a model to determine the initial phase space of a clinical electron beam from measured beam data

7/28/2019 A Model to Determine the Initial Phase Space of a Clinical Electron Beam From Measured Beam Data

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INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 46 (2001) 269286 www.iop.org/Journals/pb PII: S0031-9155(01)13960-6

A model to determine the initial phase space of aclinical electron beam from measured beam data

J J Janssen1, E W Korevaar1, L J van Battum1, P R M Storchi1,3 and

H Huizenga2

1 Daniel den Hoed Cancer Center, University Hospital Rotterdam, PO Box 5201, 3008AE Rotterdam, The Netherlands2 Joint Center for Radiation Oncology ArnhemNijmegen, University Medical Center Nijmegen,PO Box 9101, 6500 HB Nijmegen, The Netherlands

E-mail: [email protected]

Received 17 May 2000

Abstract

Advanced electron beamdose calculationmodels for radiation oncology require

as input an initial phase space (IPS) that describes a clinical electron beam. The

IPS is a distribution in position, energy and direction of electrons and photons

in a plane in front of the patient. A method is presented to derive the IPS

of a clinical electron beam from a limited set of measured beam data. The

electron beam is modelled by a sum of four beam components: a main diverging

beam, applicator edge scatter, applicator transmission and a second diverging

beam. The two diverging beam components are described by weighted sums

of monoenergetic diverging electron and photon beams. The weight factors

of these monoenergetic beams are determined by the method of simulatedannealing such that a best fit is obtained with depthdose curves measured

for several field sizes at two sourcesurface distances. The resulting IPSs are

applied by the phase-space evolution electron beam dose calculation model to

calculate absolute 3D dose distributions. The accuracy of the calculated results

is in general within 1.5% or 1.5 mm; worst cases show differences of up to

3% or 3 mm. The method presented here to describe clinical electron beams

yields accurate results, requires only a limited set of measurements and might

be considered as an alternative to the use of Monte Carlo methods to generate

full initial phase spaces.

1. Introduction

Full 3D electron beam dose calculation models like the phase-space evolution model (Janssen

etal 1994, 1997) and several Monte Carlo and macro Monte Carlo models (Nelson etal 1985,

Neuenschwander et al 1995, Kawrakow etal 1996) require as input an initial phase space (IPS)

3 Author to whom correspondence should be addressed.

0031-9155/01/020269+18$30.00 2001 IOP Publishing Ltd Printed in the UK 269


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270 J J Janssen et al

that describes a clinical electron beam. This IPS describes the applied clinical electron beam

in terms of a distribution in position, energy and direction of electrons and photons in a plane

perpendicular to the beam axis in front of the patient. This plane is called the IPS plane in this

paper.

A method of determining the IPS of a clinical electron beam is to simulate the electrontransport through the head of a clinical accelerator and register the electrons that enter the IPS

plane. The EGS4/BEAM Monte Carlo code system (Rogers et al 1995) is an implementation

of this method. The IPS produced by the EGS4/BEAM code can be characterized with a

multiple-source model (Ma et al 1997) that has the advantage that it requires substantially

less IPS storage capacity. Although good results have been presented with this method it

may be difficult to apply it in clinical practice. The method requires a detailed description

of the specific accelerator for the specific beams as well as Monte Carlo expertise. As the

specifications of the accelerator components are often not known adequately and the machines

are also tuned individually at the installation sites, it can be a real challenge to match calculated

dose distributions with measured beam data (Ma and Jiang 1999a). It is probably impossible

to apply this method routinely at every cancer clinic, although a reference machine can be

simulated accurately to derive beam model parameters that can then be adjusted on the basisof measured data for machines of the same type (Ma and Jiang 1999a).

As an alternative to the described Monte Carlo method the present work pursues a simple

method to derive the IPSs of clinical electron beams from a limited set of measured beam

data. Methods based on only a limited set of measured beam data have a great advantage in

clinical practice since they can be easily implemented in treatment planning systems and do

not require any specialist knowledge for configuring a specific beam (Storchi et al 1999). In

the present case, the parameters of a simple four-source model for the electron beam IPS are

determined based on depthdose curves measured for several field sizes and sourcesurface

distances. Although the method described is very simple and open to improvements, the

resulting calculated dose distributions have an accuracy that is clinically acceptable for use

with high-energy electron beams in three-dimensional conformal radiation therapy.

2. An initial phase space formed by four beam components

The IPS describes the electron and photon fluence F ( x , y , , , E , q ) in the IPS plane per

monitor unit (MU) differential in position, energy and direction in units of MeV1 rad2 cm2.

Here (x,y) is the position in Cartesian coordinates, (,) is the direction in polar coordinates,

E is the energy and q is the charge of the particle (0 = photons, 1 = electrons).

In the present paper the location of the IPS plane is chosen to coincide with the lower end

of the electron applicator or cutout. The choice of the location of the IPS plane at the lower

end of the electron applicator results in an IPS that is not fixed for a specific combination of

electron beam energy and applicator. Here the IPS takes account of the field shape of the

applied cutout. This is different from the EGS4/BEAM code system where the location of

the IPS plane is chosen to be just above the lower end of the applicator or cutout. The part

of the applicator or cutout just below this IPS plane, the part that determines the field shape,is considered to be a part of the patient. The choice of IPS plane just above the lower end of

the applicator results in one fixed IPS for a specific combination of electron beam energy and

applicator regardless of the field shape of the applied cutout.

It is assumed that the IPS can be adequately described by the sum of four simplified beam

components, each of which accounts for a part of theelectron or photonfluence in theIPS plane.

