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C it ThCavity Theory, Stopping-Power Ratios, pp gCorrection Factors.Alan E. Nahum PhD

Physics DepartmentPhysics DepartmentClatterbridge Centre for Oncology Bebington, Wirral CH63 4JY UK([email protected])( @ )AAPM Summer School, CLINICAL DOSIMETRY FOR RADIOTHERAPY, 21-25 June 2009, Colorado College, Colorado Springs, USA 1

3 1 INTRODUCTION

A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

3.1 INTRODUCTION

3.2 “LARGE” PHOTON DETECTORS

3.3 BRAGG-GRAY CAVITY THEORY

3.4 STOPPING-POWER RATIOS

3.5 THICK-WALLED ION CHAMBERS

3 6 CORRECTION OR PERTURBATION FACTORS3.6 CORRECTION OR PERTURBATION FACTORS FOR ION CHAMBERS

3 7 GENERAL CAVITY THEORY3.7 GENERAL CAVITY THEORY

3.8 PRACTICAL DETECTORS

23.9 SUMMARY

Accurate knowledge of the (patient) dose


g (p )in radiation therapy is crucial to clinical outcome

For a given fraction size(%

) 80

100 TCP

& N

TCP

40

60

C

Therapeutic Ratio

TCP

0

20

NTCP

Dose (Gy)20 40 60 80 100

0 Dpr

3


Detectors almost never measure dose to medium directly.

Therefore the interpretation of detector readingTherefore, the interpretation of detector reading requires dosimetry theory - “cavity theory”

4


medQ

Df

especially when converting from calibration at Q to

Qdet

medQ

D

f5

from calibration at Q1 to measurement at Q2


Also the “physics” of depth-dose curves:Also the physics of depth-dose curves:

6


Dose computation in a TPSDose computation in a TPS

Termaf Kcf. Kerma

7

First we will remind ourselves of two key results which relate the particle fluence, , to energy deposition in the medium.

Under charged-particle equilibrium (CPE) conditions, theabsorbed dose in the medium, Dmed, is related to the photon(energy) fluence in the medium, h med , by

enCPE

h

ΦKD

med

enmedmedcmed h

ΦKD

for monoenergetic photons of energy h and byfor monoenergetic photons of energy h, and by

dh)h(d enmedCPE max

ΦhD

h

dh)(dh med

enmed

0med

hD

for a spectrum of photon energies where ( /) d is the

8

for a spectrum of photon energies, where (en/)med is the mass-energy absorption coefficient for the medium in question.

For charged particles the corresponding expressions are:

colmed

.eqmδ

med

SΦD

med

where (Scol/)med is the (unrestricted) electron masscollision stopping po er for the medi m

and for a spectrum of electron energies:

collision stopping power for the medium.

EESΦDE

d)(d colmed.eqmδ

d

max

EE

D dd med0

med

Note that in the charged particle case the requirement

9

Note that in the charged-particle case the requirementis: –ray equilibrium


“Photon” Detectors

DETECTOR

MEDIUM

DETECTOR

enCPE

h

ΦKD

10

det

enmeddetcdet h

ΦKD

Thus for the “photon” detector:CPE

det

endet

CPE

det

D

and for the medium:

det

enmed

CPE

med

D

med and therefore we can write:

meden,medCPE

,med

//

Q

ΨΨD

f zz

11

detendetdet /Q

ΨDf

The key assumption is now made that the photon energy fluence in the detector is negligibly different from that

t i th di t b d di t th iti f thpresent in the undisturbed medium at the position of the detector i.e. det = med,z and thus:

dd / D deten

meden

det

,med

//

Q

D

Df z

which is the well known / ratio usually written aswhich is the well-known en/ -ratio usually written as

med

enQ

f

det

Q

f

and finally for a spectrum of photon energies:y p p g

EEE

ΦE en

Ez d)(

dd

med0

,medhmed

en

max

12EE

EΦ

E enE

z d)(d

d

det0

,medh

detmax


The dependence of the en/ -ratio on photon energy for water/medium

13


14“Electron” detectors (Bragg-Gray)


15

BRAGG-GRAY

C OCAVITY THEORY

We have seen that in the case of detectors with sensitive volumes large enough for thet bli h t f CPE th ti D /D i i b ( / ) f th f i di tlestablishment of CPE, the ratio Dmed/Ddet is given by (en/)med,det for the case of indirectly

ionizing radiation. In the case of charged particles one requires an analogous relation in terms of stopping powers. If we assume that the electron fluences in the detector and at the samedepth in the medium are given by det and med respectively then according to Equ 43 wedepth in the medium are given by det and med respectively, then according to Equ. 43 wemust be able to write

D Smed med col med ( / )

DD

SS

med med col med

coldet det det

( / )( / )

(45)

For the more practical case of a spectrum of electron energies, the stopping-power ratiomust be evaluated frommust be evaluated from

DS E EE

E

col med dmax

( ( ) / ) DD

S E EE

E

col

med o

ddet

det

max

( ( ) / )

(48)

o

where the energy dependence of the stopping powers have been made explicit and it isunderstood that E refers to the undisturbed medium in both the numerator and thedenominator. It must be stressed that this is the fluence of primary electrons only; no deltarays are involved (see next section). For reasons that will become apparent in the nextparagraph, it is convenient to denote the stopping-power ratio evaluated according to Equ.48 by sBG

d d t [12]48 by s med,det [12].

