Chapter 17

Nuclear Medicine

Diagnostic Imaging

Nuclear medicine imaging uses radioactive tracers to visualize physiological processes in the body. The key modalities are:

Positron Emission Tomography (PET)

PET exploits $\beta^+$ decay, where the emitted positron annihilates with an electron, producing two back-to-back 511 keV gamma rays:

$$e^+ + e^- \to 2\gamma \quad (E_\gamma = m_e c^2 = 511\;\text{keV each})$$

Coincidence detection of these two photons defines a line of response (LOR), enabling 3D image reconstruction. The most common PET tracer is $^{18}$F-FDG (fluorodeoxyglucose), which accumulates in metabolically active tissues such as tumors.

SPECT (Single Photon Emission CT)

SPECT uses gamma-emitting isotopes (primarily $^{99m}$Tc) with a rotating gamma camera and collimator. The 140.5 keV gamma ray from $^{99m}$Tc is ideal for imaging: high enough to penetrate tissue, low enough for efficient detection.

Gamma Cameras

A gamma camera (Anger camera) consists of a collimator, a large NaI(Tl) scintillation crystal, an array of photomultiplier tubes (PMTs), and position-computing electronics. The intrinsic spatial resolution is typically 3-4 mm, limited by the collimator geometry. The detection efficiency depends on the collimator type and the gamma-ray energy.

Therapeutic Applications

Brachytherapy

Sealed radioactive sources ($^{192}$Ir, $^{137}$Cs, $^{125}$I) are placed directly within or adjacent to the tumor. The dose rate at distance r from a point source follows the inverse-square law with attenuation:

$$\dot{D}(r) = \frac{S_K \cdot \Lambda}{r^2}\,g(r)\,\phi_{\text{an}}(\theta)$$

where $S_K$ is the air-kerma strength, $\Lambda$ is the dose-rate constant,$g(r)$ is the radial dose function, and $\phi_{\text{an}}$ is the anisotropy function.

Targeted Alpha Therapy (TAT)

Alpha-emitting radioisotopes ($^{223}$Ra, $^{225}$Ac, $^{211}$At) deliver high-LET radiation with a range of only 50-100 $\mu$m in tissue (a few cell diameters). The energy deposited per alpha particle is $\sim$5-8 MeV, causing dense ionization and irreparable DNA double-strand breaks. The advantage is high tumor-cell killing efficiency with minimal damage to surrounding tissue.

Proton Therapy

Proton beams deposit most of their energy at the Bragg peak, whose depth depends on the beam energy. The energy loss is described by the Bethe-Bloch formula:

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

Key Radiopharmaceuticals

Isotope	$T_{1/2}$	Decay Mode	Application
$^{99m}$Tc	6.01 h	IT (140 keV $\gamma$)	SPECT (bone, cardiac, renal)
$^{18}$F (FDG)	109.8 min	$\beta^+$ (97%)	PET (oncology, neurology)
$^{131}$I	8.02 d	$\beta^-$ (606 keV)	Thyroid cancer therapy
$^{177}$Lu	6.65 d	$\beta^-$ (497 keV)	PRRT, PSMA therapy
$^{223}$Ra	11.4 d	$\alpha$ (5.98 MeV)	Bone metastases therapy

$^{99m}$Tc is the workhorse of nuclear medicine, used in ~80% of all diagnostic procedures. It is obtained from a $^{99}$Mo/$^{99m}$Tc generator (the "moly cow"), where $^{99}$Mo ($T_{1/2} = 66$ h) decays to $^{99m}$Tc, which is eluted with saline solution.

Dose Calculations

The absorbed dose to a target organ from a source organ is calculated using the MIRD (Medical Internal Radiation Dose) formalism:

$$D(r_T \leftarrow r_S) = \tilde{A}(r_S) \cdot S(r_T \leftarrow r_S)$$

where $\tilde{A}(r_S)$ is the cumulated (time-integrated) activity in the source organ and $S(r_T \leftarrow r_S)$ is the dose factor (mean absorbed dose per unit cumulated activity). The cumulated activity is:

$$\tilde{A} = \int_0^\infty A(t)\,dt = \frac{A_0}{\lambda_{\text{eff}}} = A_0 \cdot \frac{T_{\text{eff}}}{\ln 2}$$

The S-factor encapsulates the radiation physics: particle energies, emission probabilities, and the geometry of energy deposition from source to target.

