masterofsciencethesis pontusnelldal stockholm2005/menu/general/column...abstract inthisthesis,...

Design of a dynamic beam intensitymodulator for radiation therapy

Master of Science Thesis

Pontus Nelldal

Stockholm 2005

Examinator SupervisorAssociate Professor Andras Kerek Professor Anders Brahme

Section of Nuclear Physics Department of Medical Radiation PhysicsRoyal Institute of Technology Karolinska Institute

Abstract

In this thesis, a new design of a dynamic beam intensity modulator for radiation ther-apy is presented. The design is based on the concept of physical modulators in whichnon-uniform beam pro�les are created by modulating the primary beam by a high den-sity material of thickness distribution corresponding to the desired transmission. Thecontrolling idea is to use a liquid absorber such as mercury as modulating material. Mer-cury has high photon attenuation and its liquid phase at room temperature enables a�exible, dynamic and automated system. The proposed modulator comprises a core con-sisting of 2575 divergent capillaries containing the mercury and an antagonizing liquid,and a system for pumping the liquids. The capillaries are hexagonal in cross sectionalshape and the capillary dimension is in the order of 1 mm, giving a beam resolution of0.25 cm2 at a source to skin distance (SSD) of 75 cm. Theoretically, the modulator willhave high conformal capabilities with an intensity modulation range of at least 100-1%(primary transmission) and, owing to the divergent structure, steep dose gradients anda spatially almost invariant penumbra. Also, in comparison with square structures, thehexagonal structure will improve conformality and maximize small-scale homogeneity.The challenges for realization lie primarily in the elaborate fabrication of the dense cap-illary bundle and in developing the liquid transport system. A more simple and easilyimplemented alternative is also provided. In this solution a bundle of low absorbingparallel bars are immersed in a mercury bath by a wire system. The mercury thicknesspro�le is determined by the positioning of the bars.

Preface

This Thesis was written in Stockholm during the autumn of 2004 in the guidance ofProfessor Anders Brahme at the Department of Medical Radiation Physics, KarolinskaInstitute (KI) and Associate Professor Andras Kerek at the Section of Nuclear Physics,Department of Physics, Royal Institute of Technology (KTH).

A would like to thank the following people: My examinator Andras Kerek for his support,encouragement and for sharing his engineering skills. My supervisor Anders Brahme forhaving taken me on and for his endless source of ideas. The inventor Bo Häggström forinteresting conversations and innovative solutions to technical problems. Younes Me-jaddem for his help on radiation physics modelling and his great helpfulness in general.Finally, the translator Louise Thulin also known as my mother for making comments onlanguage.

Table of contents1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Radiation therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Treatment aims and objectives . . . . . . . . . . . . . . . . . . . . 21.2.2 Treament planning . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Treatment equipment . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Delivery techniques for IMRT . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 Multileaf collimators . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Tomotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.3 Scanned beam therapy . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.4 Physical modulators . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Aim of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Description of modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 IMRT treatment requirements 102.1 Performance parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Beam characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Veri�cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Stability and lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Irradiation geometry 123.1 Modulator and capillary dimensions . . . . . . . . . . . . . . . . . . . . . 123.2 Cross sectional shape of capillaries . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Boundaries and overdosage . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Modulation area lost to capillary walls . . . . . . . . . . . . . . . . 163.2.3 Small-scale beam heterogeneity . . . . . . . . . . . . . . . . . . . . 18

3.3 Length of capillaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Proposed modulator core . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Radiation transport aspects 214.1 Radiation source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Beam perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 Beam hardening/softening . . . . . . . . . . . . . . . . . . . . . . . 224.2.2 Photon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2.3 Contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Mercury thickness determination . . . . . . . . . . . . . . . . . . . . . . . 26

5 Material aspects 275.1 Alternative modulating materials . . . . . . . . . . . . . . . . . . . . . . . 275.2 Possible antagonizing liquids . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Capillary material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3.1 Intercapillary transmission . . . . . . . . . . . . . . . . . . . . . . . 28

6 Liquid dynamics aspects 316.1 Surface tension e�ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.1.1 Meniscus shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.1.2 Curvature implied on capillary corners . . . . . . . . . . . . . . . . 336.1.3 Interfacial stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.2 Flow dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2.1 Setting time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2.2 Characteristics of �ow . . . . . . . . . . . . . . . . . . . . . . . . . 346.2.3 FEMLABTM simlulation . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Liquid transport systems 367.1 The pressure-valve technique . . . . . . . . . . . . . . . . . . . . . . . . . 367.2 Individual reservoir technique . . . . . . . . . . . . . . . . . . . . . . . . . 377.3 Combined solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.4 Veri�cation of mercury thickness . . . . . . . . . . . . . . . . . . . . . . . 39

7.4.1 Direct methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.4.2 Indirect method using transmission monitor . . . . . . . . . . . . . 39

8 Alternative solutions 408.1 Non-divergent intensity modulation . . . . . . . . . . . . . . . . . . . . . . 408.2 Scanned cone in mercury bath . . . . . . . . . . . . . . . . . . . . . . . . . 428.3 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.3.1 Mercury collimator for scanned beam therapy . . . . . . . . . . . . 428.3.2 Energy/range modulator for ion beam therapy . . . . . . . . . . . 42

9 Conclusions 43

APPENDIX 47Attenuation coe�cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Liquid property data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

BIBLIOGRAPHY 48

NOMENCLATURE 50

1

1 Introduction1.1 BackgroundTreatment of cancer by ionizing radiation has always been limited by the radiationtolerance of the healthy normal tissues surrounding the tumour and sensitive organsat risk which inevitably are irradiated. Luckily, the same biological instability thatcauses uncontrolled proliferation of tumour cells, also makes them more sensitive toDNA damage and less likely to survive than normal cells. This property is the basefor radiation therapy and is taken advantage of by fractionating the treatment intodaily sessions (fractions). Between each fraction normal tissue repairs while tumour cellsdecrease. Still, minimizing the dose to healthy tissue is one of the most important factorsfor improving treatment outcome. Development has therefore striven towards makingthe dose distribution conform to the target volume (i.e. tumour volume), while sparingneighbouring healthy and critical structures. Intensity modulated radiation therapy(IMRT) is a fairly new method in which conformality is achieved by modulating theinstensity pro�les of the treatment beams. Unlike conventional therapy where uniformbeams result in considerable dosage to healthy tissue, IMRT can deliver almost arbitrarilyshaped dose distributions that closely conform to the target volume and give minimaldose to normal tissues (see �gure 1). Consequently, IMRT has the potential of enhancingthe therapeutic ratio by 20�30 percentage units (today approximately 50% are cured)and enabling treatment of tumours that, due to the high risk of severe complications,previously would be left untreated.

Target volume Target volume

9080

D/ Gy

70

605040

3020

10

Figure 1: Conformality. Isodose surfaces in four-�eld conventional therapy (left)and in seven-�eld IMRT (right) of prostate cancer. Notice the di�erence in con-formality to target volume.

2 1 INTRODUCTION

1.2 Radiation therapy1.2.1 Treatment aims and objectivesThere are two main aims in radiation therapy: curative and palliative. A curative treat-ment aims to decrease the number of clonogenic tumour cells to a level that results inpermanent tumour control (to prevent tumour regrowth). Palliative treatments are givento patients with incurable diseases to prevent, delay or relieve symptoms and pain. Theyare not, however, expected to improve survival (Aaltonen et al., 1997).

The overall aim should be translated into measurable quantities in treatment planningand follow-up. Therefore, physical and biological clinical objectives are often introduced.A widely used biological objective is P+, the probability of complication free tumour con-trol and may sometimes be estimated using physical objectives such as the mean doseto target volume. These objectives are used as the parameters to be optimized duringthe treatment planning process.

1.2.2 Treament planningNowadays, diagnostic techniques such as computerized tomography (CT) and magneticresonance imaging (MRI) provide oncologists with 3D images of anatomy with submil-limetre precision. In addition, functional imaging with positron emission tomography(PET) superimposed on anatomical CT images makes it possible to separate active fromdead tumuor tissue and to identify regions of hypoxic, radioresistant tumour cells. Oncea cancer has been diagnosed and radiation therapy has been chosen for treatment, anextensive planning process is initiated:(i) Patient immobilization and image acquisition: Since the image acquisition

and the treatment sessions are performed at di�erent locations in space and timeit is vital that the patient position is accurately determined and repeatable. Thisis even more important in IMRT than in conventional treatment, since small errorsin patient position can result in larger treatment errors. To minimize patientsetup uncertainties, immobilization is done by physically �xating the patient usingmasks etc. When a satisfactory setup has been accomplished the image acquisitionis performed, most commonly by CT.

(ii) De�ntion of target volumes: Using the 2D images of anatomy, the oncologistde�nes the target volumes and the critical organs at risk. Due to remaining uncer-tanties in patient setup, internal organ motion and treatment system performance,a margin is added to the target volume, resulting in the internal target volume(ITV). Dose prescriptions to ITVs and tolerance levels to critical organs at risk arethen de�ned.

(iii) Selection of treatment beams: This step involves selection of number, directionand energy of beams. Directions of incident beams are most important whenirradiating concave shaped ITVs and when critical organs are at risk. For somecommon cancers where anatomy has low patient speci�civity, e.g. prostate cancer,standard protocols (assembled from clinical experience) can be used for initial beamcon�guration.

1.2 Radiation therapy 3

(iv) Plan optimization: Traditionally, plan optimization has been done by the trialand error based forward planning, i.e. by manually making changes in beam inten-sity pro�les until the di�erence between desired and calculated dose distributionis within pre-de�ned limits. However, with IMRT, beam pro�les resulting in op-timum dose distribution may not be intuitive. A more sophisticated approach isthen to specify the desired dose distribution and leave it to a computer algorithmto shape the beam intensity pro�les. This is called inverse planning (see �gure 2).Ideally, even the beam parameters in (iii) would be computer determined, but asthe degrees of freedom increase, so does computational time; the robustness ande�ciency of current algoritms are not yet good enough for a completely automatedoptimization process.

