Exercises — Chapter 27

Radionuclide Production

Problem 27.1 ⭐ The reaction ${}^{18}\text{O}(p,n){}^{18}\text{F}$ has $Q = -2.44\,\text{MeV}$.

(a) Calculate the threshold proton energy in the lab frame. Explain why the threshold exceeds $|Q|$.

(b) A medical cyclotron delivers 50 $\mu$A of 16 MeV protons onto an enriched ${}^{18}\text{O}$-water target with a thick-target yield of 9 GBq/$\mu$A$\cdot$h. How much ${}^{18}\text{F}$ activity (in GBq) is produced after a 2-hour bombardment?

(c) The bombardment ends. How much activity remains after a further 90 minutes (the time needed for radiochemistry and quality control)?

Problem 27.2 ⭐ Calculate the specific activity of carrier-free ${}^{99}\text{Mo}$ ($t_{1/2} = 65.94\,\text{h}$, $M = 99\,\text{g/mol}$). Express your answer in GBq/g and Ci/g.

Problem 27.3 ⭐⭐ A nuclear reactor has a thermal neutron flux of $\phi = 2 \times 10^{14}\,\text{n/cm}^2\cdot\text{s}$. The $(n,\gamma)$ cross section for ${}^{176}\text{Lu}$ at thermal energies is $\sigma = 2090\,\text{b}$.

(a) A target containing $m = 10\,\text{mg}$ of enriched ${}^{176}\text{Lu}$ (100% enrichment) is irradiated for $t_{\text{irr}} = 14\,\text{days}$. Assuming negligible burnup of the target, calculate the ${}^{177}\text{Lu}$ activity at the end of irradiation using:

$$A = N_{\text{target}} \, \sigma \, \phi \, (1 - e^{-\lambda t_{\text{irr}}})$$

(b) What is the saturation activity (infinite irradiation time)?

(c) What fraction of the saturation activity is reached after 14 days? After 7 days (one half-life)?

Problem 27.4 ⭐⭐ ${}^{11}\text{C}$ ($t_{1/2} = 20.4\,\text{min}$) is produced by the reaction ${}^{14}\text{N}(p,\alpha){}^{11}\text{C}$.

(a) Calculate the $Q$-value using atomic masses: $M({}^{14}\text{N}) = 14.003074\,\text{u}$, $M({}^{1}\text{H}) = 1.007825\,\text{u}$, $M({}^{11}\text{C}) = 11.011434\,\text{u}$, $M({}^{4}\text{He}) = 4.002603\,\text{u}$.

(b) Is this reaction exothermic or endothermic? Calculate the threshold proton energy if endothermic.

(c) Why must ${}^{11}\text{C}$ be produced by a cyclotron on-site at the hospital, rather than being shipped from a central production facility?

PET Physics

Problem 27.5 ⭐ In positron annihilation at rest, two photons are produced. Using conservation of energy and momentum:

(a) Show that each photon has energy $E_\gamma = 511\,\text{keV}$.

(b) Show that the two photons must be emitted at exactly $180°$ in the center-of-mass frame.

(c) Explain qualitatively why the angular spread in tissue is $\sim 0.5°$ FWHM rather than exactly $0°$.

Problem 27.6 ⭐⭐ A patient is injected with $A_0 = 370\,\text{MBq}$ of ${}^{18}\text{F}$-FDG at time $t = 0$. A PET scan begins at $t = 60\,\text{min}$ and lasts 20 minutes.

(a) Calculate the activity at the start and end of the scan.

(b) Calculate the total number of positron-producing decays during the scan. (The positron branching ratio of ${}^{18}\text{F}$ is 96.9%.)

(c) Each annihilation produces two 511 keV photons. If the PET scanner has a geometric efficiency of 8% and a detector efficiency of 85%, estimate the number of coincidence events recorded during the scan. (Assume 40% of photon pairs are lost to attenuation and scatter.)

(d) Comment on whether this number of counts is sufficient for a high-quality image (typical requirement: $>10^7$ coincidences).

