Radiation Oncology/Physics/Equations

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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
$ES={\frac {4\cdot area}{perimeter}}={\frac {2WL}{W+L}}$
Equivalent Square for circular field= 0.89 * D; (D=diameter)
- See reference .
- 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

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 lateraly 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)²

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)

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