IN DEGREE PROJECT MEDICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2017

Spectroscopic Study of Radiation around the Leksell Gamma Knife for Room Shielding Applications

ALEXIS HUBERT

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF TECHNOLOGY AND HEALTH

Spectroscopic Study of Radiation

around the Leksell Gamma Knife

for Room Shielding Applications

Spektroskopisk Studie av Strålning

runt Leksell Gamma Knife

för rumsavskärmningsapplikationer

Alexis Hubert

January, 2017

Supervisor: Per Kjäll

Reviewer: Massimiliano Colarieti-Tosti

Abstract

Any center planning to install a Gamma Knife radiosurgery unit hasto provide for an ecient shielding of the treatment room, to protectthe patient, the sta and the public, against undesired radiation.The shielding barrier design is controlled by national and inter-national recommendations; the reference documents for gamma rayradiotherapy facilities are the NCRPa reports 49 and 151. However,some facts highlighted in this thesis point out that NCRP methodsare ill-adapted to the Gamma Knife. Spectroscopic measurementswere performed around the Gamma Knife with a Germanium detec-tor. They revealed that the radiation eld contains few high energyphotons, is highly anisotropic, and that the leakage level is muchlower than the NCRP estimation. These observations led to the de-velopment of a new approach to determine the necessary shielding,based on the actual and directly measurable radiation eld aroundthe unit. This method would reduce the shielding oversizing in-duced by the unsuitability of the NCRP recommendations for theGamma Knife.

aNational Council on Radiation Protection and Measurements

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Acknowledgements

I would like to thank my supervisor, Per Kjäll, for providing theopportunity to work on such an interesting topic at Elekta, for hisguidance and his precious comments when writing this thesis. Mythanks also goes to Per-Erik Tégner for his interest in my work,his expertise, and for helping me throughout the rst part of theproject. Finally, I would like to thank my reviewer, MassimilianoColarieti-Tosti, and all the people that helped me in any way duringthe project.

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Table of Contents

1 Introduction 8

2 Materials and methods 10

2.1 Data acquisition around the Gamma Knife . . . . . . . . . . . . . . . . . . . . . . 102.1.1 Ambient dose equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 Spectroscopic measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Post-processing of spectroscopic data . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 Approximations on the response function . . . . . . . . . . . . . . . . . . 122.2.2 Determination of the response function . . . . . . . . . . . . . . . . . . . . 132.2.3 Unfolding of measured spectra . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Room shielding design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.1 Proposed model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Results 20

3.1 Response function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.1 Validation of the approximations on the response function . . . . . . . . . 203.1.2 Accuracy of the response matrix . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Dose load and energy distribution in the room . . . . . . . . . . . . . . . . . . . . 233.3 Shielding results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Discussion 26

5 Conclusion 30

A State-of-the-Art 32

A.1 Introduction to gamma radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.1.1 Radioactive decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.1.2 Gamma ray interactions with matter . . . . . . . . . . . . . . . . . . . . . 32

A.2 Shielding design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A.2.1 Dosimetric quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A.2.2 Dose limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.2.3 Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A.3 Measurements of radiation spectra with semiconductor detectors . . . . . . . . . 39A.3.1 Semiconductor radiation detector . . . . . . . . . . . . . . . . . . . . . . . 39A.3.2 Pulse height spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.3.3 Eciency and resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.3.4 Detector response function . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.3.5 Unfolding measured spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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B Materials and methods 48

B.1 Spectroscopic measurements conguration . . . . . . . . . . . . . . . . . . . . . . 48B.2 Coordinate system used for the measurements . . . . . . . . . . . . . . . . . . . . 50B.3 Adaptation of NCRP methods to the Gamma Knife . . . . . . . . . . . . . . . . 51

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Chapter 1

Introduction

Radiosurgery is a high precision surgery discipline that uses ionizing radiation to treat targetsin the body, without the need of any surgical incision. In order to precisely locate the targets,stereotactic methods are often used in combination with radiosurgery. The Swedish neurosurgeonLars Leksell invented the rst stereotactic device for brain surgery in 1949, at the KarolinskaInstitute [1]. Brain stereotactic radiosurgery is the operating principle behind the Gamma Knife,and Leksell's research led to the creation of Elekta in 1972.

Elekta's Gamma Knife consists of 192 sources1 of 60Co, whose beams are directed so as tomeet in a common isocenter point. Due to the intersection of the 192 beams in an isocenter andto the relatively sharp dose fall-o for single beams, a very high dose is delivered at the target,while the surrounding tissues are relatively spared. The patient's couch can be moved in relationto the isocenter to reach any point in the brain; and the shape and size of the focal volume canbe optimized thanks to a unique collimator, divided in 8 sectors. Each sector can be movedto dierent positions: `Home' is the position when the system is switched o, `4 mm', `8 mm'and `16 mm' correspond to the 3 possible treatment congurations, and `O' is an intermediaryshielding position when the system is on but the patient's couch is being moved.

Figure 1.1: The Gamma Knife collimator system, showing the sectors (double headed arrows)and the isocenter (central red dot) Image courtesy of Elekta

1Older version of the Gamma Knife system such as models B/C and 4C have 201 sources.

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As the Gamma Knife uses ionizing radiation, shielding is necessary to protect the patient,the sta, as well as the public present in areas adjacent to the treatment room. The referencedocuments to design structural shielding for gamma-ray radiotherapy facilities are the NCRP2

Reports 49 and 151. The general methodology is briey described in appendix A.2.3 and is basedon the determination of primary barriers to attenuate the photon beam emanating directly fromthe treatment unit, and secondary barriers to protect against radiation leakage and scatteredphotons by the patient and objects in the room. However, these guidelines cover a wide rangeof x- and gamma-ray radiotherapy devices, and some of the assumptions made are rather ill-adapted to the specics of the Gamma Knife. The NCRP assumes that radiation originatesfrom a single source, is isotropic, only composed of primary photons of 1.25 MeV (mean energyof the two 60Co gamma-rays), and that leakage radiation is 0.1% of the useful beam. In fact,the Gamma Knife houses massive casing and doors which shield most of the radiation from the60Co sources, and the leakage level is much lower than the estimated 0.1%. The design of theGamma Knife itself also prevents any of the primary beams to exit the unit without passingthrough the internal shielding, reducing the amount of primary photons in the treatment room.Moreover, no indication on how to sum the contribution of the 192 sources is provided by theNCRP, and the radiation eld resulting from these nearly 200 sources contained within a mas-sive shielding case is highly anisotropic. The NCRP guidelines therefore seems unsuitable to theGamma Knife system, and motivates the development of a new way to design structural shielding.

Figure 1.2: The Leksell Gamma Knife, Perfexion Image courtesy of Elekta

A rst objective of this thesis was to analyse the energy distribution of radiations in thetreatment room using spectroscopic measurements. Elekta had, together with the Instrumenta-tion Physics Division at Stockholm University, acquired a number of gamma-ray spectra aroundthe Gamma Knife for various states of the system. The spectra were acquired with the aid ofa Germanium radiation detector. These measurements had to be post-processed in order to getthe true energy distribution of the photons. If less than 50% of the photons emanating fromthe Gamma Knife are primary photons, the development of a new method to determine thenecessary shielding would be justied, and was the nal goal of this thesis.

2National Council on Radiation Protection and Measurements

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Chapter 2

Materials and methods

2.1 Data acquisition around the Gamma Knife

2.1.1 Ambient dose equivalent

Ambient dose equivalent were acquired around the Gamma Knife with a 6150AD-18 surveymeter (Automess, Germany). The energy range of 5 keV to 1.3 MeV was suitable for 60Co asno energy would exceed 1.33 MeV (highest photopeak). Figure (2.1) is a top view of the set-up.The dierent positions of the survey meter are marked with black dots. The measurements weredone without the plastic cover of the Gamma Knife, explaining why they seem to intersect inthe gure. The measurements have been interpolated to get a 11 m × 12 m rectangular map ofthe dose rate around the Gamma Knife.

