strategy for dosimetry implementation medaustron€¦ · chemical reactivity and thus affects...
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Proton beam interactions: basic
Hugo Palmans1,2
1 EBG MedAustron GmbH, Wiener Neustadt, Austria 2 National Physical Laboratory, Teddington, UK
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Overview
Interactions:
Energy loss
Scattering
Nuclear interactions
Illustrations of clinical and dosimetry issues
A few examples of other interaction mechanisms (radiolysis, ionization of air, 1-hit detector)
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Quantities and units
See book chapter
Only fluence and dose
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Quantities and units A physical quantity is a property that can be expressed as the product of a number and a unit. A unit is a selected reference sample of a quantity. There are 7 SI base units and 22 SI derived units with a special name
Quantity Unit Type of unit Symbol
Length metre SI base unit m
Mass kilogram SI base unit kg
Area metre square SI derived unit m2
Absorbed dose
gray SI derived unit with special name
Gy (= J kg-1 = m2 s-2)
Absorbed dose
Rad non-SI unit rad (= 0.01 Gy)
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Radiation field: Fluence
Fluence is defined as the number dN of particles incident on a sphere of cross-sectional area da. By using a sphere, the area perpendicular to the direction of each particle is accounted for so that all particles passing through this volume of space are included. Unit: m-2
da
da
dN
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Absorbed Dose
Absorbed dose D Where 𝑑𝜀 is the mean energy imparted to matter of mass dm. Energy imparted is the energy incident minus the energy leaving. The medium should always be specified. Current dosimetry protocols are increasingly based on absorbed dose rather than kerma. Unit: J kg-1 = Gy (gray)
dm
dD
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dm
T2
h=0.511MeV
h1
T1 e-
e+
h=
0.511MeV
= 0 – 1.022 MeV + Q
Q = h1 – 2m0c2 + 2m0c
2 = h1
Absorbed Dose
dm
dD
h1
dm
T
T’
h2
h3
h4
e-
= h1 – (h2 + h3 + T’)
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Electronic stopping power
Linear stopping power (also Linear Energy Transfer LET)
Unit: J m-1
but often quoted in keV μm-1 or MeV cm-1
(Unrestricted) Electronic Mass stopping power
Unit: J m2 kg-1 but often quoted in MeV cm2/g
dx
dES el
dx
dES el
1
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Relation fluence and absorbed dose: stopping power
For a differential fluence med,E(E) of identical charged particles in a medium (if radiative photons escape the volume of interest and secondary electrons are absorbed on the spot), the absorbed dose Dmed is given by: Where Sel/ is the mass electronic stopping power. Integrating over the fluence spectrum for a given medium,
dE
ESED
med
elEmedmed
)()(,
med
elmedmed
SD
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Stochastic and non-stochastic quantities
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Radioactivity
