LII: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 (HurterDriffield) 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 3639 of IAEA text (radiation oncology physics) Below are the Mass attenuation coefficient dependencies
 Coherent scattering ≈ Z
 Photoelectric absorption ≈ Z^{3}/E^{3}
 Compton scattering ≈ independent of Z, ≈ 1/E, ≈ electrons/gram
 Pair production ≈ Z
 Triple production ≈ Z^{2}
 Hounsfield units
 HU = 1000* (μ_{tissue}  μ_{water}) / μ_{water}
 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 buildup 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 (Q^{2} * ρ) / (v^{2} * 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
 Energy deposited by particle; depends on charge and density of medium
 Inverse square law: I_{2}/I_{1} = (r_{1}/r_{2})^{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)
 Co60 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
 Co60 ~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 falloff
Factors that affect PDD:
 Energy ==> Increases
 Field size ==> Increases
 SSD ==> Increases
 Depth ==>Decreases
D_{2} = D_{1} * (PDD_{2} / PDD_{1})
By energy at 100 cm SSD, 10x10 field, and depth of 10cm
 Co60 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 [1].
 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 PMID 15507419
Skin dose
Factors that affect Skin dose:
 Energy ==> Decreases
 SSD ==> Decreases
 Field size ==> Increases
 Bolus ==> Increases
 Oblique incidence ==>Increases
Dose Ratios
 Mayneord Ffactor:
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 = W_{0}*dose_{0} + W_{60}*dose_{60}, where W_{0} = 1W_{60}, and W_{60} = tan θ/tan 60
Penumbra
 P = s * (SSD + d  SDD) / SDD, where s is source width and SDD is sourcediaphragm/collimator distance
Superficial energies
 HVL (in Al or Cu) specifies penetrability of lowenergy 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) PMID 10863086 (2000: van Herk M, Int J Radiat Oncol Biol Phys. 2000 Jul 1;47(4):112135.)
 PTV margin = 2.5 sigma + 0.7 delta (cover CTV for 90% of patients with 95% isodose)
Electron Dosimetry
 Probability of bremsstrahlung interaction: Z^{2}
 Xray emission spectrum proportionate to kVp^{2} * mAs / d^{2}, also depends on amount of filtration
 Lead block thickness to attenuate 95%: t_{Pb} (mm) = Electron energy / 2
 Cerrobend block thickness t_{Cerr} = 1.2 * t_{Pb}
 Range
 Practical range in water: R_{p} (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
 d_{ref} = 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 = N_{0} * e^{μx}, where N is number of photons remaining, μ is linear attenuation coefficient, x is thickness of block
 Attenuation: N = N_{0} * (1/2)^{n}, where n is number of HVLs
Brachytherapy
 1 Ci = 37 x 10^{9} Bq
 Activity: A = A_{0} * e^{λt}
 Activity: A = A_{0} * (1/2)^{n}, where n is number of halflives elapsed
 Specific activity: SA = A / m = λ * (N_{a} / A_{W})
 Halflife: t_{1/2} = ln 2 / λ
 Mean (average) life: t_{avg} = 1 / λ = 1.44 * t_{1/2}
 Permanent implant: Dose_{total} = Dose rate_{0} * t_{avg}
 Temporary implant: Dose_{total} = Dose rate_{0} * t_{avg} * (1  exp(t/t_{avg}) = Dose rate_{0} * t_{avg} * (1  exp(λt))
 Exposure rate: X = Γ * Α / d^{2}
 Where Γ is gamma constant, A is activity, and d is distance from source
 Dose rate: D = S_{k} * Λ * G * F * g
 Where S_{k} is airkerma strength, Λ is doserate 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/r^{2}
 Line source: (θ_{2}  θ_{1})/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
 PatersonParker (Manchester): nonuniform needles (1/3, 1/2, 2/3 center vs periphery depending on plane size), uniform dose
 Quimby: uniform needles, nonuniform dose (higher in center)
Shielding
 Workload (W): Beamon 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, d_{iso} is the distance from the source to the isocenter, d_{wall} is the distance from the isocenter to the wall, and F is the maximum field area in cm^{2}.
 Secondary barrier leakage equation:
 where d_{head} is the minimum distance from the linac head to the wall.
Internal Sources
 Effective halflife: Accounts for physical halflife and for biologic halflife, always less than either
 t_{eff,uptake} = (t_{biol, uptake} * t_{phys}) / (t_{biol, uptake} + t_{phys})
 t_{eff,elim} = (t_{biol, elim} * t_{phys}) / (t_{biol, elim} + t_{phys})
Radiation Protection
 Dose equivalent (H): Absorbed dose (D) * W_{R} * N
 W_{R}, previously known as Q, is the quality factor
 N is geometry factor
 Unit in Sievert (Sv)
 Effective dose equivalent (H_{T}): 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 H_{T} for whole body across different tissues
 Gonads have W_{T} = 0.12 (lower than lung/breasts/stomach/bone marrow/colon)
Notes
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