Thesis

D O SIM ETRY O F VERY SM A LLPH O TO N FIELD S

b y

K am en A. Pas k alevD epartm ent of Med ical Ph ys ics

M cGill Univers ity, M ontrealM ay 2002

A th es is subm itted to th e Faculty ofGraduate Studies and Research in partial

fulfillm ent of th e req uirem ents for th edegree of M aster of Science

K am en A. Pask alev, 2002

ii

Abstract

Several dosimetric parameters were measured for three very small radiation fields

(1.5, 3, and 5 mm diameter at the machine isocenter) with a small ionization chamber and

a new type of radiochromic film. The experiments were carried out on a Clinac-18 linac

and the fields were shaped by specially manufactured collimators. When measuring dose

profiles, the ionization chamber measurements were first corrected for response variation

in off-axis direction, and then deconvolved to eliminate the blur due to the poor

resolution of the chamber. The measured data agreed with Monte Carlo simulations

within the established statistical uncertainties.

Dynamic stereotactic radiosurgery was carried out on the same accelerator using

the very small radiation beams. The dose distributions and their displacements from the

laser-defined isocenter of the linac were measured and then compared to 3-D Monte

Carlo calculations. The results proved that dynamic radiosurgery with very small beams

has potential for clinical use.

iii

Résumé

Plusieurs paramètres dosimétriques ont été mesurés pour trois très petits champs

de radiation (1.5, 3 et 5 mm de diamètre à l’isocentre de la machine de traitement) avec

l’aide d’une chambre à ionisation à petit volume et un nouveau type de film

radiochromic. Les mesures ont été effectuées avec un accélérateur linéaire Clinac-18 sur

lequel les champs de traitement sont définis par des collimateurs spécialement usinés.

Lors des mesures des profils de dose, les mesures initiales sont premièrement corrigées

pour la variation de la réponse en direction hors axe, et puis convoluées pour éliminer le

flou causé par la faible résolution de la chambre. Les données mesurées sont en accord

avec les simulations Monte-Carlo à l’intérieur des incertitudes statistiques établies.

Des irradiations de radio chirurgie stéréotaxique dynamique ont été effectuées sur

le même accélérateur en utilisant les très petits champs. Les distributions de dose et leurs

déplacements par rapport à l’isocentre, défini par les lasers de la machine de traitement,

ont été mesurés et puis comparés aux calculs 3-D de Monte-Carlo. Les résultats

démontrent que la radio chirurgie dynamique à l’aide de très petits champs est

potentiellement utilisable en clinique.

iv

Acknowledgments

I would like to express my gratitude to Dr. Ervin B. Podgorsak, firstly for being

my teacher in medical physics, and secondly for giving me the opportunity to work on

such a challenging and interesting project, as well as for being my supervisor. He has

allowed me to work independently and has encouraged me to explore any idea related to

the subject. At the same time Dr. Podgorsak has always been very helpful and open for

discussions, providing me with the real scientific point of view. He also secured all the

resources I needed to work quickly and efficiently, regardless of the expenses and the

time required to do so.

I would also like to thank Dr. Jan Seuntjens for providing me with all knowledge I

needed in the areas of precise dosimetry and Monte Carlo simulations, and for his

friendly and supportive attitude.

During my research I had very helpful discussions with Dr. Dimitre Hristov, Dr.

Pavel Stavrev, Dr. François DeBlois and my fellow students Wamied Abdel-Rahman,

Kristin Stewart, Robert Doucet, and Khalid Al-Yahya. I want to thank them and all the

staff and students at the Department of Medical Physics for being a friendly and helpful

environment

Finally, I would like to thank my family for their constant support, and all my

previous teachers for their dedication over the years.

Table of Contents

Chapter 1: Introduction............................................................................................... 1

1.1. Overview of external photon beam therapy................................................. 1

1.2. Stereotactic Radiosurgery ........................................................................... 3

1.3. Treatment planing ant the basic dosimetry functions................................... 5

1.4. Dosimetry of very small fields.................................................................... 15

1.5. Thesis objectives ........................................................................................ 21

1.6. Thesis organization..................................................................................... 22

Chapter 2: Materials and Methods – Equipment and Experimental Techniques........... 29

2.1. Radiation source......................................................................................... 29

2.2. Dosimeters ................................................................................................. 33

2.3. Experimental setups.................................................................................... 40

2.4. Dynamic stereotactic radiosurgery.............................................................. 43

Chapter 3: Materials and Methods – Monte Carlo Particle Transport Simulations....... 48

3.1. Theoretical basics....................................................................................... 48

3.2. Monte Carlo transport of electrons and photons.......................................... 51

3.3. Monte Carlo code systems for photon and electron transport ...................... 54

3.4. Monte Carlo simulation of the Clinac-18 linear accelerator......................... 60

3.5. Monte Carlo simulations in water phantoms............................................... 63

Table of Contents

vi

Chapter 4: Exradin A14P ionization chamber study.................................................... 69

4.1. Geometry of the measuring volume............................................................ 69

4.2. Deconvolution of dose profiles in water...................................................... 75

4.3. Correction factors for off-axis measurements.............................................. 83

Chapter 5: Experimental Results

5.1. Beam quality.............................................................................................. 89

5.2. Physics of dose deposition .......................................................................... 100

5.3. Off-axis ratios (dose profiles)..................................................................... 105

5.4. Central axis measurements......................................................................... 112

5.5. Dynamic stereotactic radiosurgery with the 1.5 mm and 3 mm beams......... 117

Chapter 6: Conclusions and Future Work................................................................... 123

6.1. Summary and conclusions ......................................................................... 123

6.2. Future work ................................................................................................ 126

List of Figures............................................................................................................ 128

List of Tables ............................................................................................................. 136

Bibliography .............................................................................................................. 137

Chapter 1:

Introduction

1.1. Overview of external photon beam therapy

Radiation was first used for cancer treatment soon after the discovery of x-rays by

Roentgen in 1895 and radioactivity by Becquerel in 1896. Since then the knowledge of

how ionizing radiation cures disease has grown tremendously as a result of large

improvements in technologies employed for this purpose and experience gained from

clinical work. There are several different types of treatment techniques in modern

radiation therapy, but the vast majority of cancer treatments are carried out using the

external photon beam irradiation techniques in which the radiation source is placed at

some distance (typically 100 cm) from the patient. For almost five decades since the

beginning of last century, the treatments of this kind were performed using x-ray tubes

and Van de Graaff generators. However, these devices were capable of producing x-ray

beams only in the 100 kV range, and thus were not very useful for treating tumors located

deep inside the human body1.

At the end of 1951, the first patient treatment with cobalt-60 gamma rays was

delivered. The cobalt teletherapy unit was developed by Harold Johns’ group in

Saskatoon, Saskatchewan, Canada1,2. The operation of the unit was based on a cobalt-60

radioactive isotope source, emitting gamma rays with energies of 1.17 MeV and 1.33

MeV, and having an average energy of 1.25 MeV. The cobalt unit had a great impact on

Chapter1: Introduction

2

radiotherapy at that time and was the most widely used equipment for external beam

treatments during the following twenty years.

In the beginning of the 1960s a new machine for external beam dose delivery was

introduced: a relatively small linear electron accelerator (linac), specifically designed for

cancer treatment in a clinical environment1,2. A photograph of a modern clinical linac is

shown in Fig. 1.1. The operating principle of a linac is based on a narrow high-energy

electron beam striking a target in order to produce bremsstrahlung photons. The linac has

many advantages compared to the cobalt unit, and only a few disadvantages. It is capable

of delivering photon and electron beams with energies much higher than the 1.25 MeV

gamma rays produced by a cobalt unit. Because of its considerable advantages over the

cobalt unit, the linac became the equipment of choice in radiotherapy during the past

twenty years.

Figure 1.1: Typical view of a linear accelerator used in cancer therapy.


3

1.2. Stereotactic radiosurgery

AAPM Report No. 54 defines Stereotactic Radiosurgery (SRS) as a treatment

technique that combines the use of stereotactic apparatus and energetic radiation beams to

irradiate a lesion with a single treatment3. This treatment modality is mainly used to treat

intracranial lesions, such as primary brain tumors, functional disorders (epilepsy,

Parkinson’s disease, etc.), arteriovenous malformations (AVMs), and brain metastases.

Several clinical applications have been presented by Podgorsak et al.4,5, McKenzie et al.6,

Luxton et al.7, etc. The doses delivered in a single session can be as high as 50 Gy, and

the typical dimensions of the planning target volume are on the order of several

millimeters to several centimeters1.

A digital subtraction angiography (DSA) image of a small AVM is shown in Fig.

1.2 with the dimension of the malformation on the order of 5 mm. Since the targets are

even smaller when some functional disorders are treated with radiation8,9, the sizes of

radiation beams used for radiosurgery are thus typically much smaller than those used in

conventional radiotherapy.

During the 1960s and the 1970s, photon beam radiosurgery was carried out with a

special device called the Gamma knife or Gamma unit1,10-12, based on 201 cobalt-60

sources and dedicated solely to radiosurgery. The beams were collimated in such a way

that they crossed one another at the isocenter (focus) of the unit. In the 1980s, when

linear accelerators with well-defined and precise isocenters became commercially

available, several linac-based radiosurgery techniques were introduced into clinical

practice4,5,13.


4

In the current clinical practice both the linac and the Gamma knife are used for

radiosurgery treatments. The diameter of the radiation beam is between 10 mm and 40

mm when the radiosurgery treatment is carried out with a linear accelerator, while the

Gamma unit uses four collimators defining beam diameters of 4, 8, 14 and 18 mm.

Because the fields are so small, there are three main issues that have a great

impact on the treatment accuracy: target localization, target positioning and dose

distribution calculations.

The uncertainty associated with target localization is an issue that the different

imaging modalities (MRI, CT, DSA) are dealing with. The modern equipment is capable

of determining the target position with uncertainties as low as ±1 mm.

The second issue is related to target positioning with respect to the isocenter of

the treatment machine. It introduces uncertainties firstly because of possible tissue

1 cm

Figure 1.2: DSA image of a small AVM.


5

motion between the imaging and treatment processes and secondly because of problems

in defining the isocenter of the treatment machine. The isocenter of a Gamma unit is

defined within ±0.3 mm whereas that for a linac is defined within ±1 mm1.

The third issue is related to the accuracy of dose distribution calculations, and this

represents a serious problem, especially for fields smaller than a few millimeters in

diameter. The basic dosimetry parameters for such fields, as discussed bellow, are

difficult to measure.

The first and second issues will not be addressed in this thesis; the third one, on

the other hand, motivated this work. Very small radiation fields (defined with dimensions

bellow 5 mm) have a great potential for use in treatment of functional disorders;

however, defining their parameters, be it with measurement or calculation, represents a

considerable challenge, as discussed in this work.

1.3. Treatment planning and the basic dosimetry functions

The goal of every external photon beam treatment is to deliver a dose distribution

which conforms to the planning target volume (PTV) as precisely as possible, sparing the

surrounding healthy tissue. The term planning target volume, as well as some other terms

commonly used in radiation therapy, are defined by the International Commission on

Radiation Units and Measurements (ICRU), Report No. 50 (Ref. 14) and refined in the

ICRU, Report No. 62 (Ref. 15). The typical steps associated with an external beam

treatment are shown in the block diagram in Fig. 1.3.


6

After a patient is diagnosed with cancer and a decision is made to treat the disease

with radiotherapy, a physician and a medical physicist determine the planning target

volume (PTV), taking into account margins for non-visible (microscopic) spread of the

decease and for all uncertainties in dose delivery. When the volume to be irradiated is

determined, a particular treatment technique must be chosen. The beam geometry is

selected either on a computer using 3D diagnostic data from a CT scanner (virtual

simulation) or on a low energy x-ray machine, which has the geometry that the high-

energy accelerator will have during the treatment, but incorporates a high quality imaging

system (conventional simulation). After all parameters of the treatment technique are

Figure 1.3: Typical steps in radiotherapy treatment process.

Virtual Simulation/Conventional Simulation

Treatment

Diagnosis Target localizationand PTV definition

Dose distributioncalculation Prescription


7

determined, a special software commonly called the “treatment planing software” is

used, generally by a dosimetrist, to calculate the dose distribution inside the patient’s

body. If the dose distribution conforms well to the PTV and spares the organs at risk, a

physician will prescribe the actual dose to be delivered, and the treatment will begin. The

dose distribution of a typical four-field box treatment plan superimposed on a CT axis

abdominal image is shown in Fig. 1.4.

In order to understand the relationship between radiation dosimetry and radiation

therapy, one should be familiar with the basic principles on which treatment planing

algorithms are based. The radiation dose delivered to an arbitrary point (a point-of-

interest) in a phantom or a patient is calculated by multiplying the dose to a certain point,

called a reference point, with a coefficient, which relates the dose at the reference point to

Figure 1.4: Dose distribution for a typical four-field box technique.


8

the dose at the point-of-interest. We need to know both the absolute dose delivered to the

reference point and the relationship between this dose and the dose to the point-of-

interest to perform this calculation.

The standard procedures developed for measuring absolute dose follow absolute

dose measurement calibration protocols. Examples of commonly used protocols are the

protocol that was developed by the Task Group 51 (Ref. 16) of the American Association

of Physicists in Medicine (AAPM) and the protocol developed by the International

Atomic Energy Agency (IAEA) TRS-398 (Ref. 17). Usually linacs are adjusted in such a

way that the maximum dose rate delivered over the central axis of the radiation beam for

field size of 10×10 cm2 and a source-surface distance (SSD) of 100 cm is 1 cGy/1

Monitor Unit (MU). One monitor unit corresponds to a given amount of electrical charge

collected by the monitor ionization chamber embedded in the linac treatment head. It is

an adjustable quantity, used for dose measurement instead of time, because it

compensates for the small output variations that are very typical for linacs.

The relationship between the dose to the very specific point mentioned above and

the dose to any other point within any beam geometry is derived using several tabulated

dosimetric functions, often called dosimetric parameters. These parameters are

determined by relative measurements and can be divided into two general categories: (i)

central axis functions and (ii) off-axis parameters.

For a given photon beam hν at a given source-surface distance (SSD), the dose at

point P (at depth maxd in phantom) depends on the field size A; the larger the field size,

the larger the dose. The relative dose factor RDF (often referred to as the total scatter


9

factor Sc,p or sometimes the machine output factor) is defined as the ratio of the dose at P

in phantom for field A to the dose at P in phantom for a 10×10 cm2 field, i.e.:

( )( , )

(10)P

P

D ARDF A h

Dν = , (1.1)

The geometry for measurement of the RDF(A) is shown in Fig. 1.5; part (a) for

the measurement of ( )pD A and part (b) for the measurement of (10)pD . It is important to

note that maxd might be different for different field sizes. RDF increases with the field

size and has a value of one for the standard 10×10 cm2 field. As it follows from the

definition, RDF is useful for calculating absolute dose delivered to a point at maxd .

Central axis dose distributions inside the patient or phantom are usually

Figure 1.5: Geometry for the measurement of the relative dose factor RDF(A). Thedose at point P at dmax in phantom is measured with field A in part (a)and with field 10×10 cm2 in part (b).


10

normalized to Dmax = 100% at the depth of dose maximum maxd and then referred to as

the percentage depth dose (PDD) distributions. The percentage depth dose is thus

defined as follows:

( , , , ) 100 Q

P

DPDD d A f h

Dν = × , (1.2)

with the geometry shown in Fig. 1.6. Point Q is an arbitrary point at depth d on the beam

central axis; point P represents the specific dose reference point at maxd d= on the beam

central axis. PDD depends on four parameters: depth in phantom d , field size A, source-

surface distance SSD = f , and photon beam energy hν. PDD ranges in value from 0 at

d → ∞ to 100 at maxd d= .

Figure 1.6: Geometry for percentage depth dose measurement and definition.Point Q is an arbitrary point on the beam central axis at depth d, pointP is the point at dmax on the beam central axis. The field size A isdefined on the surface of the phantom.


11

A typical PDD curve for a 10 MV photon beam (SSD of 100 cm and field size of

10×10 cm2) is shown in Fig. 1.7. Tabulated curves for different photon beams are

published in the British Journal of Radiology, Supplement No. 25 (Ref. 18). In modern

radiotherapy, the treatments are mostly done using isocentric setups, which means that

the center of the PTV is placed at the linac isocenter. The main advantage of this setup is

that several beams might be used in patient treatment with no change in patient position

on the treatment machine when switching from one beam to the next. However, as a

result of the human body not being cylindrically symmetric, the SSD will change with the

gantry rotation. Hence, using PDD, which depends on SSD, is not a very convenient way

of determining the dose along the central beam axis. Therefore another approach is taken

in case of isocentric, often called source-axis-distance (SAD), setups. The function used

0

20

40

60

80

100

120

0 2 4 6 8 10 12 14 16

Depth (cm)

PD

D

Figure 1.7: PDD curve for a 10 MV photon beam, SSD = 100 cm, 10×10 cm2.


12

for this purpose is the Tissue Maximum Ratio (TMR). The definition of TMR is as

follows:

max

( , , )

QQ

Q

DTMR d A h

Dν = ’ (1.3)

where DQ is the dose in phantom at arbitrary point Q on the beam central axis and maxQD

is the dose in phantom at depth maxd on the beam central axis. The geometry for the two

dose measurements is shown in Fig. 1.8. It has been established that TMR does not

depend on SSD over the SSD range used in clinical radiotherapy.

The most commonly used off-axis function is the Off-Axis Ratio (OAR). OAR

shows how the dose changes in lateral direction with respect to the dose delivered to the

point on the central axis at the same depth:

dmax

Figure 1.8: Geometry for the measurement of the tissue-maximum ratio, TMR(d,AQ,hν).


13

( )( )(0)

D rOAR rD

= , (1.4)

where r is the off-axis distance. This function has to be tabulated for every particular

field, including fields in which additional accessories, such as wedges, are used.

A typical algorithm for calculating the dose to a certain point-of-interest using all

the data obtained by absolute and relative measurements is given in the block diagram in

Fig. 1.9. Obviously, one needs very precise data in order to calculate a dose distribution

with very low uncertainties. Therefore it is very important to improve continuously the

dosimetry equipment and techniques in order to develop new and more precise dose

delivery methods.

