Figure 1. Positron emission and annihilation.
2. The physical principles of
PET.
2.1 Introduction
2.2 Positron emission and annihilation
2.3 Coincidence detection and electronic
collimation
2.4 Photon interactions in human tissue and correction
for gamma-ray attenuation
2.5 Types of coincidence events
After injection of a tracer compound labelled with
a positron emitting radionuclide the subject of a PET study is placed
within the field of view (FOV) of a number of detectors capable of
registering incident gamma rays. The radionuclide in the radiotracer
decays and the resulting positrons subsequently annihilate on contact
with electrons after travelling a short distance (~ 1 mm) within the
body. Each annihilation produces two 511 keV photons travelling in
opposite directions and these photons may be detected by the
detectors surrounding the subject. The detector electronics are
linked so that two detection events unambiguously occurring within a
certain time window may be called coincident and thus be determined
to have come from the same annihilation. These "coincidence events" can be stored in arrays corresponding to projections through the
patient and reconstructed using standard tomographic techniques. The
resulting images show the tracer distribution throughout the body of
the subject. This section describes the physical principles
underlying PET and will discuss some of the intrinsic advantages that
PET exhibits over Single Photon Emission Computed Tomography (SPECT)
techniques.
2.2 Positron emission and
annihilation
Proton-rich isotopes may decay via positron
emission, in which a proton in the nucleus decays to a neutron, a
positron and a neutrino. The daughter isotope has an atomic number
one less than the parent. Examples of isotopes which undergo decay
via positron emission are shown in table
2.
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Table 2. Properties of commonly used positron emitting radio-isotopes
As positrons travel through human tissue they give up their kinetic energy principally by Coulomb interactions with electrons. As the rest mass of the positron is the same as that of the electron, the positrons may undergo large deviations in direction with each Coulomb interaction, and they follow a tortuous path through the tissue as they give up their kinetic energy (figure 1).
When the positrons reach thermal energies, they start to interact with electrons either by annihilation, which produces two 511 keV photons which are anti-parallel in the positronís frame, or by the formation of a hydrogen-like orbiting couple called positronium. In its ground-state, positronium has two forms - ortho-positronium, where the spins of the electron and positron are parallel, and para-positronium, where the spins are anti-parallel. Para-positronium again decays by self-annihilation, generating two anti-parallel 511 keV photons. Ortho-positronium self-annihilates by the emission of three photons (Evans, 1955). Both forms are susceptible to the ìpick-offî process, where the positron annihilates with another electron. Free annihilation and the pick-off process are responsible for over 80% of the decay events. Variations in the momentum of the interacting particles involved in free annihilation and pick-off result in an angular uncertainty in the direction of the 511 keV photons of around 4 mrad in the observerís frame (Rickey et al 1992). In a PET camera of diameter 1m and active transaxial FOV of 0.6m this results in a positional inaccuracy of 2-3 mm.
The finite positron range and the non-collinearity of the annihilation photons give rise to an inherent positional inaccuracy not present in conventional single photon emission techniques. However, other characteristics of PET which are discussed below more than offset this disadvantage.
Figure 1. Positron emission and annihilation.
2.3. Coincidence detection and electronic collimation.
In a PET camera, each detector generates a timed pulse when it registers an incident photon. These pulses are then combined in coincidence circuitry, and if the pulses fall within a short time-window, they are deemed to be coincident. A conceptualised diagram of this process is shown in figure 2.
Figure 2. Coincidence detection in a PET camera.
A coincidence event is assigned to a line of response (LOR) joining the two relevant detectors. In this way, positional information is gained from the detected radiation without the need for a physical collimator. This is known as electronic collimation. Electronic collimation has two major advantages over physical collimation. These are improved sensitivity and improved uniformity of the point source response function (psrf).
When a physical collimator is used, directional
information is gained by preventing photons which are not normal or
nearly normal to the collimator face from falling on the detector. In
electronic collimation, these photons may be detected and used as
signal. This results in a significant gain in sensitivity (typically
a factor of 10 for 2D mode PET compared with SPECT). This increase in
sensitivity means that typical realisable image resolution in PET is
around 5-10 mm, whereas in SPECT it is around 15-20 mm.
(a) (b)
Figure 3. Variation of point source response function (psrf)
with position P in SPECT and in PET.
In SPECT (a) the FWHM of the psrf increases with increasing distance
from the collimator. In PET (b), the FWHM of the psrf varies from one
detector width at the edge of the FOV to approximately 1/2 detector
width at the centre of the FOV.
In SPECT, the full-width at half-maximum (FWHM) of
the psrf increases with increasing distance of the source from the
collimator (figure 3
(a)). This results in variable resolution
in the reconstructed images. In PET, a coincidence event may be
detected if the direction of the annihilation photons is constrained
to lie along a line-of-sight joining both detector faces. If the
annihilation photons are strictly anti-parallel, this results in a
psrf which varies in a similar way to that shown in figure
3(b). This constraint is relaxed somewhat
because of the small uncertainty in the direction of the annihilation
photons, and in practice the psrf changes only very slightly in the
central third of the FOV (Phelps
et al 1986). As a result, the
resolution of reconstructed PET images is more uniform than is the
case for SPECT images.
2.4. Photon interactions in human tissue and correction for gamma-ray attenuation.
The most important interactions which photons resulting from the positron annihilation undergo in human tissue are Compton scatter and photoelectric absorption.
