Radiation Oncology/Physics/Equations

Radiation Physics Equations Diagnostic Radiology Film , where OD is optical density, is amount of incident light, and is amount of transmitted (measured) light OD values are additive H and D curve (Hurter-Driffield) gives relationship between OD and absorbed dose. Sigmoid shape Flat region: OD independent of dose Toe region: OD increases rapidly Linear region: OD increases linearly with dose Saturation region: OD doesn't increase as function of dose Photon Dosimetry Atomic coefficient dependence Note: Probability of interation is not the same as mass attenuation coefficient Consult Page 36-39 of IAEA text (radiation oncology physics) Below are the Mass attenuation coefficient dependencies Coherent scattering ≈ Z Photoelectric absorption ≈ Z3/E3 Compton scattering ≈ independent of Z, ≈ 1/E, ≈ electrons/gram Pair production ≈ Z Triple production ≈ Z2 Hounsfield units HU = 1000* (μtissue - μwater) / μwater Heterogeneity corrections Lung: 10 cm of lung ≈ 3 cm of tissue = 3.3x Bone: 10 cm of bone ≈ 16 cm of tissue = 0.6x With higher energy, less correction necessary (since Compton effect is 1/E) With higher energy, slower build-up at lung/tumor interface, and thus possibly underdosing If no correction, higher dose at prescription point due to lower attenuation in lung LET Specific ionization: number of ion pairs formed per unit path length; depends on velocity and particle charge Energy transferred to medium per unit path length (energy gain) LET is proportionate to (Q2 * ρ) / (v2 * Z) LET = Specific ionization * W Stopping power Energy deposited by particle; depends on charge and density of medium Colisional: lost due to collisional processes (secondary electrons); predominates, especially at lower energies Radiative: lost due to radiative processes (photons, high energy secondary electrons) Restricted stopping power: energy lost by particle per unit length, locally absorbed Inverse square law: I2/I1 = (r1/r2)2 Back scatter factor (SSD setup): BSF = Exposure at surface / Exposure in air Dose = Exposure (X) * f * BSF Only applies at low energies, dmax at surface Peak scatter factor (SSD setup): PSF = Dose at dmax / Dose in air d_max Photon d_max (cm) Co-60 0.5 4MV 1.0 6MV 1.5 10MV 2.5 15MV 3.0 18MV 3.2 20MV 3.5 25MV 4.0 In most centers, we have 6MV, 10MV and 18MV so 6MV : 1.5cm 10MV : 2.5cm 18MV : 3.2cm Photon attenuation Co-60 ~4.0% per 1 cm depth 6MV ~3.5% per 1 cm depth 20MV ~2.0% per 1 cm depth PDD Percent depth dose (SSD setup): PDD = Dose at depth / Dose at dmax Two components: patient attenuation and inverse square dose fall-off Factors that affect PDD: Energy ==> Increases Field size ==> Increases SSD ==> Increases Depth ==>Decreases D2 = D1 * (PDD2 / PDD1) By energy at 100 cm SSD, 10x10 field, and depth of 10cm Co-60 56% 4MV 61% 6MV 67% 10MV 73% 20MV 80% 25MV 83% Equivalent squares Square area that has the same PDD as the rectangular field --- This is only true for W = L since Otherwise: . See, The Physics of Radiation Therapy by Khan, Chapter 9, p. 185. Equivalent Square for circular field (D=diameter) See reference . A square with side a will be equivalent to a circle with radius r when they have the same area, , so , or Elliptical fields: Equivalent diameter of elliptical fields: -- see 15507419 Skin dose Factors that affect Skin dose: Energy ==> Decreases SSD ==> Decreases Field size ==> Increases Bolus ==> Increases Oblique incidence ==>Increases Dose Ratios Mayneord F-factor: Tissue air ratio (SAD setup): TAR = Dose at depth / Dose in air Tissue phantom ratio (SAD setup): TPR = Dose at depth / Dose at reference depth Tissue maximum ratio (SAD setup): TMR = Dose at depth / Dose at dmax via inverse square correction MU Calculation Treatment time or monitor units: where OF is the output factor, WF is the wedge factor, TF is the tray factor, and ISF is the inverse square factor. Wedges Wedge angle: angle by which the isodose curve is turned by the wedge, typically at 10 cm Hinge angle: angle between the central axes of two incident beams Dose for arbitrary wedge field θ using flying wedge or dynamic wedge = W0*dose0 + W60*dose60, where W0 = 1-W60, and W60 = tan θ/tan 60 Penumbra P = s * (SSD + d - SDD) / SDD, where s is source width and SDD is source-diaphragm/collimator distance Superficial energies HVL (in Al or Cu) specifies penetrability of low-energy photon beam. HVL is determined by the combination of kVp and filtration (different combinations can give same HVL) Typically short SSD is used Compared with electrons, superficial photons have sharper penumbra, deliver higher skin dose, but also higher dose to underlying tissues Blocks Dose under 1.5 cm width block (5 HVL), in 15 x 15 cm field, 6 MV, 5 cm depth is ~15% of