With the recent rapid development in the field of microelectronics and computer

systems, a new concept for dose delivery calculations is becoming more and more

common. The idea is to use computer simulations for all physical processes taking place

in a medium through which high-energy particles are moving. Several different

algorithms for this simulation have been developed19-21 under the common name of

“Monte Carlo particle transport simulations.” There is a very strong relationship

between the computer power involved and the calculation time on one hand, and the

precision of the Monte Carlo results on the other. For this reason, the Monte Carlo

algorithms are becoming more and more popular, as modern computers become ever

faster, cheaper and more affordable. However, even if very precise simulation results are

obtained, they must be verified by real experiments, using extremely accurate dosimetry

measurement techniques. On the one hand, Monte Carlo techniques are quite suitable for

simulations of the small diameter radiosurgical beams, minimizing calculation time in

comparison with relatively large standard radiotherapy beams. On the other hand,


14

experimental verification of radiosurgical beams is considerably more difficult than that

of standard radiotherapy beams.

Relativemeasurements

Absolutemeasurements

Dose to a referencepoint in standard

field

Dose to a referencepoint in the field of

interest

Dose to the centralaxis at the depth of

interestDose to the point of

interest

RDF

PDD

OAR

Figure 1.9: Basic treatment planning algorithm.


15

1.4. Dosimetry of very small fields

The photon fields used in the conventional radiation therapy are commonly

referred to as standard or large fields. Their sizes are larger than 5×5cm2. The fields with

diameters from 1 cm to 4 cm, used for radiosurgery, are known as small photon fields. In

this work we will refer to photon fields with diameters on the order of several millimeters

as very small fields. Dosimetry of very small photon fields is a very interesting subject,

because most of the standard dosimetry functions become extremely sensitive in the

range of small and very small radiation field sizes. For example, the depth of maximum

dose maxd decreases significantly with field size for fields smaller than 1 cm. The subject

is also challenging, because there are several specific requirements for the detectors,

which are very difficult to fulfill in very small radiation fields.

Many studies have been done using various detectors for different measurements

in small fields. The usefulness of small ionization chambers for central axis and for off-

axis measurements has been studied in detail22-27. The p-type silicon diodes have also

been considered because of their small measuring volume22,23,25. Several other

dosimeters such as radiochromic film26, plastic scintillator22,28, TLD26, MOSFET26, and

liquid-filled ionization chamber22 have been used as well. In addition, Monte Carlo

simulations have often been used for theoretical verification of experimental results29.

There are two important parameters concerning all detectors used for

measurements in very small photon fields: (i) resolution and (ii) water-equivalence of the

measuring material. Usually, the dose profiles (OAR) for fields smaller than 5 mm in

diameter change drastically over off-axis distance as small as 1 mm. Thus, the resolution


16

of the detector used will have a great impact on both the central axis and off-axis dose

measurements. There are two different approaches to solving this problem: (i) we either

need a detector with a well-known geometry or (ii) we need a detector with a very small

size, at least 6 to 7 times smaller than the field size.

Evidently, it is difficult to manufacture a detector small enough and accurate at

the same time when, for example, the field size is as small as 2 mm in diameter.

However, if the exact geometry of the detector sensetive volume is known, it is possible

to perform a deconvolution procedure that is very suitable for off-axis measurements.

The real dose profile will be the result of deconvolving the measured one (Fig. 1.10).

Several extensive studies on dose profile deconvolution have been conducted30-33 and in

all of them 1-D deconvolution was performed using the Line Spread Function (LSF) of

the detector. However, in the case of very small fields the dose fall-off in all lateral

directions should be considered. Therefore, 2-D deconvolution based on the Point Spread

Function (PSF) of the detector must be performed.

The second detector-related issue is water-equivalence of the measuring material

in terms of radiation properties, since water is the most commonly used phantom

OAR1

2

Off-axis Distance

Figure 1.10: Narrow beam profiles: curve (1) represents a measured profile, curve(2) a deconvolved profile.


17

material, similar to the human body tissue. Having a water-equivalent detector is

important, because the concept of the charged particle equilibrium (CPE) does not apply

to small photon fields in a lateral direction. In order to illustrate the importance of this

concept, let us take a closer look at how the radiation dose gets deposited in the phantom

when small photon beams are used.

When a high-energy photon enters a medium, photon-electron interactions will

occur with a certain probability. There are four main types of interactions: photoelectric

effect, coherent scatter, incoherent (Compton) scatter, and pair production. Usually, the

very rare triplet production interaction is taken into account in the pair production cross-

section. Due to these interactions, a significant part of the incident photon energy might

be transferred to an electron that thereafter will be set in motion. This electron will then

ionize the medium over its track by electron-electron Coulomb interactions. The radiation

dose deposited as a result of this two-step process is expressed as follows:

LD φ

ρ =

, (1.5)

where φ is the electron fluence at the point-of-interest and ( )/L ρ is the restricted mass

stopping power averaged over the electron energy spectrum. The restricted mass stopping

power is defined as:

1L Exρ ρ

=

VV , (1.6)

where ρ is the density of the medium and EV is the energy transferred to low-energy

electrons that deposit this energy locally, over a small distance xV .


18

For large photon fields, the assumption is made that the number of electrons that

stop inside a small volume VV is equal to the number of electrons set in motion by

photons inside the same volume (Fig. 1.11). Thus, the electrons can be considered as

continuously moving through the medium, instead of stopping and being set in motion.

This assumption actually underlines the concept of CPE. The electron fluence is constant

when this concept applies.

Generally, the detectors and the phantoms are made of different materials.

According to Eq. (1.5), the relationship between the dose delivered to the detector, dD ,

and the dose delivered to the medium, medD , is:

( ) ,med

medmed d d

d

LD D φρ

=

(1.7)

where med

med

d

d

LL

Lρ

ρρ

=

and ( )med medd

d

φφφ

= .

The dose to the phantom (medium) is calculated by multiplying the dose to the

detector by two correction factors: restricted stopping power ratio ( )/med

dL ρ and electron

fluence perturbation factor ( )med

dφ for the two materials: medium and detector. The

restricted stopping power ratio accounts for the difference in the energy deposited by an

electron per unity track length in the two different materials. The electron fluence

perturbation factor corrects for the difference in the electron fluence. When the detector

size is much smaller than the electron range for a given photon energy, in which case the

Bragg-Gray or Spancer-Attix cavity theories apply34, the second correction factor is


19

considered to be equal to one. Under condition of CPE, the two factors are constant for a

given photon energy, so that, even if they are not established, relative measurements may

be carried out without adversely effecting the reliability of measurements results.

For very small photon fields, the field size is usually smaller than the electron

range in the phantom material. In this situation a very important difference appears when

comparing an electron moving in a lateral direction in a very small field to an electron

moving in the same fashion in a large field (Fig. 1.12). When an electron moving

laterally in a large photon field approaches the point where it will stop, another electron

will be set in motion by a photon in the vicinity of this point. This will provide CPE in

the irradiated volume except for the penumbra region, which is very small in comparison

to the field size. In a very small field, many electrons will be able to reach points outside

the photon beam where no photon-electron interaction occurs, and the electron fluence

will change with the increase in the off-axis distance. Thus there will be no CPE in a

lateral direction, leading to the conclusion that the correction factors defined in Eq. (1.7)

Figure 1.11: Charged particle equilibrium: number of electrons stopped in a smallvolume is equal to the number of electrons set in motion by photonsin the same volume.

VV


20

will vary causing difficulties not only for absolute measurements, but even for relative

dosimetric measurements. In order to overcome this obstacle and to obtain reliable results

one would therefore need either a phantom-equivalent detector with both correction

factors equal to one or else a detector whose correction factors are precisely known.

Obviously, the choice of a detector is not an easy one. For this reason, the

experimental results have to be compared to reliable reference data until a reliable

measurement technique is established. Monte Carlo simulations can provide such reliable

data, as they are always relevant, regardless of the field size and shape. Of course, the

size of the voxels, in which the radiation dose is calculated, has to be very small to

achieve a high resolution and this requires a lot of computer power. Another very useful

application of the Monte Carlo simulations is calculating the correction factors for

different detector positions. Establishing such correction factors for ionization chambers,

for example, has recently become a routine approach in absolute dosimetry35-38.

Large fieldVery small

field

Figure 1.12: Electrons moving in lateral direction in large and very small photon fields.


21

1.5. Thesis objectives

The dosimetry of very small radiation fields is a challenging problem, largely

unexplored to date, but of great clinical importance for further developments in

radiotherapy. It could inspire many studies in the area of basic physics of dose delivery

and radiation dosimetry and result in new techniques in treatment of functional disorders

requiring very small radiation fields.

The first objective of this thesis is to explore the usefulness of two dosimeters for

very small field measurements: a small volume ionization chamber and radiochromic

film. The chamber is the smallest commercially available ionization chamber,

manufactured under the name Exradin A14P (Standard Imaging, Middleton, WI, USA).

This chamber is important because ionization chambers in general are very precise

measuring devices. An A14P dose response study should include not only the acquisition

of accurate measurements but also development of algorithms for both the profile

deconvolution and calculation of correction factors. The second dosimeter is a new type

of radiochromic film that recently appeared on the market under the commercial name

HS GafchromicTM (International Specialty Products, Wayne, NJ, USA). This film is a

very good reference dosimeter, however it has high uncertainties. It is meant to be water

equivalent over a very large energy range in contrast to the standard radiographic film.

The second objective of the present work is to examine the mechanism of dose

delivery by studying both photon and electron fluence and energy in a water phantom

using Monte Carlo simulations.


22

The third objective of the thesis is obtaining reliable data for PDD, OAR and RDF

by A14P ionization chamber, HS film measurements and Monte Carlo calculations for

several very small field sizes that might prove useful for clinical radiosurgery.

The final objective of the thesis is to determine whether or not linac-based

radiosurgery may be useful for irradiating intracranial targets with typical sizes on the

order of few millimeters. This study involves calculation of 3-D dose distributions as well

as dose distribution measurements.

1.6. Thesis organization

The present thesis consists of six chapters. Chapter 1 started with a brief overview

of external photon beam radiation therapy, followed by the basic concepts of stereotactic

radiosurgery and an introduction to treatment planing algorithms. Physics of radiation

dose deposition in very small photon fields is discussed later in this chapter, and the main

problems of dosimetry of such fields are outlined. The objectives of the thesis are stated

at the end of the first chapter.

Both the equipment as well as the experimental techniques, used for the purposes

of this work, are described in Chapter 2. The dynamic stereotactic radiosurgery technique

is introduced briefly in this chapter as the technique that we used to perform stereotactic

radiosurgery with very small photon beams.

Chapter 3 is dedicated to Monte Carlo particle transport simulation. It starts with

theoretical introduction to the Monte Carlo method, followed by an overview of all

Monte Carlo code systems used in this work.


23

An extensive study of the Exradin A14P ionization chamber is presented in

Chapter 4 starting by calculating the active measuring volume of the A14P chamber.

Based on this result, a simple deconvolution algorithm as well as method for studying the

dose response variation in a lateral direction are described.

All results are presented in Chapter 5. The results include measurements and

Monte Carlo calculations of several dosimetric parameters associated with static photon

beams, as well as 3-D dose distributions of dynamic radiosurgery procedures.

Chapter 6 contains a brief summary of the thesis, conclusions, and a discussion on

possible future studies and developments.


24

References:

1 J. Van Dyk, The Modern "Technology of Radiation Oncology," Medical Physics

Publishing, Madison, Wisconsin, 1999.

2 H. Johns and J. Cunningham, "The Physics of Radiology," Charles C Thomas,

Springfield, Illinois, 1983.

3 M. Schell, F. Bova, D. Larson et al., "Stereotactic Radiosurgery: Report of Task

Group 42 radiation Therapy Committee," Americam Institute of Physics, 1995.

4 E. B. Podgorsak, A. Olivier, M. Pla et al., “Dynamic stereotactic radiosurgery,” Int. J.

Radiat. Oncol. Biol. Phys. 14, 115-126 (1988).

5 E. B. Podgorsak, A. Olivier, M. Pla et al., “Physical aspects of dynamic stereotactic

radiosurgery,” Appl. Neurophysiol. 50, 263-268 (1987).

6 M. R. McKenzie, L. Souhami, J. L. Caron et al., “Early and late complications

following dynamic stereotactic radiosurgery and fractionated stereotactic

radiotherapy,” Can. J. Neurol. Sci. 20, 279-285 (1993).

7 G. Luxton, Z. Petrovich, G. Jozsef et al., “Stereotactic radiosurgery: principles and

comparison of treatment methods,” Neurosurgery 32, 241-259 (1993).

8 D. Urgosik, J. Vymazal, V. Vladyka et al., “Treatment of postherpetic trigeminal

neuralgia with the gamma knife,” J. Neurosurg. (Suppl. 3) 93, 165-169 (2000).

9 A. Haas, G. Papaefthymiou, G. Langmann et al., “Gamma knife treatment of of

subfoveal, classic neovascularization in age-related macular degeneration: a pilot

study,” J. Neurosurg. (Suppl. 3) 93, 172-175 (2000).


25

10 L. Leksell, “Cerebral radiosurgery I. Gamma thalamotomy in two cases of intractable

pain,” Acta Chir. Scand. 134, 585-595 (1968).

11 D. G. Leksell, “Stereotactic radiosurgery: current status and future trends,” Stereotact.

Funct. Neurosurg. 61 (Suppl. 1), 1-5 (1993).

12 M. R. McLaughlin, B. R. Subach, L. D. Lunsford et al., “The origin and evolution of

the University of Pittsburgh Department of Neurological Surgery,” Neurosurgery 42,

893-898 (1998).

13 W. Lutz, K. R. Winston, and N. Maleki, “A system for stereotactic radiosurgery with a

linear accelerator,” Int. J. Radiat. Oncol. Biol. Phys. 14, 373-381 (1988).

14 A. Wambersie, T.G. Landberg, J. Chavaudra et al., ICRU Report No. 50, 1993.

15 T. Landberg, J. Chavaudra, J. Dobbs et al., ICRU Report No. 62.

16 P. R. Almond, P. J. Biggs, B. M. Coursey et al., “AAPM's TG-51 protocol for clinical

reference dosimetry of high-energy photon and electron beams,” Med. Phys. 26, 1847-

1870 (1999).

17 P. Andreo, D.T. Burns, K. Hohlfeld et al., IAEA TRS Report No. 398, 2000.

18 E. Arid, J. Burns, M. Day et al., “Central axis depth dose data for use in radiotherapy:

1996,” British Journal of Radiology Supll. 25 (1996).

19 I. Kawrakow, “Accurate condensed history Monte Carlo simulation of electron

transport. I. EGSnrc, the new EGS4 version,” Med. Phys. 27, 485-498 (2000).

20 D. W. Rogers, B. A. Faddegon, G. X. Ding et al., “BEAM: a Monte Carlo code to

simulate radiotherapy treatment units,” Med. Phys. 22, 503-524 (1995).


26

21 J. Sempau, S. J. Wilderman, and A. F. Bielajew, “DPM, a fast, accurate Monte Carlo

code optimized for photon and electron radiotherapy treatment planning dose

calculations,” Phys. Med. Biol. 45, 2263-2291 (2000).

22 M. Westermark, J. Arndt, B. Nilsson et al., “Comparative dosimetry in narrow high-

energy photon beams,” Phys. Med. Biol. 45, 685-702 (2000).

23 X. R. Zhu, J. J. Allen, J. Shi et al., “Total scatter factors and tissue maximum ratios for

small radiosurgery fields: comparison of diode detectors, a parallel-plate ion chamber,

and radiographic film,” Med. Phys. 27, 472-477 (2000).

24 B. E. Bjarngard, J. S. Tsai, and R. K. Rice, “Doses on the central axes of narrow 6-MV

x-ray beams,” Med Phys 17, 794-799 (1990).

25 C. McKerracher and D. I. Thwaites, “Assessment of new small-field detectors against

standard-field detectors for practical stereotactic beam data acquisition,” Phys. Med.

Biol. 44, 2143-2160 (1999).

26 P. Francescon, S. Cora, C. Cavedon et al., “Use of a new type of radiochromic film, a

new parallel-plate micro-chamber, MOSFETs, and TLD 800 microcubes in the

dosimetry of small beams,” Med. Phys. 25, 503-511 (1998).

27 Y. W. Vahc, W. K. Chung, K. R. Park et al., “The properties of the

ultramicrocylindrical ionization chamber for small field used in stereotactic

radiosurgery,” Med. Phys. 28, 303-309 (2001).

28 D. Letourneau, J. Pouliot, and R. Roy, “Miniature scintillating detector for small field

radiation therapy,” Med. Phys. 26, 2555-2561 (1999).

29 F. Verhaegen, I. J. Das, and H. Palmans, “Monte Carlo dosimetry study of a 6 MV

stereotactic radiosurgery unit,” Phys. Med. Biol. 43, 2755-2768 (1998).


27

30 P. Charland, E. el-Khatib, and J. Wolters, “The use of deconvolution and total least

squares in recovering a radiation detector line spread function,” Med. Phys. 25, 152-

160 (1998).

31 F. Garcia-Vicente, J. M. Delgado, and C. Peraza, “Experimental determination of the

convolution kernel for the study of the spatial response of a detector,” Med. Phys. 25,

202-207 (1998).

32 P. D. Higgins, C. H. Sibata, L. Siskind et al., “Deconvolution of detector size effect

for small field measurement,” Med. Phys. 22, 1663-1666 (1995).

33 C. H. Sibata, H. C. Mota, A. S. Beddar et al., “Influence of detector size in photon

beam profile measurements,” Phys. Med. Biol. 36, 621-631 (1991).

34 L. V. Spencer and F. H. Attix, “A theory of cavity ionization,” Radiat. Res. 3, 239-254

(1955).

35 J. E. Bond, R. Nath, and R. J. Schulz, “Monte Carlo calculation of the wall correction

factors for ionization chambers and Aeq for 60Co gamma rays,” Med. Phys. 5, 422-

425 (1978).

36 J. Borg, I. Kawrakow, D. W. Rogers et al., “Monte Carlo study of correction factors

for Spencer-Attix cavity theory at photon energies at or above 100 keV,” Med. Phys.

27, 1804-1813 (2000).

37 C. M. Ma and A. E. Nahum, “Monte Carlo calculated stem effect correction for

NE2561 and NE2571 chambers in medium-energy x-ray beams,” Phys. Med. Biol. 40,

63-72 (1995).


28

38 J. Mazurier, J. Gouriou, B. Chauvenet et al., “Calculation of perturbation correction

factors for some reference dosimeters in high-energy photon beams with the Monte

Carlo code PENELOPE,” Phys. Med. Biol. 46, 1707-1717 (2001).

