Diode Response Correction

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in

Large Photon Fields

Thesis for Master of Science in Medical Radiation Physics

Robert Vorbau

June 24, 2010

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1 Introduction

In Sweden it is estimated that every third person will at some point in life be diag-nosed with cancer. Half of all current cancer patients are at some point treated with radiotherapy, and of all cured patients 30% have had radiotherapy. The aim of the treatment is to sterilize the cancer cells by damaging their DNA molecules, while sparing healthy tissue by reducing its dose as far as possible. In radiotherapy the cells are damaged by energy deposited from ionizing radiation. The physical quan-tity used to describe the energy deposited per mass is called absorbed dose, with the unit Gy. For a successful treatment the uncertainty in delivered dose should be min-imized, the ICRU Report 24 (1976) recommends the delivered dose to be accurate within 5%. There are a number of sources of uncertainties in radiotherapy dosime-try, from the calibration for reference dosimedosime-try, to the uncertainty in positioning of the patient. For all steps in the treatment planning/delivery, the uncertainty should be minimized as to precisely aim for doses that can provide cure without severe side reactions.

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by Eklund and Ahnesjö (2009).

The purpose of this work was to create a prototype software implementing the models developed by Eklund and Ahnesjö (2009) to correct for the energy dependent response of a silicon diode for arbitrary eld settings. The software was developed in a combined project, as part of this work, and as part of the work by Omar (2010), for the computing environment MATLAB. The software is able to import les containing information about irregular eld specications exported from a treatment plan verication system and les containing information from measurements made with a computerized water phantom. The software was based on uence pencil beam kernel calculations to calculate spectra and the spectra based response model proposed by Eklund and Ahnesjö (2009).

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2 Theory

2.1 Diodes

This section briey describes the design and properties of silicon diodes, interested readers can nd more detailed information in e.g. Knoll (2000).

Solid materials can according to their conduction properties be classied as either conductors, semiconductors or insulators. Diodes are built from semiconducting material such as germanium or silicon, these materials have some unique properties that makes them suitable for measuring ionizing radiation.

The electron properties in a solid state material can be characterized, by means of a valence band and a conducting band. The electrons in the valence band have low energy and are bound to certain locations in the crystal, while the electrons in the conduction band have more energy and are able to move through the crystal. An energy gap separates these two bands, and no electrons are allowed in this gap. In semiconductors this energy gap is approximately 1 eV, less for conducting materials, and in insulators the energy gap is usually larger than 5 eV. If an impurity is introduced in a semiconductor it is said to be doped. There exist two types of doping, n-type and p-type. The n-type doping refers to a situation when a material with more valence electrons than the semiconductor is introduced in to the crystal, and p-type when a material with fewer valence electrons than the semiconductor is introduced. A diode consists of semiconducting material where one part of the chip is n-doped and the other is p-doped, this is known as a p-n junction. These dierent doping types creates an electric eld across the p-n junction. Diodes which are more n-doped than p-doped are called n-type, and vise versa.

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to dierent parts of the diode, the electron moves to the p-doped side and the hole moves to the n-doped side of the diode. This forms an electric signal, which both electrons and holes contribute to.

2.1.1 Silicon diodes as dosimeters in photon beams

Diodes can be used as radiation detectors since they produce an electrical signal when exposed to ionizing radiation. There are two dierent ways to use diodes as radiation detectors. One is to the analyze the pulses formed in the diode and create a pulse height spectrum. The other way is to use the diode as dosimeter by measure the total charge formed in the diode during irradiation. This work will concern the latter.

When measuring dose in water the ideal dosimeter should be made of materials similar to water (Westermark et al., 2000). The use of silicon diodes as dosimeters has many benets compared to other detectors. The active volume in silicon diodes can be kept small and still produce a high signal (without the need for external bias voltage) since the density is high. The small active volume also give rise to a high spatial resolution, which is benecial for measuring dose in regions with steep gradi-ents in e.g. penumbras and small elds. Diodes are also known to be mechanically robust, compared to ionization chambers which are more fragile. The use of silicon diodes is however suering from some problems. One problem is the damage in the crystal structure caused by the radiation, which results in change of the response as the accumulated dose increases, making it necessary to recalibrate the diodes at reg-ular intervals (Rikner and Grusell, 1987). This eect is most pronounced when the accumulated dose is low and saturates when the accumulated dose becomes larger, therefore diodes are usually pre-irradiated before taken into use, the dose given is in the order of kGy (Rikner and Grusell, 1983).