open field dose. Transmitted dose is ~3% (shielded by 5 HVL), scattered dose from open field contributes the rest Scattered dose Patient with pacemaker, if dose to pacemaker to be <5%, need to be at least 2cm from 6 MV beam edge Patient with breast tangents, ovaries 20 cm from field: dose to ovaries ~0.5% Dose at 1 m laterally from treatment beam: ~0.1% Treatment margins PTV margin PTV margin = 2.5 (quadratic sum of standard deviation of all preparation (systematic) errors) + 0.7 * (quadratic sum of standard deviation of all execution (random) errors) 10863086 (2000: van Herk M, Int J Radiat Oncol Biol Phys. 2000 Jul 1;47(4):1121-35.) PTV margin = 2.5 sigma + 0.7 delta (cover CTV for 90% of patients with 95% isodose) Electron Dosimetry Probability of bremsstrahlung interaction: Z2 X-ray emission spectrum proportionate to kVp2 * mAs / d2, also depends on amount of filtration Lead block thickness to attenuate 95%: tPb (mm) = Electron energy / 2 Cerrobend block thickness tCerr = 1.2 * tPb Range Practical range in water: Rp (cm) = Electron energy / 2 R50: depth at which dose is 50% of maximum Depth of calibration I50: Find depth of 50% ionization in water R50: Calculate R50 = 1.029 * I50 - 0.06 if <10 cm depth, R50=1.059 * I50 - 0.37 if >10 cm depth dref = 0.6 * R50 - 0.1 Energy is specified by the R50 parameter Typically treated as SSD setup No physical source in accelerator head; clinical beams appears to emerge from a "virtual source". Can be found by backprojecting beam profiles at different depths Virtual SSD shorter than actual (photon) SSD Inverse square corrections can be done on virtual SSD for large fields; for small fields effective SSD should be determined Output Dose rate = Applicator Dose rate * Back scatter factor(cutout)/Back scatter factor(Applicator)/ (SSD/SSD+SO)^2 (SSD= Source to surface distance & SO= Stand Off) Radiation Quality Half Value Layer: HVL = ln 2 / μ Tenth Value Layer: 1 TVL = 3.32 HVL Attenuation: N = N0 * e-μx, where N is number of photons remaining, μ is linear attenuation coefficient, x is thickness of block Attenuation: N = N0 * (1/2)n, where n is number of HVLs Brachytherapy 1 Ci = 37 x 109 Bq Activity: A = A0 * e-λt Activity: A = A0 * (1/2)n, where n is number of half-lives elapsed Specific activity: SA = A / m = λ * (Na / AW) Half-life: t1/2 = ln 2 / λ Mean (average) life: tavg = 1 / λ = 1.44 * t1/2 Permanent implant: Dosetotal = Dose rate0 * tavg Temporary implant: Dosetotal = Dose rate0 * tavg * (1 - exp(-t/tavg) = Dose rate0 * tavg * (1 - exp(-λt)) Exposure rate: X = Γ * Α / d2 Where Γ is gamma constant, A is activity, and d is distance from source Dose rate: D = Sk * Λ * G * F * g Where Sk is air-kerma strength, Λ is dose-rate constant, G is geometry factor (see below), F is anisotropy factor, and g is radial dose function Geometry factor G(r,θ) Point source: 1/r2 Line source: (θ2 - θ1)/Ly, where L is length of line, y is distance ICRU dose rate: Low 0.4 - 2.0 Gy/h Medium 2.0 - 12.0 Gy/h High >12.0 Gy/h Brachytherapy systems Paterson-Parker (Manchester): non-uniform needles (1/3, 1/2, 2/3 center vs periphery depending on plane size), uniform dose Quimby: uniform needles, non-uniform dose (higher in center) Shielding Workload (W): Beam-on time (in Gy at 1 m from source) Use factor (U): Fraction of time beam aimed at particular target (dimensionless) Occupancy factor (T): Fraction of time area is occupied by an individual (dimensionless) Distance (d): from isocenter to area of interest (m) Barrier transmission factor (B): amount of radiation passing through barrier Permissible dose (P): maximum dose for an area of interest (Gy) Shielding equations Primary barrier dose equation: Primary barrier shielding equation: Secondary barrier scattering equation: where α is the scattered fraction, diso is the distance from the source to the isocenter, dwall is the distance from the isocenter to the wall, and F is the maximum field area in cm2. Secondary barrier leakage equation: where dhead is the minimum distance from the linac head to the wall. Internal Sources Effective half-life: Accounts for physical half-life and for biologic half-life, always less than either teff,uptake = (tbiol, uptake * tphys) / (tbiol, uptake + tphys) teff,elim = (tbiol, elim * tphys) / (tbiol, elim + tphys) Radiation Protection Dose equivalent (H): Absorbed dose (D) * WR * N WR, previously known as Q, is the quality factor N is geometry factor Unit in Sievert (Sv) Effective dose equivalent (HT): Sum of H for a given tissue across different radiation types (e.g. for nuclear explosion) Formerly known as "equivalent" dose Effective dose (E): Sum of HT for whole body across different tissues Gonads have WT = 0.12 (lower than lung/breasts/stomach/bone marrow/colon)