Figure 2.1: Dose rate acquisition. The black dots are the positions where the dose rate wasacquired around the Gamma Knife, which is represented in grey. In the blue area, the dosewas linearly interpolated using triangulation methods, and it was radially extrapolated using thenearest value outside. The X and Y axis are dened in appendix, gure B.1

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2.1.2 Spectroscopic measurements

Ambient dose equivalent is an integrated dose quantity, and doesn't convey any information onthe energy of the photons contributing to the dose. To remedy this deciency, spectroscopicmeasurements were performed last January (2016), with a GC2020 high-purity germanium coax-ial detector (Canberra, USA). The detector is basically a germanium cylinder, whose diameter is5.35 cm and length is 5.9 cm. With the detector, a 7600SL Cryostat and a 2002CSL preamplier(Canberra, USA) were used. Such kind of photon counting detector is briey described in thestate-of-the-art A.3.1.

The spectra recorded with the detector are composed of 8192 channels. The energy calibra-tion was done using the two photopeaks of 60Co, as well as two reference sources 241Am and137Cs, whose energies are known. Channels and energies are matched using a linear regressionas shown in gure (2.2).

Figure 2.2: Energy calibration. The calibration spectrum is displayed in (a) with an upperabscissa-axis corresponding to energy, and a lower abscissa-axis to channels. The correspondencebetween both axis comes from the calibration curve in (b).

To study the radiation eld around the Gamma Knife, the detector was moved in the roomin dierent positions see gure (2.3). Besides varying distance and angle from the isocenter,dierent states of the Gamma Knife were also tested: the doors were alternatively open or closed,the patient's couch was moved inside and outside the unit, a phantom head was sometimes placedon the couch, and the collimator sectors were moved in the dierent positions (home, o, 4 mm,8 mm, 16 mm). The conguration for all the measurements is summed-up in a table in appendixB.1.

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Figure 2.3: Positions of the dierent measurements performed near the Gamma Knife. The Xand Y axis are dened in appendix, gure B.1

2.2 Post-processing of spectroscopic data

2.2.1 Approximations on the response function

The response of a detector subjected to a photon ux is modelled by its response function,introduced in appendix A.3.4. Two main approximations were made on this response function:

1. The response of the detector is independent on the distance between the source and thedetector.

2. The impact of the incident angle of photons on the response function is negligible.

In order to verify the correctness of these approximations on the incident angle and distance,measurements were done in the low-level lab in AlbaNova University Center. The walls, ceilingand roof of the lab are made of a 5 cm iron lining plus 60 cm of low-level concrete to shieldradiations from terrestrial sources. In addition to 60Co, other accurately calibrated radioactivepoint sources were used: Americium (241Am) and Cesium (137Cs) were selected for having onlyone gamma-ray 59.54 keV and 661.66 keV respectively in the range of interest. Yttrium (88Y)is not mono-energetic but has two relatively spaced rays of 898.04 keV and 1836.06 keV, and wasmostly chosen for having a higher energy ray than 60Co.

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To check the impact of the travelled distance before a photon reach the detector, the radialdistance d between the source and the detector was changed from 50 cm to 100 cm. As thegeometric eciency1 is 4 times lower when the distance is doubled, the measurements wereperformed during 2 hours at 50 cm and 8 hours at 100 cm to get comparable results. For theapproximation on the angle, the radial distance d between the source and the detector was keptconstant at 50 cm, and 5 angles were tested:

Angle θ 0 45 90 135 180

241Am × × × × ×137Cs × × × × ×60Co × × ×88Y × × ×

When the incident angle is 0, the distance travelled by photons through the detector is5.9 cm, corresponding to the length of the germanium cylinder. When photons reach the detectorfrom its side, they travel a maximum distance equal to the diameter (5.35 cm) through thegermanium. Thus, the incident angle of a photon has an impact on its probability of interactionin the detector, but this impact is found to be negligible see section 3.1.1.

Figure 2.4: Experimental setup, showing the distance d and the angle θ between the source andthe detector.

2.2.2 Determination of the response function

The only three terms composing the response function retained in this study are the intrinsiceciency, the energy resolution and the energy distribution spectrum. The method to determinethese terms and build the nal response function is described in this section.

1The geometric eciency is the ratio of the number of photons emitted towards the detector by the numberof photons emitted by the source.

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Intrinsic eciency

The intrinsic eciency plotted in gure (2.5) is calculated from the Beer-Lambert law see equa-tion A.21 in appendix. The linear attenuation coecient comes from the NIST2 database, andthe distance travelled through the detector is considered to be 5.9 cm (length of the germaniumcylinder). The underlying hypothesis is that all photons arrive perpendicular to the front face ofthe detector. For most of the measurements around the Gamma Knife, the detector was located5 meters from the isocenter, and the majority of photons probably enter through the front faceof the germanium cylinder. When the detector is only a few centimetres away from the GammaKnife cover, photons scattering in the case are more likely to reach the detector from the side ofthe cylinder. However, considerations on the incident angle of photons are disregarded in thisstudy.

Figure 2.5: Intrinsic eciency of the detector.

Energy resolution

The energy resolution is introduced in A.3.3.1 and was measured in AlbaNova by tting a Gaus-sian function on the data, after subtracting a linear background. The method is shown in gure(2.6) and was applied to all the measurements done at dierent angles, heights, distances. Theresults are presented in table (2.1) as the mean Full Width at Half Maximum (FWHM) for eachphotopeak and uncertainty (standard deviation).

Source 241Am 137Cs 60Co 88Y

Energy (keV) 59.5 661.7 1173.2 1332.5 898.0 1836.1FWHM (keV) 1.18 1.55 1.80 1.89 1.67 2.11

Uncertainty on FWHM ± (keV) 0.01 0.02 0.02 0.01 0.01 0.04

Table 2.1: Measured energy resolution

2National Institute of Standards and Technologyhttps://www.nist.gov/pml/x-ray-mass-attenuation-coefficients

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Figure 2.6: Determination of the energy resolution of the 59.5 keV peak from 241Am by tting aGaussian function on the experimental data.

The data points in table (2.1) were interpolated using a second order polynomial. Theinterpolated resolution function is plotted in gure (2.8). It can be seen that the resolution isalways higher than 1 keV. The spectra acquired with the germanium detector have 8192 channelsand one channel corresponds roughly to 0.4 keV. The data can thereby be re-sampled down tolarger energy bins without damaging the resolution.

Figure 2.7: Interpolated energy resolution of the detector

Energy deposition spectrum (EDS)

The energy deposition spectrum (EDS) is estimated using Monte Carlo methods. It was simu-lated every 50 keV for energies ranging from 150 keV to 1400 keV, and every 5 keV for energiesranging from 20 keV to 100 keV. For each simulation, no wall, ceiling or roof were included, and100 millions particles were emitted toward the detector. In accordance to the energy resolution,1 keV wide energy channels were used, meaning 100 bins for the simulation at 100 keV and 1400bins for the simulation at 1400 keV.

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Deriving the response matrix

To assemble the response function matrix from the intrinsic eciency, the energy resolution andenergy deposition, the following method was used:

1. Contrary to the Monte Carlo simulated EDS, which have rather good statistics, the ex-perimental data are too noisy for the escape peaks or escape photons to be visible. Thecorresponding peaks are therefore removed from the EDS.

2. A Gaussian broadening is applied to all channels of the simulated EDS to account for theexperimental energy resolution.

Figure 2.8: Example of Gaussian broadening for the 1150 keV energy channel, containing Ncounts. The simulated spectrum has a perfect resolution (a) and a Gaussian broadening isapplied (b) to match the actual energy resolution of the detector.