Problem 27.7 ⭐⭐ The spatial resolution of PET is limited by several factors. For ${}^{18}\text{F}$:

(a) The positron range in tissue (rms $\sim 0.6\,\text{mm}$). This blurs the annihilation point relative to the decay point. Explain why this contribution is worse for ${}^{82}\text{Rb}$ ($E_{\max}^{\beta^+} = 3.35\,\text{MeV}$) than for ${}^{18}\text{F}$ ($E_{\max}^{\beta^+} = 0.634\,\text{MeV}$).

(b) The non-collinearity of the two 511 keV photons ($\sim 0.5°$ FWHM). For a PET scanner with ring diameter $D = 80\,\text{cm}$, calculate the spatial blurring at the center of the ring due to non-collinearity.

(c) The detector crystal width is 4 mm. Estimate the overall spatial resolution by adding the three contributions in quadrature.

Problem 27.8 ⭐⭐⭐ Three-photon vs. two-photon annihilation. In vacuum, ortho-positronium ($S = 1$) annihilates into three photons with a rate $\Gamma_{3\gamma} = 7.04 \times 10^6\,\text{s}^{-1}$, while para-positronium ($S = 0$) annihilates into two photons with $\Gamma_{2\gamma} = 7.99 \times 10^9\,\text{s}^{-1}$.

(a) Calculate the lifetimes of ortho- and para-positronium in vacuum.

(b) In condensed matter, ortho-positronium undergoes "pick-off" annihilation (with an electron from the medium) at a rate $\Gamma_{\text{pick-off}} \approx 5 \times 10^9\,\text{s}^{-1}$. What fraction of ortho-positronium decays via three-photon annihilation in tissue?

(c) Why does this mean that PET imaging is unaffected by the ortho/para ratio?

The ${}^{99\text{m}}\text{Tc}$ Generator

Problem 27.9 ⭐ A ${}^{99}\text{Mo}/{}^{99\text{m}}\text{Tc}$ generator is delivered with 20 GBq of ${}^{99}\text{Mo}$ on Monday at 8 AM.

(a) Calculate the ${}^{99}\text{Mo}$ activity remaining on Wednesday at 8 AM, on Friday at 8 AM, and on the following Monday at 8 AM.

(b) The minimum useful generator activity is about 2 GBq. On which day does the generator become unusable?

Problem 27.10 ⭐⭐ Starting from the general Bateman equation for a two-member decay chain (Chapter 12), derive the expression for the ${}^{99\text{m}}\text{Tc}$ activity after milking at $t = 0$:

$$A_2(t) = f \cdot \frac{\lambda_2}{\lambda_2 - \lambda_1} \cdot A_1(0) \cdot (e^{-\lambda_1 t} - e^{-\lambda_2 t})$$

where $f = 0.876$ is the branching ratio to the metastable state.

(a) Verify that $A_2(t) \to 0$ as $t \to 0$ (immediately after milking) and as $t \to \infty$ (parent decayed away).

(b) Derive the expression for $t_{\max}$ (time of maximum daughter activity) and verify the numerical value of $\approx 22.8\,\text{h}$ given in the text.

(c) Calculate the maximum ${}^{99\text{m}}\text{Tc}$ activity achievable from a generator with 20 GBq of ${}^{99}\text{Mo}$, assuming it was milked 24 hours prior.

Problem 27.11 ⭐⭐⭐ Secular vs. transient equilibrium. The ${}^{99}\text{Mo}/{}^{99\text{m}}\text{Tc}$ system is technically in transient equilibrium (the ratio $t_{1/2,\text{parent}}/t_{1/2,\text{daughter}} = 65.94/6.01 \approx 11$, which is large but not $\gg 100$).

(a) In true secular equilibrium, $A_2 = A_1$ (daughter activity equals parent activity). In transient equilibrium, $A_2 = A_1 \times \lambda_2/(\lambda_2 - \lambda_1) \times f$. Calculate this ratio for the Mo/Tc system. By what percentage does it differ from unity?

