Medical radionuclides, generators and dose

A gamma camera is a NaI(Tl) crystal behind a lead collimator, and that pairing sets a window. Below about 80 keV the photon is absorbed in the patient before it reaches the head; above about 250 keV it crosses a 9.5 mm crystal without depositing and starts to penetrate the collimator septa. The 140.5 keV line of Tc-99m sits in the middle of that window, and it is essentially the only penetrating emission — apart from low-energy conversion and Auger electrons, there is no beta delivering dose the camera cannot use. That is the level scheme talking. The isomer is 1/2− at 142.7 keV while the Tc-99 ground state is 9/2+, so a direct photon needs L = 4 or 5 with a parity change, M4 or E5; the alternative is a 2.17 keV E3 step down to the 140.51 keV 7/2+ level. Both routes are heavily hindered — Weisskopf rates fall steeply with L and scale as E^(2L+1) — and that hindrance is exactly what buys the 6.01 h half-life. The 140.51 keV level then has a 0.19 ns half-life and emits the M1 photon you image, in 89% of decays, the rest lost to internal conversion.

Write the chain as dN/dt = A N, with A₁₁ = −λₚ, A₂₂ = −λ_d and A₂₁ = b λₚ, where b = 0.876 is the fraction of Mo-99 decays that pass through the isomer; the other 12.4% cascade to the Tc-99 ground state without passing through it, and that Tc-99, though it is eluted too, adds no usable activity because its half-life is 2.1 × 10⁵ y. For Mo-99, λₚ = ln2/65.94 h = 0.01051 h⁻¹; for Tc-99m, λ_d = ln2/6.01 h = 0.11533 h⁻¹. Saline through the alumina column strips the pertechnetate and leaves the molybdate behind, so an elution is the initial condition N_d = 0 imposed on a parent that is untouched. Between elutions the closed form is A_d(t) = k⋅Aₚ(0)⋅e^{(−λₚ t)}⋅[1 − e^{(−(λ_d − λₚ)s)}], with k = b λ_d/(λ_d − λₚ) and s the time since the last draw. The output is a sawtooth riding under a slow exponential envelope: each tooth rebuilds from zero, and each is a little shorter than the last because the Mo-99 behind it has decayed.

The half-lives differ by a factor of 11, not a factor of a thousand, so this is transient equilibrium and not secular equilibrium. Once e^{(−λ_d t)} has died the daughter tracks the parent and thereafter decays at the parent's rate, sitting at the fixed fraction k = b λ_d/(λ_d − λₚ) = 0.876 × 1.100 = 0.964 of it — below the parent, because the branch more than cancels the kinetic factor. Secular equilibrium would need λₚ ≪ λ_d and would give A_d = b Aₚ exactly. Before the ceiling is reached the buildup peaks at tₘₐₓ = ln(λ_d/λₚ)/(λ_d − λₚ) = 22.9 h, and there dN_d/dt = 0 forces A_d = b Aₚ exactly — a check worth remembering, because it is independent of the algebra used to reach it. The peak is also very flat: eluting at 24 h instead of 22.9 h costs under 0.1% of the yield, which is why hospital generators are drawn once each morning rather than on a stopwatch.

F-18 answers the same logistics question a different way. It has no long-lived parent to load onto a generator, and its 109.8 min half-life is too short to ship far, so the cyclotron sits near the scanner. It decays 96.9% by β+ with a 633.5 keV endpoint and a 250 keV mean, and the positron thermalises within roughly 0.6 mm of water before annihilating into two 511 keV photons emitted back to back. PET exploits that geometry instead of a collimator: two detectors firing inside a few-nanosecond window define a line of response, which is why PET sensitivity exceeds SPECT's by about two orders of magnitude. It also inherits two blurs no electronics can remove — the positron range travelled before annihilation, and the roughly 0.5° acollinearity from the residual momentum of the annihilating pair, contributing about 0.0022⋅D for a ring of diameter D, or 1.8 mm FWHM for an 80 cm ring. And 511 keV demands a dense, fast scintillator such as LSO, not the thin NaI a 140 keV camera uses.

Two independent routes remove the tracer: physical decay at λ_phys and biological clearance at λ_bio. Rates add, exactly as partial widths do for competing decay channels, so λ_eff = λ_phys + λ_bio and 1/T_eff = 1/T_phys + 1/T_bio, and T_eff is always shorter than either. For a Tc-99m agent with T_phys = 6.01 h and T_bio = 24 h, T_eff = 4.81 h. Dose does not care about the activity at any one instant, only about the total number of decays and where they happened, so the quantity to compute is the cumulated activity à = ∫A dt. For instantaneous uptake and a single clearance exponential, à = A₀ τ_eff with τ_eff = T_eff/ln2, so 740 MBq makes 1.85 × 10¹³ decays. Both assumptions are the weak point. Real biokinetics is a multi-compartment fit, making à a sum of exponentials with organ-specific residence times, and it is fitted to a reference person rather than measured in the patient in front of you.

Absorbed dose is energy per unit mass, D = Ã⋅Δ⋅φ/m, in joules per kilogram, named the gray. Δ is the mean energy released per decay, summed over emissions with their yields; φ is the fraction of that energy absorbed in the target mass m. For electrons, conversion electrons and positrons φ is essentially 1, because their range is a few millimetres at most; for photons it is a transport result — about 0.35 for 511 keV in a 70 kg adult — and it is where a Monte Carlo code earns its keep. Equivalent dose H = w_R D and effective dose E = Σ_T w_T H_T then follow, both in sievert. That step is not a measurement. w_R = 20 for alphas and w_T = 0.12 for red marrow are values a committee chose so that stochastic risk adds roughly linearly across a reference population, and they carry no meaning for one patient's outcome. Quote gray when you mean deposited energy, and sievert only when you mean population risk.