Biological and Effective Half-Life

A radiopharmaceutical is eliminated from the body by two independent processes: physical decay and biological clearance. The total elimination rate is the sum:

$$\lambda_{\text{eff}} = \lambda_{\text{phys}} + \lambda_{\text{bio}}$$

The effective half-life is therefore:

$$\frac{1}{T_{\text{eff}}} = \frac{1}{T_{\text{phys}}} + \frac{1}{T_{\text{bio}}} \qquad \Rightarrow \qquad T_{\text{eff}} = \frac{T_{\text{phys}} \cdot T_{\text{bio}}}{T_{\text{phys}} + T_{\text{bio}}}$$

The effective half-life is always shorter than both the physical and biological half-lives. For diagnostic imaging, short effective half-lives are desirable to minimize patient dose. For therapy, longer effective half-lives allow sustained dose delivery to the tumor.

LET and RBE

The Linear Energy Transfer (LET) describes the energy deposited per unit path length by ionizing radiation:

$$\text{LET} = \frac{dE}{dx}\bigg|_{\text{local}} \quad [\text{keV}/\mu\text{m}]$$

The Relative Biological Effectiveness (RBE) compares the biological effect of a given radiation type to a reference radiation (usually $^{60}$Co gamma rays):

$$\text{RBE} = \frac{D_{\text{ref}}}{D_{\text{test}}}\bigg|_{\text{same biological effect}}$$

High-LET radiation (alpha particles, heavy ions) produces dense ionization tracks, causing clustered DNA damage that is more difficult to repair. The RBE increases with LET up to about 100 keV/$\mu$m (optimal LET), then decreases due to "overkill" (wasted dose beyond that needed to kill a cell). The equivalent dose is:

$$H = \sum_R w_R \cdot D_R \quad [\text{Sv}]$$

where $w_R$ is the radiation weighting factor ($w_R = 1$ for photons/electrons,$w_R = 2$ for protons, $w_R = 20$ for alpha particles and heavy ions).

Python Simulation: Nuclear Medicine Physics

Radioactive decay of common radiopharmaceuticals, effective half-life calculations, Bragg peak comparison for proton therapy, and LET-RBE relationship.

Nuclear Medicine Calculations

Python

Decay curves, effective half-life, Bragg peak, and LET-RBE plots

script.py138 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# Nuclear Medicine: Dose and Decay Calculations
# ──────────────────────────────────────────────

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0a0a1a')

for ax_row in axes:
    for ax in ax_row:
        ax.set_facecolor('#0a0a1a')
        for spine in ax.spines.values():
            spine.set_edgecolor('#334155')

# ─── Panel 1: Radioactive decay of common radiopharmaceuticals ───
t = np.linspace(0, 48, 500)  # hours

isotopes = {
    'Tc-99m': {'t_half': 6.01, 'color': '#ef4444'},     # hours
    'F-18':   {'t_half': 1.83, 'color': '#f59e0b'},
    'I-131':  {'t_half': 192.5, 'color': '#22c55e'},     # ~8 days
    'Ga-68':  {'t_half': 1.13, 'color': '#3b82f6'},
    'Lu-177': {'t_half': 159.5, 'color': '#a855f7'},     # ~6.6 days
}

for name, props in isotopes.items():
    lam = np.log(2) / props['t_half']
    A = np.exp(-lam * t)
    axes[0,0].plot(t, A, color=props['color'], linewidth=2, label=name)

axes[0,0].set_xlabel('Time [hours]', color='#94a3b8', fontsize=11)
axes[0,0].set_ylabel('A(t) / A(0)', color='#94a3b8', fontsize=11)
axes[0,0].set_title('Radiopharmaceutical Decay', color='white', fontsize=13)
axes[0,0].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[0,0].tick_params(colors='#94a3b8')
axes[0,0].set_ylim(0, 1.05)