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10−3 10−2 10−1 100 101 102 10−8 10−6 10−4 10−2 100 102 104 Energy [MeV]

Mass attenuation coefficient [cm

2/g]

Photoelectric effect

Compton scatter

Pair production

Figure 2.1: Mass attenuation coecient for dierent interactions in water (blue dashed) and silicon (black solid). Data from NIST (Berger et al., 2005b)

(Westermark et al., 2000).

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2.2 Cavity theories

Since very few detectors are able to determine dose to water directly, theories for relating the dose to a cavity of a dierent medium than its surrounding have been de-veloped. In this section such theories of relevance for this work are briey described, more information can be found in e.g. Attix (1968).

2.2.1 Bragg-Gray theory

The Bragg-Gray cavity theory is the most simple theory for small cavities. This theory states that the dose in a medium can be approximated as the product of the charged particle uence (electrons) and the mass collision stopping power if

δ-particle equilibrium exist,

D = Φbg Scol ρ (2.1)

where Φbg _{is the electron uence (excluding δ-particles), and (S}

col/ρ) is the mass

collision stopping power of the electrons.

The mass collision stopping power only includes energy losses of charged particles that results from collisions, i.e. losses that results from bremsstrahlung and in-ight annihilation are excluded in this approximation.

If a small cavity is placed in a surrounding medium, then the ratio of the dose in the cavity to the dose in the surrounding medium can be written as

Dcav Dmed = Φbg Scol ρ cav Φbg Scol ρ med = Scol ρ cav Scol ρ med (2.2)

where Dcav respective Dmedare the doses in the cavity and the surrounding medium,

respectively.

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Small cavity Large cavity

Figure 2.2: Conceptual illustrations of a small and large cavity. The path of the electrons are illustrated as arrows. In the case of a small cavity all the electrons can be approx-imated as produced outside the cavity. For a large cavity the complete opposite situation prevail, all electrons can be assumed to be produced inside the cavity. rst condition is that the cavity needs to be small compared to the range of the charged particles so it doesn't aect the charged particle eld and, consequently the charged particle uence is the same in the presence and absence of the cavity. The second condition is that the absorbed dose in the cavity is assumed to be deposited entirely by charged particles crossing it (Attix, 1968), this is illustrated in Figure 2.2. Equation (2.2) is only valid for monoenergetic charged particles, if more than one energy is present the following equation can be applied,

Dcav Dmed = _¯ Scol ρ cav _¯ Scol ρ med (2.3) ¯_S_col ρ = ERmax 0 Φbg_E Scol ρ dE ERmax 0 Φbg_E dE (2.4)

where Emax is the maximal energy of the electrons and ΦbgE is the electron particle

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2.2.2 Spencer-Attix theory

A more accurate theory for small cavities is the Spencer-Attix cavity theory which in contrast to the Bragg-Gray theory takes the spectrum of δ-particles into account. The Bragg-Gray theory assumes that all the δ-particles drop all their energy on the spot and doesn't travel any distance at all. In the Spencer-Attix theory the electrons are split into two groups, fast and slow electrons (Spencer and Attix, 1955). The fast electrons have energy larger than a certain value ∆ (cuto energy) and are treated like they have enough energy to cross the cavity and they are able to deposit energy in the cavity. It's also assumed that the slow electrons (energy less than ∆) not have enough energy to cross the cavity and therefore not deposit any energy in the cavity. δ-particles generated inside the cavity with energies less than ∆ are assumed to deposit all their energy inside the cavity, and δ-particles with energies larger than

∆ are assumed to leave the cavity and deposit none of their energy in the cavity.

The ∆-value should be equal to the energy of an electron with just enough energy to cross the specic cavity.

The second terms in the nominator and denominator in equation (2.5) are called track end terms (equation (2.6)), these accounts for electrons that were produced outside the cavity and ended their tracks in the cavity. The track end terms were not part of the original equation presented by Spencer and Attix, these terms were added later by Nahum (1978), yielding

Dcav Dmed = ERmax ∆ ΦE L∆ ρ

cavdE + (T.E.)cav

ERmax ∆ ΦE L∆ ρ

meddE + (T.E.)med

(2.5) T.E. = Φ(∆) Scol(∆) ρ ∆ (2.6)

where ΦE is the electron uence spectrum (including δ-particles) and (L∆/ρ)cav,

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sur-rounding medium, and T.E. is the track end term.

2.2.3 Large cavity

The quantity kerma quanties the energy transferred to charged particles (electrons) by uncharged particles (photons and neutrons), more precise kerma is the energy transferred to charged particles by uncharged particles per mass at the location of the transfer. The kerma does not provide any information about where the charged particles deposited their energy, or in which way the energy was deposited, only from where it originates.

