Nuclear Medicine: Positron Emission Tomography Joana Rodrigues, 1263935 Nuclear and Solid State Physics
Abstract The discovery of radioactivity in 1896 by Becquerel and the hypothesis of the existence of an atomic nucleus are considered the events that were responsible for the origin of the nuclear physics. The development of the nuclear techniques during the past years allowed a huge progress in several areas of science. The achievements of nuclear physics have been particularly important in astrophysics, production of energy, in industry and even in the agriculture. However it was in medicine that the use of nuclear physics had the best impact, since it provided a better diagnosis and a more effective way of treatment of several diseases that affect our society.
The purpose of
this essay is to give a solid understanding of some of the physics and instrumentation aspects of one of the most prominent nuclear imaging techniques: Positron Emission Tomography (PET). In another words, in this essay it will be mentioned the basic physics underlying PET, such as the production of radionuclides using a cyclotron and the nuclear reactions that characterize this technique. Finally, it will be discussed the main sources of noise in PET, Coincidences, that must be correct for attenuation effects, as well as the most important applications of this technique.
1. Introduction - Nuclear Medicine Nuclear Medicine corresponds to a specialty of medicine and medical imaging that makes use of radionuclides, and it is based on the process of radioactive decay to diagnose and treat some diseases [1]. In opposition to other imaging techniques, nuclear medicine imaging techniques do not produce an anatomical map of the body. Instead they image the spatial distribution of a radiopharmaceutical that is introduced in the body. Some pathologic conditions are initiated by a change in the basic chemistry and biochemistry in the tissue. These chemical changes, with time, may lead to deficiencies in the organ function and changes in the physical properties of the tissue. Some nuclear medicine imaging techniques are sensitive to these early biochemical changes and for that reason they are able to detect early indicators of disease by imaging the uptake and biodistribution of a radioactive compound. This 1
compound is introduced in the body in small amounts, via inhalation into the lungs, direct injection into the bloodstream, subcutaneous administration or oral administration. The radioactive compounds that are introduced in the body, also called, radiopharmaceuticals, are compounds consisting of a chemical substrate linked to a radioactive element. The chemical structure of the particular radiopharmaceutical determines the biodistribution of the complex within the body, and a large number of radiopharmaceuticals are used clinically in order to target specific organs. Usually abnormal tissue distribution or an increase or decrease in the rate at which the radiopharmaceutical accumulates in a particular tissue is a strong indicator of a disease. The radiopharmaceutical within the body suffers a radioactive decay, resulting in -rays that are detected using an imaging device called gamma camera. [2] The most recently developed nuclear medicine imaging techniques is PET, which is based on positron-emitting radiopharmaceuticals. The main characteristics of this technique will be described further below. 2. PET As it was mentioned before, PET is a nuclear medicine imaging technique based on the detection of the photons generated by the positron emission of a radioactive substance inserted into the body. Before a PET scan, the patient is injected with a radioactive tracer (usually in the blood circulation), which is chosen so as to target a specific physiological or metabolic process [3]. Once the tracer is inside the body, the patient is positioned into the PET scanner. As the body processes the tracer compound, it suffers radioactive decay ( decay) through the emission of a positron. Upon its collision with an electron, a pair of
rays is produced, which are then transmitted through the body and
measured by one or more rings of detectors placed around the patient (Figure 1). Computational algorithms are then used to reconstruct images of the various organs at work within the body, based on these measurements. The fact that two gamma rays are detected, rather than one as in SPECT, allows the instrumentation used in PET to be designed to produce images with higher Signal-to-Noise Ratio (SNR) and spatial resolution than in SPECT [2]. Figure 1. The patient is placed inside the scanner. The tracer injected into his body will suffer radioactive decay.
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3. Basic nuclear physics 3.1 Radioactivity The nucleus of an atom is composed by two different types of nucleons, protons and neutrons. These particles have similar masses but different charge: protons have a positive charge while neutrons are uncharged. In the nucleus it is also possible to find electrons, which are particles with negative charge. These particles are responsible for the charge of the atom. Thus, in an uncharged atom, the number of electrons equals the number of protons [4]. The basic properties of protons, electrons, and neutrons are listed in Table 1. Table 1. Mass and Charge Properties of Nucleons, Electrons, and Positrons
Proton (p) Mass (kg) Charge (C)
Neutron (n)
Electron (
Positron
0
Isotopes are variants of atoms of a particular chemical element, which have the same number of protons, but different number of neutrons. Radioactivity corresponds to an intrinsic property of particular isotopes that have unstable nuclei and it refers to the process whereby various forms of radiation are emitted as a result of spontaneous change in the composition of the nucleus. The main reason to an isotope to be radioactive depends on the relationship between the number of protons (Z, atomic number) and the number of nucleons, the sum of number of protons and neutrons (A, atomic mass).
