PHYS365/PHYS965

PHYS365/PHYS965 – Assignment 3, 2018
Set on Mon 23
rd April, due on Mon 7
th May.
Essay on Small Field dosimetry
Recent advancements in linear accelerator and x-ray collimation technology have given the possibility to
produce photon radiation fields of few millimetres even at very high photon energy. This equipment has
been introduced recently in industry and clinical practice for irradiation of small targets. The use of MV
range photon beams in small target irradiation presents a challenge for a physicist which intends to
measure the dose distribution generated by the beam in water.
Bouchard et al. published in 2015 a remarkable publication where the theoretical concepts associated
with small field dosimetry are discussed and explored.
Part A
Using the arguments explained in this publication and in other documents of your selection (please refer
to the documents appropriately), write an essay of 2000 words answering the following questions:
1. When does a field become “small”?
2. What the factors are which make small field dosimetry different from standard field dosimetry?
3. What main parameters or characteristics a detector should have to be used in small field dosimetry?
Part B – case scenario
A 6MV industrial linear accelerator is used to irradiate small objects made of rubber (approximately
water equivalent) for sterilisation purposes. The nominal target dose distribution is 100 Gy delivered to
a sphere of 1cm diameter at 2 cm depth. The objects are rotated 360 degree to be uniformly irradiated.
The beam is 1×1 cm2 of area at isocentre and the only accessible point of measurement is at a depth of 2
cm inside the object. The physicist in charge of verifying the dose delivered to these objects has to
provide the value in real-time and has the following dosimeters available:
a. Ionisation Chamber CC013 from PTW
b. TLD carbon loaded 2x2mm2
(5mm thick)
c. Stereotactic diode EDGE from Sun Nuclear
d. PinPoint Ionisation Chamber 31014 from PTW
e. Microdiamond from PTW
Write a 1000 words essay to motivate your choice with the aim of theory, knowledge of the detectors’
working principles and drawings.
Detector dose response in megavoltage small photon beams. II. Pencil beam
perturbation effects
Hugo Bouchard, Yuji Kamio, Hugo Palmans, Jan Seuntjens, and Simon Duane
Citation: Medical Physics 42, 6048 (2015); doi: 10.1118/1.4930798
View online: http://dx.doi.org/10.1118/1.4930798
View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/42/10?ver=pdfcov
Published by the American Association of Physicists in Medicine
Articles you may be interested in
Detector dose response in megavoltage small photon beams. I. Theoretical concepts
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Absorbed dose conversion factors for therapeutic kilovoltage and megavoltage x-ray beams calculated by the
Monte Carlo method
Med. Phys. 24, 336 (1997); 10.1118/1.598084
APL Photonics
Detector dose response in megavoltage small photon beams.
II. Pencil beam perturbation effects
Hugo Boucharda)
Acoustics and Ionising Radiation Team, National Physical Laboratory, Hampton Road,
Teddington TW11 0LW, United Kingdom
Yuji Kamio
Centre hospitalier de l’Université de Montréal (CHUM), 1560 Sherbrooke Est, Montréal,
Québec H2L 4M1, Canada
Hugo Palmans
Acoustics and Ionising Radiation Team, National Physical Laboratory, Hampton Road,
Teddington TW11 0LW, United Kingdom and Medical Physics, EBG MedAustron GmbH,
Wiener Neustadt A-2700, Austria
Jan Seuntjens
Medical Physics Unit, McGill University, Montréal, Québec H3G 1A4, Canada
Simon Duane
Acoustics and Ionising Radiation Team, National Physical Laboratory, Hampton Road,
Teddington TW11 0LW, United Kingdom
(Received 27 January 2015; revised 19 August 2015; accepted for publication 29 August 2015;
published 25 September 2015)
Purpose: To quantify detector perturbation effects in megavoltage small photon fields and support
the theoretical explanation on the nature of quality correction factors in these conditions.
Methods: In this second paper, a modern approach to radiation dosimetry is defined for any detector
and applied to small photon fields. Fano’s theorem is adapted in the form of a cavity theory and
applied in the context of nonstandard beams to express four main effects in the form of perturbation
factors. The pencil-beam decomposition method is detailed and adapted to the calculation of perturbation
factors and quality correction factors. The approach defines a perturbation function which,
for a given field size or beam modulation, entirely determines these dosimetric factors. Monte Carlo
calculations are performed in different cavity sizes for different detection materials, electron densities,
and extracameral components.
Results: Perturbation effects are detailed with calculated perturbation functions, showing the relative
magnitude of the effects as well as the geometrical extent to which collimating or modulating the
beam impacts the dosimetric factors. The existence of a perturbation zone around the detector cavity
is demonstrated and the approach is discussed and linked to previous approaches in the literature to
determine critical field sizes.
Conclusions: Monte Carlo simulations are valuable to describe pencil beam perturbation effects and
detail the nature of dosimetric factors in megavoltage small photon fields. In practice, it is shown that
dosimetric factors could be avoided if the field size remains larger than the detector perturbation
zone. However, given a detector and beam quality, a full account for the detector geometry is
necessary to determine critical field sizes. C 2015 American Association of Physicists in Medicine.
[http://dx.doi.org/10.1118/1.4930798]
Key words: beam calibration, cavity theory, Monte Carlo simulations, nonstandard beams, perturbation
effects, photon beams, quality correction factors, radiation detectors, radiation dosimetry, small
fields
1. INTRODUCTION
Monte Carlo methods play an important role in radiation
dosimetry. Over the past decades, developments in electron
condensed history algorithms1–9 have led to general purpose
codes capable of achieving accuracy levels of the order of
0.1% in heterogeneous geometries when performing selfconsistency
tests on their charged particle transport methods.10,11
The sensitivity of the simulation accuracy to physical data
and geometrical parameters has also been thoroughly
investigated,12–14 and comparisons with experimental results
showed that Monte Carlo codes can reproduce ionization
chamber relative dose response to about 0.1%–0.2% accuracy
in standard reference beams.15,16Based on these results, Monte
Carlo methods have been widely used to characterize detector
response in radiotherapy beams (see Ref. 17) and play a major
role in clinical small field dosimetry.18–26
A detailed and effective method to explain the dosimetric
issues of small nonstandard beams consists of determining
detector response to pencil beams. Contrarily to classical
6048 Med. Phys. 42 (10), October 2015 0094-2405/2015/42(10)/6048/14/$30.00 © 2015 Am. Assoc. Phys. Med. 6048
6049 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6049
cavity theories, this approach addresses the problem independently
of charged particle equilibrium (CPE) and provides
realistic quantitative analysis. Bouchard and Seuntjens27 used
the pencil-beam decomposition method (PBDM) to quantify
the absorbed dose contribution to the cavity from primary
pencil beams and used the method to calculate quality correction
factors of IMRT beams. The same approach was used
by Tantot and Seuntjens,28 Gonzalez-Castaño et al.,
29 Underwood
et al.,
29 Kamio and Bouchard30 and adapted experimentally
by Looe et al.31 who confirmed the behavior of
dose response functions (DRFs) predicted by Monte Carlo
calculations.27,28,30,32
The goal of the present paper is to detail the problems
of nonstandard photon beams, with particular focus on small
fields, using Monte Carlo to simulate detector dose response
to pencil beams. The paper supports explanations described
in the accompanying paper, referred as Paper I.57 Using the
PBDM, perturbation effects of pencil beams are determined
in the form of perturbation functions and linked to dosimetric
factors. The present paper is structured as follows. Section 2
describes the theory to support the Monte Carlo approach to
radiation dosimetry using a simple, yet general, cavity theory.
In Sec. 3, the PBDM is described in detail and linked to the
calculation of quality correction factors. In Sec. 4, results of
Monte Carlo simulations are shown and analyzed in detail.
Section 5 summarizes the paper in its key elements.
2. A MODERN APPROACH TO CAVITY THEORY
2.A. Formulation of the Fano cavity theory
2.A.1. A modern interpretation of Fano’s approach
Inthefollowingapproach,MonteCarlosimulationsareused
to determine perturbation effects without the need for approximations
in conventional cavity theories. To do so, we adapt a
modern interpretation of Fano’s theorem for in-water measurements
in the form of a cavity theory by stating that an ideal
cavity and its components would be perfectly radiologically
water-equivalent for the fluence not to be perturbed, without
bringing conditions on the source. On that basis, let us define
a Fano cavity fulfilling the following two conditions:
1. The atomic properties of the cavity medium are the same
as the surrounding medium.
2. The fluence of all particle types crossing the cavity
is the same as if the cavity is filled with surrounding
medium.
In the general case where we assume these conditions to
be met, the energy loss per unit volume is proportional to
the electron density. Therefore in the context of reporting the
average absorbed dose in the cavity filled with water, one can
write the factor f (Q), defined as the ratio of the absorbed dose
at a point in water to the average absorbed dose in the detector,
under these ideal conditions as follows:
fFano(Q) = *
,
Z
A
+

