Equations

Photon energy and wavelength

E = hc/λ for x-ray and γ photons. hc ≈ 1.240 keV·nm.

A photon’s energy is Planck’s constant times its frequency, or hc divided by wavelength. In radiology we use E (keV) = 1.240 / λ (nm), so a 0.1 nm x-ray is 12.4 keV and a 12.4 pm γ-ray is 100 keV. Frequency follows from f = c/λ with c = 3×10⁸ m/s.

Compton scattered photon energy

Scattered photon energy versus scatter angle for a free electron.

In Compton scatter the photon transfers part of its energy to a loosely bound electron. The outgoing photon energy E′ falls as the scatter angle θ rises. At 180° (backscatter) E′ = E / (1 + 2E/511 keV). For 511 keV annihilation photons, backscatter is 170 keV — the origin of the 180 keV backscatter peak.

Pair production threshold

Kinetic energy shared by e⁻ e⁺ after the 1.022 MeV rest-mass cost.

Pair production converts a photon into an electron–positron pair in the Coulomb field of a nucleus. At least 2×511 keV = 1.022 MeV is consumed as rest mass; any surplus is kinetic energy of the pair (and a tiny nuclear recoil). The positron later annihilates, yielding two 511 keV photons.

Photoelectric scaling

τ/ρ scales roughly as Zⁿ / Eᵐ between absorption edges.

The photoelectric mass attenuation coefficient rises steeply with atomic number and falls with photon energy, except at absorption edges. A working rule is n ≈ 3–4 and m ≈ 3–3.5. That is why iodine (Z=53) and barium (Z=56) are x-ray contrast agents, and why bone (effective Z ~ 13) absorbs more than soft tissue (Z ~ 7.4) at diagnostic energies.

Activity from mass

A = λN with N from mass and molar mass. Carrier-free specific activity.

Activity is the number of decays per second: A = λN. For a pure radionuclide the atom count N is mass over molar mass times Avogadro’s number. Specific activity a = A/m is then λ N_A / M. Short half-life nuclides have huge specific activity (F-18, Tc-99m); long-lived ones (C-14, U-238) do not.

Mean life and decay constant

τ = 1/λ = T½ / ln 2 ≈ 1.443 T½. Mean life, not half-life.

Half-life is the time for activity to fall to 50%. Mean life τ is the average lifetime of an atom, equal to 1/λ, and is about 44% longer than T½. Cumulated activity for a pure exponential is A₀τ = 1.443 A₀ T½ — the factor that appears in MIRD.

Linear and mass attenuation

μ = (μ/ρ) ρ. HVL = ln 2 / μ, TVL = ln 10 / μ.

Mass attenuation μ/ρ is tabulated per element and energy (XCOM). Multiply by density to get the linear coefficient used in I = I₀ e^{−μx}. HVL is the thickness that halves a narrow beam; TVL reduces it by ten. For water at ~100 keV, μ/ρ ≈ 0.017 cm²/g so HVL ≈ 4 cm.

Dose from photon fluence

D = Φ E (μen/ρ) with the MeV-to-joule conversion.

Collision kerma and absorbed dose (under CPE) equal energy fluence times μen/ρ. Writing energy fluence as ΦE and converting MeV/g to J/kg gives D(Gy) = Φ E (μen/ρ) × 1.602×10⁻¹⁰. This is the bridge from Monte Carlo fluence tallies to gray.

Compton wavelength shift

Δλ = λ_C (1 − cos θ) with λ_C = 2.426 pm, independent of E.

The Compton wavelength shift depends only on scatter angle, not on incident energy. λ_C = h/(m_e c) = 2.426 pm. At 90°, Δλ = λ_C; at 180°, Δλ = 2λ_C. Energy change is large when λ is comparable to λ_C (hard x-rays and γ-rays) and tiny for optical photons.

Fluence inverse square

Primary fluence from a point source: Φ = N / (4π r²).

Photons (or particles) emitted isotropically from a point spread over a sphere of area 4πr². Fluence therefore falls as 1/r². This is the geometric origin of the inverse-square law used in radiotherapy output, HDR, and radiation protection.

Klein–Nishina cross section

Unpolarised Compton differential cross section per electron versus scatter angle.

Klein–Nishina is the quantum-electrodynamic cross section for Compton scatter from a free electron. At diagnostic energies (~30–150 keV) scatter is almost isotropic in the forward half; at MV and PET energies it is strongly forward-peaked. Integrating over angle gives the Compton attenuation coefficient τ_C = Z n_e σ_KN.

Photoelectric Z³ / E³·⁵ scaling

Relative photoelectric cross section between two materials or energies.

The photoelectric effect goes roughly as Z³–Z⁴ / E³–E³·⁵ between K-edges. That is why bone and iodine light up on kV images, why lead is an excellent kV shield, and why photoelectric contrast collapses at MV energies. The exponent 3.5 is a teaching compromise between the non-relativistic 3.5 and the high-energy 3.

Pair-production Z² scaling

Pair (and triplet) production rises as Z² above 1.022 MeV.

A photon of E > 1.022 MeV can materialise as e⁺e⁻ in the Coulomb field of a nucleus. The cross section per atom goes as Z² and, well above threshold, roughly linearly with E. In bone and in high-Z shields at 18 MV this is no longer negligible; in kV imaging it is identically zero.

Moseley Kα characteristic energy

Kα x-ray energy from Moseley’s law: E ≈ 10.2 (Z−1)² eV.

Characteristic x-rays are emitted when an outer electron fills a K-shell vacancy. Moseley treated the K-shell as hydrogen-like with screening constant 1, so E_Kα = 13.6 (Z−1)² (1 − 1/4) eV. Tungsten (Z=74) predicts ~54 keV; the measured Kα is 59.3 keV. Molybdenum Kα is 17.5 keV — the reason Mo anodes are used in mammography.

Kramers bremsstrahlung spectrum

Unfiltered thick-target intensity I(E) ∝ Z (E_max − E), E_max = kVp.

In a thick anode the electron slows from e·kVp to rest, radiating a triangular photon spectrum that is maximum at 0 keV and zero at E_max. Filtration (inherent + added Al) cuts the low-energy end, so a clinical beam peaks near E_max/2 to E_max/3 and the mean energy is ~E_max/3 to ~E_max/2. Tube output scales as Z of the anode and roughly as kVp² (after filtration, closer to kVp²–kVp³).

HVL, TVL and barrier n-value

HVL = ln 2 / μ, TVL = ln 10 / μ ≈ 3.32 HVL. n = log(1/B) / log 2 HVLs.

One half-value layer cuts intensity in half; one tenth-value layer cuts it by ten. Narrow-beam μ gives the theoretical HVL; broad-beam (with scatter) needs a larger effective HVL — the first HVL is smaller than the second because the beam hardens. Shielding reports quote TVLs of lead, concrete and steel at the design energy.

Buildup factor

Broad-beam transmission I = B I₀ e^{−μx} includes scatter that narrow-beam law omits.

A narrow-beam (good-geometry) measurement rejects scatter, so I/I₀ = e^{−μx}. In a wall, a patient, or a broad therapy field, scattered photons still reach the point of interest and the observed transmission is larger by the buildup factor B ≥ 1. B grows with optical thickness μx, field size and decreasing energy. Shielding TVLs are tabulated as broad-beam values for this reason.

Proton range (Bragg–Kleeman)

R = α E^p in water, with p ≈ 1.77 and α ≈ 0.0022 cm·MeV^{−p}.

The Bragg–Kleeman rule is the power-law fit to CSDA proton range in a given material. In water, 80 MeV protons stop near 5 cm (ocular / shallows), 160 MeV near 17 cm, 200 MeV near 26 cm — the numbers every proton physicist quotes from memory. Differentiating gives the residual-range relation used to pull a spread-out Bragg peak (SOBP) from a pristine peak.

Average LET from energy and range

Track-averaged LET ≈ E / R. Unrestricted collisional stopping power.

Linear energy transfer is the energy locally imparted per unit track length. Dividing the particle’s kinetic energy by its CSDA range gives the track-averaged unrestricted LET — a useful order-of-magnitude number. 6 MV electrons (E≈2 MeV, R≈1 cm) sit near 0.2 keV/μm (low LET); a 5 MeV α in water (R≈0.04 mm) is ~100 keV/μm (high LET, RBE ≫ 1).

Parent–daughter decay (secular / transient)

Bateman solution for daughter activity starting from a pure parent.

A parent nuclide decays to a radioactive daughter. If λ₂ ≫ λ₁ (half-life of the daughter much shorter) the system reaches secular equilibrium: A₂ ≈ A₁ after a few daughter half-lives, and they then decay together with the parent’s T½ — the ⁹⁹Mo/⁹⁹ᵐTc generator and ²²⁶Ra/²²²Rn. If the half-lives are comparable (λ₂ > λ₁ but not ≫) the equilibrium is transient: A₂ / A₁ = λ₂ / (λ₂ − λ₁) > 1 (¹³²Te/¹³²I).

Branching ratio and partial activity

Partial emission rate A_i = A × BR_i × n_i for photons, β or α of a given branch.

