Radiation Oncology/Physics/Equations

Radiation Physics Equations

Diagnostic Radiology

• Film
  • $OD=log({\frac {I_{0}}{I_{t}}})$, where OD is optical density, $I_{0}$ is amount of incident light, and $I_{t}$ is amount of transmitted (measured) light
  • $I_{t}=I_{0}\cdot 10^{-OD}$
  • 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* (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 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
• $ES={\frac {4\cdot area}{perimeter}}={\frac {2WL}{W+L}}$ --- This is only true for W = L since ${\frac {A}{P}}={\frac {w}{4}}$

• Otherwise:

• $ES={\frac {area}{perimeter}}={\frac {WL}{2(W+L)}}$.
  • See, The Physics of Radiation Therapy by Khan, Chapter 9, p. 185.

• Equivalent Square for circular field $=0.89\times D$ (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, $a^{2}=\pi \times r^{2}$, so $a=r{\sqrt {\pi }}$, or $a=0.89\times D$
• Elliptical fields:
  • Equivalent diameter of elliptical fields:
  • $D={\frac {2ab}{(a+b)}}$ -- 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 F-factor: $f=\left({\frac {SSD_{2}+d_{max}}{SSD_{1}+d_{max}}}\right)^{2}\left({\frac {SSD_{1}+d}{SSD_{2}+d}}\right)^{2}$
• $PDD_{2}=PDD_{1}\cdot f$

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

• $D_{2}=D_{1}\cdot {\frac {TMR_{2}}{TMR_{1}}}\cdot \left({\frac {SAD}{SSD+d_{2}}}\right)^{2}$ via inverse square correction

MU Calculation

Treatment time or monitor units: $MU={\frac {\mbox{dose at prescription point}}{OF\cdot PDD\cdot FSC\cdot WF\cdot TF\cdot ISF}}$

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
• $WA=90-HA/2$
• 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) PMID 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%: 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 = 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 * (N_a / A_W)
• Half-life: t_1/2 = ln 2 / λ
• Mean (average) life: t_avg = 1 / λ = 1.44 * t_1/2
• Permanent implant: Dose_total = Dose rate₀ * t_avg
• Temporary implant: Dose_total = Dose rate₀ * t_avg * (1 - exp(-t/t_avg) = Dose rate₀ * t_avg * (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 = S_k * Λ * G * F * g
  • Where S_k 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: $D=B\cdot {\frac {WUT}{d^{2}}}$
  • Primary barrier shielding equation: $B={\frac {Pd^{2}}{WUT}}$
  • Secondary barrier scattering equation: $B={\frac {P}{\alpha WT}}d_{iso}^{2}d_{wall}^{2}{\frac {400}{F}}$

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 cm².

• Secondary barrier leakage equation: $B={\frac {1000Pd_{head}^{2}}{WT}}$

where d_head 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
• 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)