w
m
Pvol, (1)
where (
Z
A
)w
m
is the ratio Z
A
, water to detector medium, and Pvol
is a factor accounting for volume averaging defined as
Pvol =
Dw
Dw,cav
, (2)
with Dw the absorbed dose to water at the point of measurement
and Dw,cav the average absorbed dose in the detector
cavity filled with water.
2.A.2. Link to Fano’s theorem
This approach can be linked to Fano’s theorem, which states
that if a medium of uniform atomic properties is irradiated
by a uniform photon source, then CPE is established and
the particle fluences are independent of the mass density in
the medium. One consequence of this statement is that the
energy loss per unit volume scales exactly with the number
of electrons per unit volume, equal to ρ
(
Z
A
)
NA. This implies
that the energy absorbed per unit mass scales with the ratio Z
A
.
While Fano’s theorem requires a uniform photon source,
the conditions presented here do not specify any condition on
the source. Therefore, the ideal detector (i.e., a Fano cavity)
is such that the interactions scale in exactly the same way as
under Fano’s conditions (i.e., CPE). Therefore, in the context
of Monte Carlo simulations, this interpretation of Fano’s theorem
is as valuable as other cavity theories on the basis that
all theories require ideal conditions, which are subsequently
corrected with additional factors. Conversely to other theories,
this adaptation is independent of the beam quality (i.e., in this
context, its energy and the field size) and the cavity size.
2.A.3. An example of a Fano cavity
It is worth giving an example where the approach is exact
for any beam quality and cavity size. Let us imagine a hypothetical
cavity constituted of heavy water, i.e., water molecules
such that all hydrogen atoms (A ≈ 1) are deuterium (A ≈ 2)
and oxygen remains the same (A ≈ 16), and let us imagine
that the electron density (in cm−3
) equals that of the liquid
water surrounding it. Clearly, such a cavity is a Fano cavity,
and therefore, the ratio of absorbed dose is
f (Q) = *
,
Z
A
+

w
hw

10
18

10
20
 ≈ 1.11,
independent of the beam quality and the cavity geometry. The
same result can be obtained intuitively considering that the
energy absorbed per unit volume in a megavoltage photon
beam would be the same in water or heavy water, but the mass
density would be scaled by a factor of about 1.11, hence the
ratio f (Q).
2.A.4. The context of direct Monte Carlo calculations
This simple, yet naïve interpretation of Fano’s approach
is used in the present study as the basis of a factorization of
perturbation factors obtained by Monte Carlo calculations.
The advantage of this choice is that the ideal conditions are
Medical Physics, Vol. 42, No. 10, October 20
6050 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6050
independent of the detector geometry and do not require
the simulation of ideal (or artificial) conditions, such as in
stopping-power ratio or mass–energy absorption coefficient
calculations. Instead, the ideal conditions defining a Fano
cavity are implicit to the material composition and are independent
of anything else. Therefore, the perturbation factor
P(Q) (defined in Paper I) includes all key effects related to
the detector, as well as the effects of the atomic properties of
the medium to dose response which are meant to be described
by stopping-power ratios. As shown in Sec. 2.B, such an
approach brings an end to the discussion on the extent to which
small fields may satisfy cavity theory requirements and allows
focusing on the perturbation effects.
2.B. Factorization of cavity absorbed dose
for general conditions
To characterize the contributions to detector dose response
under general conditions, one can decompose the overall
perturbation factor P(Q) into a product of subfactors, meant
to represent detector-specific physical effects in a given beam
quality, including small fields and modulated beams. Although
the approach of accounting for correction factors was proposed
earlier,33–37 the first study to insist on a mathematically consistent
factorization of perturbation factors (i.e., the product of all
perturbation factors must equal the overall factor) was done
by Bielajew.38 Such coherent factorization is at the basis of
recent Monte Carlo studies using direct calculations.12,32,39–44
However, as the following equation describes, these subfactors
are not independent from one another (i.e., they are correlated).
In practice, one should keep in mind that they are
not meant to be applied separately to determine f (Q), except
for volume averaging corrections (i.e., Pvol), as described in
Paper I.
To formalize the approach, let us define a sequence of
N + 1 geometries {Gi} = {G1,G2,…,GN+1} each with a single
homogeneous scoring volume (i.e., the cavity). Let us
define G1 to be the fully modeled detector in a reference water
phantom, GN the bare detector cavity filled with water, and
GN+1 a small cavity in water meant to represent the point
of measurement in the reference phantom. Let us define the
average absorbed doses in Gi as Di = D1, D2,…DN+1. Note
here that to simplify the terminology, we use the term absorbed
dose in the cavity instead of average absorbed dose
in the cavity. To be suitable in the context of describing the
perturbation effects with direct Monte Carlo calculations, the
interpretation of Fano’s approach is chosen and the following
ratio is defined:
f (Q) =
Dw
Ddet
= *
,
Z
A
+

w
m
PMCPvol. (3)
This relation is obtained with the corrective approach defined
in Paper I, choosing fideal(Q) = fFano(Q) and P(Q) = PMC. Note
the implicit quality-dependence of the perturbation factors.
The overall perturbation factor PMC, representing the global
detector perturbation factor with the exception of volume
averaging, is decomposed as the following product:
PMC =