Chapter 2:

Materials and Methods -Equipment and Experimental

Techniques

1.1. Radiation source

All experiments presented in this work were carried out using a Clinac-18 linear

accelerator (Varian Oncology Systems, Palo Alto, CA) as a radiation source. A

photograph of the linear accelerator (linac) is given in Fig. 2.1. The machine has been in

clinical service in the Radiation Oncology Department of the Montreal General Hospital

since 1977. Currently the machine is used for special procedures such as total skin

electron irradiation and stereotactic radiosurgery. The distance between the linac

isocenter and the source of radiation (source-axis distance SAD) is 100 cm and the

distance between the isocenter and the floor is 126 cm.

The Clinac-18 linac has two operating modes. It can be used as a source of high-

energy electron beams with five different energies between 6 MeV and 18 MeV, and as a

source of 10 MV photons represented by a bremsstrahlung spectrum with a maximum

energy of 10 MeV. The design of the linac treatment head in the two different modes is

shown in Fig. 2.2. In both cases a narrow beam of electrons, accelerated in a 1.4 m long

waveguide, is transported to the treatment head assembly by a 270o achromatic bending

Chapter2: Materials and Methods – Equipment and Experimental Techniques

30

magnet system. When the machine is used in the photon mode, the electrons are slowed

down and completely stopped in a target, and the resulting bremsstrahlung radiation is

flattened by a flattening filter (Fig 2.2(a)). The electron range in the target material is

smaller than the thickness of the target itself, so that no high-energy electron can pass

through it. Such a target is called a thick target. The intensity of a photon beam produced

in a thick target decreases with the increase in the off-axis distance. Therefore after the

photon beam is collimated by a primary collimator, it traverses a flattening filter. The

purpose of this filter is to provide a constant intensity of the beam in off-axis directions.

The flattened beam then passes through a monitor ionization chamber, which is used to

Figure 2.1: Clinac-18 linear accelerator.


31

count monitor units (see Chapter 1). The secondary collimator, commonly referred to as

collimator upper and lower jaws, determines the actual field size defined at SAD. There is

a tray holder on the bottom side of the treatment head to which different accessories, such

as wedges, radiosurgical collimators and electron beam cones, are attached when special

treatment techniques are performed.

When the Clinac-18 linac is used in the electron mode, the target is removed and

the flattening filter is replaced with a scattering foil (Fig. 2.2(b)). The foil spreads out the

high-energy pencil electron beam, so that flat electron fields with relatively large field

sizes (up to 25×25 cm2 at machine isocenter) can be produced at an SSD of 100 cm.

The 10 MV photon beam was used as a radiation source in our experiments: the

gantry angle was 180o, the collimator angle was 90o, and the jaws were set to a 5×5 cm2

field at the isocenter. The very small photon fields were shaped using specially

manufactured collimators. Three circular fields with diameters of 1.5 mm, 3 mm, and 5

mm at the isocenter have been studied and the 5 mm collimator is shown in Fig. 2.3. The

Figure 2.2: Treatment head design in (a) photon and (b) electron mode.

?- ?-

Target

PrimaryCollimator

MonitorChamber

SecondaryCollimator

Tray

FlatteningFilter

ScatteringFoil

(a) (b)

Electron beamcone


32

collimators were designed in such a way that they could fit into a radiosurgery holder

developed at the McGill University Health Centre for the Clinac-18 linac (Fig. 2.4). The

radiosurgery holder is attached to the tray holder of the machine and allows for adjustable

and very precise collimator positioning.

The collimators are made of 10 cm thick blocks of lead or tungsten. The bottom

side of the collimators is at 70 cm from the source and at 30 cm from the isocenter when

a collimator is placed onto the holder. The 5 mm collimator consists of two hollow

cylinders made of lead, each of them 5 cm thick. The outer diameter of both cylinders is

6.8 cm. The inner diameter of the upper cylinder is 2.95 mm, whereas the diameter of the

lower one is 3.25 mm, as shown schematically in Fig. 2.5(a). The 1.5 mm collimator is

also made of lead, but the diameter of the hole is 1 mm all the way through the collimator

(Fig. 2.5(b)). The 3 mm is also a cylinder, but is made of tungsten and the diameter of the

collimator hole is 2 mm (Fig. 2.5(c)). The metal pieces themselves are placed in plastic

boxes, so that the collimators have outer geometry corresponding to the inner diameter of

the holder.

Figure 2.3: External view of a 5 mm collimator.


33

2.2. Dosimeters

There are two main issues related to detectors that could be used for dose

measurements in small photon fields: resolution and water-equivalence. These issues and

the relevant radiation physics explanations have been discussed in Chapter 1 and an

Figure 2.4: Radiosurgery holder with a small field collimator on, attached to theClinac-18 treatment head.

2.95 mm 1 mm

3.25 mm

10 cm

Pb Pb W

(a) (b) (c)

2 mm

Figure 2.5: Design of the collimators: (a) 5 mm field, (b)1.5 mm field, and (c)3 mm field.Collimators in (a) and (b) are made of lead, in (c) of tungsten.


34

overview of the use of different detectors in several studies has been presented in Chapter

1 as well. None of the detectors discussed, except for the radiochromic film, has the

resolution required for measurements in fields smaller than 5 mm. When film is used, the

aperture of the scanner/densitometer determines the resolution. As far as water

equivalency is concerned, the only detectors that could be referred to as water-equivalent

with no additional corrections are the radiochromic film and the liquid-filled ionization

chamber. This chamber would be the best dosimeter for measurements in non-

equilibrium conditions, but it is not commercially available.

Two types of dosimeters were used for the purposes of this work: a micro

parallel-plate ionization chamber (Exradin A14P: Standard Imaging, Middleton, WI,

USA) and radiochromic film (HS GafchromcTM: International Specialty Products, Wayne,

NJ, USA). The ionization chamber is very accurate, and if the correction factors are

properly calculated, the water-equivalency problem can be resolved easily. The

radiographic film is a less precise dosimeter than the ionization chamber, however, in

terms of radiological properties, it is very close to water. In addition, this film has a very

high resolution and is therefore a very good reference dosimeter.

2.2.1. Exradin A14P ionization chamber

The Exradin A14P ionization chamber is one of the smallest commercially

available ionization chambers (Fig. 2.6) and was specifically designed for measurements

in small photon fields. Pankuch et al.1 have presented a detailed study of this chamber

and Francescon et al.2 have used this chamber for central axis measurements for fields as

small as 4.4 mm in diameter.


35

Figure 2.7 shows the schematic design of an A14P ionization chamber. The

chamber actually resembles a cylindrical chamber with two caveats: (i) the top is flat, and

(ii) the guard electrode encompasses the collecting electrode all the way from top to

bottom. Therefore, the A14P chamber is referred to as a parallel-plate ionization

chamber. It is made of Shonka air-equivalent plastic C552, with the diameter of the

collecting electrode 1.5 mm and the inner diameter of the cavity 2 mm. Both the wall and

the entrance window are 1 mm thick. The chamber is waterproof and the measuring

volume is connected to the outside air by two ventilation tubes.

Obviously, the measuring volume of the A14P chamber is very small and this

results in a very low signal. According to the manufacturer, the collecting volume is 2.3

mm3 and an exposure of 1430 R is needed for collecting a charge of 1 nC. As a result, the

radiation-induced current forms a large fraction of the total signal measured, and this

means that the polarity correction factor, as defined in the Task Group No. 51 report3, is

relatively large. We calculated a value of 1.21 based on measurements in a large field.

Leakage of the ionization chamber is another important issue because of the low signal

current. According to the specifications, the leakage current of the A14P chamber is

Figure 2.6: Exradin A14P ionization chamber.


36

lower than 10-15 A, but this value also depends on the electrometer with which the

measurements are taken.

The electrometer used for our measurements was Keithley 6517A (Keithley

Instruments Inc., Cleveland, OH, USA) and is shown in Fig. 2.8. This electrometer

incorporates an adjustable voltage source that can supply voltages between –1000 V and

+1000 V to the chamber. The device was used in charge mode and it was connected to

the A14P chamber with a shielded triaxial cable. When the chamber is used along with

this electrometer, the leakage current is on the order of 0.02 pA at 300 V, provided that

Figure 2.7: Sketch of A14P ionization chamber.

Figure 2.8: Keithley electrometer.


37

the A14P chamber is well warmed up prior to the measurement. Usually irradiation with

30 Gy to 40 Gy in an open field is enough to attain such a low leakage current. A detailed

study of the A14P ionization chamber is provided in a separate chapter in this thesis.

2.2.2. Radiochromic film

The radiochromic film HS GafchromicTM was recently released on the market. In

general, the radiochromic films are good practical dosimeters, since they are not sensitive

to visible light, and no development is needed after irradiation. Moreover, they are water-

equivalent and are useful in a large energy range, which makes them very appropriate

dosimeters for measurements under non-equilibrium conditions.

Various types of radiochromic films that were widely used before the advent of

the HS film (MD55, MD–55-2, HD) have been extensively studied and the results of

these studies are summarized by the Task Group 55 of the American Association of

Physicists in Medicine4.

The biggest advantage of the HS film over other radiochromic films is its

sensitivity; the HS film is twice as sensitive as the MD-55 film. This is very important,

because generally the radiochromic films require very high radiation doses in order to

attain optical densities at reasonable levels. The first studies with HS film have already

been done5,6, showing that the film is water-equivalent, and its response linear and

energy independent.

Figure 2.9 shows the structure of the HS GafchromicTM film. The sensitive layer

of the film is sandwiched between two sheets of clear, transparent polyester. The

sensitive layer is made of a special polymer material that changes its optical density when


38

irradiated. The total thickness of the film is about 240 µm and it is usually available in

packages that contain five 12×12 cm2 sheets. All specification and performance

parameters are available at the manufacturer’s web site7. The two most important

parameters are: (i) the spatial response uniformity and (ii) the dose response curve (H&D

curve). Deviation of the response is reported to be as low as 3.4% within two standard

deviations. However, this result was determined using 49 measurements over a 12×12

cm2 piece of film, and it may overestimate the variation of the response over a smaller

area. The maximum deviation that was registered during the measurements of single

profiles in the present study was as low as 2%. Hence, this value is accepted as a typical

uncertainty in the film response for all profile measurements in this work.

The dose response curve for the radiation source, our 10 MV photon beam, used

for the purposes of this thesis was determined. The optical density of exposed film

changes during the first 48 hours after irradiation, and then practically remains constant

after that time. The dose response curves measured at 36 hours and 25 days after

Clear Polyester - ~97 microns

Clear Polyester - ~97 microns

Active Layer - 40 microns

Figure 2.9: Structure of HS radiochromic film.


39

irradiation are presented in Fig. 2.10. The dose response is linear within the combined

uncertainty of response variation and the optical density readout (0.005).

All optical density measurements were carried out with a Nuclear Associates

Radiochromic Densitiometer, Model 37-443, and a film transport system, capable of

positioning the film with an uncertainty as low as 0.1 mm (Fig. 2.11). The densitometer

uses a red light source with a wavelength between the two absorption maxima of the

polymer dye, which are at 615 nm and 675 nm.

For all relative dose measurements the aperture of the densitometer was closed to

about 0.3 mm in order to achieve a high resolution. Obviously, when the aperture

diameter is changed from the standard value, the readouts no longer correspond to the

optical density. Therefore, before each series of measurements, the linearity of the

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30 35

Dose (Gy)

Opt

ical

Den

sity

36 hrs

3 weeks

Figure 2.10: Dose response curve for HS radiochromic film.


40

readouts was checked against the optical density, using pieces of films with known

optical density that were provided by the manufacturer. When the optical density of the

film is linear with the delivered dose, and the readouts are linear with optical density, we

can conclude that the readouts are linear with the delivered dose. All readouts were

reduced by the readout for a non-irradiated film before performing any further

calculations.

2.3. Experimental setups

All experiments with the A14P ionization chamber were carried out in a

30×30×30 cm3 water phantom. The chamber was pointing toward the radiation beam

and the source-surface distance was 100 cm (Fig. 2.12).

Figure 2.11: Densitiometer Model 37-443 and the film transport system.


41

The chamber was attached to a transport system that allowed positioning of the

chamber with an uncertainty as low as 0.1 mm. This transport system is used clinically at

the Radiation Oncology Department of the Montreal General Hospital for calibration of

linear accelerators. Some modifications were done so that the same transport system

could be used for off-axis measurements, as well as for central axis measurements (Fig.

2.13).

Prior to the measurements, the chamber was warmed up with 40 Gy in an open

field. After this irradiation, the readouts became consistent and the leakage had a stable

value between 0.5 and 1.0 pC collected over a time interval of 1 min. The leakage current

was lower than 0.2 pA. The experiments were carried out using two polarities: +300 V

and -300 V. During the experiments the chamber was irradiated with 400 MU (1.5 mm

field) and 300 MU (3 mm and 5 mm fields). The radiation dose was delivered at a rate of

400 MU/min. The leakage was checked every 5 minutes.

Waterphantom

Photon beam

A14P

SSD = 100 cm

Cable andventilation tubes

Figure 2.12: A14P ionization chamber orientation in the water phantom.


42

All experiments with the HS radiochromic film were carried out in a phantom

made of a special plastic material that was water-equivalent and known as Solid WaterTM

(Gammex RMI, Middleton, WI, USA). The sheets of HS film were cut into strips in the

direction designated by the manufacturer as the direction with the lowest response

variation. Afterwards, the strips were cut into small pieces that were used for the

experiments. This approach minimized the uncertainty due to the film response variation.

For dose profile measurements, where only a single piece of film was used, the profile

was scanned in the lowest response variation direction. The film was irradiated with 2000

MU to 6200 MU, depending on the field size.

Both dosimeters were used for: (i) central axis percentage depth dose (PDD)

measurements, (ii) off-axis ratios (OAR) measurements at a depth of 2.5 cm, and (iii)

relative dose factor (RDF) measurements. When HS film was used for PDD

measurements, 2-D profiles for different depths were scanned in order to obtain the

values over the central axis.

(a) (b)

Figure 2.13: A14P chamber positioning (a) for central axis and (b) off-axismeasurements.


43

2.4. Dynamic stereotactic radiosurgery

Dynamic stereotactic radiosurgery, often referred to as the McGill technique, is a

radiosurgery technique that was introduced in 1987 (Ref 8,9). When this dynamic

technique is performed, both the table and the gantry of the accelerator are moving

continuously and simultaneously. The irradiation starts at 330o gantry angle and –75o

table angle and it stops at 30o gantry angle and 75o table angle (Fig. 2.14). The beam trace

on the patient scull for this technique is shown in Fig. 2.15.

The dose distributions for dynamic radiosurgery carried out with both the 1.5 mm

and the 3 mm collimators were measured in a X-Y plane containing the isocenter of the

linac with a very slow radiographic film (Kodak EDR-2). The purpose of these

measurements was to determine the 50% isodose surface, which usually is the

Isocenter

330o

75o-75o

30o

Figure 2.14: Simultaneous gantry and tablerotations during a dynamicradiosurgery treatment.

Figure 2.15: Beam trace on the patient’sscull for the dynamicradiosurgery technique.


44

prescription isodose surface in a radiosurgery treatment with very small fields. The

measurements did not aim to a very high precision and therefore, the fact that the film

may slightly differ from water in terms of radiological properties was not an issue. On the

other hand, the Kodak EDR-2 film was a convenient dosimeter, because it required a

dose of only 4 Gy in order to obtain a reasonable optical density. The film was squeezed

between two hemispheres made of polystyrene (Fig. 2.16) and the phantom was

positioned into the machine isocenter using a stereotactic frame. The aperture of the

scanner used to read the film was smaller than 0.1 mm. The technique was carried out

with 1700 MU for the 1.5 mm field and with 1000 MU for the 3 mm field.

Another experiment was performed to determine the displacement between the

center of the measured dose distribution and the isocenter of the machine defined by the

room lasers. A dynamic radiosurgery procedure was carried out again, but this time the

dose distribution in the X-Y plane was imaged on a piece of radiochromic film, which is

Figure 2.16: Hemispheres made of polystyrene, used as a dynamicradiosurgery phantom.


45

not sensitive to visible light so that we were able to mark precisely the lasers by four

pinpricks.

The results are presented in Chapter 5, where they are compared with Monte

Carlo-calculated 3-D dose distributions (see Chapter 3).


46

References:

1 M. Pankuch, J. Chu, J. Spokas et al., “Characteristics of a new parallel plate

microchamber explicitly designed for high spatial resolution, Bragg-Gray cavity

measurements of small photon beams,” presented at the 2000 World Congress on

Medical Physics and Biomedical Engineering, Chicago, 2000 (unpublished).




3 P. R. Almond, P. J. Biggs, B. M. Coursey et al., “AAPM's TG-51 protocol for clinical


1870 (1999).

4 A. Niroomand-Rad, C. R. Blackwell, B. M. Coursey et al., “Radiochromic film

dosimetry: recommendations of AAPM Radiation Therapy Committee Task Group 55.

American Association of Physicists in Medicine,” Med. Phys. 25, 2093-2115 (1998).

5 I. Das and C. Cheng, “Dosimetric Characteristics of New Gafchromic-HS Film,” Med.

Phys. 28, 1244-1244 (2001).

6 J. Ashburn, A. Al-Otoom, K. Sowards et al., “Investigation of the New Highly

Sensitive Gafchromic HS and XR Films,” Med. Phys. 28, 1244-1244 (2001).

7 International Specialty Products, “GAFCHROMIC® HS Radiochromic Dosimetry

Films For High Energy Photons Configuration, Specifications and Performance

Data,” (http://www.ispcorp.com/products/dosimetry/products/).


47

8 E. B. Podgorsak, A. Olivier, M. Pla et al., “Physical aspects of dynamic stereotactic

radiosurgery,” Appl. Neurophysiol. 50, 263-268 (1987).

9 E. B. Podgorsak, A. Olivier, M. Pla et al., “Dynamic stereotactic radiosurgery,” Int. J.

Radiat. Oncol. Biol. Phys. 14, 115-126 (1988).

Chapter 3:

Materials and Methods -Monte Carlo Particle

Transport Simulations

3.1. Theoretical basics

The Monte Carlo method is a mathematical method that provides solutions to

various mathematical problems using random number generators to assign specific values

to statistical variables following a predetermined statistical distribution. This

mathematical routine is commonly referred to as sampling a given distribution. The

standard deviation of a given statistical quantity calculated with the Monte Carlo method

decreases as 1/N2, where N represents the number of samples. Monte Carlo became a

popular scientific method in 1950s following the development of first computers,

however, the method was already known and used in the 19th century with random

number generators simulated through throwing a needle onto a line grid.

