Radiotherapy Physics Quick Reference
Ryan Flynn, 7/26/2011
Disclaimer: The accuracy of the information in this reference is not guaranteed. It is not
intended for clinical use. Use at your own risk.
1. General
1.1. Important constants
Charge/electron
1.602 x 10-19 C
1 mg-Ra eq
W /e
33.97 J/C or eV/ion pair
1 U (air kerma
strength)
eV
Electron rest
mass
1 amu
Ci
Ci (defn.)
R-to-rad in air
1R
1.602 J
511 keV
8.25 x 10-4 R/h
7.227 µGy m2/h
cGy cm2/hr =
µGy m2/hr
931 MeV = 1.66 x 10-27 kg
3.7 x 1013 Bq
Activity of 1 gm of 226Ra
0.876 rad/R
2.58 x 10-4 C/kg
1.2. Isotopes: γ-emitters
Radioisotope
226
Ra
Rn
60
Co
137
Cs
192
Ir
198
Au
103
Pd
125
I
222
Half-life
1,622 y
3.83 d
5.26 y
30.0 y
73.83 d
2.7 d
16.97 d
59.4 d
Average γray energy
(keV)
830
830
1,250
660
380
412
21
28
Γx-Value
(R-cm2/mCi-hr)
8.25 (R-cm2/mg-hr)
10.15
13
3.26
4.69
2.38
1.48
1.46
1.3. Isotopes: β-emitters
•
•
•
90
89
Sr (0.546 MeV, 28.8 y) -> 90Y (2.28 MeV, 64 hr) -> 90Zr
Sr (1.46 MeV, 50 d) -> 89Y
Electron range in air: 4 m for 2 MeV
Apparent
activity
0.348
0.243
0.486
0.773
0.787
HVL (mm Pb)
12
12
11
5.5
2.5
2.5
0.008
0.025
2. Radiation protection
2.1. Dose equivalent and effective dose equivalent
H E = ∑ wT H T
T
= ∑ wT ∑ wR DT
T
R
wT = weighting factor for tissue, T [unitless]
HT = dose equivalent delivered to tissue T [Sv]
wR = relative weighting factor for radiation R [Sv/Gy]
DT = absorbed dose delivered to tissue T from radiation R [Gy]
HE = effective dose equivalent [Sv]
2.2. Permissible doses
Exposure limits
Occupational 50 mSv/yr
lens
other organs
cumulative
Public
Frequent
Infrequent
lens, other
Fetal
150 mSv/yr
500 mSv/yr
age (yrs) x
10 mSv/yr
1 mSv/yr
5 mSv/yr
50 mSv/yr
5 mSv total
0.5 mSv/mo
Shielding thicknesses (cm)
Energy
4
6
(MV)
Concrete
35
37
Steel
10
10
Lead
5.7 5.7
10
15
18
20
41
11
5.7
44
11
5.7
45
11
5.7
46
11
5.7
Transporting radioactive isotopes
White I Yellow I Yellow III
Max surface (mrem/hr)
0.5
50
200
Transport index
0
1.0
10.0
Cancer induction probability
5% per Sv (ICRP Report 60)
2.3. Shielding
αWT F
0.001 WT
WUT
Secondary: P = 2 2
Leakage: P =
Bs
B
Bp
2
d sec d sca 400
d 2
d
P = permissible dose, W = workload (500 - 1000 Gy/wk),
U = use factor = 1 for controlled area, 1/4 for ceiling, 1/4 for walls),
T = occupancy factor = 1 controlled area, 1/4 partially occupied, 1/16 occasionally occupied,
α = the scatter-to-primary ratio off the scatterer at 1 m, F = field size at scatterer
Rule of thumb: If the secondary and leakage barriers differ by at least three HVLs, then the
thicker of the two will suffice. Otherwise add one HVL onto the thicker of the two and use that.
