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LYSO and the background activity of lutetium
2.59% of the natural lutetium (Lu-175) in the LYSO is Lu-176, which decays to Hf-176 by {EQUATION()}\beta ^{-}{EQUATION} decay with a long half life ({EQUATION()}3.78\cdot 10^{10}{EQUATION} years), thus fortunately positrons are not created during the decay. When meeting an electron, positrons would annihilate to form two gamma photons, and acting as an extra ‘source’ they would confound the measurement. The antineutrinos do not cause problems, but the released electrons lose energy in different processes, causing scintillation. Moreover, Hf-176 decays further while photons with gamma energies of {\bf 88 keV, 202 keV, and 307 keV} are created. The count rate density of the material of the LYSO crystals is approximately {EQUATION()}\frac{cps}{cm^3}{EQUATION} in calculations.
::{img fileId="3344" max="400" malign="center" desc=" Figure 1.: The decay scheme of Lu-176 and the single spectrum of the LYSO. (Based on the Saint-Gobain catalogue)"}::
If we naively estimated the random coincidence resulting from this single rate, we would get a very small number (1-100). In comparison, a few thousand (approximately 3000) can be measured in the NanoPET (this depends on the coincidence mode as well), therefore our estimation is not good. The problem is that the decay of Hf-176 is a cascade decay, in about 4 ps the gamma photons with the energies of 88, 202, 307 keV are created non-uniformly in time, or more precisely, corresponding to the Poisson distribution of decays that are not independent of each other. If one of them escapes the crystal (typically one of the two with the highest energy), then it is perceived as real coincidence with a time window of a few nanoseconds. That is why there are two orders of magnitude more random coincidences than there would be in the case of a uniform temporal distribution.
::{img fileId="3345" max="400" imalign="center" desc=" Figure 2."}::
Luckily, the energy of the LYSO gamma photons in coincidence seems to be 509 keV, if the two higher-energy photons are absorbed in the same module (202+307) and the one with the energy of 88 keV is absorbed somewhere else. (However, typically the higher-energy photons can escape and it is not probable that they arrive in the same module, since their direction is not in correlation. In other pairings we see events with lower energies. Thus, the peak of the energy spectrum of the LYSO measured in coincidence is at around 2-300 keV, so it can practically entirely be filtered out of the measurement data by a lower energy gate of 400 keV.
Positron range
Positron range is a phenomenon that is inevitably present and it deteriorates spatial resolution. The positrons are created during the decay of a {EQUATION()}\beta^+{EQUATION}-decaying nucleus. They leave the nucleus with a relatively high energy (~0-2 MeV). During the {EQUATION()}\beta^+{EQUATION} decay (positron decay) a proton is converted into a neutron, while a positron and a neutrino are created due to the conservation laws.
Three particles carry away the initial energy of the proton. They can do that in many different angles, dividing the initial energy and the momentum among themselves in several ways. Therefore, the kinetic energy of the positron that is created has a continuous spectrum, it can take on any value between zero and an upper limit. The kinetic energy of the generated positron is between 0-540 keV in the case of the decay of Na-22, which is often applied during calibrations. The decaying nucleus often does not decay exclusively via a {EQUATION()}\beta^+{EQUATION} process, but also in other ways; that is why the term positron yield is used. The velocity of the positron created in the decay is too high, so it cannot undergo annihilation when it meets an electron; despite the fact that they attract each other, the positron practically flies past the electron. Since the mass of the positron is very low as compared to its charge, it rapidly loses its kinetic energy and once it slows down it interacts with an electron, and they annihilate. A distance in the order of a millimetre is needed for the slowing down of the positron, thus the annihilation does not take place exactly where the decay occurred. The blurring caused by this degrades the spatial resolution. The positron range is the average distance covered by the positron before it annihilates.
It of course depends on the atomic environment, thus on the material, but it does not have a conventional definition: in some cases it is defined using the full width at half maximum, in other cases it is given by the full width at tenth maximum, while sometimes the maximal distance the positron can travel is used in the definition. Although the maximal distance is theoretically infinite, this marginal problem is neglected. That is why the comparison of relative values appearing in the same interpretation (in the same article) makes sense. The table below contains such a collection (Thanks must go to László Balkay for collecting the data).
||isotope | half life | {EQUATION()}\beta^+{EQUATION} ratio | max. {EQUATION()}E_{\beta^+}{EQUATION} | positron range | production
C-11 | 20.4 minutes | 0.99 | 0.96 MeV | 0.4 mm | cyclotron
Na-22 | 2.6 years | 0.90 | 0.54 MeV | 0.3 mm | spallation
N-13 | 9.96 minutes | 1.00 | 1.20 MeV | 0.7 mm | cyclotron
O-15 | 123 seconds | 1.00 | 1.74 MeV | 1.1 mm | cyclotron
F-18 | 110 minutes | 0.97 | 0.63 MeV | 0.3 mm | cyclotron
Cu-62 | 9.74 minutes | 0.98 | 2.93 MeV | 2.7 mm | generator
Cu-64 | 12.7 hours | 0.19 | 0.65 MeV | 0.3 mm | cyclotron
Ga-68 | 68.3 minutes | 0.88 | 1.83 MeV | 1.2 mm | generator
Br-76 | 16.1 hours | 1.00 | 1.90 MeV | 1.2 mm | cyclotron
Rb-82 | 78 seconds | 0.96 | 3.15 MeV | 2.8 mm | generator
I-124 | 4.18 days | 0.22 | 1.50 MeV | 0.9 mm | cyclotron ||
Non-collinearity
The momentum of the electron-positron pair is not necessarily exactly zero before annihilation. Therefore, due to the conservation of momentum, the two 511 keV gamma-photons do not always travel in entirely opposite directions. The full width at half maximum of this angle distribution is accepted to be 0.5°. This means that in the case of a decay occurring in the middle of a human PET scanner with a diameter of 90 cm, a blurring with a full width at half maximum of {EQUATION()}45 \mathrm{cm}\cdot tan(\frac{0.5*2\pi}{360}) \approx 4 mm {EQUATION} will appear on the circumference (where the detectors are). Therefore, it would be useless to apply crystal needles which are significantly smaller than those in human PET systems, it would only increase manufacturing costs.
::{IMG(src="tiki-download_file.php?fileId=426&display",max="200", desc = "Figure 3.")}{IMG}::
DOI - Depth Of Interaction
In the case of the flash of a crystal needle no information is available about the exact place of the scintillation within the crystal. This causes parallax error. It is not important in the case of human PET systems because of the large diameter, but it is one of the main factors that limits resolution in small animal systems. (It would be pointless to shorten the crystals because the single rate decreases at a lower rate than coincidence sensitivity.) Fortunately it can be modelled relatively well.
There are prototype devices where it could be measured, but for now they are too complex and expensive and they require many compromises.
::{IMG(src="tiki-download_file.php?fileId=425&display",max="200", desc = "Figure 4.")}{IMG}::
Scattered events
In small animal PETs it is chiefly scattering within the detector module that matters, while in human PET applications mostly scattering within the body is of importance. The energy of the scattered events is lower, so their ratio can be reduced by energy filtering, but of course detectors with good energy resolution are needed to achieve this.