Almost no radionuclide emits a single radiation. ⁹⁹ᵐTc: 88% 140 keV γ (the imaging photon), internal conversion and a few other lines. ¹⁸F: 97% β⁺ (hence 194% of 511 keV annihilation photons per decay) and 3% EC. Dose constants, gamma constants and imaging yields always fold in the branching ratio.

Exposure and air kerma

X = Q/m_air. 1 R = 2.58×10⁻⁴ C/kg → K_air = 0.00876 Gy/R (W/e = 33.97 J/C).

Exposure X is ionisation charge per mass of dry air — the oldest radiation quantity, still on older survey-meter scales (R, mR/h). Multiplying by W/e = 33.97 J/C converts charge to energy and gives air kerma. 1 roentgen = 2.58×10⁻⁴ C/kg = 8.76 mGy air kerma. Absorbed dose in tissue is then f-factor × exposure (see the f-factor calculator).

Inverse square law

Intensity or dose rate versus distance from a point source.

Intensity (or dose rate) from a point source falls as 1/d² because the same energy is spread over a sphere of area 4πd². Doubling distance quarters the intensity. This is the geometric backbone of SSD changes, HDR dwells, and protection calculations.

Percent depth dose

Dose at depth d relative to dose at dmax on the beam axis.

Percent depth dose is the central-axis dose at depth d expressed as a percentage of the dose at dmax, measured at a fixed SSD. PDD increases with energy and field size and falls with depth beyond dmax.

TMR from PDD

Convert percent depth dose to tissue-maximum ratio.

TMR is the isocentric analogue of PDD: dose at depth in phantom divided by dose at dmax, both at the same SAD. The conversion shown applies an inverse-square factor between the SSD setup and the isocentre.

Mayneord F-factor

PDD correction when SSD changes at constant surface field size.

Mayneord F corrects PDD when SSD changes at constant surface field size. It is two inverse-square ratios: one at dmax and one at depth. Longer SSD slightly improves PDD (less inverse-square fall-off across the phantom).

Equivalent square

Map a rectangular field to a square of similar scatter (Sterling).

Sterling’s equivalent square s = 4A/P = 2ab/(a+b) maps a rectangle to a square that produces nearly the same scatter (hence similar PDD, TMR, and Sp). A 10×15 field ≈ 12 cm square.

Monitor units — SSD technique

MU for a prescribed dose at depth with an SSD setup.

Monitor units for an SSD (fixed-SSD) treatment equal prescribed dose divided by the product of calibration output, PDD/100, and the modifiers Sc, Sp, wedge and tray. If the machine is calibrated to 1 cGy/MU at dmax, 10×10, reference SSD, the formula is the clinical workhorse for non-isocentric setups.

Monitor units — SAD technique

Isocentric MU calculation using TMR.

Isocentric MU uses TMR instead of PDD and an inverse-square from the calibration distance (SCD, often SSD_cal+dmax) to SAD. Most linacs treat isocentrically; this is the matching hand calculation.

Adjacent-field gap

Skin gap so beam edges meet at a chosen depth.

When two adjacent photon fields meet at depth d, a skin gap is needed because the geometric edges diverge. g = (d/2)(L₁/SSD₁ + L₂/SSD₂). Too small a gap hot-spots the junction; too large cold-spots it — critical near cord and craniospinal junctions.

SSD ↔ SAD output conversion

Convert dose rate between SSD and SAD calibrations at dmax.

Calibration at SSD (dose to dmax at the surface distance) and at SAD (dose to dmax at isocentre) differ by the inverse-square between (SSD+dmax) and SAD. Converting lets you compare outputs quoted in different protocols.

Tissue-phantom ratio TPR

Isocentric dose at depth d relative to dose at reference depth.

Tissue-phantom ratio is the isocentric dose at depth d divided by the dose at a reference depth (often 5 or 10 cm), both with the same SAD and field size at isocentre. TPR20/10 is the IAEA TRS-398 beam-quality index for MV photons.

Field size at depth

Geometric projection of a surface field to depth d.

Photon field size grows linearly with distance from the source. At depth d the side is s₀ (SSD+d)/SSD. This projected size is what you use for scatter (Sp, TMR) lookups, not the surface size.

Geometric penumbra

Source-size penumbra at depth from collimator distance.

Geometric penumbra is the region where the finite source is partially obscured by the collimator. It widens with source size and with distance beyond the collimator, and narrows if you push the jaws closer to the patient (larger SCD… actually smaller gap beyond SCD).

Electron range rules of thumb

Rp ≈ E/2, R90 ≈ E/3.2, R80 ≈ E/2.8, R50 ≈ E/2.33 (cm, MeV, water).

Clinical electron beams in water follow robust rules of thumb: practical range Rp ≈ E/2 cm, therapeutic 90% range ≈ E/3.2 cm, and R50 ≈ E/2.33 cm, with E the most-probable energy at the surface in MeV. They come from the nearly linear CSDA range of 5–20 MeV electrons.

Co-60 source decay

Output correction for T½ = 5.271 y (~1.1% per month).

Co-60 decays with T½ = 5.271 years, about 1.09% output loss per month. Source-output tables in cobalt units must be decay-corrected from the calibration date. Two photons (1.17 and 1.33 MeV, mean 1.25 MeV) accompany each decay.

Ir-192 HDR decay

HDR source strength with T½ = 73.83 days.

Ir-192 (HDR) has T½ = 73.83 days, so source strength drops ~6.4% per week. After-loaders require a decay table or live calculation before every fraction. Air-kerma strength S_K is in U (cGy cm² h⁻¹).

Paddick conformity & gradient

CI = (TVPIV)² / (TV × PIV) and GI = PIV₅₀ / PIV.

Paddick’s conformity index rewards a prescription isodose that covers the target without spilling outside: CI = (TVPIV)²/(TV×PIV). Perfect conformal coverage scores 1. The gradient index GI = PIV₅₀/PIV describes how fast dose falls outside the target — crucial in SRS/SBRT.

Effective SSD (electrons)

Output versus air gap using the effective-SSD inverse square.

Electron output does not follow the photon inverse square from the virtual source because of scatter in the cone. An effective SSD (often 50–90 cm) is fitted so that output versus air gap g behaves as [(f+dmax)/(f+dmax+g)]².

Output factor S_c,p

Total scatter factor as collimator × phantom scatter.

The total scatter factor S_c,p factors into collimator scatter S_c (jaws, from in-air measurements) and phantom scatter S_p (from in-phantom). Output relative to 10×10 is S_c × S_p. Small fields have S_c,p well below 1.

Off-axis dose

Dose off axis from CAX dose and the off-axis ratio.

Off-axis ratio is the dose at a lateral distance x relative to the central axis, at the same depth. Horns in flattened beams can push OAR slightly above 1 just inside the field; it then falls through the penumbra.

Tissue-air ratio TAR

Dose in phantom at depth d divided by dose in air at the same point.

TAR is the original isocentric quantity: it folds inverse-square, attenuation and scatter into one number measured at a point that stays at a fixed distance from the source. TMR = TAR(d)/TAR(dmax). TAR of a 0×0 field is the primary exponential e^{−μ(d−dmax)} and is the backbone of Clarkson sector integration.

Backscatter factor BSF

BSF = TAR(dmax, s) = dose at dmax in phantom / dose in air at the same point.

Backscatter is the extra dose at dmax coming from photons scattered back by the phantom. It rises with field size and falls with energy (MV beams have BSF ≈ 1.02–1.06; orthovoltage 5×5 to 20×20 can be 1.1–1.5). Peak scatter factor (PSF) is the same quantity. Output at dmax in phantom = output in air × BSF.

Output factors Sc, Sp, Sc,p

Total output Sc,p = Sc(collimator) × Sp(phantom). Measured relative to 10×10.

Collimator scatter Sc (or Sc, head scatter) is measured in air with a miniphantom and depends on the jaws / MLC opening. Phantom scatter Sp depends on the irradiated phantom area at the level of the detector. Their product Sc,p is the in-phantom output factor used in every MU calculation. Blocking a field with MLC changes Sp more than Sc.

Wedge factor and effective wedge angle

Physical WF = D_wedge / D_open. Dynamic/virtual: tan θ_eff = (MU_w / MU) tan θ_w.

A physical wedge attenuates more on one side, producing a tilted isodose. The wedge factor (central axis) is typically 0.5–0.8 depending on angle and energy, and it is an MU multiplier. A dynamic (virtual, flying) wedge mixes an open-field segment with a wedged segment; the effective angle follows the tangent rule. EDW / OmniWedge implementations differ in MU split but the tangent mixing rule is the teaching standard.

RTOG conformity index

CI = V_RI / TV. Ideal = 1. Paddick CI also accounts for overlap.

Conformity asks how tightly the prescription isodose hugs the target. RTOG’s simple ratio V_RI / TV is 1 when they match in volume, but a sphere of prescription dose sitting next to the target still scores 1. Paddick’s index (already in the library) multiplies by the overlap squared and is the SRS/SBRT reporting standard.

Homogeneity index ICRU 83

HI = (D2% − D98%) / D50%. Zero is perfectly homogeneous.