Skyshine calculations
B D Ω1.3
B ΦΩ
Photons: D = 0.249 × 106 xs io 2
Neutrons: H = 0.84 × 10 −5 ns 2 0
di
( di d s )
D = dose equivalent rate at ground level [nSv/s],
Dio = x-ray dose rate at 1 m from target [cGy/s], Ω = solid angle of radiation beam [steradians],
Bxs = roof shielding transmission ratio, di = distance [m] from x-ray target to 2 m above the roof,
ds = distance [m] from isocenter to the point where the dose equivalent rate is D,
H = nSv/s due to neutrons at ground level, Bns = roof shielding transmission ratio for neutrons,
Φ0 = neutron fluence rate (cm-2 s-1) at 1 m from the target
Primary: P =
2.4. Scatter and secondary particles
•
Maximum energy of 90° Compton scattered photon = 511 keV
•
Photoneutron production energy threshold for photons: 10 MeV
2.5. Radioactive seed disposal
Seeds can be discarded after having decayed away for 10 half-lives. For 125I this is 594 days.
3. Monitor unit calculations
3.1. General equations
D = dose delivered
MU = monitor units
SSD = source-surface distance
SAD = source-axis distance
r = field size at surface
rd = field size at calculation point
rc = coll. setting at SAD
d = depth of calculation point
dmax = depth of maximum dose
SAD setup (TPR/TMR/TAR calculation):
⎛ SAD + d max ⎞
D = MU ⋅ TMR (rd , d ) ⋅ S c (rc ) ⋅ S c (rd ) ⋅ OF ⋅ ⎜
⎟
⎝ SSD + d ⎠
2
SSD setup (PDD calculation):
( )
D = MU ⋅ OF ⋅ PDD(r , d , SSD ) ⋅ S c (rc ) ⋅ S c rd max
3.2. Converting PDD measurements to TMR and TAR data
TMR (rd , d ) =
( )
2
PDD(r , d ) ⎛ SSD + d max ⎞ S p rdmax
⎜
⎟
100 ⎝ SSD + d ⎠ S p (rd )
2
PDD(r , d ) ⎛ SSD + d max ⎞
TAR (rd , d ) =
⎜
⎟ BSF(r )
100 ⎝ SSD + d ⎠
3.3. Converting PDDs at one SDD to PDDs at another SSD
(
(
PDD(SSD 2 , r , d ) TAR rd ,SSD2 , d
Mayneord factor:
=
PDD(SSD1 , r , d ) TAR rd ,SSD1 , d
) ⎛⎜ SSD + d
) ⎜⎝ SSD + d
2
max
1
⎞
⎟⎟
⎠
2
3.4. Scatter-air-ratio calculations
SAR (d,rd ) = TAR (d,rd ) − TAR (d ,0)
TAR = TAR (d,0) + SAR , TAR (d,0) = e − µ (d −dmax )
SAR isolates the scatter component from
TAR values. For irregular fields,
TAR is obtained and used for MU calc.
3.5. Penumbra, gap, and collimator rotation calculations
L ⎞
d ⎛ L1
s(SSD + d - SCD )
L/2
⎜⎜
+ 1 ⎟⎟
penumbra =
tan θ coll =
SCD
SSD
2 ⎝ SSD1 SSD1 ⎠
d = prescription depth, SCD = source-collimator distance, L1,2 = size of field 1 or 2, s = source
focal spot size (~3 mm for a linac), θcoll = collimator angle for craniospinal field
gap =
3.6. Linac beam characteristics for photons and electrons
6 MV
10 MV
18 MV
6 MeV
9 MeV
dmax
1.5
2.3
3.0
1.1
1.9
d90%
4.0
5.5
6.5
1.6
2.6
d80%
6.5
8.0
9.5
1.8
2.9
%DD(10) %DD(20) TPR(10)
TPR(20)
67.0
38.8
0.778
0.523
73.6
46.6
0.846
0.627
77.8
51.1
0.884
0.673
Electron beam properties:
d90% [cm] ~ E [MeV]/4, d80% ~ E/3, range ~ E/2
PDD(0)
52.8
37.9
31.6
77.3
80.5
12 MeV
15 MeV
18 MeV
21 MeV
2.5
3.0
2.4
2.1
3.5
4.4
5.5
6.0
3.8
4.9
6.2
6.9
84.7
89.6
93.4
94.8
E 0 = 2.33R0 , E p = E (1 − d / R p )
3.7. Heterogeneity corrections
CF =
TAR (d1 + ρ 2 d 2 + d 3 )
TAR (d1 + d 2 + d 3 )
CF corrects the dose calculation for heterogeneities. The MU
value will be increased by a factor of 1/CF.