It is assumed that by adequately deriving parameter values and weights for these components,

a description of the IPS emerges that is adequate as input for electron beam dose calculations.


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Initial phase space of clinical electron beams 271

main

diverging beam

applicator

scatter

applicator

transmission

second

diverging beam

IPS-plane

(a) (b) (c) (d)

Figure 1. A schematic view of the directional distribution of the four beam components that are

used to model theIPS of a clinical electron beam. Thesecond divergingbeam is a rough approach to

model the particles (scattered from photon jaws, scrapers and applicator walls) that are not covered

by the other three beam components.

First we will describe these components qualitatively, then we will present the mathematical

description and finally we will explain how the parameters are derived from measured beam

data of a specific accelerator. The four beam components (figure 1) are:

(a) The first component, the main diverging beam, models the electrons and bremsstrahlung

photons that do not interact with any other part of the accelerator head besides scattering

foils, monitor chamber and air before passing through the open part of the applicator

diaphragm.

(b) The second component, the applicator scatter, models the electrons that scatter from the

edge of the lower applicator as the result of the impact of electrons on the lower applicator.

(c) Thethirdcomponent, theapplicator transmission, models thephoton transmission throughthe lower applicator as the result of the impact of electrons and photons on the lower

applicator.

(d) The fourth component, the second diverging beam, models the electrons and photons that

have interacted with various other parts of the accelerator head before passing through the

applicator diaphragm, and which are not modelled by the first three components.

The first, second and third components should closely model particles that are present in the

electron beam. The fourth component, however, is a rough approach to model the particles

that are not covered by the first three components. To model these remaining particles

with a second diverging beam is a rough approach, since these particles have scattered from

photon jaws, scrapers and applicator walls, and most certainly do not originate from one point

source.

The first component is assumed to form the prominent part of a clinical electron beam(figure 1(a)). It is assumed that the directional distribution of the electrons and photons of

the main diverging beam is adequately described by the directional distribution of a diverging

beam with an angular variance (Huizenga and Storchi 1987). It is also assumed that this

main diverging beam remains identical with respect to its direction and energy distribution for

different electron beam applicators, for a specific clinical accelerator energy. Electron beam

applicators,possibly combined with a device tomodifybeam shape,such asa cerrobendcutout,

will only affect the beam shape and the weight of the main diverging beam. The weight or


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fluence per monitor unit of the main diverging beam depends on the amount of backscattering

towards the monitor chamber from the photon jaws, whose settings are associated with the

applied applicator. Furthermore it is assumed that the most of the possible non-flatness of an

electron beam is caused by the non-flatness of the main diverging beam. This non-flatness can

be incorporated into the main diverging beam by correcting the fluence over the surface of theIPS plane.

The second component reflects the fact that the edge of the lower applicator or cutout is an

important source of scattered electrons and possibly photons (figure 1(b)). It is assumed that

the applicator scatter is adequately described by integrating a point spread kernel of electrons

and photons along the diaphragm edge. One point spread kernel describes the direction and

energy distribution of scattered particles, and is applied to simulate the scattered particles at

every position along the edge. Differences between scattered particles at different positions

along the edge are neglected.

The third component reflects the fact that the lower applicator or cutout does not stop all

electrons and photons outside the open part of the applicator diaphragm (figure 1(c)). Photon

transmission in particular can account for a few per cent of the dose in the region outside the

main beam area. It isprobablysufficientto simulate thedirectional distributionof thesephotonswith a diverging beam originating from an isotropic point source. For convenience, however,

the directional distribution of the photons of the applicator transmission is described by the

directional distribution of a diverging beam with an angular variance. This is the same type

of directional distribution that is applied by the main diverging beam, but now this directional

distribution is applied for the area outside the open part of the diaphragm. This simplification

will hardly affect the resulting calculated dose distributions.

The fourth component, the second diverging beam, is a rough approach to describe the

remaining electrons and photons that pass through the open part of the diaphragm in the IPS

plane (figure 1(d)). These electrons and photons do not originate from a single point source

but come from sources like photon jaws, scrapers, applicator walls etc. However, we attempt

to model these electrons with a second diverging beam with a directional distribution that is

identical to the directional distribution of the main diverging beam. This second diverging

beam has an adjustable beam weight that is supposed to correct for the inherent inadequacy ofthe second diverging beam to simulate the intended electrons and photons. Further, the weight

of this second diverging beam depends on the applied applicator, on the applied cutout and on

the actual sourcesurface distance (SSD).

Cutouts are applied to shape the field of a clinical electron beam. They will mainly block

the direct electrons and photons that we simulate with our main diverging beam. Cutouts will

also block the electrons that scatter from various parts of the accelerator head. We simulate

these scattered electrons in a crude way with our second diverging beam. The blocking by a

cutout of the scattered electrons from the applicator walls will result in a decrease in fluence

at the surface of the phantom (figure 2(a)). A diverging beam like our second diverging beam

cannot simulate this decrease in fluence without decreasing its weight. Therefore, we allow

the weight of the second diverging beam to be adjustable. We assume that this weight depends

on the shielding length (in centimetres) of the electron beam applicator (figure 2(b)).The shielding length is the length of the lower part of an electron beam applicator that is

blocked from sight by an applied cutout if one looks upwards from the patient surface. The

idea is that each part of the applicator wall (including photon jaws and scrapers) is a potential

source of scattered particles, and that for an increasing shielding length the number of these

scattered particles that reach the patient surface decreases because they are blocked by the

cutout. The shielding length of the electron beam applicator depends on the equivalent square

field size of the applied cutout, the size or width of the applied applicator and the actual SSD.