When a particle is moving through a medium, various interactions might take

place, each of them with certain probability. These interactions will result in different

outcomes: for example, the initial particle might get scattered, or another particle might

be created or set in motion. These events, again, will occur with certain probabilities,

following a certain distribution. For instance, in Compton scatter we can determine the

Chapter 3: Materials and Methods - Monte Carlo Particle Transport Simulations

49

scattering angle and the energy of the scattered photon by sampling a Klein-Nishina

distribution. Hence, a particle can be transported through a medium by simulating all

interactions that this particle undergoes in the medium.

In general, there are three main requirements for performing Monte Carlo particle

transport simulations: (i) the cross-sections and the probability distributions that describe

all possible interactions must be known, (ii) a reliable random number generator must be

used for sampling, and (iii) adequate computing power must be available. Fortunately

theoretical physics has provided enough information about the interactions of different

particles with one another. In addition, random number generators and sampling

techniques have been developed since the first computing machines were introduced.

Lack of adequate computing power has always been the most serious obstacle when it

comes to Monte Carlo particle transport simulations. Recently, however, microelectronics

and computer technologies have been improved to such an extent that there are now

extremely fast and at the same time affordable computers on the market. For this reason,

the Monte Carlo simulations have become a very popular tool for dosimetric calculations.

The basic idea of Monte Carlo particle transport is presented in Fig 3.1. Let us

consider a volume of interest and an initial particle entering this volume. All interactions

that the particle undergoes, in addition to the path-length between two consecutive

interactions, may by simulated by sampling distributions, as described above.

Furthermore, all secondary particles either created or set in motion by the interactions

also should be transported through the volume of interest. Each particle is accounted for

until either it leaves the volume of interest or its energy becomes lower than a certain

cutoff energy. All particles with energies below the cutoff energy are considered to


50

deposit their energy locally. The Monte Carlo simulation goes on until there is no particle

left in the volume of interest. The initial particle is called history, because it creates the

secondary particles to be transported through the medium.

The standard deviation of all stochastic quantities calculated in the volume of

interest during the history transport, such as energy deposited, particle fluence, etc.,

depends on the number of events taking place in the volume of interest. There are three

fundamental parameters that have impact on uncertainties: (i) the number of simulated

histories, (ii) the dimensions of the volume of interest, and (iii) the cutoff energy. The

statistical uncertainties are lower when more histories are transported and when the cutoff

energies are lower, but this approach automatically increases the amount of computing

power required. Another way for decreasing uncertainties is by increasing the volume of

interest. Provided that in Monte Carlo simulations the phantoms are considered as sets of

voxels, increasing the voxel size will result in decreasing the spatial resolution, and this

often is not acceptable.

Figure 3.1: The basic idea of Monte Carlo particle transport: a particle istransported until either it leaves the volume of interest or itsenergy becomes less then the cutoff energy.

E<Ecutoff

E<Ecutoff

Initialparticle

Volume ofinterest


51

Another approach to the accuracy versus computing power compromise is to use

the variance reduction techniques. The idea behind these techniques is to change the

physical reality during the simulation so that more interactions would take place. For

example, if we increase the interaction cross-section, we will get a higher number of

interactions for the same number of histories. Of course, when these techniques are

applied the final results are statistically precise, but incorrect, because of the changed

reality. Therefore the results should be corrected by factors that take in account the

deviation from reality during the simulation.

First computer algorithms for Monte Carlo particle transport were developed at

the end of 1950s1, although the idea had been worked on for a long time before that time.

Nowadays, a variety of Monte Carlo algorithms for transport of different particles

(electrons, photons, protons, neutrons, etc.) are available. Each of these algorithms

usually applies to certain type of particles, within a certain energy range.

3.2. Monte Carlo transport of electrons and photons

In conventional radiation therapy the dose is delivered by either photon or

electron sources. Electrons, being charged particles, ionize the medium that they are

traversing, and lose energy through Coulomb interactions with orbital electrons and

nuclei of the absorber. A portion of the energy lost by the electrons is deposited in the

absorbing medium and a portion is transformed into bremsstrahlung. When photons are

used, they create high-energy electrons that deposit the dose to the medium. Therefore,


52

the Monte Carlo transport of photons and electrons is very important for practical

medical physics applications.

The photon transport is relatively easy to simulate, because photons undergo only

a few interactions while passing through a medium and the cross-sections for the various

possible photon-electron interactions were extensively studied in the past and are

tabulated for materials with atomic numbers from 1 to 100 (Ref. 2). The distributions

describing the outcomes of the interactions are well known as well. Hence, all

information needed to simulate a single interaction is available and the transport might be

carried out on a single interaction basis with no need for large computing power.

Unlike photons, high-energy electrons undergo about 105 interactions before they

stop in the medium. Obviously, if this physical process is simulated on a single

interaction basis, the simulation will take much longer than the photon transport. This

obstacle has been overcome by “condensing” multiple scattering events into a single

electron step, known as the condensed history (CH) technique3. This technique is based

on the multiple scattering theories and relevant electron multiple scattering distributions

that are sampled in order to determine the change in both position and direction of a

scattered electron. The theory that inspired the development of electron transport

algorithms was the Molière’s multiple scattering theory4. The energy that an electron

loses per single step in general is calculated using tabulated stopping power values.

There are two types of CH techniques, known as the class I and class II

techniques. In the class I techniques the electrons are moving on a predetermined energy

loss grid, and the secondary particles are added sampling a statistical distribution. This

approach was believed to be very accurate, but it turned out that it did not provide a good


53

correspondence between energy loss and production of secondary particles. In class II

algorithms all secondary particles are created in a very natural way as part of the primary

particle transport algorithm, considering the catastrophic events3.

Due to multiple scattering events, electrons are not moving over straight lines

during the CH steps. Therefore, every CH technique considers path-length corrections

(PLC) that account for the difference between the length of the real electron trajectory

and the step size, because this length is crucial for energy loss. Another correction that

CH techniques consider is the so-called lateral correction (LC) that accounts for lateral

deflection from the original direction again due to multiple scattering.

Often electrons have to be transported through inhomogeneous phantoms.

However, the multiple scattering theories apply to homogeneous media, and electron

transport algorithms have to take care of each electron crossing a boundary between two

different media.

In summary, the main characteristics of a Monte Carlo electron transport

algorithm are: (i) the multiple scattering theory used, (ii) the class of the CH technique,

(iii) the path-length corrections, (iv) the lateral corrections, and (v) the boundary

crossing algorithm. Generally, the precision of electron transport depends on the electron

step, commonly defined as fraction of energy lost over the step. The smaller the step, the

more accurate the simulation and the more computing power required. On the other hand,

electron steps must not be very small so as not to violate the restriction of the multiple

scattering theories. This issue is very important, especially for the boundary crossing

algorithms, and it will be discussed bellow.


54

3.3. Monte Carlo code systems for photon and electron transport

Many Monte Carlo codes for photon and electron transport were developed, but

just few of them have become popular and have been studied and tested in detail over the

years. Special attention will be paid to the codes that have been used for the purposes of

this work.

3.3.1. Electron Gamma Shower (EGS4/EGSnrc)

EGS4 code system was introduced in 1985 (Ref. 5) and was developed in order to

extend the low energy limits of the previous version, EGS3. The EGS4 code has become

the most popular and the most tested Monte Carlo code. The system is written in the

MORTRAN structured language. After preprocessing, the MORTRAN macros are

translated into a FORTRAN code, which is to be compiled and executed. Generally, the

system consists of both an electron and a photon transport routine, as well as a routine

that handles a huge database with detailed information about radiological properties of

various materials (PEGS data).

Initially the EGS4 electron transport algorithm had some weaknesses, such as not

performing lateral corrections. Therefore, two years after its introduction a new electron

transport algorithm was developed in the new EGS4 version referred to as

EGS4/PRESTA6. PRESTA (Parameter Reduced Electron-Step Transport Algorithm) is

an algorithm based on the Molière’s multiple scattering theory, using a class II CH

technique. It performs both path-length and lateral displacement corrections.


55

The most interesting feature of the PRESTA algorithm is the variable electron

step introduced in order to handle the boundary crossing problem, without slowing down

the calculations. The idea is presented in Fig 3.2. Since the multiple scattering theory

works for semi-infinite homogeneous media, the step at which an electron crosses the

boundary between two media will introduce some error. Hence, the algorithm aims to

minimize this step. Let us consider an electron to be transported through a voxel of

interest. Prior to each step, the minimum distance between the current electron position

and the boundary is calculated, and this distance limits the step so that the electron could

not reach the boundary. This procedure is repeated until the distance between the electron

and the boundary becomes smaller than a given minimum step mint . At this point the

lateral displacement correction is switched off and the electron crosses the boundary. The

minimum step cannot be smaller than the limit established by the underlying multiple

scattering theory.

EGS4/PRESTA was the most popular Monte Carlo code in the beginning of

1990s and was used and tested for many different applications with a continuously

Boundary

Medium 1

tmin

Medium 2

Figure 3.2: Boundary crossing algorithm in PRESTA.


56

increasing demand for more accurate results under complicated dosimetry conditions. In

order to facilitate these demanding applications, a new electron transport algorithm called

PRESTA II was developed by the National Research Council of Canada in Ottawa,

resulting in a new EGS4 version referred to as the EGSnrc7. Instead of Molière’s theory,

PRESTA II uses multiple scattering based on the screened Rutherford single cross section

that leads to a better estimate of the lateral displacement corrections. Another very

important advantage of PRESTA II is its boundary crossing algorithm. When the distance

between an electron and the boundary becomes smaller than mint (Fig. 3.2), PRESTA II

switches off the multiple scattering and the electron is transported through the boundary

by single elastic scattering events, providing better results than the original PRESTA

algorithm. Obviously in PRESTA II the value of the minimum step is not crucial, since it

is only related to calculation efficiency. Series of tests have proved that PRESTA II is a

better electron transport algorithm than the original RPESTA, especially at low energies.

The EGSnrs code is incorporated into various user codes that calculate

different quantities. Every user code is responsible for: (i) reading both the geometrical

information and the Monte Carlo simulation parameters from a user-defined input file;

(ii) initializing EGSnrc system by setting several parameters; and (iii) creating an output

from the scoring information returned by the EGSnrc code. The user codes that have been

used in this work are: (i) DOSXYZnrc performing dose calculations in phantoms with

rectangular geometry; (ii) DOSRZnrc performing dose calculations in phantoms with

circular geometry; and (iii) FLURZnrc performing energy and fluence calculations in

circular phantoms.


57

3.3.2. BEAM/EGS4

Monte Carlo simulations were first introduced into radiation therapy as a 3-D

dose calculation tool. Initially, the radiation beams were described by their spectra and

these beam models were used as radiations sources for the simulations. However, a better

approach was developed consisting of simulation of the treatment unit as well as the

phantom in which the dose distribution was calculated. The only problem in this

approach was that the treatment machines had a complicated design that was difficult to

describe precisely in the user code input files. This problem was successfully resolved by

the Monte Carlo code based on the EGS4/PRESTA system called BEAM8. The concept

of BEAM is simple: it supports a set of geometrical elements called component modules

(CMs). Some CMs as SLABS (stack of slabs) and CONESTAK (stack of cones) are

more general. Other CMs, such as JAWS and MIRROR are developed specially to

facilitate treatment machine simulations. A treatment unit is described in the user input

file as a series of CMs. The dimensions and the material should be specified for each of

the CMs as well.

The diagram in Fig. 3.3 illustrates how the BEAM code works. First, the

treatment unit is built of different CMs and the model is compiled resulting in an

executable program. This program is run, given both the input data (CMs dimensions,

materials, and incident particles) and physical properties of the materials (PEGS data).

When all the input information is loaded, BEAM invokes EGS4/PRESTA routines to

transport incident particles through the model of the treatment machine. As a result

BEAM scores all particles coming out of the treatment head into a certain plane in a file.

The scoring plane is called a phase space and the generated files are phase space files.


58

The more particles in the phase space, the better the phase space describes the simulated

treatment unit. Another very useful feature of the BEAM software is the LATCH option.

This option keeps on record the CM that each particle in the phase space originates from.

LATCH is very helpful for determining where the contamination electrons come from,

when studying linear accelerators, for example.

BEAM is software that works on Linux platforms. Recently a graphical interface

was added to the system, and since then it has become easy to use the software. Moreover

there are auxiliary programs, such as BEAMDP, that can analyze the phase space data

and determine different quantities (spectra, mean energies, etc.). Some results obtained

with BEAMDP are presented in Chapter 5. BEAM also generates a graphical output that

might be used to view the simulated treatment unit. With all these features BEAM/EGS4

has become the most popular tool for precise linear accelerator simulations.

User input:- Geometry- Radiation

source

Phase Space

Medium dataEGS4/PRESTA MonteCarlo simulations

Treatment unit, buildof CMs

Figure 3.3: The concept of the BEAM/EGS4 Monte Carlo code.


59

3.3.3. Voxel Monte Carlo Code (VMC, XVMC)

The Monte Carlo particle transport algorithms have undergone continuous

improvement since their introduction and they have become a very accurate and reliable

dosimetric tool. Despite the fact that computing power is very cheep nowadays, the

precise Monte Carlo simulations are unfortunately still not fast enough to be employed

for real treatment planning. Therefore, researchers have been looking for simplified

Monte Carlo algorithms that would be much faster, keeping the uncertainties in

calculated results within acceptable limits.

A fast Monte Carlo code for electron beam dose calculations, referred to as Voxel

Monte Carlo (VMC) code9, was introduced in 1995. VMC is based on the fact that in

radiation therapy the dose is delivered to materials with low density and low atomic

number (Z) by electrons within certain energy range. Thus, VMC was developed in a way

that it would be accurate for low Z materials with densities between 0 and 3 g/cm3 and

for electron energies between 1 and 30 MeV.

The most important changes that make the VMC code faster in comparison to

other Monte Carlo codes are: (i) some simplifications; (ii) new material data source; and

(iii) multiple use of each simulated history. Simplifications are made in all distributions

derived from the electron multiple scattering theory resulting in faster sampling. The

distributions are changed in such a way that the introduced errors are negligible within

the energy and material limits listed above. Some simplifications are also made in the

boundary crossing algorithm.


60

VMC takes CT patient data as a set of voxels that represent the phantom.

Unlike the other codes, instead of reading the material properties from a special database,

it just calculates all quantities required for electron transport (density, stopping power,

etc.) based on the CT numbers. The calculations are performed by very simple

relationships, valid only within the VMC material and energy limits.

The multiple history use is the technique that speeds up the VMC code the most.

Every time a history is calculated, it is applied to several different regions in the

phantom. These regions should be away from one another so that any event that occurs in

one of them could not influence the events taking place in the others.

VMC Monte Carlo code proved to be a very fast and accurate code for electron

beam treatment planning and because of that, shortly after its advent, Monte Carlo photon

transport was added to VMC. The new code was called the XVMC10 code. In addition to

all material data that VMC calculates using the CT numbers, XVMC also calculates the

effective atomic number and electron density for each voxel in order to perform photon

transport.

XVMC, like all EGSnrc user codes, can use a phase space, obtained with BEAM,

as a radiation source. It has been established that XVMC is about 20 times faster than the

EGS4/PRESTA algorithm using similar transport parameters.

3.4. Monte Carlo simulation of the Clinac-18 linear accelerator

Various dosimetric quantities were calculated in water phantoms for the purpose

of this work. Given that dosimetry of small photon beams is very complicated, the only


61

scientific approach was to use the phase spaces below the small collimators as radiation

sources for all calculations in phantoms.

First, the treatment head of the Clinac-18 linear accelerator was simulated

according to the technical documentation provided by the manufacturer without any

small field collimator in place, using the BEAM/EGS4 code. The jaws were set to a

10×10 cm2 field at the isocenter of the machine. The model is shown in Fig. 3.4. Then

1,700,000,000 electrons, all with energy of 10 MeV, were simulated to strike the target.

All particles crossing a horizontal plane at SSD = 100 cm were scored in a phase space

file. This simulation, as well as all other Monte Carlo simulations presented in this work,

was performed without using any variance reduction techniques and with very low cutoff

Figure 3.4: Treatment head of the Clinac-18 linear accelerator: BEAM/EGS4 model.


62

energies (0.01 MeV for both photons and electrons) in order to keep the simulations as

close as possible to reality. In total 4,700,000 particles were scored in the phase space.

The calculation time was approximately 1.5 sec per 1000 histories on a Pentium III 450

MHz processor. All calculations described below had approximately the same speed.

The phase space at SSD = 100 cm for a 10×10 cm2 field was used to calculate the

central axis percentage depth dose curve and a dose profile (off-axis ratios OAR) at a

depth of 10 cm in a water phantom. The calculations were performed with the

DOSXYZnrc user code. Figure 3.5 shows that after 6,100,000,000, histories the Monte

Carlo simulated PDD curve perfectly agrees with the measured curve. The profile

simulation was not that precise, but Fig. 3.6 clearly shows that the 50% width of the

profile simulated at depth of 10 cm is very close to 11 cm, which corresponds to a width

of 10 cm at the surface of the phantom. These two results validated the model of the

0

20

40

60

80

100

120

0 5 10 15 20 25Depth (cm)

PD

D

Monte Carlo

Measured

Figure 3.5: Monte Carlo simulated and measured PDD curves for 10×10 cm2 field sizeand SSD = 100 cm.


63

Clinac-18 linear accelerator. Since the model was validated, the jaws were then closed to

a field of 5×5 cm2 at the isocenter and a phase space was calculated for each of the three

small collimators. The collimators were simulated using the geometry described in

Chapter 2. The phase space files were scored at 70 cm from the source, just bellow the

collimators, within a circular area with radius of 3 cm. The phase space results are

presented in Chapter 5.

3.5. Monte Carlo simulations in water phantoms

For each of the three small fields four different types of Monte Carlo simulations

were carried out in this work: (i) percentage depth dose curve at SSD = 100 cm; (ii) dose

profile at the same SSD and a depth of 2.5 cm; (iii) fluence and energy calculations at the

same depth; and (iv) A14P chamber response in case of off-axis measurements. All

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-10 -5 0 5 10

Off-axis distance (cm)

OA

R

Figure 3.6: Monte Carlo calculated dose profile (OAR) for 10×10 cm2 field size,SSD = 100 cm and depth of 10 cm.


64

calculations were carried out with the DOSRZnrc user code. As it has been pointed out

above, no variance reduction techniques were used and the cutoff energy was set to 0.01

MeV for both electrons and photons. The phase spaces obtained with the BEAM/EGS4

were used as radiation sources.

The PDD calculations were performed in a phantom, shown schematically in Fig.

