Physics of Radioactivity & Principles of Nuclear Medicine notes

Nuclides

• Nuclei with different numbers of protons and/or neutrons are called nuclides.
• There are about 1500 nuclides (natural + artificial).
• The binding energy of a particle refers to the energy required for the escape of such a particle from the nucleus (e.g., BEp, BEn, BE_α).
• The nucleus, like the atom, has discrete energy levels. During the transition of a nuclide from one energy level to another level with lower energy, the difference in energy is emitted in the form of a photon.
• Nuclides at a high energy level may emit a whole particle to achieve a more favorable energy state.
• Only 280 nuclides belonging to 83 elements are stable in nature.

Radionuclides

• The property of some nuclides (unstable) to transit to a lower energy state by emitting energy in the form of particle or photon radiation is called radioactive decay (or disintegration).
• The unstable nuclides are called radionuclides.
• The decay rate of a radionuclide is ‘constant’ and does not change with any physical or chemical means.

Decay Constant

• Decay constant λ (s^{-1}) of a radionuclide is the “probability” that a particular nucleus of a radioactive substance solely consisting of that radionuclide will decay per unit time.
• λ (s^{-1}) is characteristic of the radionuclide.
• The number of nuclei that decay in a given time interval is proportional to the total number of undecayed nuclei: \Delta N = -λ \Delta t N(t).
• The number of decays per unit time is called activity: A = \frac{\Delta N}{\Delta t} = λN. (The minus sign indicates the number of original nuclei decreases over time).

Radioactive Decay Law

• For a given substance of radioactive material, the total number N of radioactive nuclei remaining after time t is given by the equation: N = N0 e^{-λt}, where N0 is the number of undecayed nuclei at t=0 and N is the number of undecayed nuclei at time t, and λ is the decay constant of the radionuclide.
• Also, λ = \frac{ln2}{T{1/2}} = \frac{0.693}{T{1/2}}.
• The (physical) half-life of a radionuclide is defined as the time for half of the original number of radioactive nuclei to decay.

Activity (Radioactivity)

• The activity A is defined as the magnitude of the decay rate: A = \frac{\Delta N}{\Delta t}.
• The units used for the measurement of the activity are:
  • Bequerel (Bq) = 1 decay/sec
  • Curie (Ci) = 3.7 \times 10^{10} decays/sec (1 mCi = 37 MBq).
• It is easily proved that the law of exponential decay applies also to the activity: A = A0 e^{-λt}, where A0 is the initial activity of a radioactive substance and A is the activity after time t.
• Example Question: Tc-99m is a radionuclide with T_{1/2} = 6 h. A quantity of Tc-99m has an activity of 32 MBq at time t = 0 s. We measure the activity of this amount after 12 h. What will we measure?

Statistics of Radioactive Decay

• For a radioactive sample, the number dN of decays that occur in a time interval dt follows the Poisson distribution.
• Let a huge number (infinite) of samples with the same number of undecayed nuclei N_0 at t=0.
• The number of decays observed within time Δt in each sample differs.
• The number of undecayed nuclei N in each sample after time t will be different between samples.
• The mean number of undecayed nuclei of all samples after time t will be equal to the value N provided by the radioactive decay law.
• 67% of samples will be observed with a number of undecayed nuclei between (N - \sqrt{N}) and (N + \sqrt{N}).
• When we measure activity (remember A = λN) we have a statistical uncertainty.
• This statistical uncertainty is trivial due to the gigantic number of undecayed nuclei in typical radioactive samples used in nuclear medicine.