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Shielding

length (cm)

(a) (b)

Figure 2. Blocking of the scattered electrons from the applicator walls by a cutout ( a) and

correspondingshieldinglength (b). Theshielding lengthof theapplicator dependson theequivalent

square field size of the cutout and the actual SSD.

A weight factor look-up table is determined for each applicator. This weight factor look-up

table will show a decreasing weight of the second diverging beam for an increasing shielding

length. The weight factors for arbitrary cutouts and SSDs are determined from this look-up

table based on the actual shielding length of the electron beam applicator.

The four beam components form the total IPS of the electron beam denoted by:

F ( x , y , , , E , q ) = Fmain(x , y , , , E , q ) + Fscatter(x , y , , , E , q )

+Ftransmission(x , y , , , E , q ) + Fsecond(x , y , , , E , q ). (1)

Each of the four beam components is described with a functional form. The functional forms

of the main diverging beam, applicator transmission and second diverging beam are almost

identical. They are described by the product of a weight factor, an energy distribution, a

directional distribution of a diverging beam and a field shape function. The functional form ofthe main diverging beam is extended with a fluence correction function that allows correction

for not-flat beam profiles. The applicator edge scatter is described by the product of a weight

factor, a point spread kernel and a field edge function. The functional forms that describe the

four beam components for a specific electron beam energy of an accelerator are:

Fmain(x , y , , , E , q ) = w(A)em(E,q)d(x,y; ,)oin(C; x,y)c(x,y)

Fscatter(x , y , , , E , q ) = w(A)k(, + edge(C; x,y),E,q)ledge(C; x , y )

Ftransmission(x , y , , , E , q ) = w(A)et(E,q)d(x,y; ,)oout(C; x , y )

Fsecond(x , y , , , E , q ) = w2(A; C, SSD)e2(A; E,q)d(x,y; ,)oin(C; x,y). (2)

Here A is the applied applicator, C is the applied cutout and SSD is the actual sourcesurface

distance. C equals A for applicators without a cutout. The functions are as follows:

w(A) and w2(A; C, SSD) are weight factors.

em(E,q), et(E,q) and e2(A; E , q ) are energy distributions of electrons and photons in

units of MeV1.

d(x,y; , ) is a position-dependent directional distribution function in units of rad2

that describes a diverging beam with a constant energy-dependent angular variance at the

level of the IPS plane. This function is defined over an infinite plane without restricting

itself to a specific area. Field area functions restrict this function to a specific area.


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Cutout C

(a)

oin(C;x,y)

(b)

oout(C;x,y)

(c) (d)

ledge(C;x,y)

edge(C;x,y)

Figure 3. A cerrobend cutout (a) and corresponding field shape functions, the inside (b)and outside (c) field area functions, the diaphragm edge (d) (outline) and the azimuth anglesperpendicular to the diaphragm edge (d) (arrows).

oin(C; x , y ) and oout(C; x , y ) are field area functions in units of cm2 that respectivelydescribe the area inside and outside the open part of the diaphragm at the level of the IPS

plane. oin is defined by: oin(C; x , y ) = 1 (cm2) for (x,y) inside the open part of the

diaphragm, 0 otherwise. oout is defined by: oout(C; x , y ) = 1 (cm2) for (x,y) outside

the open part of the diaphragm, 0 otherwise (see figures 3(b) and 3(c)).

c(x,y) is a fluence correction function in arbitrary units that allows correction for any

non-flatness in the profiles of clinical electron beams at the IPS plane level.

k( , , E , q ) is a point spreadkernel of electronsandphotons, differential in direction and

energy in units of MeV1 rad2. This kernel is not circularly symmetric in azimuth angle.

The majority of the electrons scatter from the field edge with an azimuth angle towards

the diaphragm opening. The field edge azimuth angle function edge(C; x , y ) points this

kernel in the right direction.

edge(C; x , y ) is a field edge azimuth angle function in radians that describes the azimuthangle perpendicular to the field edge in the direction of the diaphragm opening. edgeis defined for all x and y coordinates of the field edge at the level of the IPS plane (see

figure 3(d)).

ledge(C; x , y ) is a field edge function that describes the field edge at the level of the IPS

plane in units of cm2. ledge is defined by: ledge(C; x , y ) = (x xI(u)). (y yI(u))

where xI(u) and yI(u) are the field edge coordinates (see figure 3(d)).

This functional form implies that different IPSs for different applicators or cutouts for one

specific electron beam energy have a lot in common. The functions that are independent of

the applied applicator or cutout are:

The directional distribution of the main diverging beam, applicator transmission and

second diverging beam.

The energy distribution of the main diverging beam.

The fluence correction function of the main diverging beam.

The point spread kernel of the applicator scatter.

The energy distribution of the applicator transmission.

Since these functions are oblivious to the field shape they are combined with functions that

take the shape of the cutout diaphragm into account. The functions that depend solely on the

shape of the cutout diaphragm are (figure 3):


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The field area functions.

The field edge function.

The field edge azimuth angle.

Only a few functions depend on the applied applicator: The weight factors for all four beam components.

The energy distribution of the second diverging beam.

2.1. Required measured beam data

When applying this model in treatment planning, the functions that depend solely on the shape

of the cutout diaphragm are determined during planning for each specific beam. The other

functionsare determined duringcommissioning of theplanning systemfrom a prescribed setof

measurements. Most of the required measurements are absolute central axis depth dose curves

in water in units of dose per monitor unit (cGy MU1). In this paper the abbreviation ADD

is applied for absolute central axis depthdose curve. A few measurements are required for

uncollimated electron beams. The uncollimated electron beam is produced by an acceleratorwithout an electron applicator and with the photon jaws in their outermost positions. The

basic beam data set that is required for the commissioning of a specific energy of a specific

accelerator consists of:

The measured ADDs in water at SSDs of 100 cm and 110 cm for each applicator without

cutout.