# ─── Panel 2: Effective half-life ───
t_bio = np.linspace(0.5, 50, 200)  # biological half-life (hours)
t_phys_vals = [6.01, 1.83, 192.5]  # Tc-99m, F-18, I-131
colors_eff = ['#ef4444', '#f59e0b', '#22c55e']
labels_eff = ['Tc-99m (6h)', 'F-18 (1.83h)', 'I-131 (193h)']

for tp, c, lab in zip(t_phys_vals, colors_eff, labels_eff):
    t_eff = (t_bio * tp) / (t_bio + tp)
    axes[0,1].plot(t_bio, t_eff, color=c, linewidth=2, label=lab)

axes[0,1].set_xlabel('Biological half-life [hours]', color='#94a3b8', fontsize=11)
axes[0,1].set_ylabel('Effective half-life [hours]', color='#94a3b8', fontsize=11)
axes[0,1].set_title('Effective Half-Life', color='white', fontsize=13)
axes[0,1].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[0,1].tick_params(colors='#94a3b8')

# ─── Panel 3: Bragg peak for proton therapy ───
depth = np.linspace(0, 30, 500)  # cm in water

def bragg_curve(depth, R, sigma=0.5):
    """Approximate Bragg curve with Gaussian peak"""
    # Entrance plateau + Bragg peak
    dose = np.zeros_like(depth)
    mask = depth < R
    # Approximate 1/v behavior before peak
    dose[mask] = 1.0 + 0.5 * (depth[mask] / R)**2
    # Gaussian peak at range R
    dose += 3.0 * np.exp(-(depth - R)**2 / (2*sigma**2))
    # Kill dose after range
    dose[depth > R + 3*sigma] = 0
    return dose / dose.max()

for E, R, c, lab in [(150, 15.6, '#ef4444', '150 MeV'),
                       (200, 25.7, '#3b82f6', '200 MeV'),
                       (100, 7.7, '#22c55e', '100 MeV')]:
    dose = bragg_curve(depth, R, sigma=0.4)
    axes[1,0].plot(depth, dose, color=c, linewidth=2, label=f'p ({lab})')

# Photon (exponential falloff for comparison)
dose_photon = np.exp(-0.05 * depth) * (1 - np.exp(-0.5 * depth))
dose_photon /= dose_photon.max()
axes[1,0].plot(depth, dose_photon, color='#94a3b8', linewidth=1.5,
               linestyle='--', label='Photon (6 MV)')

axes[1,0].set_xlabel('Depth in water [cm]', color='#94a3b8', fontsize=11)
axes[1,0].set_ylabel('Relative dose', color='#94a3b8', fontsize=11)
axes[1,0].set_title('Bragg Peak (Proton Therapy)', color='white', fontsize=13)
axes[1,0].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
axes[1,0].tick_params(colors='#94a3b8')

# ─── Panel 4: LET vs RBE ───
LET = np.linspace(0.1, 200, 300)  # keV/um
# Simplified RBE model
RBE = 1.0 + 0.5 * np.sqrt(LET) * np.exp(-LET/100)
RBE_max = 1.0 + 2.5 * np.exp(-(np.log(LET/100))**2 / 2)

axes[1,1].plot(LET, RBE, color='#ef4444', linewidth=2.5, label='RBE model')
axes[1,1].axhline(y=1, color='#64748b', linewidth=0.8, linestyle='--')
axes[1,1].axvline(x=100, color='#f59e0b', linewidth=0.8, linestyle='--', alpha=0.5)
axes[1,1].text(105, 3.5, 'Optimal LET', color='#f59e0b', fontsize=9)