Modes of Radioactive Decay

• A nuclide before disintegration is called a parent nuclide and that after disintegration is called a daughter nuclide. P \rightarrow D + \text{disintegration}
• If the daughter nucleus is stable, then the decay process stops.
• Some radionuclides remain energetically unstable even after disintegration, which means that the original radionuclides have transformed into other types of radionuclides.
• If this daughter nucleus is itself radioactive, it will further decay into another element, and the process of successive decays will only seize when a stable daughter nucleus is finally produced.
• In a series of successive radioactive decays, the decay mechanisms may be different from each other.
• The main modes of radioactive decay are:
  • Alpha (α) Decay
  • Beta-minus (β-) Decay
  • Beta-plus (β+) decay
  • Electron capture
  • Gamma (γ) decay

Gamma (γ) Decay

• Gamma (γ) decay is a radioactive decay mode where an unstable atomic nucleus releases a gamma ray (i.e., a high-energy electromagnetic photon). Unlike alpha or beta decay, gamma decay does not alter the atomic or mass numbers but serves to release excess energy and stabilize the nucleus.
• The daughter nucleus of a radioactive decay (α, β-, β+, EC) is usually in an excited state.
• The de-excitation with gamma-ray emission usually occurs immediately after the decay.
• However, if the residence time in this state is long (> 10^{-12} sec) the state is called metastable. The de-excitation, through gamma-ray emission, of a nucleus that is in a metastable state is called isomeric transition.
• For example, Tc-99m (Technetium-99m) is a daughter radionuclide of Molybdenum-99 beta emission.
• Tc-99m decays to 99Tc (half-life=6 h).
• Tc-99m is the most important radionuclide in diagnostic Nuclear Medicine procedures.

α - Decay

• Alpha decay is a radioactive process where an unstable atomic nucleus emits an alpha particle, consisting of two protons and two neutrons (α particle = nucleus of helium). This emission reduces the atomic number by 2 and the mass number by 4, transforming the original element into a new one with increased stability.
• Alpha decay occurs in elements with high atomic numbers, such as uranium, radium, and thorium.
• Example: \text{}^{226}{88}Ra \rightarrow \text{}^{222}{86}Rn + \text{}^{4}{2}He (T{1/2} = 1600 years)
• Alpha particles emitted from the decay of a specific radioactive nucleus are monoenergetic.
• Emitted alpha particle energy is defined by the energy release of the decay and the mass fraction of the daughter nucleus and alpha particle.

β Minus Decay

• Beta minus decay is a type of radioactive decay in which an atomic nucleus emits a fast energetic electron, transforming into an isobar of that nuclide.
• A neutron of the parent nucleus transforms into a proton by the emission of an electron accompanied by an antineutrino.

β Plus Decay

• Beta plus decay is a type of radioactive decay in which an atomic nucleus emits a fast energetic positron, transforming into an isobar of that nuclide.
• A proton is converted into a neutron by the emission of a positron with a neutrino in so-called positron emission β-decay.
• The total energy released by the decay process is divided between 3 particles (the electron, the antineutrino, and the recoiling daughter nucleus in beta minus or the positron, the neutrino, and the recoiling daughter nucleus in beta plus decay).
• Therefore, the beta particles emitted during a beta decay are not monoenergetic. The beta spectrum, or distribution of energy values for the beta particles, is continuous from 0 to a maximum energy E{max}. The mean energy of emitted beta is about 1/3 E{max}.

Electron Capture

• Electron capture is a process in which the proton-rich nucleus of an electrically neutral atom absorbs an inner atomic electron, usually from the K or L electron shells.
• This process thereby changes a nuclear proton to a neutron and simultaneously causes the emission of an electron neutrino.
• The resulting daughter nuclide is usually in an excited state and transitions to its ground state by emitting a gamma ray.

Decay Chains (or Decay Series)

• There are 3 decay chains, also known as radioactive decay series, that occur in nature.
• Example: from unstable radioactive \text{}^{232}Th to stable \text{}^{208}Pb

Nuclear Medicine

• The use of radionuclides for therapeutic and diagnostic purposes led to the creation of the medical specialty called ‘Nuclear Medicine’.
• Nuclear Medicine basis: the unstable isotopes (radionuclides) have the same chemical properties as the stable ones, so a stable isotope of an element can be replaced with the corresponding unstable one.
  • For example, I-127 is stable and I-131 is unstable (beta minus decay).
  • Iodine I is concentrated in the thyroid regardless of whether it is I-127 or I-131.
• The therapeutic applications of Nuclear Medicine are based on the biological effects of radiation emitted by the administered radionuclide on the neoplasm that has developed in a specific tissue or organ.
• The diagnostic applications of Nuclear Medicine rely on the emission of gamma radiation from the tissue or organ selectively taken up by a chemical compound containing a radionuclide.
• Diagnostic Nuclear Medicine examinations are divided into two categories:
  • A. In vivo examinations
  • B. In vitro examinations