3.7. The phantom is shown in the R-Z geometry, following the convention for describing

phantoms in the DOSRZnrc input files. The phantom starts with a 30 cm thick air slab.

This slab is needed to transport the phase space particles from the bottom side of the

collimator that is at 70 cm from the source to the surface of the water phantom that is

at100 cm from the source. First fifteen water slabs have a thickness of 2 mm in order to

obtain good resolution in the first 3 cm of the PDD curve. Beyond the depth of 3 cm, the

Figure 3.7: A phantom used for PDD curve Monte Carlo calculations with DOSRZnrc.

Air

Water

2 mm

5 mm

Centralaxis

Radius

PDD scoringvoxels


65

slabs are 5 mm thick. The last slab is 20 cm thick and its purpose is to serve as a

backscatter material that is not taken into account in the PDD curve.

In radial directions all slabs are divided into two regions. The first region has a

very small radius and it defines the small voxels at the central axis where the radiation

dose is calculated. The radius of these voxels varies with the field size: 0.5 mm for the 5

mm field, 0.3 mm for the 3 mm field, and 0.15 mm for the 1.5 mm field. The second

region has a radius of 15 cm and it is used as side-scattering material.

Figure 3.8 illustrates the phantom used for dose profile calculations at a depth of

2.5 cm. The phantom again starts with a 30 cm air slab, followed by a water slab that has

a thickness of 2.25 cm. The radiation dose is calculated in the next slab with a thickness

Figure 3.8: A phantom used for dose profile (OAR) Monte Carlo calculations with DOSRZnrc.

Air

Water

5 mm

0.5 mm

Centralaxis

Radius

0.2 mm

OAR scoringvoxels


66

of only 5 mm. There is again a thick backscatter slab at the bottom. In radial directions

the phantom consists of twenty-five rings with thicknesses of 0.2 mm and six rings with

thicknesses of 0.5 mm, and a 14.2 cm thick side-scatter ring. Resolution of 0.2 mm within

the field, and 0.5 mm outside the field is considered to be adequate. The same phantom

was used to calculate the photon and electron fluence, as well as photon and electron

energy as functions of the off-axis distance. These calculations were performed with the

FLURZnrc user code.

A modified version of the DOSRZnrc user code, that has the capability to displace

the phase space in lateral directions, was used to study the A14P chamber response as a

function of the off-axis distance. This study is presented in Chapter 4.

3.6. 3-D dose distribution calculation for dynamic stereotactic radiosurgery

One of the goals of this work is to explore the precision of dynamic radiosurgery

performed with very small photon beams by calculating the 3-D dose distribution and

carrying out the procedure with the Clinac-18 linear accelerator. The technique has been

described in Chapter 2. Unfortunately, the treatment planning software used at the McGill

University Health Centre could not handle calculations in phantoms with voxel size as

small as 0.5 mm. On the other hand, we really need very small voxels in order to

calculate a reliable dose distribution for radiation beams as small as few millimeters in

diameter. Therefore, the dose distributions for the dynamic radiosurgery carried out with

the 1.5 mm and the 3 mm collimator were calculated with the XVMC Monte Carlo code.

When this dynamic technique is performed, both the table and the gantry of the


67

accelerator are continuously moving. The irradiation starts at 330o gantry angle and –75o

table angle, and it stops at 30o gantry angle and 75o table angle (see Chapter 2). The

XVMC code facilitates gantry rotation over a user-defined arc during the irradiation, but

the table angle should be fixed. Therefore, we slightly modified the software, adding few

lines of code that dynamically change the table angle, as a function of the gantry angle.

The relation is then as follows:

90 0.5 oTable Angle Gantry Angle= − × (3.1)

The calculations were performed in a sphere made of water, centered at the

isocenter of the machine. The phase spaces for the 1.5 mm and the 3 mm collimator were

used as radiation sources. The voxel size for both beams was 0.5 mm. It seems that this

resolution is not high enough to perform calculations with the 1.5 mm collimator.

However, we are interested in the approximate size of the 50% isodose surface, which is

expected to be about 2.5 mm. Moreover, the auxiliary software that analyses the XVMC

calculated dose distribution performs interpolation between the voxel dose values in

order to obtain the isodose surfaces. This procedure improves the resolution of the

isodose distribution. The calculated results for the 1.5 mm and the 3 mm collimators are

presented in Chapter 5, where they are compared with measured distributions.


68

References:

1 J.C. Butcher and H. Messel, “Electron number distribution in electron-photon

showers,” Phys. Rev. 112, 2096-2106 (1958).

2 J. H. Hubbell, J. M. Berger, and S. M. Seltzer, “X-ray nad Gamma-ray cross sections

and attenuation coefficients,” National Bureau of Standards Standard Reference

Database, 1985.

3 M.J. Berger, “Methods in computational physics, ” Academic Press, New York, 1963.

4 H. A. Bethe, “Molière's theory of multiple scattering,” Phys. Rev. 89, 1256-1266

(1953).

5 W. R. Nelson, H. Hirayama, and D. O. Rogers, Report No. SLAC-265, 1985.

6 A. F. Bielajew and D. W. O. Rogers, Report No. PIRS-0042, 1987.

7 I. Kawrakow, “Accurate condensed history Monte Carlo simulation of electron

transport. I. EGSnrc, the new EGS4 version,” Med. Phys. 27, 485-498 (2000).

8 D. W. Rogers, B. A. Faddegon, G. X. Ding et al., “BEAM: a Monte Carlo code to

simulate radiotherapy treatment units,” Med. Phys. 22, 503-524 (1995).

9 I. Kawrakow, M. Fippel, and K. Friedrich, “3D electron dose calculation using a

Voxel based Monte Carlo algorithm (VMC),” Med. Phys. 23, 445-457 (1996).

10 M. Fippel, “Fast Monte Carlo dose calculation for photon beams based on the VMC

electron algorithm,” Med. Phys. 26, 1466-1475 (1999).

Chapter 4:

Exradin A14P IonizationChamber Study

4.1. Geometry of the measuring volume

The two detector-related issues associated with dose measurements in small

photon fields, namely (i) resolution and (ii) water-equivalency, have been discussed in

Chapter 1. Obviously, the A14P parallel-plate ionization chamber is neither a high-

resolution detector nor a water-equivalent one. Its resolution is not sufficiently high

because the diameter of the collecting electrode is 1.5 mm (see Chapter 2). Therefore, the

diameter of the sensitive chamber volume which has at least this value, is large in

comparison to sizes of the fields studied in this work (1.5, 3, and 5 mm). The sensitive

measuring material for this chamber is air with a density of 1.293×10-3 g/cm3 at 0o C and

101.3 kPa.

In order to handle both the resolution and water-equivalency problems, we must

know the exact geometry of the measuring volume of the chamber. A good approach to

its determining is to determine the lines of the electric field inside the chamber cavity,

and this is equivalent to finding the lines of constant potential. The electric potential in

the cavity may be obtained by solving the Laplace equation, assuming that the charge

density in the chamber is zero:

Chapter4: Exradin A14P Ionization Chamber Study

70

∇ =2 0U , (4.1)

where U is the electric potential inside the cavity. If we want to solve this equation

numerically, particularly for the A14P chamber, we have to define the boundaries of the

cavity first and then set the boundary conditions over the boundaries. We can simplify the

geometry of the cavity slightly by neglecting the air gap between the collecting and the

guard electrodes (Fig 4.1(a)).

The Laplace equation written in cylindrical coordinates is given as follows:

∂ ∂ ∂ ∂+ + + =∂ ∂ ∂φ ∂

2 2 2

2 2 2 2

1 10

U U U Ur r r r z

, (4.2)

where the space variables r , z and φare defined in Fig 4.1(b). Because of the rotational

symmetry of the A14P ionization chamber, the potential does not depend on φ. Hence

2 2/ 0U∂ ∂φ = and the Laplace equation simplifies to the following:

1 mm

3 mm

4 mm

Z axis

CentralElectrode

Cap

(a) (b)

Z axis

r1

r2

(2)

(1) φ

z

r

s

Figure 4.1: Exradin A14P ionization chamber: (a) simplified geometry and (b) areawhere the equation is solved.


71

2 2

2 2

10

U U Ur r r z

∂ ∂ ∂+ + =∂ ∂ ∂

, (4.3)

which is equivalent to:

2 2

2 2 0U U U

rr z r

∂ ∂ ∂+ + = ∂ ∂ ∂ . (4.4)

Equation (4.4) represents a 2-D problem and the boundaries are much easier to

define. Figure 4.1(b) shows the contour that envelops the air cavity in the R-Z geometry

and the boundary conditions must be defined over this contour. The potentials of both the

cap and the central electrode are fixed during the measurements; hence, it is convenient

to use the Direchlet boundary conditions and this implies the use of the value of the

electric potential over the contour as a boundary condition.

Usually, the cap is grounded when voltage is applied to the chamber. Thus we can

assume that the potential of the cap is equal to zero and the potential of the electrode

maxU is equal to the applied voltage. The potential over the segment (1) in Fig. 4.1(b)

( 0r = ) is given as follows:

max 1z

U Us

= − , (4.5)

where z is the distance form the collecting electrode and s is the separation between the

two electrodes of the chamber. Essentially, Eq. (4.5) represents the potential between two

infinite surfaces and applies over the central axis in this particular case because of

symmetry considerations. The segment (2) (Fig 4.1(b)) is considered to be far enough

from the edge of the electrode and thus the potential over this segment is given by the

formula for the potential between two infinite coaxial cylinders:


72

1max

2

1

ln1

ln

rr

U Urr

= −

. (4.6)

The basic numerical methods for solving partial differential equations may be

found in many reference books1. If we consider the Taylor expansion of the function

( ),U r z at points ( , )r h z+ and ( ),r h z− located at a small distance h from the point

( ),r z we obtain:

( ) ( ) ( ) ( )2 2

2

, ,, , ...

2U r z U r z hU r h z U r z h

r r∂ ∂

+ = + + +∂ ∂ (4.7)

and

( ) ( ) ( ) ( )2 2

2

, ,, , ...

2U r z U r z h

U r h z U r z hr r

∂ ∂− = − + +

∂ ∂(4.8)

Neglecting all terms above the second order and adding Eq. (4.7) and Eq. (4.8), we find:

( ) ( ) ( ) ( )2

2 2

, , , 2 ,U r z U r h z U r h z U r zr h

∂ + + − −=

∂. (4.9)

The uncertainty associated with Eq. (4.9) depends on 4h . If we now subtract Eq. (4.8)

from Eq. (4.7) we obtain for the first derivative:

( ) ( ) ( ), , 2 ,2

U r z U r h z U r h zr h

∂ + − −=

∂, (4.10)

where the uncertainty goes with 3h . By analogy with Eq. (4.9) we can find the second

partial derivative with respect to z :

( ) ( ) ( ) ( )2

2 2

, , , 2 ,U r z U r z h U r z h U r zz h

∂ + + − −=

∂. (4.11)


73

Finally, if we substitute all derivatives into Eq. (4.4) with Eq. (4.9), (4.10) and

(4.11), we get:

( ) ( ) ( ) ( ) ( )

( ) ( )

1, , , , ,4

, , .8

U r z U r h z U r h z U r z h U r z h

h U r h z U r h zr

= + + − + + + − +

+ + + − (4.12)

Equation (4.12) gives a relationship between the value of the electric potential at a

given point inside the air cavity, and the values of four adjacent points at distance h in

both r and z directions from the given point.

For the purposes of this study a value of 0.02 mm was assigned to the step h , so

that there were 50 points per millimeter in each direction. After that, the value of the

electric potential at each point was calculated using Eq. (4.12), starting from the contour.

The calculations were performed using the MATLAB software package and they were

repeated iteratively until average relative difference between two consecutive runs

became as low as 0.01%. The data was divided into two matrices to speed up the

calculations. The matrix (A) contained all points in the separation region above the

collecting electrode (in Fig. 4.2: from segment (1) to the side wall of the cap). The other

matrix (B) contained all points in the side region between the side walls of the cap and

the electrode (in Fig 4.2: above the segment (2)). The guard electrode that engulfs the

collecting electrode will collect all charges created by radiation in the side region (matrix

(B)). Therefore only the electric field in the top region (matrix (A)) is important for

determining the measuring volume.

Figure 4.3 shows the lines of constant potential and a line of the electric field that

determines the measuring volume in the top region. The figure is drawn in the R-Z

geometry. The line of the electric field that determines the measuring volume is chosen


74

to start at the middle of the air gap. The measuring volume calculated according to Fig.

4.3 is 3.3 ±0.2 mm3. The uncertainty is mainly due to the fact that dimensions of the air

gap and the guard electrode are not very well defined. The measuring volume was

verified by measurements with a 10 MV photon beam at maxd (2.5 cm) in a water

phantom in a standard field (SSD = 100 cm, 10×10 cm2), immediately after a routine

check of the output of the machine, when the dose delivered to water was very well

known. Using values of 0.001197 g/cm3 for the density of air at 22 oC and 101.3 kPa, and

33.97 J/C for the ionization potential of air, we obtained 3.4 ±0.1 mm3 for the chamber

sensitive volume. The uncertainty in this result is due firstly to about 1% uncertainties in

measurements, and secondly to not accounting for any correction factor for the A14P

ionization chamber.

The geometry of the measuring volume is important for developing the algorithms

described below.

Z axis

(B)

(2)

(1)(A)

Figure 4.2: Calculation regions.


75

4.2. Deconvolution of dose profiles in water

The idea of using deconvolution to eliminate the blur of a low-resolution detector

is not a new one. Several studies have been done with the aim to increase the resolution

in the penumbra region for relatively large fields2-5. Figure 4.4(a) shows a detector placed

at the edge of a large square field from a beam’s eye of view. Obviously in this region the

radiation dose decreases in the X direction and stays constant in the Y direction.

Therefore the studies mentioned above have considered 1-D deconvolution using the

detector Line-Spread Function (LSF). For very small circular fields though, the radiation

dose changes in both directions (Fig 4.4(b)), making the measured dose profile a result of

a 2-D convolution between the dose field and the detector Point-Spread-Function (PSF)

and requiring that a 2-D deconvolution be performed.

Guardelectrode

Measuring Volume

Collectingelectrode

Cap

Z axis

Figure 4.3: Electric field in the air cavity.


76

The first problem to be solved is to determine the PSF of our detector.

Theoretically this could be done by taking measurements in a X-Y plane of a very narrow

beam, mathematically presented as a 2-D δfunction. In reality, however, even if such a

photon beam is collimated, the electrons set in motion in the phantom will spread out,

and the electron fluence will not follow the 2-D δfunction pattern any more.

Another, much more practical, approach is to model the PSF of the detector. If the

measuring material has the same properties everywhere in the measuring volume, the

response of the detector for a given point in the X-Y plane is equal to the thickness of the

measuring volume at that point. The shape of the detector PSF, resulting from the

measuring volume geometry that is obtained from the electric field lines, is shown in Fig.

4.5(a). However, it turns out that the deconvolution algorithm described below, also

works well even for a slightly simplified PSF, shown in Fig. 4.5(b).

The two functions are almost identical in terms of the resolution of the

deconvolved profile that this study aims for. The underlying assumption is that the

chamber has a constant response over a circle with a diameter of 2 mm and zero response

X

Y

(b)

X

Y

(a)

Figure 4.4: Detector in the penumbra region of (a) a large square field, and (b) asmall circular field.


77

elsewhere. When the PSF is determined and the measured profile is obtained, there are

two different ways of performing deconvolution. The first is to perform direct

deconvolution, often referred to as filtering. This is usually done in the Fourier space,

where this operation corresponds to a simple division: the Fourier transform of the

function to be filtered is divided by the Fourier transform of the PSF, known as the

system transfer function (STF). After this, the result of division undergoes an inverse

Fourier transformation that yields the deconvolved profile. The block diagram in Fig.

4.6(a) illustrates the whole deconvolution procedure. The measured profiles are

designated as 2-D functions, whereas they are usually one-dimensional. For circular

fields this is appropriate, because of their circular symmetry. It is clear that the measured

profiles do not represent the real dose values, since they incorporate some errors.

However, the PSF model introduced is not perfect and the final result of the filtering

could be very unrealistic, occasionally resulting even in negative doses.

The second algorithm for performing deconvolution is more practical and gives

more reliable results; however, it also requires more computing power. The block

diagram in Fig. 4.6(b) presents this algorithm. It starts from some arbitrary values for the

X

PSF(X,Y)

Y

(?)

X

PSF(X,Y)

Y

(b)

Figure 4.5: Point-Spread Function (PSF) of (a) A14P chamber and (b) a simplified PSF.


78

real dose profile (the deconvolved one) and performs convolution with the PSF for all

points at which measurements are taken. After that, the calculated results are compared

with the measured ones and a new real profile is generated. This routine is repeated until

the difference between the calculated and the measured profile becomes small enough

and a given condition is satisfied. This kind of algorithms are referred to as minimization

algorithms. The method by which the real profile is generated will be discussed below,

but it is important to note that, when the profile is generated, certain restrictions might

apply. These restrictions keep the dose profile within a set of acceptable solutions. The

second approach (Fig. 4.6(b)) was chosen for the purposes of this study and a simple

minimization algorithm, whose description follows, was developed.

Convolution2-D PSF

2-DDeconvolved

Profile ConvolvedProfile

Comparisonwith the

MeasuredProfile

2-D PSF

Measured2-D Profile

Filtering

Fourier Space

2-DDeconvolved

Profile

(a)

(b)

Figure 4.6: Direct deconvolution by (a) filtering and (b) a minimization algorithm.


79

Let us consider a geometrical model of a radiation field with a circular symmetry

(Fig 4.7). The field is divided into very small pixels, so that the dose delivered to a pixel

is constant throughout the pixel. The radiation dose jD is delivered to all pixels, which

are at distance jr from the central axis. Different detector positions are shown in the

figure as well. According to the model, for a given position i , the measured value iM is:

i ij jj

M c p D= ∑ , (4.13)

where: ijp is the number of pixels receiving the dose jD within the measuring volume

when the detector is at position i ; c is a constant that relates the dose measured in Gy on

the equation’s right hand side to the measurements in nC on the left hand side. This

constant incorporates the density of air, the ionization potential of air, the pixel size, and

Dj

rj

M1 Mn

Ion chamberpositions

Figure 4.7: Geometrical model of taking measurements in a small circular field.


80

the separation of the ionization chamber. However, the study aims to obtain a profile

normalized to the central axis value, so the constant can be neglected. Then Eq. (4.13)

becomes:

i ij jj

M p D= ∑ , (4.14)

actually representing a discreet convolution of the dose field with the detector PSF.