Activity A -dN : number of nuclear transformations in time interval dt Unit: s-1 Special name for the unit of activity is the becquerel (Bq) The primary standard of activity is the beta-gamma-coincidence counter.
𝐴 = λ𝑁 = −𝑑𝑁
𝑑𝑡 𝑁(𝑡) = 𝑁 0 𝑒−λ𝑡
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Radioactivity
Parent-Daughter:
𝑑𝑁𝐷
𝑑𝑡= λ𝑃𝑁𝑃 0 𝑒
−λ𝑃𝑡 − λ𝐷𝑁𝐷 𝑡
𝑁𝐷(𝑡) = 𝑁𝑃 0λ𝑃
λ𝐷−λ𝑃𝑒−λ𝑃𝑡 − 𝑒−λ𝐷𝑡
Nuclear activation
𝑑𝑁𝐷
𝑑𝑡=
𝜎Φλ𝐷
λ𝐷−𝜎Φ𝑁𝑃 0 𝑒−𝜎Φ𝑡 − 𝑒−λ𝐷𝑡
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Radioactivity
Parent-Daughter:
𝑑𝑁𝐷
𝑑𝑡= λ𝑃𝑁𝑃 0 𝑒
−λ𝑃𝑡 − λ𝐷𝑁𝐷 𝑡
𝑁𝐷(𝑡) = 𝑁𝑃 0λ𝑃
λ𝐷−λ𝑃𝑒−λ𝑃𝑡 − 𝑒−λ𝐷𝑡
Nuclear activation
𝑑𝑁𝐷
𝑑𝑡=
𝜎Φλ𝐷
λ𝐷−𝜎Φ𝑁𝑃 0 𝑒−𝜎Φ𝑡 − 𝑒−λ𝐷𝑡
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a: the classical atomic radius
b: the classical impact parameter
Interaction of the proton with the atom
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“Soft” Collisions (b >> a)
• The particle’s Coulomb force field affects the atom as a whole
Atomic distortion / polarization
Excitation
Ionizing by ejecting a valence electron
• Small amount of energy transferred to the atom (a few eV)
• Most probable interaction accounts for ± half of total energy transfer
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Hard (or “Knock-On”) Collisions (b ~ a)
• Interaction primarily with a single atomic electron -ray electron ejected from atom -ray energetic enough to be ionizing dissipates energy along separate track (spur)
• First approximation interaction models neglect binding energy, i.e. electron treated as free
• Few interactions with high energy loss: total energy transfer comparable as soft collisions
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Relative probability soft/hard collision
Figure: GSI
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Secondary electron production
d𝜎
d𝐸𝑒=4𝜋𝑎0
2
𝐸𝑝
𝑚𝑝
𝑚𝑒
𝑅∞
𝐸𝑒
2 (3.28)
𝐸𝑒,𝑚𝑎𝑥 = 4 𝑚𝑒 𝑚𝑝 𝐸𝑝
d𝜎
d𝐸𝑒=4𝜋𝑎0
2
𝐸𝑝
𝑚𝑝
𝑚𝑒
𝑅∞
𝐸𝑒
21 − 𝛽2
𝐸𝑒
𝐸𝑒,𝑚𝑎𝑥 (3.30)
𝐸𝑒,𝑚𝑎𝑥 =𝛽2
1−𝛽2𝑚𝑝𝑐
2 1
1−𝛽2+1
2
𝑚𝑝
𝑚𝑒+𝑚𝑒
𝑚𝑝
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Rutherford and Bhabha cross sections
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Electron slowing down spectrum
(Medin and Andreo 1997, Phys Med Biol 42:89-105)
200 MeV proton beam, z = 20 cm
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
energy / MeV
parti
cle
flu
en
ce p
er i
ncid
en
t p
ro
ton
/ M
eV
-1 c
m -2
proton
electrons
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Coulomb interaction with nucleus (b << a) • Mostly elastic scattering • Small fraction of proton’s energy lost in
momentum transfer • Mainly source of deflection • Multiple scattering • In Monte Carlo, energy loss usually treated
uncorrelated to directional deflection • Differential elastic-scattering cross section
proportional to Z²
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EM energy loss – stopping power
θ
b
𝑣𝑒,⊥ = 𝐹⊥
𝑚𝑑𝑡
𝑡𝑖𝑛𝑡0
= 1
𝑚
𝑧𝑒2
4𝜋 0
1
𝑏2+𝑥2𝑏
𝑏2+𝑥2
𝑑𝑥
𝑣𝑝
+∞
−∞
=2𝑧𝑒2
4𝜋 0𝑚𝑒𝑏𝑣𝑝
𝐸𝑒 =2𝑧2𝑒4
4𝜋𝜀02𝑚𝑒𝑏
2𝑣𝑝2
𝑥
b
db
Δx
d𝐸𝑒 = −d ∆𝐸𝑝 =2𝑧2𝑒4
4𝜋𝜀02𝑚𝑒𝑏
2𝑣𝑝2ρ𝑍
𝐴2𝜋bdb∆𝑥
𝑆𝑒𝑙ρ= −
1
ρ
∆𝐸𝑝∆x
=𝑍
𝐴
𝑧2𝑒4
4𝜋 𝜀02𝑚𝑒𝑣𝑝
2 ln𝑏𝑚𝑎𝑥𝑏𝑚𝑖𝑛
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EM energy loss – stopping power
First order: 1/v2 – Bragg peak:
0.0
0.2
0.4
0.6
0.8
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Depth (cm)