The kerma can be split into two terms, collision kerma and radiative kerma,

K = dtr

dm = Kcol+ Krad (2.7)

where tr is energy transfered to charged particles by uncharged particles, dm is the

mass of the small volume element where the energy transfers took place, Kcol is the

collision kerma and Krad is the radiative kerma.

The collision kerma only includes the part of the energy transfered to charged par-ticles to be spent by the charged parpar-ticles in collision interactions, and the radiative kerma is the part to be spent in radiative interactions (e.g. bremsstrahlung). If charged particle equilibrium (CPE) exists at a point, the dose in that point can be approximated as the collision kerma. In the case when photon beams irradiate a water phantom, CPE is established after the build-up region provided that the eld is large enough to provide lateral electron equilibrium (LEE). The collision kerma is usually modelled (in the case of photon beams) as the product of the photon energy uence and the mass energy absorption coecient,

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where ΨE is the photon energy uence spectrum and Emax is the maximum photon energy.

The dose is larger than the collision kerma (after the build region) since most of the electrons are traveling a distance in the forward direction and deposit their energy at a depth deeper than the depth where the energy transfer took place.

A cavity can be considered as a large cavity if the cavity not eects the photon uence, CPE exist, and if all energy deposit in the cavity is made by charged particles produced within the cavity. A large cavity is illustrated in Figure 2.2. The ratio of the dose to a large cavity and the dose to the surrounding medium can be calculated by the following equation,

Dcav Dmed = ERmax 0 ΨE µen ρ cav dE ERmax 0 ΨE µen ρ med dE . (2.9)

2.2.4 Burlin cavity theory

Burlin cavity theory can be applied to intermediate cavity sizes. This theory is a linear combination of Spencer-Attix theory and the theory for large cavities,

Dcav Dmed = d _E_R_max ∆ ΦE L∆ ρ

cavdE + (T.E.)cav

+ (1− d) ERmax 0 ΨE µen ρ cav dE d _E_R_max ∆ ΦE L∆ ρ

meddE + (T.E.)med

+ (1_{− d)} ERmax 0 ΨE µen ρ med dE (2.10) where d is a mixing factor, 0 ≤ d ≤ 1.

Burlin (1966) suggested that the mixing factor d should be determined by the atten-uation of the electron spectrum in the cavity. If the size of the cavity is very small

d approaches 1 and equation (2.10) is reduced to Spencer-Attix theory (equation

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equation (2.9), i.e. the theory for large cavities.

2.3 Response model for silicon diodes

2.3.1 Yin et al. response model

The cavity theories described in the previous sections are traditionally used for other detectors than silicon diodes. Yin et al. (2002, 2004) proposed a minor modication of Burlin cavity theory that allowed it to be applied to silicon diodes in megavoltage photon beams.

Silicon diodes have a much higher response to photons of low energy compared to photons of high energy (see section 2.1.1). The photon spectrum of a megavoltage beam in water contains both high energy photons (mostly primary) and low energy photons which includes primary photons, compton scatter and annihilation photons. Yin et al. (2002, 2004) assumed that the high energy part of the spectrum would interact with the diode as if it was a small cavity, while the low energy part would interact with the diode as if it was a large cavity. They also assumed that the scattered photons fullled the conditions for the large cavity theory and that the primary photons fullled the conditions for Spencer-Attix cavity theory, leading to

DSi= EZmax ∆ Φp_E L∆ ρ Si dE + Φp(∆) Scol(∆) ρ Si ∆ + EZmax 0 Ψs_E µen ρ Si dE (2.11)

where DSiis the dose to silicon, ΦpE is the primary electron particle uence spectrum

(including δ-particles), and Ψs

E is the scattered photon energy uence spectrum.

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the deviation arised from lack of CPE in the large cavity term.

2.3.2 Eklund and Ahnesjö response model

The model suggested by Eklund and Ahnesjö (2009) treats the radiation more ac-cording to their energy and not completely acac-cording to their origin as in the Yin et al. (2002, 2004) model, this can be seen in the following equation,

DSi= EZmax ∆ Φ[EA,Emax],p E L∆ ρ Si dE+(T.E.)Si+ EZmax 0 K(E) Ψ[0,EA],p E + Ψ s E µen ρ Si dE (2.12) where Φ[EA,Emax],p

E is the primary electron particle uence spectrum caused by

inci-dent photons with energies between EAand Emax, and Ψ[0,EE A],pis the primary photon

energy uence spectrum caused by incident photons with energies lower than EA.