ICRU 83 replaced the old ±5% of ICRU 50 with a DVH-based spread between near-max (D2%) and near-min (D98%), normalised to median dose. IMRT/VMAT PTVs typically sit at HI ≈ 0.05–0.15; a simultaneous integrated boost is allowed to be hotter. SBRT PTVs are deliberately inhomogeneous (HI 0.2–0.4) because the prescription isodose is 70–80%.

Gradient index

GI = PIV_{50%} / PIV. Lower is a sharper fall-off (SRS/SBRT).

Gradient index measures how fast dose falls outside the target — the quantity that predicts V12Gy in brain SRS and chest-wall dose in lung SBRT. A single-iso 6 MV VMAT SRS plan often lands at GI 3–4; a large target or a low-energy cone can exceed 5. CyberKnife / Gamma Knife with many entries do better (GI ~2.5–3).

van Herk PTV margin

M = 2.5 Σ + 0.7 σ so that 90% of patients get ≥95% dose to the CTV.

Systematic error Σ (preparation: delineation, setup bias, image registration) shifts the whole treatment and must be covered generously (2.5 Σ ≈ 90% of the population). Random error σ (daily setup, organ motion) blurs the dose and is covered by 0.7 σ, which restores the 95% isodose to the CTV. Add a small penumbra term 1.64 σ_p if you start from a 50% instead of 95% edge — omitted here.

Equivalent uniform dose (Niemierko)

EUD = (Σ v_i D_i^a)^{1/a}. For a single hot/cold volume: EUD = D v^{1/a}.

Niemierko’s generalised EUD compresses a DVH into the uniform dose that would cause the same biological effect. The exponent a is large and negative for tumours (cold-spot sensitive, a ≈ −10), near 1 for parallel organs (mean dose, a ≈ 1), and large positive for serial organs (hot-spot sensitive, a ≈ 16 for cord). gEUD is the input to many TCP/NTCP models.

Electron monitor units

MU = D / [(D/MU)_ref × cutout × insert × SSD factor].

Electron output is calibrated at a reference cone (often 10×10 or 15×15) and dmax, 1 cGy/MU. End-user cutouts change output (small cuts under-scatter and can drop 10–20%). An insert/cone factor accounts for the applicator. Extended SSD uses an effective-SSD inverse-square factor, not the nominal 100 cm, because of scatter from the cone.

HDR dwell time (point source)

t = D / [S_K Λ (r₀/r)² g(r) F]. S_K in U, Λ in cGy h⁻¹ U⁻¹.

TG-43 writes dose rate as air-kerma strength × dose-rate constant × geometry × radial × anisotropy. For a point-source approximation G(r) = (r₀/r)² with r₀ = 1 cm, and F≈1 on the transverse axis. Ir-192 Λ ≈ 1.12 cGy h⁻¹ U⁻¹. A 10 Ci source is ~40 700 U and delivers ~12.7 cGy/s at 1 cm — dwells are seconds, not hours.

Proton RBE-weighted dose

Clinical proton dose: D_RBE = 1.1 D_phys (ICRU 78 constant RBE).

Proton centres prescribe in Gy(RBE) using a constant factor 1.1 on physical dose. That hides a real distal-end RBE rise (LET climbs as the proton slows, variable RBE models predict 1.1 → ~1.3–1.6 in the last millimetres). Planning practice is to keep serial organs just beyond the distal fall-off or to use a variable-RBE model when available.

Radiological path length

Effective water depth d_eff = Σ ρ_i Δx_i (physical-density or relative-stopping-power scaling).

Heterogeneity correction in a 1-D ray: each slab of thickness Δx and relative density (or relative stopping power, RSP, for protons) contributes ρ Δx to the water-equivalent path. Lung (ρ≈0.25) shortens the effective depth — dose goes up, range goes longer. Bone (ρ≈1.4–1.8) does the opposite. MV photon Batho/equivalent-TAR and proton RSP all start from this sum.

Photon dmax versus energy

Teaching rule: dmax (cm) ≈ 0.25 E(MV) for flattening-filter beams, clamped 0.5–5 cm.

Build-up depth increases with energy because the first collision electrons are thrown further forward. Clinical numbers every trainee memorizes: Co-60 0.5 cm, 6 MV 1.5 cm, 10 MV 2.5 cm, 18 MV 3.3 cm. FFF beams have slightly shallower dmax (more contamination). Field size, SSD and a tray all pull dmax up (contamination) or down.

Geometric magnification

Image magnification from SID over SOD.

Geometric magnification M = SID/SOD = SID/(SID−OID). The image is larger than the object because the beam continues to diverge beyond the object. Magnification also enlarges focal-spot blur.

Geometric unsharpness

Focal-spot blur from effective spot size and distances.

Geometric unsharpness (focal-spot blur) U_g = f × OID/SOD. Large foci, large object-to-image gaps, and short SOD all blur the image. This is why a small focal spot is used for magnification mammography.

Entrance skin dose

Estimate ESD from tube output, kVp, mAs, and FSD.

Entrance skin dose estimates the air kerma (with backscatter) at the patient’s skin from tube output Y, kVp, mAs and focus-to-skin distance. It is the quantity compared with deterministic skin thresholds in fluoroscopy.

Weighted and volumetric CTDI

CTDI_w and CTDI_vol from center, periphery, and pitch.

CTDI_w weights the centre and periphery of a CTDI phantom (1/3 centre + 2/3 periphery) to represent average dose in the slice. Dividing by pitch gives CTDI_vol, the standard dose index displayed on the console.

DLP and effective dose

Dose-length product converted to effective dose with a k-factor.

Dose-length product DLP = CTDI_vol × L (mGy·cm) summarises the whole acquisition. Effective dose E ≈ k × DLP with k from ICRP/AAPM body-region coefficients — a screening estimate, not a personal dose.

HVL and attenuation

Compute μ, HVL, and transmitted intensity through thickness x.

Under narrow-beam geometry, intensity falls exponentially: I = I₀ e^{−μx}. The half-value layer ln2/μ is the thickness that halves the beam — a practical measure of quality (hardness) for kV x-ray beams.

Exposure vs mAs and kVp

Rule of thumb: exposure ∝ mAs × (kVp)^n.

Receptor exposure scales with mAs and approximately with (kVp)^n. n ≈ 2 for air kerma and 4–5 for film optical density. The 15% kVp rule is the n ≈ 5 special case (1.15^5 ≈ 2).

Pixel size and Nyquist frequency

Pixel size from FOV and matrix, plus Nyquist frequency.

Pixel size Δx = FOV / N and the Nyquist frequency 1/(2 Δx) is the highest spatial frequency you can sample without aliasing. A 400 mm FOV on a 512 matrix → 0.78 mm pixels → 0.64 lp/mm Nyquist.

Hounsfield unit

CT number from linear attenuation relative to water.

The Hounsfield unit rescales linear attenuation so water is 0 and air is −1000: HU = 1000 (μ−μ_w)/μ_w. Fat is negative (~ −80 to −100), soft tissue ~0–50, contrast-enhanced vessels hundreds, cortical bone hundreds to 1000+.

15% kVp rule

mAs that keeps exposure constant when kVp changes (film-screen n≈5).

The 15% rule: raising kVp by 15% roughly doubles film exposure (n ≈ 5), so you may halve mAs to keep density. In digital radiography it is a dose-management tool, not a density tool — check EI/DI.

Grid ratio

Grid ratio r = h/D and Bucky factor for mAs increase.

Grid ratio r = h/D (lead height over interspace) sets how well the grid rejects oblique scatter. Higher ratio → better cleanup, more primary loss, higher Bucky factor and more mAs.

Dose-area product

DAP = K_a × area, converted to effective dose with a conversion factor.

Dose-area product (kerma-area product) is incident air kerma times field area and is independent of distance (kerma falls as 1/d² while area grows as d²). Conversion coefficients f turn DAP into a rough effective dose.

CT noise scaling

Pixel noise scales as 1/√(mAs · slice · dose).

In quantum-limited CT, pixel noise σ scales as 1/√(dose × slice thickness × pixel area). Halving mAs multiplies noise by √2 ≈ 1.41. Thinner slices look noisier at the same mAs.

CT pitch

Pitch = table feed per rotation / total collimation.

Pitch is table travel per rotation divided by the total nominal beam collimation N×T. Pitch 1 means no overlap and no gap at isocentre; pitch > 1 is faster and lower dose; pitch < 1 overlaps.

Subject contrast

Fractional difference in transmitted intensity between two regions.

Subject contrast is the relative difference in transmitted fluence between two regions, before the detector. It is driven by photoelectric Δμ at low kVp and by density differences at high kVp.

Air kerma inverse square

Air kerma at a new distance from a measured value.

In air, kerma follows inverse square from a point focal spot (plus a small extrafocal component). Use it to move a measured air-kerma value from the chamber position to the skin or to another SID.

Bucky factor from transmission

B = (primary + scatter in) / (primary + scatter out of the grid).

The Bucky factor is how much you must raise mAs when inserting a grid: incident (primary+scatter) over what the grid transmits. It also yields scatter-to-primary ratios before and after the grid.

Water-equivalent diameter

WED = 2 √(A_w / π) from the water-equivalent area of the CT slice.

Size-specific dose estimate needs a patient-size metric. Water-equivalent diameter converts the slice’s mean HU to the diameter of a water cylinder that would attenuate the same. It is more accurate than geometric diameter for a barrel-chested or dense abdomen. SSDE = f(WED) × CTDIvol.

