1. C it Th
Cavity Theory,
Stopping-Power Ratios,
pp g
Correction Factors.
Alan E. Nahum PhD
Physics Department
Physics Department
Clatterbridge Centre for Oncology
Bebington, Wirral CH63 4JY UK
(alan.nahum@ccotrust.nhs.uk)
( @ )
AAPM Summer School, CLINICAL DOSIMETRY FOR RADIOTHERAPY,
21-25 June 2009, Colorado College, Colorado Springs, USA 1
2. 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 FACTORS
3.6 CORRECTION OR PERTURBATION FACTORS
FOR ION CHAMBERS
3 7 GENERAL CAVITY THEORY
3.7 GENERAL CAVITY THEORY
3.8 PRACTICAL DETECTORS
2
3.9 SUMMARY
3. Accurate knowledge of the (patient) dose
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
g (p )
in radiation therapy is crucial to clinical
outcome
For a given fraction size
(%)
80
100 TCP
&
NTCP
40
60
C
Therapeutic Ratio
TCP
0
20
NTCP
Dose (Gy)
20 40 60 80 100
0
Dpr
3
4. 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, the interpretation of detector reading
requires dosimetry theory - “cavity theory”
4
5. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
med
Q
D
f
especially when converting
from calibration at Q to
Q
det
med
Q
D
f
5
from calibration at Q1 to
measurement at Q2
6. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
Also the “physics” of depth-dose curves:
Also the physics of depth-dose curves:
6
7. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
Dose computation in a TPS
Dose computation in a TPS
Terma
f K
cf. Kerma
7
8. 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
en
CPE
h
Φ
K
D
med
en
med
med
c
med h
Φ
K
D
for monoenergetic photons of energy h and by
for monoenergetic photons of energy h, and by
dh
)
h
(
d en
med
CPE max
Φ
h
D
h
dh
)
(
dh med
en
med
0
med
h
D
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.
9. For charged particles the corresponding expressions are:
col
med
.
eqm
δ
med
S
Φ
D
med
where (Scol/)med is the (unrestricted) electron mass
collision stopping po er for the medi m
and for a spectrum of electron energies:
collision stopping power for the medium.
E
E
S
Φ
D
E
d
)
(
d col
med
.
eqm
δ
d
max
E
E
D d
d med
0
med
Note that in the charged particle case the requirement
9
Note that in the charged-particle case the requirement
is:
–ray equilibrium
10. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
“Photon” Detectors
DETECTOR
MEDIUM
DETECTOR
en
CPE
h
Φ
K
D
10
det
en
med
det
c
det h
Φ
K
D
11. Thus for the “photon” detector:
CPE
det
en
det
CPE
det
D
and for the medium:
det
en
med
CPE
med
D
med
and therefore we can write:
med
en
,
med
CPE
,
med
/
/
Q
Ψ
Ψ
D
f z
z
11
det
en
det
det /
Q
Ψ
D
f
12. 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 th
present in the undisturbed medium at the position of the
detector i.e. det = med,z and thus:
d
d /
D
det
en
med
en
det
,
med
/
/
Q
D
D
f z
which is the well known / ratio usually written as
which is the well-known en/ -ratio usually written as
med
en
Q
f
det
Q
f
and finally for a spectrum of photon energies:
y p p g
E
E
E
Φ
E en
E
z
d
)
(
d
d
med
0
,
med
h
med
en
max
12
E
E
E
Φ
E en
E
z
d
)
(
d
d
det
0
,
med
h
det
max
13. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
The dependence of the en/ -ratio on
photon energy for water/medium
13
14. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
14
“Electron” detectors (Bragg-Gray)
15. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
15
16. BRAGG-GRAY
C O
CAVITY THEORY
We have seen that in the case of detectors with sensitive volumes large enough for the
t bli h t f CPE th ti D /D i i b ( / ) f th f i di tl
establishment 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 same
depth in the medium are given by det and med respectively then according to Equ 43 we
depth in the medium are given by det and med respectively, then according to Equ. 43 we
must be able to write
D S
med med col med
( / )
D
D
S
S
med med col med
col
det det det
( / )
( / )
(45)
18. For the more practical case of a spectrum of electron energies, the stopping-power ratio
must be evaluated from
must be evaluated from
D
S E E
E
E
col med d
max
( ( ) / )
D
D
S E E
E
E
col
med o
d
det
det
max
( ( ) / )
(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 sBG
d d t [12]
48 by s med,det [12].
20. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
The Spencer-Attix “solution”
medium
gas
gas
E
D
L E E S
E
E
E col
E
med
med med
d
max
( ( ) / ) ( )( ( ) / )
20
D
L E E S
E
E
E col
d
det
det det
max
( ( ) / ) ( )( ( ) / )
21. 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-Gray theory effectively assumed that all collision losses resulted in energy
original Bragg 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,
id d b f h fl i id h i All
were now considered to be part of the fluence spectrum incident on the cavity. 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 model
stopping power restricted to losses less than , L (see lecture 3). This 2 component model
leads to a stopping-power ratio given by [12,13]:
Emax
E
med
col
tot
E
med
med
tot
E
med S
E
E
L
E
L
/
)
(
)
(
d
/
)
(
)
(
max
max
gas
col
tot
E
gas
med
tot
E
gas
S
E
E
L
E
/
)
(
)
(
d
/
)
(
)
(
a
22. BRAGG-GRAY CAVITY
Th S i P R i
E
E
S
E
d
)
/
)
(
(
max
The Stopping-Power Ratio smed, det :
E
E
S
E
E
S
D
D
E
col
E
d
)
/
)
(
(
d
)
/
)
(
( med
o
det
med
max
E
E
Scol
E d
)
/
)
(
( det
o
det
The Spencer-Attix formulation:
med
med
d
)
/
)
(
)(
(
d
)
/
)
(
(
max
col
E
E
E S
E
E
L
D
The Spencer Attix formulation:
det
det
det
med
)
/
)
(
)(
(
d
)
/
)
(
(
max
col
E
E
E S
E
E
L
D
D
23. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
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 y
one condition which must be fulfilled:
– The cavity must not disturb the charged particle
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
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
24. A second condition is generally added:
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
A second condition is generally added:
The absorbed dose in the cavity is deposited
ti l b th h d ti l i it
entirely by the charged particles crossing 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 to fulfil the first condition then the build-up
g 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
particle fluence will differ from that in the undisturbed
medium for this very reason.
24
25. A third condition is sometimes erroneously
added:
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 q
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.
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).
26. Do air-filled ionisation chambers function as Bragg-Gray
Do air filled ionisation chambers function as Bragg Gray
cavities at KILOVOLTAGE X-ray qualities?
26
27. Ma C-M and Nahum AE 1991 Bragg-Gray theory and ion chamber dosimetry for photon
beams Phys. Med. Biol. 36 413-428
28. 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:
29. Ma C-M and Nahum AE 1991 Bragg-Gray theory and ion chamber dosimetry for photon
beams Phys. Med. Biol. 36 413-428
30. 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 FACTORS
3.6 CORRECTION OR PERTURBATION FACTORS
FOR ION CHAMBERS
3 7 GENERAL CAVITY THEORY
3.7 GENERAL CAVITY THEORY
3.8 PRACTICAL DETECTORS
30
3.9 SUMMARY
34. Depth variation of the Spencer-Attix water/air stopping-power
ratio, sw,air, for =10 keV, derived from Monte Carlo generated
l 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)
30
25
20
18
14
10
7
5
1 3
ir
1.05
1.10
50
40
30
r
ratio,
s
w,a
1.00
ing-power
0.95
stoppi
0 4 8 12 16 20 24
0.90
depth in water (cm)
37. “THICK-WALLED” ION CHAMBERS
A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
med
en
D
D
)
C
(
)
C
(
CPE
wall
Δ
L
D
D
)
C
(
)
C
(
&
37
wall
wall
med D
D
)
C
(
)
C
(
air
Δ
air
wall D
D
)
C
(
)
C
(
&
38. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
med
en
wall
Δ
med L
D
Q
f
)
C
(
wall
air
air
D
Q
f
)
C
(
Thick-walled cavity chamber free-in-air
where the air volume is known precisely
where the air volume is known precisely
(Primary Standards Laboratories):
'
)
( K
L
D
C
K
air
en
wall
air
i
38
1
)
( K
g
C
K
wall
air
air
air
39. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
CORRECTION FACTORS
FOR ION CHAMBERS
FOR ION CHAMBERS
(measurements in phantom)
( p )
39
40. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
Are real practical Ion Chambers
Are real, practical Ion Chambers
really Bragg-Gray cavities?
y gg y
40
41. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
i
med
air
Δ
air
C
med P
L
D
Q
D
/
,
i
,
d
stem
cel
wall
repl
med
air
Δ
air
C
med P
P
P
P
L
D
Q
D
/
,
41
FARMER chamber (distances in millimetres)
42. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
THE EFFECT OF THE CHAMBER WALL
42
43. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
43
44. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
The Almond-Svensson (1977) expression:
med
wall
med
1
Δ
Δ
en L
L
med
air
air
wall
wall
Δ
L
P
44
air
Δ
45. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
45
46. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
46
47. THE EFFECT OF THE FINITE VOLUME
OF THE GAS CAVITY
A. E. Nahum: Cavity Theory,
Stopping-Power Ratios, Correction
Factors
47
48. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
48
49. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
49
50. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
FLUENCE PERTURBATION?
FLUENCE PERTURBATION?
Electrons
50
51. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
P
Φ
Φ P
z
E
fl
cav
med P
Φ
Φ
eff
P
Johansson 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 E
z
E
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
z
E z
E
at = 10 MeV, and to ≈ 0.997 at = 20 MeV.
52. A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.
52
53. GENERAL CAVITY THEORY
We have so far looked at two extreme cases:
i) detectors which are large compared to the electron
i) detectors which are large compared to the electron
ranges in which CPE is established (photon radiation only)
ii) d t t hi h ll d t th l t
ii) detectors which are small compared to the electron
ranges and which do not disturb the electron fluence (Bragg-
Gray cavities)
Many situations involve measuring the dose from photon (or
neutron) radiation using detectors which fall into neither of the
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
developed as an approximation.
54. In essence these theories yield a factor which is a weighted
f
mean of the stopping-power ratio and the mass-energy
absorption coefficient ratio:
det
det
det
det
det
1
en
Δ
d
L
d
D
D
med
med
med
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)
