration of the acquisition. It should be noted that the randoms total is proportional to the integral of the product of the singles rates, and not simply the product of the integrated singles rates. Failure to account for this leads to an error of about 4% when the scan duration T is equal to the isotope half-life T-, and about 15% when T = 2Ti.

For coincidence-based transmission scans, where positron-emitting sources are rotating in the field of view, f(t) becomes a complicated function dependent on position as well as time, and equation (4) is no longer valid. However, in principle the total number of randoms could still be obtained by sampling the singles rates with sufficiently high frequency.

The singles rates used for calculating randoms should ideally be obtained from data that have already been qualified by the lower energy level discriminator - they are not the same as the singles rates that determine the detector dead-time. Correction schemes have been implemented which use detector singles rates prior to LLD qualification [4], but the differences between the energy spectrum of events giving rise to randoms and that giving rise to trues and scatter must be carefully taken into account. These differences are dependent upon the object being imaged and upon the count-rate, since pulse pile up can skew the spectra [5].

In its simplest form this method does not account for the electronics dead-time arising from the coincidence processing circuitry (to which the randoms in the coincidence data are subject).

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