3. The now broadened EDS are interpolated for all the energies that were not simulated, butrequired for the response matrix. For example, a spectrum at 1110 keV EDS would beinterpolated from the ones at 1100 keV and 1150 keV (EDS were simulated every 50 keVfor energies ranging from 150 keV to 1400 keV). The interpolation method is explained ingure (2.9).

Figure 2.9: Interpolation method. The Compton continuum is interpolated between channelscorresponding to the same scattering angle (orange). From the Compton edge to the photopeak,the spectra are interpolated on a channel-by-channel basis (blue). The photopeaks are directlyinterpolated (yellow).

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4. The obtained spectra, after being normalized to the intrinsic eciency corresponding totheir energies, are assembled in column in the response matrix. Thus, the sum of the1200 keV column's coecients is about 0.8, meaning that an incident photon of 1200 keVhas 80% change of being detected by the detector.

2.2.3 Unfolding of measured spectra

If the energy resolution is of importance, the channels must be rather small. With a bin widthof 1 keV, the spectra to unfold have more than a thousand bins. Optimization methods suchas the Tikhonov or entropy deconvolution are computationally too expensive with a thousandvariables to be conceivable. In that case a folding iterative method such as the one described inthe state-of-the-art A.3.5.3 seems the best option.

In this study, the spectral data were used to calculate the wall thickness needed in treatmentrooms. The suggested model is presented in the next section 2.3.1, and the experimental datawere re-sampled in 10 keV wide bins, ranging from 40 keV to 1400 keV i.e. 137 bins. The usualapproach consists in using only one photon energy of 1.25 MeV. The proposed intermediate so-lution between 1 bin and more than a thousands seemed a realistic approach for this study.

The nal method selected to unfold our experimental spectra is a L2 regularization (Tikhonovmethod). As there are statistical uncertainties in the measurement, a certain amount of freedomin the tting of the experimental data is allowed to prevent any over-tting. The objective of themethod is to provide the smoothest solution compatible with the experimental data and expecteduncertainties. The smoothest solution is found by minimizing ||T ′||2 and the compatibility isinsured with the following constraints:

∑i12

((M−RT )i

σi

)2≤ 137 i = 1..137∑

i12

((M−RT )i

σi

)2= 0 i ∈ Photopeaks

Ti ≥ 0 i = 1..137

(2.1)

With:

M Measured spectrum (experimental data) T True spectrum (the one being unfolded) R Response matrix σ Estimated statistical uctuations on M , set equal to the square-root of the number ofcounts.

The rst constraint is the χ2 statistic. The second constraint was introduced to prevent anysmoothing of the photopeaks, and the last constraint ensures the positivity of the deconvolvedspectrum. Figure (2.10) is an example of regularization.

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Figure 2.10: Inverse vs Tikhonov deconvolution.

2.3 Room shielding design

2.3.1 Proposed model

A new shielding design approach based on dose rate and spectroscopic data is suggested in thissection as an alternative to the NCRP methods. In this new method, three states of the GammaKnife are taken into account:

1. O : The Gamma Knife is in a an idle state, the collimators are in `home' positions to blockradiations and the doors are closed. All the radiation in `O' mode arises from leakagethrough the shielding case of the Gamma Knife.

2. Trans: The transition time corresponds to the time where the patient is lying on the bed,and moved in and out of the treatment position. During the transition time, the collimatorsare in the intermediate `O' position between the 4 mm and 16 mm positions.

3. On: Treatment time. The patient is in treatment position, and the collimators are eitherin 4 mm, 8 mm, or 16 mm positions. As the dose load is higher for the 16 mm position,we will consider any treatment to be performed in this conguration (worst-case scenario).

Let's dene D(P ) the dose load at the point protected P (X,Y ) where X and Y are thecoordinates of point P . D(P ) is determined using the dose rate data D(P ) from 2.1.1 and theestimated time T for the o, trans and on states, in hours per week.

D(P ) =

Doff

Dtrans

Don

=

Toff · Doff

Ttrans · Dtrans

Ton · Don

From all the unfolded spectra, a contribution-to-the-dose fraction matrix C(P ) can be derived.

The coecient c1 i is the contribution of channel i to the dose in o state. The columns of C(P )thereby correspond to the contribution of the dierent energy bins of the unfolded spectra, andthe rows to o, trans, o states.

C(P ) =

c1 1 c1 2 . . . c1 137c2 1 c2 2 . . . c2 137c3 1 c3 2 . . . c3 137

← off← trans← on

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The contribution of each energy bin to the total dose load is calculated with the productDTX. The dose of each energy channel is attenuated dierently depending on the Half-ValueLayer (HVL) for the corresponding energy. Let hi the HVL of bin i, and t the required wallthickness to reach the shielding goal. We can dene a vector H corresponding to the wallattenuation for the dierent energy bins:

H =

2− t

h1

2− t

h2

...

2− t

h137

The design goal DG is met if:

DTCH < DG (2.2)

This problem can be solved easily as an optimization problem: nd the minimum wall thick-ness t that satisfy the constraint in equation (2.2).

2.3.2 Testing

The proposed method was tested on a 6.5 m × 4.5 m room. A working week is 40 hours (5 daysof 8 hours), corresponding to the maximum time a person can be present outside the treatmentroom. In this application, it was considered that 3 patients a day undergo radiosurgery with theGamma Knife, leading to 15 patients per week, with an average treatment time of 12 minutes3

per patient. The transitioning time (accounting for moving the couch in and out of the treatmentposition, and switching between collimator settings) is assumed to be 3 minutes per treatment.The leakage is always present i.e. 40 hours a week. The shielding design goal DG used is theone recommended by NCRP: 0.02 mSv/week for an unrestricted area (to achieve the 1 mSv/yearlimit). The times in the dierent phases o, trans, on per week therefore are:

Toff = 40 hTtrans = 3 min× 15 patients = 0.75 hTon = 12 min× 15 patients = 3 h

The dose rates and fraction matrix results presented in section 3.2 were used for D and C.The required thickness to achieve the design goal is calculated for every point outside the room(or actually, every 2 cm). As the wall cannot have a varying thickness, the maximum thicknessis to be used to build the barrier.

3This average time is for a newly loaded Gamma Knife, with a total activity of 6600 Ci

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Chapter 3

Results

3.1 Response function

3.1.1 Validation of the approximations on the response function

Source-detector distance

The rst approximation was the independence of the response function on the distance betweenthe source and the detector. A few measurements were done to verify the correctness of thisstatement using two distances. The spectra obtained at 50 cm and 100 cm are superimposedin gure (3.1), and some disparities can be seen at low energy. The maximum dierence in thenumber of counts in a channel is about 40% for the three sources, and located at the backscat-tering peak. Here is a possible explanation for these disparities at low energy: the source at100 cm being closer to the back wall of the room, the number of backscattered photons reachingthe detector is increased, modifying the energy distribution measured by the detector.

Figure 3.1: Impact of the source-detector distance for 137Cs (a), 60Co (b) and 88Y (c). Thespectra at 50 cm are plotted in blue, and the ones at 100 cm in orange. The backscatteringpeaks where the maximum dierence is are indicated with green arrows.

Incident Angle

The second approximation was that the incident angle of photons has a negligible impact onthe response function. Out of the measurements performed with 5 dierent angles to check thisapproximation, the ones at 180 were directly rejected for contrasting with the other measure-ments. An angle of 180 means the detectors is turned away from the source. Photons enterthe detector from its back side, and they may scatter in the cover, electronics, etc. producing apeculiar spectrum. The spectra for the remaining angles, normalized to the number of counts aresuperimposed in gure (3.2). Normalizing prevents any disparity in the number of counts dueto variations in the geometric eciency: the detector being cylindrical, the plane frontal area

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visible at the source is circular for 0and 180, and rectangular at 90. Moreover, the positioningof the detector is not so accurate as it was done manually, and the source-detector distance mayvary of a few millimetres due to this imprecision.