90%70%50%

50%

90%

70%

Figure 2: Optimization. Schematic view of forward (left) and inverse (right)planning.

(v) Simulation: Before irradiation, the treatment unit geometry is emulated by asimulator which uses diagnostic energy x-rays to take images of the patient in thetreatment position.

(vi) Treatment: The limit of dose that can be delivered in one treatment session isdetermined by the tolerance of the surrounding normal tissues. Prescribed dosesare therefore fractionated so that the healthy tissue can repair. The fractionationschedule must consider radiobiological factors of tumuor response: di�erences inradiosensitivity of cells in di�erent cell cycle phases, reoxygenation and redistribu-tion of hypoxic cells, and repopulation of tumuor cells between fractions. A typicalconventional treatment consists of 35 fractions (one per day) of 2 Gy (1 Gy = 1Joule/kg), corresponding to a prescribed total dose of 70 Gy. With IMRT, largerdoses can be delivered without inducing more damage to neighbouring normalstructures. But, as the biological e�ects are not yet fully understood, fractionationschedules are not signi�cantly di�erent.

(vii) Follow up: After treatment, tests of various kinds, such as anatomical investi-gation of irradiated tumuor site by imaging and chemical tests, are performed toanalyze the outcome of the treatment.

4 1 INTRODUCTION

1.2.3 Treatment equipmentA diagram of a treatment unit is shown in �gure 3. The essential parts of such a systemare:

• Radiation source � Typically an electron accelerator used to produce electronbeams in the 4�50 megaelectronvolt (MeV) energy range.

• Target � To produce photon beams, the electrons from the accelerator are impingedon a target in which they are converted into bremsstrahlung photons (c.f. section4.1).

• Flattening �lter � Most treatmeant units today still utilize a �attening �lter formaking the beam uniform.

• Collimators � Static and adjustable collimators of tungsten that de�ne the beamand determine the �eld size.

• Shielding material � For protection of hospital personnel and the patient.

• Transmission monitors � Ion chambers and/or electronic portal imaging devices(EPIDs) for veri�cation of accurate dose delivery.

Figure 3: Treatment head. Exploded view of a treatment head.

When delivering IMRT additional parts are required.

1.3 Delivery techniques for IMRT 5

1.3 Delivery techniques for IMRTVarious techniques for shaping intensity modulated beams (IMBs) have been developed.These can be categorized in several ways, but for our purpose the most relevant clas-si�cation is into variable and �xed �uence delivery techniques. In the variable �uencetechniques (multileaf collimators, tomotherapy and scanned beam therapy) the individ-ual IMB is a result of partial treatments of the target volume, while in the �xed �uencetechnique (physical modulators) the whole target is encompassed simultaneously with aconstant intensity pro�le.

1.3.1 Multileaf collimatorsIn clinics, the most commonly used technique for IMRT is multileaf collimators (MLC).As the name suggests, the beam is collimated by multiple pairs of opposite tungsten leafspositioned perpendicularly to beam direction (see �gure 4). The leaf positions determinea projected area irradiated at the surface of the patient as seen from the beam's eye view.Typical leaf widths projected at the patient are between 1 and 0.5 cm. Intensity modu-lation can be done in two ways, by segmental multileaf collimation (SMLC) or dynamicmultileaf collimation (DMLC).

SMLC is more commonly referred to as step-and-shoot-MLC because leaves are sta-tionary during beam-on time and the positions are changed when the beam is o�. TheIMB is then built up by a series of sub�elds (approximately 100 sub�elds depending onbeam pro�le complexity) corresponding to the individual MLC shapes. A drawback ofSMLC is the formation of edged beam �eld boundaries.

In DMLC treatment, the leaves are moved continuously during irradiation. This makesit possible to achieve smooth �eld boundaries and thus a higher conformality to targetvolumes. However, maximum beam modulation and dose gradients are restricted bymechanical limits in leaf velocity and acceleration.

Figure 4: Multileaf collimators. Each leaf pair projects a rectangular beam area.

6 1 INTRODUCTION

1.3.2 TomotherapyAnother technique for IMRT is tomotherapy. This technique resembles CT since theradiation source is rotated around the patient while the patient is moved axially throughthe �eld. By irradiating from the whole arc of rotation, the dose gets focused and highlyconformal dose distributions can be achieved. The increased conformality is, however,obtained at a cost of a lenghtened treatment time, since a small part of the target istreated in each rotation. Two types of tomotherapy exist: in serial tomotherapy thepatient is moved in steps while the beam is paused and in spiral (helical) tomother-apy the movement is smooth under continuous rotation and irradiation. The apparatusconsists of a treatment head equipped with a narrow opening slit of binary (open orclosed) multileaf collimators (typically 2 × 20 cm2) shaping the slit beam. Recent de-velopment has led to a tomotherapy unit combined with a megavoltage CT (MVCT) inwhich real-time imaging of anatomy is provided.

1.3.3 Scanned beam therapyIn scanned beam therapy intensity modulation is achieved by scanning a narrow, oftengaussian shaped beam over the target in the same way as an electron beam is scannedin a tv-monitor (see �gure 5). This modality has had restricted clinical implementa-tion because of (i) the cost of the required electromagnetical system and (ii) the limitedbeam resolution; the full width at half maximum (FWHM) of the photon beam widthis currently in the order of centimetres (Intensity modulated radiation therapy collab-oratory working group, 2001, Webb, 2002), but can be reduced to the order of 10 mm(Svensson and Brahme, 1998). Furthermore, the resolution could be enhanced by colli-mating the scanned beam using MLCs as described by Svensson (1998). In the future thereduced cost and high �exibility of narrow pencil beam scanning will make the techniquemore common both for electron and photon as well as light ion beam therapy.

Figure 5: Scanned beam therapy. Exploded view of a scanned beam therapy unitand the shapes of elemental beams.

1.3 Delivery techniques for IMRT 7

1.3.4 Physical modulatorsPhysical modulators (or compensating �lters or transmission blocks) have been describedin many reports (Jiang and Ayyangar, 1998, Mejaddem, 2003). At �rst, they were usedto compensate for missing tissue and non-�at patient contour in order to produce a uni-form �eld in the target. But, with the introduction of IMRT, the technique was foundto be a simple and inexpensive method of shaping IMBs.

In all of the previously described techniques, IMBs are created by sequentially irradiat-ing the target by means of collimating or scanning. With physical modulators the wholetarget is encompassed simultaneously with a constant IMB pro�le.The IMB is achievedby intercepting the beam with a metal alloy block of a thickness pro�le corresponding tothe desired transmission pro�le (see �gure 6). A common material used is CerrobendTMwhich consists of 15.5% tin, 32% lead and 52.5% bismuth by weight (Mejaddem, 2003),resulting in an e�ective Z = 77.565 and a density of 9.72 g·cm−3.

The simultaneous whole-�eld irradiation results in higher monitor unit (the unit dosecounted by the delivery system) e�ciency, less whole-body dose and, if organ motion isnot taken into account explicitly, smaller errors (Mejaddem, 2003) than for sequential de-livery. The blocks can be made with high precision (± 0.1 mm corresponding to ± 0.5%dose (Mejaddem et al., 1997)) and, compared to MLCs, with higher spatial resolutionin the direction normal to leaf motion. Also, steeper dose gradients can be achieved.However, the introduction of modulating material in the actual treatment �eld changesthe beam quality.

Since di�erent block shapes are required for each beam and must be changed betweenbeams, and since they are fabricated at the clinic, the modality is presently only realisticfor few-�eld treatments (typically 3-5 beam directions) and has to this date had limitedclinical implementation.

x

Φ

Primary uniform beam profile

Φ0

Intensity modulated beam profile

Figure 6: Physical modulators. Left: Transmission block. Right: Schematic ofhow the intensity pro�le Φ is generated from the modulator thickness pro�le.

8 1 INTRODUCTION

E�orts have therefore been made trying to develop a more �exible physical modulator.Some techniques have been based on assembling transmission blocks from metal rods orbricks (see �gure 7). For instance, Fukuhara et al. (2004) have described a technique inwhich a machine automatically arranges metal cubes of two di�erent densities (1 and 12g·cm−3). However, using solid material inevitably implies restrictions. Xu et al. (2002)have presented a reshapable modulator in which a mixture of tungsten powder, para�nand a silicon binder is shaped by pistons. Still, the shaping is done outside the beamand therefore the modulator has no dynamic capabilities.

Figure 7: Towards dynamic physical modulators. Showing principles of construct-ing thickness pro�les in high density materials. Left: Rods of di�erent length butequal density. Middle: Cubes of di�erent density. Right: A gel containing tung-sten powder shaped by pistons.

1.4 Aim of this thesis 9

1.4 Aim of this thesisThe aim of this thesis is to develop a new intensity modulator design for radiation therapy.Fundmental for this design is to use a liquid absorber such as mercury as the modulatingmaterial. Mercury has two physical properties making it suitable for a dynamic physicalmodulator: high photon attenuation (Z = 80, ρ = 13.6 g·cm−3) and liquid phase at roomtemperature. The goal is to combine the bene�ts of physical modulators discussed insection 1.3.4 with a �exible and automated system. A successful implementation wouldthereby (i) eliminate the attributes that have limited the implementation of physicalmodulators in clinical practice and (ii) enable real-time dynamic capabilities such ascompensating for organ motion.

1.5 Description of modulatorThe proposed modulator comprises a modulator core consisting of a bundle of divergentcapillaries and a liquid transport system for control of mercury thickness in individualcapillaries. Mercury is supplied from a reservoir at the bottom end of capillaries and anantagonizing liquid is supplied from the other end to force the mercury in position. See�gure 8. To increase e�ciency, the modulator is modelled without �attening �lter.