Mathematically, this calculation is one-dimensional because of the circular symmetry of

the radiation field.

The main input parameters for the minimization algorithm are: (i) all detector

positions, (ii) all measured values and (iii) the radius of the detector. The algoritm starts

by calculating all ijp . Then the first approximate dose profile is calculated using the

formula:

2

1meas

ij ii

jij

i

p MD

R p=

π

∑∑ , (4.15)

where R is the detector radius in number of pixels and measiM are the measured values.

Equation (4.15) represents a weighted sum of all measured values in which jD takes

place, divided by the number of pixels that fit within the detector. This is the initial

profile from which the minimization algorithm starts and is very important. This profile

could be an arbitrary one, but if it is close to the ideal solution, the minimization is much

faster and the final result is very reliable. Several different ways of calculating the initial

profile have been tested, and Eq. (4.15) shows the most successful approach.

When the initial profile is determined, the minimization starts as indicated in Fig.

4.6(b). The values iM are calculated according to the geometrical model (Eq. (4.14)) and


81

they are compared to the measured values measiM . The minimization criterion, referred to

as the objective function, is given as:

21 meas

i imeas

i i

M Mn M

−

∑ , (4.16)

representing the average relative error. In the beginning of every new run the dose profile

is modified. The new values are obtained by multiplying the values from the previous run

with coefficients jk defined as follows:

( )1 0,1jk = + σΝ , (4.17)

where σ is an input parameter usually close to 0.1 and ( )0,1Ν is a random function that

generates numbers following the normal distribution, with zero mean and unity variance.

After the values are generated, several restrictions are applied allowing the algorithm to

distinguish the meaningful solutions. Many different restrictions have been tested and a

final set of three of them that were found useful is as follows (Fig 4.8): (i) radiation dose

should decrease with off-axis distance; (ii) the second derivative of the profile should be

negative before a given point (inflexion point) and positive beyond this point; (iii) over a

small extent in the tail of the profile the dose is constant.

The first restriction is natural, because there is no physical reason for an increase

in the dose with off-axis distance for the very small fields that we studied. The second

restriction takes care of the smoothness of the profile, because otherwise some strange

edges due to errors in the measured values might appear. The inflexion point, an input

parameter for the algorithm, is easy to determine. Usually, it is the point of the 50% value

that is the same for both the convolved and deconvolved profiles. Even if it is not very

precisely determined, this point is not crucial for the minimization algorithm. The third


82

restriction is not needed, however, it speeds up the calculations. This restriction is applied

to a small region in the tail where the differences between the measurements are smaller

than their uncertainties.

The minimization algorithm, as described above, was coded in the C

programming language. The C-routine that was used for the function ( )0,1Ν (Eq. (4.17))

was obtained from a book called “Numerical Recipes in C”6. The size of the pixel for all

calculations was 0.2 mm and the radius of the A14P measuring volume was 5 pixels. The

same resolution was used in all Monte Carlo simulations (see Chapter 3).

The software was tested using computer-generated profiles. Another software was

developed to generate fake measured profiles given the “ideal” ones, according to Eq.

(4.14). Then the minimization algorithm was used to retrieve the ideal profiles. The result

of one these tests is shown in Fig 4.9. The average relative error, as defined in Eq. (4.16),

in this particular case is 0.03%. The deconvolved profiles for the three different fields

studied in this thesis are presented in Chapter 5.

OAR

Off-axis distance (pixels)

Second derivative control:Inflexion point

Tail

Figure 4.8: Restrictions applied to the profiles.


83

4.3. Correction factors for off-axis measurements

The problem of the water-equivalence of a detector used for measurements in

very small photon fields has already been discussed in Chapter 1. The A14P ionization

chamber is not a water-equivalent detector because the measuring material in the

chamber sensitive volume is air. The standard relationship between the dose delivered to

water waterD and the dose delivered to air airD is given as follows:

( )water

waterwater air air

air

LD D =

φρ

, (4.18)

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25

Off-axis distance (pixel)

OA

RIdeal profile

Deconvolved profile

Figure 4.9: Test result for the minimization algorithm.


84

where the correction factors ( )/water

airL ρ and ( )water

airφ were defined and discussed in detail

in Chapter 1. The restricted mass stopping power correction factor is defined for the same

electron fluence in two different mediums. This correction factor is also relevant when

the electron fluence in the mediums is not the same, but the electron energy spectra are

identical. For this more complicated case the fluence perturbation correction factor is

introduced. However, when the fluence perturbation is very drastic, it is questionable if

the two correction factors are still independent, because the electron energy spectrum

might change as well. Therefore, in cases where the dosimetry and the underlying physics

are not well known, it is a better approach to combine the two correction factors into one,

so that Eq. (4.18) becomes:

( )waterwater air air

D D D= , (4.19)

where ( )water

airD is the total correction factor. In general, this total correction factor depends

on the beam geometry, the detector design, and the detector position. Therefore ( )water

airD

has to be calculated for each particular field and each particular detector position. The

best way to obtain ( )water

airD is by using Monte Carlo simulations because they are relevant

under any conditions. Moreover, when using simulations, there is no need of simplifying

the reality in order to apply some theoretical model. In other words, when the basic

physics of dose delivery is not well understood, it is difficult to say which theoretical

model is relevant and which one is not.

In this work ( )water

airD correction factors of the A14P chamber were calculated at

different off-axis distances for each of the studied very small photon fields. The

calculations were performed using the DOSRZnrc Monte Carlo code from the EGSnrc


85

package. This code has been discussed briefly in Chapter 3. It is used for Monte Carlo

particle transport in phantoms with cylindrical symmetry. The A14P chamber was

simulated as an air cavity that consists of two coaxial cylinders inside a water phantom.

Figure 4.10 shows the geometry of the cavity. Not only the measuring volume,

but the whole air cavity of the ionization chamber, was simulated because the size of the

cavity had an impact on the fluence perturbation. The measuring volume is modeled as a

cylinder with a radius of 1 mm. The geometry shown in Fig 4.10 is somewhat different

from the results obtained from the electric field (Fig 4.3). The real geometry could be

simulated using the DOSXYZnrc code, but the number of voxels in this case would

increase substantially. On the other hand, this simulation did not aim to very high

precision, therefore the DOSRZnrc code was chosen. The only problem that appeared

was that it was impossible to move the cavity into off-axis positions in a DOSRZnrc

phantom. Therefore another approach was taken: the code was slightly modified so that

instead of moving the air cavity the phase space was moved. The phase space contains all

2 mm1 mm

Z axis

1 mm

Figure 4.10: Geometry of the air cavity in the Monte Carlo simulations.


86

pertinent information about particles below the collimator (see Chapter 3). Therefore, if a

small displacement is added to the X coordinate of each particle, this will be equivalent to

moving the air cavity in the opposite direction, provided that the water phantom is

sufficiently large (Fig 4.11).

During the simulations, 20,000,000 to 100,000,000 particles were sent through the

phantom, depending on field size, and uncertainties as low as 1% were achieved at zero

off-axis distance. Uncertainties in the tail (off-axis distance of 7 to 8 mm) were about 6%.

Each simulation was performed twice: once for the air cavity and once for the cavity

filled with water.

DOSRZnrc code reports the radiation dose deposited in a voxel normalized to the

initial number of high-energy electrons striking the accelerator target. For a given

Figure 4.11: Monte Carlo simulated geometry for off-axis correction factors calculation. Thephase space is moved over the phantom, instead of moving the air cavity.

Air cavity

Water phantom

Z axis

Phase space


87

electron energy (10 MeV for our particular case) the number of electrons that have

striked the target corresponds to the practical dosimetric quantity Monitor Unit (MU)

used for dose delivery with linacs. Therefore, as a result of this normalization, the

reported doses are comparable even if a different number of particles is sent through the

phantom.

The ( )water

airD correction factors were established by dividing the dose calculated

in the measuring volume when filled with water to the dose to the measuring volume in

the air cavity for each different off-axis position. The results were used to correct the

A14P chamber measurements prior to applying the deconvolution algorithm, and are

presented in Chapter 5.


88

References:

1 W. Cheney and D. Kincaid, “Numerical Mathemetics and Computing,” Wadsworth,

Inc., Belmont, CA, USA, 1985.

2 P. Charland, E. El-Khatib, and J. Wolters, “The use of deconvolution and total least

squares in recovering a radiation detector line spread function,” Med. Phys. 25, 152-

160 (1998).

3 F. Garcia-Vicente, J. M. Delgado, and C. Peraza, “Experimental determination of the


202-207 (1998).

4 P. D. Higgins, C. H. Sibata, L. Siskind et al., “Deconvolution of detector size effect

for small field measurement,” Med. Phys. 22, 1663-1666 (1995).

5 C. H. Sibata, H. C. Mota, A. S. Beddar et al., “Influence of detector size in photon

beam profile measurements,” Phys. Med. Biol. 36, 621-631 (1991).

6 W. Press, S. Teukolski, W. Vetterling et al., “Numerical Recipes in C,” Cambridge

University Press, Cambridge, MA, USA, 1992.

Chapter 5:

Experimental Results

5.1. Beam quality

The quality of small radiation beams differs from the quality of standard clinical

photon beams with field sizes on the order of 10×10 cm2 even when both the small and

the standard beams are produced under the same linac operating conditions. The radiation

beam quality changes when the beam is collimated with the small field collimators

because of the photon interactions that take place inside the collimator. The beam quality

for three small photon fields (1.5, 3 and 5 mm) shaped with the collimators described in

Chapter 2 were studied with Monte Carlo simulations.

The particle transport through the Clinac-18 treatment head and the collimators

was simulated using the BEAM/EGS4 code, as described in Chapter 3. All particles

below the collimators were scored and their parameters, such as energy, charge, position

and direction were stored in three phase-space files; one file for each collimator. These

files contained all particles collected in a circular area with a radius of 3 cm at 70 cm

from the source, just bellow the collimators, and they were used for all Monte Carlo

simulations in various phantoms presented in this work.

Various beam quality parameters were analyzed using the BEAMDP code from

the BEAM/EGS4 package. The pictures presenting the X-Y scatter of the particles in the

phase-space files are shown in Fig. 5.1. The phase-space data for the 1.5 mm field


90

contains 50,284 (electrons and photons) particles and 49,735 or 98.9% of them are

photons (Fig. 5.1(a)). This particle fluence was produced by almost 3,900,000,000 initial

high-energy electrons, striking the target of the linac. Figure 5.1(b) shows 13,795

particles (electrons and photons) collected below the 3 mm collimator, with the photon

fraction of 99%. About 6,000,000,000 high-energy electrons hit the linac target in order

to produce this fluence. For the 5 mm collimator a simulation of 5,000,000,000 initial 10

MeV electrons produced 81,229 particles bellow the collimator. The X-Y particle scatter

is shown in Fig. 5.1(c). The photon fraction is again 99%.

In all experiments and simulations the collimator jaws of the accelerator were set

to project a 5 ×5 cm2 field at the isocenter of the machine, and this corresponded to a field

of 3.5×3.5 cm2 at 70 cm from the source, where the phase-space files were determined.

The transmission radiation with this field size is evident in Fig. 5.1(a) for the 1.5 mm

collimator and in Fig. 5.1(c) for the 5 mm collimator. For the 3 mm collimator the

transmission radiation cannot be distinguished in Fig 5.1(b), because this collimator is

made of tungsten, whereas the other two are made of lead. Since all collimators have a

thickness of 10 cm and the density of tungsten is about 1.8 times higher than that of lead,

we can expect a considerably lower photon transmission through the tungsten-made

collimator.

Collimator # of initial electronsstriking the target

# of particles inthe phase space

# of photons in thephase space

1.5 mm 3 900×106 50 285 49 735

3 mm 6 000×106 13 795 13 654

5 mm 5 000×106 81 229 80 484

Table 5.1: Phase space data for the three collimators used in our study.


91

The total numbers of particles listed in Table 5.1 above differ for the various

phase-space files, firstly because of the differences in field size, and secondly because of

the different transmissions through the collimators. However, the particle densities inside

Figure 5.1: Photon fluence for the (a) 1.5 mm, (b)3 mm, and (c) 5 mm fields. The calculationsare performed below the collimators at 70 cm from the radiation source.

(a) (b)

(c)


92

the radiation fields are similar, as shown from the following analysis. The number of

photons in a small circular region with a diameter of 1 mm is: 1460 particles for the 1.5

mm field, 2100 particles for the 3 mm field, and 2000 particles for the 5 mm field. This

particle fluence is considered sufficiently large to obtain good statistical results for any

further dose calculations in the phantom, although the total number of photons is not high

enough to obtain good results for the photon fluence analysis.

Figure 5.2 shows the photon fluence spectra for the three different fields, obtained

with the BEAMDP analyzing software. The spectra are not smooth because the total

numbers of photons analyzed are low, resulting in high statistical uncertainties. The

spectra are normalized to their maximum values beyond the energy of 0.511 MeV, where

there are sharp peaks in all spectra. These peaks result from electron-positron

annihilations that take place in the collimators. The positrons are produced by pair

production interactions; interactions with a relatively high probability for photons with an

energy on the order of several MeV in high-atomic number materials. The peak at 0.511

MeV is higher for the 1.5 mm (Fig. 5.2(a)) and 5 mm (Fig. 5.2(c)) collimators that are

made of lead in comparison with for the 3 mm collimator (Fig. 5.2(b)), made of tungsten.

This effect is mainly due to two reasons. Firstly, the atomic cross-section for a pair-

production interaction at a given photon energy depends on the square of the atomic

number of the attenuator. Since the atomic number of lead is 82 and the atomic number

of tungsten is 74, the probability for pair production interaction in lead is higher than that

for tungsten. Secondly, tungsten has a higher density and, since the collimators have the

same thickness, the 3 mm collimator attenuates more photons with the relatively low

energy of 0.511 MeV.


93

Figure 5.2: Normalized photon fluence density vs. energy for the (a) 1.5 mm, (b) 3 mm, and(c) 5 mm fields. The calculations are performed below the collimators at 70 cmfrom the radiation source.

(a)

(b)

(c)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10

Energy (MeV)

Nor

mal

ized

flue

nce

dens

ity

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10

Energy (MeV)

Nor

mal

ized

flue

nce

dens

ity

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10

Energy (MeV)

Nor

mal

ized

flue

nce

dens

ity

5 mm

3 mm

1.5 mm


94

The 0.511 MeV peak is slightly higher for the 1.5 mm collimator in comparisons

to the 5 mm collimator. This must be due to the fact that the 1.5 mm collimator covers a

larger area of the 5×5 cm2 radiation field and thus offers more lead for interaction

purposes.

Figure 5.3 shows the energy fluence spectra for the three radiation fields. The

0.511 MeV peaks are present again, but here they are lower compared to those in Fig.

5.2, because in this case the spectra are weighted by energy. Therefore, the low energy

part of the spectrum diminishes and the peaks are not as high. The uncertainty in the 3

mm energy fluence spectrum is very high because the phase-space file for this field

contains only 13,800 particles.

Figure 5.4 shows the change in photon fluence with the off-axis distance in the

scoring plane below the three collimators. The photon fluence is normalized to the central

axis value. When performing such an analysis, the BEAMDP software divides the area,

over which the phase space file has been collected into coaxial rings with equal surfaces.

Therefore the radial resolution is poor close to the central axis, and increases with the off-

axis distance. For all graphs, shown in Figure 5.4, the size of the first bin of the histogram

is 2.12 mm, which corresponds to a circular region with diameter of a 4.14 mm. This

region is larger than any of the fields at 70 cm from the source, where the analysis is

performed. Thus, there is not enough information about the photon fluence drop off at the

edge of the field. However, the transmission radiation is well presented.

Figure 5.4(b) shows a negligible transmission for the 3 mm collimator compared

to the 1.5 mm (Fig. 5.4(a)) and 5 mm (Fig. 5.4(c)) collimators. The transmission,

normalized to the central axis fluence, is much higher for the 1.5 mm collimator than for


95

Figure 5.3: Normalized photon energy fluence density vs. energy for the (a) 1.5 mm, (b) 3mm, and (c) 5 mm fields. The calculations are performed below the collimators at70 cm from the radiation source.

(a)

(b)

(c)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10

Energy (MeV)

Nor

mal

ized

ene

rgy

fluen

ce

dens

ity

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10

Energy (MeV)

Nor

mal

ized

ene

rgy

fluen

ce

dens

ity

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10

Energy (MeV)

Nor

mal

ized

ene

rgy

fluen

ce

dens

ity

5 mm

3 mm

1.5 mm


96

Figure 5.4: Normalized photon fluence vs. off-axis distance for the (a) 1.5 mm, (b) 3 mm, and(c) 5 mm fields. The calculations are performed below the collimators at 70 cmfrom the radiation source.

(a)

(b)

(c)

0.0

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0 1 2 3


Nor

mal

ized

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97

the 5 mm one, because the central axis fluence increases with the collimator opening,

whereas the transmission through a 10 cm thick lead block is the same. In addition, there

is an averaging effect over the first bin of the histogram, and this effect lowers the central

axis value for the 1.5 mm field much more than the central axis value for the 5 mm field.

For all fields the photon fluence starts decreasing at the edge of the field as determined by

the setting of the collimator jaws (3.5 cm at 70 cm from the source), although this effect

is not visible for the 3 mm field (Fig. 5.4(b)).

The energy fluence as a function of the off-axis distance, normalized to the central

axis value, is shown in Fig. 5.5. All considerations about the histogram resolution pointed

out for Fig. 5.4 also apply to Fig. 5.5. A very important observation is that energy fluence

for the different fields (Fig. 5.5) has a similar behavior to the corresponding photon

fluence (Fig. 5.4). Therefore, the average photon energy does not change significantly

with the off-axis distance for any of the fields.

Figure 5.6 shows the average photon energy as a function of the off-axis distance

for the three fields. For all fields the average energy has a value around 3 MeV on the

central axis. The averaging effect due to the low resolution of the histogram, as discussed

above, can be noticed in Fig. 5.6(a). Most probably, because of this effect, the central

axis value is lower than 3 MeV. The value of 3 MeV is in good agreement with the

average photon energy in an open field calculated according the rule of thumb as one

third of the maximum photon energy. For all collimators, the average energy drops to

about 2.5 to 2.7 MeV outside of the field. There are scattered photons, as well as

transmission photons in this region. After an off-axis distance of 1.75 cm, which

corresponds to the setting of the collimator jaws, the average energy starts diminishing,


98

Figure 5.5: Normalized photon energy fluence vs. off-axis distance for the (a) 1.5 mm, (b) 3mm, and (c) 5 mm fields. The calculations are performed below the collimators at70 cm from the radiation source.