N
i=1
Pi
, (4)
where each subfactor Pi
is defined as
Pi =
Di+1
(
Z
A
)
i+1
(
Z
A
)
i
Di
, (5)
with (
Z
A
)
i
taken for the medium constituting the cavity of the
geometry Gi
. Based on this definition, the overall perturbation
factor PMC can be written as
PMC =
DN+1
(
Z
A
)
w
(
Z
A
)
m
D1
(6)
and
Pvol =
DN+1
(
Z
A
)
w
(
Z
A
)
w
DN
=
Dw
Dw,cav
. (7)
Note that the values Pi depend on the definition of the
geometry sequence Gi
, but their product equals to PMC no
matter what sequence is chosen. Making the assumption that
the intrinsic detector dose response is unaffected by the change
in beam quality, the relation between the subfactors and the
quality correction factor which corrects for the differences
between Q1 and Q2 is given by
kQ2
,Q1 =
PMC(Q2)Pvol(Q2)
PMC(Q1)Pvol(Q1)
=

N
i=1
Pi(Q2)
Pi(Q1)
·
Pvol(Q2)
Pvol(Q1)
. (8)
This equation states that the quality correction factor is equal
to the product of the ratio of subfactors for beam quality Q2-
to-Q1.
In this formalism, the volume averaging factor Pvol is
defined separately from the direct Monte Carlo approach.
As discussed in Paper I, simulating dose in a volume small
enough to represent absorbed dose at a point in water can
be highly inefficient and the choice of elemental volume size
can be arbitrary. Analytical approaches, such as deconvolution
methods,12 allow determining these effects independently of
the detector characteristics other than the shape of its cavity.
2.C. Perturbation factors
2.C.1. Key effects
From the point of view of this modern approach, the main
characteristics responsible for the fact that f (Q) is detectorand
quality-dependent can be summarized as follows.
Regarding condition (1) of the Fano cavity:
• the atomic properties of the detector cavity medium are
not the same as water which affects the cavity dose
response and perturbs the particle fluence crossing it.
Medical Physics, Vol. 42, No. 10, October 2015
6051 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6051
Regarding condition (2) of the Fano cavity:
• the electron density of the detection medium relative to
water scales the interaction coefficients and perturbs the
particle fluence;
• the presence of extracameral components in the detector,
such as a wall, electrode, or any other nonsensitive
components, causes particle interactions to be different
from the situation where the detector is a bare cavity in
the absence of such components.
Regarding the requirement to report absorbed dose at a point
in water:
• the finite size of the detector, even if it was constituted
of water, causes volume averaging effects compared to
absorbed dose at a point due to in-depth and lateral
gradients in the beam fluence.
To represent these effects into subfactors, it is worth defining a
set of geometries whose differences are reflected by these main
characteristics.
2.C.2. Proposed series of calculation with Monte Carlo
There exist many possible ways to define these subfactors,
depending on how the geometries {Gi} are defined and in what
order they are constructed. In the context of traditional detectors,
five key geometries (i.e., {Gi} = {G1,G2,G3,G4,G5}) are
defined to describe the main physical effects responsible for
large quality correction factors in small fields. Two calculation
series (or chains) are proposed (see paths A and B in Fig. 1)
and the geometries are (1) the fully modeled detector, (2) the
bare detector volume filled with detector medium, i.e., the
detector without its extracameral (nonsensitive) components,
(3) the bare detector volume filled with artificial medium,
being either water with the electron density of the detector
medium (path A), or the detector medium with the electron
density of water (path B), (4) the bare detector filled with water,
F. 1. Illustration of the present approach to characterize the main effects
governing detector dose response in selfcoherent ways. Although volume
averaging is not treated explicitly in this present paper, the effect is shown
for completion with respect to the standard definition of absorbed dose at a
point in water.
and (5) a volume of water small enough to represent absorbed
dose at a point in water. In both cases, the series gives rise to
the following subfactors: the extracameral perturbation factor
Pext, the atomic properties perturbation factor Pmed, the density
perturbation factor Pρ, and the volume averaging perturbation
factor Pvol. Figure 1 illustrates these two calculation series and
includes the point of measurement in water for completeness
(Pvol is not evaluated in this work).
This leads to the following definitions of absorbed doses:
• Ddet: the absorbed dose in the detector being fully constituted
(paths A and B);
• Dm,cav: the absorbed dose in the bare detector cavity
(paths A and B);
• Dw∗,cav: the absorbed dose in the detector cavity filled
with water having the electron density of the detection
medium (path A);
• Dm∗,cav: the absorbed dose in the detector cavity filled
with detection medium having the electron density of
water (path B);
• Dw,cav: the absorbed dose in the cavity filled with water
(paths A and B);
• Dw: the absorbed dose to water at the point of measurement
(shown in paths A and B for completeness).
The subfactors can then be defined as follows:
Pext =
Dm,cav
(
Z
A
)
m
(
Z
A
)
m
Ddet
(paths A and B), (9)
Pmed =




Dw∗,cav
(
Z
A
)
w∗
(
Z
A
)
m
Dm,cav
(path A),
Dw,cav
(
Z
A
)
w
(
Z
A
)
m∗
Dm∗,cav
(path B),
(10)
Pρ =




Dw,cav
(
Z
A
)
w
(
Z
A
)
w∗
Dw∗,cav
(path A),
Dm∗,cav
(
Z
A
)
m∗
(
Z
A
)
m
Dm,cav
(path B).
(11)
Note that m, m∗
, w, and w∗
stand for the cavity medium, the
cavity medium having the electron density of water, water,
and water having the electron density of the cavity medium;
therefore, (
Z
A
)
m∗ =
(
Z
A
)
m
and (
Z
A
)
w∗ =
(
Z
A
)
w
.
To verify the coherence of the method, the overall perturbation
factor PMC can be decomposed into a series of subfactors
using Eq. (5) as follows (path A of Fig. 1, excluding absorbed
dose at a point):
PMC =*
,
Z
A
+

m
w

Dw,cav
Ddet 
=







Dm,cav
(
Z
A
)
m
(
Z
A
)
m
Ddet














Dw∗,cav
(
Z
A
)
w∗
(
Z
A
)
m
Dm,cav














Dw,cav
(
Z
A
)
w
(
Z
A
)
w∗
Dw∗,cav







=PextPmedPρ, (12)
Medical Physics, Vol. 42, No. 10, October 2015
6052 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6052
or alternatively as follows (path B of Fig. 1, except for absorbed
dose at a point):
PMC =*
,
Z
A
+