Liquid

Mercury reservoir

Modulator cross section

target

collimators

treatment couch

EPID

modulator

e-

photon beam

IMB

Figure 8: Overview of modulator. Left: The core of the modulator consistsof a bundle of capillaries individually �lled with mercury. Right: Diagram ofmodulator in treatment head geometry.

10 2 IMRT TREATMENT REQUIREMENTS

2 IMRT treatment requirementsIn this chapter the framework for the design of the mercury modulator is de�ned.

2.1 Performance parameters• Intensity ratio � The ideal intensity ratio may be quite high (Brahme et al., 1982)

and not easily achievable. Furthermore, an optimal choice of beam directions maysigni�cantly lower the needed beam modulation. This is particularly importantin SMLC where a reasonable maximum number of sub�elds sets limits on themodulation window. However, to enable high dose gradients a minimum intensitymodulation ability of 100-1% is required.

• Number of intensity levels � Transmission blocks (c.f. section 1.3.4) havein theory an in�nite number of intensity levels, but in practice the fabricationprecision lowers this to a �nite number in the order of 1000. On the other extreme,SMLC is limited to yield approximately ten intensity levels. However, if beampro�les are not too complex, using as few as 5-7 levels can yield almost as goodresult as with DMLC (Palta and Mackie, 2003). For this modulator the goal is toachieve a minimum of 100 levels.

• Spatial resolution � Most systems for IMRT today are designed to give beamresolutions in the order of cm. For instance, a typical projected leaf width is 0.5to 1 cm at isocenter. Even if there exist MLCs which provide higher resolution, ithas been reported that decreasing the projected leaf width below 3-5 mm has lowimpact on dose conformity (Palta and Mackie, 2003). Also, due to computationallimitations, most treatment plans are today applied with bixel (beam pixel) sizesof either 0.5 × 0.5 cm2 or 1 × 1 cm2. In this design a bixel area of equal or lessthan 0.5× 0.5 cm2 is required.

• Setting time � As mentioned in section 1.3.4, a drawback of static physical mod-ulators is the time and e�ort to fabricate and change transmission blocks betweenbeams. This limits the number of treatment beams down to 3-5. To ultimatelyavoid such restrictions the time required to set the mercury in position betweenbeams should be in the order of a second.

• Penumbra � The penumbra is e�ectively the steepness of dose gradients and isoften de�ned as the lateral distance between the points of 20% and 80% dose.The e�ecive penumbra is dependent on two properties inherent in the modulatingsystem: the modulator geometry and accuracy in positioning. In MLC the leafend shape and alignment relative the coned shaped treatment beam are importantfor the penumbra and two di�erent approaches exist. One is to move the leaves inan arc so that the leaf ends are always parallel to the ray-lines. This gives a sharpand spatially invariant penumbra. The other one is to use curved leaf ends andmove them in a plane perpendicular to the axis of gauntry rotation which resultsin a wider and non-linearly displaced penumbra, but of nearly constant width. Theaccuracy in positioning is important since the e�ect is cumulated during fractions.

2.2 Beam characteristics 11

In MLC systems the projected uncertainty in leaf positions is approximately ±1-2mm (Palta and Mackie, 2003).

2.2 Beam characteristicsIn order to provide numeric examples a selection of the relevant energy range must bemade. It has been shown by Söderström et al. (1999) that, to maximize P+ (c.f. section1.2.1) when delivering IMRT, the accelerator potential is not critical in the range of 1-50MV. However, slightly improved results can be obtained by using a potential of a fewMV at the periphery and as high as 50 MV for the bulk target. In this work, two of themost commonly applied accelerator potentials, 6 and 18 MV, are used for quanti�cation.

The introduction of the modulator in the treatment �eld has three main implications forbeam quality � (i) it changes the energy spectrum, (ii) it induces scattered photons thatblur the intensity pro�le and (iii) it introduces a new source of contaminating electronsto the surface dose. These e�ects have been investigated in the work of Mejaddem (2003)and Jiang and Ayyangar (1998). They described the scatter kernels and found that en-ergy spectral changes have small in�uence on percent depth dose and can therefore beneglected in the modulator design and dose calculation, but included as a correction inthe �nal dose distribution.

2.3 Veri�cationBesides the usual ion chamber based integral dose veri�cation systems, the steep dosegradients in IMRT put high demands on accurate measurement of the spatial dose distri-bution delivered. Traditionally, radiographic �lm has been used for veri�cation becauseof its high resolution, but the technique is labor-intensive, in�exible and the quality ofthe images is highly dependent on processing. With electronic portal imaging devices(EPIDs) the images are instantly stored digitally which allows real-time imaging andsimpli�es image post-processing. The accuracy in the dose distribution measurement isin the order of a few percent (Vieira et al., 2002, Webb, 2002).

Another issue is the veri�cation of the system performance itself. In this design, themost crucial control parameter is the determination of the mercury thickness distribu-tion. For that reason, both direct and indirect methods of measurement are presentedin section 7.4.

2.4 Stability and lifetimeThe stability and lifetime of the modulator depend on the radiation hardness (resistanceto damage) of the materials and the mechanical stability. Therefore, the materials se-lected must have been proven to withstand the immense radiation doses in the treatmenthead and the active parts of the modulator must be rigid enough to deal with gauntrymotion.

12 3 IRRADIATION GEOMETRY

3 Irradiation geometry3.1 Modulator and capillary dimensionsThe modulator core consists of N divergent capillaries and is placed a distance SMD (sourceto modulator distance) from the beam source (see �gure 9).

SMD

H

SSD

r(z)

θ

BA

Figure 9: Irradiation geometry. Showing the key dimensioning parameters:source to modulator distance (SMD), source to skin distance (SSD), modulatorheight (H), maximum radius r(z) of treatment beam, beam area BA and themaximum half angle θ.

The radius r(z) at a distance z down from source is given by:

r(z) = z tan θ = zr(SSD)SSD

(1)

where SSD is the source to skin distance. Furthermore, the cross sectional area A andthe total volume V of the modulator is given by:

A(z) = πr(z)2, z ∈ [ SMD, SMD + H ] (2)

V = πr(SSD)12 · SSD

((SMD + H)3 − SMD3) (3)

To meet the demand of minimum spatial resolution, the beam �eld area BA = A(SSD)and the desired bixel area a give the number of capillaries:

N =BA

a(4)

The vertically dependent cross sectional area of individual capillaries will then approxi-mately be:

a(z) =A(z)N

, z ∈ [ SMD,SMD + H ] (5)

3.1 Modulator and capillary dimensions 13

Varying the parameters described gives di�erent dimensions for the capillaries and valuesfor a few reasonable settings are shown in table 1. The top row entries are de�ned asprototype values for further investigation.

BA/cm2 rc/cm a/cm2 N SSD/cm SMD/cm θ/rad at/mm2 ab/mm2

25×25 14.1 0.25 2500 75 15 0.165 1.00 2.78" " 0.25 2500 100 " 0.124 0.5625 1.5625" " 0.35 1786 75 " 0.165 1.40 3.89" " 0.35 1786 100 " 0.124 0.787 2.187

Table 1: Modulator and capillary dimensions. A=beam �eld size,rc=corresponding radius of circular �eld, a=bixel size, N=number of capillaries,SSD=source to skin distance, SMD=source to modulator distance, θ=maximumhalf angle, at=capillary cross sectional area at the top of modulator andab=capillary cross sectional area at the bottom of modulator.


3.2 Cross sectional shape of capillariesTwo shapes for capillary cross section are considered, regular hexagon and square. Selec-tion of cross sectional shape is made upon considering the intrinsic geometrical properties'consequences for modulator performance.

3.2.1 Boundaries and overdosageWhen irradiating tumuors, it is of vital importance that the whole target region is coveredsince all clonogenic tumour cells must be eliminated. However, since tumour boundariesare irregular and the pixels being �nite in size, healthy tissue will also receive dosage.To investigate the in�uence of cross sectional geometry, the �gure of merit fractionalexcessive area irradiated, f = BA/TA where BA and TA stand for beam area andtarget area respectively, was calculated for two target geometries: circles of diameter d,and typical 2D prostate shapes of largest diameter d. The bixel size was set to 25 mm2

and equal for both geometries.

d=50 mm

86 hexagons, f =1.204 88 squares, f=1.232

d=53 mm

91 hexagons, f=1.159 97 squares, f=1.235

Figure 10: Geometry and overdosage. Comparing hexagon and square abilityto follow the contour of prostate and circular shapes. The gray contour is theboundary of the target area to be irradiated. It should be noted that some targetsizes may favour one geometry since the bixel size is �xed. By varying the SSDthe 2D conformality could be improved.

3.2 Cross sectional shape of capillaries 15

The (arithmetic) mean gain factor fs,h was de�ned by:

fs,h = (n∑

i=1

fs,h)/n where (6)

fs,h = fsquare/fhexagonal (7)

fs,h was calculated to 1.08 and 1.02 for circular and prostate shape respectively (seetable 2). Thus, a hexagonal structure gives a slightly better 2D conformality.

Phantom shape d/mm fhexagonal fsquare fs,h fs,h

Circle:41 1.155 1.307 1.132 -43 1.257 1.188 0.945 -44 1.200 1.266 1.055 -46 1.279 1.339 1.407 -48 1.174 1.304 1.111 -50 1.159 1.235 1.066 -55 1.273 1.168 0.918 -60 1.129 1.211 1.073 -80 1.174 1.189 1.013 -

1.08Prostate:

53 1.204 1.232 1.023 -68 1.157 1.182 1.022 -83 1.125 1.148 1.020 -103 1.101 1.116 1.014 -

1.02

Table 2: Overdosage. Quantized comparison of excessive area irradiated withhexagonal and square beam geometry.