Figure 3.2: Impact of the angle between the source and the detector for 241Am (a), 137Cs (b),60Co (c) and 88Y (d). Angle 0 is in blue, 90 in red, 45 in green and 135 in yellow. The countsper channel were normalized to have the total number of counts equal 1.

To quantify the impact of the angle on the response function, the variation in the number ofcounts per channel in the region of interest (from 40 keV to the photopeak) has been studied.Both the standard deviation per channel due to the changing angle, and statistical uncertaintieswere calculated, and are represented as boxplots in gure (3.3). These boxplots show that thevariation in the number of counts in a channel for dierent angles is less than 8% for 75% ofthe channels, which is in the same order of magnitude as the uncertainty in the number ofcounts in a channel (5.6% in average). Table (3.1) gathers the mean standard deviation, and themean statistical uncertainty in the number of counts per channels for the four sources. Except for241Am, no inuence of the angle on the response function is noted as the dierence in the numberof counts may be explained by statistical uctuations due to the variability of radioactive decayand photon counting. The approximation that the incident angle of photon does not impact theresponse of the detector therefore seems justiable.

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Figure 3.3: Variation in the number of counts due to the angle vs. statistical uncertainties for137Cs. Boxplot (a) represents the distribution of the standard deviation per channel (in %) forthe four angles, and boxplot (b) the statistical uncertainty per channel (in %).

Source 241Am 137Cs 60Co 88Y

Mean standard deviation (%) 20.3 6.5 5.4 11.5Mean statistical uncertainty (%) 4.0 5.6 5.3 13.2

Table 3.1: Dispersion of measurements due to the changing angle, and statistical uncertaintiesfor the four sources.

3.1.2 Accuracy of the response matrix

To evaluate the accuracy of the response matrix, simple spectra of 137Cs and 60Co, measuredin the low level lab in AlbaNova, were compared to their theoretical spectra. The theoreticalspectra are composed of one photopeak for 137Cs or two photopeaks for 60Co, plus a backgrounddue to photons scattering in the wall, ceiling, oor. In gure (3.4), the experimental spectraare compared to the folded theoretical spectra i.e. R× Ttheory, where R is the response matrix.The Pearson's correlation coecients were calculated to assess the similarity between theoryand measurement, and found to be 0.9960 for 60Co and 0.9988 for 137Cs. Both coecients arevery close to 1, meaning an excellent matching. It appears the response matrix describe quiteaccurately the response of the detector, and provides a sound tool to fold or unfold spectra.

Figure 3.4: Comparison of measured and theoretical spectra for 137Cs (a) and 60Co (b)

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3.2 Dose load and energy distribution in the room

The dose rate maps for the three states of the Gamma Knife (idle, transitioning, treatment) areshown in gure (3.5). The highest dose rate is logically present during treatment, in front of theGamma Knife, at an angle θ of approximately 30. On the contrary, when the system is o, themaximum leakage appear to be behind the Gamma Knife.

Figure 3.5: Dose load in the room for o, trans, on states in µGy/h. The dose is given in theXY plane at 1 m from the oor see appendix B.1 for the coordinate system used.

The contribution-to-the-dose fraction matrix X is displayed in gure (3.6). A curve repre-sents the fraction (in %) of the dose due to the counts recorded in channel of energy e, e beingthe abscissa of the graph.

Figure 3.6: Relative contribution to the dose (in %) in the front area (a) and behind (b) theGamma Knife

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The dose rate at the back of the Gamma Knife is essentially composed of leakage radiationthrough the cover, and does not vary much between treatment and rest state. The proportionof high energy photons contributing to the dose is however higher at the back than the front ofthe Gamma Knife. During treatment in the front area, 81% of the dose is composed of photonswith an energy lower than 1150 keV i.e. less than 20% of the dose is due to primary photons.

3.3 Shielding results

Figure 3.7: Required thickness to achieve a shielding goal of 0.02 mSv/year outside the concretewalls. The dotted line is the calculated wall thickness for every point outside the room, and theplain line is the maximum thickness on six dierent wall sections.

24

Chapter 4

Discussion

Spectroscopic measurements showed that, during treatment, 81% of the dose in the area in frontof the Gamma Knife is composed of photons with an energy lower than 1150 keV. Less than 20%of the dose is due to primary photons, and the NCRP assumption that the dose load is onlycomposed of primary photons appears to be rather incorrect. A new approach to design shieldingbarriers that includes considerations on the photons energy has been suggested in this report1,as an alternative to the NCRP reference method. The NCRP general method is described in thestate-of-the-art A.2.3. An adaptation of the general equations to the specic case of the GammaKnife is appended in B.3, and was applied to the same room used in 2.3.2. Results for bothmethods are compared and discussed in this chapter.

Figure 4.1: Concrete thickness required to shield radiation during treatment, for every pointoutside the treatment room (a) and nal thickness value (maximum) selected for the wall (b).The red line corresponds to the NM, NCRP thickness is in blue, and standard Gamma Knifesite planning in green.

1The method suggested in this report is referred as the `New Method' (NS) in this section.

26

During treatment, the patient is inside the unit, and all beams intersect in the intracranialtarget volume. The radiation eld in the room is essentially composed of scattered radiationfrom the patient, and the dose rate at the barrier was determined using equation (B.1). Theworkload is 600 Gy/week (15 patients, with a maximum dose delivered per patient of 40 Gy), andthe occupancy factor is 0.075, corresponding to three hours of treatment per week. As shown ingure (4.1) above, the NCRP method results in slightly smaller barrier thickness than the NM:the front wall is 20% smaller and the side wall 7% smaller. An explanation for the dierence isthat scattered radiation in the wall, the cover of the Gamma Knife, etc. also contribute to thedose load during treatment, but are not taken into account in the NCRP calculation.

The barrier transmission for leakage when the Gamma Knife is o was estimated using equa-tion (B.2). The resulting wall thickness is displayed in gure (4.2). According to NCRP method,more than 50 cm of concrete is required to shield 40 hours of leakage a week, when only a quar-ter of this thickness is indicated by the NM. There is no venture in saying that NCRP methodprobably overestimates the leakage level around the Gamma Knife. The factor 10−3 in equation(B.2) arises from the assumption that leakage radiation is 0.1% of the useful beam. Contraryto other x- and gamma-ray radiotherapy devices, massive casings and doors are present in theGamma Knife, and the leakage when the unit is o is very low. Even if a lower factor than 0.1%could be used in the calculation, it would still be rather incorrect to estimate the leakage as afraction of the workload. Ambient dose equivalent and spectroscopic measurements indeed showthat the leakage distribution vary considerably around the Gamma Knife, with the highest doseload behind the unit. Both the assumptions of an isotropic leakage of a fraction of the workload,and inverse square law used in the NCRP leakage model are experimentally not veried.

Figure 4.2: Concrete thickness required to shield leakage for every point outside the treatmentroom (a) and nal thickness value (maximum) selected for the wall (b). The red line correspondsto the NM, NCRP thickness is in blue, and standard Gamma Knife site planning in green.

27

To overcome the unsuitability of the NCRP methods for the Gamma Knife, Elekta providesmatrix of dose rates around the Gamma Knife during treatment and in the o state in its siteplanning guide [2]. These data are to be used by centres installing a Gamma Knife for a betterestimation of the dose load and leakage around the unit than with NCRP purely geometricconsiderations. A standard way to use the o state data to compute leakage barrier is to replacethe dose load 10−3W

d2by its actual value provided in the matrix. The wall thickness calculated

with this method is shown in green in gure (4.2). This method is derived from NCRP anddoes therefore not consider the energy distribution of photons in the room. Due to the massiveinternal shielding in the Gamma Knife, most of the leakage photons scatter at least once whileescaping the unit, and have a lower energy than 1.25 MeV (the standard value recommended byNCRP). The new method suggested in this report couples the dose load data with spectroscopicmeasurements, and shows that reducing the wall thickness of about 20% see gure (4.3) compared to the standard site planning method would still provide a sucient protection tomeet the dose limit recommendation.