In this model primary photons with energies less than a partition energy (EA), and

all scattered photons interacts with the cavity as if it were large. All scattered photons are included in the large cavity term, since the contribution from scattered

photons with energies higher than EAcan be neglected (Eklund and Ahnesjö, 2009).

The primary photons with energies higher than EA interacts with the cavity as if

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2.3.3 The response factor

The response factor (RF) for a silicon diode surrounded by water can be dened as the ratio of the dose to the silicon cavity to the dose to water,

RF = DSi

Dw

, (2.13)

assuming that the signal produced in the diode is proportional to the dose in silicon. A theoretical response factor can be calculated by the use of the cavity theory for silicon diodes (equation (2.12)) combined with equation (2.13),

RFcalc ₌ ERmax ∆ Φ[EA,Emax],p E L∆ ρ SidE + (T.E.)Si+ ERmax 0 K(E) Ψ[0,EA],p E + ΨsE µen ρ Si dE ERmax ∆ Φ[EA,Emax],p E L∆ ρ wdE + (T.E.)w+ ERmax 0 Ψ[0,EA],p E + ΨsE µen ρ w dE . (2.14) In the denominator of equation (2.14) (the water dose) the correction factor, K(E) has been approximated to unity since the dierence between the water collision kerma and the dose to water is assumed to be neglectable for low energetic photons. This approximation is valid because the electrons produced by low energetic photons have a short range, i.e. they deposit their energy locally. The response factor depends on both the electron and photon spectrum and varies with the position of the detector and aperture conguration.

The normalized response factor (RFnorm) can be dened as the ratio of the response

factor at position r in the eld A to the response factor in the reference position rref

in the reference eld Aref,

RFnorm =

RF (r, A)

RF (rref, Aref) (2.15)

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calculations of the spectra have to be performed. The complexity and computation time of the spectra calculations depend on several factors, e.g. which method is employed, the geometry and the accuracy needed in the measurement.

Notice that the denitions of the response factor (equation (2.13)) and the nor-malized response factor (equation (2.15)) can be used for values obtained both by theoretical calculations and experiments, and that r = (x, y, z) describes not only the depth (d) but also the coordinates (x, y) in the lateral plane.

To correct the energy dependent response and with that removing the over response, the relative readout of the diode need to be multiplied with the normalized correction

factor. The normalized correction factor (CFnorm) is dened as the inverse of the

normalized response factor,

CFnorm=

1

RFnorm

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3 Methods

A prototype software was developed to correct the energy dependent response of silicon diodes. For evaluation of the software, corrected relative diode readouts were compared with relative ionization chamber readouts. The ionization chamber read-outs were only corrected with the Spencer-Attix stopping power ratio (see section 2.2.2), since the eect of perturbation factors in these relative measurements were assumed to approximately cancel out each other. Measurements were also carried out with shielded diodes to investigate how this new approach with corrected diodes perform versus the routine method with shielded diodes. This work was focused on investigation of this new method for large square elds (≥ 10 cm x 10 cm) and large irregular elds. The unshielded diode and the ionization chamber readouts were normalized and corrected according to the following equations:

Ddiode_norm,corr(A, r) = diode readout(A, r)

diode readout(A, rref)· S

diode

cp (A, rref)· CFcalcnorm(A, r) (3.1)

DIC_norm,corr(A, r) = IC readout(A, r)

IC readout(A, rref) · S

IC

cp(A, rref)·

sw,air(A, r)

sw,air(Aref, rref) (3.2)

Scp =

readout(A, rref)

readout(Aref, rref) (3.3)

where Scp(A, rref)is the output factor in water at position rref in eld A, CFcalcnorm(A, r)

is the silicon diode correction factor (see equation (2.16)) at position r in eld A

and sw,air(A, rref)is the Spencer-Attix water to air stopping power ratio at position

rref in eld A.

The reference conditions (Aref, rref)was set to 10 cm depth at CAX in a 10 cm x 10 cm

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3.1 Calculations

3.1.1 Correction of silicon diodes

The software uses a response model proposed by Eklund and Ahnesjö (2009), this model was described in section 2.3.2. The model requires the knowledge of both the electron particle uence and the photon energy uence spectra to all dose measure-ment points to predict the diode response factors and thereby the correction factors.