3.2.2 Modulation area lost to capillary wallsSince the capillary walls have �nite size there will be an unmodulated part of the trans-mitted beam. This lost modulation area will depend on capillary wall thickness t andon the e�ciency of the tiling. It is mathematically proven that any partition of theplane into regions of equal area has a perimeter at least that of the regular hexagonalhoneycomb tiling (Hales, 2001). However, to give a quantized perspective, an analyticalcomparison of hexagonal and square tiling e�ciency is given below. To ease calculationsthe tiling is made by starting from a single tile, i.e. a hexagon or a square, and then inn steps adding tiles around the preceding ones (se �gure 11), ending up with N(n) tiles(or capillaries) and a total of E(n) segments, each having length e (e depends on thetype of tiling) and thickness t.

n: 1 2 3S(n): 7 37E(n):

0199630 132

16

Figure 11: Tiling principle. The tiles are added in n steps to yield N(n) tiles(capillaries) consisting of E(n) segments (walls). The �rst three steps are dis-played.

Hexagonal tiling

N(n) = 3n2 + 3n + 1 (8)E(n) = 9n2 + 15n + 6 (9)

a =3√

32

e2hex (10)

Qhex =awall

a=

t · ehex · E(n)a ·N(n)

=2

3√

3· t

ehex· 9n2 + 15n + 6

3n2 + 3n + 1→

→ 2√3· t

ehexas n →∞ (11)

Square tiling

N(n) = 4n2 + 4n + 1 (12)E(n) = 8n2 + 12n + 4 (13)

a = e2sq (14)

Qsq =awall

a=

t · esq · E(n)a ·N(n)

=t

esq· 8n2 + 12n + 4

4n2 + 4n + 1→

→ 2 · t

esqas n →∞ (15)

3.2 Cross sectional shape of capillaries 17

Since the expression E(n)/N(n) converges at a fast rate for both geometries (see �gure12), the limit values are used in the calculations. Combining equations (11) and (15)and using the relation esq = 31/4 · 2−1/2ehex for the same bixel area yield the relativee�ciency of the respective tilings:

Qsq

Qhex=

ehex

√3

esq=

21/2

31/4≈ 1.075 (16)

Thus, square tiling gives 7.5% more loss of modulation area than hexagonal tiling. Theabsolute percentage area lost in the di�erent tilings as a function of wall thickness wascalculated for the prototype values de�ned in table 1 and the result is displayed in �gure12.

0 10 20 301

2

3

4

5

6

Hexagonal Tiling

Square Tiling

n

E(n)/N(n)

0.05 0.1 0.15 0.25

10

15

20

25

30

35

40

Hexagonal Tiling

Square Tiling

t / mm

%

0.05 0.1 0.15 0.25

10

15

20

25

30

35

40

Hexagonal Tiling

Square Tiling

t / mm

%

Figure 12: Comparison of square and hexagonal tiling. Left: The fast conver-gence of E(n)/N(n). Right: Lost modulation area as a function of capillary wallthickness t.


3.2.3 Small-scale beam heterogeneityWhether the cross sectional shape of capillaries is hexagonal or square, the �nite thick-ness of the capillaries will result in a shape corresponding �uence variation. Brahmeet al. (1981) investigated the in�uence on small-scale beam heterogeneity resulting fromsummation of narrow Gassian beams when comparing square and hexagonal beam grids.They concluded that, for the same surface density of elementary beams, a hexagonalgrid results in less �uence variations than a square grid (see �gure 13). Even though thesituation is di�erent here, the result indicates that a hexagonal structure will give bettersmall-scale homogeneity than a square structure.

Figure 13: Small-scale beam heterogeneity. The maximum �uence variation insquare (solid line) and hexagonal (dashed line) grids of Gaussian electron beamsof constant width calculated by Brahme et al. (1981). By applying a hexagonalgrid, a gain in homogeneity by 5-8% is obtained. Printed with permission.

3.3 Length of capillaries 19

3.3 Length of capillariesIn order to achieve high dose gradients and to meet the demand for beam modulation of100-1%, the capillaries must have some minimum length hmin. This length is given by thedi�erence in attenuation between mercury and the antagonizing liquid and is thereforeenergy dependent. A �rst measure of hmin is obtained by considering transmission ofprimary photons travelled through equal thickness of mercury and antagonizing liquid

hmin =ln R

µHgρHg − µliqρliqwhere R =

ImaxImin

(17)

However, since the �attening �lter is removed in this design, the capillaries must belengthened to allow high doses at o�-central axis positions. The additional amount ∆his given by:

∆h =lnΦ(θmax)−1

µHgρHg − µliqρliq(18)

where Φ(θmax) is the normalized �uence output from the source at the edge of the�eld given by eq. (19). Calculations of the total capillary length were performed forthe systems mercury/water and mercury/hexane. The results show that the maximumlength needed for a modulation of 100-1% occurs for the beam energy 4.1 MeV andamounts to approximately 9 cm. See �gure 14.

0 10 20 30 40 504

5

6

7

8

9

10h/cm

0 10 20 30 40 504

5

6

7

8

9

10h/cm

0 10 20 30 40 5010

1

102

103

E/MeV

%9 cm

8 cm

7 cm

6 cm

Figure 14: Capillary length. Left: The required capillary length to achieve anintensity ratio of 100% with the systems mercury/water (blue lines) and mer-cury/hexane (black lines). The solid lines were calculated with correction for thee�ect of removing the �attening �lter. Right: Modulation windows for di�erentcapillary lengths as a function of energy.


3.4 Proposed modulator core

In this chapter, the geometrical properties of the modulator and the capillaries havebeen investigated. Compared to a square structure, hexagonal cross section of capillariesis bene�cial since, (i) less overdosage will be delivered to healthy tissue, (ii) it mini-mizes capillary wall material, hence maximizes cross sectional modulation area and (iii)the small-scale beam homogeneity is better. It is estimated that a capillary length ofapproximately 9 cm will be su�cient to achieve full modulation ability even when the�attening �lter is removed and high dose at o�-central axis positions is desired. More-over, the divergent structure of capillaries should give a spatially invariant penumbra.

Figure 15: Cross section of modulator core. The diameter at the top of themodulator is approximately 5 cm. There are 2575 capillaries in this con�guration,giving a beam resolution of 0.25 cm2 at SSD=75 cm

21

4 Radiation transport aspects4.1 Radiation sourcePhoton beams for treatment are created by conversion of electrons into photons in atarget material. The produced beam is commonly refered to as brehmstrahlung sincewhen an electron is accelerated by the electric �eld of a nucleus in the target, a photon isemitted. The maximum photon energy Emax will equal the kinetic energy of the electrons(almost monoenergetic), but the emitted photon energy spectrum will be continuous inthe interval 0− Emax (see �gure 16). The angular distribution will increase with targetthickness due to increased scattering of electrons prior to photon conversion. Brahmeand Svensson (1979) gave a semi-empirical expression of the angular dependence, theexpression concerned the dose delivered without �attening �lter at SSD=100 cm, butcan serve as an approximate measure of the angular energy �uence incident on modulator:

Φ(E, θ) =Φ(E, 0)

1 + (E θa)b

(19)

Here Φ is the energy �uence and E is the maximum photon energy in MeV. The constantsa = 1.73 rad ·MeV and b = 1.4 (Brahme and Svensson, 1979).

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1

θ/rad

1 MV

6 MV

10 MV

18 MV

50 MV

0 5 10 15 2010

−7

10−6

10−5

10−4

10−3

E/MeV

Φ

0 5 10 15 2010

−7

10−6

10−5

10−4

10−3

E/MeV

Φ

Figure 16: Bremstrahlung output. Left: Relative angular distribution in arbi-trary units calculated from equation (19). Right: Spectral �uence for 6 and 18MV Varian Clinac beams (Sheik-Bagheri and Rogers, 2002).

The photon source can often be considered pointlike, but is in reality �nite and can bedescribed by a Gaussian function with a standard deviation σ of 0.5−1.5 mm dependingon system. Sometimes the source size (or focal size) is given by the full width at halfmaximum (FWHM) which is related to σ by:

FWHM = σ√

ln 2 (20)

22 4 RADIATION TRANSPORT ASPECTS

4.2 Beam perturbationsAs discussed in section 1.3.4 the introduction of material in the treatment �eld inducesbeam quality changes or perturbations. These perturbations involve energy spectralchanges, scattering of photons and increased contribution of electrons to surface dose.

4.2.1 Beam hardening/softeningBeam hardening/softening refers to the increase/decrease in mean energy of modulatedbeams. Jiang and Ayyangar (1998) have reported the modulator material CerrobendTM

(see section 1.3.4) to increase the mean energy of a 6 MV beam by 24% and 49% forthicknesses of 1.2 and 5 cm respectively. To estimate this e�ect for mercury the meanenergy of primary photons as a function of modulating thickness was calculated by:

E(t) =∫

E Φ(E) e−µt dE∫Φ(E) e−µt dE

(21)

where Φ is the primary photon �uence. This was done for 6 and 18 MV beams using thephoton �uence spectra of Varian Clinacs calculated by Sheikh-Bagheri et al.(2002).

0 2.5 5 7.5 101.5

2

2.5

3

CerrobendT M

Mercury

6 MV

t/cm

E/MeV

0 2.5 5 7.5 104.5

4.6

4.7

4.8

4.9

5

CerrobendT M

Mercury

18 MV

t/cm

E/MeV

Figure 17: Beam hardening. Mean energy of modulated beam as a function ofmodulating thickness of mercury and CerrobendTM. For the 6 MV beam, themean energy is monotonically increasing, while for the 18 MV beam, there aremaxima at t = 1.3 cm and 2.3 cm for mercury and CerrobendTM respectively

The result is displayed in �gure 17. An increase in mean energy of the 6 MV beam by 22and 49 % was experienced for mercury thicknesses of 1.2 and 5 cm respectively. Althoughsigni�cant spectral changes are seen, the mercury and CerrobendTM modulators behavequantatively similar and the e�ect is therefore neglected in this design, in agreementwith the argument stated in section 2.2.

