PHYSICS IN NUCLEAR MEDICINE
Electrons occupy discrete energy levels and can transition between them by absorbing or emitting energy in
the form of photons. Electrons in the defined orbits remain stable without radiating energy, overcoming
classical physics' limitations.
BASIC ATOMIC & NUCLEAR PHYSICS
A
A nuclide is represented as Z X , where:
𝐴: Mass number (total number of protons and neutrons)
𝑍: Atomic number (number of protons)
𝑋: Element symbol
Isotopes are variants of the same element with the same 𝑍 but different 𝐴, due
to a varying number of neutrons. Excited states are temporary configurations of
A
a nucleus with higher energy. Metastable states (denoted as m ZX ) are longer-
lived excited states with a measurable lifetime (greater than 10 -12 seconds).
Isomers are nuclides where one is a metastable version of the other.
The line of stability is a curve on a chart plotting the number of neutrons (𝑁)
versus the number of protons (𝑍) in nuclei. It represents the region where nuclei are stable and not prone to
radioactive decay. Stable nuclei for lighter elements (e.g., carbon, oxygen) have a neutron-to-proton ratio close
to 1:1. For heavier nuclei, stability requires more neutrons than protons to counteract the increasing
electrostatic repulsion among protons:
Neutron-rich nuclei (above the line): Tend to undergo β - decay, converting neutrons into protons to
move toward stability.
Proton-rich nuclei (below the line): Tend to undergo β + decay (positron emission) or electron capture
to convert protons into neutrons.
During decay by β- emission, a neutron in the nucleus converts into a proton, emitting an electron (e −, the beta
particle) and an antineutrino (v). This results in 𝑛 → 𝑝+ + 𝑒− + 𝜐 + 𝑒𝑛𝑒𝑟𝑔y. The mass number (𝐴) remains
unchanged, while the atomic number (𝑍) increases by 1, forming a new element. The emitted beta particle
carries kinetic energy. The antineutrino accounts for the rest of the energy, ensuring conservation of energy
and momentum.
During decay by β+ emission, a proton in the nucleus converts into a neutron,
emitting a positron (e+, the beta particle) and a neutrino (ν). This results in 𝑝 +
→ 𝑛 + 𝑒 + + 𝜐 + 𝑒𝑛𝑒𝑟𝑔y. The mass number (A) remains unchanged, while the
atomic number (Z) decreases by 1, forming a new element. The emitted
positron interacts with an electron, resulting in annihilation and producing two
gamma photons traveling in opposite directions.
DECAY OF RADIOACTIVITY
Radioactive decay is a stochastic process where the number of undecayed nuclei decreases exponentially over
time. The general formula is: N(t) = N0e−λt, where:
N(t): Number of undecayed nuclei at time 𝑡
𝑁0: Initial number of nuclei at 𝑡 = 0
𝜆: Decay constant, a measure of the probability of decay per unit time
𝑡: Time elapsed
The poisson distribution models the probability of a given number of decay events occurring in a fixed time
period, assuming these events are independent and occur at a constant average rate.
A cyclotron is a type of particle accelerator that uses electromagnetic fields to accelerate charged particles,
such as protons, to high energies along a spiral path. These high-energy particles are then directed to collide
with a target material, producing radioactive isotopes. Cyclotrons are typically located near nuclear medicine
Page | 1
,facilities because the isotopes they produce often have very short half-lives. Rapid transport and processing
are critical to ensure the isotopes are active when administered to patients.
INTERACTION OF RADIATION WITH MATTER
The transmission of a photon beam through a material follows the
ΔI
exponential attenuation law: =−μ Δ x , where:
I
𝐼: Intensity of the photon beam before interaction
Δ𝐼: Change in intensity over a small distance Δ𝑥
𝜇: Linear attenuation coefficient, which depends on the
material's composition and photon energy
Δ𝑥: Thickness of the material through which the photons
travel
The linear attenuation coefficient (𝜇) describes the fraction of photons removed from the beam per unit
thickness due to interactions like photoelectric absorption, Compton scattering, or pair production.