In Vitro Nuclear Medicine Examinations: Radio-Immuno-Assay Tests (RIA)

• Radioimmunoassay (RIA) is a sensitive laboratory test for quantifying hormones, drugs, or other substances in the blood.
• It involves using radioactively labeled substances and specific antibodies to measure the amount of a target substance.
• RIA is crucial in medical diagnostics and research for detecting and monitoring various conditions.
• Radioimmunoassay aims to determine the concentration of a specified substance (antigen) in a biological fluid (usually blood) of the subject.
• They rely on mixing:
  • biological fluid sample e.g., blood serum containing a substance (antigen) whose concentration we want to measure (e.g., TSH, T3).
  • radioactive substance (the same antigen labeled with I-125).
  • non-radioactive substance (antibody) that has the ability to bind to the antigen we want to measure.
• Radiolabeled and unlabeled antigens compete to bind to antibodies.
• How many radiolabeled molecules will make it is determined by the concentration of the antigen in the patient's serum.
• After the free antigens are removed, the activity of the sample is measured by a special γ-radiation detector (γ-counter).
• The result of a RIA examination is numerical and corresponds to the concentration of the substance in the biological fluid sample.

Basic Principle of RIA Test

• Mix antigen + I-125 (high concentration) with a blood sample from the patient.
• Add a sample of the antibody with known concentration.
• Remove free antigens (Centrifugation).
• Measure & compare.

In Vivo Examinations

• They are based on administering radioactivity to the patient's body. A specific chemical substance labeled with a suitable radionuclide (radiopharmaceutical: Rph) is administered to the subject.
• The labeled substance (Rph) is absorbed (mainly but not exclusively) into the under-study tissue or organ through normal metabolic processes.
• After a specified time interval (5 min-4 h depending on the examination), the radiation emitted by the patient is detected, and the spatial distribution of the radiopharmaceutical in the tissues is determined using a suitable detection system (gamma camera).
• The result of such an examination is a series of images, and the diagnostic information obtained is related to the morphology and functionality of the tissue or organ being imaged.
• It is a method of imaging the functionality of tissues.

Radiopharmaceuticals

• A chemical substance labeled with a radioactive nuclide, capable of selectively accumulating in a specific tissue or organ of the human body, is called a radiopharmaceutical.
• \text{CHEMICAL COMPOUND + RADIOISOTOPE = RADIOPHARMACEUTICAL}
• The chemical (non-radioactive) substance is usually provided in powder form in special vials. Specific radioisotope activity (mCi) is introduced as a solution into the vial, resulting in the radiopharmaceutical (through mixing, heating, etc.). The activity of the radiopharmaceutical (usually in a syringe) is measured using suitable detectors.
• Radiopharmaceuticals are administered to the patient intravenously, orally, or by inhalation, depending on the type of examination.

Properties of a Radiopharmaceutical for Diagnostic Examination

• The selection of the substance is made so that the produced radiopharmaceutical:
  • selectively accumulates in the examined or treated tissue/organ.
  • is safe and does not cause allergies or toxic reactions.
  • is manufactured under sterile microbial conditions.
  • is free from toxins and pyrogenic substances.
• The radiochemical purity of a radiopharmaceutical is defined as the % of total radioactivity present in the chemical form stated in the formulation.
• The radioisotopic purity is defined as the % of radioactivity attributed to the radionuclide stated in the formulation.