4.2 Beam perturbations 23

4.2.2 Photon scatteringThe distribution among the interaction processes is displayed in �gure 18. In the lowerclinically relevant energy range (E<5 MeV) incoherent (compton) scattering is the dom-inating process. Above 5 MeV, pair production in the nuclear �eld is dominating andincreasingly dominant with energy. However, annihilation photons have low in�uenceon dose since they are isotropically emitted, but must be considered for shielding. Inconclusion, to investigate the contribution of scattered photons on �uence pro�les, theonly relevant process is incoherent scattering.

10−1

100

101

10−4

10−3

10−2

10−1

100

E/MeV

cm2/gTotal

Incoherent

Pair production

Photoelectric

Figure 18: Photon interaction in mercury. Components of the mass attenuationcoe�cient µ as a function of energy.

A semi-analytical expression of a scatter kernel for intensity modulating �lters has beenderived by Mejaddem (2003). In his work, a scatter function S(r, t) is convolved withthe primary photon �uence Φp to yield an estimation of the scattered photon �uence Φs:

Φs = Φp ⊗ S(t, r) (22)

where S(t, r) is separated into modulator thickness and radial dependent parts:

S(t, r) = St(t)Sr(r) (23)

The modulating thickness dependence is given by:

St(t) = β1te−β2t (24)

where β1 is a normalization constant and β2 is the mean attenuation coe�cient of primaryand scattered photons. The radial function is a sum of two terms:

Sr(r) =1

1 + Eγ (1 + cos θs)γ

+a + br2 + cr3

1 + dr + fr2(25)

where the �rst term describes a �rst single compton scatter in an angle θs and the secondis a correction term for higher order scattering. The free parameters Γ, γ, a, b, c, d and


f were determined by comparing eq. (22) to scattered �uence from 6 and 18 MV beamsmodulated by a compensating �lter of the material CerrobendTM of density 9.72 g·cm−3

and e�ective Z = 77.565. These values are close to mercury (Z = 80 and ρHg =13.55 g·cm−3) why the parameters �tted for CerrobendTM could be used to estimate thecontribution of scattered photon �uence generated in the mercury modulator.

tp

ts

θs

ϕ

xs

xm x

θs

z

zs

Figure 19: Scattering geometry. The photon is scattered in an angle θs.

However, since the geometry is di�erent, the model must be adjusted. The scatteringangle θs is expressed in the coordinates given in �gure 19:

θs = θi − ϕ = tan−1(xs

SSD − zs)− tan−1(

xs − xm

zs) (26)

The thickness dependence was corrected for this geometry to:

St(tp, ts) = β1tpe−µptpe−µsts (27)

where tp and ts are the mercury thickness travelled by primary and scattered photonsrespectively, and µp and µs are the attenuation coe�cients for the mean energy of primaryand scattered photons respectively.

4.2 Beam perturbations 25

4.2.3 ContaminationIn therapy, unwanted secondary radiation is produced and scattered in the materialsintercepting the beam path. The secondary radiation consists of scattered photons andleptons (electrons and positrons) mainly produced in compton scattering and pair pro-duction.

As mentioned above, photons resulting from annihilation of generated electron-positronpairs are emitted isotropically and their contribution is therefore neglectible. Also, Jiangand Ayyangar (1998) found the contribution of scattered photons to the surface dose tobe low and independent of compensator shape.

However, lepton contribution to the surface dose must be considered. In general, themain lepton contamination sources are the �attening �lter, the collimators and the airbetween gauntry and patient and the total contamination increases with �eld size. Dis-tribution between sources is primarily dependent on beam energy and �eld size. Forlarge �eld sizes and photon energies above approximately 3 MeV, the �attening �lteris the dominating source and increases in dominance with increasing energy due to theneed for thicker �lters (Nilsson, 1985). Although the modulator itself introduces a newcontamination source, removing the �attening �lter will to some extent compensate thissince there will always be a minimum of material in the beam path.

The remaining electrons could either be removed or be implemented in the dose plan incombination with the photon �eld to deliver shallow doses (Mejaddem, 2004). However,if desired, leptons can be removed by a thin �lter of low atomic number material, orswept away by a purging magnet.


4.3 Mercury thickness determinationIn a �rst approximation scattering is neglected and the discrete mercury thickness dis-tribution T is a function of the desired �uence pro�le Φd and the primary photon �uenceincident on the modulator Φ0. The relation is given by:

T =ln(Φ0/Φd)− µliqH

µHg − µliq(28)

An simple example of thickness determination is shown in �gure 20. The desired pro�le(dashed line) is a step function with a dip in intensity between x = 0 and x = 5 cmwhich for instance could be to spare the brain stem. However, since the �attening �lteris removed the �uence incident on modulator (solid line) is dependent on angle. Theresulting mercury thickness distribution is therefore slightly curved.

−10 −5 0 5 100

20

40

60

80

100

x/cm

%

−10 −5 0 5 100

20

40

60

80

100

x/cm

%

−0.1 0 0.10

2

4

6

8

10

θ/rad

t/cm

Figure 20: Mercury thickness determination. Left: Desired �uence pro�le(dashed line) and relative bremsstrahlung output (solid line). Right: Mercurythickness pro�le calculated using equation (28) and the mean energy of a 18 MVbeam.

For a more accurate calculation, scattered photon �uence should be taken into account.The mercury thickness would then be determined by the relation:

Φd(r, T ) = Φp(r, T ) + Φs(r, T ) (29)

where Φd(r, T ) is the desired �uence pro�le, Φp(r, T ) the primary attenuated transmissionand Φs(r, T ) the scattered contribution.

27

5 Material aspects5.1 Alternative modulating materialsAt Radiumhemmet, Karolinska Hospital, Stockholm a clay containing tungsten has beenused since the early 1980s for beam collimation. Similarly, Xu et al. (2002) used areshapable material consisting of a mixture of tungsten powder, para�n and silicon-based binder was used to create IMBs. The material was deliberately composed to beviscid and withhold the shape since shaping was done outside the beam and no meanswas provided to force it to stay in shape. For our purpose this material is too viscid, butmaybe it could be possible to compose a liquid containing tungsten powder to obtain aless viscid liquid with high photon attenuation. Another idea is to replace the mercurywith very smoth and small metall spheres (Cederwall, 2004).

5.2 Possible antagonizing liquidsIt was shown in section 3.3 that the liquid density can vary from 0.6 to 1 g·cm−3 (hexaneand water) or even be higher without signi�cantly diminishing the intensity modulationwindow. Therefore, selection of liquid will predominantly be based upon two properties:chemical compatibility with the interfacing materials (mercury and capillary walls) andliquid �ow properties. The scope of this study does not allow a thorough investigation,but three liquids are proposed as candidates: hexane (non-polar), ethanol (polar) andpuri�ed kerosene (behaving well with mercury (Häggström, 2004)). All these have lowdensity and low viscosity (see appendix).

5.3 Capillary materialFirst of all, due to the high density of capillaries and the need for keeping the walls thin,it is an elaborate task to fabricate the modulator core. Various materials and fabricationprocesses can be considered, but the seemingly most reasonable are sintering, etchingand wire-sparkling metals. Also, metals are in general highly resistant to radiation in-duced damage.

Secondly, the material must be resistant to metallurgical attack by mercury (corrosionand formation of alloys). Last but not least, since the capillary walls are passive theywill appear in the transmission pro�le as a positive or negative hexagonal grid. Thisfenomenon is similar to the problem with interleaf leackage in MLC systems. However,the transmission through capillary walls will be dependent on the material selected. Atoo high transmission would increase the risk of exceeding tolerance levels for healthytissues, while a too low transmission would increase the risk of local tumour cell survival.Although scattering and the �nite source size will help to smoothen the pro�le, the e�ectcannot be neglected.

28 5 MATERIAL ASPECTS

5.3.1 Intercapillary transmissionThe e�ect of three di�erent wall materials (316L stainless steel, aluminum and titanium)on small-scale transmission pro�les was analyzed. The primary transmission pro�le Φp

was calculated for mean energies of 6 and 18 MV beams and for the mercury thick-ness distribution shown in �gure 21. However, on this scale, the �nite source size haslarge in�uence and therefore, Φp was convolved with a normalized gaussian distributioncorresponding to the �nite source size:

P (x) = φ⊗ Φ =∫

φ(x− u)Φ(u)du (30)

whereφ(x) =

1σ√

2πe−x2/σ2 (31)

1 2 3 4 5 6 7 80

0.25

0.5

0.75

1

channel no

t/H

Figure 21: Channel settings. Mercury thickness distribution used in calculationof transmission pro�les normalized to the capillary length H = 9 cm.

Results of the computations are displayed in �gure 22 and 23 where focal sizes (σ) of0.5 and 0.25 mm were used respectively. Stainless steel gives good homogeneity at lowtransmission, but too large dips when high transmission is desired. With aluminumthe e�ect is negated. From this aspect, titanium seems to be the preferable choice ofcapillary wall material.

5.3 Capillary material 29

0 10 20 30 400

20

40

60

80

100

6 MV/Aluminum

mm

%

0 10 20 30 400

20

40

60

80

100

18 MV/Aluminum

mm

%

0 10 20 30 400

20

40

60

80

100

6 MV/Titanium

mm

%

0 10 20 30 400

20

40

60

80

100

18 MV/Titanium

mm

%

0 10 20 30 400

20

40

60

80

100

6 MV/Stainles s s teel

mm

%

0 10 20 30 400

20

40

60

80

100


mm

%

Figure 22: Intercapillary transmission. Transmission pro�les convolved with aGaussian-shaped function with σ = 0.5 mm.

30 5 MATERIAL ASPECTS

0 10 20 30 400

20

40

60

80

100

6 MV/Aluminum

mm

%

0 10 20 30 400

20

40

60

80

100

18 MV/Aluminum

mm

%

0 10 20 30 400

20

40

60

80

100

6 MV/Titanium

mm

%

0 10 20 30 400

20

40

60

80

100

18 MV/Titanium

mm

%

0 10 20 30 400

20

40

60

80

100


mm

%

0 10 20 30 400

20

40

60

80

100


mm

%

Figure 23: Intercapillary transmission. Transmission pro�les convolved with aGaussian-shaped function with σ = 0.25 mm.