(b) Plot (or calculate at $t = 0, 6, 12, 18, 24, 48, 72\,\text{h}$) both the ${}^{99}\text{Mo}$ and ${}^{99\text{m}}\text{Tc}$ activities for a generator milked at $t = 0$ with $A_{{}^{99}\text{Mo}}(0) = 20\,\text{GBq}$. Show that the daughter activity exceeds the parent activity at transient equilibrium.

(c) Compare your result to the secular equilibrium approximation. At what level of precision does the distinction between secular and transient equilibrium matter clinically?

Bragg Peak and Therapy Physics

Problem 27.12 ⭐ Using the approximate range formula $R \approx 0.0022 \times T^{1.77}\,\text{cm}$ (with $T$ in MeV):

(a) Calculate the range in water for protons of kinetic energy 70, 100, 150, 200, and 230 MeV.

(b) A tumor is located at a depth of 20 cm. What proton energy is required to place the Bragg peak at this depth?

(c) The distal edge of the tumor extends to 24 cm depth. What is the highest proton energy needed?

Problem 27.13 ⭐⭐ The Bethe-Bloch stopping power for a charged particle with charge $ze$ and velocity $v$ in a medium with electron density $n_e$ and mean excitation energy $I$ is (non-relativistic limit):

$$-\frac{dE}{dx} = \frac{4\pi z^2 e^4 n_e}{m_e v^2} \ln\left(\frac{2m_e v^2}{I}\right)$$

(a) Show that $-dE/dx \propto z^2 / v^2$ for a given medium. What happens to $dE/dx$ as the particle slows down?

(b) For a proton ($z = 1$) and a carbon ion ($z = 6$) at the same velocity, what is the ratio of their stopping powers?

(c) Using the approximation $R \propto v_0^4 / z^2$ (from integrating the stopping power), show that at the same initial velocity, the range of a carbon ion in water is approximately $R_C \approx (A_C/z_C^2) / (A_p/z_p^2) \times R_p = 3 R_p$.

Problem 27.14 ⭐⭐ A single fraction of a photon treatment plan delivers 2 Gy to a deep-seated tumor. Because of the exponential dose profile, the skin (entrance) receives 1.8 Gy and the tissue behind the tumor (exit) receives 0.8 Gy. A proton plan delivers the same 2 Gy to the tumor, with entrance dose 0.7 Gy and zero exit dose.

(a) Calculate the total energy deposited in 1 kg of tissue for each plan if the irradiated tissue column is 30 cm long and 10 cm $\times$ 10 cm cross-section ($\rho = 1\,\text{g/cm}^3$). (Hint: approximate the photon dose as linear from entrance to exit; approximate the proton dose as linear from entrance to the Bragg peak at 15 cm, then zero beyond.)

(b) What is the ratio of integral (total) dose for photons vs. protons?

(c) Discuss why this reduction in integral dose is particularly important for pediatric patients.

Problem 27.15 ⭐⭐⭐ The spread-out Bragg peak (SOBP). A tumor extends from depth $d_1 = 10\,\text{cm}$ to $d_2 = 15\,\text{cm}$. A treatment planner must create a uniform dose across this 5 cm target.

(a) Using the range formula, calculate the proton energies needed to place Bragg peaks at depths of 10, 11, 12, 13, 14, and 15 cm.

(b) Explain qualitatively why the contribution from the deepest Bragg peak (at 15 cm) must have the highest weight, and the shallowest (at 10 cm) the lowest weight.

(c) The dose at the entrance (surface) from a single pristine Bragg peak is approximately 30% of the peak dose. If six weighted Bragg peaks are summed to give a uniform dose of 2 Gy across the target, estimate the approximate entrance dose. Why is this entrance dose unavoidable?

Targeted Radionuclide Therapy

Problem 27.16 ⭐ ${}^{131}\text{I}$ emits $\beta^-$ particles with $E_{\max} = 606\,\text{keV}$ and $\overline{E}_\beta = 182\,\text{keV}$.

(a) Calculate the mean range of the $\beta^-$ particles in tissue using the empirical formula $R_{\beta}\,(\text{cm}) \approx 0.412 \times E^{1.265-0.0954\ln E}$, where $E$ is in MeV (use $E = \overline{E}_\beta$).