The measured ADDs in water at SSDs of 100 cm and 110 cm for each applicator with

three different cutouts.

The ADD in water at a SSD of 100 cm for the uncollimated electron beam.

Ion chamber readings in air, at several distances from the assumed source of the

uncollimated electron beam.

A profile measurement in air at the level of the IPS plane for the uncollimated electron

beam.

2.2. The directional distribution d(x,y; , ) of the diverging beam

The directional distribution of the diverging beam is applied in three components, i.e. the main

diverging beam, the applicator transmission and the second diverging beam. This directional

distribution is based on the directional distribution of the electrons that do not interact with

any other part of the accelerator head besides scattering foils, monitor chamber and air before

passing through theopen part of theapplicator diaphragm. These electrons form theprominent

part of a clinical electron beam. A good approximation of the directional distribution of these

electrons of a clinical electron beam at the level of the IPS plane within the diaphragm is the

directional distribution of a diverging electron beam with an angular variance originating from

scattering in air denoted by (Huizenga and Storchi 1987)

d(x,y; x , y ) =1

2 x yexp

(x x/z0)2

22x

exp

(y y/z0)2

22y

[rad2]. (3)

This directional distribution function is defined by projected angles x and y . 2x

and 2ydenote the angular variance in x and y respectively (

2x

= 2y ). z0 is the distance from

the IPS plane to the focal point of the main diverging beam. The directional distribution

d(x,y; , ) that is defined in polar and azimuth angles and is also required.


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The distribution in the polar interval k (

k < < +k ) and azimuthal interval l

(l < < +l ) can be derived by integration ofd (x , y , x , y ) (equation (3)) denoted by

d(x,y; k

, l) =

+x (k,l)

x (k,l)

+y (k,l)

y (k,l)

d(x,y; x

, y

) dy

dx

. (4)

Application ofx = arctan(tan cos ) and y = arctan(tan sin ) allows the transformation

of this integration over projected angles x and y into an integration over polar and azimuthal

angles and denoted by:

d(x,y; k, l ) =

+kk

+ll

1

2 2x

1

1 + (tan cos )21

1 + (tan sin )2tan

cos2

exp

[arctan(tan cos ) x/z0]2

22x

exp [arctan(tan sin ) y/z0]222x

d d. (5)After the determination of the values for the parameters x and z0 the directional distribution

d(x,y; , ) is solved numerically.

The angular variances can be determined from measurements of profiles in air,

perpendicular to the beam axis (Hogstrom et al 1981, Huizenga and Storchi 1987). A value

of2x = 13T(E) [rad2] is adequate for almost all accelerators (van Battum and Huizenga

1999), where T(E) is the scattering power in air in units of rad2 cm1 for an electron beam

of energy E MeV. The focal point z0 can be taken as the source of the electron beam, but it is

preferable to define a virtual source (ICRU 1984) which is determined by applying the inverse

square law to ion chamber readings made in air, at several distances from the assumed source

of the uncollimated electron beam.

The photons of the main diverging beam are produced in the scattering foils asbremsstrahlung. Forconvenience, it is assumedthatthedirectional distributionof these photons

is identicalto thedirectionaldistribution of theelectrons of themain divergingbeam (Hogstrom

et al 1981). This implies that for the angular variance in the direction of the photons the same

values are used as for the angular variance of the electrons in the main diverging beam. In

reality, it is expectedthat theangularvarianceof thesephotons is negligible. This simplification

will hardly affect the resulting dose distributions calculated by any electron beam model since

x and y are relatively small (e.g. 0.026 rad at 15 MeV), and the energy deposition by the

photons created as bremsstrahlung in an accelerator head is relatively small compared with the

energy deposition by the electrons in that beam, even at 50 MeV.

2.3. The fluence correction function c(x,y) of the main diverging beam

The dose profiles of clinical electron beams should ideally be flat at the surface. This is not

usually the case. If the non-flatness of an electron beam does not vary in time it may be

possible to incorporate this non-flatness within the IPS of the clinical electron beam. The

fluence correction function c(x,y) can be determined by in air profile measurements of the

uncollimated beam at the level of the IPS plane. The fluence correction function is normalized

such that the fluence in the beam centre in the IPS plane is 1 (cm2), thus c(0, 0) = 1. So far,

only circularly symmetric fluence correction functions c(r) are applied where r =

x2 + y2.


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2.4. The energy distribution em(E,q) of the main diverging beam

The main diverging beam models the electronsand bremsstrahlung photons that do not interact

with any other part of the accelerator head besides scattering foils, monitor chamber and air

before passing through open part of the applicator diaphragm in the IPS plane. Directlymeasuring dose distributions of this sole clean main diverging beam is practically impossible

since this would require the dismantling of the accelerator head except for the scattering foils

andmonitor chamber. Thenearestpossibleclean machine for whichmeasurementsarepossible

is the uncollimated electron beam, an electron beam without an electron applicator and with

the photon jaws in their outermost positions. We assume that the IPS of this uncollimated

electron beam is described by a sole diverging beam, denoted by

Fun(x , y , , , E , q ) = wuneun(E,q)d(x,y; ; )c(x, y). (6)

d(x,y; , ) and c(x,y) are the already defined directional distribution and fluence correction

functions. wun and eun(E,q) are the weight factor and energydistribution for the uncollimated

electron beam. The beam shape function is not required to describe the uncollimated beam

since oin(un; x , y ) = 1 [cm2] for all x and y. The energy distribution eun(E,q) and weight

factor wun are determined from the measured absolute central axis depthdose curve (ADD) ofthe uncollimated electron beam in water at an SSD of 100 cm. The idea is that this measured

ADD can be fitted with a weighted sum of calculated ADDs of diverging monoenergetic

electron and photon beams with a fluence in the IPS plane that is defined by the product

d(x,y; ,)c(x,y).