28

Wall New Method (cm) Standard SP (cm) Reduction (%)

Wall 1 21.4 26.9 20.5Wall 2 26.8 33.1 19.1Wall 3 17.4 23.1 24.7Wall 4 24.5 28.7 14.7Wall 5 15.6 20.8 24.9Wall 6 20.6 29.0 20.7

Figure 4.3: Comparison of barrier thickness resulting from standard site planning method (green)and the NM (red) for wall 1 to 6.

29

Chapter 5

Conclusion

The aim of this thesis was to analyse the energy distribution of radiation around the LeksellGamma Knife using spectroscopic measurements, in order to devise a new model to designshielding barriers for treatment rooms. A response matrix was derived to unfold the experi-mental spectra acquired around the Gamma Knife, and shown to give satisfactory results. Theenergy distribution of photons obtained after the unfolding process conrmed that few highenergy photons are present in the treatment room, and less than 20% of the dose is due toprimary photons during treatment. The NCRP recommendations to design structural shieldingfor gamma-ray radiotherapy are ill-adapted for the Gamma-Knife, and an alternative methodhas been developed in this thesis. The new suggested method sizes the wall based on the actualradiation eld measured in the room, rather than purely geometric considerations. The casingsand doors of the Gamma Knife providing an ecient protection, the leakage measured in theroom is much lower than the one estimated using NCRP principles. This new method therebyenables treatment rooms to be designed with much less shielding material, reducing costs, andincreasing the number of suitable locations for a treatment room in a hospital.

30

Appendix A

State-of-the-Art

A.1 Introduction to gamma radiation

A.1.1 Radioactive decay

During a radioactive decay, an unstable atomic nucleus transforms into another more stablenucleus, emitting particles and energy in the process. Dierent types of radioactive decay existbut only the β− is briey introduced here, as being the decay of 60Co the radioactive isotope ofcobalt used in the Gamma Knife. In β− decay, a neutron-rich isotope converts one of its neutroninto a proton, and emits an electron and an antineutrino. In the case of 60Co, the decay processis written schematically:

6027Co→ 60

28Ni∗ + e− + ν

After the decay, the daughter nucleus is in an excited state 60Ni∗ and is transitioned to lower-lying nuclear levels with the emission of two gamma rays of 1.17 MeV and 1.33 MeV. The totaldecay energy is 2.8 MeV corresponding to the mass dierence between 60Co and 60Ni. The decayscheme is presented in gure (A.1) below.

Figure A.1: Decay scheme for 6027Co Simplied from [3]

A.1.2 Gamma ray interactions with matter

Gamma-ray photons such as the ones emitted during radioactive decay, may interact with matterwhen travelling through the surrounding environment. In the range of energy of 60Co, threemain kinds of interactions are possible: Photoelectric absorption, Compton scattering and PairProduction they are presented in the following sections. The signicance of these eects ischaracterized by the interaction cross section1 σ, and depends on the energy E of the gammaray and the atomic number Z of the material.

1The interaction cross section is an eective area that can be interpreted as a probability for interaction.

32

A.1.2.1 Photoelectric absorption

During photoelectric absorption, a photon is absorbed by an atom and a photoelectron is emittedwith a kinetic energy Ee− [4]:

Ee− = E −B (A.1)

With:

E energy of the incident photon B binding energy of the electron to its original shell, which is emitted as characteristicX-rays (or Auger electron).

Photoelectric absorption dominates at low energies, and its cross section is [4]:

σP ∝ Zn

(E)3(A.2)

With:

n coecient varying between 4 and 5 Z atomic number E energy of the incident photon

A.1.2.2 Compton scattering

In Compton interaction, a gamma photon of energy E is deviated by an atomic electron. Theincident photon is absorbed and re-emitted in the direction Ω = (θ,Φ) as a secondary photon ofenergy E′, transferring some of its energy to the electron in the process. The relation betweenE′ and θ is [4]:

E′ = E1

1 + Emec2

(1− cos θ)(A.3)

The angular distribution is described by the Klein-Nishina Formula [5]:

dσC

dΩ=

1

2r2e

(E′

E

)2( EE′

+E′

E− sin2(θ)

)(A.4)

With:

re classical electron radius value θ polar angle of scattering Φ azimuthal angle E energy of the incident photon E′ energy of the secondary photon

The cross-section for the Compton scattering phenomenon can be calculated by integratingthe Klein-Nishina formula over Ω, and is proportional to Z and E−1 [4]:

σC ∝ Z

E(A.5)

However, the Klein-Nishina formula is derived for free electrons at rest, and is a roughapproximation of Compton scattering interactions. In reality, electrons are moving with a certainmomentum, which gives rise to the Doppler broadening of the Compton line. Some more complexmodels exist and are used in recent Monte Carlo transport codes [6].

33

A.1.2.3 Pair Production

In Pair Production, a photon annihilates to create an electron-positron pair near a nucleus. Thisphenomenon is possible when the gamma-ray energy E exceeds twice the rest-mass energy of anelectron 2mec

2 = 1022 keV. Pair production is thereby a high energy phenomenon.

The energy 2mec2 is required to create the pair, the rest of the energy is shared in the form

of kinetic energy by the electron and positron [4]:

Ee+ + Ee− = E − 2mec2 (A.6)

The positron is an unstable particle and will combine quickly with an electron, producingtwo annihilations photons of energy mec

2.

The cross section for pair production is σpp ∝ Z2 ·f(Z,E) [4] where f(Z,E) is some complexfunction that depends on the energy and atomic number.

34

A.2 Shielding design

A.2.1 Dosimetric quantities

Fluence and energy uence

The uence Φ and energy uence Ψ are useful quantities to dene before introducing any otherdosimetric quantity. The photon uence is the number of photons passing through a unit cross-sectional area; and the energy uence is the amount of energy passing through this unit area [7].The photon uence and energy uence are integrated quantities (Eq. A.8). When photons ofdierent energies are present (poly-energetic source, scattered mono-energetic beam, etc.) onemay prefer considering the distribution of uence ΦE and energy uence ΨE with respect toenergy instead [8]:

ΦE =dΦ

dEΨE =

dΨ

dE(A.7)

And:

Φ =

∫ΦE dE Ψ =

∫ΨE dE (A.8)

Kerma

The Kerma K is an acronym for Kinetic Energy Released in MAtter. It is dened as the kineticenergy transferred to charged particle (such as electrons) by the photons [7] and expressed inthe unit gray (Gy) corresponding to Jkg−1. The Kerma is usually calculated from the energydistribution of the energy uence [8]:

K =

∫ΨE E

µtrρdE (A.9)

Where µtrρ is the mass energy transfer coecient. This coecient depends on the material and

the photon energy, and it characterizes the amount of energy transferred to charged particle. Ifdoes not include the energy of scattered photons escaping the interaction site as their energy isnot transferred to charged particles.

Doses

The absorbed dose D is the quotient of dε by dm, where dε is the mean energy imparted byionizing radiation to matter of mass dm [9]:

D =dε

dm(A.10)

Same as the Kerma, the absorbed dose can be derived from the energy distribution of the energyuence:

D =

∫ΨE E

µenρdE (A.11)

µenρ is the mass energy absorption coecient and diers only slightly from µtr

ρ . Not all the energytransferred to charged particles will be locally absorbed in the matter. Indeed, electrons mayproduce x-rays (bremsstrahlung radiation) that might escape the interaction site, inducing a lossin the absorbed energy compared to the transferred energy. If the amount of bremsstrahlungloss is negligible, the absorbed dose and Kerma are equal.