PD
D
100 MeV protons
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Energy loss – stopping powers
Bethe equation for mass electronic stopping power:
𝑆𝑒𝑙𝜌=4𝜋𝑟𝑒
2𝑚𝑒𝑐2
𝛽21
𝑢
𝑍
𝐴𝑧21
2𝑙𝑛
2𝑚𝑒𝑐2𝛽2𝑊𝑚
1 − 𝛽2− 𝛽2 − 𝑙𝑛 𝐼
with
𝑊𝑚 =2𝑚𝑒𝑐
2𝛽2
1 − 𝛽21 +
2
1 − 𝛽2
𝑚𝑒𝑚𝑝+𝑚𝑒𝑚𝑝
2
≈ 4𝐸𝑝𝑚𝑒𝑚𝑝
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Corrections to Bethe formula
𝑆𝑒𝑙
𝜌=4𝜋𝑟𝑒
2𝑚𝑒𝑐2
𝛽21
𝑢
𝑍
𝐴𝑧2
1
2𝑙𝑛
2𝑚𝑒𝑐2𝛽2𝑊𝑚
1−𝛽2− 𝛽2 − 𝑙𝑛 𝐼 −
𝐶
𝑍−𝛿
2+ 𝐵1 + 𝐵2
(3.34)
−𝐶
𝑍 : shell correction (electrons not stationary)
−𝛿
2 : Fermi density effect (electrons shielded due polarisation)
𝐵1 : Barkas or Barkas-Andersen correction (z3 term)
𝐵2 : Bloch correction (z4 term)
Bethe-Barkas-Andersen-Bloch formula
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r o,p = 1mm r o,e = 1mm
Stopping powers – protons versus
electrons
100
101
102
103
10-6 10-4 10-2 100 102
t = E k /E rest
S co
ll / (
MeV
cm
2
g -1
)
proton water
protons air
electrons water
electrons air
0.90
1.00
1.10
1.20
1.30
s w
,air
sw,air protons
sw,air electrons
ICRU 49 ICRU 37
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High-LET component: 200 MeV protons
Kempe et al. (2007) Med. Phys. 34:183-92
0.0
5.0
10.0
15.0
20.0
0 5 10 15 20 25 30
depth in water / g cm -2
en
erg
y lo
ss p
er u
nit
dep
th /
MeV
cm
2
g -1
total 1 H
primary 1 H
secondary 1 H secondary
1 H <10 eV/nm
> 2 eV/nm
> 6 eV/nm
> 10
eV/nm > 20
eV/nm
< 2 eV/nm
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How good do we know stopping powers?
(Paul 2006 NIM-B 247:166)
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How good do we know stopping powers?
-> differences up to 1% in peak to plateau ratio
(Paul 2006 NIM-B 247:166)
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Relation stopping power / range
𝑟0 𝐸𝑝 = −𝑆
𝜌
−1d𝐸
0
𝐸𝑝 (3.41)
(range in g cm-2)
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I values and range uncertainty
Andreo (2009) Phys. Med. Biol. 54:N205-14
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I values and range uncertainty
Andreo (2009) Phys. Med. Biol. 54:N205-14
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I values and range uncertainty
Andreo (2009) Phys. Med. Biol. 54:N205-14
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Sel/ρ as function of HU units
Schneider et al. (1996) Phys. Med. Biol. 41:111
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Sel/ρ as function of HU units
Schneider et al. (1996) Phys. Med. Biol. 41:111
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Stopping powers compounds
Bragg’s rule: 𝑆𝑒𝑙
𝜌 𝑐𝑜𝑚𝑝= 𝑤𝑖
𝑆𝑒𝑙
𝜌 𝑖𝑖 (3.36)
Equivalent:
𝑍
𝐴 𝑐𝑜𝑚𝑝= 𝑤𝑖
𝑍
𝐴 𝑖𝑖 and 𝑙𝑛 𝐼𝑐𝑜𝑚𝑝 =
𝑤𝑖𝑍
𝐴 𝑖𝑙𝑛 𝐼𝑖𝑖
𝑍
𝐴 𝑐𝑜𝑚𝑝
(3.37) &
(3.38)
But: mean excitation energy I, density correction & Barkas correction consistent with ICRU 49
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Example PMMA
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Example: alanine
0.97
0.98
0.99
1.00
1.01
1 10 100 1000
energy / MeV
s a
lan
ine,w
ate
r
alanine
alanine pellet (NPL) Bragg
Bragg
ICRU49 recommend.
ICRU49 recommend.
0.99
1.00
1.01
1.02
1.03
1 10 100 1000
energy / MeV
s a
lan
ine,P
MM
A
alanine
alanine pellet (NPL)
Bragg
Bragg
ICRU49 recommend.
ICRU49 recommend.