The phase-space evolution (PSE) model (Janssen et al 1994, 1997) is applied to calculate

the ADDs for the required set of monoenergetic electron beams. The beam energies of the

set of monoenergetic beams are 0.5, 1, 1.5, 2, 2 .5, . . . , 30 MeV. The PSE model is also

equipped with a photon transport and dose calculation model to calculate the dose distribution

resulting from bremsstrahlung created in the patient and the accelerator head. Since the energy

deposition by photons is relatively small compared with the energy deposition by electrons, a

simple photon transport model was implemented in the PSE model. This model is based on

the linear attenuation of photon energy fluence during photon transport. The linear attenuation

coefficient that controls this photon energy fluence transport depends on the mean energyE0 of the electron beam. Consequently, the PSE model does not calculate a set of ADDs

for monoenergetic photon beams, but calculates one ADD for a photon energy fluence in the

reference plane that is defined by the product d(x,y; ,)c(x,y).

The method of simulated annealing (Metropolis et al 1953) is applied to determine the

weight factors of the monoenergetic electron beams and the weight factor of the photon beam.

The weight factors are determined such that the deviation between the measured ADD and the

weighted sum of calculated ADDs is minimized in a root mean square sense according to

=

15 cmz=0.5 cm

ADD(m, z) w(p)ADD(p,z)

30 MeVe=0.5 MeV

(w(e)ADD(e,z))

2. (7)

Here z is the depth (z = 0.5, 1, . . . , 15 cm) of measured and calculated ADDs. ADD(m, z) is

the measured dose at depth z, ADD(e,z) is the calculated dose at depth z for electron beams ofe MeV (e = 0.5, 1, . . . , 30 MeV), ADD(p,z) is the calculated dose at depth z for the photon

energy fluence, w(e) are the electron beam weight factors and w(p) is the photon beam weight

factor. Simulated annealing is applied to minimize by adapting these weight factors.

wun and eun(E,q) can be determined from these calculated weight factors ifeun(E,q) is

normalized according to

0

eun(E, 1) dE = 1. (8)


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(a)

0.0

0.1

0.2

0 5 10 15 20

Energy (MeV)

Weight(#)

scattered electrons

direct electrons

(b)

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

Depth (cm)

Dose(cGy/MU)

measured un-collimated beam

calculated photons only

calc. direct e- and photons

calc. direct+scattered e- and photons

Figure 4. Energy distributions of the main diverging beam (a) and corresponding ADDs (b) of

the uncollimated 20 MeV Varian Clinac 2300 electron beam. eun(E,q) contains the direct and

scattered electrons and the photons. The photonsare not shown in (a). em(E,q) contains the directelectrons and photons.

Thederivedenergydistributioneun(E,q) probablycontainsa small fractionofelectronand

photon contamination that has scattered from various remaining parts of the accelerator head.

The energy distribution of the main diverging beam em(E,q) is estimated by cleaning up

the possible scattered electrons from eun(E,q). Figure 4(a) shows the product wuneun(E, 1)

(wun = 0.45). In general three parts are distinguishable in eun(E,q) (figure 4(a)): (a) a large

amount of electron fluence around the nominal beam-energy (the direct electrons), (b) some

smaller amounts of electron fluence in the region with a lower energy (the scattered electrons)

and (c) a large amount of photon energy fluence (not shown in figure 4(a)). The energy

distribution of the main diverging beam em(E,q) is found by removing the scattered electrons

from eun(E,q), thus leaving the direct electrons and the photons. This cleaning up is based onthe assumption that the energy distribution of the electrons after passing through the scattering

foils and monitor chamber is still rather monoenergetic. Simulations of accelerator heads with

the BEAM Monte Carlo code system (Rogers et al 1995) support this assumption. BEAM

simulations show that there area large numberof directphotons andthat theenergy distribution

of the direct electrons is very monoenergetic. In the example shown in figure 4(a) the scattered

electrons form about 4% of the total (direct plus scattered) electron energy. The associated

photon component is such that 24% of the total (electron plus photon) energy is carried by

photons. These photons contribute only a few per cent of dose to the ADD; however, they are

the major contributors to the photon-tail (figure 4(b)). The IPS of equation (6) is applicable

for full 3D absolute (cGy MU1) dose calculations. Figure 4(b) shows the match between

measured and PSE calculated ADDs of the uncollimated electron beam. Figure 4(b) also

shows the PSE calculated ADD based on the IPS of equation (6) where eun(E,q) is replacedby the cleaned energy distribution em(E,q).

2.5. The weight factorw(A)

The weight factor w(A) represents the fluence per monitor unit of the main diverging beam.

The fluence per monitor unit depends on the amount of backscattering towards the monitor

chamber from thephoton jaws, the settings of which are associated with the applied applicator.


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Therefore w(A) can be different for each applied applicator. However, we assume that the

effect of this backscattering towards the monitor chamber from the photon jaws is negligible,

and that the weight factor w(A) of the main diverging beam is equal to the weight factor wunof the uncollimated electron beam as determined in the previous section for each applicator A

(see the discussion).

