# 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 ≈ 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 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 (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
* 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)
* 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
D$_2$ = D$_1$ * (PDD$_2$ / PDD$_1$)
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 .
** 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 = W$_0$*dose$_0$ + W$_{60}$*dose$_{60}$, where W$_0$ = 1-W$_{60}$, and W$_{60}$ = 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: Z$^2$
* X-ray 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 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 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 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/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
** 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 cm$^2$.
   * 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)