m
w

Dw,cav
Ddet 
=







Dm,cav
(
Z
A
)
m
(
Z
A
)
m
Ddet














Dm∗,cav
(
Z
A
)
m∗
(
Z
A
)
m
Dm,cav














Dw,cav
(
Z
A
)
w
(
Z
A
)
m∗
Dm∗,cav







=PextPρPmed. (13)
Note that the sub-factors in Eqs. (9)–(11) are correlated
and therefore have a certain degree of arbitrariness by the
choice of the intermediate scoring volumes. One way is to
define an intermediate volume made of water having the
electron density of the detection medium (path A). This has
the advantage of linking Pρ to Fano’s theorem. The other
approach is to define an intermediate volume made of detection
medium having the electron density of water (path B).
This has perhaps the advantage of a more intuitive characterization
of Pmed. Overall, both ways (paths A and B)
must yield the same factor PMC. Nonetheless, for the purpose
of describing the cavity perturbation effects explicitly, it is
helpful to decompose these factors and evaluate their relative
magnitudes.
2.D. Link to conventional theory
Based on the current standard cavity theory formalism, the
ratio f (Q) defined in Paper I is written as45
f (Q) = *
,
L
ρ
+

w
m
P(Q), (14)
with (
L/ρ)w
m
the Spencer–Attix stopping power ratio and P(Q)
the overall perturbation factor accounting for the breakdown of
the theory in realistic conditions. Using the extended AAPM
notation of Bouchard et al.,
43 Eq. (14) can be written with
explicit perturbation factors as follows for ionization chambers:
f (Q) = *
,
L
ρ
+

w
m
PflPρPvolPwallPcelPstem. (15)
Using the current approach [i.e., Eq. (3)], the explicit notation
is written as
f (Q) = *
,
Z
A
+

w
m
PmedPextPρPvol. (16)
By analogy to the standard approach, the overall perturbation
factor PMC = PmedPextPρ corrects for the fact that the detector
is not a Fano cavity. Based on these equations, the present
approach can be linked to standard theory as follows:
Pmed = *
,
Z
A
+

m
w
*
,
L
ρ
+

w
m
Pfl (17)
and
Pext = PwallPcelPstem. (18)
The factors Pρ and Pvol are defined identically as in their
original definition.12,43
3. PENCIL-BEAM PERTURBATION EFFECTS
3.A. Quantification of perturbation factors with
the pencil-beam decomposition method
A judicious approach to quantify perturbation effects in
detail is the PBDM. The rationale behind the idea is that a finite
extent photon beam can be decomposed into pencil beams.
In the context of perturbation factors, the approach proposed
by Bouchard and Seuntjens27 is used, assuming that the beam
is constituted of primary photons only. This PBDM defines
the absorbed dose in the cavity as the sum of absorbed doses
from pencils present in the beam accounting for the individual
modulation of each pencil in the photon beam fluence. The
absorbed dose to the cavity by a single pencil beam as a
function of its position with respect to the cavity is called dose
response function (DRF) and must be evaluated by an accurate
Monte Carlo technique.
It is worth noting that the PBDM is equivalent to a superposition
method (in a single cavity) that approximates the beam to
be constituted of primary photons only prior to be transported
in the phantom. Among other examples, the method is exact
for beams entirely constituted of primary photons having the
same spectral distribution (i.e., no collimator scatter) and the
direction of the pencils being unique for a given position
(x,y) (e.g., an isotropic source or a parallel beam). Although
scattered radiation from the beam collimator is ignored, all
physical interactions of the primary beam with the phantom,
including the production of scattered photons and bremsstrahlung
radiation, are modeled. Note that source occlusion is not
accounted in the present approach and would require adapting
the method.
The PBDM can be written as follows:27
Dcav =
 ∞
−∞
 ∞
−∞
dcav(x,y)F(x,y)dxdy. (19)
Here, Dcav is the absorbed dose per primary fluence (in Gy cm2
/
hist) in the cavity filled either with water or the detector medium.
The function dcav(x,y) is the DRF of the cavity and is
defined as the cavity absorbed dose per primary photon (in
Gy/hist) in a pencil beam positioned at (x,y) on the primary
fluence plane (i.e., defined by the user). F(x,y) is the fluence
weight distribution (dimensionless) of the beam, which
corresponds to the amount of primary photons passing through
(x,y) relative to an open beam, such that F(x,y) ∈ [0,1] and
F(x,y) = 1 if the beam is uncollimated at (x,y).
In a recent paper by Kamio and Bouchard,30 the PBDM
was used to establish a criterion for using detectors in nonstandard
beams without the need for applying correction factors.
The approach is based on expressing the quality correction
factor as a function of the detector perturbation function as
follows:
Medical Physics, Vol. 42, No. 10, October 2015
6053 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6053
k
fF, fref
QF,Qref

(PMC)
fF
QF
(PMC)
fref
Qref
=







1−
 ∞
−∞
 ∞
−∞
hdet(x,y)F(x,y)dxdy
(

fF, fref
QF,Qref
)
w







−1
, (20)
using IAEA–AAPM notation.19 Here, (

fF, fref
QF,Qref
)
w
is the field
factor defined as
(

fF, fref
QF,Qref
)
w
=
D
fF
w,QF
D
fref
w,Qref
, (21)
with D
fF
w,QF
and D
fref
w,Qref
the absorbed dose per primary fluence
in the water cavity (in Gy cm2
/hist) in the fields fF
and fref, respectively. Note here that fF could represent any
field, including an arbitrary nonstandard field fns. It is worth
noting that in the IAEA–AAPM formalism, absorbed doses
are strictly considered at a point, and therefore, the present
notation extends the approach to average absorbed doses in
the cavity. Also note that (

fF, fref
QF,Qref
)
w
acts as a normalization
factor on the integral and therefore on the magnitude of F(x,y).
Indeed, in the limits of a zero field size (i.e., a pencil beam), in
this convention the absorbed dose per fluence is zero (since the
fluence is infinity), but so is the field factor, while the quality
correction factor remains finite.
By definition, for any static field the fluence weight distribution
is given by the following binary function:
F(x,y) =




1 for all (x,y) in the collimator opening,
0 elsewhere.
(22)
Note that if source occlusion were accounted for, in static
beams F(x,y) would no longer be binary and would takes
continuous values between 0 and 1 in the penumbra region.
The perturbation function (in cm−2
) is defined as follows:
hdet(x,y) =
dw(x,y)
D
fref
w,Qref

ddet(x,y)
D
fref
det,Qref
. (23)
Note that the present definition of hdet(x,y) is subtly different
from the original one introduced by Kamio and Bouchard30
which introduces a multiplicative constant (i.e., the area of the
field) which is not needed here. It follows that the perturbation
function integrates to zero,