In the photoelectric effect process, a photon interacts with an atom, transferring all its energy to an inner-shell
electron. The inner-shell electron is ejected from the atom, becoming a photoelectron. This leaves a vacancy in
the electron shell, which is filled by an outer-shell electron, releasing characteristic X-rays. In the Compton
scattering process, a photon interacts with a loosely bound outer-shell electron, transferring some of its
energy to the electron. The electron is ejected (recoil electron), and the photon is deflected with reduced
energy.
Photoelectric effect Compton scattering
At low energies, the photoelectric effect dominates. As the energy increases, Compton scattering takes over.
At very high energies, pair production becomes significant.
DETECTION OF RADIATION
A scintillation detector is a device that detects ionizing radiation
(like gamma rays or X-rays) by using a scintillator material that
emits light (scintillates) when it absorbs energy from radiation.
This emitted light is then converted into an electrical signal, which
can be measured. A typical scintillation detector consists of
several key components:
Scintillator material: This is a material that absorbs the
energy from the incoming radiation and then emits
visible or ultraviolet light.
Photomultiplier tube (PMT): The light emitted by the scintillator is too weak to be directly measured,
so it is amplified by a photomultiplier tube (PMT).
o The PMT converts the light (photons) into an electrical signal (electrons). The intensity of the
light is proportional to the energy of the radiation that caused the scintillation.
Electronics: After the PMT converts light into an electrical signal, it is processed and recorded by a
computer or counting system, allowing the energy and intensity of the radiation to be measured.
Page | 2
,When ionizing radiation (such as a gamma ray) interacts with the scintillator material, it excites the atoms or
molecules in the material. The excited atoms/molecules return to their ground state by emitting light
(photons), which is the scintillation. The emitted light is detected by the PMT, which amplifies the signal and
converts it into an electrical pulse. The electrical pulse is processed by electronics to provide information about
the energy and number of events (radiation detected).
A multichannel analyzer (MCA) is used to analyze the energy spectrum of detected radiation by sorting and
categorizing the signals (counts) from radiation detectors, such as scintillation detectors, into energy channels.
The MCA allows to measure the energy distribution of detected radiation. The incoming electrical pulses
(representing the energy of individual radiation events) are sorted into discrete energy channels (bins). These
channels are then used to construct an energy spectrum of the radiation source.
Light sharing is used in scintillation detectors to determine the position of radiation interactions within the
detector. When ionizing radiation interacts with the scintillation crystal, it produces a burst of light
(scintillation photons). These photons spread out and are detected by multiple photomultiplier tubes (PMTs)
or photodiodes arranged in an array beneath the scintillation crystal. The light is shared among the PMTs or
photodiodes, with the intensity of the detected light decreasing as the distance from the interaction point
increases.
IMAGE RECONSTRUCTION
Backprojection is a method used to reconstruct an image from its projections. A projection represents the sum
or integral of radiation intensities along a specific direction through an object (e.g., a patient) at a given angle.
During imaging (e.g., CT or SPECT), multiple projections are acquired from different angles around the object.
Each projection records the cumulative intensity of radiation along the path through the object at a specific
angle. Each projection is distributed (backprojected) across the image plane along the direction it was
measured. This assumes that the recorded intensity is evenly distributed along the corresponding ray path.
The contributions from all angles are summed to form the reconstructed image.
Simple backprojection leads to blurred images because the data from projections are smeared evenly along
the ray paths. This uniform distribution assumption creates artifacts and reduces image sharpness. The goal of
filtering is to enhance image quality by suppressing low-frequency components (blurring effects) and
amplifying high-frequency components (sharp details and edges). For example, the Hamming filter suppresses
very high-frequency noise components, while maintaining image clarity.
Filtered backprojection (FBP) combines:
1. Filtering:
The raw projection data (sinogram) is mathematically processed to enhance sharpness and
reduce artifacts.
A filter (like a ramp filter) is applied to the projection data in the frequency domain to
counteract the blurring effect of backprojection.
2. Backprojection:
The filtered projections are backprojected across the image plane to reconstruct a high-
quality image.
Backprojection Filtered backprojection
Page | 3
, Iterative reconstruction is a computational approach to reconstruct an image from projection data. Unlike
direct methods like filtered backprojection (FBP), iterative reconstruction refines the image step by step,
comparing the projections of the reconstructed image to the actual measured data.
Start with an initial estimate of the image (e.g., a uniform image or one based on FBP).
Simulate projection data from the current image
estimate by modeling the imaging system's
physics and geometry.