2.6. The edge scatter kernel k(,,E,q) of the applicator scatter

The main diverging beam models the majority of electrons and photons that pass through the

open part of the applicator or cutout diaphragm. It is assumed that the majority of electrons

and photons that are incident on the closed part of the applicator or cutout diaphragm are

similar to the electrons and photons that are modelled by the main diverging beam. The PSE

model is applied to determine the point spread kernel k(,,E,q) by calculating the impact

of the main diverging beam on the applicator or cutout edge placed in the beam centre and

registering all electrons and photons which leave the applicator via its edge. This method

has been described in the literature (Ebert and Hoban 1995, Asell and Ahnesjo 1997). In the

calculation of the point spread kernel k(,,E,q) the transport of other electrons or photonsbesides those of the main diverging beam through the applicator edge is neglected and the

effect of the divergence of the main diverging beam is ignored for off-axis applicator edges.

2.7. The energy distribution et(E,q) of the applicator transmission

The energy distribution function et(E,q) models the photon (q = 0) as well as electron

(q = 1) transmission. So far, only photon transmission has been considered and electron

transmission is ignored. Again, it is assumed that the majority of electrons and photons that

are incident on the closed part of the applicator or cutout diaphragm are similar to the electrons

and photons that are modelled by the main diverging beam. The PSE model is applied to

determine the photon energy fluence leaving the applicator by calculating the transport of the

main diverging beam through a fully closed applicator or cutout and registering all photons

which leave the applicator. In the calculation of the photon energy fluence the transport ofother electrons or photons besides those of the main diverging beam through the applicator is

neglected and the derived photon energy fluence is assumed to be constant outside the open

part of the diaphragm.

2.8. The energy distribution e2(A; E , q ) of the second diverging beam

So far, three beam components, i.e. the main diverging beam, applicator scatter and applicator

transmission, are fully determined and their contribution to any measured ADD can be

calculated. The difference between any measured ADD and the associated calculated ADD

contribution by the three known beam components shouldyield theIPS of thesecond diverging

beam (see equation (2)):

Fsecond(x , y , , , E , q ) = w2(A; C, SSD)e2(A; E,q)d(x,y,,)oin(C; x,y). (9)

The energy distribution e2(A; E , q ) is determined based on this difference between the

measured and calculated ADDs of an electron beam collimated by applicator A without cutout

in water at an SSD of 100 cm. The energy distribution e2(A; E , q ) and associated weight

factor w2(A; A, 100) are determined with a method similar to the determination of the energy

distribution eun(E,q) and weight factor wun of the uncollimated beam (see section 2.4). Now

theidea is that this difference between ADDs canbe fitted with a weightedsumof thecalculated


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(a)

0.0

0.1

0.2

0 5 10 15 20

Energy (MeV)

Weight(#)

main diverging beam

second diverging beam

(b)

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

Depth (cm)

Dose(cGy/MU)

measured 10x10transmission

edge scatt er

second diverging beam

main diverging beam

sum of 4 components

Figure 5. Energy distributions of the two diverging beam-components (a) and corresponding

ADDs of all the beam-components (b). The measured ADD is of a 10 10 cm2 20 MeV Varian

Clinac 2300 electron beam.

ADDs of diverging monoenergetic electron and photon beams with a fluence in the IPS plane

that is defined by the product d(x,y; , ) oin(A; x , y ).

The PSE model is applied to calculate the ADD contribution of the three known beam

components plus the ADDs for the required set of monoenergetic beams. The method of

simulated annealing is applied to determine the weight factors that result in the best fit of

the difference in ADDs. Again, w2(A; A, 100) and e2(A; E , q ) can be determined from

these calculated weight factors ife2(A; E , q ) is normalized in a way similar to equation (8).

Figure 5(a) shows the product w2(A; A, 100)e2(A; E, 1) for the 10 10 cm2 applicator

(A = 10 10) (w2(A; A, 100) = 0.116). In this case the associated photon component is

negligibly small. Figure 5(b) shows the corresponding contributions per beam component for

this 20 MeV 10 10 cm2 electron beam. The main diverging beam contributes about 80%

of the total electron beam. This contribution agrees well with EGS4/BEAM results (Rogers

et al 1995). The sum of the four beam components gives a perfect match with the measured

ADD.

2.9. The weight factorw2(A; C,SSD) of the second diverging beam, when using cutouts

Theweightof thesecond divergingbeam depends on theshielding length(in centimetres)of the

applicator (figure 2(b)). This shielding length depends on the equivalent squarefield sizeof the

applied cutout and on the actual SSD. w2 is a kind of look-up table that contains the shielding

lengths andassociated weight factors for a predefinedsetof cutoutandSSDcombinations. The

weight factor for an arbitrary shielding length of the applicator is determined by linear interpo-

lation between the available weight factors in the look-up table. A weight factor look-up tablethat consists of eight weight factors and associated shielding lengths of the applicator is found

to be sufficient. At the moment our look-up tables consist of weight factors for the applicator

without cutout and for the applicator with three different cutouts at SSDs of 100 and 110 cm.