(b) Compare this range to the diameter of a thyroid follicle ($\sim 100$–$300\,\mu\text{m}$). Is the beta radiation well-confined to the thyroid tissue?

(c) ${}^{131}\text{I}$ also emits a 364 keV gamma ray (81.7% per decay). What is the purpose of this gamma ray in the clinical setting?

Problem 27.17 ⭐⭐ A patient with differentiated thyroid cancer receives $A_0 = 5.55\,\text{GBq}$ (150 mCi) of ${}^{131}\text{I}$ orally. Thyroid remnant tissue ($m = 5\,\text{g}$) concentrates 20% of the administered activity. Assume all iodine uptake occurs instantaneously and clearance is purely by physical decay ($t_{1/2} = 8.02\,\text{d}$).

(a) Calculate the cumulated activity in the thyroid remnant.

(b) Calculate the absorbed dose to the remnant from $\beta^-$ emission alone ($\overline{E}_\beta = 182\,\text{keV}$, $\phi_\beta = 1$). Express in Gy.

(c) The dose needed to ablate thyroid remnants is typically 300–400 Gy. Does this treatment achieve ablation?

(d) Discuss why the assumptions (instantaneous uptake, no biological clearance) give an upper bound on the dose.

Problem 27.18 ⭐⭐ An alpha particle from ${}^{225}\text{Ac}$ ($E_\alpha = 5.83\,\text{MeV}$) traverses a cell nucleus of diameter $d = 8\,\mu\text{m}$.

(a) If the LET at this energy is approximately $90\,\text{keV}/\mu\text{m}$, how much energy does the alpha particle deposit in a single cell nucleus?

(b) Convert this energy to gray, assuming the cell nucleus has a mass of $m_{\text{nuc}} \approx 5 \times 10^{-13}\,\text{kg}$.

(c) Compare this single-track dose to the typical lethal dose for a cell ($\sim 2$–$8\,\text{Gy}$). How many alpha traversals are needed to kill the cell?

(d) Calculate the recoil energy of the daughter ${}^{221}\text{Fr}$ after the alpha emission. Express in keV.

Problem 27.19 ⭐⭐⭐ Daughter redistribution in targeted alpha therapy. ${}^{225}\text{Ac}$ decays through a chain producing four alpha particles before reaching stable ${}^{209}\text{Bi}$. After the first alpha emission, the daughter ${}^{221}\text{Fr}$ recoils with $\sim 106\,\text{keV}$ and is released from the targeting molecule.

(a) The recoil range of a 106 keV francium atom in tissue is approximately 90 nm. Is this sufficient for the daughter to escape from a tumor cell ($\sim 10\,\mu\text{m}$ diameter)?

(b) ${}^{221}\text{Fr}$ has $t_{1/2} = 4.9\,\text{min}$. Using the diffusion equation, estimate how far a free francium ion can diffuse in blood before it decays. Assume a diffusion coefficient $D \approx 10^{-9}\,\text{m}^2/\text{s}$. The root-mean-square displacement is $\langle r^2 \rangle^{1/2} = \sqrt{6Dt}$.

(c) Discuss the clinical implications of this result for off-target toxicity. Which organs might be at risk?

(d) Propose two strategies (from the literature or your own reasoning) to mitigate the daughter redistribution problem.

Dosimetry

Problem 27.20 ⭐ Define the following quantities and give their SI units: absorbed dose, equivalent dose, effective dose, dose rate, cumulated activity, residence time.

Problem 27.21 ⭐⭐ A point source of ${}^{192}\text{Ir}$ (used in HDR brachytherapy) has an activity of 370 GBq. The dose-rate constant for ${}^{192}\text{Ir}$ in water is $\Lambda = 1.108\,\mu\text{Gy}\cdot\text{m}^2/(\text{GBq}\cdot\text{h})$.

(a) Calculate the dose rate at a distance of 1 cm from the source.

(b) Calculate the dose rate at 5 cm.

(c) An HDR treatment delivers the source to a position 1 cm from the tumor surface for 300 seconds. What absorbed dose is delivered at 1 cm? At 5 cm?

