Chapter 27 — Nuclear Medicine: Diagnosis, Therapy, and Theranostics

"The tracer principle is based on the fact that radioactive atoms are indistinguishable in their chemical behavior from stable atoms of the same element." — George de Hevesy, Nobel Lecture (1944)

Chapter Overview

Every day, in hospitals around the world, nuclear physics saves lives. A patient lies inside a PET scanner while fluorine-18 atoms — produced hours earlier in a cyclotron via the ${}^{18}\text{O}(p,n){}^{18}\text{F}$ reaction — undergo positron emission inside her body. Each positron annihilates with a nearby electron, producing two 511 keV photons that fly apart at almost exactly $180°$. A ring of scintillation detectors records these photon pairs in coincidence, and a computer reconstructs a three-dimensional image showing precisely where the radioactive glucose analog accumulated — in the tumor, which consumes glucose at ten times the rate of normal tissue. The oncologist studies the image. The tumor is localized. Treatment planning begins.

In another wing of the same hospital, a proton beam — accelerated to 230 MeV in a cyclotron — enters a patient's skull and deposits nearly all its energy in a sharp Bragg peak at the depth of a brain tumor, sparing the healthy tissue beyond. The nuclear physics is the same Bethe-Bloch formula you studied in Chapter 16, except now the target is a human being and the stakes could not be higher.

This chapter traces the nuclear physics through every step of modern nuclear medicine: from the production of medical radionuclides (Section 27.1), through diagnostic imaging with PET and SPECT (Sections 27.2–27.3), to radiation therapy with external beams (Section 27.4), sealed sources (Section 27.5), and targeted radionuclides that seek out tumor cells with molecular precision (Section 27.6). We conclude with the theranostic paradigm (Section 27.7) — the idea that a single molecular platform can carry a diagnostic isotope for imaging and a therapeutic isotope for treatment — and the dosimetry calculations (Section 27.8) that connect nuclear decay data to clinical radiation doses.

Throughout, we will connect every clinical application back to the fundamental nuclear physics of Parts I–IV. PET is positron emission (Chapter 14) plus pair annihilation (Chapter 16). The ${}^{99\text{m}}\text{Tc}$ generator is secular equilibrium (Chapter 12). Proton therapy is the Bragg peak (Chapter 16). Targeted alpha therapy is alpha decay (Chapter 13) with a biological homing device. Nuclear medicine is not a separate subject — it is nuclear physics, applied where it matters most.

🏥 Anchor Example Returns: The ${}^{18}\text{F}$-FDG / PET system was introduced as a teaser in Chapter 1 and has appeared in Chapter 12 (positron emission) and Chapter 16 (511 keV detection). In this chapter, it receives its full treatment — from the cyclotron target to the reconstructed image.

Spaced Review — Chapter 12 Connection: Recall that activity $A(t) = \lambda N(t) = A_0 e^{-\lambda t}$, that secular equilibrium requires $\lambda_{\text{parent}} \ll \lambda_{\text{daughter}}$, and that the daughter activity in secular equilibrium approaches the parent activity: $A_2 \approx A_1$. These results are essential for understanding the ${}^{99}\text{Mo}/{}^{99\text{m}}\text{Tc}$ generator (Section 27.3) and for all dosimetry calculations (Section 27.8).

Spaced Review — Chapter 16 Connection: Recall the Bethe-Bloch formula for charged-particle energy loss, the characteristic Bragg peak at end-of-range, and the three photon interaction mechanisms (photoelectric, Compton, pair production). Section 27.4 will apply these directly to understand why proton beams are superior to photon beams for many tumors.

27.1 Production of Medical Radionuclides

Every nuclear medicine procedure begins with a radionuclide — an unstable nucleus whose decay provides either the signal for imaging or the energy for therapy. The ideal medical radionuclide has a half-life matched to the clinical timescale (long enough for the procedure, short enough to limit patient dose), emits radiation of the right type and energy, and can be attached to a biologically relevant molecule. Producing such radionuclides in sufficient quantity and purity is itself a triumph of applied nuclear physics.

27.1.1 Cyclotron Production

Medical cyclotrons — compact machines with proton energies of 11–30 MeV, far smaller than the research accelerators of Chapter 30 — produce the short-lived positron emitters used in PET imaging. The key reactions are:

Let us examine the most important of these — the production of fluorine-18 — in detail.

The ${}^{18}\text{O}(p,n){}^{18}\text{F}$ reaction. A medical cyclotron accelerates protons to 11–18 MeV and directs them onto a target of ${}^{18}\text{O}$-enriched water (typically $>97\%$ enrichment). The nuclear reaction is:

$${}^{18}\text{O} + p \to {}^{18}\text{F} + n$$

The $Q$-value is:

$$Q = [M({}^{18}\text{O}) + M({}^{1}\text{H}) - M({}^{18}\text{F}) - m_n]c^2 = -2.44\,\text{MeV}$$

This is an endothermic reaction — the proton must supply at least 2.44 MeV in the center-of-mass frame to proceed. The threshold energy in the lab frame is:

$$T_{\text{thresh}} = -Q\left(1 + \frac{m_p}{m_{{}^{18}\text{O}}}\right) = 2.44 \times \frac{19}{18} = 2.58\,\text{MeV}$$

The cross section rises from threshold, peaks near 5 MeV ($\sigma \approx 500\,\text{mb}$), and then decreases. At 18 MeV, the thick-target yield is approximately 9 GBq/$\mu$A$\cdot$h (240 mCi/$\mu$A$\cdot$h), meaning a 50 $\mu$A beam running for two hours produces roughly 900 GBq (24 Ci) of ${}^{18}\text{F}$ — enough for dozens of patient scans, even accounting for the 110-minute half-life and the time required for radiochemistry and transport.

The ${}^{18}\text{F}$ is recovered from the target water, purified, and then incorporated into deoxyglucose through a rapid automated synthesis to produce ${}^{18}\text{F}$-fluorodeoxyglucose (FDG). The entire process — from cyclotron bombardment to quality-controlled, injectable FDG — takes approximately 90 minutes.

💡 Why ${}^{18}\text{F}$? Among PET radionuclides, ${}^{18}\text{F}$ occupies a sweet spot: its 110-minute half-life is long enough to allow synthesis, transport (to hospitals without cyclotrons), and a 60–90 minute imaging session, yet short enough that the patient's radiation exposure is modest. The positron kinetic energy is low (maximum 634 keV, mean 250 keV), giving a short positron range in tissue ($\sim 0.6\,\text{mm}$ rms) and hence excellent spatial resolution. And fluorine forms a strong bond to carbon, allowing stable attachment to biomolecules.

27.1.2 Reactor Production

Nuclear reactors — with their intense neutron fluxes ($10^{13}$–$10^{15}\,\text{n/cm}^2\cdot\text{s}$) — produce neutron-rich radionuclides through $(n,\gamma)$ and $(n,f)$ reactions. The most important reactor-produced medical radionuclide is ${}^{99}\text{Mo}$, the parent of the ${}^{99\text{m}}\text{Tc}$ generator (Section 27.3).

${}^{99}\text{Mo}$ production:

$${}^{98}\text{Mo}(n,\gamma){}^{99}\text{Mo} \quad \text{(neutron capture)}$$

or, for higher specific activity:

$${}^{235}\text{U}(n,f) \to {}^{99}\text{Mo} \quad \text{(fission product, yield} \approx 6.1\%)$$

The fission route produces carrier-free ${}^{99}\text{Mo}$ (no stable Mo mixed in), giving much higher specific activity — essential for compact generators. However, it requires handling irradiated highly enriched uranium (HEU), raising nonproliferation concerns (Chapter 28). The nuclear medicine community is actively transitioning to low-enriched uranium (LEU) targets and accelerator-based production methods.

