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• Film
• ${\displaystyle OD=log({\frac {I_{0}}{I_{t}}})}$, where OD is optical density, ${\displaystyle I_{0}}$ is amount of incident light, and ${\displaystyle I_{t}}$ is amount of transmitted (measured) light
• ${\displaystyle I_{t}=I_{0}\cdot 10^{-OD}}$
• 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

• 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
• ${\displaystyle ES={\frac {4\cdot area}{perimeter}}={\frac {2WL}{W+L}}}$ --- This is only true for W = L since ${\displaystyle {\frac {A}{P}}={\frac {w}{4}}}$

• Otherwise:

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

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

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

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

MU Calculation

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

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
• ${\displaystyle WA=90-HA/2}$
• 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%: 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)

• 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: ${\displaystyle D=B\cdot {\frac {WUT}{d^{2}}}}$
• Primary barrier shielding equation: ${\displaystyle B={\frac {Pd^{2}}{WUT}}}$
• Secondary barrier scattering equation: ${\displaystyle B={\frac {P}{\alpha WT}}d_{iso}^{2}d_{wall}^{2}{\frac {400}{F}}}$
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: ${\displaystyle B={\frac {1000Pd_{head}^{2}}{WT}}}$
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)