Random Coincidences

As already mentioned, prompt coincidence counts include random or accidental coincidences that raise the background on the images. Random events 

Efforts have been made to minimize random events by using the faster electronics and shorter time window, e.g., BGO system (12 ns), GSO and Nal(Tl) systems (8 ns), and LSO system (6ns). Still, further corrections are needed to improve the image contrast. Note that crystals with shorter scintillation decay time also reduce the random coincidences, e.g., random events are less in LSO (70 ns) than in BGO (300ns). 

A common method of correction for random events is to employ two coincidence circuits - one with the standard time window (e.g., 6 ns for LSO) and another with a delayed time window (say, 50-56 ns) of the same energy window. The counts in the standard time window include both the randoms plus trues, whereas the delayed time window contains only the randoms. For a given source, the random events in both time windows are the same within statistical variations. Delayed window counts are subtracted from the standard window counts to obtain the true coincidence counts, which are essentially free of any systematic errors associated with the PET scanner because they cancel by subtraction. 

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A significant number of the detected events are invalid, as either they correspond to a "random coincidence" between two unrelated y-rays or else one or both of the y-rays has been scattered before detection. The PEPT algorithm attempts to discard these invalid events, using an iterative procedure in which the centroid of the events is calculated, the y-rays passing furthest from the centroid are discarded, and this process is repeated until a specified fraction / of the original events remains. The optimum value of / depends on the mass of material between the tracer and the detectors, which adds to the number of scattered events. For example, when studying flow inside a vessel with 15 mm thick steel walls, 80% of the detected events must be discarded, so that the fraction/ of useful events is 0.2. The precision A of a PEPT location is given approximately by... 

A total of 20 cases of type I diabetes melhtus suspected to be induced by MMR immunization have been reported to Behringwerke, Marburg, Germany, probably due to the mumps component (79). The earliest case occurred 3 days after receiving the vaccine and the latest 7 months after immunization. Twelve cases were diagnosed within 30 days of immunization. The investigators considered the cases of diabetes meUitus to have a temporal relation to the immunization. For every 5 million children immunized against mumps 50 spontaneous cases of diabetes meUitus are to be expected by random coincidence within... 

There might also be systematic errors like, for example, erroneous assignment of mother-daughter decay correlations or random coincidences. Such problems are... 

Figure 3.1. (a) True coincidence events (b) Random coincidence events detected by two detectors connected in coincidence along the dotted line. The two 511-keV photons originated from different positron annihilations, (c) Scattered coincidence events. Two scattered photons with little loss of energy originating from two annihilation events may fall within PHA window and also within coincidence time window to be detected as a coincidence event by two detectors. 

The projection data acquired in the form of sinograms are affected by a number of factors, namely variations in detector efficiencies between detector pairs, random coincidences, scattered coincidences, photon attenuation, dead time, and radial elongation. Each of these factors contributes to the sinogram to a varying degree depending on the 2D or 3D acquisition and needs to be corrected for prior to image reconstruction. These factors and their correction methods are described below. 

Describe the methods of correction for random coincidences and scatter coincidences in the acquired data for PET images. 

The measured proton energy spectrum from the (d,p) reaction in coincidence with the fission fragments (after subtraction of random coincidences) is shown in O Fig. 5.7a in terms of the excitation energy of the compound nucleus Pu. The spectrum is proportional to the product of the fission probability and the known smoothly varying (d,p) cross section, which shows no fine structure (Specht et al. 1969). 

The measured high-resolution excitation-energy spectrum of is shown in Fig. 5.22 as a function of the excitation energy of the compound nucleus in the region of = 4.0-. 8 MeV together with the contribution of the random coincidence events (dashed line). 

Summing of events, either a result of coincident emission of gamma rays in the decay chain of the nuclide of interest or of random coincident emissions, can lead to significant losses or potential additions to an otherwise clean peak (De Bruin and Blaauw 1992 Becker et al. 1994). While coincidence losses are not an issue for comparator NAA, calibration, and/or computational correction must be applied (Debertin and Helmer 1988 Blaauw and Celsema 1999) to arrive at true peak areas for other methods of calibration. 

In PGAA, the sample-to-detector distance is usually large, which makes the summing of true and random coincidences negligible. 

Such random coincidence is undesirable because it causes counts to be lost from the full energy peaks in the... 

In Chapter 4, I discussed random summing in connection with the pile-up rejection circuitry in amplifiers. We came to the conclusion that even with pile-up rejection there must be some residual random coincidences. There is then, whether or not pile-up rejection is available, a need to be able to correct for random summing in high count rate spectra. In some circles, there seems to be an assumption that pile-up rejection is 100% effective... 

A pulse will be involved in a summing whenever it is not preceded or followed by a certain period of time. This time, T, is the resolution time of the electronic system. Using the Poisson distribution, it can easily be demonstrated that the probability of a random coincidence, Pq, within T is ... 

CCFs often dominate the unreliability of redundant systems by virtue of defeating the random coincident failure feature of redundant protection. Consider the duplicated system in Figure 5.2. The failure rate of the redundant element (in other words the coincident failures) can be calculated using the formula developed in Table 5.1, namely 2X MDT. Typical failure rate figures of 10 per million hours (10 per hr) and 24hrs down time lead to a failure rate of 2 x 10 x 24 = 0.0048 per million hours. However, if only one failure in 20 is of such a nature as to affect both channels and thus defeat the redundancy, it is necessary to add the series element, shown as X2 in Figure 5.3, whose failure rate is... 

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