Other reactor-produced therapeutic radionuclides include:

27.1.3 Generator Systems

For radionuclides with half-lives too short for shipping, a generator system provides the solution: a long-lived parent is shipped to the hospital, and the short-lived daughter is "milked" on demand. The physics is the decay chain equilibrium of Chapter 12 — applied to save lives.

The ${}^{68}\text{Ge}/{}^{68}\text{Ga}$ generator ($t_{1/2}^{\text{Ge}} = 270.8\,\text{d}$, $t_{1/2}^{\text{Ga}} = 67.7\,\text{min}$) is increasingly important for PET imaging of neuroendocrine tumors and prostate cancer, providing a PET radionuclide without a cyclotron.

The ${}^{99}\text{Mo}/{}^{99\text{m}}\text{Tc}$ generator — the workhorse of nuclear medicine — is so important that it deserves its own section (Section 27.3).

27.2 PET Imaging: Positron Annihilation Meets Coincidence Detection

Positron emission tomography is, in terms of nuclear physics, the most elegant of all medical imaging modalities. It exploits a fundamental symmetry of nature — the annihilation of a particle with its antiparticle — to produce images of exquisite sensitivity.

27.2.1 The Physics of Positron Emission and Annihilation

A proton-rich nucleus undergoes $\beta^+$ decay (Chapter 14):

$$p \to n + e^+ + \nu_e$$

The emitted positron travels a short distance through tissue (the positron range), losing kinetic energy through the same Coulomb interactions that govern charged-particle stopping in Chapter 16. For ${}^{18}\text{F}$, the maximum positron energy is 634 keV and the mean is 250 keV, giving a root-mean-square range in water of approximately 0.6 mm.

Once the positron has slowed to thermal energies, it encounters an electron and forms a short-lived bound state — positronium — which annihilates within $\sim 10^{-10}\,\text{s}$ (para-positronium, singlet state) or $\sim 10^{-7}\,\text{s}$ (ortho-positronium, triplet state; in tissue, pick-off annihilation reduces this to $\sim 10^{-10}\,\text{s}$). The annihilation produces photons:

$$e^+ + e^- \to 2\gamma \quad (\text{dominant: 99.7\% in tissue})$$

Conservation laws dictate the photon properties. Consider the annihilation in the center-of-mass frame of the nearly-at-rest $e^+e^-$ pair:

• Energy conservation: The total energy is $2m_e c^2 = 1.022\,\text{MeV}$, shared equally between two photons:

$$E_\gamma = m_e c^2 = 511\,\text{keV}$$

• Momentum conservation: With the pair nearly at rest, the two photons must have equal and opposite momenta. They are emitted back-to-back — at $180°$ to each other. (In practice, the residual momentum of the pair causes a small angular spread of $\pm 0.25°$, or about 0.5° FWHM, which contributes a spatial resolution limit of $\sim 2\,\text{mm}$ at the center of a typical PET scanner.)

• Timing: Both photons are produced simultaneously. This is the key to PET imaging.

🔬 Note on Three-Photon Annihilation: Ortho-positronium (triplet state, $S=1$) in vacuum annihilates into three photons (two-photon annihilation is forbidden by charge conjugation symmetry for $S=1$). In condensed matter, however, the positron preferentially undergoes "pick-off" annihilation with an electron of opposite spin from the surrounding medium, producing two photons. In tissue, the two-photon channel dominates overwhelmingly.

27.2.2 Coincidence Detection

A PET scanner consists of a ring (or multiple rings) of scintillation detectors — typically lutetium oxyorthosilicate (LSO) or lutetium-yttrium oxyorthosilicate (LYSO) crystals coupled to photomultipliers or silicon photomultipliers (SiPMs) — surrounding the patient.

When two detectors register 511 keV photons within a narrow coincidence time window (typically 2–10 ns), the electronics record a line of response (LOR): the annihilation event occurred somewhere along the line connecting the two detectors.

Accumulating millions of LORs from all detector pairs, reconstruction algorithms (filtered back-projection, or more commonly, iterative methods such as ordered subset expectation maximization, OSEM) produce a three-dimensional image of the positron-emitting radionuclide distribution.

The power of coincidence detection: Unlike SPECT (Section 27.3), PET does not require a physical collimator. The coincidence requirement acts as an electronic collimator — it selects only photon pairs that are geometrically consistent with a single annihilation event. This gives PET a fundamental sensitivity advantage of roughly 10–100 times over SPECT.

Types of coincidence events. Not all coincidence events are useful. The three categories are:

• True coincidences: Both photons from a single annihilation reach detectors without scattering. These carry accurate spatial information.
• Scattered coincidences: One or both photons undergo Compton scattering in the body before detection, causing the recorded LOR to be mispositioned. The fraction of scattered coincidences is typically 30–40% in 3D PET, and correction algorithms (based on the scatter physics of Chapter 16) are essential.
• Random (accidental) coincidences: Two photons from different annihilation events happen to arrive within the coincidence window. The random rate scales as $R_{\text{random}} = 2\tau S_1 S_2$, where $\tau$ is the coincidence window width and $S_1, S_2$ are the singles rates on the two detectors. Randoms are subtracted using a delayed-window technique.

Time-of-flight (TOF) PET. Modern PET scanners exploit the finite speed of light to further constrain the annihilation position. If detector A records a photon $\Delta t$ before detector B, the annihilation occurred not at the midpoint of the LOR but offset by:

$$\Delta x = \frac{c \, \Delta t}{2}$$

With current timing resolution of $\sim 200$–$400\,\text{ps}$, the position is localized to a $\sim 3$–$6\,\text{cm}$ segment of the LOR. While this does not directly improve spatial resolution (which is still limited by detector size and positron range), it dramatically improves the signal-to-noise ratio of the reconstructed image — equivalent to increasing the effective sensitivity by a factor of $D / \Delta x$, where $D$ is the patient diameter. For a 35 cm torso and 300 ps timing, the SNR improvement is $\sim 35 / 4.5 \approx 8 \times$, or an effective sensitivity gain of $\sim 60\times$. This is why TOF-PET enables faster scans, lower doses, and better image quality — all from a more precise measurement of when each photon arrives.

27.2.3 ${}^{18}\text{F}$-FDG: The Complete Physics

Let us trace the complete nuclear physics of an ${}^{18}\text{F}$-FDG PET scan, step by step — the anchor example that has been building since Chapter 1.

Step 1: Production. A medical cyclotron accelerates protons to $\sim$16 MeV. They strike a target of ${}^{18}\text{O}$-enriched water:

$${}^{18}\text{O}(p,n){}^{18}\text{F} \qquad Q = -2.44\,\text{MeV}$$

The ${}^{18}\text{F}^-$ ions produced in the water are trapped on an anion exchange resin, eluted, and used in an automated radiosynthesis to produce ${}^{18}\text{F}$-FDG. Total production-to-injection time: $\sim 2\,\text{hours}$.

Step 2: Biodistribution. FDG is a glucose analog — it enters cells via glucose transporters (GLUT1, GLUT3) and is phosphorylated by hexokinase to FDG-6-phosphate. Unlike normal glucose-6-phosphate, FDG-6-phosphate cannot proceed further through glycolysis (the 2-hydroxyl group is replaced by ${}^{18}\text{F}$, blocking the next enzymatic step). It is metabolically trapped in the cell. Tumor cells, which have upregulated glucose metabolism (the Warburg effect), accumulate FDG at rates 5–10 times higher than most normal tissues.