Figure 2. The number of neutrons plotted against the number of protons for all the stable nuclei. Note how the neutron/proton ratio increases for the heavier elements.
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The nucleus is held by two opposing forces. The strong force is an attractive force between nucleons and is balanced by the repulsive coulomb (electrical) force between the positively charged protons. If a nucleus has either an excess number of protons or neutrons, it is unstable and prone to a radioactive decay, which will lead to a change in the number of protons or neutrons in the nucleus and a more stable configuration. Nuclei that decay in this manner are known as radionuclides and the isotopes that are unstable and which undergo radioactive decay are known as radioisotopes of that element. The radioactivity
of a radionuclide is defined as the number of nuclear disintegrations per
unit time. For N nuclei of a particular element, radioactivity is given by: (1) where
is the decay constant. Radioactivity is measured in units of Curies (Ci), or more
conveniently, in millicuries, where 1 Ci equals 3.7x1010 disintegrations per second. Equation 1 can be solved in order to give (2) where
is the number of nuclei at the time t=0. The half life
of a particular element
corresponds to the time required for the radioactivity to decrease one-half of its current value. The value of the half life depends on the value of N: (3) The calculation of the time dependence of radioactivity within the body for a nuclear medicine scan takes also into consideration the biological half-life of the radionuclide. In many cases, excretion of the radionuclide from biological tissue is also an exponential process, which can be characterized by a decay constant (
and a correspondent half-life
. The effective half-life
of radioactivity within the body is given by equation 4. It is possible to observe in the equation 4 that the value of
is always less than the shorter of the two half-lives
and
. (4)
3.2 Radionuclides
As it as mention before, radionuclides are atoms characterized by an unstable nucleus. The most common radionuclides that are used in PET are 18F, 11C, 15O and 13N, which are incorporated into biologically active molecules before being injected into the patient or inhaled by him.
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Most of the radionuclides used in PET are produced by a device called cyclotron that in general is in close proximity to the PET imaging facility, due to the short half-lives of most of the radionuclides. The radionuclides can all be produced from relatively small cyclotrons using protons with an energy of ~
or deuterons with an energy ~
[2]. Some of the typical
reactions that can be observed in the production of the radionuclides for PET are shown in Figure 3. After the production of a particular radionuclide, it must be incorporated, via rapid chemical synthesis, into the corresponding radiopharmaceutical. Nowadays, the most commonly used radiopharmaceutical in clinical PET scanning is FDG (fludeoxyglucose), an analogue of glucose [2].
Figure 3. Production of radionuclides for PET [2].
3.3 Cyclotron
The Cyclotron is a type of particle accelerator. This device is characterized by a frequency, also called gyrofrequency, which corresponds to the frequency of a charged particle moving perpendicularly to the direction of a uniform magnetic field (magnetic field with a constant magnitude and direction). The cyclotron frequency is well defined, since the motion of the particle is always circular. During several decades, cyclotrons were considered one of the best sources of high-energy beams for nuclear physics experiments. Nowadays, cyclotron beams are used to bombard other atoms to produce short-lived positron-emitting isotopes suitable for PET imaging technique. The pioneer of this technology was Ernest Lawrence from University of California, Berkeley that in 1931 built the first cyclotron. Due to his work, Ernest Lawrence received the Nobel Prize for physics in 1939 [5].
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3.3.1 How the cyclotron works?
The cyclotron is localized into a vacuum chamber and it consists of two D-shaped electrodes (also called “dees”) placed in a magnetic field that it is perpendicular to the electrodes. In this device, a high-frequency alternating voltage is applied across the D electrodes, which alternately attracts and repels charged particles. The particles, injected near the center of the magnetic field, increase in speed (and therefore energy) only when passing through the gap between the electrodes. The magnetic field combined with the increasing energy of the particles will force them to travel in a spiral path. If there is not a change in the energy, the particles in a magnetic field, will follow a circular path. In the cyclotron, energy is applied to the particles as they cross the gap between the dees and so they are accelerated (at the typical sub-relativistic speeds used) and they will increase in mass as they approach the speed of light. Either of these effects (increased velocity or increased mass) will increase the radius of the circle and thus the path will be a spiral. The radius will increase until the particles hit a target at the perimeter of the vacuum chamber. Different kinds of materials may be used as a target, and the collisions will origin secondary particles that may be guided outside of the cyclotron and into instruments for posterior analysis [5].