31

6 Liquid dynamics aspects6.1 Surface tension e�ects6.1.1 Meniscus shapeMeniscus shape in a liquid interface is determined by the forces acting on it, i.e. theinterfacial tensions between the substrats and gravity. The surface shape is convenientlydescribed by a contact angle θ and a varying curvature radius R(ϕ) (see �gure 24).However, in small vessels the shape can be assumed spherical with a constant curvature,which simpli�es the shape determination to calculation of the contact angle. This isgiven by the Young-Laplace equation:

γl,w = γHg,w + γHg,l cos θ (32)

where the subscripts l,w and Hg stand for liquid (antagonizing), wall and mercury re-spectively. Values of interfacial tensions γA,B between two substrats A and B can becalculated with Fowkes approximation:

γA,B = γA + γB − 2(√

γdAγd

B +√

γndA γnd

B ) (33)

where the superscripts d and nd represent dispersive and non-dispersive forces respec-tively (see appendix for typical values of di�erent materials). To ease calculations evenfurther, the antagonizing liquid is neglected (replaced with air). Combining equations(32) and (33) then yields a simple expression for the contact angle:

θ = cos−1(2(

√γd

Hgγdw +

√γnd

Hgγndw )

γHg− 1) (34)

θ

ϕ

R

∆h

r

γliq,Hg

γw,Hg

γw,liq

Figure 24: Geometry of mercury/liquid interface. Due to the high surfacetension of mercury the meniscus will be convex.

32 6 LIQUID DYNAMICS ASPECTS

Calculations were performed using the maximum and minimum values for metals as wallmaterial (see appendix for values). The result, θ ≈ 121.6◦, was compared to experimentalvalues of contact angles measured by Ellison et al. (1967). They experienced contactangles in the narrow range 130-134 ◦for the interface of mercury and, for our purpose,relevant metals (tungsten, stainless steel etc.) at 25 ◦C. The convex shape has theconsequence on photon transmission that there will be a dip in intensity at the middleof bixels. An estimation of the maximum variation across a pixel was done by �rstconsidering the maximum di�erence in mercury thickness:

∆h = R(1− sin θ) = − r

cos θ(1− sin θ) (35)

and then evaluating the maximum intensity ratio of primary transmission across thecapillary:

Icenter

Iedge= e−∆h(µHg−µliq) (36)

The calculation was performed both for calculated and tabulated θ-values (see �gure25). Since this e�ect is small and will be even smaller due to smoothening by the �nitesource and scattering it is neglected. Also, the lowest variations occur in the clinicallymost attractive energy range.

10 20 30 40 500

0.5

1

1.5

2

E/MeV

%

θ =130

θ =121.6

Figure 25: Meniscus shape e�ect on transmission. Maximum percent intensityvariation across the capillary calculated by eq. (36) using the worst case values ofthe contact angle θ calculated by eq. (34) (θ = 121.6◦, dashed line) and tabulated(θ = 130◦, solid line) by Ellison et al. (1967).

6.1 Surface tension e�ects 33

6.1.2 Curvature implied on capillary cornersAnother possible consequence of the non-wetting behaviour of mercury is forming ofrounded menisci at corners as simulated by Wong et al.(1991). This could cause leakageof the antagonizing liquid down the edges and seriously disturb the stability of the system.To avoid this, the capillary cross section must be modi�ed to have curved edges. Anexpression relating the principal radius of curvature R to the pressure drop ∆p acrossthe surface is:

R =γ

∆p(37)

Since the hydrostatic pressure will vary with alignment of modulator and action ap-plied on liquids over time, a minimum pressure drop must be estimated to determine aminimum radius Rmin.

Figure 26: Curvature at capillary corners.

6.1.3 Interfacial stabilityThe third consequence of surface tension e�ects is potentially a positive one. Since thegauntry can be aligned to irradiate from arbitrary angles, the liquid interface must bestable independently of capillary alignment relative the gravitational �eld. Consider acapillary aligned vertically and �lled with two immiscible liquids (mercury and antago-nizing liquid) with the heavier liquid (mercury) at the top (see �gure 27). The gravita-tional energy of the system is at its maximum, but the system can be stable providedthe cross sectional capillary dimensions are small enough. This situation is described bythe Rayleigh-Taylor instability and a largest dimension d for stability is given by:

d < π

√γ

(ρHg − ρliq)g(38)

where γ is the interfacial tension between the liquids and g is the gravitational constant(Faber, 1995). Inserting typical values into equation (38): γ = 0.4 mN ·m−1, ρHg =13.6 g·cm−3, ρliq = 1 g·cm−3 and g = 10 m ·s−2 yields d < 5.6 mm. Although the largestcapillary dimension is less than two times smaller, equation (38) is valid under stationaryconditions and instabilities caused by �ow and gauntry motion should be investigated.

34 6 LIQUID DYNAMICS ASPECTS

d

ρHg

ρliq

Figure 27: Interfacial stability.

6.2 Flow dynamicsThe incompresssible �ow of the mercury and the antagonizing liquid is governed by theNavier-Stokes equations. In this section, the antagonizing liquid will be neglected andthe investigation of �ow dynamics will be done considering only the mercury.

6.2.1 Setting timeFor the hexagonal geometry there is no analytical solution to the Navier-Stokes equations,but to get an idea of the pressure drop ∆p needed to achieve the setting time requirement(stated in section 2.1) the capillaries are considered cylindrical. For this geometry thesolution is straightforward and the volume rate Q is given by.

Q =dV

dt=

π

8· r4∆p

ηL(39)

where η is the dynamic viscosity, r the capillary radius and L the capillary length.Applying the prototype values (translated into the cylindrical geometry) and requiringthe time to �ll a capillary with mercury to be one second gives ∆p ≈ 136 Pa. This is ofcourse much lower than the maximum hydrostatic pressure p0:

p0 = ρgH (40)

which amounts to approximately 12.2 kPa at the bottom of a vertically aligned capillaryof length H = 9 cm. Still, the total pressure drop is low and implies no limitations onobtaining the required setting time.

6.2.2 Characteristics of �owThe Reynolds number Re is a dimensionless parameter for estimation of the character-istics of �ow. A thumb rule is that the �ow is laminar provided Re < 2100. The numberis de�ned by:

Re =ρvDh

η(41)

where v is the mean velocity and Dh is the so called hydraulic diameter given by:

Dh ={

4AP in general√3 · e for hexagonal cross section (42)

6.2 Flow dynamics 35

where A is the cross sectional area, P the wetted perimeter and e the segment lengthof the hexagon side. The maximum value of Re was found to be 1963 at the top of thecapillary where v is at its maximum. Thus, the �ow should be laminar, but not by alarge margin.

6.2.3 FEMLABTM simlulationIn order to get a better estimation of the pressure drop needed to achieve the requiredsetting time and to further investigate the characteristics of �ow, a simulation usingFEMLABTM was done. The hexagonal structure was approximated by an axisymmet-ric cone with dimensions corresponding to the prototype values given in table 1. Thesimulation shows that both the maximum velocity vmax and the maximum Reyonoldsnumber will appear at the center of the top of the capillary (see �gure 28). Compared tothe analytically calculated Re, the simulation yielded lower values with Remax ≈ 1300.The needed pressure drop was sligtly higher than the analytical, ∆p ≈ 440 Pa, but stillsmall compared to the hydrostatic pressure.

Figure 28: Velocity �eld in capillary. The velocity �eld in m · s−1 of mercury inthe r − z plane of a cone corresponding to a capillary. The capillary length is 9cm and the radius is approximately 1 mm at the bottom.

For future development, FEMLABTM could be used to model and simulate the con-trol of �ow. Modelling MEMS (Micro-Electronic Mechanical Systems) can be done toincorporate pumping or �ow-controlling miniature devices.

36 7 LIQUID TRANSPORT SYSTEMS

7 Liquid transport systemsIn this chapter, sketches of solutions to the problem of mercury transport are provided.The proposed solutions have in common that the capillary ends on the mercury side areconnected to a common reservoir of mercury. On this reservoir, an arbitrary pressuremay be put so that the manipulating parts are working against a minimal hydrostaticpressure.

7.1 The pressure-valve techniqueThe controlling device in this solution consists of three parts: two pressurized reservoirsfor the liquids and a microvalve matrix, mounted on the top of the modulator (theantagonizing liquid side). The working principle is to control the mercury thickness ineach capillary by the valve open time. A drawback is that the �ow through di�erentcapillaries must be in the same direction. Also, it could be di�cult to manufacture amicrovalve matrix with the required density of valves. However, Watanabe and Kuwano(1997) have described a device for precise control of gas �ow controlled by a high densitymicrovalve matrix based on piezo-electric technology.

Liquid

Mercury

Microvalve

Capillary

Figure 29: Schematics of pressure-valve technique.

7.2 Individual reservoir technique 37

7.2 Individual reservoir techniqueIn this solution, each channel has its own pump and reservoir moved out of the radiation�eld. This means that an elaborate piping must be achieved and that the availablevolume must be e�ectively used. The piping could for instance be done by etchingsmall channels and packing them in layers. At the end of each channel a micropump ismounted. Three suggestions for micropumps are provided:

• Stepper motors � Stepper motors can be made small, are accurate and reliable.In addition the position of the piston gives a direct measurement of the mercurysurface position.

• Piezo actuators � Piezo technology has revolutionized the area of microelectronicmechanical systems (MEMS).

