Cavity
C
it Th
Theory,
Stopping-Power
pp g
Ratios,
Correction Factors.
Alan E. Nahum PhD
Physics Department
Clatterbridge 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
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
31
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
36
3.6
CORRECTION OR PERTURBATION FACTORS
FOR ION CHAMBERS
37
3.7
GENERAL CAVITY THEORY
3.8
PRACTICAL DETECTORS
3.9
SUMMARY
2
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
Accurate knowledge
g of the (p
(patient)) dose
in radiation therapy is crucial to clinical
outcome
For a given fraction size
TCP & NTCP (%)
100
TCP
80
60
Therapeutic Ratio
40
NTCP
C
20
0
20
40
60
Dpr
Dose (Gy)
80
100
3
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
Detectors almost never measure dose to
medium directly.
Therefore, the interpretation of detector reading
Therefore
requires dosimetry theory - “cavity theory”
4
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
 Dmed 

f Q   
 Ddet  Q
especially when converting
from calibration at Q1 to
measurement at Q2
5
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
Also the “physics”
physics of depth-dose curves:
6
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
Dose computation in a TPS
Terma
cf.
f Kerma
K
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, the
absorbed dose in the medium, Dmed, is related to the photon
(energy) fluence in the medium, h med , by
CPE
Dmed  K c med  h Φmed
  en

 


 med
for monoenergetic photons of energy h, and by
CPE
Dmed 
h max
dΦmed
0 h dh
 en (h ) 

 dh
 
 med
for a spectrum of photon energies
energies, where (en/)medd is the
mass-energy absorption coefficient for the medium in question.
8
For charged particles the corresponding expressions are:
δ  eqm .
Dmed  Φmed
 S col 


   med
where (Scol/)med is the (unrestricted) electron mass
collision stopping power
po er for the medium.
medi m
and for a spectrum of electron energies:
δ  eqm. Emax
Dmedd 

0
dΦmed
dE
 S col ( E ) 

 dE
   med
Note that in the charged-particle
charged particle case the requirement
is:
–ray equilibrium
9
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
“Photon” Detectors
MEDIUM
DETECTOR
DETECTOR
CPE
Ddet  K c det  h Φmed
  en

 


 det
10
Thus for the “photon” detector:
 en 
Ddet  det  
  det
CPE
and for the medium:
Dmed
 en 
 med  
  med
CPE
and therefore we can write:
f Q  
Dmed, z
D det
CPE

Ψ med, z  en /  med
Ψ det
en /  det
11
The key assumption is now made that the photon energy
fluence in the detector is negligibly different from that
presentt in
i the
th undisturbed
di t b d medium
di
att the
th position
iti off the
th
detector i.e. det = med,z and thus:
f Q  
Dmedd , z
D det

en /  medd
en /  det
which is the well-known
well known en/ -ratio
ratio usually written as
  en
f Q   
 
med



 det
and finallyy for a spectrum
p
of p
photon energies:
g
Emax
  en

 

med

 

 det

E h
dΦmed , z   en ( E ) 

 dE
dE    med
E h
dΦmed , z   en ( E ) 

 dE
dE    det
0
Emax

0
12
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
The dependence of the en/ -ratio on
photon energy for water/medium
13
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
“Electron” detectors (Bragg-Gray)14
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
15
BRAGG-GRAY
C
CAVITY
THEORY
O
We have seen that in the case of detectors with sensitive volumes large enough for the
establishment
t bli h
t off CPE,
CPE the
th ratio
ti Dmed/Ddet is
i given
i
by
b (
( en/)
/ )med,det for
f the
th case off indirectly
i di tl
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 same
depth in the medium are given by det and med respectively,
respectively then according to Equ.
Equ 43 we
must be able to write
Dmed  med ( Scol /  ) med

Ddet
 det ( Scol /  ) det
(45)
17
For the more practical case of a spectrum of electron energies, the stopping-power ratio
must be evaluated from
Emax
Dmed