Figure 4. Diagram of cyclotron operation.
So if there is a charged particle, characterized by a certain velocity, in the left electrode, it will stay under influence of the perpendicular magnetic field, B. This field creates a magnetic force that keeps the particle velocity constant, provoking only the alteration of its direction. As a consequence, the particle is forced to describe a circular path, with radius r. ⃗⃗⃗⃗
⃗
(5)
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Before entering in the right electrode, the particle will suffer the influence of a uniform electric field created by both electrodes. This field accelerates the particle, thus increasing the velocity. ⃗
The centripetal force, given by
(6)
provided by the transverse magnetic field B, and the
force on a particle travelling in a magnetic field (which causes it to be angularly displaced, i.e. spiral) is equal to ⃗⃗⃗⃗
⃗⃗⃗
⃗
⃗
(7)
Therefore: ⃗
where
⃗
is equal to angular velocity,
(8)
, so ⃗
(9)
Since the angular frequency corresponds to (10) as a result one obtained ⃗
(11)
The frequency of the driving voltage is simply the inverse of this frequency so that the particle crosses between the dees at the same point in the voltage cycle. ⃗
(12)
This shows that for a particle of constant mass, the frequency does not depend upon the radius of the particle's orbit. As the beam spirals out, its frequency does not decrease, and it must continue to accelerate, as it is travelling more distance in the same time. However, as particles approach the speed of light, they acquire additional mass, requiring modifications to the frequency, or the magnetic field during the acceleration. For these cases it was necessary to take into account some relativistic corrections [6]. In another words the non-relativistic expression for the cyclotron frequency (Equation 9) show that particles could continue to be accelerated by the constant angular frequency, but in fact that is
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not true. Large oscillations occurred at high frequencies and the particles did not continue to accelerate. The radius of curvature for a particle moving relativistically in a static magnetic field is given by: (13) where
is the Lorentz factor,
√
(14)
( )
The frequency is thus given by √
( )
(15)
It is clear by equation 15 that the frequency is no longer constant, depending on the velocity. Its value changes as the particles accelerated.
3.4 Types of radioactive decay
Radioactive elements can decay via a number of mechanisms, being
-particle decay,
-
particle emission, -ray emission and electron capture the most common ones in a nuclear medicine imaging technique. Within -particle emission, in which a -particle (an electron or a positron) is emitted from an atom, two different approaches can be considered: beta minus
(electron) and beta plus
(positron) decays.. And decay by positron emission is the basis for PET imaging [2].
Figure 5. Illustration of the Positron Emission decay
3.4.1
Decay
In this type of radioactive decay, a proton in the nucleus of the atom is converted into a neutron ( ) and a positron (
). The positron is the antiparticle of the electron, having the same
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mass but opposite electric charge (Table 1). The positron is ejected from the nucleus, along with a neutrino ( ), that is not detected [7]. This decay can be represented by the equation 16. (16) This reaction is not spontaneous, since the mass of the neutron is greater than the mass of the proton. So, unlike β- decay, β+ decay cannot occur in isolation, because it requires energy, which is given to the proton. Due to this reason, this decay can only happen inside the nuclei, when the value of the binding energy of the mother nucleus is less than that of the daughter nucleus [7]. Positrons are therefore emitted with a range of energies, from zero up to a maximum endpoint energy,
This endpoint energy is determined by the difference in atomic masses between
the parent atom and the daughter atom, taking into account -ray emission from excited states that may occur if the transition is not between the ground states of the two nuclei. The mean kinetic energy of the emitted positrons is, approximately,
. An example of a
radionuclide that decays by positron emission is 11C [4]. (17) 3.5 Annihilation
The positron that is ejected in the
decay is characterized by a very short lifetime in an
electron-rich material, such as human tissue. It rapidly loses its kinetic energy in inelastic interactions with atomic electrons in the tissue, and as soon as most of its energy is dissipated (typically 10-1 to 10-2 cm), it will combine with an electron to form a hydrogen-like state known as the positronium [4]. This state lasts only 1010 seconds before annihilation occurs, in which the mass of the electron and the positron is converted into electromagnetic energy. Because the positron and the electron are almost at rest when this occurs, the energy released comes largely from the mass of the particles and can be computed from Einstein’s mass-energy equivalence as: (18) where
corresponds to the mass of the electron,
speed of light ( into account that
to the mass of the positron and c to the
m/s). Inserting the values presented in Table 1 in equation 18, and taking , one obtained a value of energy
of 1.022 MeV, which is released in the form of high-energy photons. The positron and the electron are almost at rest when the annihilation occurs, resulting in a net momentum close to zero. Because both momentum and energy have to be conserved, it is not in general possible for annihilation to result in the emission of a single photon. Figure 6. Positron Emission and PositronElectron Annihilation [4].