Compare the simulated projections to the actual
measured projections from the scanner.
Calculate the difference (residual error).
Adjust the image estimate to reduce the error,
incorporating corrections derived from the
difference.
Repeat the process (iterate) until the error
between simulated and measured data is
minimized or meets a predefined threshold.
Iterative reconstruction causes improved image quality, because it reduces noise and artifacts (e.g. streak
artifacts in CT), and provides better spatial resolution and contrast.
In PET, the radioactive tracer emits positrons (𝛽+). Each positron travels a short distance in the tissue before
annihilating with an electron. This annihilation produces two gamma photons of 511 keV energy, which are
emitted in opposite directions (180° apart) to conserve momentum.
Coincidence detection refers to the process of detecting these two
gamma photons simultaneously (within a few nanoseconds) to
determine the location of the annihilation event along a line.
In PET, the two 511 keV photons produced by positron annihilation
travel in opposite directions and are detected by detectors around the
patient. The line of response (LOR) represents the straight path along
which the annihilation occurred but does not provide the exact point of
interaction. While standard PET assumes the annihilation occurred
somewhere along the LOR, time of flight (TOF) PET measures the
difference in arrival times of the two photons to calculate a more
precise location of the annihilation event.
Spatial resolution defines how small a detail can be detected in an image. Higher spatial resolution means the
system can distinguish finer details, while lower resolution results in blurred or merged objects.
There are different types of events recorded during PET imaging, which can affect the quality and accuracy of
the reconstructed image:
1. True coincidence events: True events occur when two photons produced by the same annihilation
event are detected simultaneously (within the coincidence time window) by opposite detectors.
These events define a correct LOR and are essential for accurate image reconstruction.
2. Random coincidence events: Random events occur when two unrelated photons, originating from
different annihilation events, are detected simultaneously within the coincidence time window.
These events create false LORs, introducing noise into the image and degrading image
quality.
3. Scatter coincidence events: Scatter events occur when one or both photons undergo Compton
scattering before reaching the detector, altering their original trajectory.
Scatter events result in incorrect LORs, causing artifacts and reducing image contrast.
True coincidence Random coincidence Scatter coincidence
Page | 4
Electrons occupy discrete energy levels and can transition between them by absorbing or emitting energy in
the form of photons. Electrons in the defined orbits remain stable without radiating energy, overcoming
classical physics' limitations.
BASIC ATOMIC & NUCLEAR PHYSICS
A
A nuclide is represented as Z X , where:
𝐴: Mass number (total number of protons and neutrons)
𝑍: Atomic number (number of protons)
𝑋: Element symbol
Isotopes are variants of the same element with the same 𝑍 but different 𝐴, due
to a varying number of neutrons. Excited states are temporary configurations of
A
a nucleus with higher energy. Metastable states (denoted as m ZX ) are longer-
lived excited states with a measurable lifetime (greater than 10 -12 seconds).
Isomers are nuclides where one is a metastable version of the other.
The line of stability is a curve on a chart plotting the number of neutrons (𝑁)
versus the number of protons (𝑍) in nuclei. It represents the region where nuclei are stable and not prone to
radioactive decay. Stable nuclei for lighter elements (e.g., carbon, oxygen) have a neutron-to-proton ratio close
to 1:1. For heavier nuclei, stability requires more neutrons than protons to counteract the increasing
electrostatic repulsion among protons:
Neutron-rich nuclei (above the line): Tend to undergo β - decay, converting neutrons into protons to
move toward stability.
Proton-rich nuclei (below the line): Tend to undergo β + decay (positron emission) or electron capture
to convert protons into neutrons.
During decay by β- emission, a neutron in the nucleus converts into a proton, emitting an electron (e −, the beta
particle) and an antineutrino (v). This results in 𝑛 → 𝑝+ + 𝑒− + 𝜐 + 𝑒𝑛𝑒𝑟𝑔y. The mass number (𝐴) remains
unchanged, while the atomic number (𝑍) increases by 1, forming a new element. The emitted beta particle
carries kinetic energy. The antineutrino accounts for the rest of the energy, ensuring conservation of energy
and momentum.