Step 3: Radioactive decay. The ${}^{18}\text{F}$ nucleus decays:

$${}^{18}\text{F} \to {}^{18}\text{O} + e^+ + \nu_e \qquad t_{1/2} = 109.8\,\text{min}, \quad Q = 0.634\,\text{MeV}$$

The positron is emitted with a continuous energy spectrum (Chapter 14), with $E_{\max} = 634\,\text{keV}$ and $\langle E \rangle \approx 250\,\text{keV}$.

Step 4: Positron thermalization and annihilation. The positron loses energy through Coulomb interactions with atomic electrons, traveling a mean distance of $\sim 0.6\,\text{mm}$ in tissue. It then annihilates with an electron:

$$e^+ + e^- \to \gamma(511\,\text{keV}) + \gamma(511\,\text{keV})$$

The two photons are emitted at $180° \pm 0.25°$.

Step 5: Detection. The 511 keV photons traverse the body (undergoing some attenuation and Compton scattering — corrections for which are essential and based on the photon interaction physics of Chapter 16) and reach opposing detectors in the PET ring. If both are detected within the coincidence window, a LOR is recorded.

Step 6: Reconstruction. From $\sim 10^8$ recorded coincidence events, iterative algorithms reconstruct a 3D image with spatial resolution of $\sim 3$–$5\,\text{mm}$ (limited by positron range, detector size, non-collinearity, and reconstruction).

Quantitative example. A typical FDG PET scan involves injection of $A_0 = 370\,\text{MBq}$ (10 mCi) of ${}^{18}\text{F}$-FDG. After an uptake period of 60 minutes, the residual activity is:

$$A(60\,\text{min}) = 370 \times e^{-0.693 \times 60/109.8} = 370 \times e^{-0.379} = 370 \times 0.685 = 253\,\text{MBq}$$

During a 20-minute scan starting at $t = 60\,\text{min}$, the total number of decays in the patient is:

$$N_{\text{decay}} = \int_{60}^{80} A(t)\,dt = \frac{A_0}{\lambda}\left[e^{-\lambda t_1} - e^{-\lambda t_2}\right]$$

$$= \frac{370 \times 10^6}{1.052 \times 10^{-4}\,\text{s}^{-1}}\left[e^{-0.379} - e^{-0.505}\right]$$

$$= 3.517 \times 10^{12} \times [0.685 - 0.604] = 3.517 \times 10^{12} \times 0.081 = 2.85 \times 10^{11}\,\text{decays}$$

Of these, about $2.85 \times 10^{11} \times 0.969 \approx 2.76 \times 10^{11}$ produce positrons (the rest decay by electron capture, which produces no annihilation radiation). Each annihilation produces a 511 keV photon pair. After attenuation in the body, geometric acceptance of the detector ring ($\sim 5$–$10\%$), and detector efficiency ($\sim 80\%$), the scanner records on the order of $10^8$ coincidence events — sufficient for a high-quality image.

🔗 Cross-Reference: The energy spectrum of the emitted positron follows the Fermi theory of $\beta$ decay (Chapter 14). The 511 keV photon energy is simply $m_e c^2$ — a direct measurement of the electron mass via pair annihilation.

27.2.4 Other PET Radionuclides

While ${}^{18}\text{F}$-FDG dominates clinical PET ($>90\%$ of all PET scans worldwide), other radionuclides offer unique advantages:

• ${}^{11}\text{C}$ ($t_{1/2} = 20.4\,\text{min}$): Can replace ${}^{12}\text{C}$ in any organic molecule without altering its chemistry. Used in neuroscience research (${}^{11}\text{C}$-raclopride for dopamine receptors, ${}^{11}\text{C}$-PiB for amyloid plaques). The short half-life demands an on-site cyclotron.

• ${}^{68}\text{Ga}$ ($t_{1/2} = 67.7\,\text{min}$): Available from a ${}^{68}\text{Ge}$/${}^{68}\text{Ga}$ generator (no cyclotron needed). Used as ${}^{68}\text{Ga}$-DOTATATE for neuroendocrine tumor imaging and ${}^{68}\text{Ga}$-PSMA-11 for prostate cancer imaging. The theranostic partner of ${}^{177}\text{Lu}$ (Section 27.7).

• ${}^{89}\text{Zr}$ ($t_{1/2} = 78.4\,\text{h}$): Long enough to track monoclonal antibodies, which take 2–5 days to accumulate at tumor sites. Used in immuno-PET to verify that a therapeutic antibody reaches its target before committing to expensive therapy.

• ${}^{82}\text{Rb}$ ($t_{1/2} = 1.27\,\text{min}$): From a ${}^{82}\text{Sr}/{}^{82}\text{Rb}$ generator. A potassium analog used for myocardial perfusion PET — the patient is scanned during infusion.

27.3 SPECT Imaging and the ${}^{99\text{m}}\text{Tc}$ Generator

27.3.1 The Physics of SPECT

Single-photon emission computed tomography (SPECT) uses gamma-emitting radionuclides and a rotating gamma camera equipped with a physical collimator — a lead plate with many small holes that select only photons traveling in specific directions. The camera records a 2D projection at each rotation angle, and tomographic algorithms reconstruct a 3D image.

Compared to PET, SPECT has lower sensitivity (the collimator rejects $>99.99\%$ of emitted photons) and lower spatial resolution ($\sim 8$–$12\,\text{mm}$ vs. $\sim 3$–$5\,\text{mm}$ for PET). But SPECT has two enormous practical advantages: the radionuclides are cheaper and more widely available, and the gamma cameras are less expensive than PET scanners.

27.3.2 The Ideal SPECT Radionuclide: ${}^{99\text{m}}\text{Tc}$

Technetium-99m is the single most important radionuclide in nuclear medicine. Over 30 million diagnostic procedures per year use ${}^{99\text{m}}\text{Tc}$-labeled compounds — roughly 80% of all nuclear medicine scans worldwide.

Why is ${}^{99\text{m}}\text{Tc}$ ideal?

1. Gamma energy: 140.5 keV. This is in the sweet spot for gamma camera imaging — high enough to escape the body without excessive attenuation, low enough to be efficiently stopped in the NaI(Tl) scintillation crystal of a gamma camera. For comparison, if the gamma energy were 30 keV (too soft), most photons would be absorbed before exiting the patient; if it were 500 keV (too hard), the collimator efficiency would plummet and the spatial resolution would degrade.

2. Half-life: 6.01 hours. Long enough for radiopharmaceutical preparation, injection, biodistribution, and a complete imaging study. Short enough that the patient's radiation exposure is modest. After 24 hours ($\sim 4$ half-lives), only $\sim 6\%$ of the injected activity remains.

3. Isomeric transition. The "m" in ${}^{99\text{m}}\text{Tc}$ stands for metastable — it is an excited nuclear state. The decay is an isomeric transition (Chapter 15):

$${}^{99\text{m}}\text{Tc} \xrightarrow{\text{IT}} {}^{99}\text{Tc} + \gamma(140.5\,\text{keV})$$

Because this is a nuclear de-excitation (not a $\beta$ or $\alpha$ decay), there are no emitted charged particles — the patient receives radiation dose only from the 140.5 keV gamma ray and the internal conversion electrons. This is the most "dose-efficient" possible imaging scenario: all the radiation is the signal.