Size-specific dose estimate SSDE

SSDE = f(WED) × CTDIvol. f from AAPM 204 exponential fits.

CTDIvol is reported to a 16-cm (head) or 32-cm (body) phantom, not to the patient. A paediatric or slim adult is under-represented by a 32-cm CTDIvol (true dose is higher); an extra-large patient is over-represented. The conversion f(d) maps phantom CTDIvol onto the dose to a water cylinder of diameter d = WED.

Nyquist frequency

f_N = 1 / (2 Δx). Spatial frequencies above f_N alias.

A pixel of width Δx can faithfully represent at most one line-pair every two pixels. That limiting frequency is Nyquist. Sampling a bar pattern finer than f_N produces Moiré / aliasing — the classic grid-line artefact on CR, or wrap of high-frequency contrast in MRI. Detector MTF is usually already small at f_N, which is why we get away with it.

Rose model SNR

SNR = C √(N A) for a large-area object against a Poisson background.

Albert Rose asked how many quanta you need to see an object of contrast C and area A. With Poisson statistics the signal is C·N·A and the noise is √(N A), so SNR = C √(N A). Empirically a human observer needs SNR ≈ 5 (the Rose criterion) to detect a low-contrast lesion reliably. This is the ancestor of every detective-quantum-efficiency argument.

Detective quantum efficiency DQE

DQE(f) = SNR_out² / SNR_in² = MTF²(f) / (u² NPS) × (incident quanta).

DQE is the fraction of incoming Poisson information that the detector actually uses. A perfect detector has DQE=1; a real CsI DR panel is 0.6–0.7 at low frequency and falls with MTF². CR is ~0.2–0.3. Raising DQE is how manufacturers cut dose at equal image quality — it is the number to ask for, not just pixel size.

Mean glandular dose (Dance)

AGD = K g c s. Incident air kerma × conversion factors for thickness, composition, spectrum.

European and UK mammography dose is mean glandular dose, not entrance air kerma. Dance factored the conversion into g (glandularity 50%, HVL and thickness), c (composition other than 50%) and s (spectrum: Mo/Mo = 1, Mo/Rh, W/Rh, W/Ag, W/Al). Typical screening AGD is 1–2 mGy per view; EUREF achievable is < 2.0 mGy at 4.5 cm 50% glandularity.

Optical density

OD = log₁₀(I₀ / I). A factor-of-10 drop in transmission is one OD unit.

Film (and now some display QC) uses logarithmic optical density so that equal perceptual steps are equal OD steps. Base+fog ≈ 0.15–0.20, a chest film mid-grey ≈ 1.2, a black border ≈ 3. H&D characteristic curves plot OD versus log exposure. A 0.3 OD change is a factor of 2 in transmission — coincidentally one doubling of exposure in the linear portion of a film curve (γ ≈ 1).

Grey levels and bit depth

N = 2^n distinct levels. Contrast resolution cannot exceed 1/N of the full scale.

An n-bit ADC maps the detector’s analogue range onto 2^n integers. 12-bit CT is 4096 HU levels (the historical −1024 to +3071 window). 14-bit mammography is 16384. Human observers distinguish far fewer grey levels on a display (~8–9 bits after the LUT); extra bits are for processing headroom and to keep quantisation noise below quantum noise.

Scatter-to-primary ratio

SPR = S/P. Grid contrast improvement = (1+SPR)/(1+SPR/K).

In a thick abdomen most of the photons reaching the detector are Compton scatter, not primary. SPR of 4–7 is typical without a grid; a well-used grid (selectivity Σ = K, often 5–10) brings it down and restores subject contrast. The price is a Bucky factor of 3–6 in dose. This is why grids are mandatory on Bucky tables and usually retracted on thin extremities and paediatrics.

K-edge energy

K-edge ≈ 1.17 × Moseley Kα. Iodine 33.2 keV, barium 37.4, gadolinium 50.2, tungsten 69.5.

The K-edge is the binding energy of the K-shell: photoelectric absorption jumps by a factor of 3–5 just above it. Iodine (33.2 keV) and barium (37.4 keV) are matched to the 60–80 kVp spectra of fluoroscopy and GI work. Gadolinium K-edge (50.2 keV) is why Gd can be a CT contrast when iodine is contraindicated. Tungsten K-edge (69.5 keV) is why a 70 kVp beam barely makes W characteristic x-rays.

Fluoroscopy air-kerma rate

K̇ at IRP from displayed K̇ and inverse-square, plus cumulative KAP → time.

Interventional reference point (IRP) is 15 cm toward the tube from isocentre on a C-arm. Displayed air-kerma rate is legally capped (typical US: 88 mGy/min in normal fluoro, 176 mGy/min in high-level). Skin dose is this number × backscatter × f-factor × a table-attenuation correction, and it is what triggers the 2, 5, 10 Gy substantial-radiation-dose-level alerts.

AEC mAs versus thickness

mAs₂ / mAs₁ = exp(μ Δx) at fixed kVp, or the 4–5 cm doubling rule.

Automatic exposure control holds detector dose constant, so mAs tracks e^{μ x} as the patient thickens. A teaching rule of thumb: double mAs for every extra 4–5 cm of soft tissue at fixed kVp (μ ≈ 0.15–0.2 cm⁻¹ in the 80 kVp range). Prefer raising kVp (15% rule) when the mAs would otherwise become huge — that is the thorax vs abdomen technique split.

Radioactive decay

Activity remaining after time t from the half-life.

Radioactive decay is exponential: A = A₀ e^{−λt} with λ = ln2 / T½. After one half-life half remains; after ten half-lives about 0.1% remains. This is the first calculation in any nuclear-medicine or HDR workflow.

Effective half-life

Combine physical decay and biological clearance.

Effective half-life combines physical decay and biological clearance as reciprocals: 1/T_e = 1/T_p + 1/T_b. T_e is always shorter than both. It governs the time integral of activity in an organ (MIRD).

Time between two activities

Elapsed time from A₀, A, and half-life.

Inverting the decay law gives the time between two assayed activities: t = (T½/ln2) ln(A₀/A). Useful when a labelled syringe was assayed earlier and you need the activity ‘now’ — or to date an unknown sample.

Standardized uptake value

SUV from activity concentration, injected dose, and body weight.

SUV is activity concentration in a voxel divided by injected activity per body mass. A uniform distribution would give SUV = 1 g/ml. Tumours, brain and myocardium run higher; lung and fat run lower. SUV_max is the usual PET reporting figure.

Counting efficiency

Well-counter or detector efficiency from count rate, activity, and yield.

Counting efficiency ε = R / (A γ) is counts per photon emitted. A well counter for Tc-99m can exceed 80%; a gamma camera with a collimator is 100–200 cps/MBq (a few 10⁻⁴ as a fraction).

Mo-99 / Tc-99m generator

Daughter activity in transient equilibrium after ingrowth time.

A Mo-99/Tc-99m generator is a classic transient-equilibrium parent–daughter pair. Tc-99m grows in with T½ = 6.02 h while Mo-99 (66 h) slowly decays; activity peaks around 24 h after a perfect elution. Branching to Tc-99m is ~86%.

Cumulated activity

Time-integral of activity to infinity: Ã = A₀ / λ_e.

Cumulated activity à is the time integral of A(t). For a single exponential to infinity, à = A₀/λ_e = 1.443 A₀ T_e. Multiply by the S-value to get organ dose (MIRD).

Pediatric activity (Clark)

Scale adult administered activity by body weight.

Clark’s rule scales adult activity by child/adult weight (reference 70 kg). It is a starting point only — EANM and SNMMI paediatric cards use more sophisticated (often class/weight) schedules and minima.

Exposure rate from Γ

Ẋ = Γ A / d² for a point source in air.

The specific gamma-ray constant Γ converts activity and distance into exposure rate in air: Ẋ = Γ A / d². It is the health-physics companion of inverse square, with the nuclide’s photon yield and energy baked into Γ.

Non-paralyzable dead time

True rate n = m / (1 − mτ) from observed rate and dead time.

Detectors need a finite time τ to process a count. In the non-paralyzable model the true rate is n = m/(1−mτ). Losses of a few percent already appear at tens of kcps in older cameras; PET and well counters have their own τ.

Counting statistics

Poisson: σ = √N, %SD = 100/√N. Time to reach a %SD at rate R.

Nuclear counting is Poisson: variance equals the mean, so σ = √N and the percent standard deviation is 100/√N. 10 000 counts give 1% SD. Time needed is N/R.

Gamma-camera spatial resolution

System FWHM from intrinsic and collimator (geometric) terms.

System spatial resolution of a gamma camera combines intrinsic (crystal + electronics) and collimator geometric resolution in quadrature: R_s = √(R_i² + R_g²). Collimator term usually dominates at 10 cm.

Collimator geometric resolution

R_g ≈ d (L_eff + b) / L_eff for a parallel-hole collimator.

Parallel-hole collimator resolution worsens linearly with distance: R_g = d (L_eff + b) / L_eff. Longer, narrower holes improve resolution and kill sensitivity. L_eff = L − 2/μ accounts for septal penetration.