# Mark particle types
axes[1,1].annotate('Photons/e-', xy=(0.5, 1.0), fontsize=9, color='#94a3b8',
                   xytext=(5, 1.5), arrowprops=dict(arrowstyle='->', color='#94a3b8'))
axes[1,1].annotate('Protons', xy=(10, 1.5), fontsize=9, color='#22c55e',
                   xytext=(20, 2.5), arrowprops=dict(arrowstyle='->', color='#22c55e'))
axes[1,1].annotate('Alpha/C ions', xy=(100, 3.2), fontsize=9, color='#ef4444',
                   xytext=(130, 3.8), arrowprops=dict(arrowstyle='->', color='#ef4444'))

axes[1,1].set_xlabel('LET [keV/um]', color='#94a3b8', fontsize=11)
axes[1,1].set_ylabel('RBE', color='#94a3b8', fontsize=11)
axes[1,1].set_title('LET vs RBE', color='white', fontsize=13)
axes[1,1].tick_params(colors='#94a3b8')
axes[1,1].set_xlim(0, 200)
axes[1,1].set_ylim(0, 5)

plt.tight_layout()
plt.savefig('nuclear_medicine.png', dpi=130, bbox_inches='tight',
            facecolor=fig.get_facecolor())

print("=== Nuclear Medicine Parameters ===")
print()
for name, props in isotopes.items():
    t_h = props['t_half']
    lam = np.log(2) / t_h
    unit = 'hours' if t_h < 48 else 'days'
    t_disp = t_h if t_h < 48 else t_h/24
    print(f"{name}: T_1/2 = {t_disp:.2f} {unit}, lambda = {lam:.4f} /hr")

print()
print("Effective half-life examples (T_bio = 24 hr):")
T_bio = 24.0
for name in ['Tc-99m', 'F-18', 'I-131']:
    T_phys = isotopes[name]['t_half']
    T_eff = (T_bio * T_phys) / (T_bio + T_phys)
    print(f"  {name}: T_eff = {T_eff:.2f} hours")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Implementation: Dose Calculator

MIRD-based dose calculations for radiopharmaceutical therapy, including effective half-life computation, absorbed dose estimates, and LET/RBE data tables.

Nuclear Medicine Dose Calculator

Fortran

Radiopharmaceutical properties, dose calculation, and radiation quality factors

program.f90116 lines

program nuclear_medicine
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: pi = 3.14159265358979323846_dp
  real(dp), parameter :: ln2 = 0.693147180559945_dp

integer :: i, j
  real(dp) :: T_phys, T_bio, T_eff, lambda_phys, lambda_bio, lambda_eff
  real(dp) :: A0, A_t, dose, cum_dose, t, dt
  real(dp) :: S_factor, mass_tumor, dose_rate
  integer, parameter :: n_steps = 1000

write(*,'(A)') '=================================================='
  write(*,'(A)') ' Nuclear Medicine Dose Calculations'
  write(*,'(A)') '=================================================='
  write(*,'(A)') ''

! ─── Radiopharmaceutical Properties ───
  write(*,'(A)') '--- Common Radiopharmaceuticals ---'
  write(*,'(A)') ''
  write(*,'(A20,A12,A12,A16,A20)') 'Isotope', 'T_phys(h)', &
       'T_bio(h)', 'T_eff(h)', 'Application'
  write(*,'(A)') '─────────────────────────────────────────────────' // &
       '────────────────────────────'

! Tc-99m
  T_phys = 6.01_dp
  T_bio = 24.0_dp
  T_eff = (T_phys * T_bio) / (T_phys + T_bio)
  write(*,'(A20,F12.2,F12.1,F16.3,A20)') 'Tc-99m', T_phys, T_bio, &
       T_eff, 'SPECT imaging'

! F-18 FDG
  T_phys = 1.83_dp
  T_bio = 12.0_dp
  T_eff = (T_phys * T_bio) / (T_phys + T_bio)
  write(*,'(A20,F12.2,F12.1,F16.3,A20)') 'F-18 FDG', T_phys, T_bio, &
       T_eff, 'PET imaging'

! I-131
  T_phys = 192.5_dp
  T_bio = 80.0_dp * 24.0_dp  ! 80 days in hours
  T_eff = (T_phys * T_bio) / (T_phys + T_bio)
  write(*,'(A20,F12.2,F12.1,F16.3,A20)') 'I-131', T_phys, T_bio, &
       T_eff, 'Thyroid therapy'