Ddet
  E (Scol ( E ) /  ) med dE
o
Emax
  E (Scol ( E ) /  ) det dE
(48)
o
where the energy dependence of the stopping powers have been made explicit and it is
understood that E refers to the undisturbed medium in both the numerator and the
denominator. It must be stressed that this is the fluence of primary electrons only; no delta
rays are involved (see next section). For reasons that will become apparent in the next
paragraph, it is convenient to denote the stopping-power ratio evaluated according to Equ.
48 by sBGmed,det
[12]
d d t [12].
The problem with -rays
medium
e
e
e-
e-
gas



19
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
The Spencer-Attix “solution”
medium
gas


Emax
Dmed

Ddet


 E (L ( E ) /  ) med dE   E (  )(Scol ( ) /  ) med  

Emax


 E (L ( E ) /  ) det dE   E (  )(Scol ( ) /  ) det  

20
In the previous section we have neglected the question of delta-ray equilibrium, which is a
pre-requisite for the strict validity of the stopping-power ratio as evaluated in Equ. 43. The
original Bragg
Bragg-Gray
Gray theory effectively assumed that all collision losses resulted in energy
deposition within the cavity. Spencer and Attix proposed an extension of the Bragg-Gray
idea that took account, in an approximate manner, of the effect of the finite ranges of the
delta rays [11]. All the electrons above a cutoff energy , whether primary or delta rays,
were now considered
id d to be
b part off the
h fluence
fl
spectrum iincident
id on the
h cavity.
i All energy
losses below  in energy were assumed to be local to the cavity and all losses above were
assumed to escape entirely. The local energy loss was calculated by using the collision
stopping power restricted to losses less than , L (see lecture 3). This 2-component
2 component model
leads to a stopping-power ratio given by [12,13]:
Emax
 L

 

med

 

 gas
 

Emax
a
tot
E
 

(E)
tot
E
 L ( E ) /  
(E)
med

 L ( E ) /  
med





tot
d
E


E (  )S col (  ) /  med 
med
tot
d
E


E (  )S col (  ) /  gas 
gas
BRAGG-GRAY CAVITY
Th S
The
Stopping-Power
i P
Ratio
R i smed, det :
Emax
Dmed

Ddet

E
o
Emax

E
( S col ( E ) /  ) med dE
( S col ( E ) /  ) det dE
o
The Spencer
Spencer-Attix
Attix formulation:
Emax
Dmedd

Ddet


 E (L ( E ) /  ) med dE  E ()(Scol () /  ) med 

Emax



E
(L ( E ) /  ) det dE   E ()( S col () /  ) det  
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
When is a cavity
y “Bragg-Gray?
gg
y
In order for a detector to be treated as a
gg
y (B–G)
(
) cavity
y there is really
y onlyy
Bragg–Gray
one condition which must be fulfilled:
– The 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
p
to the electron ranges,
g ,
be small compared
and in the case of photon beams, only gasfilled cavities,, i.e. ionisation chambers,, fulfil
this.
23
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
A second condition is generally added:
The absorbed dose in the cavity is deposited
entirely
ti l by
b the
th charged
h
d particles
ti l crossing
i it.
it
This implies that any contribution to the dose due to
photon interactions in the cavity must be negligible.
Essentially it is a corollary to the first condition. If the cavity
is small enough
g to fulfil the first condition then the build-up
p
of dose due to interactions in the cavity material itself must
be negligible; if this is not the case then the charged
particle fluence will differ from that in the undisturbed
medium for this very reason.
24
A third condition is sometimes erroneously
added:
Charged Particle Equilibrium must exist in the
absence of the cavity.
Greening
g ((1981)) wrote that Gray’s
y original
g
theory
y required
q
this.
In 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.
Gray and other early workers invoked this CPE condition
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).
25
Do air-filled
air filled ionisation chambers function as Bragg-Gray
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 beam
is the clearest case of a Bragg-Gray cavity. However for typical ion chamber dimensions for
kilovoltage x-ray beams, the percentage of the dose to the air in the cavity due to photon
interactions 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
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
31
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
36
3.6
CORRECTION OR PERTURBATION FACTORS
FOR ION CHAMBERS
37
3.7
GENERAL CAVITY THEORY
3.8
PRACTICAL DETECTORS
3.9
SUMMARY
30
STOPPING-POWER
STOPPING
POWER RATIOS
31
air
adipose tissue
bone (compact)
Graphite
LiF
photo-emulsion
PMMA
Silicon
(S col/ )-ratio, water to m
medium
1.6
1.5
1.4
1.3
12
1.2
1.1
1.0
0.9
0 01
0.01
0 10
0.10
1 00
1.00
10 00
10.00
100 00
100.00
Electron energy (MeV)
32
Mass collisiion stopping
M
g-power rattio, water/air
1.20
unrestricted
1.15
delta=10 keV
1.10
1.05
1.00
0.95
0.90
0.85
0.80
0.1
1
10
100
Electron kinetic energy (MeV)
33
Depth variation of the Spencer-Attix water/air stopping-power
ratio, sw,air, for =10 keV, derived from Monte Carlo generated
electron
l t
spectra
t
for
f
monoenergetic,
ti
plane-parallel,
l
ll l broad
b
d
electron beams (Andreo (1990); IAEA (1997b)).
1.15
stoppiing-powerr ratio, sw,air
1
3 5
7
10
electron energy (MeV)
14
18 20
1 10
1.10
25
30
40
50
1.05
1.00
0.95
0.90
0
4
8
12
16
depth in water (cm)
20
24
S
35
36
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
“THICK-WALLED” ION CHAMBERS
med
wall
  en 
 LΔ 