(a)

(b)

(c)

0.0

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ene

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ce

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ce

5 mm

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1.5 mm


99

Figure 5.6: Average photon energy vs. off-axis distance for the (a) 1.5 mm, (b) 3 mm, and (c)5 mm fields. The calculations are performed below the collimators at 70 cm fromthe radiation source.

(a)

(b)

(c)

0.0

0.5

1.0

1.5

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rage

ene

ry (M

eV)

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0 1 2 3


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eV)

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)

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1.5 mm


100

because, in general, there are only scattered photons in this area. The average energy for

the 3 mm field (Fig. 5.6(b)) has large statistical uncertainties, especially in the region

beyond 2 cm from the central axis. This is due to the low transmission of this collimator,

resulting in a very low number of photons available for analysis.

5.2. Physics of dose deposition

The basic physics of dose deposition in absorbing medium is essentially the same

for small and large photon fields: the photon fluence creates electron fluence and the

electron fluence deposits radiation dose to the medium. The dose deposition depends on

both the electron fluence and the electron energy. In large fields, under the condition of

charged particle equilibrium, there is a strong relationship between the photon fluence

and the electron fluence on one hand, and the photon energy and the electron energy on

the other. Thus, it is not very difficult to predict the dose deposition in a homogeneous

phantom, when the quality parameters of the photon beam are known. In small fields,

due to the lack of charged particle equilibrium in lateral direction, the relationships

between the photon and electron parameters are not well established. Moreover, these

relationships depend not only on the photon beam quality but also on the field size.

In this work the photon and the electron fluence, as well as the photon and the

electron average energy were studied using Monte Carlo simulations. The calculations

were performed at depth of 2.5 cm in a water phantom with the FLURZnrc code from the

EGSnrc package, as described in Chapter 3.


101

The average photon and electron energies with respect to the off-axis distance for

the three fields are shown in Fig. 5.7. The average photon energies are calculated with

very high statistical uncertainties, because of the low number of photons in the phase-

space files. The average electron energies on the other hand, are calculated with

uncertainties as low as 0.1%. However, due to the photon energy uncertainties there are

additional inherent uncertainties in the electron energies, which are difficult to assess. In

general, the behavior of the average photon energy at a depth of 2.5 cm in water

resembles the graphs shown in Fig. 5.7, representing the average photon energy in air,

just below the collimators. For all fields the average photon energy has a value of about 3

MeV inside the field and it starts dropping at the geometrical edge of the field to level off

at about 2 MeV. After that due to the transmission photons, that have high energies, the

average energies of the 1.5 mm field (Fig 5.7(a)) and the 5 mm field (Fig5.7(c)) start to

increase. For the 3 mm field (Fig. 5.7(b)) the average photon energy keeps decreasing

because of the very low transmission through the tungsten collimator.

The average electron energies, presented in the same graphs, follow a similar

pattern, but their variations are smaller. They have values between 1.7 MeV and 1.8 MeV

inside the fields, and drop slightly at the field edges. The average electron energies of the

1.5 mm field (Fig 5.7(a)) and the 5 mm field (Fig5.7(c)) start to increase again, whereas

the energy of the 3 mm field (Fig. 5.7(b)) keeps decreasing beyond the field edge. An

important observation that is relevant to dose deposition in the medium is that, for all

fields, the average electron energy is always within a region where the electron stopping

power in water is almost constant.


102

Figure 5.7: Average electron and photon energies vs. off-axis distance for the (a) 1.5 mm, (b)3 mm, and (c) 5 mm fields. The calculations are performed at a depth of 2.5 cm ina water phantom.

(a)

(b)

(c)

0

1

2

3

4

0 2 4 6 8Off-axis distance (mm)

Ave

rage

ene

rgy

(MeV

)

Photons

Electrons

0

1

2

3

4


Ave

rage

ene

rgy

(MeV

)

Photons

Electrons

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4


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rgy

(MeV

)

Photons

Electrons

5 mm

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1.5 mm


103

Figure 5.8 shows the photon and electron fluence against the off-axis distance for

the three fields. The fluence is normalized to the central axis value. Since the photon

fluence is determined with high uncertainties, due to reasons discussed above, the

normalization value is determined using an interpolation between the points close to the

central axis. The photon fluence in the three fields drops sharply at the geometrical edge

of the field and stays very low beyond the edge. This is in good agreement with the

average photon energy increase, shown in Fig. 5.7(a) and Fig. 5.7(c); since the photon

fluence is very low, the transmission photons represent a significant fraction of the total

fluence, and given that they have a relatively high energy, the average energy increases.

However, the electron fluence, shown in the same figure, does not follow the photon

fluence: the electron fluence profiles are significantly wider, because the electron range

in water is larger than the actual field sizes.

In the same figure, a Monte Carlo-calculated dose profile at a depth of 2.5 cm is

shown for each field. The dose profiles follow the electron fluence profiles almost

perfectly. This observation corresponds to the conclusion that electron energy stays

within a range where the stopping power is constant: the radiation dose delivered to the

medium is then simply equal to the electron fluence multiplied by the electron stopping

power (see Chapter 1). Therefore, the normalized electron fluence profiles are identical to

the normalized dose profiles for the three very small fields that we studied.


104

Figure 5.8: Normalized photon fluence, electron fluence and dose profiles vs. off-axis distancefor the (a) 1.5 mm, (b) 3 mm, and (c) 5 mm fields. The calculations are performedat a depth of 2.5 cm in a water phantom.

(a)

(b)

(c)

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105

5.3. Off-axis ratios (dose profiles)

The dose profiles for the three small photon fields were measured with the A14P

ionization chamber and the HS GafchromicTM film at a depth of 2.5 cm in a water

phantom using the equipment and the setups described in Chapter 2. Profiles under the

same conditions were also calculated using EGSnrc Monte Carlo code (see Chapter 3).

The profiles measured with the A14P chamber were corrected by Monte Carlo-

calculated ( )/wat airD D correction factors in order to correct for the response variation

problem, because the chamber is not a water-equivalent dosimeter. The corrected profiles

underwent a 2-D deconvolution in order to eliminate the blur due to the low resolution

(large dimensions) of the chamber. These two procedures have been extensively

discussed in Chapter 4.

The Monte Carlo results for ( )/wat airD D correction factors as a function of the

off-axis distance in the three fields are shown in Figure 5.9. The statistical uncertainties

are as low as 2% on the central axis, because of the high particle density of the three

phase-space files close to the center of the field, and they increase with the off-axis

distance, due to the decrease in the number of analyzed particles. The ( )/wat airD D

correction factors for the 1 mm field (Fig. 5.9(a)) were obtained by simulating

100,000,000 histories for each chamber position. The value on the central axis is

1.58(1 ±0.015) and it drops with off-axis distance having a value close to unity outside of

the photon field. The central axis values decrease with field size: they are 1.52(1±0.012)

for the 3 mm field (Fig. 5.9(b)) and 1.3(1±0.02) for the 5 mm field (Fig. 5.9(c)). The

simulations for the last two fields were performed with 20,000,000 histories.


106

Figure 5.9: ( )/wat airD D correction factors vs. off-axis distance for the (a) 1.5 mm, (b) 3 mm,and (c) 5 mm fields. The calculations are performed at a depth of 2.5 cm in awater phantom.

(a)

(b)

(c)

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1.6

0 1 2 3 4 5 6 7 8Off-axis distance (mm)

Dw

a

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107

The central axis values of the ( )/wat airD D correction factors are much higher than

the stopping power ratio value that equals 1.12 in a 10 MV photon beam for a large field

size. When the air gap inside the chamber has dimensions close to the actual size of the

field, the fraction of the dose delivered by in-scattered electrons decreases. At the same

time the electrons passing through the cavity go over straight-line trajectories, and this

also decreases the dose delivered to the cavity. These two effects result in an electron

fluence perturbation in the cavity that has to be taken into account along with the

stopping power ratio value, and therefore, the total correction factor is larger than the

stopping power ratio value. Finally, the radiation dose delivered to the air cavity is

significantly smaller than the dose delivered to water on the central axis of the radiation

beam.

The correction factors for all fields are close to unity away from the central axis.

In this region the dose delivered to the air cavity is close to the dose delivered to water,

due to the higher electron fluence in the cavity that compensates for the lower stopping

power in air. For example, a stopping electron that would travel several micrometers in

water has enough energy to cross a 2 mm air cavity, resulting in increase of the electron

fluence in the cavity. This effect has a large impact on the ( )/wat airD D values, because

outside the photon field, the stopping electrons, coming from the field are a much higher

fraction of the total electron fluence than the electrons set in motion locally by either the

transmission or scattered photons. The conclusion based on graphs in Fig. 5.9 is that the

A14P chamber under-responds close to the center of the field and over-responds outside

the fields for photon fields smaller than 5 mm in diameter.


108

Figure 5.10 shows dose profiles of the three fields measured with the A14P

ionization chamber. The raw data represents measurements at two-polarities, corrected

for leakage current. The uncertainties due to the linac output variations and signal

variations are very low: less than 1%. The uncertainties due to ( )/wat airD D corrections

are less than 2% of the central axis value of the OAR curve. The uncertainties due to the

deconvolution procedure are about 1%. An assessment of the uncertainties is presented in

Table 5.2 below.

Figure 5.10 shows that the importance of both the deconvolution and the

( )/wat airD D correction increases with a decreasing field size: the difference between the

three profiles is significant for the 1.5 mm field, whereas for the 5 mm field the profiles

are very close to each other. Figure 5.9 has already illustrated that the smaller is the field

size, the large is the variation in the ( )/wat airD D correction. As far as the convolution is

concerned, it is clear that the smaller is the field size, the larger is the blur effect due to

the detector size. Based on the data presented in Fig. 5.10, we can make the conclusion

that both the ( )/wat airD D correction and the deconvolution are extremely important when

the A14P chamber or any other air-filled ionization chamber is used for off-axis

measurements in very small photon fields.

The profiles (corrected and deconvolved) measured with the A14P chamber are

compared with HS radiochromic film measurements and Monte Carlo simulations in Fig.

5.11. The Monte Carlo simulations were performed with the DOSRZnrc user code, as

described in Chapter 4. About 450,000,000 histories were sent through a water phantom

for each field in order to attain statistical uncertainties less than 1%. Table 5.2 shows the

uncertainties for the various profiles.


109

Figure 5.10: Dose profiles (OAR) measured with the A14P chamber at a depth of2.5 cm in awater phantom for the (a) 1.5 mm, (b) 3 mm, and (c) 5 mm fields. The measured,corrected and deconvolved profiles for each field are presented.

(a)

(b)

(c)

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110

Figure 5.11: Dose profiles (OAR) for the (a) 1.5 mm, (b) 3 mm, and (c) 5 mm fields at a depthof 2.5 cm.

(a)

(b)

(c)

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111

Profile 1.5 mm 3 mm 5 mm

A14P chamber

Measurements:

Correction*:

Deconvolution:

Total*:

HS film

Measurements*:

Non-linearity:

Spatial response variation:

Total*:

Monte Carlo Calculations:

0.5% - 1.0%

0.5% - 1.5%

1.7 %

1.0% - 4%

1.0%

1%

2%

0.8% - 4.0%

0.7%

0.5% - 1.0%

0.2% - 1.2%

0.4%

1.0 % - 2.5%

1.0%

1%

2%

0.6% -4.0%

0.4%

0.5% - 1.0%

0.6% - 2.1%

1.1%

2.5% – 4.2%

1.0%

1%

2%

0.7% -4.0%

0.2%

The total uncertainties are reported as a percentage of the profile value on the

central axis, which is unity as a result of normalization. They are less than 1% far away

from the central axis and 2.5% to 4% close to the central axis. Figure 5.11 shows an

agreement between the measurements and the Monte Carlo simulations within ±3% of

the value on the central axis, which is within the reported uncertainties. Partially these

discrepancies are due to imperfections in the setup, such as the collimator geometry and

positioning that is very difficult to account for.

In the steepest part of the profiles a mispositioning may introduce errors as high

as 5%. However, these uncertainties are usually systematic (type B) resulting in a profile

shift that is very easy to eliminate. In general, the determined profiles are adequate for

stereotactic radiosurgery dose calculations. Moreover, the agreement between

* Percentage of the OAR value on the central axis

Table 5.2: Assessment of the uncertainties associated with the variousmeasurements and calculations of the dose profiles.


112

measurements and Monte Carlo calculations validates the three phase spaces, obtained by

simulating treatment head of the linac along with the radiosurgical collimators.

5.4. Central axis measurements

Percentage depth doses (PDDs) and relative dose factors (RDFs) were determined

for the three different fields. The experimental techniques have been discussed in

Chapter 2. The Monte Carlo calculations for the same quantities were performed with

DOSRZnrc user code, as explained in Chapter 3.

The measured and calculated percentage depth doses are presented in Fig. 5.12.

We had some problems with measuring the PDD curves with the A14P ionization

chamber, because the positioning was not precise enough and it was difficult to keep the

chamber on the central axis as it was moving down in the water tank. As the profiles

show, even a lateral displacement as small as 0.2 mm might lower the PDD value at

certain depth by several percent, and obviously the smaller the field, the more serious the

problem becomes. For these reasons the agreement between the measurements and the

Monte Carlo calculations is best for the 5 mm field and worst for the 1.5 mm field. For

all fields the uncertainties of the film measurements are about 5%, taking into account the

spatial response variation, densitometer error, and uncertainty in the film dose-response

curve.

The uncertainties in the ionization chamber measurements are manly due to the

positioning problem, as mentioned above: considering the profiles, we can estimate

uncertainties of about 1% for the 5 mm field, 3% for the 3 mm field, and 5% for the 1.5


113

mm field. Initially, the complete PDD curves were measured with two polarities,

verifying that within the stated uncertainties there was no difference between the two

polarities beyond the depth of the maximum dose maxd . As a result, we have presented

only the positive polarity data beyond maxd for the 1.5 mm and the 3 mm field in Fig 5.12

in order to minimize the mispositioning errors.

Monte Carlo simulations were performed using between 450,000,000 and

500,000,000 histories resulting in uncertainties in PDD values of less than 0.5%. The

various calculated and measured PDD curves agree within the stated uncertainties, and

this again validates the three phase spaces used as particle sources for the Monte Carlo

simulations.

The most interesting effect, considering PDD measurements for small and very

small photon fields, is the well-known maxd shift toward the surface. We determined the

following maxd values: 1.0 cm for the 1.5 mm field, 1.2 cm for the 3 mm field, and 1.4 cm

for the 5 mm field. These values are much lower than the maxd values for large clinical

photon fields that are on the order of to 2.5 cm for 10 MV x-ray beams, but match the

radiosurgical data by Sixel and Podgorsak, who measured maxd values in the range of

field diameters from 10 mm to 30 mm (Ref. 1).

The surface doses for the three fields were measured with the HS radiochromic

film. The following values were determined: 15% for the 1.5 mm field, 11% for the 3

mm field, and 9% for the 5 mm field. However, it is difficult to make any conclusion,

based on these values, because the surface dose depends on the collimator design, and in

our study we have used collimators made of two different materials (the 1.5 mm and the

5 mm collimators are made of lead, and the 3 mm collimator is made of tungsten).


114

Figure 5.12: Percentage depth dose curves for the (a) 1.5 mm, (b) 3 mm, and (c) 5 mm fields.

(a)

(b)

(c)

0

20

40

60

80

100

120

0 2 4 6 8 10 12Depth (cm)

PD

D

A14P chamber

Monte Carlo

HS film

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120

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D

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Monte Carlo

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5 mm

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1.5 mm


115

The presented PDD curves prove that the A14P chamber may be useful for

central axis dose measurements in very small circular photon fields, provided that a very

precise setup is employed. In order to verify this statement, we conducted a Monte Carlo

based study to evaluate the importance of the detector size in relative central axis

measurements in very small fields. Using the phase-space of the 1.5 mm collimator, we

calculated several PDD curves in a water phantom with different diameter of the scoring

region. The results are shown in Fig 5.13. Three curves, calculated with scoring regions

of diameters of 0.3 mm, 0.6 mm, and 1.5 mm, agree perfectly within the statistical

uncertainty of the simulations. This result proves that PDDs of small circular photon

fields may be measured successfully with a circularly symmetrical detector with a

diameter close to that of the field because the small fields have a negligible divergence.

Of course, another issue, that of the water equivalence of the detector must also be

considered. Since the radiation quality changes with depth in phantom, it is very likely

0

20

40

60

80

100

120

0 2 4 6 8 10 12Depth (cm)

PD

D

1.5 mm

0.6 mm

0.3 mm

Figure 5.13: Monte Carlo-calculated percentage depth dose curves for the1.5 mm field with various diameters of the scoring region.


116

that the response of a non-water equivalent detector will also change, and this will have

an impact on the PDD measurements. However, effects like this are minor and a very

precise setup is required to study them.

The results for the relative dose factor (RDF) (see Chapter 1) measurements and

calculations for the three collimators are presented in Fig 2.14. The ionization chamber

measurements are corrected for ( )/wat airD D , using the correction factors on the central

axis for the three small fields presented above and a ( )/wat airD D correction of 1.12 for

the 10×10 cm2 field. The value of 1.12 is equal to the ratio of the electron stopping

powers of water and air for a 10 MV photon beam, and its considered to be very close

(within 2%) to the total ( )/wat airD D correction factor for chambers as small as the

Exradin A14P chamber. The RDF increases with field size as expected and the RDF

values are presented in Table 5.3.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5 6Diameter of the field (mm)

RD

F

Monte CarloA14P chamberHS Film

Figure 5.14: Relative dose factor (RDF) vs. the diameter of the field.


117

The Monte Carlo calculated RDFs are very close to the film measurements.

However, the A14P chamber measurements are consistently lower than the other two

values, and the discrepancy increases with the decrease in field size. This effect is due to

the relatively large diameter of the A14P chamber collecting volume (about 2 mm: see

Chapter 4). Our measured and calculated RDFs are consistent with some results that

have been already published2-5.

5.5. Dynamic stereotactic radiosurgery with the 1.5 mm and 3 mm photon beams

Radiosurgery represents the main practical application for the very small photon

beams that were studied here. Since we measured the basic dosimetric parameters of

three very small photon beams, we were able to explore two issues of great practical

importance for the potential use of these beams in radiosurgery: (i) the dose distribution

of a radiosurgical technique carried out with the very small beams, and (ii) the capability

of a linear accelerator to carry out the radiosurgery with the very small beams.