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Instead, two photons are emitted at the same time, but in opposite directions (180ยบ apart), along a straight line of coincidence, also known as the line of response, or LOR. Each one of the photos carries an energy equal 5.11keV (1.022 MeV/2), which ensures that the momentum and energy are conserved. 3.6 Positron range and noncolinearity
There are two main effects in PET imaging systems responsible for errors in determining the line along which a positron-emitting radionuclide is to be found. These effects affect the spatial resolution attainable with PET and manifest themselves as a blurring of the reconstructed image. The mentioned effects correspond to Positron Range and Noncolinearity (Figure 7) [4].
Figure 7. Phenomenon of Positron Range or Non-Colinearity [4].
The first one corresponds to the perpendicular distance from the site of positron emission to the line defined by the annihilation photons. In rest, the mass of the positron is equal to the mass of the electron; and thus positrons follow a tortuous path in tissue, undergoing multiple directionchanging interactions with electrons prior to annihilation. In the end, the total path length the positron travels is considerably longer than the positron range. The positron only suffers annihilation when its kinetic energy equals to zero [4]. As it was mentioned before, radionuclides differ in the energy of emitted positrons, and thus some radionuclides emit, on average, higher energy positrons than others. The choice of the radionuclides that are used in PET has to take into
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consideration the amount of energy they emit during the decay, since higher energy corresponds to a higher range (Table 2). Table 2 shows the values of water range for some radionuclides, since water is the main compound of human body. The positron range should not exceed 1 or 2 mm, in order to obtain an image that shows the exact location of a tumor, but also doesn’t provoke any damage to the healthy tissues that surround the tumor.
Table 2. Values of Endpoint Energy and Water Range of the radionuclides most used in PET
Radionuclides 11
Endpoint Energy (MeV) Range in water (mm)
C
0.96
1.1
13
N
1.20
1.4
15
O
1.73
1.5
0.63
1.0
18
F
The second effect comes from the fact that the positron and electron are not completely at rest when they annihilate. The small net momentum of these particles means that the annihilation photons will not occur at exactly 180° and will, in fact be emitted with a distribution of angles around 180°. This phenomenon is known as noncolinearity. This effect is independent of radionuclide because the positrons must lose most of their energy before they can annihilate, thus the initial energy is irrelevant. It is assumed that the distribution of emitted angles is roughly Gaussian in shape, with a full width at half maximum (FWHM) of 0.5°. After detecting the annihilation photons, PET assumes that the emission was exactly back to back, resulting in a small error in locating the line of annihilation. Taking into account the distribution mentioned before and using the fact that the angles are small, the blurring effect associated to the noncolinearity,
, is given by: (19)
where
is the diameter of the PET scanner. The error increases linearly as the diameter of the
PET scanner increases. Knowing that in general PET diameter is 80 cm, it is possible to determine the deviation of the angle due to photon noncolinearity. Using the equation 19, one obtained: (20) The angle can be then calculated using the expression above: (21)
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3.7 Interactions with matter
After the electron-positron annihilation, the
emitted photons
interact with the tissue surrounding them, with the detector material of the PET scanner, among others. The two major mechanisms by which 511-keV photons interact with matter is the Photoelectric Effect and Compton Scattering. Relatively to the Photoelectric Effect, the 511-keV photon will
Figure 8. Photoelectric Effect [4].
interact with an atom in the surrounding medium, transferring all of its energy to it. The electron is then ejected from the atom, but it is quickly absorbed by material nearby. In the Compton Effect, the incident
photon transfers part of its energy to an electron,
causing it to change direction. Here, the photon scatters off a free or loosely bound electron in the medium, transferring some of its energy to the electron and changing direction in the process. Imposing conservation of momentum and energy leads to a simple relationship between the energy of the original photon (E), the energy of the scattered photon (Esc) and the angle through which it is scattered: (22)
where
is the mass of the electron and is the speed of light. The term
is equal to
. In PET, the incoming photon of interest has an energy level of
, thus the
equation, therefore, can be reduced: (23) The recoil energy that is transferred to the electron, equal to
, which is dissipated in the medium, is
: (24)
If one substitutes again the
for the value of 511keV, the final expression is given by: (25)
4.