4. Versatile chemistry. Technetium, a transition metal, forms stable complexes with a wide variety of ligands, allowing it to be attached to molecules that target specific organs: ${}^{99\text{m}}\text{Tc}$-MDP for bone scans, ${}^{99\text{m}}\text{Tc}$-sestamibi for cardiac perfusion, ${}^{99\text{m}}\text{Tc}$-MAG3 for kidney function, and many more.

27.3.3 The ${}^{99}\text{Mo}/{}^{99\text{m}}\text{Tc}$ Generator: Secular Equilibrium in Action

The generator is a beautiful application of the secular equilibrium you studied in Chapter 12.

The decay chain:

$${}^{99}\text{Mo} \xrightarrow[\beta^-]{t_{1/2} = 65.94\,\text{h}} {}^{99\text{m}}\text{Tc} \xrightarrow[\text{IT}]{t_{1/2} = 6.01\,\text{h}} {}^{99}\text{Tc} \xrightarrow[\beta^-]{t_{1/2} = 2.11 \times 10^5\,\text{yr}} {}^{99}\text{Ru}\,(\text{stable})$$

Since $t_{1/2}({}^{99}\text{Mo}) = 65.94\,\text{h} \gg t_{1/2}({}^{99\text{m}}\text{Tc}) = 6.01\,\text{h}$, the ratio $\lambda_1/\lambda_2 \ll 1$ and the system reaches secular equilibrium (Chapter 12, Section 12.6). In secular equilibrium, the daughter activity equals the parent activity:

$$A_{{}^{99\text{m}}\text{Tc}}(t) \approx A_{{}^{99}\text{Mo}}(t) \quad \text{(after several daughter half-lives)}$$

More precisely, the approach to secular equilibrium for the daughter activity after a complete separation (milking) at $t = 0$ is:

$$A_{{}^{99\text{m}}\text{Tc}}(t) = \frac{\lambda_2}{\lambda_2 - \lambda_1} A_{{}^{99}\text{Mo}}(0) \left(e^{-\lambda_1 t} - e^{-\lambda_2 t}\right)$$

Since the branching ratio of ${}^{99}\text{Mo}$ decay to the metastable state is 87.6% (the remaining 12.4% goes directly to the ground state of ${}^{99}\text{Tc}$), the effective yield is:

$$A_{{}^{99\text{m}}\text{Tc}}(t) = 0.876 \times \frac{\lambda_2}{\lambda_2 - \lambda_1} A_{{}^{99}\text{Mo}}(0) \left(e^{-\lambda_1 t} - e^{-\lambda_2 t}\right)$$

The ${}^{99\text{m}}\text{Tc}$ activity reaches its maximum at:

$$t_{\max} = \frac{\ln(\lambda_2/\lambda_1)}{\lambda_2 - \lambda_1} = \frac{\ln(65.94/6.01)}{(0.1155 - 0.01051)\,\text{h}^{-1}} = \frac{2.396}{0.1050\,\text{h}^{-1}} \approx 22.8\,\text{h}$$

The physical generator is a small, shielded column of alumina ($\text{Al}_2\text{O}_3$) onto which ${}^{99}\text{Mo}$ (as molybdate, $\text{MoO}_4^{2-}$) is adsorbed. When the column is "milked" by passing sterile saline solution through it, the ${}^{99\text{m}}\text{Tc}$ (as pertechnetate, $\text{TcO}_4^-$) is eluted while the ${}^{99}\text{Mo}$ remains bound. The generator can be milked every 6–24 hours for about 1 week (limited by the 66-hour half-life of ${}^{99}\text{Mo}$).

Numerical example. A generator is delivered on Monday morning with $A_{{}^{99}\text{Mo}}(0) = 20\,\text{GBq}$. It was last milked 24 hours prior. The ${}^{99\text{m}}\text{Tc}$ activity available at elution is:

$$A_{{}^{99\text{m}}\text{Tc}}(24\,\text{h}) = 0.876 \times \frac{0.1155}{0.1050} \times 20\,\text{GBq} \times \left(e^{-0.01051 \times 24} - e^{-0.1155 \times 24}\right)$$

$$= 0.876 \times 1.100 \times 20 \times (0.777 - 0.0633) = 0.963 \times 20 \times 0.714 = 13.8\,\text{GBq}$$

This is enough for approximately 20–40 patient doses (typical dose: 370–740 MBq per scan).

27.3.4 Other SPECT Radionuclides

• ${}^{201}\text{Tl}$ ($t_{1/2} = 72.9\,\text{h}$, 68–80 keV X-rays): Potassium analog for myocardial perfusion imaging. Largely replaced by ${}^{99\text{m}}\text{Tc}$-sestamibi but still used in some protocols.

• ${}^{123}\text{I}$ ($t_{1/2} = 13.2\,\text{h}$, 159 keV gamma): Thyroid imaging, DaTscan for Parkinson's disease (dopamine transporter imaging). Cyclotron-produced.

• ${}^{111}\text{In}$ ($t_{1/2} = 2.80\,\text{d}$, 171 and 245 keV gammas): Infection imaging, tumor receptor studies. Being replaced by ${}^{68}\text{Ga}$ and ${}^{177}\text{Lu}$ in theranostic protocols.

27.4 External Beam Radiation Therapy: Photons, Protons, and Heavy Ions

27.4.1 The Goal

Radiation therapy exploits a fundamental asymmetry: rapidly dividing tumor cells are, on average, less efficient at repairing DNA damage than normal tissue. By delivering a lethal dose to the tumor while minimizing the dose to surrounding healthy tissue, we can kill the cancer while the patient survives.

The nuclear physics challenge is dose conformity: depositing the prescribed dose in the tumor volume and as little as possible elsewhere. This is where the physics of radiation interactions with matter (Chapter 16) becomes a matter of life and death.

27.4.2 Photon Therapy

Clinical photon beams are produced by linear accelerators (linacs) that accelerate electrons to 6–18 MeV and direct them onto a high-$Z$ target (tungsten), producing bremsstrahlung X-rays with a continuous spectrum. The maximum photon energy equals the electron kinetic energy.

The dose-depth profile of a megavoltage photon beam in water (or tissue) has a characteristic shape:

1. Build-up region (0 to $d_{\max}$): Dose increases with depth as forward-scattered secondary electrons build up. For a 6 MV beam, $d_{\max} \approx 1.5\,\text{cm}$; for 18 MV, $d_{\max} \approx 3.5\,\text{cm}$.

2. Exponential attenuation: Beyond $d_{\max}$, the dose decreases approximately exponentially. For a 6 MV beam, the depth at which the dose drops to 50% of maximum is approximately 15 cm.

The problem is immediately apparent: to treat a deep-seated tumor, the photon beam must traverse a large volume of healthy tissue, depositing dose all along the way. Even with multiple beam angles and intensity modulation (IMRT), there is always an entrance dose and an exit dose.

27.4.3 Proton Therapy: The Bragg Peak Advantage

Protons interact with matter through the same Coulomb interactions described by the Bethe-Bloch formula (Chapter 16):

$$-\frac{dE}{dx} = \frac{4\pi z^2 e^4 n_e}{m_e v^2}\left[\ln\frac{2m_e v^2}{I} - \ln(1-\beta^2) - \beta^2\right]$$

where $z = 1$ for protons, $n_e$ is the electron density, $v$ is the proton velocity, and $I$ is the mean excitation energy of the medium. Two features of this formula shape the clinical advantage:

1. The $1/v^2$ dependence: As the proton slows down, $dE/dx$ increases. The proton deposits more and more energy per unit path length as it approaches the end of its range.