(d) Explain why the steep $1/r^2$ falloff is clinically advantageous in brachytherapy.

Problem 27.22 ⭐⭐ A patient receives $A_0 = 7.4\,\text{GBq}$ of ${}^{177}\text{Lu}$-PSMA-617. Imaging shows that 0.8% of the activity localizes in the kidneys (total mass 310 g) with a biological half-life of 48 h. The physical half-life of ${}^{177}\text{Lu}$ is 159.6 h.

(a) Calculate the effective half-life and effective decay constant.

(b) Calculate the cumulated activity in the kidneys.

(c) Calculate the absorbed dose to the kidneys from $\beta^-$ particles ($\overline{E}_\beta = 133\,\text{keV}$, $\phi_\beta = 1$).

(d) The kidney dose limit for peptide receptor radionuclide therapy is typically 23 Gy (cumulative over all cycles). How many treatment cycles can be administered before reaching this limit?

Problem 27.23 ⭐⭐⭐ Full MIRD calculation. ${}^{177}\text{Lu}$ emits $\beta^-$ particles ($\overline{E}_\beta = 133\,\text{keV}$) and two principal gamma rays: 113 keV (6.2% per decay) and 208 keV (10.4% per decay).

(a) Calculate $\Delta_\beta$ (mean $\beta^-$ energy per decay in joules).

(b) Calculate $\Delta_{\gamma,113}$ and $\Delta_{\gamma,208}$ (mean gamma energy per decay for each line).

(c) For a tumor of mass 20 g, the absorbed fraction for $\beta^-$ is $\phi_\beta \approx 1.0$ and for both gamma lines $\phi_\gamma \approx 0.03$. Calculate the $S$-value $S(\text{tumor} \leftarrow \text{tumor})$.

(d) If the cumulated activity in the tumor is $\tilde{A} = 4 \times 10^{13}\,\text{decays}$, calculate the absorbed dose from both $\beta$ and $\gamma$ contributions. What fraction of the total dose comes from the gamma rays?

Theranostics and Clinical Applications

Problem 27.24 ⭐⭐ A theranostic protocol for prostate cancer uses ${}^{68}\text{Ga}$-PSMA-11 for PET imaging followed by ${}^{177}\text{Lu}$-PSMA-617 for therapy.

(a) List the nuclear decay properties of ${}^{68}\text{Ga}$ and ${}^{177}\text{Lu}$ that make them suitable for their respective roles.

(b) ${}^{68}\text{Ga}$ and ${}^{177}\text{Lu}$ are different elements. Explain why the chelator (DOTA or HBED-CC) is critical for ensuring that the biodistribution of the diagnostic and therapeutic agents is similar.

(c) An alternative approach uses isotopes of the same element: ${}^{64}\text{Cu}$ (PET, $t_{1/2} = 12.7\,\text{h}$, $\beta^+$ 17.5%) and ${}^{67}\text{Cu}$ ($\beta^-$ therapy, $t_{1/2} = 61.8\,\text{h}$). What is the advantage of this "matched-pair" approach over the ${}^{68}\text{Ga}$/${}^{177}\text{Lu}$ combination?

Problem 27.25 ⭐⭐ Compare the physical characteristics of the three main approaches to treating a 3 cm tumor at 8 cm depth:

(a) External photon beam (6 MV linac) (b) Proton beam (variable energy) (c) ${}^{177}\text{Lu}$-labeled targeted therapy

For each, describe: (i) the dose distribution in the body, (ii) the minimum number of treatment sessions, (iii) the types of tumors for which this approach is most appropriate.

Problem 27.26 ⭐⭐⭐ The global ${}^{99}\text{Mo}$ supply crisis. As of 2025, approximately 95% of the world's ${}^{99}\text{Mo}$ is produced by just five aging research reactors, several of which have experienced extended shutdowns.

(a) Explain why ${}^{99}\text{Mo}$ cannot be stockpiled (use a quantitative argument based on its half-life).

(b) The fission production route uses HEU targets. Calculate the number of fissions per week needed to supply a generator facility producing 10,000 patient doses/week, assuming 6.1% fission yield of ${}^{99}\text{Mo}$ and an average patient dose of 740 MBq (allow for a factor-of-5 loss from production to injection due to decay during processing and transport).