Aref
hdet(x,y)dxdy = 0, (24)
where Aref is the area of the reference beam, i.e., where the
function F(x,y) equals 1. It is worth noting that the perturbation
function defined here differs from that of Underwood
et al.,
32 who chose the unnormalized difference of the DRFs,
and from Looe et al.,
31 who chose the absolute value of the
unnormalized DRF difference.
In a similar way, the perturbation subfactors defined in
Eq. (5) can be expressed as follows:
(Pi)
fF
QF
=
(Pi)
fref
Qref
1−
 ∞
−∞
 ∞
−∞hi
(x, y)F(x, y)dxdy
(

fF, f
ref
QF,Qref
)
i+1
, (25)
with the functions hi(x,y) defined as
hi(x,y) =
di+1(x,y)
D
fref
i+1,Qref

di(x,y)
D
fref
i,Qref
, (26)
and the factors (

fF, fref
QF,Qref
)
i
defined as
(

fF, fref
QF,Qref
)
i
=
D
fF
i,QF
D
fref
i,Qref
. (27)
Finally, from the above definitions, it is worth writing the
following property:
hdet(x,y) =

i
hi(x,y)
= hext(x,y)+hmed(x,y)+hρ(x,y). (28)
This equation states that perturbation functions are additive.
3.B. Monte Carlo simulations
To determine the DRF of each scoring volume, calculations
are performed with the EGSnrc user-code cavity7 distributed
with the egs++ library.46 A circular parallel photon beam of
1.25 MeV, corresponding to the mean photon energy emitted
by cobalt-60, is used to determine the cavity absorbed doses in
a reference beam. Pencil beams with perpendicular incidence
on the phantom surface are used to determine the DRFs. A
30 × 30 × 30 cm3 water phantom is used as base geometry
where a cavity is centered at 10 cm depth on the central axis
of the phantom being aligned with the central axis of the
reference beam. Several cavity media are used during simulations
and are summarized in Table I. The choices of materials
and dimensions are such that perturbation effects addressed
herein can be clearly described while assuring a reasonable
simulation efficiency. The cavity is oriented in such a way
that the axes of the cylindrical shape and the pencil beams are
parallel. A cross section data file (i.e., pegs4 file) is defined
for these media with a production threshold of 1 keV for
photons and 512 keV total energy for electrons and positrons.
For these media, such as water-density air, air-density water,
silicon-density water, and water-density silicon, the polarization
corrections (i.e., the density effect parameter47) and the
I-value of the medium’s natural state are used to determine
the interaction cross sections and stopping powers. To quantify
the extracameral perturbation effects, a simple graphite or
aluminum wall is added around the cavity. Simulation transport
parameters are set to default with the use of range rejection
up to a kinetic energy of 10 keV (ESAVE = 0.521 MeV). To
evaluate DRFs under Fano’s conditions (CPE), simulations are
performed using the Fano cavity test (i.e., calculation type
= Fano). The user code cavity is also modified to calculate
the charged particle fluence in the cavity and the mean kinetic
energy of electrons crossing it.
Medical Physics, Vol. 42, No. 10, October 2015
6054 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6054
T I. Summary of media used in the simulations. Some of them are artificial to demonstrate the effect of
density as well as medium properties on cavity response. The elemental composition of dry air used is C: 0.01%,
N: 75.53%, O: 23.18%, and Ar: 1.28%.
Medium name Composition Z
A Mass density (g/cm3
) Electron density (cm−3
)
Vapor water H2O 0.5551 1.000 × 10−03 3.343 × 10+20
Sparse water H2O 0.5551 1.000 × 10−01 3.343 × 10+22
Water H2O 0.5551 1.000 × 10+00 3.343 × 10+23
Dense water H2O 0.5551 1.000 × 10+01 3.343 × 10+24
Air dry air 0.4992 1.205 × 10−03 3.622 × 10+20
Silicon Si 0.4984 2.330 × 10+00 6.994 × 10+23
Air-density water H2O 0.5551 1.084 × 10−03 3.622 × 10+20
Water-density air dry air 0.4992 1.112 × 10+00 3.343 × 10+23
Silicon-density water H2O 0.5551 2.092 × 10+00 6.994 × 10+23
Water-density silicon Si 0.4985 1.114 × 10+00 3.343 × 10+23
Graphite C 0.4995 1.700 × 10+00 5.114 × 10+23
Aluminum Al 0.4818 2.699 × 10+00 7.831 × 10+23
4. SIMULATIONS ANALYSIS
4.A. Density perturbation effects
4.A.1. CPE calculations
DRFs of 1.25 MeV photons are simulated using a pencil
beam with Fano calculations, i.e., the regeneration technique48,49
(calculation type = Fano) such that Fano’s condition
(CPE) can be achieved by combining all pencil beams.
Note that under these conditions, the functions are referred
to as Fano dose response functions. The scoring volume is a
cylindrical cavity either filled with vapor water (0.001 g/cm3
),
sparse water (0.1 g/cm3
), liquid water (1 g/cm3
), or dense
water (10 g/cm3
). Results are shown in Fig. 2 and dimensions
are specified in the caption. A rather drastic dependence of
the DRF on the mass density can be observed, as predicted
and described by Fig. 3 of Paper I. At off-axis distances where
pencil beams cross the cavity, the DRF increases with increasing
mass density. This is mainly caused by the fact that the
amount of electrons produced in the cavity increases with
increasing density. At off-axis distances where the pencil beam
avoids the cavity, the DRF decreases with increasing mass
F. 2. The Fano DRF of a cylindrical cavity of 5 mm radius by 2 mm
thickness in a 1.25 MeV photon beam, calculated using water having different
mass densities: 0.001, 0.1, 1.0, and 10.0 g/cm3
.
density. This is mainly explained by the average path length
(and consequently, the fluence) of electrons being higher at
low densities, hence the fluence increasing with decreasing
cavity mass density. It must be kept in mind that using Fano
calculations causes CPE to exist in broad beams and constrains
the integral of Fano DRFs to be independent of the
cavity mass density. Therefore, there exists a compensation
effect across the boundary of the cavity; the (under/over)dose
response of the detector to pencil beams directed at the cavity
is compensated by an over- or under-response to pencil beams
not directed at the cavity. This is what occurs in CPE.
4.A.2. Realistic calculations
In realistic calculations, where beam attenuation, scattered
photons, and radiation processes are accounted for, the same
overall density perturbation effect on the DRFs exists, despite
some differences caused by photon attenuation and scatter.
A more detailed characterization of the density perturbation
effect is achieved by repeating the simulations for different
densities but calculating the fluence and the mean energy of
electrons crossing the cavity instead of absorbed dose. Results
are shown in Fig. 3. It can be observed that the behavior of
electron fluence is the same as expected in CPE, with the
exception of the fluence being lower in dense water than in
liquid water at off-axis distances where the pencil beam crosses
the cavity. This can be explained by the attenuation of the
pencil beam, which becomes significant at such densities (10.0
g/cm3
), reducing the amount of electrons crossing the cavity
as compared to liquid water. However, at off-axis distances
where the pencil beam avoids the cavity, the behavior of the
fluence is similar to what occurs in CPE. Overall, the trend
of the dose compensation effect across the cavity boundary
occurring in CPE also exists in realistic conditions but yields
a density perturbation factor different from unity.
Analyzing the mean energy of electrons crossing the cavity,
it can be observed that electrons have a lower kinetic energy
when they are produced at off-axis distances where the pencil
beam is not directed at the cavity in comparison to distances
where it is directed at the cavity. This effect is expected since
Medical Physics, Vol. 42, No. 10, October 2015
6055 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6055
F. 3. Properties of electrons crossing the cavity as a function of the pencil beam position in water cavities of different mass (and electronic) densities: (a)
fluence (in hist/cm2
) and (b) mean kinetic energy (MeV). The photon beam energy is 1.25 MeV and the functions are calculated using realistic conditions