Residence time

τ = Ã / A₀. The MIRD source-organ residence time.

Residence time τ = Ã / A₀ is the MIRD source-organ time, in hours, that 1 Bq of administered activity spends decaying in that organ. Multiply by S (mGy/MBq·s) after converting hours to seconds, or use consistent tables.

Percent uptake

Organ uptake as a percentage of injected activity, decay-corrected.

Percent uptake is the decay-corrected organ activity as a fraction of injected activity. Classic thyroid uptake uses a probe at 24 h for I-131 or 4–6 h / 24 h for I-123.

Carrier-free specific activity

a = λ N_A / M for a pure radionuclide.

Carrier-free specific activity is λ N_A / M — the Bq per gram if every atom is the radionuclide. F-18 is enormous (TBq/µg); I-131 is high; U-238 is tiny. Real products are diluted by stable carrier and other isotopes.

PET time-of-flight localisation

Δx = c Δt / 2. A 400 ps coincidence window localises the annihilation to ~6 cm FWHM.

In PET the two 511 keV photons are born together. The time difference Δt between their arrivals encodes the position along the line of response: the source is closer to the earlier crystal by Δx = c Δt / 2. Conventional PET (Δt ~ 2–4 ns) only tells you the LOR; TOF PET (200–400 ps) shrinks the uncertainty to a few centimetres and gains SNR roughly as √(D / Δx) where D is the patient diameter.

Noise-equivalent counts NEC

NEC = T² / (T + S + 2R) (or k=1 for a delayed window). The figure of merit for PET count-rate.

Raw coincidence rate is a poor figure of merit because scatter and randoms add variance without adding signal. NEC is the true-coincidence rate that, in a scatter- and random-free scanner, would give the same SNR. Peak NEC (typically 20–40 kcps on a clinical torso phantom) is how vendors compare count-rate performance; it occurs well below the dead-time peak.

Energy resolution

% resolution = 100 × FWHM / E. NaI 140 keV is ~9–10%; CZT is 3–5%; HPGe < 1%.

Scintillator energy resolution is set by photoelectron statistics (and intrinsic scintillator non-proportionality). A 9% NaI camera at 140 keV has a 12.6 keV FWHM photopeak, so a 20% window is ±14 keV and still accepts a good fraction of Compton-scattered photons. Better resolution (CZT, LaBr₃) lets you tighten the window and reject scatter without losing photopeak counts.

PET random coincidences

R = 2 τ S₁ S₂ for a pair of detectors; τ is the coincidence window.

A random coincidence is two unrelated photons arriving within the window τ. Rate is the product of the singles rates times the window (the factor 2 counts both time orderings). Randoms grow as activity squared and become the count-rate killer at high dose; TOF (smaller effective τ) and a well-shielded ring are the remedies. Delayed-window subtraction measures R directly.

SUV lean-body mass (James)

SUV_LBM = C × LBM / A_inj. James LBM from height, weight and sex.

Body-weight SUV overestimates uptake in obese patients because fat takes up almost no FDG. PERCIST prefers SUV normalised to lean body mass (SUL). The James formula is the most common: LBM from weight (kg) and height (cm), different coefficients for male and female. SUL_peak of the background liver is the PERCIST reference.

Mo-99 breakthrough

USP / NRC limit: ≤ 0.15 μCi Mo-99 per mCi Tc-99m (0.15 kBq/MBq) at administration.

A ⁹⁹Mo/⁹⁹ᵐTc generator can leak parent into the eluate. Mo-99 emits 740 keV γ that degrade image quality and add patient dose, so every elution is assayed in a lead pig that shields Tc-99m (140 keV) but not Mo-99. The limit is 0.15 μCi Mo per mCi Tc at the time of administration — note that the ratio grows with time because Tc decays faster (6 h vs 66 h).

Gamma-camera sensitivity

S = C / A (cps/MBq). System sensitivity includes collimator, crystal and window.

System sensitivity is the count rate per unit activity for a known source geometry (typically a Petri dish at 10 cm with a LEHR collimator). A modern NaI camera is ~70–100 cps/MBq on LEHR at 140 keV; a high-sensitivity collimator can double that at the cost of resolution. Daily floods track uniformity, not absolute S — S is a NEMA acceptance / annual test.

Attenuation correction factor

PET: ACF = exp(∫ μ ds) along the LOR. SPECT: exp(μ d) for a broad-beam μ.

A 30-cm abdomen at 511 keV (μ ≈ 0.096 cm⁻¹) attenuates a coincidence by e^{μ D} ≈ 18 — that is why uncorrected PET looks skin-bright and liver-dark. CT-based AC measures μ at ~60–80 keV and bilinearly scales it to 511 keV (or to the SPECT window). A stale μ-map (misregistration, truncation, metal) is the most common AC artefact.

Recovery coefficient

RC = C_measured / C_true. Partial-volume loss for lesions smaller than ~3× FWHM.

A sphere smaller than the reconstructed PSF spills counts into the background (spill-out) and the measured SUV underestimates truth. Recovery coefficient vs diameter is the NEMA image-quality curve: RC ≈ 0.3 at 10 mm, ≈ 0.8 at 22 mm, ≈ 1 at 37 mm on a typical TOF PET. Partial-volume correction multiplies by 1/RC or deconvolves the PSF.

Positron range (empirical)

FWHM range in water ≈ 0.35 × E_mean(MeV) mm — a blur on top of the 2-mm annihilation physics.

The positron is not born at rest: it travels a random-walk millimetres before annihilating. That range is a fundamental blur, independent of the scanner. F-18 (E_mean 250 keV) ~0.2 mm RMS — negligible next to a 4 mm PET voxel. Rb-82 (E_mean 1.5 MeV) ~2–3 mm FWHM, which you can see as extra blur on a cardiac Rb scan. Ga-68 sits in between (~1 mm).

Marinelli thyroid dose

D(Gy) ≈ 0.034 × C(μCi/g) × T_eff(d) × Ē(MeV) for a uniformly distributed β/γ emitter.

The Marinelli formula is the ancestor of MIRD: equilibrium dose in a large organ from a uniformly distributed emitter is activity concentration × effective half-life × mean energy per decay, with a unit-conversion constant. For I-131 thyroid (Ē_β ≈ 0.19 MeV, T_eff ≈ 6 d, uptake U, mass m) it becomes the classic 90–110 Gy from 3–4 MBq/g retained. Modern thyroid dosimetry uses OLINDA / voxel S-values, but Marinelli is how every textbook still introduces the idea.

Coincidence window width

τ ≥ 2 D / c plus timing FWHM. A 70 cm ring needs ≥ 4.7 ns of flight plus jitter.

The coincidence window must be wide enough that a true pair born at the edge of the FOV still meets, but narrow enough that randoms (∝ τ) stay tolerable. Flight time across a 70 cm detector ring is 2.3 ns one-way, so 4.7 ns round-trip; add a few FWHM of timing jitter. Clinical windows are 4–6 ns (BGO, LYSO) and can be 2–3 ns on a fast LSO/LYSO TOF system.

Time, distance, and shielding

Dose from a reference rate with time, inverse square, and transmission.

The three levers of external protection are time, distance and shielding: D = Ḋ₀ t (d₀/d)² B. Cut time, increase distance (inverse square), and insert a barrier of transmission B. ALARA in one line.

Stay time

Maximum time to reach a dose limit at a known rate.

Stay time is dose limit divided by the ambient rate: t = D_lim / Ḋ. It is the clock you give a worker (or a visitor) in a known field, before other controls.

Transmission from HVL and TVL

Barrier transmission for thickness x.

A slab of HVL halves a narrow beam; a TVL reduces it tenfold. B = 2^{−x/HVL} = 10^{−x/TVL}. TVL = HVL × log₂(10) ≈ 3.32 HVL for a monoenergetic beam.

Required thickness

Barrier thickness that yields transmission B.

Invert transmission to get thickness: x = HVL × log₂(1/B) = TVL × log₁₀(1/B). A required B of 0.01 is two TVLs or about 6.6 HVLs.

Required barrier transmission

B from weekly dose limit, workload, use and occupancy factors.

NCRP 151 primary-barrier transmission: B = P d² / (W U T). P is the weekly design goal at the point of interest, W the workload (Gy/week at 1 m), U the use factor (fraction of beam pointing that way), T occupancy.

Thickness from TVL₁ and TVLₑ

First tenth-value layer differs from equilibrium TVL for broad beams.

Broad MV beams harden in the first tenth-value layer, so TVL₁ > TVL_e (equilibrium). Thickness is n×TVL₁ if n≤1, else TVL₁ + (n−1) TVL_e with n = −log₁₀ B.

Equivalent dose

H = D × w_R. Radiation weighting turns gray into sievert.

Equivalent dose H = D w_R puts different radiations on a sievert scale. ICRP 103: photons and electrons w_R = 1, protons 2, α and heavy ions 20, neutrons a function of energy (≈ 2.5–20).

Effective dose (single tissue)

E = w_T H_T for one tissue; sum w_T = 1 over the whole body.

Effective dose E = Σ w_T H_T folds organ doses into one number for stochastic risk. ICRP 103 weights: 0.12 (colon, lung, stomach, breast, remainder, red marrow), 0.08 gonads, 0.04 bladder/liver/thyroid/oesophagus, 0.01 brain/skin/salivary/bone surface.