Effective Half-Life of a Radiopharmaceutical

• Natural half-life (T_n) of a radiopharmaceutical is the time required for an amount of this Rph to decrease its activity into half.
• Biological half-life (T_b) is the time needed for biological mechanisms to eliminate half of a chemical substance administered to a patient.
• The effective half-life (T{eff}) of a radiopharmaceutical is a concept that combines both the physical decay (natural half-life Tn) of the radionuclide and its biological clearance from the body (biological half-life T_b). It represents the time it takes for the activity of the radiopharmaceutical in the patient's body to decrease by half, considering both processes.
• It is proven that the effective half-life depends on both the natural and biological half-life:
  • T{eff} = \frac{Tn \cdot Tb}{Tn + T_b}

Radionuclides for Diagnostic Examinations

• The radionuclides (radioisotopes) used for in vivo diagnostic examinations should:
  • have a relatively short half-life to ensure the recipient's total dose is tolerable and allow completion of the examination (ideal time ⁓a few hours).
  • emit only γ radiation.
  • emit γ with energy such that emitted γ radiation is not heavily absorbed by tissues and, simultaneously, can be efficiently detected by the measuring system (ideal energy⁓150 keV).
  • be easily and cost-effectively produced.
• Technetium Tc-99m, with a half-life of 6 hours and energy of emitted γ of E=140 keV, is ideal for diagnostic purposes, covering almost 80-90% of diagnostic applications.

Radionuclides for Therapy

• The radionuclides used for therapeutic purposes should:
  • emit beta or alpha radiation (emission of gamma radiation allows imaging, facilitating dose distribution verification but simultaneously increases the dose to healthy tissues).
  • have a sufficiently long half-life (several days) so that the total dose received by the treated organ/tissue is significant.
  • the energy of the emitted particles should be high enough to sufficiently penetrate the pathological tissue and produce a uniform dose distribution in the treated tissue/organ.

Production of Radionuclides

• Radioactive nuclei can be produced technically by the following methods:
  • A) Bombardment of stable nuclei with neutrons: Co-59 + n → Co-60
  • B) In a cyclotron, bombardment of stable nuclei with protons: p + Zn-68 → Ga-67 + 2n
  • C) Through fission of heavy nuclei: U-235 → Mo-99 + Sn-134 + 3n
  • D) In generators for radioisotope production: Mo-99 → Tc-99m + β-
• The half-life of radioisotopes should be relatively short, making it challenging to transport such isotopes from an external laboratory to a hospital's laboratory. Additionally, having a cyclotron in hospitals for on-site radioisotope production is prohibitive due to installation and maintenance costs.
• The most common solution is radioisotope generators. The radionuclide generator contains a parent radionuclide with a relatively long half-life. Upon decay, it produces the short-lived radionuclide used for clinical applications.

The Μο-99/Τc-99m Generator

• The Μο-99/Τc-99m generator contains Μο-99 (parent radionuclide with 66 h half-life). Upon decay, it produces Τc-99m used for clinical applications.
• The quantity of Tc-99m contained in the generator is determined by the decay rate of Mo-99.

Usage of Mo-99/Tc-99m

• The collection of Tc-99m (the generator's "harvest") occurs daily.
• The collected activity decreases daily due to the decay rate of Mo-99.
• The half-life of Mo-99 allows the use of the generator for one week.

Gamma-Camera

• The imaging system of diagnostic (in-vivo) Nuclear Medicine is the gamma camera.
• The gamma camera produces a 2D image representing the projection of the 3D distribution of radioactivity in the patient's body onto a plane (the plane defined by the detector).
• The main components of the gamma camera head are:
  • The collimator
  • The NaI(Tl) scintillation crystal detector
  • The photomultiplier tubes
  • The electronics for data analysis

The Collimator

• Gamma camera collimators play a crucial role in determining the spatial resolution and sensitivity of the imaging system by controlling the entry of gamma rays into the detector.
• The role of the collimator is to strictly define the projection direction of the 3D distribution of the radiopharmaceutical that has been shaped by intercepting photons incident on it at an angle.

Types of Collimators

• Parallel-hole Collimator: consists of arrays of parallel holes that allow gamma rays to pass through. This design is simple and commonly used for general imaging applications.
• Convergent Collimator: have converging holes, providing better resolution in one direction. They are suitable for imaging structures with elongated or linear shapes.
• Pinhole Collimator: use a single small hole to focus on a specific area, providing high-resolution images. They are often used for small organ imaging, such as thyroid or breast imaging.
• Diverging Collimator: have holes that diverge away from each other, allowing for a larger field of view. They are suitable for imaging larger anatomical structures.
• The choice of collimator depends on the specific imaging requirements and the type of study being conducted. Different collimators offer a trade-off between spatial resolution, sensitivity, and field of view. The selection is based on optimizing these factors for the particular clinical or research application.