The weight factors that form the look-up table are determined in a similar way to the

determination of the weight factor w2(A; A, 100) in the previous section, but now the energy

distribution e2(A; E , q ) is already known. Again, the difference between a measured ADD

and the associated calculated ADD contribution by the main diverging beam, applicator


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Figure 6. Weight factorlook-up tablesof seconddiverging beams (a) anda subsetof corresponding

ADDs (b). The weight factor w2(A; C, SSD) is shown as a function of the shielding length for the12 and 20 MeV 10 10 cm2 Varian Clinac 2300 electron beams. The ADDs shown are 12 MeV

beams at an SSD of 110 cm2 and 20 MeV beams at an SSD of 100 cm for field sizes 10 10, 5 5,4 4 and 3 3 cm2.

scatter and applicator transmission should yield the second diverging beam. Now, this

difference in ADDs is fitted with the weighted ADD of the second diverging beam for the

associated cutout and SSD with a fluence in the IPS plane that is defined by the product

e2(A; E , q )d (x , y; , ) oin(C; x , y ). Only the weight factor w2(A; C, SSD) needs to be

determined. Figure 6(a) shows the weight factor look-up tables for two energies of a

10 10 cm2 applicator (see the discussion). Figure 6(b) shows a comparison between a subset

of corresponding measured ADDs and PSE calculated ADDs based on the four components

and the weight factor look-up tables.

The IPS as denoted by equation (2) is now fully determined and is applicable for full 3Dabsolute (cGy MU1) dose calculations for a specific energyof a clinical electron beam shaped

by an arbitrary cutout and applicator at an arbitrary SSD.

3. Results

The described method to determine the IPS of clinical electron beams is applied for 12 and

20 MeV electron beams of a Varian Clinac 2300 accelerator. The measurements required for

the IPS configuration are provided by Anna Samuelsson and Karl-Axel Johansson from the

Department of Radiation Physics, University of Goteborg, Sahlgrenska University Hospital,

Sweden. As already shown in figure 6(b) there is a good agreement between the measured and

PSE-calculated ADDs. However, one has to bear in mind that each of the measured ADDs

shown in figure 6(b) is applied as input for the IPS configuration method.The IPS configuration method is also applied for a 14 MeV Siemens MXE accelerator.

The required measurements are provided by Bruce Faddegon from the Department of Physics,

Toronto-Sunnybrook Regional Cancer Centre, Toronto, Canada. The weight factor look-up

table w2(A; C, SSD) is determined based on only four measured ADDs. Figure 7 shows this

look-up table based on the measured ADDs of the 10 10 and 3 3 cm2 field sizes at SSDs

of 100 and 110 cm. The IPS configuration results are shown in figure 8. Here the measured

ADDs at SSDs of 100 and 110 cm are applied as input for the IPS con figuration method.


14/18


Figure 7. Weight factor look-up table of the second diverging beam for the 14 MeV 10 10 cm2

Siemens MXE electron beam. The weight factor w2(A; C, SSD) is shown as a function of the

shielding length.

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8

Depth (cm)

Dose(cGy/MU)

(b)

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8

Depth (cm)

Dose(cGy/MU)

Figure 8. Measured (full curves) and PSE-calculated (crosses) ADDs of a Siemens MXE 14 MeV

electron beam for the 10 10 cm2 applicator without cutout (a) and for the 3 3 cm2 cutout (b)

at SSDs of 100, 110 and 120 cm.

The measured ADDs at the SSD of 120 cm are not applied as input for the IPS configuration

method; here the weight of the second diverging beam is determined by linear interpolationbetween the available weights in the weight factor look-up table (figure 7). The respective

shielding lengths for the1010 and 3 3 cm field sizes at SSD of 120 cmare 1.3 and 62.7 cm.

The calculated ADD of the 10 10 cm2 field at SSD 120 cm is in good agreement with the

measured ADD. For the 3 3 cm2 field size at SSD 120 cm the differences are too large.

This difference demonstrates the limitations of the IPS configuration model for its application

for shielding lengths outside the range of the weight factor look-up table. However, the range

of the weight factor look-up table can be extended to include the required shielding lengths


15/18


Cerrobend cutout

Figure 9. A 4 cm off-axis 4 10 cm2 cutout for use in a 15 15 cm2 applicator.

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

Depth (cm)

Dose(cGy/MU)

(b)

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8

X-axis (cm)

Dose(cGy/MU)

Figure 10. Measured (full curves) and PSE-calculated (crosses) ADDs (a) and profiles at 2.1 cmdepth (b) of a 21 MeV Siemens KD2 electron beam, SSD = 100 and 110 cm. The standard

15 15 cm2 applicator with a 4 cm off-axis cerrobend cutout of 4 10 cm2.

by incorporating measurements at an SSD of 120 cm into the IPS configuration method, thus

increasing the accuracy of the results at the SSD of 120 cm.

We also applied the described method for determining the IPS of clinical electron beams

to 21 MeV electron beams of a Siemens KD2 accelerator. Again there is a good agreement

between measurementsandPSEcalculations, as there is fora 4 cm off-axiscutoutof 410cm2

in a 15 15 cm2 applicator shown in figure 9. Figure 10 shows the associated measured and

PSE calculated dose distributions. Here the measurements have not been applied as input for

the IPS configuration method. The results at the SSD of 110 cm2 of this off-axis cutout showthat for this worse case scenario the accuracy decreases to about 3% or 3 mm.

The results shown are in good agreement with the measurements except for the 3 3 cm2

field size at 120 cm SSD shown in figure 8(b). But the results for this specific case can

be improved by incorporating measurements at an SSD of 120 cm into the IPS configuration

method. In general the accuracy is within 1.5% or 1.5 mm; worst cases are within 3% or 3 mm.

These results are quite accurate if one considers the simplicity of this four-beam-component

model and the applied assumptions.


16/18


4. Discussion and conclusions

The resulting calculated dose distributions based on the IPS configuration method are rather

good, although the derived beam components may not fully agree with the real physical beam

components of a clinical electron beam. Some aspects of this beam model may require furtherinvestigation.