2. The finite range: Unlike photons, which are exponentially attenuated (some always penetrate to arbitrary depth), protons have a well-defined range $R$ determined by their initial kinetic energy. Beyond $R$, the dose is essentially zero.

Together, these produce the Bragg peak: a sharp maximum in dose deposition at the end of the proton's range, followed by an abrupt drop to zero.

Clinical proton energies and ranges in water:

For a 200 MeV proton beam, the range in water is approximately 26 cm — sufficient to reach deep tumors. The Bragg peak width (a few millimeters to ~1 cm) is much smaller than the tumor, so a clinical treatment uses a spread-out Bragg peak (SOBP): the beam energy is modulated to superimpose Bragg peaks at different depths, creating a uniform dose distribution across the tumor volume.

⚠️ The range formula. A useful approximation for the range $R$ of protons in water (valid for $T \lesssim 250\,\text{MeV}$) is:

$$R \approx 0.0022 \times T^{1.77}\,\text{cm}$$

where $T$ is in MeV. This follows from the power-law energy dependence of the stopping power. For carbon ions, the range scales as $R_C \approx R_p \times (A_C / A_p) \times (z_p / z_C)^2 \approx R_p / 3$ at the same velocity (same energy per nucleon).

Why protons are better than photons for many tumors: Consider a 5 cm tumor centered at 10 cm depth. A photon beam delivers dose throughout its entire path — from the skin surface through the tumor and beyond. The integral dose to healthy tissue is large. A proton beam, with its Bragg peak placed at 10 cm depth and energy-modulated to cover the 5 cm tumor width, delivers a high dose to the tumor, a lower entrance dose, and no exit dose. The integral dose to healthy tissue can be reduced by a factor of 2–3 compared to photons.

Integral dose comparison — a quantitative example. Consider a 5 cm tumor centered at 15 cm depth, treated with a single anterior field. For a 6 MV photon beam normalized to deliver 2 Gy at the tumor center, the entrance dose is approximately 1.7 Gy and the exit dose at 30 cm depth is approximately 0.5 Gy. The integral dose (total energy deposited in the patient) can be estimated by integrating the dose-depth curve over the beam cross-section:

$$E_{\text{integral, photon}} = \rho \int_0^{L} D(z) \, A_{\text{beam}} \, dz$$

For a $10 \times 10\,\text{cm}^2$ field and a 30 cm patient thickness, the integral dose is approximately $\sim 4\,\text{J}$. For a proton beam with a spread-out Bragg peak covering the same tumor, the entrance dose is approximately 1.0 Gy, the tumor dose is 2.0 Gy, and the exit dose is zero. The integral dose is approximately $\sim 2\,\text{J}$ — a reduction by roughly half. With multiple beam angles, the reduction is even greater.

This factor-of-two reduction in integral dose translates directly into clinical outcomes: fewer second cancers (the risk of radiation-induced malignancy scales roughly linearly with integral dose), less fatigue during treatment, and better quality of life. For a 5-year-old child treated for a brain tumor who may live another 70 years, the cumulative risk reduction from halving the integral dose is substantial.

This matters most for: - Pediatric cancers: Children's growing tissues are sensitive to radiation; reducing dose to healthy organs limits long-term side effects (second cancers, growth impairment, cognitive effects). The Children's Oncology Group now recommends proton therapy as the standard of care for many pediatric tumors. - Tumors near critical structures: Brain tumors near the brainstem, spinal cord tumors, base-of-skull tumors where millimeters matter. - Re-irradiation: When a previously irradiated region must be treated again and dose tolerance is limited.

27.4.4 Carbon Ion Therapy: Higher LET, Higher RBE

Carbon ions (${}^{12}\text{C}^{6+}$) carry the Bragg peak advantage further. Because $dE/dx \propto z^2$ (Bethe-Bloch), carbon ions ($z=6$) have a stopping power $6^2/1^2 = 36$ times that of protons at the same velocity. This produces:

1. A sharper Bragg peak: The lateral and longitudinal dose fall-off is steeper, improving dose conformity.

2. Higher linear energy transfer (LET): At the Bragg peak, carbon ions deposit energy at a rate of $\sim 100$–$200\,\text{keV}/\mu\text{m}$, compared to $\sim 5$–$10\,\text{keV}/\mu\text{m}$ for protons at the same depth. This dense ionization track produces complex, clustered DNA damage — multiple strand breaks within a few nanometers — that is far more difficult for the cell to repair.

3. Higher relative biological effectiveness (RBE): The RBE is the ratio of photon dose to particle dose required to produce the same biological effect:

$$\text{RBE} = \frac{D_{\text{photon}}}{D_{\text{ion}}}\bigg|_{\text{same effect}}$$

For protons, RBE $\approx 1.1$ (a clinical convention — the actual value varies with LET, dose, and biological endpoint). For carbon ions at the Bragg peak, RBE $\approx 2$–$4$, meaning the biological effect is 2–4 times greater than the same physical dose of photons.

4. Reduced oxygen enhancement ratio (OER): Hypoxic tumor cells (common in large tumors with poor blood supply) are resistant to low-LET radiation because the oxygen fixation mechanism is needed to make DNA damage permanent. High-LET carbon ions are effective against hypoxic cells regardless — the OER drops from $\sim 3$ (photons) to $\sim 1.5$ (carbon ions at high LET).

The trade-off: Carbon ion therapy requires larger, more expensive accelerators (synchrotrons delivering $\sim 430\,\text{MeV/nucleon}$, compared to $\sim 230\,\text{MeV}$ cyclotrons for protons). There are approximately 15 carbon ion therapy centers worldwide (compared to $\sim 120$ for proton therapy), primarily in Japan, Germany, Italy, Austria, and China. The clinical evidence is strongest for radio-resistant tumors: chordomas, chondrosarcomas, hepatocellular carcinoma, locally advanced pancreatic cancer, and some sarcomas.

📊 A comparison of clinical modalities:

Property	Photons (6 MV)	Protons (200 MeV)	Carbon Ions (350 MeV/n)
Dose profile	Exponential falloff	Bragg peak	Sharp Bragg peak
Exit dose	Yes	No	No (minimal fragmentation tail)
LET at tumor (keV/$\mu$m)	0.2–2	2–10	50–200
RBE (at tumor)	1.0 (reference)	$\sim$1.1	2–4
OER	$\sim$3.0	$\sim$2.5	$\sim$1.5
Facilities worldwide	$\sim$15,000	$\sim$120	$\sim$15
Cost per treatment room	\$2–5M | \$25–50M	\$50–100M

27.4.5 Nuclear Fragmentation: A Complication for Heavy Ions

When a carbon ion undergoes a nuclear interaction with a target nucleus in tissue, it can fragment into lighter ions (boron, beryllium, lithium, helium, hydrogen) plus neutrons. These fragments travel beyond the Bragg peak of the primary beam, creating a low-dose "tail" beyond the tumor. For carbon ions, the fragmentation tail contributes $\sim 10$–$15\%$ of the peak dose at depths beyond the Bragg peak. This is a trade-off that must be accounted for in treatment planning — the dose beyond the Bragg peak is not truly zero, unlike the idealized single-particle picture.

The nuclear fragmentation cross sections are calculated using models from nuclear reaction theory (Chapter 17) — specifically, the abrasion-ablation model and Glauber theory adapted for heavy-ion collisions. This is nuclear physics directly informing clinical treatment plans.