The problem with -rays

medium

e-

e-

gas

e-

e-

19


The Spencer-Attix “solution”medium

gasgas

E

D

L E E SE

E

E col

Emed

med meddmax

( ( ) / ) ( )( ( ) / )

20 D

L E E SE

E

E colddet

det det

max

( ( ) / ) ( )( ( ) / )

In the previous section we have neglected the question of delta-ray equilibrium, which is apre-requisite for the strict validity of the stopping-power ratio as evaluated in Equ. 43. Theoriginal Bragg-Gray theory effectively assumed that all collision losses resulted in energyoriginal Bragg Gray theory effectively assumed that all collision losses resulted in energydeposition within the cavity. Spencer and Attix proposed an extension of the Bragg-Grayidea that took account, in an approximate manner, of the effect of the finite ranges of thedelta rays [11]. All the electrons above a cutoff energy , whether primary or delta rays,

id d b f h fl i id h i Allwere now considered to be part of the fluence spectrum incident on the cavity. All energylosses below in energy were assumed to be local to the cavity and all losses above wereassumed to escape entirely. The local energy loss was calculated by using the collisionstopping power restricted to losses less than , L (see lecture 3). This 2-component modelstopping power restricted to losses less than , L (see lecture 3). This 2 component modelleads to a stopping-power ratio given by [12,13]:

Emax

E

medcoltotEmedmed

totEmed SEELE

L

/)()(d/)()(

max

max

gascol

totEgasmed

totE

gas SEELE

/)()(d/)()(a

BRAGG-GRAY CAVITY

Th S i P R i

EESE

d)/)((max

The Stopping-Power Ratio smed, det :

EES

EES

DD

E

colE

d)/)((

d)/)(( medo

det

medmax

EEScolE d)/)(( deto

det

The Spencer-Attix formulation:

medmedd

)/)()((d)/)((max

colE

E

E SEELD

The Spencer Attix formulation:

detdetdet

med

)/)()((d)/)((max

colE

E

E SEELDD


When is a cavity “Bragg-Gray?In order for a detector to be treated as a Bragg–Gray (B–G) cavity there is really only

y gg y

gg y ( ) y y yone condition which must be fulfilled:– The cavity must not disturb the charged particleThe cavity must not disturb the charged particle

fluence (including its distribution in energy) existing in the medium in the absence of the cavity.

In practice this means that the cavity must be small compared to the electron ranges, p g ,and in the case of photon beams, only gas-filled cavities, i.e. ionisation chambers, fulfil , ,this. 23

A second condition is generally added:


A second condition is generally added: The absorbed dose in the cavity is deposited

ti l b th h d ti l i itentirely by the charged particles crossing it. This implies that any contribution to the dose due tophoton interactions in the cavity must be negligible.Essentially it is a corollary to the first condition. If the cavityis small enough to fulfil the first condition then the build-upg pof dose due to interactions in the cavity material itself mustbe negligible; if this is not the case then the chargedparticle fluence will differ from that in the undisturbedparticle fluence will differ from that in the undisturbedmedium for this very reason.

24

A third condition is sometimes erroneouslyadded:

Charged Particle Equilibrium must exist in the absence of the cavity.

Greening (1981) wrote that Gray’s original theory required this. g ( ) y g y qIn fact, this “condition” is incorrect but there are historical reasons for finding it in old publications.

CPE is not required but what is required, however, is that the stopping-power ratio be evaluated over the charged-particle (i.e. electron) spectrum in the medium at the position of the detector.electron) spectrum in the medium at the position of the detector.

Gray and other early workers invoked this CPE condition because they did not have the theoretical tools to evaluate the

25

because they did not have the theoretical tools to evaluate the electron fluence spectrum (E in the above expressions) unless there was CPE. (but today we can do this using MC methods).

Do air-filled ionisation chambers function as Bragg-GrayDo air filled ionisation chambers function as Bragg Gray cavities at KILOVOLTAGE X-ray qualities?

26

Ma C-M and Nahum AE 1991 Bragg-Gray theory and ion chamber dosimetry for photon beams Phys. Med. Biol. 36 413-428

The commonly used air-filled ionization chamber irradiated by a megavoltage photon beamis the clearest case of a Bragg-Gray cavity. However for typical ion chamber dimensions forkilovoltage x-ray beams, the percentage of the dose to the air in the cavity due to photoninteractions in the air is far from negligible as the Table, taken from [9], demonstrates:

Ma C-M and Nahum AE 1991 Bragg-Gray theory and ion chamber dosimetry for photon beams Phys. Med. Biol. 36 413-428