4.1. The main diverging beam

The determination of the weight factor w(A) and energy distribution em(E,q) of the main

diverging beam is not very precise. The cleaning method ofeun(E,q) (figure 4(a)) is rather

arbitrary. Perhaps some of the scattered electrons should have been left in, and some of the

direct electrons should have been removed from em(E,q). Also, the weight factor w(A) is

kept constant between applicators (section 2.5) thus ignoring the effect of the position of the

photon jaws on this weight factor.

In an earlier version of the IPS configuration model the cleaning methodwas notapplied,

thus em(E,q) = eun(E,q), and the weight factor was assumed constant w(A) = wun. In this

version the weight factor of the second diverging beam was also a constant w2

(A; C, SSD) =

w2(A; A, 100). In order to test this model Bruce Faddegon (Department of Physics, Toronto-

Sunnybrook RegionalCancerCentre, Toronto, Canada) provided us with measured data from a

Siemens MXE accelerator. There wasa good agreementbetween measured andPSE calculated

dose distributions for a clinical 14 MeV 10 10 cm2 electron beam at SSDs of 100, 110 and

120 cm. But, the PSE calculation overestimated the dose by 10%, 15% and 20% for a cutout

of 3 3 cm2 at respective SSDs of 100, 110 and 120 cm. The main diverging beam alone

(without applicator scatter, applicator transmission or second diverging beam) delivered about

10% more dose in the case of an SSD of 120 cm. The first conclusion was that either the

energydistribution or the weight factor, or both, of the main diverging beam had to be reduced.

However, in the IPS configuration method, a reduction of the main diverging beam results in

an increase of the second diverging beam. Thus, the reduction of the main diverging beam

would have no effect on the PSE calculated dose distributions. The second conclusion was

that for the second diverging beam also, either the energy distribution or the weight factor, orboth, had to be reduced. The result of these two conclusions is found in this paper: the energy

distribution of the main diverging beam is cleaned, the weight factor of the main diverging

beam is still a constant, the energy distribution of the second diverging beam is not changed,

and the weight factor of the second diverging beam is determined via a look-up table that

depends on the shielding length of the applicator.

The weight factor of the main diverging beam is still kept constant. Possible effects due to

variable positions of the photon jaws as associated with different applicators are still ignored.

The change in jaw position affects the number of backscattered electrons from the photon jaws

to the monitor chamber. It also affects the volume of the air column that helps to form the main

diverging beam. The change in backscatter towards the monitor chamber may be relatively

smallandcan probablybe ignored, but thechange involume of theair columncannot be ignored

that easily (Ma and Jiang 1999b). This means that it might be better to vary the weight of themain diverging beam per applied applicator. Therefore, a simple method should be developed

to determine this change in beam weight. If possible this method should be based on some

simplemeasurements, analogously to thesimplicity of the IPS configuration modeldescribed.

4.2. The second diverging beam

Theindirectparticlesin a clinical electron beam aresimulated rathercrudelywith theapplicator

scatter, applicator transmission and second diverging beam. In particular the simulation of the


17/18


remainderof indirectparticleswitha seconddivergingbeamthat originates fromonefocalpoint

is quite an approximation. Perhaps there is an alternative beam component (or a combination

of components) that is more suitable for this task. This alternative beam component would

probably have to be used with a kind of ray tracing method in order to discard the electrons

that are blocked by the cutout. This ray tracing is now performed by the variable weight factorof the second diverging beam.

The shielding-length-dependent curve that determines this weight factor is seldom a

smooth function (figure 6(a)). This non-smoothness is caused by the fact that equal shielding

lengths can be associated with several combinations of cutout diaphragms and SSDs that each

requires an alternative weight factor. It is easier to determine a set of smooth weight factor

curves for separate SSDs, also at an extended SSD of 120 cm, but this is not implemented in the

current IPS configuration method. An impression of the possible error caused by choosing the

wrong weight factor can be seen by considering that about of 20% of the total electron beam is

carried by the second diverging beam (figure 5(b)) and the weight factor alters the contribution

of the second diverging beam by somewhere between 75% and 100% (figure 6(a)). Therefore

the dose range that is controlled by the weight factor of the second diverging beam is limited

to about 0.05 cGy MU

1

or 5%.This IPS configuration method shows that a model based on four beam components is

sufficient for the calculation of the IPSs of clinical electron beams. This method also shows

that the parameter values that define such IPSs can be derived from a limited set of measured

electron beam data.

Acknowledgments

The authors would like to thank students Lars Wittebrood, Robert-Jan Westerduin and Astrid

van der Horst for investigating topics related to the subject of this paper during their practical

year. We thank our colleagues Remco van Vliet and Hafid Akhiat for testing and commenting

on the first user friendly version of the IPS configuration program. We also thank our

colleagues Dick Bax and Philip Verlinde for helping to link the PSE and IPS software to the

CadPlan planning system. We thank Anna Samuelsson and Karl-Axel Johansson (Department

of Radiation Physics, University of Goteborg, Sahlgrenska University Hospital, Sweden) for

providing the measured data from a Varian Clinac2300 accelerator. We thank Bruce Faddegon

(Department of Physics, Toronto-Sunnybrook Regional Cancer Centre, Toronto, Canada) for

providing us with measured data that showed us that a predecessor of the presented IPS

configuration method without an adaptable weight for the second diverging beam did not

function well for small cutouts. Further we thank Charlie Ma and Steve Jiang (Department

of Radiation Oncology, Stanford University School of Medicine, Stanford, USA) for the

discussions during the Electron and Photon Transport Theory Workshop held in Indianapolis,

USA. We thank the Dutch Cancer Society (NKB 92-94) and Varian Medical Systems Finland

for funding this research project.

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