The dose distributions for dynamic stereotactic radiosurgery with the 1.5 mm

and the 3 mm collimators were calculated using the XVMC Monte Carlo code, as

Collimator Monte Carlo A14 chamber HS film

1.5 mm 0.26±0.005 0.15±0.01 0.28±0.02

3 mm 0.42±0.005 0.38±0.02 0.42±0.02

5 mm 0.56±0.01 0.53±0.02 0.57±0.03

Table 5.3: RDF of the three different collimators.


118

described in Chapter 3. The calculations were performed in a spherical water phantom,

using the validated phase spaces of the two beams. The dose distributions were also

measured in an X-Y plane containing the isocenter of the linac with a very slow

radiographic film (Kodak EDR-2) (see Chapter 2).

Dose distributions for the two collimators obtained by both Monte Carlo

simulations and measurements are shown in Fig 5.15. The 10%, 50% and 90% isodose

surfaces, normalized to 100% at the isocenter, are presented. The 50% isodose surface is

the most important one in case of small targets, as it would most likely serve as the

prescription isodose surface. The Monte Carlo-calculated 50% isodose surface of the 1.5

mm collimator has a diameter of 2.3 mm, as shown in Fig. 5.15(a). Compared to this

result the maximum deviation of the measured 50% isodose surface is -0.3 mm (Fig

5.15(b)). This discrepancy is due to imperfections in both the collimator and the setup, as

well as the uncertainty in the linac isocenter. The discrepancy between the calculated and

measured 10% isodose lines is larger mainly due to film positioning but does not exceed

1 mm.

The Monte Carlo-calculated 50% isodose surface of the 3 mm collimator has a

diameter of 3.8 mm (Fig. 5.15(c)), and the maximum discrepancy between the calculated

and the measured 50% isodose surface is again -0.3 mm. The 10% line for of the 3 mm

collimator is closer to the isocenter compared to the 10% line of the 1.5 mm collimator,

likely as a result of the 3 mm collimator being made of tungsten, that has a very low

transmission.


119

The last question to be answered was whether or not the measured dose

distributions were shifted with the respect to the isocenter, as defined by the lasers in the

1 cm

1.5 mm field: Kodak EDR-2 film1.5 mm field: Modified XVMC Monte Carlo code

5090

1050

9010

3 mm field: Kodak EDR-2 film

5090

10

3 mm field: Modified XVMC Monte Carlo code

5090

10

Figure 5.15: Isodose distributions obtained with (a) and (b) the 1.5 mmcollimator, and (c) and (d) the 3 mm collimator.

(a)

(d)(c)

(b)


120

treatment room. This issue actually determines the feasibility of radiosurgery with very

small beams, because in practice one must be interested not only in the dose distributions

but also in the target positioning. A dynamic radiosurgery procedure was carried out with

the 1.5 mm collimator in order to determine the position of the laser-defined isocenter

with respect to the delivered dose distribution. A small piece of HS radiochromic film

with four marks, coinciding with the treatment room lasers, was irradiated. The film was

positioned in the same way as the films used for dose distribution measurements. The

result is shown in Fig 5.16. Four pinpricks were used to designate the lasers on the film.

The 50% isodose surfaces for both 1.5 mm and 3 mm collimators, as established earlier,

are also presented in this figure. The distance between the center of the dose distribution

(point of maximum dose) and the isocenter defined by the lasers are: 0.3 to 0.4 mm in X

1 cm

Figure 5.16: Position of the dose distribution obtained with the 1.5 mmcollimator with respect to the lasers.


121

direction and 0.6 to 0.7 mm in Y direction. These results are in good agreement with the

results of a quality assurance study, carried out previously with the same linear

accelerator at the McGill University Health Centre6.

Both the isodose distributions and the isocenter check shown above prove that the

dynamic linac-based radiosurgery technique carried out with very small photon beams

may be used clinically with confidence that the dose will be delivered with an accuracy

of better than 1 mm, assuming of course the linac is in an excellent mechanical condition.


122

References:

1 K. E. Sixel and E. B. Podgorsak, “Buildup region of high-energy x-ray beams in


2 B. E. Bjarngard, J. S. Tsai, and R. K. Rice, “Doses on the central axes of narrow 6-MV

x-ray beams,” Med. Phys. 17, 794-799 (1990).




4 F. Verhaegen, I. J. Das, and H. Palmans, “Monte Carlo dosimetry study of a 6 MV

stereotactic radiosurgery unit,” Phys. Med. Biol. 43, 2755-2768 (1998).

5 X. R. Zhu, J. J. Allen, J. Shi et al., “Total scatter factors and tissue maximum ratios for


and radiographic film,” Med. Phys. 27, 472-477 (2000).

6 T. Falco, M. Lachaine, B. Poffenbarger et al., “Setup verification in linac-based


Chapter 6:

Conclusions and Future Work

6.1. Summary and conclusions

Detailed studies on dosimetry of very small photon fields produced by a linear

accelerator as well as physical aspects of dynamic radiosurgery with these fields have

been presented in this thesis. The studies concerning dosimetry of static beams may be

divided into two main groups: (i) beam quality and basic physics of dose deposition and

(ii) dosimetric parameters of very small fields. The properties of three very small

radiation beams (1.5 mm, 3 mm, and 5 mm in diameter) were studied experimentally and

with Monte Carlo simulations. The beams were produced with special collimators

attached to a 10 MV linac. The results showed that the radiation spectra are similar to the

spectra of large fields. The only noticeable difference was a strong spectral line at the

energy of 0.511 MeV resulting from pair production interactions in the collimators. The

average photon energy slightly decreases at the beam edges and it drops significantly at

the field edge defined by the collimator jaw setting.

The basic physics of dose deposition was studied by Monte Carlo calculations in a

water phantom. We saw that the photon fluence drops at the geometrical edge of a very

small photon field, whereas the electron fluence profile is much wider, because the sizes

of our very small fields were smaller than the electron range in water for 10 MV photon

beams. Moreover, the dose profiles almost perfectly matched the electron fluence

Chapter 6: Conclusions and Future Work

124

profiles, because the average electron energies do not change significantly off-axis and

the electron stopping power is almost constant in this energy range. The dose profiles,

percentage depth dose (PDD) curves, and relative dose factors (RDFs) were measured

with two dosimeters: a micro parallel-plate ionization chamber (Exradin A14P: Standard

Imaging, Middleton, WI, USA) and radiochromic film (HS GafchromicTM: International

Specialty Products, Wayne, NJ, USA).

The profiles measured with the ionization chamber were first corrected with

Monte Carlo-calculated (Dwat/Dair) correction factors, and then deconvolved to eliminate

the blur due to the poor resolution of the chamber. Based on the results, we conclude that

the A14P (and any other) ionization chamber is not a reliable detector for off-axis

measurements in very small fields, unless the response variation (Dwat/Dair correction

factors) is taken into account. This issue is relevant not only to measurements in very

small fields, but also in any field that is under non-equilibrium conditions (for instance

dynamic IMRT fields). Moreover, convolution of the real dose profile with the point

spread function (PSF) of the detector results in blur effects that cannot be neglected in

case of very small photon beams.

On the other hand, the A14P chamber is a good dosimeter for relative central axis

measurements beyond maxd in fields as small as 5 mm in diameter. The chamber is not

suitable for RDF measurements in fields smaller than 5 mm in diameter, even if

(Dwat/Dair) correction factors are considered, because of its poor resolution.

The HS GafchromicTM is a reliable dosimeter for any kind of measurements in

very small photon fields. It is not very precise, but may be used under non-equilibrium


125

conditions. In terms of radiological properties, moreover, its sensitivity to radiation is

about twice as high as that of other radiochromic films.

Since the basic dosimetric quantities were measured with adequate precision for

the three very small fields we decided to explore the usefulness of those fields for

dynamic stereotactic radiosurgery. We investigated two main aspects of the very small

fields for radiosurgery: (i) the 3-D dose distributions for dynamic radiosurgery and (ii)

the displacement of the dose distributions with respect to the laser-defined isocenter of

the linear accelerator.

The Monte-Carlo calculated 3-D dose distributions were in a good agreement

with measurements, proving that a 50% isodose surface as small as 3 mm may be attained

using a linac-based radiosurgery technique. Moreover, the displacements between the

center of the dose distributions and the isocenter of the linac, as defined by the room

lasers, were on the order of 0.6 mm or less. These results were very encouraging, because

they show that the dynamic radiosurgery might be useful for irradiating intracranial

targets on the order of 2 mm providing an option for radiosurgical treatments of

functional disorders that generally require very small radiation fields. Until now these

fields could only be provided by a Gamma knife, a specially designed radiosurgical unit

that is based on 201 stationary cobalt-60 sources. However, the Gamma knives are very

expensive and dedicated only to radiosurgery.


126

6.2. Future Work

The results presented in this thesis and the tools (software, Monte Carlo models,

experimental setups, etc.) developed during the research may be a good starting point for

a series of other studies.

Testing the usefulness of various detectors for measurements in very small fields,

or more generally, under non-equilibrium conditions, will be a study of a great practical

importance. For example, various types of radiographic silver-halide films are widely

used for dose delivery verification in dynamic IMRT fields. These fields are known as a

typical example of non-equilibrium radiation fields and therefore it is important to study

the response variation of silver-halide films in such fields. The best approach for these

studies is by Monte Carlo simulations.

Developing a more sophisticated profile deconvolution algorithm would be

another interesting project. The algorithm presented in this thesis applies only in the case

of very small photon fields with circular symmetry. However, very small irregular photon

beams, shaped by micro multi-leaf collimators are often used in modern radiosurgery.

Thus it is important to have a tool for deconvolving 2-D profiles, when the detector does

not have an adequate resolution. Such an algorithm will allow us to carry out quality

assurance procedures not only with film, but also with other, more precise dosimeters that

do not have the required resolution, such as diodes and liquid-filed ionization chambers.

As far as linac-based dynamic radiosurgery is concerned, the research can also be

extended. We have already calculated 3-D dose distributions using Monte Carlo

simulations and have validated these results by measurements. The next step would be to


127

develop an in-house Monte Carlo treatment planing system (TPS) for dynamic

radiosurgery. Software of this kind will calculate the 3-D dose distributions, using the 3-

D CT data as a phantom, and therefore the exact geometry of the phantom as well as its

non-homogeneity will be automatically taken into account with no further

simplifications. The biggest problem that Monte Carlo-based treatment planing systems

face in conventional radiotherapy is modeling the treatment units with their great variety

of settings used for defining the various radiation fields. In our radiosurgery TPS this

would not be a problem, since we have a predetermined set of clinically used collimators.

Therefore, we would have to do only one simulation of the treatment unit along with each

of the collimators and store the appropriate phase spaces. These phase spaces could be

used as radiation sources in the actual treatment planning.

The three future projects outlined above are feasible, however they would require

some research resources. The significant amount of work that has been done recently at

McGill University Health Centre in the areas of Monte Carlo treatment planning,

response calculation for different detectors, and verification of IMRT fields, would serve

as excellent base for new studies.

List of Figures

Figure 1.1: Typical view of a linear accelerator used in cancer therapy.

Figure 1.2: DSA image of a small AVM.

Figure 1.3: Typical steps in radiotherapy treatment process.

Figure 1.4: Dose distribution for a typical four-field box technique.

Figure 1.5: Geometry for the measurement of the relative dose factor RDF(A). The

dose at point P at dmax in phantom is measured with field A in part (a) and

with field 10×10 cm2 in part (b).

Figure 1.6: Geometry for percentage depth dose measurement and definition. Point Q

is an arbitrary point on the beam central axis at depth d, point P is the

point at dmax on the beam central axis. The field size A is defined on the

surface of the phantom.

Figure 1.7: PDD curve for a 10 MV photon beam, SSD = 100 cm, 10×10 cm2.

List of Figures

129

Figure 1.8: Geometry for the measurement of the tissue-maximum ratio,

TMR(d,AQ,hν).

Figure 1.9: Basic treatment planning algorithm.

Figure 1.10: Narrow beam profiles: curve (1) represents a measured profile, curve (2) a

deconvolved profile.

Figure 1.11: Charged particle equilibrium: number of electrons stopped in a small

volume is equal to the number of electrons set in motion by photons in the

same volume.

Figure 1.12: Electrons moving in lateral direction in large and very small photon fields.

Figure 2.1: Clinac-18 linear accelerator.

Figure 2.2: Treatment head design in (a) photon and (b) electron mode.

Figure 2.3: External view of a 5 mm collimator.

Figure 2.4: Radiosurgery holder with a small field collimator on, attached to the

Clinac-18 treatment head

List of Figures

130

Figure 2.5: Design of the collimators: (a) 5 mm field, (b)1.5 mm field, and (c)3 mm

field. Collimators in (a) and (b) are made of lead, in (c) of tungsten.

Figure 2.6: Exradin A14P ionization chamber.

Figure 2.7: Sketch of A14P ionization chamber.

Figure 2.8: Keithley electrometer.

Figure 2.9: Structure of HS radiochromic film.

Figure 2.10: Dose response curve for HS radiochromic film.

Figure 2.11: Densitiometer Model 37-443 and the film transport system.

Figure 2.12: A14P ionization chamber orientation in the water phantom.

Figure 2.13: A14P chamber positioning (a) for central axis and (b) off-axis

measurements.

Figure 2.14: Simultaneous gantry and table rotations during a dynamic radiosurgery

treatment.

List of Figures

131

Figure 2.15: Beam trace on the patient’s scull for the dynamic radiosurgery technique.

Figure 2.16: Hemispheres made of polystyrene, used as a dynamic radiosurgery

phantom.

Figure 3.1: The basic idea of Monte Carlo particle transport: a particle is transported

until either it leaves the volume of interest or its energy becomes less then

the cutoff energy.

Figure 3.2: Boundary crossing algorithm in PRESTA.

Figure 3.3: The concept of the BEAM/EGS4 Monte Carlo code.

Figure 3.4: Treatment head of the Clinac-18 linear accelerator: BEAM/EGS4 model.

Figure 3.5: Monte Carlo simulated and measured PDD curves for 10×10 cm2 field

size and SSD = 100 cm.

Figure 3.6: Monte Carlo calculated dose profile (OAR) for 10×10 cm2 field size, SSD

= 100 cm and depth of 10 cm.

Figure 3.7: A phantom used for PDD curve Monte Carlo calculations with

DOSRZnrc.

List of Figures

132

Figure 3.8: A phantom used for dose profile (OAR) Monte Carlo calculations with

DOSRZnrc.

Figure 4.1: Exradin A14P ionization chamber: (a) simplified geometry and (b) area

where the equation is solved.

Figure 4.2: Calculation regions.

Figure 4.3: Electric field in the air cavity.

Figure 4.4: Detector in the penumbra region of (a) a large square field, and (b) a small

circular field.

Figure 4.5: Point-Spread Function (PSF) of (a) A14P chamber and (b) a simplified

PSF.

Figure 4.6: Direct deconvolution by (a) filtering and (b) a minimization algorithm.

Figure 4.7: Geometrical model of taking measurements in a small circular field.

Figure 4.8: Restrictions applied to the profiles.

Figure 4.9: Test result for the minimization algorithm.

List of Figures

133

Figure 4.10: Geometry of the air cavity in the Monte Carlo simulations.

Figure 4.11: Monte Carlo simulated geometry for off-axis correction factors

calculation. The phase space is moved over the phantom, instead of

moving the air cavity.

Figure 5.1: Photon fluence for the (a) 1.5 mm, (b)3 mm, and (c) 5 mm fields. The

calculations are performed below the collimators at 70 cm from the

radiation source.

Figure 5.2: Normalized photon fluence density vs. energy for the (a) 1.5 mm, (b) 3

mm, and (c) 5 mm fields. The calculations are performed below the

collimators at 70 cm from the radiation source.

Figure 5.3: Normalized photon energy fluence density vs. energy for the (a) 1.5 mm,

(b) 3 mm, and (c) 5 mm fields. The calculations are performed below the


Figure 5.4: Normalized photon fluence vs. off-axis distance for the (a) 1.5 mm, (b) 3

mm, and (c) 5 mm fields. The calculations are performed below the


List of Figures

134

Figure 5.5: Normalized photon energy fluence vs. off-axis distance for the (a) 1.5 mm,

(b) 3 mm, and (c) 5 mm fields. The calculations are performed below the


Figure 5.6: Average photon energy vs. off-axis distance for the (a) 1.5 mm, (b) 3 mm,

and (c) 5 mm fields. The calculations are performed below the collimators

at 70 cm from the radiation source.

Figure 5.7: Average electron and photon energies vs. off-axis distance for the (a) 1.5

mm, (b) 3 mm, and (c) 5 mm fields. The calculations are performed at a

depth of 2.5 cm in a water phantom.

Figure 5.8: Normalized photon fluence, electron fluence and dose profiles vs. off-axis

distance for the (a) 1.5 mm, (b) 3 mm, and (c) 5 mm fields. The

calculations are performed at a depth of 2.5 cm in a water phantom.

Figure 5.9: ( )/wat airD D correction factors vs. off-axis distance for the (a) 1.5 mm, (b)

3 mm, and (c) 5 mm fields. The calculations are performed at a depth of

2.5 cm in a water phantom.

Figure 5.10: Dose profiles (OAR) measured with the A14P chamber at a depth of2.5

cm in a water phantom for the (a) 1.5 mm, (b) 3 mm, and (c) 5 mm fields.

List of Figures

135

The measured, corrected and deconvolved profiles for each field are

presented.

Figure 5.11: Dose profiles (OAR) for the (a) 1.5 mm, (b) 3 mm, and (c) 5 mm fields at

a depth of 2.5 cm.

Figure 5.12: Percentage depth dose curves for the (a) 1.5 mm, (b) 3 mm, and (c) 5 mm

fields.

Figure 5.13: Monte Carlo-calculated percentage depth dose curves for the 1.5 mm field

with various diameters of the scoring region.

Figure 5.14: Relative dose factor (RDF) vs. the diameter of the field.

Figure 5.15: Isodose distributions obtained with (a) and (b) the 1.5 mm collimator, and

(c) and (d) the 3 mm collimator.

Figure 5.16: Position of the dose distribution obtained with the 1.5 mm collimator with

respect to the lasers.

List of Tables

Table 5.1: Phase space data for the three collimators used in our study.

Table 5.2: Assessment of the uncertainties associated with the various measurements

and calculations of the dose profiles.

Table 5.3: RDF of the three different collimators.

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Thesis

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charged particle

a14p ionization

dynamic stereotactic

beam central

small radiation

small photon