Radioactive Decay in Positron Emission Tomography Scans
- Page ID
- 418940
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- I: F3a - Radioactive half-lives are unique and may be used to identify what isotopes are present.
- I: F3b - Radioactivity may occur via alpha, beta, gamma or other decay events and the type of radioactivity observed is often an important component of identifying the decaying isotope.
- V: A3a - Stoichiometric calculations of chemical reactions are based on mole ratios determined from balanced chemical equations.
- V: A3b - Because chemical equations provide mole ratios, stoichiometry problems involving masses require the use of molar mass conversions.
- IX: H5 - Radioactive chemicals require additional knowledge related to risks and use of radioactive materials must include appropriate protective measures.
Introduction
What is a PET scan?
PET scans or Positron Emission Tomography scans use nuclear chemistry to qualitatively measure the metabolic activity of cells in our body. Metabolism or metabolic activity is "the total of all chemical changes that take place in a cell or an organism to produce energy and basic materials needed for important life processes."1 Through comparative analysis, we can use these measurements to determine the presence of cancer, heart disease, and neurodegenerative diseases early-on unlike the commonly known MRI or Computed Tomography scans.
Figure 1. PET Scan comparisons between a healthy brain (center) and one with Alzhiemers Disease (right).2
Contextual Chemistry
The Positron and Antimatter Chemistry
A radioactive isotope (e.g., Fluorine-18) undergoes positron emission to reach a higher level of stability, and as a result a positron is emitted as demonstrated below:
\(^{18}_{9}\)F→\(^{18}_{8}\)O+\(^{0}_{1}\)e
The positron is the antimatter particle of the electron. So, when the positron collides and reacts with an electron, a gamma ray (photon or light) is produced as demonstrated below:
\(^{0}_{1}\)e + \(^{0}_{-1}\)e → \(^{0}_{0}\)Y
Application of Positron Antimatter Chemistry in PET Scans
PET scans use Positron Antimatter Chemistry to track the metabolic activity of the cells in the body. Using this metabolic data, physicians can determine if the tissue/organ under observation is experiencing abnormal metabolic activity and further confirm or disaffirm the presence of a disease. The step-by-step mechanism of PET scans is outlined below.
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A chemical tracer (radioactive atom such as Fluorine-18) is attached to a sugar molecule, and injected into the patient’s body. The most commonly-used molecule in PET scans is Fluorodeoxyglucose (FDG).
Figure 2. The Chemical Structure of an FDG molecule
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Cells break down this sugar molecule through a process called metabolism.
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While cells are breaking down this sugar molecule, the radioactive isotope is also breaking down by undergoing nuclear decay through positron emission. As illustrated above these emitted positrons immediately react with the surplus of electrons found in the body to produce gamma rays or light.
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The sites in the body that undergo the greatest amount of metabolic activity require the greatest amount of sugar to metabolize. Therefore, the sugar molecules with their attached radioactive isotopes will accumulate at these activity-rich sites. Therefore, the sites that undergo the greatest amount of metabolic activity will be illustrated as the brightest and most colorful areas on the PET scan images.
Example of PET Scan Application:
Cancer cells undergo unusually high amounts of metabolic activity. A PET scan will illustrate this by displaying a cancer-infected area as abnormally bright. This result will indicate an abnormal metabolic activity rate at the given site and signal the probable presence of cancer. In the image below, the brain tumor is highlighted in red.
Figure 3. A PET Scan of a tumor-infected brain.3
Clinical Chemistry
Radiation Exposure in Clinical Settings
The radionuclides that are injected into a patient’s body for PET scans have very short half lives.4 Thus, after 24 hours, the amount of radionuclides in the patient’s body are so low that patients traveling on public transportation won’t even be able to activate a radiation detector.4
Although 18F-FDG is the most common radionuclide employed in clinical settings, other radionuclides are used as well for different purposes in PET scans. Copper radionuclides, such as 60Cu, 61Cu, and 64Cu, can be sent to researchers in various parts of the country because of its 12.8-hour half-life.5 64Cu is used for PET scans that examine copper metabolism, nutrition, and copper transport.5 66Ga is also commonly used because it has a medium half-life of 9.45 hours.5 Additionally, because the decay of 66Ga produces positrons with very high energies, it is very useful in imaging.5 86Y is also useful because it is used in dosimetry,5 the “process of relating the administered amount of radioactivity to the absorbed radiation dose in tumors, organs, or the whole body.”6 Non-fluorine tracers with long half lives are not used for PET scans and are rarely employed in research.4
It is very important that physicians are aware of the half-lives of the radionuclides they are injecting into patients so they can assess (a) how long the radionuclide will remain in the system of the patient and (b) the radiation individuals are exposed to when standing near the patient. Through half-life equations and stoichiometry, we can calculate the time it takes for a certain percentage of 18F-FDG to decay as well as the moles of positrons (radioactive material) released during this time. To understand how long a radionuclide will remain in the system of the patient and the amount of radioactive material released in the patient's body, the following calculations are performed.
Application of the Integrated Rate Law to PET Scan Chemical Tracers
Calculate the time it will take for 95% of 18F-FDG to decay when injected into a patient given that t1/2 of 18F-FDG is 1.83 hours.
- Answer
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Because positrons are released during PET scans, we can assume that 18F-FDG is first order.