27.5 Brachytherapy: Sealed Sources Inside the Patient

Brachytherapy ("short-distance therapy," from Greek brachys) places sealed radioactive sources directly in or adjacent to the tumor, delivering a high local dose while sparing distant tissues through the $1/r^2$ geometric attenuation.

27.5.1 Physics Principles

The dose rate at distance $r$ from a point source of activity $A$ emitting photons of energy $E_\gamma$ is:

$$\dot{D}(r) = \frac{A \, \Gamma}{r^2}$$

where $\Gamma$ is the dose-rate constant (or air-kerma rate constant), which depends on the photon energy spectrum and can be calculated from nuclear decay data:

$$\Gamma = \frac{1}{4\pi}\sum_i f_i \, E_i \left(\frac{\mu_{\text{en}}}{\rho}\right)_{E_i}$$

where $f_i$ is the frequency (photons per decay) and $(\mu_{\text{en}}/\rho)_{E_i}$ is the mass energy-absorption coefficient of tissue at energy $E_i$.

For realistic source geometries (cylindrical seeds, wires), the dose calculation uses the TG-43 formalism (AAPM Task Group 43), which parameterizes the dose distribution using tabulated factors derived from Monte Carlo simulations.

27.5.2 Clinical Brachytherapy Sources

Prostate brachytherapy with ${}^{125}\text{I}$ seeds is one of the most common applications. Approximately 50–100 tiny titanium-encapsulated seeds (each $4.5\,\text{mm}$ long, $0.8\,\text{mm}$ diameter, containing 0.3–0.5 mCi of ${}^{125}\text{I}$) are implanted directly into the prostate gland under ultrasound guidance. The low-energy photons (27–35 keV) from the electron capture decay of ${}^{125}\text{I}$ are absorbed within a few centimeters, confining the therapeutic dose to the prostate and immediate surroundings. Over a period of several months ($\sim 6$ half-lives $= 1$ year), the seeds deliver a cumulative dose of $\sim 145\,\text{Gy}$ to the prostate — lethal to the tumor but tolerable because the dose rate is low (the cells have time to repair sublethal damage, and prostate cancer cells are particularly slow at repair).

High dose-rate (HDR) brachytherapy with ${}^{192}\text{Ir}$ uses a single, intensely radioactive source ($\sim 370\,\text{GBq}$, 10 Ci) that is driven by a computer-controlled wire through catheters placed in or near the tumor. The source dwells at programmed positions for calculated times, sculpting the dose distribution to match the tumor shape. A typical treatment takes 10–20 minutes, and several fractions are delivered over days.

27.6 Targeted Radionuclide Therapy: Molecular Missiles

The idea is simple and powerful: attach a radioactive atom to a molecule that specifically seeks out tumor cells, inject it into the patient, and let the molecule carry the radiation directly to every tumor deposit in the body — including metastases that surgery cannot reach and external beams cannot target.

27.6.1 ${}^{131}\text{I}$ for Thyroid Cancer: The Original Targeted Therapy

Radioiodine therapy for thyroid cancer, introduced by Saul Hertz and Arthur Roberts in 1941, is the oldest and most successful targeted radionuclide therapy. The targeting mechanism requires no engineered molecule — it exploits the fact that the thyroid gland naturally concentrates iodine via the sodium-iodide symporter (NIS).

${}^{131}\text{I}$ decays by $\beta^-$ emission:

$${}^{131}\text{I} \to {}^{131}\text{Xe} + e^- + \bar{\nu}_e \qquad t_{1/2} = 8.02\,\text{d}$$

The maximum $\beta^-$ energy is 606 keV (mean 182 keV), giving a mean range in tissue of $\sim 0.4\,\text{mm}$ — comparable to the diameter of thyroid follicles. The beta particles deposit their energy locally, destroying the thyroid tissue that accumulated the iodine. ${}^{131}\text{I}$ also emits a 364 keV gamma ray (81.7% per decay), which allows post-therapy SPECT imaging to verify the distribution of the therapeutic agent — an early form of theranostics.

Clinical protocol: After surgical removal of the primary thyroid tumor (thyroidectomy), patients receive a large oral dose of ${}^{131}\text{I}$ (typically 1.1–7.4 GBq, or 30–200 mCi). The radioiodine is absorbed from the gut, enters the blood, and is concentrated by any remaining thyroid cells — including distant metastases. The $\beta^-$ particles destroy these cells. Five-year survival rates for well-differentiated thyroid cancer exceed 95%, in large part because of this elegant application of nuclear physics.

27.6.2 Peptide Receptor Radionuclide Therapy: ${}^{177}\text{Lu}$-DOTATATE

Neuroendocrine tumors (NETs) overexpress somatostatin receptors on their cell surfaces. By labeling a somatostatin analog (the peptide DOTATATE) with ${}^{177}\text{Lu}$, we create a molecular missile that binds to these receptors and is internalized by the tumor cell, delivering the $\beta^-$ radiation directly to the cancer.

${}^{177}\text{Lu}$ decays by $\beta^-$ emission:

$${}^{177}\text{Lu} \to {}^{177}\text{Hf} + e^- + \bar{\nu}_e \qquad t_{1/2} = 6.65\,\text{d}$$

The $\beta^-$ endpoint energy is 498 keV (mean 133 keV), giving a mean tissue range of $\sim 0.3\,\text{mm}$ — ideal for irradiating small tumor deposits. ${}^{177}\text{Lu}$ also emits gamma rays at 113 keV (6.2%) and 208 keV (10.4%), allowing SPECT imaging for dosimetry and treatment monitoring.

The NETTER-1 clinical trial (2017) demonstrated that ${}^{177}\text{Lu}$-DOTATATE (Lutathera) significantly improved progression-free survival in patients with advanced midgut NETs, compared to standard therapy. FDA approval followed in 2018.

27.6.3 PSMA-Targeted Therapy: ${}^{177}\text{Lu}$-PSMA-617

Prostate-specific membrane antigen (PSMA) is overexpressed on prostate cancer cells. ${}^{177}\text{Lu}$-PSMA-617 — a small molecule labeled with ${}^{177}\text{Lu}$ that binds to PSMA — was approved by the FDA in 2022 (as Pluvicto) for metastatic castration-resistant prostate cancer, based on the VISION trial showing a significant survival benefit.

The theranostic pairing is explicit: patients are first imaged with ${}^{68}\text{Ga}$-PSMA-11 (PET) to confirm that their tumors express PSMA. Only PSMA-positive patients proceed to ${}^{177}\text{Lu}$-PSMA-617 therapy. This is personalized medicine built on nuclear physics.

27.6.4 Targeted Alpha Therapy: The Most Precise Cancer Treatment

Alpha particles are uniquely effective at killing cells:

• High LET: $\sim 100\,\text{keV}/\mu\text{m}$ — each alpha traversing a cell nucleus deposits $\sim 0.5$–$1\,\text{MeV}$ in the nucleus (diameter $\sim 5$–$10\,\mu\text{m}$).
• Short range: $50$–$80\,\mu\text{m}$ in tissue — a few cell diameters. The alpha particle kills the targeted cell and perhaps its immediate neighbors, but cells further away are untouched.
• High RBE: 3–7 (depending on the endpoint), compared to 1.0 for photons.
• Independent of oxygen: High-LET alpha particles produce irreparable clustered DNA damage regardless of oxygenation status (OER $\approx 1$).

