1

Proton beam interactions: basic

Hugo Palmans1,2

1 EBG MedAustron GmbH, Wiener Neustadt, Austria 2 National Physical Laboratory, Teddington, UK

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)

3

Quantities and units

See book chapter

Only fluence and dose

4

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)

5

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

6

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

7

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’)

8

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

9

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

10

Stochastic and non-stochastic quantities

11

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

12

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

13

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

14

a: the classical atomic radius

b: the classical impact parameter

Interaction of the proton with the atom

15

“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

16

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

17

Relative probability soft/hard collision

Figure: GSI

18

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

19

Rutherford and Bhabha cross sections

20

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

21

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

22

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

23

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

24

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

25

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

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

27

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

28

How good do we know stopping powers?

(Paul 2006 NIM-B 247:166)

29

How good do we know stopping powers?

-> differences up to 1% in peak to plateau ratio

(Paul 2006 NIM-B 247:166)

30

Relation stopping power / range

𝑟0 𝐸𝑝 = −𝑆

𝜌

−1d𝐸

0

𝐸𝑝 (3.41)

(range in g cm-2)

31

I values and range uncertainty

Andreo (2009) Phys. Med. Biol. 54:N205-14

32

I values and range uncertainty

Andreo (2009) Phys. Med. Biol. 54:N205-14

33

I values and range uncertainty

Andreo (2009) Phys. Med. Biol. 54:N205-14

34

Sel/ρ as function of HU units

Schneider et al. (1996) Phys. Med. Biol. 41:111

35

Sel/ρ as function of HU units

Schneider et al. (1996) Phys. Med. Biol. 41:111

36

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

37

Example PMMA

38

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.

39

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)

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

41

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

42

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/

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

44

Energy loss straggling – PRaVDa results

Price et al 2015

JINST 10:P05013

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…

46

Ionization of gasses

𝑊gas =𝐸p

𝑁 (3.44)

47

Ionization of air

Grosswendt and Baek: 𝑁 𝐸p = 𝑛𝑔𝑎𝑠 𝜎𝑡 𝐸 𝑓 𝐸

𝑆𝑒𝑙 𝐸d𝐸

𝐸p𝐸I

48

Single scattering

d𝜎

d𝛺=𝑍2𝑒2

4𝜋 0

1

𝑚𝑝𝑐2𝛽2𝑠𝑖𝑛4 𝜃 2

(3.52)

49

Multiple scattering

50

Multiple scattering - Molière

51

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

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)

53

Radiation length

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

55

Cavity simulations

electrons protons

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

57

Scattering – protons versus photons

(Palmans 2006, Scope 15:5-12)

58

Nuclear interactions

59

60

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

61

Contribution to dose – 200 MeV protons

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…

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…

64

Secondary particles – secondary proton halo – frame experiment (protons and ions)

Pedroni et al. 2005 Phys. Med. Biol. 50:541-61, 2005

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

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

67

Data points: Paganetti and Gottschalk 2003 Med Phys 30:1926

Benchmark nuclear reaction cross sections, range and straggling

1

2

3

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

69

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

70

In vivo-dosimetry : PET and prompt gamma imaging for range verification

Pshenichnov et al. Phys. Med. Biol. 52: 7295-312

Biegun, TU Delft

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

72

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