(i.e., not Fano calculations).
the maximum energy transfer to electrons occurs where their
resulting momentum is parallel to the beam, in such case they
are unlikely to reach the cavity if they are produced in the
lateral region outside of it. The variation of kinetic energy as a
function of pencil beam off-axis position across the boundaries
appears more pronounced for low density cavities than their
high density counterpart. This has the effect of yielding more
absorbed dose per electron from photons directed at the cavity
with low density, and therefore, the behavior of DRFs at the
boundaries is slightly more abrupt than the fluence shown
in Fig. 3. Figures 4 and 5 show DRFs in two cylindrical
cavities composed of media having the same properties but
different densities. A similar behavior, described numerically
by Bouchard and Seuntjens,27 Tantot and Seuntjens,28 and
Kamio and Bouchard,30 as well as experimentally by Looe
et al.,
31 suggests that the density perturbation effect is dominant
in DRFs. This statement is consistent with the previous
literature on perturbation factors.30,32,43,44
4.B. Atomic composition perturbation effects
To characterize the effect of atomic properties on DRFs,
simulations are performed in two cylindrical cavities either
filled with air, water, or silicon. A summary of atomic compositions
is provided in Table I. In each comparison, the mass
densities are adjusted such that electron density is identical
in both media. It is worth emphasizing again that electron
densities are matched rather than mass densities in order to
avoid a dependence on the proportion of neutrons, which are
inert (or quasi-inert) in megavoltage beams due to the small
(or negligible) probability of nuclear interactions. By matching
electron densities, both media have the same density of
interaction sites per unit volume and their cross sections vary
only due to the atomic (or electronic) properties of the medium.
Calculated DRFs are shown in Figs. 6 and 7. In these graphs,
DRFs are normalized to Z/A to remove the dependence on the
number of neutrons in the materials.
Results show that the effect of atomic properties on dose
perturbation is significantly smaller in comparison to density
perturbation effects. In Fig. 6, subtle differences can be
observed between air and water, which are mostly explained
by the dependence of the density effect parameter on water
density being ignored here when scaling the mass density of
media. This would have the effect of yielding a slightly larger
mass stopping power in water-density air than in water, and
a smaller mass stopping power in air-density water than in
air.47 For silicon, Fig. 7 shows that differences in DRFs relative
to water, which are more noticeable than for air, which can
be attributed to the difference in atomic number (or effective
atomic number), via the effects of photoelectric interactions
or pair production. Moreover, the I-value of silicon is significantly
different from the one of water, i.e., 173 eV versus 75 eV,
F. 4. DRFs of a 5 mm radius and 2 mm thick cavity filled with media having the same atomic properties but different mass densities: (a) water and air-density
water (path A); air and water-density air (path B). The photon beam energy is 1.25 MeV.
Medical Physics, Vol. 42, No. 10, October 2015
6056 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6056
F. 5. DRFs of a 2 mm radius and 2 mm thick cavity filled with media having the same electron densities: (a) water and silicon-density water (path A); (b)
silicon and water-density silicon (path B). The photon beam energy is 1.25 MeV.
which has the consequence of a smaller mass stopping power
normalized to Z/A in silicon or water-density silicon than in
silicon-density water or water.
4.C. Extracameral perturbation effects
To characterize the perturbation effects of extracameral
components, i.e., not being part of the sensitive volume,
the DRFs are evaluated by calculating dose in cavities surrounded
by a 1 mm wall being either composed of graphite or
aluminum, for the air and silicon cavity, respectively. Figure
8 shows the calculated functions. For the air cavity, results
show that the DRF is generally higher in the cavity surrounded
by a wall than without it, except for pencil beams directed
outside the wall hence beyond the furthest edge of the detector.
This can generally be explained by the additional amount of
electrons being produced in the graphite wall, compared to
water, when the pencil is directed at the wall. When the pencil
beam is directed outside the wall, electrons are more likely
to be absorbed and scattered if there is a wall than by water,
and therefore, the function is lower when the wall is present.
For the silicon cavity, the same trend is observed and the
effect at the edge of the cavity is even more noticeable than
for the graphite wall. This can be explained by the increased
contribution of low-energy electrons from photoelectric effects
occurring in the aluminum wall and being scattered sideways
into the cavity.
The example shows that the effect of extracameral components
can be significant and comparable to density perturbation
effects, especially if their medium composition has significantly
higher interaction probability than water, either due to
higher mass density or higher cross sections. While a 1 mm
graphite wall is realistic for ionization chambers, a 1 mm
aluminum wall is perhaps over-representative of these effects
for realistic detectors, especially when the cavity is small.
However, because some detectors contain high-Z components,
this example shows their potential impact on detector dose
response.
4.D. Perturbation functions
To demonstrate in detail the relative importance of the
effects of extracameral components, atomic properties, and
density perturbations effects, the analysis of DRFs shown
in Subsections 4.A–4.C can be expanded using perturbation
functions. These functions characterize the relative contribution
of a given pencil beam position to a perturbation factor
of interest. As described by Eq. (25), perturbation factors can
be expressed as a function of the integral of these functions
weighted by the fluence weight distribution F(x,y) of the field.
As shown in Fig. 9, perturbation functions for extracameral
components, atomic properties, and density perturbation in
both cavities are either positive or negative at a beam position
of 0 (i.e., the center of the cavity) and become of opposite
F. 6. DRFs of a 5 mm radius and 2 mm thick cavity filled with media having the same atomic properties: (a) air and air-density water (path A); (b) water and
water-density air (path B). The photon beam energy is 1.25 MeV.
Medical Physics, Vol. 42, No. 10, October 2015
6057 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6057
F. 7. DRFs of a 2 mm radius and 2 mm thick cavity filled with media having the same electron densities: (a) silicon and silicon-density water (path A);
(b) water and water-density silicon (path B). The photon beam energy is 1.25 MeV.
sign once the beam position is beyond the cavity boundary.
At larger distances from the cavity within the size of the
reference beam, the perturbation function becomes very small
in magnitude and remains small up to the edges of the reference
beam. The overall behavior of the function with beam position
is such that its integral over the area of a reference beam is 0,
as stated by Eq. (24).
For small static fields, F(x,y) = 1 is a binary function
(i.e., 0 or 1) defined by the collimation [see Eq. (22)]. For
modulated beams, the function can take any value in F(x,y)
∈ [0,1]. If the field size is large enough such that the integral
of the perturbation function is nearly 0 outside the collimator
opening, the field’s perturbation factor is very close to that of a
reference beam. However, if the field edges are near to regions
where the perturbation function is significant, the perturbation