35

The ICRP2 has established weighting factors to take into account the biological impact ofradiations, they are presented in gure (A.2). The equivalent dose HT for a specic tissue T dueto radiations of type R [9] is dened as:

HT =∑R

wRDT,R (A.12)

Radiation type Radiation weighting factor wRPhotons 1Electrons 1Protons 2Alpha particles, heavy ions 20Neutrons A continuous function of neutron energy

Figure A.2: Radiations weighting factors adapted from ICRP 103 [9]

The eective dose received by the whole body is dened as a weighted sum of the equivalentdoses for all the dierent organs and tissues. The tissue weighting factors determined by theICRP are in gure (A.3).

Deff =∑T

wTHT (A.13)

Tissue wT ΣwTBone-marrow, Colon, Lung, Stomach, Breast, Remainder tissues 0.12 0.72Gonads 0.08 0.08Bladder, Oesophagus, Liver, Thyroid 0.04 0.16Bone Surface, Brain, Salivary glands, Skin 0.01 0.04

Total 1.00

Figure A.3: Tissue weighting factors adapted from ICRP 103 [9]

The special name for the unit of equivalent dose and eective dose is the Sievert (Sv), where1 Sv = 1 Jkg−1. For gamma rays, as wR = 1, the absorbed dose and equivalent dose are equaland 1 Gy = 1 Sv.

A.2.2 Dose limits

The limits for the body-related protection quantities equivalent dose and eective dose rec-ommended by the ICRP are presented in gure (A.4). The Swedish Radiation Safety Authority Strålsäkerhetsmyndigheten follows exactly the ICRP's recommendations [10] but other nationalauthorities requirements for radiation protection may dier from ICRP's.

Type of limit Personnel Public

Eective dose 20 mSv per year 1 mSv per yearAnnual equivalent dose in:- Lens of the eye 150 mSv 15 mSv- Skin 500 mSv 50 mSv- Hands and feet 500 mSv

Figure A.4: Recommended dose limits adapted from ICRP 103 [9]

2International Commission on Radiological Protection

36

A.2.3 Shielding

The reference document for shielding design of Gamma Knife facilities is the Report No. 151of the U.S. National Council on Radiation Protection and Measurements [11]. As treatmentcentres must comply with national authorities requirements, they don't have much room formanoeuvre, and very few publications on radiation shielding for the Gamma Knife can be foundin the literature. However, a publication from P.N. McDermott suggests a special method toadapt the general principles outlined in Report No. 151 to the case of the Gamma Knife [12].

A.2.3.1 Transmission factors

The basic concept to determine barriers thickness from Report No. 151 is based on the computa-tion of transmission factors for all the area outside the treatment room [11]. The primary beamis composed of photons emanating directly from the treatment unit. The transmission factor ofthe primary barrier is dened as equation (2.1) in [11] :

Bpri =DG d

2L

WUT(A.14)

Secondary barriers are to be designed to protect individuals against secondary radiation re-sulting from scattering of the primary beams by the patient and surfaces in the room, plusradiation leakage through the case of the Gamma Knife.

Barrier transmission factor for scattered radiation equation (2.7) in [11]:

Bsca =DG

aWTd2scad

2sec

400

F(A.15)

Barrier transmission factor for leakage radiation equation (2.8) in [11]:

BL =DGd

2L

10−3WT(A.16)

With:

Bpri transmission factor of the primary barrier DG shielding design goal expressed as a dose equivalent3 beyond the barrier dL distance from the source to the point protected dsca distance from the source to the scattering object (patient or surface) dsec distance from the scattering object to the point protected W workload weekly absorbed dose from photons delivered at the isocenter (due to patienttreatment, quality and control checks, calibrations, etc.)

U use factor fraction of the workload directed toward the barrier being designed T occupancy factor fraction of the workweek that a person is present beyond the barrier a fraction of the primary-beam that scatters from the object at a particular angle F Field area at mid-depth of the patient at 1 m

A.2.3.2 Wall thickness

The thickness t of a barrier is calculated using its transmission factor B, and the tenth-valuelayer TV L for the material used. The TV L of a material corresponds to the thickness requiredto absorb 90% of the incident radiation. For ordinary concrete and 60Co sources, the TVL isaround 21 cm [11].

t = − log(B)× TV L (A.17)

3Out-of-date quantity similar to the equivalent dose, but using a quality factor instead of weighting factors

37

At this point, it is important to note that the TV L is dened for a material and a specicenergy. For 60Co sources, the average energy of 1.25 MeV between the two gamma rays of 1.17MeV and 1.33 MeV is usually used. That is assuming all the photons contributing to the doseload have either an energy of 1.17 MeV or 1.33 MeV, and in the same proportions. In practice, theambient dose equivalent, H*(10), measured with dosimeters, is used to estimate the dose load.This operational quantity is an integrated quantity that measures both primary and secondaryphotons. The photons contributing to the dose load thereby have an energy ranging from 0 MeVto 1.33 MeV and each and every one of them would have dierent TVL as shown in gure (A.5).

Figure A.5: Tenth-value layer for concrete (qualitative plot) adapted from NCRP 151 [11]

In China, OUR International Scientic and Technological Development Co invented their veryown Gamma Knife for body applications. On this equipment, a study of the energy distributionof the photons contributing to the dose load has been carried out by X. Xie et al. [13]. A methodfor structural shielding design not based on a single TVL is described; but no similar study forthe Leksell Gamma Knife can be found in the literature.

38

A.3 Measurements of radiation spectra with semiconductor de-

tectors

A.3.1 Semiconductor radiation detector

Semiconductors are a kind of crystals, with specic electrical characteristics. They have an elec-trical conductivity that increases with temperature and is intermediate between that of a metaland an insulator. The band-gap between the valence band4 and the conduction band5 is smallenough (for example 0.7 eV for germanium) for electrons to easily jump from the valence bandto the conduction band. In the process, a vacancy called a hole in the valence band appears.Contrary to a metal where the charge carriers are free electrons, the charge carriers in a semi-conductor are electron-hole pairs.

When a photon interacts with a semiconductor material, and deposits energy, many electron-hole pairs are produced within a few picoseconds where the photon is absorbed. Semiconductordetectors measure the number of charge carriers set free by the radiation to determine its energy.To achieve that, an electric eld is present throughout the detector's volume and drifts holesand electrons apart in opposite directions towards two electrodes. These charge carriers arecollected and converted into a voltage pulse by an integral charge-sensitive preamplier. Thecharge collected is proportional to the energy deposited in the detector by the incoming photon[4].

A.3.2 Pulse height spectra

When gamma photons are travelling through a photon counting detector, they may interactin the active volume of the detector or cross it without any interaction. If they interact, theydeposit some energy through one of the three mechanisms photoelectric absorption, Comptonscattering or Pair Production and a burst of current is generated. The pulses recorded for eachburst of current are counted and usually displayed in a pulse height spectrum. The abscissa isthe energy deposited in the detector, and the ordinate the number of events detected for thecorresponding energy.

Figure A.6: Pulse height spectrum for a 2760Co source

In gure (A.6) is an example of pulse height spectrum for a Cobalt-60 radiation source. Onthis gure, we can observe dierent features such as photopeaks, Compton continuum, Comptonedge, escape peaks.

4The valence band corresponds to the outer-shell electrons that are bound to specic lattice sites within thecrystal

5The electrons in the conduction band are free to travel through the crystal

39

Photopeaks

The photopeaks correspond to full deposition of the incident photons energy. Two peaks for thetwo gamma rays of 1173 keV and 1332 keV emitted by 60Co. There is two possibilities for anevent to be counted in the photopeak:

1. Photoeletric absorption. The incident photon is fully absorbed and all its the energyis deposited in the detector.

2. Multiple events. If Compton scattering or Pair Production occur, one or more secondaryphotons are produced. If these photons don't escape the detector and are further absorbedthrough photoelectric eect, all the energy of the incoming photon is eventually got back,and the event is counted in the photopeak too.

