Radiation Oncology/Physics/Equations

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Radiation Physics Equations


Diagnostic Radiology[edit]

  • Film
    • OD = log (I0 / It), where OD is optical density, I0 is amount of incident light, and It is amount of transmitted (measured) light
    • It = I0 * 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[edit]

  • Probability of interaction:
  • 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[edit]

Photon d_max (cm)

  • Co-60 0.5
  • 4MV 1.0
  • 6MV 1.5
  • 10MV 2.5
  • 15MV 3.0
  • 18V 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[edit]

  • 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[edit]

  • Square area that has the same PDD as the rectangular field
  • ES = 4 x area / perimeter = 2ab / (a + b)
  • Equivalent Square for circular field= 0.89 * 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 = πr^2, so a = r √π, or a = 0.89 * D (D=diameter)
  • Elliptical fields:
    • Equivalent diameter of elliptical fields: D = 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

Mayneord F-factor = ( ((SSD2 + dmax) / (SSD1 + dmax)) * ((SSD1 + d) / (SSD2 + d)) )2

  • PDD2 = PDD1 * 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

  • D2 = D1 * (TMR2 / TMR1) * (SAD / SSD + d2)2 via inverse square correction

Treatment time: MU = dose at prescription point / OF * PDD * FSC * WF * TF * ISF

Wedges

  • Wedge angle: angle by which the isodose curve is turned by the wedge, typically at 10 cm
  • Hinge angle: angle between 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 lateraly from treatment beam: ~0.1%

Treatment margins[edit]

  • 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[edit]

  • 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[edit]

  • 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)2

Brachytherapy[edit]

  • 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[edit]

  • 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
    • Dose Equation: Dose = B * WUT / d2
    • Shielding Equation: B = P * d2 / WUT


Internal Sources[edit]

  • 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[edit]

  • 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)
  • Effective dose (E): Sum of HT for whole body across different tissues
    • Highest is gonads, with WT = 0.2