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
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
 * Atomic coefficient dependence
 * 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* (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
Two components: patient attenuation and inverse square dose fall-off
 * Percent depth dose (SSD setup): PDD = Dose at depth / Dose at dmax

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.
 * 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
Tissue air ratio (SAD setup): TAR = Dose at depth / Dose in air
 * 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 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%: 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

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

Radiation Protection

 * 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)
 * Formerly known as "equivalent" dose
 * Effective dose (E): Sum of HT for whole body across different tissues
 * Gonads have WT = 0.12 (lower than lung/breasts/stomach/bone marrow/colon)