Dmed (C)  Dwall (C) 
& Dwall (C)  Dair (C)  

   wall
 37  air
CPE
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
wall
Dmed (C)  LΔ 
f Q  
  
Dair (C)    air
med
  en 


  

 wall
Thick-walled cavity chamber free-in-air
where the air volume is known precisely
(Primary Standards Laboratories):
Dair
K airi (C ) 
1  g air
wall
 L 
 
  
  air
air
  en 

 K'
  

 wall 38
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
CORRECTION FACTORS
FOR ION CHAMBERS
((measurements in phantom)
p
)
39
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
Are real,
real practical Ion Chambers
really
y Bragg-Gray
gg
y cavities?
40
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

Dmed ,C Q   D air LΔ / 
 P
med
i
air
i

Dmed ,C Q   D air LΔ / 
FARMER chamber

medd
air
Prepl Pwall Pcel Pstem
(distances in millimetres)
41
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
THE EFFECT OF THE CHAMBER WALL
42
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
43
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
The Almond-Svensson (1977) expression:
Pwall
med
   med  L  wall 
 LΔ 
en
Δ
     1    
 
   wall   air 
  air

med
 LΔ 
 
  air
44
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
45
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
46
THE EFFECT OF THE FINITE VOLUME
OF THE GAS CAVITY
A. E. Nahum: Cavity Theory,
Stopping-Power Ratios, Correction
Factors
47
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
48
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
49
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
FLUENCE PERTURBATION?
Electrons
50
Ez
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
Φmed Peff   Φcav Pfl
Johansson et al used chambers of 3, 5 and 7-mm
radius and found an approximately linear relation
between (1(1 Pfl) and cavity radius (at a given energy)
energy).
Summarising the experimental work by the
Wittkämper, Johansson and colleagues, for the
NE2571 cylindrical
li d i l chamber
h b Pfl increases
i
steadily from ≈ 0.955 at E z = 2 MeV, to ≈ 0.980
at E z = 10 MeV
MeV, and to ≈ 0.997
0 997 at E z = 20 MeV
MeV.
51
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
52
GENERAL CAVITY THEORY
We have so far looked at two extreme cases:
i)
detectors which are large compared to the electron
ranges in which CPE is established (photon radiation only)
ii) detectors
d t t
which
hi h are smallll compared
d tto th
the electron
l t
ranges and which do not disturb the electron fluence (BraggGray cavities)
Many situations involve measuring the dose from photon (or
neutron) radiation using detectors which fall into neither of the
above categories. In such cases there is no exact theory.
However, so-called General Cavity Theory has been
developed as an approximation.
approximation
In essence these theories yield a factor which is a weighted
mean off the stopping-power ratio and the mass-energy
absorption coefficient ratio:
Ddet
d
Dmed
det
det
 LΔ 
 en 
   1  d  

   med
   med
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)
Paul Mobit
EGS4
CaSO4 TLD
discs, 0.9
mm thick
Photon
beams
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
SUMMARY OF KEY POINTS
56
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
57
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
59
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
60
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
61
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
62
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
Thank you for your attention
63