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𝑆𝑒𝑙
𝜌=4𝜋𝑟𝑒
2𝑚𝑒𝑐2
𝛽21
𝑢
𝑍
𝐴𝑧2
1
2𝑙𝑛
2𝑚𝑒𝑐2𝛽2𝑊𝑚
1−𝛽2− 𝛽2 − 𝑙𝑛 𝐼 −
𝐶
𝑍−𝛿
2+ 𝐵1 + 𝐵2
(3.34)
Restricted stopping power
𝐿∆
𝜌=4𝜋𝑟𝑒
2𝑚𝑒𝑐2
𝛽21
𝑢
𝑍
𝐴𝑧2
1
2𝑙𝑛
2𝑚𝑒𝑐2𝛽2𝛥
1−𝛽2− 𝛽2 − 𝑙𝑛 𝐼 −
𝐶
𝑍−𝛿
2+ 𝐵1 + 𝐵2
(3.35)
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40
Ion chambers – Spencer-Attix stopping power ratios
Goma et al 2013 Phys Med Biol 58:2509
< Nahum 1978 Phys Med Biol 23:24
1.125
1.130
1.135
1.140
1.145
1.150
1.155
1.160
0 50 100 150 200 250
Eeff (MeV)
(S/
) air
Janni (1982)
ICRU report 49 (1993)
Medin and Andreo (1997)
w
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Ionization chamber perturbations
0.995
1.000
1.005
1.010
1.015
1.020
0 5 10 15
Chamber #
D
w,N
E2
57
1
/D
w
,Ch
C-C
&PTW30002
A150-Al &NE2581
PMMA-Al &PTW30001
Nylon66-Al
IC18
ExrT2
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Stopping power data sources for protons
Janni 1982 Atomic Data Nucl. Data Tables 27:147–529
ICRU report 49
http://www.nist.gov/pml/data/radiation.cfm
http://dedx.au.dk/
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43
Energy loss straggling
Landau distribution:
𝑓 ∆, 𝑠 ≈1
2𝜋𝑒−1
2
∆−𝑎∆𝑚𝑝
𝑠ξ+𝑒
−∆−𝑎∆𝑚𝑝
𝑠ξ
Thick targets (Fano): 𝑓 ∆, 𝑠 ≈1
2𝜋𝑒−∆−𝑠
𝑆𝜌
2
2𝛺2
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44
Energy loss straggling – PRaVDa results
Price et al 2015
JINST 10:P05013
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45
Ionization
Stopping power combines all energy transfers (ionization and excitation)
Interest in ionization: most dosimeters and processes detect ionization only (main exception is calorimetry)
But: to complicate matters more, excitation increases chemical reactivity and thus affects radiation chemistry and biological effects…
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46
Ionization of gasses
𝑊gas =𝐸p
𝑁 (3.44)
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47
Ionization of air
Grosswendt and Baek: 𝑁 𝐸p = 𝑛𝑔𝑎𝑠 𝜎𝑡 𝐸 𝑓 𝐸
𝑆𝑒𝑙 𝐸d𝐸
𝐸p𝐸I
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48
Single scattering
d𝜎
d𝛺=𝑍2𝑒2
4𝜋 0
1
𝑚𝑝𝑐2𝛽2𝑠𝑖𝑛4 𝜃 2
(3.52)
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49
Multiple scattering
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Multiple scattering - Molière
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H. Bichsel, ‘Multiple scattering of protons,’ Phys. Rev. 112 (1958) 182-185 Protons (0.77-4.8 MeV) on targets of Al, Ni, Ag and Au His detector was a tilted nuclear track plate. He fitted his measurements with the Molière form at the appropriate B, adjusting only the characteristic angle θ0 . The results agreed with theory to ±5%.
Multiple scattering – Molière / experimental confirmation
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52
Multiple scattering – Molière / approximations
𝑓 𝜃 𝜃d𝜃 =𝜃
𝜒𝑐2𝐵2𝑒−𝜃2
𝜒𝑐2𝐵 +𝑓(1)
𝜃
𝜒𝑐 𝐵
𝐵+𝑓(2)
𝜃
𝜒𝑐 𝐵
𝐵2d𝜃 (3.54)
≈2𝜃
𝜃02 𝑒−𝜃2
𝜃02d𝜃
with
𝜃0 = 𝜒𝑐 𝐵 − 1.2 (Hanson et al)
or
𝜃0 = 𝜃2 𝑡 =20MeV
𝑝𝑣
𝑡
𝐿𝑅1 +
1
9𝑙𝑜𝑔10
𝑡
𝐿𝑅
(Highland)
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53
Radiation length
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54
Multiple scattering protons/electrons
Moliere parameter:
𝜃0~𝜒𝑐 𝐵~𝑡𝐵
𝑝𝑣
𝜃0 𝑝
𝜃0 𝑒≈
𝑝𝑣 𝑒
𝑝𝑣 𝑝≈
𝛾𝑒𝑚𝑒
𝛾𝑝𝑚𝑝 𝛽𝑝2
For 110 MeV p / 20 MeV e
𝜃0 𝑝
𝜃0 𝑒≈
40×0.511
1.12×938×0.20≈ 0.10
Hollmark 2004 Phys Med Biol 49(14):16
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55
Cavity simulations
electrons protons
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56
Alanine – stack in PMMA (Palmans 2003 Technol Cancer Res Treat. 2:579) experimental data from Onori et al 1997 Med. Phys. 24:447
0.0
2.0
4.0
6.0
8.0
10.0
0.5 1.0 1.5 2.0 2.5 3.0
depth (cm)
D (
MeV
g
-1 )
Experiment
McPTRAN.RZ
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57
Scattering – protons versus photons
(Palmans 2006, Scope 15:5-12)
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58
Nuclear interactions
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59
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Attenuation – nonelastic nuclear interactions
0.0
0.2
0.4
0.6
0.8
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Depth (cm)
PD
D
0.0
0.2
0.4
0.6
0.8
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 N
um
ber o
f pro
ton
s
100 MeV protons
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61
Contribution to dose – 200 MeV protons
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62
How good do we know nuclear interaction cross sections?