3.8. Wedges
3.8.1. Hinge angles
θ wedge = 90° − ϕ hinge / 2
θwedge = wedge angle
φ = hinge angle = angle between central axes of wedged fields
3.8.2. Generating wedge angles with a universal wedge
Dtot = D0 + DW = MU0 ⋅ OF + MU W ⋅ OF ⋅ WF,
A = D0 /Dtot = MU0 ⋅ OF/Dtot
B = D0 /Dtot = MU W ⋅ OF ⋅ WF/Dtot .
A =1− B
tan θ E θ E
B=
≅
tan θW θW
MU 0 = (1 − B ) ⋅ Dtot / OF
B ⋅ MU 0
B ⋅ Dtot
MUW =
=
(1 − B )WF OF ⋅ WF
Dtot = total dose prescribed, D0 = dose from open field, DW = dose from universal wedged field
θW = universal wedge angle, θE = desired effective wedge angle, OF = open field output factor,
WF = wedge factor, MU0 = MU from open field, MUW = MU from wedged field
3.9. Timer error for superficial x-rays or 60Co
M 1 = nM (tshort + Δt ) = M (t tot + nΔt )
M 2 = M (t tot + Δt )
Δt =
t tot (M 2 − M 1 )
M 1 − nM 2
M = charge collection rate, n = number of short measurements (10 at least),
tshort = short measurement time, ∆t = timer error, ttot = ntshort = total measurement time,
M1 = total charge collected over n short measurements,
M2 = total charge collected over one measurement session of length ttot.
3.10. Patient thickness vs. dose uniformity
Energy
60
Co
4 MV
10 MV
24 MV
Max/midline dose for
30 cm thickness
1.4
1.25
1.15
1.05
See Khan 3rd Ed., p.211-212
Dose ratios start at 1.0 for 10 cm thickness
Dose ratios increase approximately as thickness squared
3.11. Compensators/bolus/spoilers
Compensators: account for missing tissue while maintaining buildup region
Spoilers: increase angular spread of phase space of electrons, degrade e- beam energy
Bolus: shift PDD buildup region toward skin surface
3.12. Photon compensator and lead shield thicknesses determination
tc = photon compensator thickness
TD = tissue deficit (cm)
τ = thickness ratio, function of distance between compensator and absorber
ρc = density of compensator material (Al is 2.7 g/cm3)
Lead shield thickness for e- beams: 0.5 mm / MeV. Cerrobend must be 20% thicker than Pb.
τ
t c = TD
ρc
4. Dosimetry
4.1. TG-51
General: The TG-51 protocol yields absorbed dose in water, in Gy, at the point of measurement
of the absent ion chamber. Reference dosimetry must be performed in a water phantom with
dimensions of at least 30 cm x 30 cm x 30 cm.
For photon beams with energies between 60Co and 50 MeV
60
M = M raw Pion Ppol Pelec PTP
DwQ = MkQ N D,Co
w
Ppol =
−
+
M raw
− M raw
PTP =
+/−
2M raw
Pion =
H
M raw
1 − VH / VL
L
/ M raw
− VH / VL
273.15 + T 760
273.15 + 22 P
DwQ = dose to water for beam of quality, Q, determined by %dd(10)x,
Mraw = raw charge measurement, kQ = quality factor,
Correction factors: Pion = recombination, Ppol = polarity, Pelec = electrometer, PTP = temp/press.