Head leakage dose

Leakage is limited to 0.1% of useful-beam dose rate at 1 m.

IEC and NCRP cap linac head leakage at 0.1% of the useful-beam dose rate at 1 m from the source. Leakage loads secondary barriers and the maze. Inverse square from the target still applies.

Patient scatter at 1 m

Scattered dose ≈ α (A/400) D / d² with α ~ 0.001 for MV photons.

Patient scatter at 1 m is about 0.1% of the primary dose for a 400 cm² field (α ~ 10⁻³), scaling with field area and 1/d². It dominates secondary-barrier B for many walls beside the couch.

Quality factor (older H = QD)

Legacy ICRU equivalent dose H = Q D. Q ≈ 1 photons, ~10 neutrons, 20 α.

Older ICRU dose equivalent H = Q D uses a quality factor Q (≈ 1 photons, ~10 neutrons, 20 α) instead of ICRP w_R. Numerically similar for many fields; legally, new reports should use ICRP 103.

Weekly design limit P

Convert an annual effective-dose limit to a weekly shielding design goal.

Shielding design goals are weekly. Divide the annual limit by 50 working weeks: public 1 mSv/y → 0.02 mSv/week; many clinics design controlled areas to 5 mSv/y (0.1 mSv/week) even though the occupational limit is 20 mSv/y.

Workload, use and occupancy WUT

W = dose at 1 m × patients × fractions. Design quantity is W U T.

Workload W is the weekly source output at 1 m (Gy/week or mA·min/week). Use factor U is the fraction of time the beam points at this barrier (1 for floors, 0.25 for walls, 1/16 for a rarely used wall). Occupancy T is the fraction of the week a person spends in the area (1 control room, 1/5 corridors, 1/20 outdoor). Shielding thickness is sized so WUT × B / d² ≤ P (weekly design limit).

Primary-barrier transmission B

B = P d² / (W U T). Then n_TVL = log₁₀(1/B).

Rearrange the shielding inequality WUT B / d² ≤ P to find the transmission the wall must provide. Convert B to thickness with the first and equilibrium TVLs of the material at the design energy: x = TVL₁ + (n−1) TVL_e. This is the entire primary-barrier calculation in one line; scatter + leakage is a separate secondary-barrier problem.

Thickness from HVL / TVL

x = TVL₁ + (n − 1) TVL_e with n = log₁₀(1/B). Broad-beam NCRP stacking.

The first TVL is smaller than later ones because the beam hardens and scatter builds up; NCRP therefore publishes TVL₁ and an equilibrium TVL_e. After the first tenth-value, every extra decade of attenuation costs TVL_e. Concrete TVL_e at 6 MV is ~33–37 cm; lead is ~5.7 cm. Never mix narrow-beam μ with a broad-beam B.

I-131 patient-release dose

NRC 35.75: release if D(∞) to the public ≤ 5 mSv. D(∞) ≈ 34.6 Γ A₀ T_p E / r².

A thyroid-ablation patient is a walking source. NRC permits release when the total dose to any other individual is not likely to exceed 5 mSv, using D(∞) = 34.6 Γ A₀ T_p E / r² with occupancy E ≈ 0.25 at 1 m. The 34.6 converts mR/h × days to mR (24×1.44). Typical release threshold is ~1.2 GBq (33 mCi) if no extra instructions, higher with written instructions and a measured dose rate.

Lead equivalence

x_Pb = x_mat × (μ_mat / μ_Pb). Teaching conversion between concrete, steel and lead.

Lead equivalence is how many millimetres of lead would match a given slab of concrete, steel, glass or gypsum at a stated kV. It is energy-dependent: 1 mm Pb ≈ 80–100 mm concrete at 100 kV, but the ratio shrinks at MV because Compton (∝ electron density) takes over from photoelectric (∝ Z³). Aprons are specified in mm Pb-eq at 80–100 kV.

Dose-limit remaining

ICRP 103: 20 mSv/y occupational (averaged 5 y, 50 mSv cap), 1 mSv/y public, 1 mSv embryo.

Occupational effective-dose limit is 20 mSv per year averaged over 5 years, not exceeding 50 mSv in any single year. Lens of eye 20 mSv/y (after ICRP 118), extremities 500 mSv/y, public 1 mSv/y, embryo-fetus 1 mSv after declaration. This calculator subtracts year-to-date dose from a chosen cap so you can see remaining budget — it is not a legal record.

Skyshine (NCRP 151 approximation)

Ḣ ≈ 2.5×10⁻² (B_roof Ḋ_0 Ω^{1.3}) / d² (empirical, mSv/h, teaching form).

Skyshine is photon (and neutron) radiation that goes through a thin roof, scatters in air, and comes down in the car park. It dominates when walls are thick and the roof is the cheap path. NCRP 151 / McGinley give empirical formulas in Ω (solid angle of the roof as seen from the source) and distance d from the isocentre to the outdoor point. This calculator uses a scaled teaching form: you supply a reference rate and it inverse-squares and scales Ω^{1.3}.

Photoneutron TVL (concrete)

n_TVL = log₁₀(Ḣ / P). Concrete TVL for giant-resonance neutrons ~ 20–30 cm.

Above ~8–10 MV the linac grows photoneutrons in the target and jaws (giant dipole resonance). Neutrons do not care about lead — they care about hydrogen (concrete, polyethylene, borated mix). A primary photon wall that is already 2 m of concrete is usually enough for neutrons too; a lead door is not, and needs a hydrogenous core. This calculator turns an unshielded neutron Ḣ into a concrete thickness.

Survival fraction — LQ model

Cell survival after a single dose or n fractions.

The linear-quadratic model writes cell kill as S = exp[−n(αd + βd²)]. The linear term αd dominates at low dose per fraction; the quadratic βd² (double-strand breaks from two tracks) grows with fraction size. α/β is low for late-reacting tissues (~3 Gy) and higher for most tumours (~10 Gy).

Biologically effective dose

BED for comparing fractionation schedules.

Biologically effective dose BED = nd (1 + d/(α/β)) is the LQ dose that would produce the same log cell kill if given in infinitely small fractions. It lets you compare 2 Gy × 30 with 2.67 Gy × 20 on one scale.

Equivalent dose in 2 Gy fractions

Convert a schedule to the equivalent dose in 2 Gy fractions.

EQD₂ converts a schedule to the total dose in 2 Gy fractions that is isoeffective at a given α/β: EQD₂ = D (d+α/β)/(2+α/β). It is BED divided by (1 + 2/(α/β)).

BED with time factor

Subtract repopulation after kick-off time Tk.

Protracted schedules lose BED to tumour repopulation after a kick-off time T_k (often ~21 days for H&N). The subtraction is (ln2)/(α T_p) × (T−T_k). Accelerated regimes recover that term.

Relative biological effectiveness

Reference dose over test-radiation dose for the same effect.

Relative biological effectiveness is the photon (or reference) dose divided by the test-radiation dose that produces the same effect. Protons are assigned RBE 1.1 in the clinic; carbon ions and neutrons are higher and vary with LET, dose, and endpoint.

Oxygen enhancement ratio

Hypoxic dose over aerated dose for the same effect.

Oxygen enhancement ratio is hypoxic dose over aerated dose for the same survival. Photons have OER ~ 2.5–3 at high dose; it falls toward 1 at very low dose and for high-LET radiation (a rationale for carbon ions in hypoxic tumours).

Tumour control probability

Poisson model: TCP = exp(−N₀ S).

Poisson TCP = exp(−N₀ S) is the probability that zero clonogens survive. If N₀ S = 1 (on average one survivor), TCP = 37%. The curve is a steep sigmoid once you plot vs dose through S(d).

Isoeffect dose (Withers)

Change fraction size at constant BED: D₂ = D₁ (d₁+α/β)/(d₂+α/β).

Withers’ isoeffect: if BED is held constant, total dose scales as D₂ = D₁ (d₁+α/β)/(d₂+α/β). Changing from 2 Gy to 3 Gy fractions reduces the total dose more for late tissues (low α/β) than for tumours (high α/β).

α/β from two isoeffective schedules

If two schedules are isoeffective, α/β = (D₁ d₁ − D₂ d₂)/(D₂ − D₁).

Two isoeffective schedules fix α/β: α/β = (D₁ d₁ − D₂ d₂)/(D₂ − D₁). This is how historical fraction-size trials estimated tissue α/β (and why prostate estimates sit near 1.5 Gy).

Surviving fraction at 2 Gy (SF2)

SF2 = exp(−2α − 4β). A common in-vitro radiosensitivity index.

SF2 is surviving fraction after a single 2 Gy dose, a standard in-vitro radiosensitivity index: SF2 = e^{−2α−4β}. Typical tumour SF2 is 0.4–0.6; more sensitive lines sit lower.

Lea–Catcheside g-factor

Incomplete-repair factor for a continuous irradiation of duration T.

The Lea–Catcheside g-factor reduces the quadratic dose term when irradiation is protracted, because some sublethal damage is repaired during the exposure. g → 1 for an acute dose and g → 0 for a very long LDR treatment. BED = D (1 + g D/(α/β)).