During decay by β+ emission, a proton in the nucleus converts into a neutron,
emitting a positron (e+, the beta particle) and a neutrino (ν). This results in 𝑝 +
→ 𝑛 + 𝑒 + + 𝜐 + 𝑒𝑛𝑒𝑟𝑔y. The mass number (A) remains unchanged, while the
atomic number (Z) decreases by 1, forming a new element. The emitted
positron interacts with an electron, resulting in annihilation and producing two
gamma photons traveling in opposite directions.
DECAY OF RADIOACTIVITY
Radioactive decay is a stochastic process where the number of undecayed nuclei decreases exponentially over
time. The general formula is: N(t) = N0e−λt, where:
N(t): Number of undecayed nuclei at time 𝑡
𝑁0: Initial number of nuclei at 𝑡 = 0
𝜆: Decay constant, a measure of the probability of decay per unit time
𝑡: Time elapsed
The poisson distribution models the probability of a given number of decay events occurring in a fixed time
period, assuming these events are independent and occur at a constant average rate.
A cyclotron is a type of particle accelerator that uses electromagnetic fields to accelerate charged particles,
such as protons, to high energies along a spiral path. These high-energy particles are then directed to collide
with a target material, producing radioactive isotopes. Cyclotrons are typically located near nuclear medicine
Page | 1
,facilities because the isotopes they produce often have very short half-lives. Rapid transport and processing
are critical to ensure the isotopes are active when administered to patients.
INTERACTION OF RADIATION WITH MATTER
The transmission of a photon beam through a material follows the
ΔI
exponential attenuation law: =−μ Δ x , where:
I
𝐼: Intensity of the photon beam before interaction
Δ𝐼: Change in intensity over a small distance Δ𝑥
𝜇: Linear attenuation coefficient, which depends on the
material's composition and photon energy
Δ𝑥: Thickness of the material through which the photons
travel
The linear attenuation coefficient (𝜇) describes the fraction of photons removed from the beam per unit
thickness due to interactions like photoelectric absorption, Compton scattering, or pair production.
In the photoelectric effect process, a photon interacts with an atom, transferring all its energy to an inner-shell
electron. The inner-shell electron is ejected from the atom, becoming a photoelectron. This leaves a vacancy in
the electron shell, which is filled by an outer-shell electron, releasing characteristic X-rays. In the Compton
scattering process, a photon interacts with a loosely bound outer-shell electron, transferring some of its
energy to the electron. The electron is ejected (recoil electron), and the photon is deflected with reduced
energy.
Photoelectric effect Compton scattering
At low energies, the photoelectric effect dominates. As the energy increases, Compton scattering takes over.
At very high energies, pair production becomes significant.
DETECTION OF RADIATION
A scintillation detector is a device that detects ionizing radiation
(like gamma rays or X-rays) by using a scintillator material that
emits light (scintillates) when it absorbs energy from radiation.
This emitted light is then converted into an electrical signal, which
can be measured. A typical scintillation detector consists of
several key components:
Scintillator material: This is a material that absorbs the
energy from the incoming radiation and then emits
visible or ultraviolet light.
Photomultiplier tube (PMT): The light emitted by the scintillator is too weak to be directly measured,
so it is amplified by a photomultiplier tube (PMT).
o The PMT converts the light (photons) into an electrical signal (electrons). The intensity of the
light is proportional to the energy of the radiation that caused the scintillation.
Electronics: After the PMT converts light into an electrical signal, it is processed and recorded by a
computer or counting system, allowing the energy and intensity of the radiation to be measured.
Page | 2
,When ionizing radiation (such as a gamma ray) interacts with the scintillator material, it excites the atoms or
molecules in the material. The excited atoms/molecules return to their ground state by emitting light
(photons), which is the scintillation. The emitted light is detected by the PMT, which amplifies the signal and
converts it into an electrical pulse. The electrical pulse is processed by electronics to provide information about
the energy and number of events (radiation detected).
A multichannel analyzer (MCA) is used to analyze the energy spectrum of detected radiation by sorting and
categorizing the signals (counts) from radiation detectors, such as scintillation detectors, into energy channels.
The MCA allows to measure the energy distribution of detected radiation. The incoming electrical pulses
(representing the energy of individual radiation events) are sorted into discrete energy channels (bins). These
channels are then used to construct an energy spectrum of the radiation source.