factor diverges significantly from the one of a reference beam.
For modulated fields, the fluence weight distribution F(x,y)
can have a large gradient. The same logic is applied here;
if the modulation is important across the region where the
perturbation function is significant, perturbation factors can be
significantly different from reference conditions. Beams with
a constant fluence weight distribution F(x,y) over the region
of significant perturbation yield perturbation factors close to
reference conditions. Otherwise, detector dose response requires
correction.
It is worth verifying the validity of the theoretical concepts
addressed in Paper I and comparing the magnitude of the
effects in Fig. 9. For the air cavity, it can be observed that the
density is the dominating perturbation effect over all pencil
beam positions, as the magnitude of the perturbation function
related to that detector characteristic is the highest of all three.
The presence of the wall also yields significant perturbation
effects, but clearly less than density and higher than atomic
properties, which overall appears negligible. While perturbation
functions are additive with respect to Eq. (28), it is worth
noting that the density and extracameral perturbation effects
are partially in the same direction (i.e., either positive or negative)
from the cavity up to the inside region of the wall. This
can be explained as follows. A cavity of density smaller than
water under-responds to pencil beams directed at the cavity
and over-responds to pencil beams not directed at the cavity,
as described in Sec. 4.A. When the wall is present, the absorbed
dose contribution from the edges to the air cavity is much
higher than in the absence of wall, which gives less relative
importance to the cavity response to pencil beams directed
at the cavity, hence an apparent under-response, and more
importance to the cavity response to pencil beams directed at
the inner region of the wall where the cavity apparently overresponds
with respect to a reference beam. Furthermore, the
air cavity apparently under-responds to pencil beams incident
beyond the wall since secondary electrons are less likely to
reach the cavity when the wall is present, whose higher density
yields higher electron stopping power in the wall compared
to water, hence reducing the number of electrons reaching
F. 8. DRFs in cavities of 2 mm thickness with and without the presence of a 1 mm wall: (a) a 5 mm radius air cavity and a graphite wall; (b) a 2 mm radius
silicon cavity and an aluminum wall. The photon beam energy is 1.25 MeV.
Medical Physics, Vol. 42, No. 10, October 2015
6058 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6058
F. 9. Perturbation functions calculated in two cavities using the scoring volumes described by Fig. 1 to demonstrate extracameral, atomic composition, and
density perturbation effects: (a) a 5 mm radius and 2 mm thickness air cavity with 1 mm graphite wall; (b) a 2 mm radius and 2 mm thickness silicon cavity
with 1 mm aluminum wall. The photon beam energy is 1.25 MeV. The two vertical dotted lines represent the position of the cavity and detector boundary,
respectively, the latter including the wall.
the cavity. The reader should be reminded that the “apparent”
response refers to detector dose response in a pencil beam relative
to a reference beam. That is, an apparent under-response
corresponds to hdet > 0 and an apparent over-response corresponds
to hdet < 0, with respect to Eq. (23).
For the silicon cavity, Fig. 9 shows that the magnitude of
the extracameral component (i.e., wall) perturbation effects
is higher than the density perturbation effects. Indeed, here
a rather extreme effect is observed due to the wall thickness
being comparable to the cavity size (i.e., half its radius) and
its high-Z composition with respect to water (i.e., aluminum).
It is clear that the presence of the wall makes the detector
dose response of the silicon cavity relative to a reference beam
higher than without a wall for pencil beams directed at the
region of the wall near the lateral edge of the cavity, due to
the over-production of electrons in comparison to water. This
has the effect of inducing an apparent under-response of the
silicon cavity to pencil beams directed at it compared to no
wall. The density also presents significant perturbation effects,
with an over-response of the silicon detector to pencil beams
directed at its cavity, and an under-response to pencil beams
not directed at its cavity, as described in Sec. 4.A. Despite
being the least important, the atomic composition perturbation
effects are also noticeable, and the magnitude of the effect can
be attributed to the properties of silicon being significantly
different from water.
4.E. How small is small?
A true challenge arises when it comes to rigorously defining
what field size can be considered small. In previous
studies,50–52 a small field was identified as one for which the
edges are near the dimensions of the detector with respect to
the electron range. In more recent work,53–55 the notion of
LCPE was used to delimit what field size should be considered
small. In a newer paper by Kamio and Bouchard,30 which uses
the PBDM described herein, the approaches are somewhat
unified by defining a criterion on the field size with respect to
a perturbation zone, which definition is based on the behavior
of the perturbation functions. This zone (or region) represents
the area where the beam must be uncollimated to not require
significant correction with respect to reference beam dosimetry,
the significance being determined by a tolerance on the
quality correction factor. For small field dosimetry, this zone
can be interpreted as the limiting area within which a field
size is considered small and requires correction above the
tolerance. For instance, during quality assurance procedures
one could tolerate a k
fclin, fmsr
Qclin,Qmsr
within 0.995 and 1.005 without
the need for correcting the measurement. By definition, the
perturbation zone is associated to a detector, a beam energy,
and a given tolerance on the quality correction factor and can
be entirely determined using the perturbation function.30
Using this approach, it might be worth representing the
perturbation function varying in terms of distance from the
edge of the detector which electrons can travel. In Fig. 10,
detector perturbation functions (hdet) are presented as a function
of the pencil distance from the cavity edge in terms of
number of continuous slowing down approximation (CSDA)
F. 10. Detector perturbation functions calculated in two cavities: a 5 mm
radius and 2 mm thickness air cavity surrounded by a 1 mm graphite wall
and a 2 mm radius and 2 mm thickness silicon cavity surrounded by a 1 mm
aluminum wall. The x-axis represents the distance of the pencil beam from
the cavity edge (excluding the wall) in terms of maximum electron CSDA
range, taken here as 0.572 cm for electrons set in motion by the 1.25 MeV
photon beam.
Medical Physics, Vol. 42, No. 10, October 2015
6059 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6059
F. 11. Tolerance on kQ factors as a function of the field edge distance from the detector edge in number of CSDA ranges for five radiation detectors used in
the axial position (parallel to the beam). Data are shown for two photon spectra obtained with a Varian CLINAC 21EX linear accelerator model using 10×10
cm2
jaws settings: (a) 6 MV and (b) 25 MV. The data analyzed are taken from the work of Kamio and Bouchard (Ref. 30). Note that the reference beam was
defined as a 4×4 cm2
.
ranges potentially traveled by electrons with the highest
kinetic energy. For 1.25 MeV photon pencil beams, it is shown
that the perturbation function decreases by at least two orders
of magnitude from its maximum value when the pencil beam
position is approximately at a distance comparable to one
CSDA range, taken here as 0.572 cm and corresponding to