Parallel Hole Collimator

• The collimator consists of a grid with sides made of lead defining gaps (holes). The gaps determine a direction.
• Photons traveling in the direction of the holes pass through the collimator, while photons with a different direction are blocked and do not reach the crystal.

Why a Collimator is Needed

• Isotropic radiation emission from a source incident on a detector results in a low-resolution image.
• Isotropic radiation emission from a source incident on a detector after passing through a parallel hole collimator results in an improved-resolution image.
• The collimator sets the projection direction of the 3D distribution of the R/ph in the plane defined by the detector (planar imaging).

The Detector as a Scintillator

• It is a square crystal of iodized sodium with thallium doping (NaI:Tl).
• Upon photon incidence on the crystal, scintillation occurs (momentary emission of visible light), and the intensity depends on the energy of the incident photon.
• NaI crystal has a high atomic weight to efficiently absorb gamma radiation and is transparent in the visible frequency produced during scintillation.
• The crystal thickness is typically 1-3 cm. The dimensions of the scintillation crystal determine the useful imaging field (~ 40 x 40 cm).

Photomultipliers

• In optical contact with the crystal of a gamma camera, there is a group of photomultipliers (up to 100) arranged in a hexagonal or square pattern.
• Each photomultiplier receives a fraction of the photons produced during scintillation and produce an electric pulse.

The Photomultiplier Function

• A photomultiplier is an electronic tube that converts light pulses into electrical pulses.
• The height of the pulse at the end (anode) is proportional to the incident light pulse.
• The height of the pulse at the output of a photomultiplier depends on the quantity and energy of the photons striking the photocathode.

Photomultipliers of Gamma Camera

• The quantity of light falling on each photomultiplier depends on the distance of the photomultiplier from the point where the scintillation occurred.
• The closer a photomultiplier is to the point of a scintillation, the higher is the output signal.

Determination of the Incidence Point of γ

• The ratio of pulses (X+ = C+D, X- = A+B) determines the x coordinate of the incidence point.
• The ratio of pulses (Y+ = A+C, Y- = B+D) determines the y coordinate of the incidence point.

γ-Camera Electronics

• The electronic circuit of the gamma camera system generates four pulses (X+, X-, Y+, Y-) that convey information about the position of the scintillation (γ incidence point).
• The sum of these pulses (pulse Z) is proportional to the energy of the incident photon that struck the crystal and initiated the scintillation.
• Pulse Z is fed into a pulse height analyzer, determining whether the event (scintillation) will be accepted or rejected concerning the formation of the scintigraphic image.

How the Scintigraphic Image is Generated

• The dimensions of the scintigraphic image are determined by the size of the NaI crystal and the type of collimator used.
• The image consists of a predetermined number of square image elements (picture elements: pixels). Scintigraphic images are typically acquired with matrices ranging from 64x64 to 256x256.
• ➢ A photon striking the crystal at a specific position generates a scintillation.
• ➢ The position of scintillation is determined and the scintillation is checked if it corresponds to expected γ. If yes, then the content of the image element corresponding to this position increases by 1.
• ➢ Recording the photons (usually 100,000-500,000) incident on the crystal during the examination the scintigraphic image is created.
• ➢ The image represents the projection of the three-dimensional distribution of the radiopharmaceutical in the patient's tissues at the crystal plane (planar scintigraphic imaging).

Energy of Scintillations

• In the crystal, scintillations of lower energy than the energy of the photons emitted by the radiopharmaceutical inside the patient’s body can be produced due to:
  • detection of photons scattered in the patient's body
  • detection of photons scattered in the crystal
• Scattered photons reaching the crystal convey erroneous information regarding the site of emission and are undesired.
• Scintillation energy = energy of photons emitted by the radiopharmaceutical in patient tissues.
• Scintillation energy < energy of photons emitted by the radiopharmaceutical in patient tissues.