Light sharing is used in scintillation detectors to determine the position of radiation interactions within the
detector. When ionizing radiation interacts with the scintillation crystal, it produces a burst of light
(scintillation photons). These photons spread out and are detected by multiple photomultiplier tubes (PMTs)
or photodiodes arranged in an array beneath the scintillation crystal. The light is shared among the PMTs or
photodiodes, with the intensity of the detected light decreasing as the distance from the interaction point
increases.
IMAGE RECONSTRUCTION
Backprojection is a method used to reconstruct an image from its projections. A projection represents the sum
or integral of radiation intensities along a specific direction through an object (e.g., a patient) at a given angle.
During imaging (e.g., CT or SPECT), multiple projections are acquired from different angles around the object.
Each projection records the cumulative intensity of radiation along the path through the object at a specific
angle. Each projection is distributed (backprojected) across the image plane along the direction it was
measured. This assumes that the recorded intensity is evenly distributed along the corresponding ray path.
The contributions from all angles are summed to form the reconstructed image.
Simple backprojection leads to blurred images because the data from projections are smeared evenly along
the ray paths. This uniform distribution assumption creates artifacts and reduces image sharpness. The goal of
filtering is to enhance image quality by suppressing low-frequency components (blurring effects) and
amplifying high-frequency components (sharp details and edges). For example, the Hamming filter suppresses
very high-frequency noise components, while maintaining image clarity.
Filtered backprojection (FBP) combines:
1. Filtering:
The raw projection data (sinogram) is mathematically processed to enhance sharpness and
reduce artifacts.
A filter (like a ramp filter) is applied to the projection data in the frequency domain to
counteract the blurring effect of backprojection.
2. Backprojection:
The filtered projections are backprojected across the image plane to reconstruct a high-
quality image.
Backprojection Filtered backprojection
Page | 3
, Iterative reconstruction is a computational approach to reconstruct an image from projection data. Unlike
direct methods like filtered backprojection (FBP), iterative reconstruction refines the image step by step,
comparing the projections of the reconstructed image to the actual measured data.
Start with an initial estimate of the image (e.g., a uniform image or one based on FBP).
Simulate projection data from the current image
estimate by modeling the imaging system's
physics and geometry.
Compare the simulated projections to the actual
measured projections from the scanner.
Calculate the difference (residual error).
Adjust the image estimate to reduce the error,
incorporating corrections derived from the
difference.
Repeat the process (iterate) until the error
between simulated and measured data is
minimized or meets a predefined threshold.
Iterative reconstruction causes improved image quality, because it reduces noise and artifacts (e.g. streak
artifacts in CT), and provides better spatial resolution and contrast.
In PET, the radioactive tracer emits positrons (𝛽+). Each positron travels a short distance in the tissue before
annihilating with an electron. This annihilation produces two gamma photons of 511 keV energy, which are
emitted in opposite directions (180° apart) to conserve momentum.
Coincidence detection refers to the process of detecting these two
gamma photons simultaneously (within a few nanoseconds) to
determine the location of the annihilation event along a line.
In PET, the two 511 keV photons produced by positron annihilation
travel in opposite directions and are detected by detectors around the
patient. The line of response (LOR) represents the straight path along
which the annihilation occurred but does not provide the exact point of
interaction. While standard PET assumes the annihilation occurred
somewhere along the LOR, time of flight (TOF) PET measures the
difference in arrival times of the two photons to calculate a more
precise location of the annihilation event.
Spatial resolution defines how small a detail can be detected in an image. Higher spatial resolution means the
system can distinguish finer details, while lower resolution results in blurred or merged objects.
There are different types of events recorded during PET imaging, which can affect the quality and accuracy of
the reconstructed image:
1. True coincidence events: True events occur when two photons produced by the same annihilation
event are detected simultaneously (within the coincidence time window) by opposite detectors.
These events define a correct LOR and are essential for accurate image reconstruction.
2. Random coincidence events: Random events occur when two unrelated photons, originating from
different annihilation events, are detected simultaneously within the coincidence time window.
These events create false LORs, introducing noise into the image and degrading image
quality.
3. Scatter coincidence events: Scatter events occur when one or both photons undergo Compton
scattering before reaching the detector, altering their original trajectory.
Scatter events result in incorrect LORs, causing artifacts and reducing image contrast.
True coincidence Random coincidence Scatter coincidence
Page | 4