Coincidences
During a PET scan a large number of annihilation coincidences are detected. Some of these are true coincidences, but there are many mechanisms by which “false� coincidences can be 12
recorded. If we were under ideal circumstances, only true coincidences would be recorded. True coincidences correspond to events where two detected annihilation photons originate from the same radioactive decay and that have not suffer a change in their direction or lost any energy before being detected. However, in general some of the measured coincidences are contaminated with undesirable events that can be related to the limitations of the detectors used in PET, as well as interactions of the photons in the body before they reach the detector. These undesirable events have a degrading effect on the measurement and they need to be corrected to produce an image that represents as closely as possible the true radioactivity concentration [2]. These events include random, scattered and multiple coincidences and detail about them will be given further (Figure 9).
Figure 9. Illustration of the four main coincidence event types.
4.1 Random Coincidences When
electron-positron
annihilation
occurs,
both
511-keV
photons
are
emitted
simultaneously. Ideally the detectors should detect these photons also simultaneously. However, because of the finite time resolution of the detectors it is possible that two unrelated single annihilation photons are detected and registered as a true coincidence. These unrelated events are referred to as random or accidental events. Since the random events are produced by photons emitted from unrelated isotope decays, they do not carry any spatial information about the activity distribution. As a result, they produce an undesired background in the final images. If the individual photon detection rates (counts per second) in a pair of detectors are given by , the rate of random coincidences, per second,
and
is given by:
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(26) where
is the width of the logic pulses produced when a photon is absorbed in the detector. The
term
is also known as the coincidence timing window.
4.2 Scattered coincidences Scattered coincidences are events that in essence correspond to true coincidences, but one or both of the two annihilation photons has undergone a Compton Scatter interaction and changed direction before they reach the detector pair. Using the coincidence detection technique, it is assumed that all detected coincidence events originate from an annihilation which, in turn, originates from a position anywhere on a line connecting the detector pairs. However, because of the the change in direction of the photon(s) in a scattered event, the event is assigned to an incorrect LOR. The scattered events result in a low spatial frequency background that reduces contrast and this is probably the most difficult correction to perform in PET. 4.3 Multiple Coincidences In general only two detectors are required to be activated within the coincidence time window to register a valid coincidence. However, at high count-rates, three or more detectors can be involved and in this case sometimes it becomes uncertain where the event should be positioned. Because of this ambiguity, usually multiple coincidences are discarded. The problem is that sometimes they contain information about the quantity and spatial location of positron emissions, since these events are often composed of a true coincidence together with a single photon from an unrelated decay. In this situation, up to three possible LORs can intersect the field of view and only one of them is correct. In some circumstances, it may be better to randomly select one of the possible LORs rather than completely remove the event [2].
5. Applications
PET is both a medical and research tool. This technique is widely used in clinical oncology, mainly for the diagnosis, staging, and evaluation of treatment efficacy for tumors in the breast, lung, head and neck, and also colorectal cancer. Besides tumor imaging, this technique also provides quantitative information on metabolic and physiological changes in cardiac studies and brain imaging [3]. Figure 10. PET scan of the human brain [3].
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6. Conclusion This essay allowed a better understanding about some of physics and instrumentation behind one of the most promising nuclear medicine techniques. Positron Emission Tomography corresponds to the most recently developed nuclear medicine imaging, which provides useful information about the viability of different body organs. In opposition to other imaging techniques, PET images show the spatial distribution of a radiopharmaceutical that is introduced in the body. It has a major role in diagnosing tumors. The production of the radiopharmaceuticals is possible due to the use of a cyclotron and the decay by positron emission (
decay) is the basis for PET imaging. Coincidences are considered
the main sources of noise in PET and for that reason they have to be correct in order to be possible to obtain an image that represents as closely as possible the true radioactivity concentration In conclusion nuclear medicine has a huge impact in the quality of life of our society and due to several advances made in this area it is possible nowadays to diagnose and treat patients in a more effective and less invasive way. 7. References
[1]. Retrieved on 25 November 2011 from, http://en.wikipedia.org/wiki/Nuclear_medicine [2]. Webb, A. (2003). Introduction to Biomedical Imaging. Hardbound, IEEE Computer Society Press. ISBN: 0471237663 [3].Retrieved
on
25
November
2011
from,
http://en.wikipedia.org/wiki/Positron_emission_tomography [4]. Michael E. Phelps (2006). PET: physics, instrumentation, and scanners. Springer. pp. 8– 10.ISBN 0387349464. [5]. Retrieved on 25 November 2011 from, http://en.wikipedia.org/wiki/Cyclotron [6].
Retrieved
on
25
November
2011
from,
http://hyperphysics.phy-
astr.gsu.edu/hbase/magnetic/cyclot.html [7]. Retrieved on 25 November 2011 from, http://en.wikipedia.org/wiki/Beta_decay
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