3 1 INTRODUCTION


3.1 INTRODUCTION

3.2 “LARGE” PHOTON DETECTORS

3.3 BRAGG-GRAY CAVITY THEORY

3.4 STOPPING-POWER RATIOS

3.5 THICK-WALLED ION CHAMBERS

3 6 CORRECTION OR PERTURBATION FACTORS3.6 CORRECTION OR PERTURBATION FACTORS FOR ION CHAMBERS

3 7 GENERAL CAVITY THEORY3.7 GENERAL CAVITY THEORY

3.8 PRACTICAL DETECTORS

303.9 SUMMARY

STOPPING-POWER RATIOSSTOPPING POWER RATIOS

31

1.6 airadipose tissue

1.4

1.5

med

ium

bone (compact)GraphiteLiFphoto-emulsion

1 2

1.3

o, w

ater

to m PMMA

Silicon

1.1

1.2

(Sco

l/ )-r

ati

0.9

1.0

0 01 0 10 1 00 10 00 100 00

32

0.01 0.10 1.00 10.00 100.00

Electron energy (MeV)

r1.15

1.20tio

, wat

er/a

iunrestricted

delta=10 keV

1.00

1.05

1.10

g-po

wer

rat

0.90

0.95

1.00

ion

stop

ping

0.80

0.85

0.1 1 10 100Mas

s co

llisi

Electron kinetic energy (MeV)

M

33

Depth variation of the Spencer-Attix water/air stopping-powerratio, sw,air, for =10 keV, derived from Monte Carlo generatedl t t f ti l ll l b d

electron spectra for monoenergetic, plane-parallel, broad electron beams (Andreo (1990); IAEA (1997b)).

1 10

1.15

electron energy (MeV)

302520181410751 3

ir

1.05

1.10

5040

30

r rat

io, s

w,a

1.00

ing-

pow

er

0.95stop

pi

0 4 8 12 16 20 240.90

depth in water (cm)

“THICK-WALLED” ION CHAMBERSA. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

med

enDD

)C()C(

CPE wall

ΔLDD

)C()C(&37wall

wallmed DD

)C()C(air

Δairwall DD

)C()C(&


med

en

wall

Δmed LDQf

)C(

wallairairDQf

)C(

Thick-walled cavity chamber free-in-air where the air volume is known preciselywhere the air volume is known precisely (Primary Standards Laboratories):

')( KLDCKair

en

wall

airi

381

)( Kg

CKwallairair

air


CORRECTION FACTORS FOR ION CHAMBERSFOR ION CHAMBERS (measurements in phantom)( p )

39


Are real practical Ion ChambersAre real, practical Ion Chambers really Bragg-Gray cavities?y gg y

40


imed

airΔairCmed PLDQD /, i

,

d stemcelwallreplmed

airΔairCmed PPPPLDQD /,

41FARMER chamber (distances in millimetres)


THE EFFECT OF THE CHAMBER WALL

42


43


The Almond-Svensson (1977) expression:

medwallmed

1

ΔΔen LL

medairairwall

wall

ΔLP

44air

Δ


45


46

THE EFFECT OF THE FINITE VOLUME OF THE GAS CAVITY

A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction

Factors

47


48


49


FLUENCE PERTURBATION?FLUENCE PERTURBATION?

Electrons

50


PΦΦ P

zE

flcavmed PΦΦ effPJohansson et al used chambers of 3, 5 and 7-mm radius and found an approximately linear relation between (1 P ) and cavity radius (at a given energy)between (1- Pfl) and cavity radius (at a given energy).

Summarising the experimental work by the Wittkämper, Johansson and colleagues, for the NE2571 li d i l h b P i

E EzE

NE2571 cylindrical chamber Pfl increases steadily from ≈ 0.955 at = 2 MeV, to ≈ 0.980 at = 10 MeV and to ≈ 0 997 at = 20 MeV

51

zE zEat = 10 MeV, and to ≈ 0.997 at = 20 MeV.


52

GENERAL CAVITY THEORYWe have so far looked at two extreme cases: i) detectors which are large compared to the electroni) detectors which are large compared to the electronranges in which CPE is established (photon radiation only) ii) d t t hi h ll d t th l tii) detectors which are small compared to the electronranges and which do not disturb the electron fluence (Bragg-Gray cavities) Many situations involve measuring the dose from photon (orneutron) radiation using detectors which fall into neither of theneutron) radiation using detectors which fall into neither of theabove categories. In such cases there is no exact theory.However, so-called General Cavity Theory has been developed as an approximationdeveloped as an approximation.

In essence these theories yield a factor which is a weightedfmean of the stopping-power ratio and the mass-energy

absorption coefficient ratio:

detdet

detdet

det 1

enΔ dLdDD

medmedmed

D

where d is the fraction of the dose in the cavity due to electrons from the medium (Bragg-Gray part),

and (1 - d) is the fraction of the dose from photon interactions in the cavity (“large cavity”/photon detector part)part)

Paul Mobit

EGS4

CaSO4 TLD discs, 0.9 mm thick

Photon beams


SUMMARY OF KEY POINTSSUMMARY OF KEY POINTS

56


57


59


60


61


62


Thank you for your attention

63

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