(c) Accelerator-based alternatives use the reaction ${}^{100}\text{Mo}(\gamma,n){}^{99}\text{Mo}$ with high-energy bremsstrahlung. Discuss the advantages and disadvantages of this approach compared to reactor production.

Research and Synthesis Problems

Problem 27.27 ⭐⭐⭐⭐ Flash radiotherapy. Recent research suggests that delivering the entire radiation dose in less than 1 second (dose rates $> 40\,\text{Gy/s}$, compared to conventional $\sim 0.03\,\text{Gy/s}$) dramatically reduces normal tissue damage while maintaining tumor-killing efficacy. This is called the "FLASH effect."

(a) At what dose rate (in Gy/s) does a conventional linac operate, delivering 2 Gy in 60 seconds?

(b) A FLASH proton beam must deliver 20 Gy to a tumor volume of 100 cm$^3$ in 0.5 seconds. Estimate the required beam current (protons per second) at the Bragg peak, assuming 200 MeV protons with a peak $dE/dx \approx 80\,\text{MeV}\cdot\text{cm}^2/\text{g}$.

(c) Current clinical proton beams operate at $\sim 1$–$5\,\text{nA}$. Comment on the technical challenge of achieving FLASH dose rates.

Problem 27.28 ⭐⭐⭐⭐ Design a theranostic protocol. A new ligand has been developed that binds with high specificity to fibroblast activation protein (FAP), which is overexpressed in the stroma of many solid tumors.

(a) Choose a diagnostic radionuclide for PET imaging and a therapeutic radionuclide for treatment. Justify your choices based on half-life, decay mode, and practical availability.

(b) Outline the clinical workflow: production, radiolabeling, patient imaging, patient selection criteria, therapy delivery, and post-therapy monitoring.

(c) Identify the dose-limiting organs likely to be at risk and explain how you would calculate their doses.

(d) Discuss whether an alpha-emitting therapeutic radionuclide (e.g., ${}^{225}\text{Ac}$) might be preferable to a beta emitter for this application. Under what circumstances would the alpha emitter be advantageous?

Problem 27.29 ⭐⭐⭐⭐ Literature review. Read the VISION trial (Sartor et al., New England Journal of Medicine 385:1091, 2021) or the NETTER-1 trial (Strosberg et al., NEJM 376:125, 2017).

(a) Summarize the trial design: What was the patient population? What was the intervention? What was the primary endpoint?

(b) Identify the nuclear physics at each step of the treatment: radionuclide production, molecular labeling, biodistribution, decay mechanism, and dose deposition.

(c) What was the magnitude of the clinical benefit? Express in terms of median progression-free survival and/or overall survival.

(d) Discuss the role of the diagnostic imaging companion (${}^{68}\text{Ga}$-PSMA-11 for VISION, ${}^{68}\text{Ga}$-DOTATATE for NETTER-1) in patient selection.

Problem 27.30 ⭐⭐⭐ Ethics and access. Proton therapy costs 2–3 times more than conventional photon therapy per course of treatment. Targeted alpha therapy with ${}^{225}\text{Ac}$ is limited by the global supply of actinium-225 (estimated total production capacity in 2024: $\sim 2\,\text{Ci/year}$, enough for only a few thousand patients).

(a) Estimate how many patients could be treated annually with the current global ${}^{225}\text{Ac}$ supply, assuming a therapeutic dose of 100 kBq/kg for a 70 kg patient, given every 8 weeks for 4 cycles.

(b) ${}^{225}\text{Ac}$ can be produced via the reaction ${}^{226}\text{Ra}(p,2n){}^{225}\text{Ac}$ in a cyclotron. Discuss the nuclear physics advantages and radiological safety challenges of this production route.

(c) Should proton therapy be reserved for pediatric patients and tumors near critical structures, where the dosimetric advantage is clearest? Or should it be available to all cancer patients? Construct a brief argument for each position, grounding your reasoning in the physics of Section 27.4.