Logistic NTCP

NTCP = 1 / (1 + exp(−4 γ₅₀ (D − TD₅₀)/TD₅₀)).

A logistic NTCP is a sigmoid in dose: 50% complications at TD₅₀, with normalised slope γ₅₀ (percent response per percent dose at the 50% point). It is a lightweight stand-in for the Lyman-Kutcher-Burman model when you have a single equivalent uniform dose.

Lyman–Kutcher–Burman NTCP

t = (D − TD₅₀(v)) / (m TD₅₀(v)), TD₅₀(v) = TD₅₀ / v^n, NTCP = Φ(t).

LKB is the classic normal-tissue complication model. TD₅₀ is the uniform whole-organ dose that causes 50% complications, m is the slope (smaller m = steeper), and n is the volume exponent (n→1 parallel organ, n→0 serial). A partial volume v is converted to an equivalent whole-organ dose via the power law, then a probit Φ(t) gives NTCP. Modern practice often prefers gEUD + a logistic, but LKB parameters are still widely published (QUANTEC).

Single-hit multi-target SF

SF = 1 − (1 − e^{−D/D₀})^n. Shoulder characterised by n and Dq = D₀ ln n.

Before LQ, clonogenic survival was fit by a single-hit multi-target model: n targets must each take a hit. The curve has a shoulder (width Dq) then an exponential of slope −1/D₀. n is the extrapolation number (2–5 for many lines). LQ replaced it because the shoulder is better described by βD² and because the multi-target model predicts zero initial slope, contradicting low-dose data. Still used to read D₀ off a published curve.

Quasi-threshold Dq

Dq = D₀ ln n = D₀ ln(n). Width of the survival-curve shoulder.

Dq is the intercept of the terminal exponential with the SF=1 axis — a one-number summary of the shoulder. Broad shoulders (large Dq) mean the tissue (or cell line) repairs sublethal damage well and will be spared by fractionation; narrow shoulders (late-responding? actually the opposite in LQ language — late tissues have small α/β i.e. more curvature). Use it when reading an old paper that quotes D₀ and n instead of α, β.

Ellis NSD (historical)

NSD = D N^{−0.24} T^{−0.11}. Historical isoeffect, superseded by LQ.

Nominal standard dose was the 1960s language of isoeffect: a 60 Gy / 30 fx / 6 week course is 1800 ret. The N^{−0.24} term is fractionation, T^{−0.11} is overall time (repopulation). It treated all tissues as one, overweighted time for late effects, and is not used for prescription today. You will still meet it in old notes and in TDF tables — compute it so you can translate, then redo the problem in BED/EQD2.

LQ-linear (high dose per fraction)

LQ is applied only up to D_t = 2 α/β; beyond that the survival curve is linear with slope γ = α + 2 β D_t.

LQ overestimates cell kill (and BED) at the large doses per fraction of SBRT because the log-survival curve is seen to straighten. LQ-L (or USC, universal survival curve) keeps LQ up to a transition dose D_t ≈ 2 α/β and then a linear tangent. For α/β = 10 Gy, D_t ≈ 20 Gy so a 18 Gy × 3 SBRT scheme is only mildly affected; for α/β = 3 Gy, D_t ≈ 6 Gy and a 10 Gy fraction is well into the linear region.

Logistic TCP

TCP = 1 / (1 + exp(−4 γ₅₀ (D − TCD₅₀)/TCD₅₀)). Population dose–response.

A logistic (or probit) of dose is how empirical tumour-control series are fit when you do not want to commit to a clonogen number. TCD₅₀ is the dose that controls 50% of tumours; γ₅₀ is the normalised slope at that point (typical 1.5–3). It is the tumour-side twin of the logistic NTCP already in the library. Poisson TCP (the other calculator) is more mechanistic but needs N₀ and α.

SBRT / hypofraction EQD2 check

EQD2 = D (d + α/β) / (2 + α/β) applied to a high d, with a warning when d > 2 α/β.

The same EQD2 algebra as the dedicated calculator, flagged for the SBRT regime where its assumptions creak. Late-tissue α/β = 3 Gy: 10 Gy × 5 = 50 Gy physical → EQD2 = 130 Gy — a number that should make you reach for LQ-L or at least a second model. Tumour α/β = 10: the same course is EQD2 = 75 Gy, closer to a 2 Gy-equivalent radical dose. Always quote α/β with the number.

Incomplete-repair BED (two fractions)

θ = e^{−μ Δt}. BED = n d [1 + g d/(α/β)] with g = 1 + 2θ/(n−…) for two doses: 2d(1 + (d(1+θ))/(α/β)).

When two fractions (or two HDR pulses, or two daily IMRT beams many hours apart? usually not) are separated by a gap Δt comparable to the repair half-time, some sublethal damage remains and the β term is inflated by 1+θ, θ = e^{−μ Δt}. Repair T½ is ~0.5–1 h for early tissues and ~1.5–4 h for late. This is why twice-daily treatments need ≥6 h and why pulsed-dose-rate HDR is not equivalent to a single LDR dwell of the same total dose.

Exponential attenuation

Narrow-beam transmission: I = I₀ e^{−(μ/ρ) ρ x}.

Narrow-beam attenuation uses the mass coefficient: I = I₀ exp[−(μ/ρ) ρ x]. μ/ρ comes from XCOM; multiply by density for μ. HVL = ln2 / μ. This is the same physics as the diagnostic HVL equation, written in mass form for any material.

Absorbed dose — TG-51

Water dose from a fully corrected chamber reading and kQ.

AAPM TG-51: dose to water in a high-energy photon or electron beam is the fully corrected chamber reading times the Co-60 absorbed-dose calibration coefficient times the quality conversion k_Q. It replaced TG-21 (air-kerma) in North American clinics.

Temperature-pressure correction

Open-air chamber correction to 22 °C and 101.33 kPa.

Open-air ionisation chambers are calibrated at 22 °C and 101.33 kPa (TG-51). Density of the cavity gas scales as P/T, so k_TP = (273.2+T)/295.2 × 101.33/P. A hot, high-altitude clinic has k_TP > 1 (less gas, you multiply the reading up).

MIRD absorbed dose

Target-organ dose from cumulated activity and the S-value.

MIRD absorbed dose in a target organ is cumulated activity in the source times the S-value (mean dose per nuclear transformation, including self-dose and cross-dose). D = Ã S, with consistent units.

Bragg-Gray relation

Medium dose from gas dose and the mass stopping-power ratio.

Bragg–Gray cavity theory: if a gas cavity is small enough not to perturb the electron fluence, dose to the medium is dose to the gas times the stopping-power ratio medium/gas. It is the ancestor of TG-21 and of every ion-chamber factor.

Electron range in water

Approximate range: E/2 rule of thumb or Katz-Penfold by energy.

Electron CSDA range in water is nearly linear with energy above a few MeV: R ≈ 0.53E − 0.106 cm (Katz–Penfold style). Below 2.5 MeV a power law is used. The clinical ‘E/2 cm’ rule of thumb sits close to practical range Rp.

Exposure to dose in air

D_air = X · (W/e). 1 R ≈ 8.76 mGy in air.

Exposure X (ionisation in air) converts to dose in air by W/e = 33.97 J/C. With 1 R = 2.58×10⁻⁴ C/kg this is 8.76 mGy/R in air. The f-factor then takes you from air to tissue.

Kerma from energy fluence

K = Ψ (μtr/ρ). Collision kerma uses μen/ρ = μtr/ρ (1−g).

Kerma is kinetic energy released per unit mass: K = Ψ (μtr/ρ). Collision kerma K_col = Ψ (μen/ρ) = K(1−g) excludes radiative losses (bremsstrahlung). Under CPE, absorbed dose equals collision kerma.

Roentgen-to-dose f-factor

f_med = 0.876 (μen/ρ)_med / (μen/ρ)_air rad/R. In SI, D = X (W/e) (μen/ρ ratio).

The historical f-factor converts exposure in roentgen to absorbed dose in a medium: f = 0.876 (μen/ρ)_med/(μen/ρ)_air rad/R. For muscle near 100–150 keV, f ≈ 0.94; for bone it is much higher at photoelectric energies.

TG-43 point-source dose rate

Ḋ(r) = S_K Λ (r₀/r)² g(r) F, with r₀ = 1 cm.

AAPM TG-43 writes brachytherapy dose rate as S_K × Λ × geometry × g(r) × F(r,θ). For a point source the geometry is (r₀/r)² with r₀ = 1 cm. Λ for Ir-192 is about 1.12 cGy h⁻¹ U⁻¹.

Recombination P_ion (pulsed)

Two-voltage technique for pulsed beams (linac): P_ion from V_H, V_L, M_H, M_L.

Ion recombination is measured with the two-voltage method. For pulsed linac beams TG-51 uses P_ion = [1−(V_L/V_H)²]/[M_L/M_H − (V_L/V_H)²]. Continuous beams (Co-60, kV) use a linear voltage ratio instead of squares.

W/e and ion charge to dose

Energy spent per ion pair in dry air is 33.97 J/C. D_air = (Q/m)(W/e).