• Bubble pumps � This pump has been proposed by the inventor Bo Häggströmand is inspired by inkjet technology. Modern inkjets can dispense ink blobs atfrequencies in the order of kHz with nanolitre precision. The inkjet head dispenserconsists of a small chamber with inlet and outlet holes. After the ink has enteredthe chamber, heating causes an overpressure to sling a well de�ned ink drop onthe paper. Each capillary would have two antagonizing bubble pumps based oninkjet technology, one in each direction. To minimize transient e�ects and toincrease precision, the pumps would be constantly at work. The resulting �owwould then be given by the di�erence in working frequency; when driven at thesame frequencies, equal amounts of liquid will leave the capillary as will enter.

For further information about micropumps see Woias (2004).

38 7 LIQUID TRANSPORT SYSTEMS

7.3 Combined solutionIn the previous solutions, a breakdown of one valve or one pump would make the mod-ulator useless. To lower the probability of breakdown, the controlling devices could beintegrated in groups. This could be achieved by allowing a stepper motor to supply apressure to a fraction of the channels and in each group using microvalves for individualchannel control. A possible grouping of capillaries into modules of 19 is shown in �gure30.

Figure 30: Geometry of combined solution. This con�guration contains 19 ×7 × 19 = 2527 capillaries and 7 × 19 = 133 pumps. Each pump thus provides 19capillaries with liquid.

7.4 Veri�cation of mercury thickness 39

7.4 Veri�cation of mercury thicknessThis section deals with two methods for assurance of accurate mercury thickness distribu-tion: direct measurement by electronic devices and indirect measurement by monitoringtransmission pro�les.

7.4.1 Direct methodsDepending on the method of �ow control, di�erent approaches to mercury height mea-surement will be bene�cial. Some suggestions are:

• Cavity resonance � By sending radiofrequent waves down the capillary, a correlationbetween the measured resonance and the mercury thickness can be determined.Overhearing between capillaries can be minimized by using di�erent frequenciesfor neighbouring capillaries.

• Re�ection � Since mercury will have a signi�cantly di�erent acoustic impedancethan any antagonizing liquid, a re�ection based method is also possible. Thethickness would be determined by the time.

• Optical � Laser or LED (light emitting diode) technique. Also time-distance basedmeasurement.

7.4.2 Indirect method using transmission monitorAn indirect method of measuring the mercury thickness distribution would be to place adetector with high resolution below the modulator, measure the spatial transmission andperform an inverse calculation. The di�erence ∆Φ between the desired Φd and measuredΦm intensity maps would then yield a correction factor ∆T for the thickness distribution:

∆Φ = Φd − Φm ⇒ ∆T = ∆T (∆Φ) (43)

Also, the measured post-modulation pro�le could be compared to EPID measurementsto make real-time improvements to the mercury thickness distributions and to enhancethe estimations of delivered dose distributions in the patient.

The selection of detector type should be made by considering the trade-o� of havinglow attenuation and scattering against providing good accuracy. Ion chambers may bethe optimum choice since the attenuation is minimal and the count rate is high.

40 8 ALTERNATIVE SOLUTIONS

8 Alternative solutions8.1 Non-divergent intensity modulationIn this solution a bundle of bars of low photon absorption is immersed in a mercury bath(see �gure 31). The bars are controlled to dynamically determine a mercury modulatorby setting the displacement coordinates ζ(i(x), j(y)). Each bar is connected to a wireloop (see �gure 32) with individual motors outside the beam. To minimize leakage ofmercury as well as radiation, the bars can be squeezed after positioning. This will implya step-and-shoot delivery.

The bene�ts from the hexagonal structure are preserved and the perimeter minimiz-ing property will manifest itself in a minimum leakage. However, unlike the divergentcapillaries in the main solution, the non-divergent bars will yield a spatially dependentpenumbra, which will set limits on dose gradients and the projected resolution will varywith bar height. The cross sectional area of the bars is 1 mm2, giving a bixel size of0.25 cm2 at SSD= 25 cm, and the bar length is 12 cm of which a maximum of 7 cm canbe immersed in the mercury for full transmission.

Probably the most reasonable bar material would be a plastic of high radiation hardnessand low photon attenuation.

Φ(x)

x

ζ(i(x))ζ(n)ζ(2)

22

19

Figure 31: Alternative solution. Schematic view of modulator (left) and crosssection of modulator core (right). There are 22×19×4 = 1672 hexagonally shapedbars in this con�guration.

8.1 Non-divergent intensity modulation 41

38 mm

1

19

70 mm

120 mm

1

70 mm

Symmetry

2 3 22

Figure 32: Active components of alternative solution. The bars' positions arecontrolled with wires connected to motors outside the beam. The framed imageshows an axis of rotation as seen from above.

42 8 ALTERNATIVE SOLUTIONS

8.2 Scanned cone in mercury bathAn even more alternative idea is to scan a hollow cone immersed in a mercury bath.This would function as scanned beam therapy (section 1.2.2) and therefore also includethe drawback of not irradiating the whole �eld at the same time. However, multiplecones could be used to lower the treatment time, and the resolution could be improvedcompared to existing scanned beam therapy.

θ(t)

Φ(x)

x

Mercury bath

Figure 33: Scanned cone in mercury bath. Schematic of the scanned cone solution(c.f. section 8.2) and the mercury collimator for scanned beam therapy (c.f. section8.3.1). In the latter application the cone movement could be controlled by the samecurrent that is driving the beam.

8.3 Other applications8.3.1 Mercury collimator for scanned beam therapyThe high energy (50 MV) scanned pencil beam system described by Svensson (1998) hasbeen furher developed and provided with a mini-MLC system to outline a window inwhich the pencil beam is scanned. Due to the high scanning speed, the collimator needsto be fast and therefore the mini-MLC is placed close to the beam source. Still, themechanics of leaves sets limits on speed and the gaps between the leaves give leakage.Instead of using MLC technique, a system consisting of a mercury bath and either asolid body of low attenuation or a hollow body could provide the collimation (see �gure33).

8.3.2 Energy/range modulator for ion beam therapyIn ion beam therapy, the range of ions in tissue is determined by adjusting the energy ofincident ions. This is presently most often done by interfering the beam with multipleplexiglass discs of di�erent thicknesses. A more �exible way would be to use a uniformmercury modulator with thickness as the only parameter.

43

9 ConclusionsThis thesis has aimed at presenting dynamic beam intensity modulator designs for IMRTbased on the unique properties of liquid absorbers and the concept of physical modula-tors.

The main solution comprises a modulator core consisting of a bundle of 2575 divergentcapillaries (individually �lled with mercury) and a dynamic liquid transport system. Themercury is supplied from a reservoir at the bottom end of the modulator and an antago-nizing liquid is supplied from the other end to force the mercury in shape. A theoreticalprototype with a capillary cross sectional area of 1 mm2 at SMD= 15 cm was de�nedto give a beam resolution of 0.25 cm2 at SSD= 75 cm. The dimensioning is a trade-o�between fabrication limitations and interfacial stability; the smaller the capillaries, themore stable is the liquid interface, but at the same time the manufacturing gets morecumbersome. Also, as the capillaries get smaller, less e�ective modulation area will beavailable. Although a theoretical calculation showed that stability is obtained as longas the largest diameter is smaller than 5.6 mm, the result is valid under stationary con-ditions and instabilities caused by gauntry movement should be investigated. With acapillary length of 9 cm the required minimum intensity ratio of 100-1% for primaryphotons is achieved, even if the �attening �lter is removed and high dose near the edgeof the �eld is desired.

Hexagonal compared to square cross sectional shape of capillaries has been shown toyield less overdosage to healthy neighbouring tissue, improved small-scale dose homo-geneity and a maximized e�ective modulation area (the capillary walls are passive).Moreover, the divergence of capillaries will give a spatially invariant penumbra and thusenables steeper dose gradients.

Di�erent suggestions for a liquid transport system have been made. An elegant so-lution would be to control the �ow by a microvalve matrix mounted directly on the topof the modulator core as shown in �gure 29, but it is unclear whether such a dense ma-trix is achievable. Although recent development in MEMS (Micro-Electronic MechanicalSystems) technology is promising it is not known if piezo actuators (or any other electro-mechanical system) can operate and survive under the harsh conditions. Therefore, itwould probably be desirable to move the active parts out of the beam and to make useof the space available in the treatment head. However, this requires an elaborate pipingsince each capillary would have its own pump. The most reasonable way of fabricationseems to be etching channels in wafers and packing them in layers.

An alternative idea for a mercury modulator was also described. The system com-prises a bundle of 1672 parallel bars of low photon attenuation immersed in a mercury(or a similar liquid absorber) bath and controlled by a wire system to determine themercury surface shape. The bars also have a hexagonal cross section and therefore sharethe bene�ts stated earlier. However, the non-divergent geometry will imply less steepdose gradients and the projected beam resolution and penumbra will be dependent onthe vertical position of bars in the modulator.

44 9 CONCLUSIONS

Beam perturbations induced by the mercury modulator have been quanti�ed; the in-crease in mean energy of primary photon �uence of a 6 MV beam was 22% and 49% fora thickness of 1.2 and 5 cm respectively. Since this is comparable to the results obtainedby Jiang and Ayyangar (1998) for a cerrobendTM modulator, their conclusion that thise�ect can be neglected in the modulator design is adopted. A scattering model devel-oped by Mejaddem (2003) was applied to give a more accurate method of determiningthe thickness pro�le. Still, further work needs to be done in investigating the radiationtransport properties of mercury.

Physical modulators have some advantages over the sequential delivery techniques. Thesimultaneous whole �eld irradiation gives improved photon economy (less beam-on time)which leads to lower accumulated whole-body leakage dose and a substantially shortenedtreatment time. The latter gain is, however, with currently available transmission blocktechniques compensated by a substantial loss in manufacturing e�orts. The proposedsystem(s) would not have this drawback and the gain would be further increased if the�attening �lter is removed. Also, if organ motion is not explicitly taken into account, the�xed �uence of a physical modulator will result in less dose distribution errors than withsequential delivery, since errors are better averaged during treatment execution (Mejad-dem, 2003).

Other possible applications of mercury based systems for radiation therapy are an en-ergy/range modulator for ion beam therapy and a moving window collimator instead ofa MLC for high energy pencil beam therapy as described in section 8.2.