The physics is clear: if you can deliver an alpha-emitting radionuclide specifically to tumor cells, the alphas will kill those cells with extraordinary efficiency while leaving neighboring healthy cells unharmed. This is the promise of targeted alpha therapy (TAT).

Microdosimetry of alpha particles. The conventional macroscopic quantity "absorbed dose" (gray) loses much of its meaning at the cellular scale for alpha particles. An alpha particle traversing a cell nucleus of diameter $d \approx 8\,\mu\text{m}$ deposits approximately $E_{\text{dep}} = \text{LET} \times d \approx 90 \times 8 = 720\,\text{keV}$. The mass of the cell nucleus is approximately $m_{\text{nuc}} \approx 5 \times 10^{-13}\,\text{kg}$. The specific energy (dose) from a single traversal is therefore:

$$z_1 = \frac{E_{\text{dep}}}{m_{\text{nuc}}} = \frac{720 \times 1.602 \times 10^{-16}\,\text{J}}{5 \times 10^{-13}\,\text{kg}} \approx 0.23\,\text{Gy}$$

Since the lethal dose for a single cell is typically 2–8 Gy (depending on cell type and repair capacity), and DNA double-strand breaks from high-LET radiation are largely irreparable, just 1–3 alpha traversals through the nucleus are sufficient to guarantee cell death. This is radically different from low-LET radiation, where thousands of photon interactions are needed to deliver the same lethal dose — and many of the individual DNA breaks are repaired between fractions. For alpha therapy, the question is not "how much dose?" but "did an alpha particle cross this cell's nucleus?" — a fundamentally stochastic, single-event question that demands microdosimetric analysis rather than conventional macroscopic dosimetry.

Key alpha-emitting therapeutic radionuclides:

Actinium-225 has emerged as one of the most promising alpha emitters. Its decay chain produces four alpha particles before reaching stable ${}^{209}\text{Bi}$:

$${}^{225}\text{Ac} \xrightarrow{\alpha} {}^{221}\text{Fr} \xrightarrow{\alpha} {}^{217}\text{At} \xrightarrow{\alpha} {}^{213}\text{Bi} \xrightarrow{\beta^-/\alpha} {}^{209}\text{Tl}/{}^{209}\text{Pb} \to {}^{209}\text{Bi}$$

Four alpha particles per decay means four times the cell-killing potential per atom — but also four chances for radioactive daughters to redistribute away from the tumor before they decay, causing off-target toxicity. This "daughter redistribution problem" is one of the central challenges in TAT and is a direct consequence of nuclear physics: the recoil energy from alpha emission ($\sim 100\,\text{keV}$, far exceeding any chemical bond energy of $\sim 1$–$5\,\text{eV}$) breaks the daughter free from whatever molecule the parent was attached to.

💡 The Recoil Problem, Quantitatively. When ${}^{225}\text{Ac}$ ($A = 225$) emits an alpha particle ($A = 4$) with kinetic energy $T_\alpha = 5.83\,\text{MeV}$, conservation of momentum gives the daughter ${}^{221}\text{Fr}$ a recoil energy of:

$$T_{\text{recoil}} = T_\alpha \times \frac{m_\alpha}{m_{\text{daughter}}} = 5.83 \times \frac{4}{221} = 0.106\,\text{MeV} = 106\,\text{keV}$$

This is roughly $10^4$–$10^5$ times the strength of any chemical bond. The daughter atom is ejected from its molecular carrier with certainty. Managing these free daughters — confining them near the tumor through encapsulation in nanoparticles, or choosing short-lived daughters that decay before they migrate far — is an active area of research at the intersection of nuclear physics, radiochemistry, and nanotechnology.

${}^{225}\text{Ac}$-PSMA-617 (the alpha-emitting analog of ${}^{177}\text{Lu}$-PSMA-617) has shown remarkable efficacy in compassionate-use cases for metastatic prostate cancer, achieving biochemical responses in patients who had exhausted all other options. Clinical trials are expanding rapidly.

27.7 Theranostics: See It and Treat It

The theranostic paradigm unifies diagnosis and therapy in a single molecular platform. The concept is straightforward: if a molecule specifically targets a tumor, label it with a diagnostic radionuclide (for PET or SPECT imaging) to locate the disease, and then label the same molecule with a therapeutic radionuclide (a $\beta^-$ or $\alpha$ emitter) to treat it.

27.7.1 The Theranostic Pairs

27.7.2 The Nuclear Physics Logic

Theranostics is not merely a clinical convenience — it reflects a deep nuclear physics principle. The molecular targeting platform (the ligand that binds to the tumor) determines where the radionuclide goes. The nuclear decay properties determine what it does when it gets there:

• Diagnostic isotope: Emits penetrating radiation (positrons for PET, gamma rays for SPECT) that exits the body and reaches external detectors. Minimal energy deposition in tissue.

• Therapeutic isotope: Emits short-range, high-LET radiation ($\beta^-$ particles with range $\sim 0.1$–$1\,\text{mm}$, or $\alpha$ particles with range $\sim 50$–$80\,\mu\text{m}$) that deposits all its energy in or near the targeted cell. Maximum local cell killing.

The ideal theranostic pair often involves isotopes of the same element (e.g., ${}^{123}\text{I}$/${}^{131}\text{I}$, ${}^{64}\text{Cu}$/${}^{67}\text{Cu}$) because identical chemistry guarantees identical biodistribution. When isotopes of different elements are used (e.g., ${}^{68}\text{Ga}$/${}^{177}\text{Lu}$), the chelator (the molecular cage that holds the metal ion) must be carefully designed so that the biodistribution is the same for both.

27.7.3 Personalized Dosimetry Through Imaging

The diagnostic scan provides more than a pretty picture — it enables patient-specific dosimetry. By quantifying the radioactivity concentration in each tumor and each organ from the PET or SPECT image, and combining this with the known decay properties of the therapeutic radionuclide, physicists can calculate the expected dose to each tumor and to each dose-limiting organ (typically kidneys and bone marrow) before therapy begins.

This is the medical equivalent of measuring the fission cross section before designing a reactor: you characterize the system first, then optimize the intervention. The dosimetry formalism is described in Section 27.8.

27.8 Dosimetry: From Nuclear Data to Patient Dose

27.8.1 Absorbed Dose

The absorbed dose $D$ is the energy deposited per unit mass of tissue:

$$D = \frac{dE}{dm} \qquad \text{[unit: gray (Gy) = J/kg]}$$

For external beams, the dose is determined by the beam parameters and the physics of radiation transport. For internal radionuclide therapy, the dose depends on:

1. How much radioactivity accumulates in each tissue (the biodistribution).
2. How long it stays there (determined by both physical decay and biological clearance).
3. How much energy the emitted radiation deposits locally versus escaping (the absorbed fractions).

27.8.2 Cumulated Activity and Residence Time

The total number of decays in a source region $r_S$ over all time is the cumulated activity (or time-integrated activity):

$$\tilde{A}(r_S) = \int_0^{\infty} A(r_S, t) \, dt$$

If the activity in the source region follows a multi-exponential model:

$$A(r_S, t) = \sum_i A_i \, e^{-\lambda_{\text{eff},i} \, t}$$

where $\lambda_{\text{eff},i} = \lambda_{\text{phys}} + \lambda_{\text{biol},i}$ is the effective decay constant (physical plus biological clearance), then:

$$\tilde{A}(r_S) = \sum_i \frac{A_i}{\lambda_{\text{eff},i}}$$

The residence time is $\tau(r_S) = \tilde{A}(r_S) / A_0$, where $A_0$ is the total administered activity.