W/e is the average energy spent per unit charge of ionisation in dry air, 33.97 J/C. Dose to air is (Q/m)(W/e). This is how a primary standard converts charge to gray before k_Q and stopping-power ratios take you to water.

Exposure to air dose

D_air = X (W/e) = 0.00876 Gy/R under charged-particle equilibrium.

Under CPE the energy spent creating ion pairs in air is exactly the absorbed dose in air: multiply exposure (C/kg) by W/e. 1 R = 2.58×10⁻⁴ C/kg × 33.97 J/C = 8.76 mGy. Collision kerma equals this dose under CPE; without CPE (build-up region, MV in air without a cap) kerma and dose part company.

Spencer–Attix cavity dose

D_med = D_gas × (L̄/ρ)_gas^med × P. Restricted stopping-power ratio with cutoff Δ.

Bragg–Gray assumes the cavity does not perturb the electron fluence and uses unrestricted stopping powers. Spencer–Attix is the practical version: δ-rays above a cutoff Δ (matched to cavity size, ~10 keV for a Farmer) are treated as part of the electron field, and restricted stopping powers L_Δ are used. Every TG-51 / TRS-398 kQ is, under the hood, a Spencer–Attix ratio evaluated in Monte Carlo.

Displacement correction Pdis

Effective point of measurement: photons Pdis ≈ 1 − 0.4 r / d? TG-51 uses 0.6 r_cav upstream for electrons; photons use a gradient (Pgr) instead.

A cylindrical cavity samples the electron fluence not at its centre but somewhere upstream, because more electrons enter from the higher-fluence side. TG-51 shifts the effective point 0.6 r_cav upstream for electrons (and uses a gradient correction Pgr at dmax for photons instead of a shift). Parallel-plate chambers are assigned Pdis = 1 (reference point at the inner surface of the front window).

Polarity correction Ppol

Ppol = |(M+ − M−)| / 2M_used, or the TG-51 form (M+ − M−) / 2M.

Collecting charge at +300 V versus −300 V does not give equal magnitudes: Compton current, extra-cameral charge and a true polarity effect in the cavity differ. TG-51 takes the signed difference over twice the reading at the polarity you will use. For a well-behaved Farmer at Co-60, Ppol is 1.000 ± 0.002; for plane-parallel electron chambers it can be a percent and must be measured at the user’s energy.

IAEA TRS-398 absorbed dose

D_w = M N_{D,w} k_Q. The IAEA twin of TG-51; k_Q is tabulated versus TPR20,10.

TRS-398 is the internationally used absorbed-dose-to-water protocol. The chamber carries N_{D,w} at Co-60 from a PSDL/SSDL; you correct the reading M for pT, recombination, polarity, electrometer and apply k_Q for the user’s beam quality. Photon quality is TPR20,10 (not %dd(10)x of TG-51). Numerically D_w agrees with TG-51 to ~0.5–1% when both are done carefully.

Water-to-air stopping-power ratio

Teaching fit: (s/ρ)_w,air ≈ 1.102 + 0.003 × (10 − TPR20,10)×10 for MV photons (order of magnitude).

The Spencer–Attix water/air stopping-power ratio falls slowly as the beam hardens (more forward, higher-energy secondaries). Co-60 ≈ 1.133, 6 MV ≈ 1.127, 18 MV ≈ 1.100. This linear teaching fit around TPR20,10 is for intuition — clinical kQ tables already fold SPR, μen and perturbations together. Electrons have a separate, depth-dependent SPR.

CEMA (converted energy per mass)

C = Φ (S_col/ρ). Charged-particle analogue of kerma; equals dose under δ-ray equilibrium.

Kerma is for uncharged particles (photons, neutrons): energy transferred to charged particles per mass. CEMA is the same idea for the charged particles themselves: fluence × mass collisional stopping power. Under δ-ray equilibrium, D = C. The distinction matters in a small cavity (Spencer–Attix Δ-cutoff) and at an interface where equilibrium fails.

Humidity correction k_h

k_h ≈ 0.997 at 50% RH, 20 °C, 101.3 kPa — a 0.3% air-density / W-value mix.

Calibration coefficients are specified for dry air, but the chamber is used in humid air. Water vapour changes both the mass of the cavity gas (density) and W/e (a bit). The net correction is only ~0.3% at typical 50% relative humidity and is often absorbed into the stated N_{D,w}. You will meet k_h in a careful TRS-398 uncertainty budget.

Larmor frequency

Nuclear resonance frequency: f = (γ/2π) B₀.

Spins precess at the Larmor frequency f = (γ/2π) B₀. For ¹H, γ/2π = 42.577 MHz/T, so 63.9 MHz at 1.5 T and 127.7 MHz at 3 T. Every RF pulse, slice select and readout is tuned to this number (plus a tiny chemical-shift offset).

Ernst angle

Flip angle that maximises signal for given TR and T1.

For a spoiled GRE with short TR, the flip angle that maximises steady-state signal is the Ernst angle θ_E = arccos(e^{−TR/T1}). Too low wastes signal; too high saturates long-T1 tissues (CSF).

T1 recovery and T2 decay

Mz after saturation and Mxy after a 90° pulse.

After a 90° pulse, longitudinal magnetisation recovers as M_z = M₀(1−e^{−t/T1}) and transverse magnetisation decays as M_xy = M₀ e^{−t/T2}. T1 is seconds-to-hundreds of ms (tissue, field); T2 is tens of ms and always ≤ T1.

Chemical shift

Water–fat frequency offset from a ppm shift.

Chemical shift is a ppm offset of Larmor frequency due to electronic shielding. Water–fat is ~3.5 ppm, i.e. Δf ≈ 224 Hz at 1.5 T and 448 Hz at 3 T. That offset is both a diagnostic tool (in/out of phase) and an artefact (India-ink, spatial misregistration).

Relative SNR

SNR scales with voxel volume, √(NEX · N_y / BW), and B₀.

MRI SNR scales with voxel volume, √(NEX × N_phase / bandwidth) and roughly with B₀ (more spins, more magnetisation). Halving slice thickness halves SNR; doubling NEX gains only √2.

Pixel and voxel size

FOV over matrix, plus voxel volume.

Pixel size is FOV/matrix in each in-plane direction; voxel volume multiplies by slice thickness. Small voxels buy resolution and cost SNR (see relative SNR).

Slice thickness from gradient

Δz = BW_rf / (γ G_ss). Thinner slices need more gradient or less RF bandwidth.

Slice thickness is RF bandwidth divided by the slice-select gradient (in Hz per metre). Steeper G_ss or narrower RF pulses give thinner slices, at the cost of more eddy currents / heating and a longer pulse.

FOV from readout gradient

FOV = BW / (γ G_read). Higher gradient or lower bandwidth shrinks FOV.

Readout FOV = receiver bandwidth / (γ G_read). A stronger readout gradient or a lower bandwidth shrinks FOV (and may alias if anatomy is larger). Pixel bandwidth is BW/N_x.

Chemical-shift pixels

Water–fat shift in pixels = Δf / (BW / N_x).

The water–fat spatial misregistration in the readout direction equals Δf divided by Hz/pixel. Low bandwidth / high field / high matrix all worsen the India-ink and fat-shift artefacts.

In-phase / opposed-phase TE

TE_in = n/Δf and TE_out = (n+½)/Δf for water–fat.

Water and fat go in and out of phase every 1/Δf. At 1.5 T (Δf ≈ 224 Hz) opposed-phase TE ≈ 2.3 ms and in-phase ≈ 4.5 ms; at 3 T the times halve. Opposed-phase images show India-ink at fat–water borders and are used to detect intracellular fat.

T2* from T2 and field inhomogeneity

1/T2* = 1/T2 + γ ΔB. Extra dephasing from local field offsets.

T2* is shorter than T2 because of extra dephasing from field offsets: 1/T2* = 1/T2 + γ ΔB. GRE, SWI and BOLD contrast live on T2*. A 0.5 µT offset already pulls T2* down hard.

Contrast-agent relaxivity

R1 = R10 + r1 C, so T1 falls as gadolinium concentration rises.

Gadolinium shortens T1 (and T2) linearly: R1 = R10 + r1 C. Typical r1 is ~4 s⁻¹ mM⁻¹ at 1.5 T. After a standard dose the blood T1 can drop from ~1200 ms to a few hundred ms, bright on T1-w GRE.

Spin-echo signal

S ∝ ρ (1 − e^{−TR/T1}) e^{−TE/T2} for a 90°–180° SE sequence.

The classic spin-echo signal S ∝ ρ (1−e^{−TR/T1}) e^{−TE/T2} is the product of a T1 recovery term and a T2 decay term. It is still the mental model for choosing TR/TE even when the sequence is TSE/FSE.

Diffusion b-value

b = (γ G δ)² (Δ − δ/3) for a pair of rectangular pulses (Stejskal–Tanner).

The b-value is the diffusion-weighting dose: how hard the sequence punishes moving spins. Two matched gradient lobes around the 180° (spin-echo) or of opposite sign (STEAM) give b ∝ G² δ² (Δ − δ/3). Clinical DWI uses b = 0 and b = 800–1000 s/mm²; oncology and body DWI go to 1500–2000. ADC maps are a fit of ln S versus b.

Apparent diffusion coefficient