Compton continuum and Compton edges

When a photon undergoes Compton scattering, its energy is shared between the secondary pho-ton and the recoil electron depending on the scattering angle. The energy of the scattered photonhas been stated in equation (A.3) and is displayed on gure (A.7)

Figure A.7: Energy of the scattered photon and recoil electron for Compton scattering

For the case where the scattering angle θ is close to zero, little energy is transferred to theelectron and the photon is only slightly deviated. For the other extreme case where θ = π thephoton is backscattered, and transfers the maximum energy possible Eedge to the electron.

Eedge = E − E′ = E − E 1

1 + hνmec2

(1− cosπ)(A.18)

If the secondary photon escapes the detector, only the energy of the recoil electron, rangingfrom 0 to Eedge is deposited in the detector. Many Compton scattering would eventually producea continuum in the energy spectra ; the maximum limit being called the Compton edge andcorresponding to Eedge

40

Escape peaks

The escape peaks on gure (A.6) are the small peaks at around 151, 310, 821 and 662 keV. Theycorrespond to the escape of one or two of the annihilation photons resulting from pair production:

1. The single escape peak. If one of the photons is absorbed and the other escapes. Theescape photon induces a loss of energy of mec

2 = 511 keV and would appear as a peak 511keV before the photopeak.

2. The double escape peak. Both photons escape the detector, resulting in a 1022 keVloss in the deposited energy. This case would appear as another peak 1022 keV before thephotopeak.

Nevertheless, as the two gamma-rays of 60Co are just above the threshold of 1022 keV forpair-production, that kind of interaction should not be dominating in our example. The escapepeaks might not be visible or very small if the signal is noisy in a measured spectrum.

A.3.3 Eciency and resolution

A.3.3.1 Energy resolution

The energy resolution of a detector is a measure of its ability to resolve two peaks that are closetogether in energy. The energy resolution is easily assessed with the width of the peaks in thespectrum. Indeed, if two close peaks are wide, they overlap and are not easily distinguished. Amore formal denition of the energy resolution uses the Full Width at Half Maximum (FWHM)of a peak divided by the centroid energy of that peak [14] as shown in gure (A.8):

Resolution (%) =FWHM

E0(A.19)

Figure A.8: Denition of detector resolution from [4]

For a normal distribution, the FWHM can be determined from the standard deviation σ:

FWHM = 2σ√

ln 2 (A.20)

41

The most common way in the literature to determine the FWHM is based on the least-squarest of a Gaussian distribution to the measured peak after subtracting the background. This mayhowever produce errors if the detectors exhibits asymmetry on the low-energy side. Anothersimple method is to nd the peak height and peak center by quadratic interpolation and theFWHM by linear interpolation [14].

A.3.3.2 Eciency

A basic denition of absolute photon detection eciency is [4]:

εabs =Number of detected photons

Number of photons emitted by the source

The absolute eciency largely depends on the geometry and the fraction of the emittedphotons that actually reach the detector. Another useful quantity is the intrinsic eciency,dened as [4]:

εint =Number of detected photons

Number of photons incident on the detector

The intrinsic eciency is the probability that a gamma ray photon entering the detectoractually interact in the detector's volume and give rise to a pulse. The probability for a photonto be transmitted over a distance d is e−µd (Beer-Lambert law) with µ the linear attenuationcoecient. The probability for the photon to be absorbed is therefore:

εint = 1− e−µd (A.21)

A.3.4 Detector response function

A.3.4.1 Denition

A photon spectrum T (E) reaching a detector is recorded as a pulse-height spectrum M(h). Thetwo spectra are related by [15]:

M(h) =

∫ Emax

0R(h,E) · T (E)dE (A.22)

M(h) is the pulse-height distribution recorded by the detector or Measured spectrum

h is the pulse height T (E) is the energy distribution of the incident photon ux or True spectrum

E is the incident photon energy and Emax the maximum incident energy

R(h,E) is the response function of the detector and characterises the intrinsic propertiesof the detector. R(h,E) can be written as the convolution of an energy deposition spectrumD(e, E) and a resolution function G(h, e), multiplied by the detection eciency εint(E) [15]:

R(h,E) = εint(E)

∫ E

0G(h, e) ·D(e, E)de (A.23)

εint(E) is the intrinsic eciency, or the probability that a photon of energy E entering thedetector actually interacts in the active volume.

D(e, E) is the deposition energy spectrum. Amongst the photons that interact in thedetector, some will through photoabsorption and deposit an energy equal to E. Someother photons may scatter and be counted at any lower energy e between 0 and E. D(e, E)is the probability that an E energy photon deposits an energy e in the detector.

42

G(h, e) is the Gaussian-shaped resolution function. As the resolution of the detector is notperfect, a photon that deposits an energy e in the detector may be counted at a slightlydierent height h in the pulse-height distribution.

If we discretize equation (A.22) over small energy intervals, we get:

M(hi) =

Emax∑Ej=0

R(hi, Ej) · T (Ej) or Mi =

jmax∑j=0

RijTj (A.24)

We can write equation (A.24) in a simple matrix formulation:

M = R× T or

m1

m2...mn

=

r11 r12 · · · r1n

r22 · · · r2n...

. . ....

(0) · · · rnn

×

t1t2...tn

(A.25)

A.3.4.2 Determination of the response function

In this part are presented some of the methods commonly used to estimate the response functionof a detector. Two main approaches exist:

1. The response function of the detector can be measured for selected energies using mo-noenergetic gamma-ray sources. The two drawbacks of this approach are the very limitednumber of available monoenergetic sources, and the sensitivity of the measurement to theenvironment. Indeed, some photons emitted from the source can scatter in the walls andobjects near the detector and appear in the Compton background even if they are not dueto scattering in the detector.

2. The response function can also be simulated using Monte-Carlo methods6. This approachalso contains high uncertainties coming from the modelling of the detector (the amount ofinformation on materials and dimensions of detectors made available by the manufacturersare sometimes rather small) and the method itself (interaction cross-section used, or randomnumbers generated with predictable algorithms for example).

In both cases, some means of interpolating between these measured or simulated spectra forall energies are to be used. The gure (A.9) shows the interpolation of a spectrum at 1050 keV (inorange) from two simulated spectra at 1000 keV and 1100 keV. Dierent techniques to interpolatespectra can be found in the literature and the most usual ones are summed-up hereunder.

1. A most simple and commonly used method is to assume the Compton continuum to bea at background extending from 0 to the Compton Edge. The only parameter neededis the ratio between the photopeak and the Compton background called the Peak-to-Compton ratio. This parameter can be determined from the measured/simulated spectraand interpolated for every energy. The method is for example used by Love and Nelsonin [16].

2. A second method consists in parametrizing a general response function, tting this functionto the measured/simulated spectra to determine the parameters, and interpolating thoseparameters as function of the energy [17]. This method is the one used in the Gadrassoftware for example. Gadras7 is a fairly used software developed by Sandia National

6Monte Carlo methods are well-accepted algorithms to simulate radiation transport, based on the use of randomnumbers and probability distributions.

7Gadras is available on the Nuclear Energy Agency https://www.oecd-nea.org/tools/abstract/detail/

psr-0610/

43

Figure A.9: Interpolation of a spectrum at 1050 keV

Laboratories to compute the response of gamma-ray detectors to incident radiations [18][19].

3. A segmentation of the measured/simulated spectra in dierent parts to match up theposition and common features of the spectrum can also be used. The photopeak, comptoncontinuum, escape peaks are tted separately with sine-series or polynomial functions ona channel-by-channel basis [20]. The method is described in considerable details in [21].