! Lu-177
  T_phys = 159.5_dp
  T_bio = 48.0_dp * 24.0_dp
  T_eff = (T_phys * T_bio) / (T_phys + T_bio)
  write(*,'(A20,F12.2,F12.1,F16.3,A20)') 'Lu-177', T_phys, T_bio, &
       T_eff, 'PRRT/PSMA therapy'

! Ra-223
  T_phys = 11.43_dp * 24.0_dp
  T_bio = 100.0_dp * 24.0_dp
  T_eff = (T_phys * T_bio) / (T_phys + T_bio)
  write(*,'(A20,F12.2,F12.1,F16.3,A20)') 'Ra-223', T_phys, T_bio, &
       T_eff, 'Alpha therapy'

write(*,'(A)') ''

! ─── Dose Calculation Example ───
  write(*,'(A)') '--- Absorbed Dose Calculation ---'
  write(*,'(A)') ''
  write(*,'(A)') 'I-131 thyroid therapy example:'

A0 = 3.7e9_dp     ! 3.7 GBq = 100 mCi initial activity
  T_phys = 192.5_dp  ! hours (8.02 days)
  T_bio = 120.0_dp * 24.0_dp  ! biological half-life ~120 days
  T_eff = (T_phys * T_bio) / (T_phys + T_bio)
  lambda_eff = ln2 / T_eff
  mass_tumor = 20.0_dp  ! grams

! S-factor for I-131 beta in thyroid (approximate)
  S_factor = 2.2e-12_dp  ! Gy per (Bq*s) -- MIRD S-value

! Cumulated activity: A_tilde = A0 / lambda_eff
  cum_dose = A0 / lambda_eff * 3600.0_dp  ! convert hr^-1 to s^-1

! Absorbed dose D = A_tilde * S
  dose = cum_dose * S_factor

write(*,'(A,ES10.2,A)') '  Initial activity: ', A0, ' Bq (3.7 GBq)'
  write(*,'(A,F8.2,A)') '  Effective half-life: ', T_eff, ' hours'
  write(*,'(A,F8.2,A)') '                     = ', T_eff/24.0_dp, ' days'
  write(*,'(A,ES10.2,A)') '  Cumulated activity: ', cum_dose, ' Bq*s'
  write(*,'(A,F10.2,A)') '  Absorbed dose: ', dose, ' Gy'
  write(*,'(A)') ''

! ─── LET and RBE Table ───
  write(*,'(A)') '--- LET and RBE for Different Radiations ---'
  write(*,'(A)') ''
  write(*,'(A20,A15,A10)') 'Radiation', 'LET(keV/um)', 'RBE'
  write(*,'(A)') '─────────────────────────────────────────────'
  write(*,'(A20,A15,F10.1)') 'Co-60 gamma', '0.3', 1.0_dp
  write(*,'(A20,A15,F10.1)') '250 kV X-rays', '2.0', 1.0_dp
  write(*,'(A20,A15,F10.1)') '10 MeV protons', '4.7', 1.1_dp
  write(*,'(A20,A15,F10.1)') '150 MeV protons', '0.5', 1.1_dp
  write(*,'(A20,A15,F10.1)') '2.5 MeV alphas', '166', 20.0_dp
  write(*,'(A20,A15,F10.1)') 'Neutrons (14 MeV)', '12', 3.0_dp
  write(*,'(A20,A15,F10.1)') 'Carbon ions', '70-200', 3.0_dp
  write(*,'(A)') ''

! ─── Proton therapy range ───
  write(*,'(A)') '--- Proton Range in Water ---'
  write(*,'(A)') ''
  write(*,'(A15,A15)') 'Energy(MeV)', 'Range(cm)'
  write(*,'(A)') '──────────────────────────────'
  write(*,'(F15.0,F15.2)') 70.0_dp, 4.1_dp
  write(*,'(F15.0,F15.2)') 100.0_dp, 7.7_dp
  write(*,'(F15.0,F15.2)') 150.0_dp, 15.6_dp
  write(*,'(F15.0,F15.2)') 200.0_dp, 25.7_dp
  write(*,'(F15.0,F15.2)') 230.0_dp, 32.4_dp

end program nuclear_medicine

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

← Nucleosynthesis Radiation Detection →