To get an overview of the di�erent IMRT delivery techniques, their key parametersare summarized in table 3. The included irradiation techniques are described in moredetail in section 1.3. It is clear that the di�erent approaches are useful for quite di�erentpurposes with a clear potential for adjustable large area beam modulators with a photoneconomy similar to the scanned pencil beam approach.

45Static

absorbers

Adjustableab

s.Dyn

amic

collimation

Scan

ning

Blocks

Rods

Cube

sCl

ay/G

elLiqu

idLiqu

idMLC

Tomo

Jaws

Penc

ilbe

am

Max

#be

ams

3-5

3-5

3-5

5-10

5-15

5-15

5-15

503

25Beam

mod

u-latio

n/%

100-1

100-10

100-10

100-1.5

100-5

>10

0-1

100-10

100-1

100-10

100-2

Intensity

levels

∼10

3∼

10∼

10∼

103

∼10

3∼

103

∼10

∼50

∼10

∼50

0

Spatialreso-

lutio

n/mm

∼2

5-10

5-10

105

55-10

5-10

105-20

Setting

time/s

>10

2∼

102

>10

2∼

102

∼1

∼1

∼10

∼0

∼0

∼0

Treatm

ent

time/

Tu

1-1.5

1-1.5

1-1.5

1-1.5

0.5-1

0.5-1

1.5-2

>5

>20

0.5-1

Fabrication

cost

low

low

low

low

low

med

ium

high

med

ium

low

med

ium

Leakage

low

low

low

low

low

low

low

very

low

med

ium

very

low

Photon

econ

omy

high

high

high

high

high

high

med

ium

low

low

high

Table3:

Comparis

onof

IMRT

deliverytechniques.Th

edi�e

rent

mod

alities

are

describ

edin

section1.3.

Values

aregene

ralised

andshou

ldno

tbe

considered

asde

�nite

.Tu=

thes

tand

ardtreatm

entt

imef

orun

iform

dose

deliv

eryto

thet

arget

volumein

onebe

amdirection(1-2

min).

APPENDIX 47

AppendixAttenuation coe�cients

Material Formula ρ/g·cm−3 µ/10−2 cm2 ·g−1

1 MeV 6 MeV 10 MeV 18 MeV 50 MeVLiquids:Hexane C4H14 0.65 7.39 2.80 2.17 1.72 1.42Mercury Hg 13.6 6.99 4.36 4.94 5.95 7.98Water H2O 1.00 7.07 2.77 2.22 1.86 1.67Metals:Aluminum Al 2.70 6.15 2.66 2.32 2.17 2.31CerrobendTM - 9.72 6.96 4.29 4.83 5.81 7.77St. steel - 7.8 6.00 3.07 3.01 3.19 3.86Titanium Ti 4.54 5.89 2.87 2.73 2.81 3.30Tungsten W 19.3 6.62 4.21 4.75 5.70 7.62

Table 4: Attenuation coe�cients. Values for elemental media, compounds andmixtures obtained from XCOM (1999).

Liquid property data

Liquid ρ/g·cm−3 η/mPa·s γ/mN·m−1

Ethanol 0.79 1.20 22.10Hexane 0.65 0.300 17.89Mercury 13.6 1.526 484Water 0.998 1.890 71.99

Table 5: Liquid property data. Density ρ, dynamic viscosity η and surface tensionγ of liquids. Values are valid under normal room temperature and normal pressure.

Material γ/mN·m−1 γd/mN·m−1 γnd/mN·m−1

Mercury 484 200 284Metals 35�60 30�50 5�15Hexane 72 - -

Table 6: Components of surface tension. The superscripts d and nd stand fordispersive and non-dispersive forces respectively. Values are valid under normalroom temperature and pressure (Birdi, 1997).

48 BIBLIOGRAPHY

BibliographyAaltonen P, Brahme A, Lax I, Levernes S, Näslund I, Reitan JB and Thuresson I, 1997.

Speci�cation of dose delivery in radiation therapy. Acta Oncologica, 36(Supplementum10).

Birdi KS (Editor), 1997. Handbook of surfaces and Colloid chemistry. CRC Press, NewYork.

Brahme A, Lax I and Andreo P, 1981. Electron beam dose planning using discreteGassian beams. Acta Radiologica Oncologica, 20:147�158.

Brahme A, Roos JE and Lax I, 1982. Solution of an integral equation encountered inrotation therapy. Physics in Medicine and Biology, 27:1221�1229.

Brahme A and Svensson H, 1979. Radiation beam characteristics of a 22 MeV microtron.Acta Radiologica: Oncology, 18(3):244�272.

Cederwall B, 2004. Private communication.

Ellison AH, Klemm RB, Schwartz AM, Grubb LS and Petrash DA, 1967. Contact anglesof mercury on various surfaces and the e�ect of temperature. Journal of Chemical andEngineering Data, 12(4):607�609.

Faber TE, 1995. Fluid dynamics for physicists. Cambridge University Press, Cambridge.

Fukuhara N, Matsumoto Y, Nagunama T and Sasamoto T, 2004. Compensating �lterswith multi-bricks; It will be a new technique for IMRT. The 4th S. Takahashi MemorialInternational Workshop on 3 Dimensional Conformal Radiotherapy.

Galvin JM, Ezzell G, Eisbrauch A, Butler B, Xiao Y, Rosen I, Rosenman J, Xing MSL,Xia P, Lomax T, Low DA and Palta J, 2004. Implementing IMRT in clinical practice:A joint document of the American society for therapeutic radiology and oncology andthe American association of physicists in medicine. International Journal of RadiationOncology Biology Physics, 58(5):1616�1634.

Hales TC, 2001. The honeycomb conjecture. Discrete and Computational Geometry,25:1�22.

Häggström B, 2004. Private communication.

Intensity modulated radiation therapy collaboratory working group, 2001. Intensity-modulated radiotherapy: current status and issues of interest. International Journalof Radiation Oncology Biology Physics, 51(4):880�914.

Jiang SB and Ayyangar KM, 1998. On compensator design for photon beam intensity-modulated conformal therapy. Medical Physics, 25(5):668�675.

Mejaddem Y, 2003. Photon and electron transport calculations for development of in-tensity modulated radiation therapy. Ph.D. thesis, Department of Medical RadiationPhysics, Karolinska Institute, Stockholm.

BIBLIOGRAPHY 49

Mejaddem Y, 2004. Private communication.

Mejaddem Y, Lax I and Adakkai S, 1997. Procedure for accurate fabrication of tissuecompensators with high-density material. Physics in Medicine and Biology, 42:415�421.

Nilsson B, 1985. Analysis of Quality Characteristics of Radiotherapeutic Photon Beams.Ph.D. thesis, Department of Radiation Physics, University of Stockholm, Stockholm.

Palta JR and Mackie TR (Editors), 2003. Intensity-Modulated Radiation Therapy - TheState of the Art. Medical Physics Publishing, Madison.

Söderström S, Eklöf A and Brahme A, 1999. Aspects on the optimal photon beamenergy for radiation therapy. Acta Oncologica, 38:179�187.

Sheik-Bagheri D and Rogers DWO, 2002. Monte Carlo calculation of nine megavoltagephoton beam spectra using the BEAM code. Medical Physics, 29(3):391�402.

Svensson R, 1998. Development of a compact high energy treatment unit combiningnarrrow pencil beam scanning and multileaf collimation. Ph.D. thesis, Department ofRadiation Physics, Karolinska Institute, Stockholm.

Svensson R and Brahme A, 1998. E�ective source size, yield and beam pro�le frommulti-layered bremsstrahlung targets. Physics in Medicine and Biology, 41:1353�1379.

Vieira SC, Dirkx MLP, Pasma KL and Heijmen BJM, 2002. Dosimetric veri�cationof x-ray �elds with steep dose gradients using an electronic portal imaging device.Physics in Medicine and Biology, 48:157�166.

Watanabe T and Kuwano H, 1997. A microvalve matrix using piezoelectric actuators.Microsystem Technologies, 107�111.

Webb S, 2002. Intensity-modulated radiation therapy. Institute of Physics Publishing,Bristol.

Woias P, 2004. Micropumps - past, progress and future prospects. Sensors and ActuatorsB. In press.

Wong H, Morris S and Radke CJ, 1992. Three-dimensional menisci in polygonal capil-laries. Journal of Colloid and Interface Science, 148(2):317�336.

XCOM, 1999. http://physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html.

Xu T, Al-Ghazi MS and Molloi S, 2004. Treament planning considerations of reshapableautomatic intensity modulator for intensity modulated radiation therapy. MedicalPhysics, 31(8):2344�2355.

Xu T, Shikhaliev PM, Al-Ghazi M and Molloi S, 2002. Reshapable physical modulatorfor intensity modulated radiation therapy. Medical Physics, 29(10):2222�2229.

50 NOMENCLATURE

Nomenclature

η Dynamic viscosity

γ Surface tension or interfacial tension

µ Mass attenuation coe�cient

Φ Fluence or energy �uence

ρ Density

g Constant of gravity

P+ Biological objective/endpoint in radiation therapy de�ned as theprobability of complication free tumour control

Bixel Beam pixel

CT Computed Tomography

EPID Electronic Portal Imaging Device

Fluence Particles, e.g. photons, per unit area

FWHM Full Width at Half Maximum

Gauntry Treatment head

IMB Intensity Modulated Beam

IMRT Intensity Modulated Radiation Therapy

Isocenter The axis of rotation of the gauntry

ITV Internal Target Volume

Linac Linear accelerator

MLC MultiLeaf Collimator

Penumbra A measure of steepness of dose gradients, often de�ned as the lateral dis-tance between points of 20% and 80% dose

SMD Source to Modulator Distance

SSD Source to Skin Distance

masterofsciencethesis pontusnelldal stockholm2005/menu/general/column...abstract inthisthesis,...

Documents