Kempe et al. 2007 Med Phys 34:183
0.0
5.0
10.0
15.0
20.0
0 5 10 15 20 25 30
depth in water / g cm -2
en
erg
y l
oss p
er u
nit
dep
th /
MeV
cm
2
g -1
total 1 H
primary 1 H
secondary 1 H secondary 1 H <10 eV/nm
> 2 eV/nm
> 6 eV/nm
> 10
eV/nm
> 20
eV/nm
< 2 eV/nm
ICRU report 63: uc(σn) 10% uc(σprod) 30-40%
-> 2-5% uncertainty on peak to plateau ratio…
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63
How good do we know nuclear interaction cross sections?
Kempe et al. 2007 Med Phys 34:183
0.0
5.0
10.0
15.0
20.0
0 5 10 15 20 25 30
depth in water / g cm -2
en
erg
y l
oss p
er u
nit
dep
th /
MeV
cm
2
g -1
total 1 H
primary 1 H
secondary 1 H secondary 1 H <10 eV/nm
> 2 eV/nm
> 6 eV/nm
> 10
eV/nm
> 20
eV/nm
< 2 eV/nm
ICRU report 63: uc(σn) 10% uc(σprod) 30-40%
-> 2-5% uncertainty on peak to plateau ratio…
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64
Secondary particles – secondary proton halo – frame experiment (protons and ions)
Pedroni et al. 2005 Phys. Med. Biol. 50:541-61, 2005
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65
Secondary particles – secondary proton halo – frame experiment (protons and ions)
Pedroni et al. 2005 Phys. Med. Biol. 50:541-61, 2005 Inaniwa et al. 2009 Med. Phys. 36:2889-97, 2009
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66
Benchmark nuclear reaction cross sections, range and straggling
Paganetti and Gottschalk 2003 Med Phys 30:1926
Kunert et al. 2013 Proc Cyclotrons 2013 Vancouver
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67
Data points: Paganetti and Gottschalk 2003 Med Phys 30:1926
Benchmark nuclear reaction cross sections, range and straggling
1
2
3
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68
Data points: Paganetti and Gottschalk 2003 Med Phys 30:1926
Benchmark nuclear reaction cross sections, range and straggling
1
2
3
1: Geant4 - Paganetti and Gottschalk 2003 Med Phys 30:1926
2: FLUKA - Rinaldi et al. 2011 Phys Med Biol 56:4001
3: SHIELD-HIT07 - Henkner et al. 2009 Phys Med Biol 54:N509
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Scaling of nuclear interaction data with HU
Palmans and Verhaegen, Phys. Med. Biol. 50:991-1000, 2005 Dmed/Dw: Paganetti, Phys. Med. Biol. 54:4399-421, 2009
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70
In vivo-dosimetry : PET and prompt gamma imaging for range verification
Pshenichnov et al. Phys. Med. Biol. 52: 7295-312
Biegun, TU Delft
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71
Radiolysis of water
0,0
1,0
2,0
3,0
LET (keV.mm-1)
G (
100 e
V-1
)
10-1 100 101 102
H+ OH•
e¯aq
H•
OH¯
H2
HO2
H2O2
60Co (100 MeV) (1MeV) protons
Affects any system of which the response is the result of chemical reactions See example water calorimetry
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Conclusions
• Definitions of quantities relevant to proton therapy physics
• Electromagnetic interactions with electrons and the nucleus
• Stopping power theory
• Ionization
• Nuclear interactions
• Single and multiple scattering
• Aqueous radiation chemistry