Thus,
ln[A] = ln[A]0 - kt where t1/2 = \(\dfrac{ln2}{k}\). Therefore, ln[A] = ln[A]0 - \(\dfrac{ln2}{t_\tfrac{1}{2}}\)t → ln\(\dfrac{A}{A_0}\) = -\(\dfrac{ln2}{t_\tfrac{1}{2}}\)t → t= (-ln\(\dfrac{A}{A_0}\)) \(\dfrac{t_\tfrac{1}{2}}{ln2}\) = -(ln0.05)\(\dfrac{1.83\,hrs}{ln2}\) = 7.909 hours
How many moles of positrons were emitted in the patient’s body during this time given that 100 milligrams of 18F-FDG were injected into the patient’s body?
- Answer
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95 mg x \(\dfrac{1 g}{1000\,mg}\) x \(\dfrac{1\,mol\, ^{18}F}{18.00\,g}\) x \(\dfrac{1\,mol\,β^+}{ 1\,mol\,^{18}F}\)= 0.00527 moles β+
Oftentimes, physicians would like to know the radiation given off by the patient to determine whether it is safe to be beside the patient at a given time period.
Thus, the cumulative radiation exposure equation allows us to use a single equation to calculate the radiation exposure.
The cumulative radiation exposure X= \(\dfrac{nΓ}{d^2}\), where Γ is the exposure rate constant of the radionuclide with the units \(\dfrac{R-cm^2}{mCi-hr}\). Ci, a curie, is the number of decays a radionuclide undergoes per second. R, a roentgen, “is defined by the amount of γ or x-ray radiation that produces 2.58 x 10-4 C of charge per kilogram of air."7
d is the distance in cm from the radioactive source,7 and n is equal to the cumulative activity, Ac.
The cumulative activity Ac is Ac=A0(\(\dfrac{1-e^{-λt}}{λ}\)) where “A0 is the initial administered activity, λ is the decay constant of the radionuclide in question, and t is the time.”7
Cumulative Radiation Exposure in PET Scans
In the hypothetical example below, let’s imagine an adult patient receives a hypothetical dose of 5mCi (185 MBq) of 18F-FDG (“the recommended dose of 18F-FDG for a 70kg adult is within the range of 185-370 MBq [5-10 mCi], intravenous injection”).8
Calculate the cumulative exposure a family member receives from this patient while standing near this patient for 2 hours since the injection at a distance of 2 meters. t1/2 of 18F-FDG is 1.83 hours, and Γ for 18F is \(\dfrac{6.96\,R-cm^2}{mCi-hr}\).
Thus, λ for 18F is \(\dfrac{0.693}{1.83\,hr}\) = 0.3787 hr-1
- Answer
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The cumulative activity Ac=A0(\(\dfrac{1-e^{-λt}}{λ}\)) = 5 mCi x \(\dfrac{1 - e^{-0.3787\,hr^{-1} x\, 2 \,hr}}{0.3787\,hr^{-1}}\) = 7.012 mCi-hr
The cumulative exposure X at 2 meters = \(\dfrac{7.012\,mCi-hr\,\,x\,\, \dfrac{6.96\,R-cm^2}{mCi-hr}}{(200\,cm)^2}\) = 0.00122 R = 1.22 mR.
The family experiences quite a low cumulative exposure of only 1.22 mR. According to Fazel and colleagues, low exposure to radiation is under 3 mSV per year (342.1 mR).9
Bibliography
(1) https://www.cancer.gov/publications/.../def/metabolic. www.cancer.gov. https://www.cancer.gov/publications/.../def/metabolic.
(2) Institut Douglas. PET scan of an healthy brain compared to a brain at an early stage of Alzheimer’s disease. https://www.flickr.com/photos/instit...-DBx1Sm-87AM7E.
(3) TattwamasiB. PET CT in Management of Patients https://commons.wikimedia.org/wiki/F...f_Patients.jpg.
(4) Radiation protection of patients during PET/CT scanning https://www.iaea.org/resources/rpop/...et-ct/patients.
(5) Mcquade, P.; Mccarthy, D.; Welch, M. 1 Metal Radionuclides for PET Imaging *.
(6) Dosimetry - an overview | ScienceDirect Topics https://www.sciencedirect.com/topics...stry/dosimetry (accessed 2021 -09 -06).
(7) Saha, G. B. Basics of PET Imaging : Physics, Chemistry, and Regulations; Springer: Cham, 2016.
(8) Fludeoxyglucose F 18 Injection (FDG): Side Effects, Interactions, Warning, Dosage & Uses https://www.rxlist.com/fludeoxyglucose-drug.htm.
(9) Paddock, C. Cumulative Radiation Exposure From Imaging Scans Should Be Weighed Against The Benefits Say Researchers https://www.medicalnewstoday.com/articles/162170#1 (accessed 2022 -11 -11).
(10) Mayo Clinic. Positron emission tomography scan - Mayo Clinic https://www.mayoclinic.org/tests-pro...t/pac-20385078.
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(12) Johns Hopkins Medicine. Positron Emission Tomography (PET) https://www.hopkinsmedicine.org/heal...tomography-pet.
(13) ANSTO News. PET Scan Animation. YouTube, 2015.
(14) Li, Z.; Conti, P. S. Radiopharmaceutical Chemistry for Positron Emission Tomography. Advanced Drug Delivery Reviews 2010, 62 (11), 1031–1051. https://doi.org/10.1016/j.addr.2010.09.007.
(15) Shukla, A.; Kumar, U. Positron Emission Tomography: An Overview. Journal of Medical Physics 2006, 31 (1), 13. https://doi.org/10.4103/0971-6203.25665.
(16) Paans, A. Positron Emission Tomography. Methods 2002, 27 (3), 193–194. https://doi.org/10.1016/s1046-2023(02)00074-9.