Incident Photon Energy Resolution

• The pulse height (Z) provides a measure of the energy of each photon incident on the crystal.
• Even for incident photons of the same energy, the produced pulse may vary due to the electronic circuit noise involved in pulse generation.
• So, we end up with a spectrum rather than a narrow peak of photon energy.
• A narrower peak indicates better energy resolution (ER) of the system.
• The Full Width at Half Maximum (FWHM) is an index of energy resolution: ER = \frac{FWHM}{h}
• In modern gamma cameras, the energy resolution is typically 10-15%. To capture all 140 keV photons reaching the detector, a window of 20% or 15% width is used.

The Energy Spectrum of Detected Photons

• The Tc-99m window, with 20% (126-154 keV), contains photons that may have been scattered at angles up to 50⁰.
• Scattered photons intruding the photopeak window are undesirable as they convey inaccurate information about their emission position.

Scintigraphic Image Dimensions and Pixel Size

• The size of the crystal determines the physical size of the image (Useful Field of View ≡ UFOV).
• The dimension of each image element is calculated by dividing the image dimension (UFOV) by the size of the utilized matrix. For example, if UFOV=45cm x 45cm and the matrix is 128x128, then 1 pixel = 45/128 x 45/128 = 0.35cm x 0.35cm.
• The number of pixels is determined by the chosen matrix. For instance, if the matrix is 128x128, then the number of pixels = 128x128 = 16384.
• A large display matrix (e.g., 512x512 as in computed tomography imaging) offers the possibility of a clearer representation of details compared to a small display matrix (e.g., 64x64).
• In computers, a small pixel size (pixel dimension) is a sample of a good screen.

Statistical Uncertainty of Pixel Value

• The radioactive decay of a multitude of radioactive nuclei follows the Poisson law.
• The standard deviation (σ or error) of the number N of nuclei that decayed in a certain time is \sqrt{N} .

Matrix Size and Statistical Uncertainty of Pixel Values

• The value N of each pixel (i.e., N detected photons in that position of the image) is on the order of 100 for an average scintigraphic image, leading to a significant error solely due to the statistics of the radioactive decay phenomenon.
• Large matrices require very high numbers of detected photons to maintain low noise levels, achieved either by increasing the administered dose or prolonging the acquisition time.
• Matrices of 64x64, 128x128, and 256x256 are used in scintigraphic imaging.

Therapy with Radionuclides

• They are based on administering relatively high amounts of a radioactive substance to the patient's body to treat medical conditions, particularly cancer.
• A radionuclide or a specialized chemical substance labeled with a suitable radionuclide emitting beta or alpha particles is administered to the patient.
• The administered radionuclide/radiopharmaceutical is absorbed (mainly but not exclusively) into the pathological tissue/organ through normal metabolic processes.
• The radiation released by the administered radioactive substance is mostly absorbed in the vicinity of tissues/organ with high uptake, damaging or destroying the nearby cells, especially rapidly dividing cancer cells.
• The aim is to deliver targeted radiation dose to specific tissues or organs affected by disease. This approach is known as targeted radionuclide therapy and aims to minimize damage to surrounding healthy tissues while effectively treating the diseased cells.
• Common radionuclides used in therapy include iodine-131, yttrium-90, and lutetium-177.

Removal of Residual Thyroid Tissue

• I-131 is administered after the surgical removal of the thyroid to eliminate any remaining thyroid tissue (thyroid remnants).
• The administered dose is usually 50-200 mCi.
• The patient stays in specially shielded rooms for 1-3 days.
• The patient is released from the hospital if the radiation exposure level at 1 meter from his/her body is <40 μSv/h, as measured by the medical physicist in charge.
• The patient receives specific radiation protection instructions to follow for 2 weeks after therapeutic administration of I-131.
• Apart from e-, I-131 emits photons so imaging of the treated region is possible.

X-Ray Production, X-Ray Spectrum

• CATHODE
• ANODE
• CROOKES TUBE
• COOLIDGE TUBE

Main parts of the X-Ray tube

• e * V = 1/2 * mm * v^2