For electron beams with energies between 4 and 50 MeV
⎧1.029I 50 − 0.06 cm for 2 ≤ I 50 ≤ 10 cm
60
d ref = 0.6 R50 − 0.1 cm
R
=
DwQ = Mkecal k R50 ' PgrQ N D,Co
⎨
50
w
1
.
059
I
−
0
.
37
cm
for
10
<
I
50
50
⎩
2
E p,0 = 0.22 + 1.98Rp + 0.0025Rp [MeV]
M (d + 0.5rcav )
PgrQ =
E d = E0 (1 − d / R p )
M (d )
E0 = 2.33 R50 [MeV]
kecal = photon-electron quality conversion factor (chamber-dependent),
k R50 ' = electron beam quality conversion factor (chamber- and beam energy-dependent),
R50 = depth of 50% dose wrt maximum, rcav = cylindrical ion chamber radius
I50 = depth of 50% ionization wrt maximum, after upstream shift of 0.5rcav,
dref = reference depth at which parallel plate to cylindrical chamber conversion factor valid
PgrQ = electron gradient correction factor, Rp = practical electron range, Ep,0 = most probable
energy at depth 0 cm, d = depth in phantom, E d = mean electron energy at depth d
4.2. Cross-calibration of a parallel plate chamber for electron dosimetry
•
•
Determine depth of dref (cm)
Measure charge collected with both the cylindrical and parallel plate chambers with point
of measurement at dref.
•
Q
Co
Calculate kecal N D ,Co
w = Mk ecal k R50 ' Pgr N D , w
(
60
) (
60
) /(Mk ')
cyl
pp
R50
4.3. Measuring electron PDD curves
•
•
•
•
•
Measure ionization curve
Shift curve upstream if necessary
Correct for inverse square
Determine average electron energy with depth
Scale curve by stopping power ratios evaluated at each depth
5. Radiation Biology
5.1. Biological effective dose
N = number of fractions
d = dose per fraction (Gy)
α/β = linear/quadratic factors = 3 for typical tumors, 10 normal tissue
2
⎡
⎛ α ⎞
1 ⎢ α
α / β ⎤⎥ d2 = dose/fraction for new number of fractions
d 2 = − + ⎜⎜ ⎟⎟ + 4 ⋅ BED1
BED1 = original BED
2 ⎢ β
β ⎠
N 2 ⎥ N = new number of fractions
⎝
2
⎣
⎦
⎛
d ⎞
BED = Nd ⎜⎜1 +
⎟⎟
⎝ α / β ⎠
6. Film
6.1. Optical density measurement
I
OD = log10 0
I
OD = optical density
I0 = light without film
I = light transmitted through film
6.2. Film dosimetry for electrons
•
•
Kodak XV-2 film is linear up to 50 cGy
When irradiating a film parallel to the central axis of the beam, it is very important to
ensure that as much air as possible is removed from between the phantom slabs and that
the film is flush with the surface of the phantom that the beam will be aimed at.
7. Brachytherapy
7.1. TG-43 dose calculation formalism
r = radius from center of mass of source [cm]
θ = angle between source axis and calculation point [degrees]
2
G (r , θ )
D = Sk Λ
g (r )F (r , θ ) r0 = 1 cm, θ0 = 90°, Sk = air kerma strength [U] = [cGy cm / hr]
Λ = dose rate constant at (r0, θ0) [cGy/U],
G (r0 , θ 0 )
G(r,θ) = geometry factor [-], g(r) = radial dose function [-],
F(r, θ) = anisotropy function [-]
2
⎧1 / r
point source approximation β = angle subtended by source [radians]
G(r , θ ) = ⎨
L = length of source
⎩β / Lr sin θ line source approximation
g (r ) =
D (r , θ 0 ) / G(r , θ 0 )
D (r , θ ) / G(r , θ )
0
0
0
0
F (r , θ ) =
D (r , θ ) / G(r , θ )
D (r , θ 0 ) / G(r , θ 0 )