In Compton scatter a photon interacts with an electron in the absorber material. In the process the kinetic energy of the electron is increased, and the direction of the photon is changed. The energy of the photon after interaction is given by (Evans, 1955):
where E is the energy of the incident photon, Eí is the energy of the scattered photon, m0c2 is the rest mass of the electron and q is the scattering angle. Equation 1 implies that quite large deflections can occur with quite small energy loss - for example, for 511 keV photons, a Compton scattering event in which 10% of the photon energy is lost will result in a deflection of just over 25 degrees.
In photoelectric absorption a photon is absorbed by an atom and in the process an electron is ejected from one of its bound shells. The probability of photoelectric absorption increases rapidly with increasing atomic number of the absorber atom, and decreases rapidly with increasing photon energy (Evans, 1955). In water, the probability of photoelectric absorption decreases with roughly the 3rd power of the photon energy and is negligible at 511 keV (Johns and Cunningham 1983).
The total probability that a photon of a particular energy will undergo some kind of interaction with matter when travelling unit distance through a particular substance is called the linear attenuation coefficient (m ) of that substance. If I0 is the initial intensity of a parallel beam of monoenergetic photons, then the intensity I(x) at a distance x through some attenuating object will be given by (Evans 1955):
(2)
provided scattered photons are removed from the beam. This relation has important consequences for PET. Consider a small volume v in an attenuating object, located at a distance x' along an LOR joining two detectors in the FOV of a PET camera (figure 4). Let the volume v contain some positron emitting substance, so that there is a flux of 511 keV photons along the line of response joining detector 1 and detector 2. If the linear attenuation coefficient at a point x along the LOR is m (x), and a is the distance between detectors 1 and 2, we can state the following:
Probability of a photon reaching detector 1 from
v is :
Probability of a photon reaching detector 2 from v is :
The probabilities are independent of each other, and both photons must reach the detectors for a coincidence to be recorded. The probability of a coincidence Pc, is the product of P1 and P2 :
(3)
So the quantity (1 -Pc ), which is the attenuation factor of the photons travelling along the LOR from v, is the same for any position along the line of response. By measuring the coincidence signal as a positron-emitting source is moved around the object within the FOV, it is possible to obtain attenuation factors for each LOR. In principle, this enables quantitative measurements of isotope distribution to be made. In SPECT techniques, where the attenuation factors increase with increasing distance from the detectors, there is no simple way to correct for photon attenuation.
Figure 4. Coincidence detection in an attenuating object
For 511 keV photons in human tissue the half-value
layer (the distance a beam of photons must travel before 50% have
interacted) is around 7 cm. Attenuation factors in human studies can
rise to around 50 for LORs crossing large dense areas, for example
those crossing the shoulders perpendicularly to the sagittal
plane.
2.5 Types of coincidence events.
Coincidence events in PET fall into 4 categories: true, scattered, random and multiple. The first three of these are illustrated in figure 5.
True coincidences occur when both photons from an annihilation event are detected by detectors in coincidence, neither photon undergoes any form of interaction prior to detection, and no other event is detected within the coincidence time-window.
A scattered coincidence is one in which at least one of the detected photons has undergone at least one Compton scattering event prior to detection. Since the direction of the photon is changed during the Compton scattering process, it is highly likely that the resulting coincidence event will be assigned to the wrong LOR. Scattered coincidences add a background to the true coincidence distribution which changes slowly with position, decreasing contrast and causing the isotope concentrations to be overestimated. They also add statistical noise to the signal. The number of scattered events detected depends on the volume and attenuation characteristics of the object being imaged, and on the geometry of the camera.
Figure 5. Types of coincidences in PET.
Random coincidences occur when two photons not arising from the same annihilation event are incident on the detectors within the coincidence time window of the system. The number of random coincidences in a given LOR is closely linked to the rate of single events measured by the detectors joined by that LOR and the rate of random coincidences increase roughly with the square of the activity in the FOV. As with scattered events, the number of random coincidences detected also depends on the volume and attenuation characteristics of the object being imaged, and on the geometry of the camera. The distribution of random coincidences is fairly uniform across the FOV, and will cause isotope concentrations to be overestimated if not corrected for. Random coincidences also add statistical noise to the data.
A simple expression relating the number of random coincidences assigned to an LOR to the number of single events incident upon the relevant detectors can be derived as follows:
Define t , the coincidence resolving time of the system, such that any events detected with a time difference of less than t are considered to be coincident (see section 5.4). Let r1 be the single event rate (singles rate) on detector channel 1. Then in one second, the total time-window during which coincidences will be recorded is 2t r1. If the singles rate on detector channel 2 is r2, we can say that the number of random coincidences R12 assigned to the LOR joining detectors 1 and 2 is given by
R12 = 2t r1 r2 (4)
This relation is true provided that the singles rate is much larger than the rate of coincidence events, and that the singles rates are small compared to the reciprocal of the coincidence resolving time t , so that dead-time effects can be ignored.
Multiple coincidences occur when more than two photons are detected in different detectors within the coincidence resolving time. In this situation, it is not possible to determine the LOR to which the event should be assigned, and the event is rejected. Multiple coincidences can also cause event mis-positioning (see section 5.6).
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Revision date:
12 Jan 1999
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