27.8.3 The MIRD Formalism

The Medical Internal Radiation Dose (MIRD) formalism provides a systematic framework for calculating absorbed dose from distributed radionuclide sources:

$$\boxed{D(r_T) = \sum_{r_S} \tilde{A}(r_S) \times S(r_T \leftarrow r_S)}$$

where: - $D(r_T)$ is the absorbed dose to target region $r_T$ - $\tilde{A}(r_S)$ is the cumulated activity in source region $r_S$ - $S(r_T \leftarrow r_S)$ is the $S$-value: the dose to target $r_T$ per unit cumulated activity in source $r_S$

The $S$-value encodes all the nuclear physics and radiation transport:

$$S(r_T \leftarrow r_S) = \frac{1}{m_{r_T}} \sum_i \Delta_i \, \phi_i(r_T \leftarrow r_S)$$

where: - $m_{r_T}$ is the mass of the target region - $\Delta_i = E_i \times n_i$ is the mean energy emitted per decay for radiation type $i$ (energy $E_i$, yield $n_i$ per decay) - $\phi_i(r_T \leftarrow r_S)$ is the absorbed fraction: the fraction of energy emitted from $r_S$ that is absorbed in $r_T$

For $\beta^-$ particles and alpha particles with ranges shorter than the target organ dimensions, $\phi \approx 1$ (all energy absorbed locally). For penetrating gamma rays, $\phi$ is typically $\ll 1$ and must be calculated via Monte Carlo simulation.

27.8.4 Dosimetry Example: ${}^{177}\text{Lu}$-DOTATATE

A patient with neuroendocrine tumors receives $A_0 = 7.4\,\text{GBq}$ of ${}^{177}\text{Lu}$-DOTATATE. Post-therapy SPECT imaging (using the 208 keV gamma ray) reveals that approximately 3% of the injected activity localizes in a liver tumor ($m = 50\,\text{g}$) and remains with a biological half-life of $t_{1/2,\text{biol}} = 96\,\text{h}$.

Step 1: Effective half-life.

$$t_{1/2,\text{eff}} = \frac{t_{1/2,\text{phys}} \times t_{1/2,\text{biol}}}{t_{1/2,\text{phys}} + t_{1/2,\text{biol}}} = \frac{159.6 \times 96}{159.6 + 96} = \frac{15,322}{255.6} = 59.9\,\text{h}$$

$$\lambda_{\text{eff}} = \frac{\ln 2}{59.9\,\text{h}} = 0.01157\,\text{h}^{-1}$$

Step 2: Cumulated activity.

$$\tilde{A}_{\text{tumor}} = \frac{A_{\text{tumor}}(0)}{\lambda_{\text{eff}}} = \frac{0.03 \times 7.4 \times 10^9\,\text{Bq}}{0.01157\,\text{h}^{-1} \times (1/3600)\,\text{s}^{-1}/\text{h}^{-1}}$$

$$= \frac{2.22 \times 10^8}{3.214 \times 10^{-6}} = 6.91 \times 10^{13}\,\text{decays}$$

Step 3: Energy deposited by $\beta^-$ particles. The mean $\beta^-$ energy per decay of ${}^{177}\text{Lu}$ is $\overline{E}_\beta = 133\,\text{keV} = 2.13 \times 10^{-14}\,\text{J}$. Assuming all beta energy is absorbed locally ($\phi_\beta \approx 1$ for a 50 g tumor):

$$E_{\text{dep}} = \tilde{A} \times \overline{E}_\beta = 6.91 \times 10^{13} \times 2.13 \times 10^{-14}\,\text{J} = 1.472\,\text{J}$$

Step 4: Absorbed dose.

$$D = \frac{E_{\text{dep}}}{m} = \frac{1.472\,\text{J}}{0.050\,\text{kg}} = 29.4\,\text{Gy}$$

Over 4 treatment cycles (the standard protocol), the cumulative tumor dose is approximately $4 \times 29 \approx 117\,\text{Gy}$ — well above the threshold for tumor control. The kidneys (the dose-limiting organ) typically receive 3–5 Gy per cycle, staying below tolerance limits.

⚠️ Important Caveat. The calculation above is simplified — it assumes instantaneous uptake, monoexponential clearance, and complete local absorption. Clinical dosimetry uses multi-compartment pharmacokinetic models, voxel-level dose calculations from SPECT/CT images, and Monte Carlo transport for the gamma-ray component. But the nuclear physics — the decay data, the energy spectra, the particle ranges — is the essential input.

27.9 Summary: Nuclear Physics Saves Lives

This chapter has traced the nuclear physics through every major application of nuclear medicine:

1. Production. Medical radionuclides are produced by cyclotrons (${}^{18}\text{F}$, ${}^{11}\text{C}$, ${}^{68}\text{Ga}$), reactors (${}^{99}\text{Mo}$, ${}^{131}\text{I}$, ${}^{177}\text{Lu}$), and generators (${}^{99\text{m}}\text{Tc}$, ${}^{68}\text{Ga}$). The choice of production route is dictated by the nuclear reaction cross sections and the required specific activity.

2. Diagnostic imaging. PET exploits positron annihilation (two back-to-back 511 keV photons, coincidence detection). SPECT uses gamma-emitting radionuclides with physical collimators. The ${}^{99}\text{Mo}/{}^{99\text{m}}\text{Tc}$ generator provides the world's most-used medical isotope through secular equilibrium.

3. External beam therapy. Protons and carbon ions exploit the Bragg peak — the sharp dose maximum at end-of-range — to deliver dose to tumors while sparing healthy tissue beyond the target. The physics is the Bethe-Bloch stopping power formula. Carbon ions additionally offer higher biological effectiveness through dense ionization tracks.

4. Brachytherapy. Sealed sources deliver high local doses through geometric proximity ($1/r^2$ falloff) and appropriate photon energies.

5. Targeted radionuclide therapy. Molecules that bind specifically to tumor cells carry $\beta^-$ emitters (${}^{131}\text{I}$, ${}^{177}\text{Lu}$) or $\alpha$ emitters (${}^{225}\text{Ac}$, ${}^{211}\text{At}$, ${}^{223}\text{Ra}$) directly to every tumor deposit in the body. Alpha particles — with their short range, high LET, and high RBE — offer the most precise form of cancer treatment ever devised.

6. Theranostics. The same molecular platform carries a diagnostic isotope (for imaging) and a therapeutic isotope (for treatment), enabling personalized medicine: image first, confirm the target is present, then treat with confidence.

7. Dosimetry. The MIRD formalism connects nuclear decay data (energies, yields, half-lives) to patient-specific absorbed doses through the cumulated activity and absorbed fractions.

Every one of these applications rests on the nuclear physics of Parts I–IV: decay laws (Chapter 12), positron emission (Chapter 14), photon interactions (Chapter 16), charged-particle stopping (Chapter 16), nuclear reactions (Chapter 17), and the properties of specific nuclides across the chart. Nuclear medicine is not an application bolted onto nuclear physics — it is nuclear physics, practiced in hospitals instead of laboratories, saving millions of lives every year.

🔬 Looking Ahead. Chapter 28 turns to a more sobering application of nuclear physics: nuclear security, nonproliferation, and the forensic science that identifies the origin of nuclear materials. The same nuclear reactions that produce medical isotopes can, in the wrong hands, produce weapons-usable materials. The physics is the same; the consequences are vastly different.

Key Equations Reference