4. The last method presented is rather similar to the previous one, but instead of tting theCompton background on a channel-by-channel basis, some [22] claim that a more judiciousapproach would be to interpolate the Compton background between channels correspondingto the same gamma-ray scattering angle rather than same energy channels.

A.3.5 Unfolding measured spectra

Events counted at a lower energy than the full energy of the incident photon occur rather fre-quently. For a Canberra coaxial GC2020 germanium detector8, the Peak-to-Compton ratiospecied in the manufacturer data-sheet is 46 to 1. Around 50% of the counts in the spectrumarise from Compton-scattering in the detector.

Therefore, one needs to unfold or deconvolve the measured pulse-height spectrum i.e. applya correction in order to get the true energy distribution of the photon ux reaching the detector.A review of the usual methods found in the literature to unfold energy spectra is presented inthe following sections.

8This detector is the one used in this study to measure gamma-ray energy distribution around the GammaKnife.

44

A.3.5.1 The inverse matrix method

Unfolding a spectrum is a typical inverse problem. The inverse matrix method is probably themost straightforward method to unfold a spectrum, and is described in a certain amount ofpapers such as [23], [24] and [25]. It consists in inverting the response matrix R in order to getthe true energy distribution of the incident photons from a measured spectrum:

T = R−1 ×M (A.26)

However, the unfolding problem is ill-conditioned and small inconsistencies in the responsematrix lead to strong oscillations in the unfolded spectrum. Some other methods have beendeveloped to provide more stable solutions.

A.3.5.2 The stripping technique

The stripping technique uses the response function R of the detector to calculate the Comptonbackground due to a given photopeak. This background is then subtracted or `stripped' fromthe spectrum. This procedure is described in [20] and [16].

From [20]:

Stripping the Compton background from one-dimensional spectra is straightforward.A loop is made from the highest to the lowest energy channel. At each step thecontribution to all the lower channels of the Compton background corresponding tothe contents of the current channel is calculated and subtracted from them.

This methods relies on the assumption that the counts cj in a photopeak of height hj areonly due to full energy absorption of primary photons. The contribution of the primary photonsto a lower energy hi is Rij · cj . Thus, the Compton background for all lower energies from i = 1to j − 1 cam be calculated using the response function of the detector, and removed from thespectrum.

As there is no contribution to the response function above the highest photopeak, by proceed-ing channel-by-channel towards lower energies we can assume only primary photons counts areleft in the channels and the Compton scattering eect in the detector is removed from the spectra.

A.3.5.3 The folding iteration method

The folding iteration method was, to my knowledge, rstly described in an unpublished reportfrom the U.S. Naval Radiological Laboratory Report [26] but has been discussed in some publi-cations such as [27] and [22].

The iteration method takes advantage of the fact that folding calculating the measuredspectrum from the true energy distribution by multiplying by R is a lot easier than unfolding.This method proceeds in three steps to estimate the unfolded spectrum U using the measuredspectrum M and the response function of the detector R using trial functions.

1. The rst trial function used is the measured spectrum U0 = M2. The rst folded spectrum F0 = RU0 is calculated thanks to the response matrix and

compared to the measured spectrum M3. The next trial function is obtained by correcting the original trial function: U1 = U0 +

(M − F0) (dierence approach) or U1 = U0 × (MF0) (ratio approach)

45

The iteration method can be considered as a regularisation method for the inverse prob-lem. The procedure is terminated as soon as the folded spectrum F agrees with the measuredspectrum M . This early interruption prevents an overtting of the observed data. After manyiterations, the unfolded spectrum would tend to the exact matrix inversion (T = R−1 ×M) andshow more and more oscillations. In [22] it is suggested that around 10 iterations usually givesatisfactory results for the unfolded spectrum.

Other iterative algorithms than the one presented here exist. The Van Cittert algorithms(Eq. A.27) or the Gold algorithm which is an extension of Van Cittert [28][29] can for examplebe mentioned.

Uk+1 = Uk + µ((RTRRT )M − (RTRRTR)Uk

)(A.27)

With µ a relaxation factor.

A.3.5.4 Tikhonov regularization

The Tikhonov regularisation is one of the most common regularisation method for ill-posedinverse problem. The objective is to nd the `smoothest' solution to the inverse problem byimposing some conditions on the derivative of T . This method is a generalization of the least-squares minimisation problem of ||M −RT ||2, with the introduction of a regularisation term:

||M −RT ||2 + α||T ′||2 (A.28)

α is the regularisation parameter that tune the smoothing strength.

A.3.5.5 Maximum entropy deconvolution method

The maximum entropy deconvolution is a maximum-likelihood Bayesian regularisation method[30]. The basic idea is to minimize the entropy S, which is often described as a measure of theamount of information known, while remaining compatible with the data.

S = −∑j

(Tj log

(TjT0

)+ T0 − Tj

)(A.29)

With:

Tj is the unfolded spectrum S the Shannon entropy as dened by Jaynes and Skilling [31] T0 the initial guess for the unfolded spectrum based on a priori information σi the estimated standard uncertainty Ω a parameter set by the user

The objective is to nd T that maximize the entropy S of the distribution in order to getthe most likely and realistic solution. Same as the Tikhonov regularisation, the problem can besolved by introducing a regularization term in the least-squares minimisation problem:

||M −RT ||2 + α||S||2 (A.30)

Maximum entropy deconvolution can also be approached with a constrained minimization ofS. In that case, a constraint such as the one in equation (A.31) must be respected to guaranteethe compatibility of the solution with the measured spectrum [30][32][33].

∑i

(Mi −

∑j RijTj

σi

)2

≤ Ω (A.31)

46

The advantage of entropy deconvolution is the sound mathematical background of informa-tion theory to justify its use, and the possibility to include standard uncertainties and other apriori known information in the calculation. It is however a computationally expensive methodcompared to an early interruption iterative regularisation method.

47

Appendix B

Materials and methods

B.1 Spectroscopic measurements conguration

48

49

B.2 Coordinate system used for the measurements

Figure B.1: Coordinate system used in this report. Most of the graphs in this report are givenin the XY plane. Image courtesy of Elekta.

Figure B.2: Coordinate system often used with the Leksell Gamma Knife. Image courtesy ofElekta.

50

B.3 Adaptation of NCRP methods to the Gamma Knife

The methods presented in the NCRP reports 49 and 151 are all based on radiation therapydevices with single beams, and with the leakage and scatter eld in the room always relatedto the eld during Beam On. If one modify the formalism to sum the contribution of the 192beams, the methods described in appendix A.2.3.1 can be adapted to the Gamma Knife. Thedose rate at the barrier for scattered radiation coming from equation (A.15) is changed to:

Dsca =wT

d2sec

F

400

192∑i=1

aid2sca,i

(B.1)

And the dose rate for leakage from equation (A.16) becomes:

DL = 10−3wT192∑i=1

1

d2L,i(B.2)

Where modied quantities from equations (A.15) and (A.16) are:

w = W/192 is the workload per beam. dsca,i is the distance between source i and the target volume see gure (B.3). dL,i is the distance between source i and the point protected. ai is the fraction of beam i that scatters at a specic angle αi. αi is the scattering angle between the central ray of the primary beam i and the pointprotected as shown shown in gure (B.4).

Figure B.3: Position of the 192 60Co sources relative to the isocenter (origin of the coordinatesystem). The sources colour is alternatively red and blue for the dierent sectors of the collimator.

51

Figure B.4: Top view of the 192 beams of the Gamma Knife. The collimator sectors are in redand blue. In grey are drawn the extrapolation of the central axes of the beams if they passedunobstructed through the primary shielding. These grey lines are a purely geometrical depictionto show the denition of the angle αi between a primary beam i and the point protected.

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