1.25 MeV electrons.56 This can be explained by the fact that
in typical fields comparable to a reference beam, most of the
energy absorbed in the detector comes from primary photon
interactions with water. Therefore, pencil beams incident from
distances larger than the maximum CSDA range of electrons
do not contribute to absorbed dose directly by secondary electrons;
energy is rather deposited by the secondary electrons
of scattered photons. The contribution of the latter electrons
to common detector absorbed dose could be expected to be
reasonably small for clinical photon energies and field sizes
below the size of typical reference beams.
Based on this rationale, one could attempt to define some
critical zone to be the area delimited by a distance of 0.572 cm
from the detector edge. In this critical zone, the primary beam
deposits most absorbed dose and the contribution of scattered
beam radiation can be neglected. Within these reasonable
assumptions, such critical zone could be used as a simple criterion
to define a size below which a field is consider to be small,
compared to a more detailed analysis requiring Monte Carlo
simulations, such as the tolerance- and detector-dependent
perturbation zone of Kamio and Bouchard.30
However, such a simplistic rule is not so trivial to apply in
realistic clinical photon beams. Indeed, the sum of the detector
edge position and the maximum electron CSDA range in water
(i.e., taking the maximum electron kinetic energy) is predictably
much larger than the size of perturbation zones allowing
a correction of a few tenths of a percent.30 Most beams are in
practice constituted of continuous spectra (with the exception
of characteristic emissions or photon annihilation), due to the
nature of photon beam production or to spectrum degradation
occurring in radioactive sources. Hence, electrons produced
in the phantom have on average a smaller lateral range than
the ones with the maximum kinetic energy. For instance, it
is unlikely that electrons from 25 MV photons incident at
distance of 11 cm from the detector edge, corresponding
roughly to the CSDA range of 25 MeV electrons,56 would
reach its cavity. The same logic is valid for 6 MV photons,
which would necessitate a margin of 3 cm around the detector,
corresponding roughly to the CSDA range of 6 MeV
electrons,56 to have full LCPE. The resulting limits on the
field size are clearly unrealistic to describe what small is, and
therefore, the rule based on the CSDA range of maximum
energy electrons cannot be correct. To determine the lateral
electron range, one should consider the spectrum of electrons
and the forward bias in their angular distribution, which is
more pronounced at higher energies and more isotropic at
lower energies, as well as electron elastic scattering. Estimating
such quantities accurately considering the full beam
spectrum would require Monte Carlo methods. But since all
physical effects are accounted for in these simulations, one
might as well consider the presence of the detector and predict
the size of the perturbation zone accurately.
In practice, the PBDM shows that the detector cannot be
ignored when providing an accurate definition of what field
size should be considered small for a given beam quality. In
Fig. 11, the tolerances of Kamio and Bouchard30 on quality
correction factors for 6 and 25 MV are plotted for different
radiation detectors as a function of the distance from the detector
cavity edge. While the tolerances decrease with field
size and depend on the beam energy, they strongly depend on
the detector geometry. This suggests that only a fully modeled
detector dose response can provide a rigorous definition of
how small a field should be considered. The theory defined by
Kamio and Bouchard30 and in the present work provides tools
to address this problem.
5. SUMMARY
The present paper describes in detail the nature of the
perturbation effects involved in radiation dosimetry and shows
how they arise in megavoltage small fields. A modern approach
to radiation dosimetry is developed by interpreting Fano’s
Medical Physics, Vol. 42, No. 10, October 2015
6060 Bouchard et al.: Detector response in small beams. II. Perturbation effects 6060
theorem in the form of a cavity theory applicable under any
conditions and adapted for Monte Carlo simulations. The
approach is detailed in a formalism factorizing the ratio Z
A
water-to-medium
Z
A
w
m
, the overall perturbation factor PMC,
and the volume averaging factor Pvol. The overall perturbation
factor is decomposed into three perturbation subfactors,
i.e., extracameral Pext, atomic composition Pmed, and density
Pρ, which are calculated using hypothetical intermediate cavities
for which the composition and geometry is changed in
three steps. The PBDM is applied to express the correction
factor for each perturbation effect as the integral of the perturbation
function, weighted by the fluence distribution of the
beam. This way, the method allows calculating quality correction
factors for any field size, given the perturbation functions.
Volume averaging is not investigated explicitly in this work
and it is recommended to correct this effect independently of
the other effects with analytic methods.
To provide examples on the importance of perturbation
effects in small fields, perturbation functions are calculated
by simulating detector dose response to pencil beams in two
simplistic detectors and their hypothetical cavities. Monte
Carlo simulations are performed using Fano calculations
which, by allowing CPE in broad parallel beams, quantitatively
support the theoretical demonstration of the density
perturbation effect provided in Paper I. DRFs and perturbation
functions are also calculated with Monte Carlo in realistic
conditions (i.e., not Fano calculations) and compared between
different scoring volumes to highlight the relative importance
of the three perturbation effects. It is concluded from the
analysis that the density perturbation factor is potentially most
sensitive to beam collimation or modulation, especially for airfilled
detectors, and that extracameral components can play
an important role if their composition is significantly different
from water and their dimensions are comparable to the cavity
size. Perturbations due to the atomic properties of the detector
are shown to be the least important effect for fields smaller
than a reference beam. However, it is pointed out that due
to the significant difference in atomic number with respect
to water, such as in silicon, it is unlikely that one can draw
a general conclusion on the importance of such perturbation
effects.
Calculations also demonstrate the existence of a critical
zone, also referred to as perturbation zone, near the edges
of the detector cavity where the perturbation function can
vary by several orders of magnitude over a short distance.
This feature is at the basis of defining to what extent, for a
given detector and beam energy, one can reduce a field size or
modulate its fluence and keep correction factors insignificant
within a given tolerance. Therefore, it is concluded that despite
the potential usefulness of lateral charged particle equilibrium
(LCPE) to define at what size a field is considered to be small,
it is unlikely that a simplistic rule can be attributed to all
detectors. Indeed, since the size of the perturbation zone varies
with the detector type, a more rigorous definition, such as the
one of Kamio and Bouchard,30 might be necessary. Further
investigation would be required to validate this in clinical
situations and we encourage the community to do so using
realistic detectors.
ACKNOWLEDGMENTS
We wish to thank Professor Jeffrey Williamson from VCU
for his support and fruitful comments on this work. Partial
funding by the UK National Measurement Office is also gratefully
acknowledged.
a)Electronic address: hugo.bouchard@npl.co.uk
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