The software uses TPR20,10 as beam quality specier and creates (by interpolation

of published spectra (Mohan et al., 1985)) an incident photon spectrum from this value to be used in the calculations. The software also requires specication of the eld shape, this information was provided through les exported from the verica-tion system MOSAIQ. The software then uses superposiverica-tion of polyenergetic pencil beam uence kernels (Eklund and Ahnesjö, 2008) to calculate the spectra, a detailed description of the algorithm used by the software can be found in Appendix A. The spectra (photon and electron) will be presented only for a few situations to illustrate.

3.1.2 Correction of ionization chambers

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Spencer-Attix stopping power ratios was calculated for seven dierent TPR20,10:s and compared with published data (Andreo, 1994).

3.2 Experiments

Measurements were performed at an Elekta Precise linear accelerator. The dose rate used during the measurements was 500 monitor units (MUs) per minute. The water phantom used was a Scanditronix RFA-300 scanner. All measurements were normalized to a 10 cm x 10 cm eld, 10 cm depth at CAX. Four types of detectors were used during the measurements, pin point chamber (PTW 31014), Scanditronix RK chamber, unshielded diode (Scanditronix EFD-3G) and two shielded diodes (Scanditronix PFD, 2730 and 1990). The electrometers used were a Scanditronix RFA-MCU and a Therados RDM 1F.

3.2.1 Depth doses and dose proles

Preceding the actual measurements with each detector, proles at 10 cm and 30 cm depth were measured in two perpendicular directions (inline and crossline) and the center of these proles were determined. Then the phantom was adjusted, and the procedure was repeated until the centers at the two depths agreed. These mea-surements were performed to ensure that the detectors were moving along the CAX when the depth doses were measured. This technique was used since it was believed that most geometric factors were taken in consideration, including the gantry angle. It is not crucial for large square elds to perform this procedure, since these elds have plateau region, the procedure was however performed to maintain accuracy for the irregular elds (see Figure 4.6).

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Figure 3.1: Illustration of the detectors (EFD, PFD and pin point chamber) orientations during the measurements of the dose proles and depth doses. The upper arrow indicates the direction of incident photons and the lower double arrow indicates the scan direction of the dose proles. The orientations were the same during the measurements of the depth doses, but the scanning direction was along the direction of the CAX (perpendicular to the double arrow).

The scan direction of the proles was perpendicular to the direction of the leafs in the MLC. This direction was chosen to avoid distortions in penumbra that rounded collimator leafs ends can cause. The scan direction of the pin point chamber was perpendicular to the chamber symmetry axis (see Figure 3.1), this scan direction was chosen to gain maximum spatial resolution. The bias applied to the pin point chamber was -400 V. A reference diode was used during these measurements to compensate for the linear accelerators uctuations in output. This diode was placed just beneath the treatment head (in air), in the outer regions of the elds, away from the penumbra regions.

3.2.2 Output factors in water

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were measured at 10 cm depth at CAX.

During the measurements it was noticed that the sensitivity for the unshielded diode was increasing as the accumulated dose was increasing, therefore the reference situation was measured for each output factor. This increase in sensitivity was believed to arise from insucient preirradiation which can cause this kind of behavior (Rikner and Grusell, 1983). This eect was too small to aect the dose proles and depth doses. All diode output factors were measured with zero bias.

The ionization chamber output factors for the square elds were measured with a RK chamber. This chamber was chosen over the pin point chamber since the pin point chamber suers from a signicant stem eect due to its low signal. However, Agostinelli et al. (2008) showed that the stem eects inuence on relative depth doses and proles (with local normalization) is small. The RK chambers large volume is not a problem since these measurements were only done at the center of large elds. No polarity correction was made for the RK chamber since this eect was neglectable, the bias used was -200 V. The ionizing chamber output factors for the irregular elds was measured with a pin point chamber, this chamber was chosen over the RK mainly because of its small volume. To minimize the stem eect of the pin point chamber it was oriented in such a way that the MLC (and the jaws) covered as much as possible of its cable. The pin point chamber was polarity corrected since a signicant dierence between positive and negative (- 400 V and + 400 V) polarity was observed. To nd the center of the elds two proles were scanned (crossplane and inplane). The ionization chambers used for measuring output factors were also preirradiated with at least 2 Gy before the measurements begun.

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4 Results

This section is divided in two parts, calculations and experiments. The calculation part was intended to give an insight in the dierent calculation steps performed by the software to calculate the correction factors. In the experimental part, the corrected relative unshielded diode readouts are compared with established methods in relative dosimetry.

4.1 Calculations

The incident photon energy uence spectrum created for the two beam qualities used in the experiments (see section 4.2) were extracted from the software and shown in Figure 4.1. The resulting photon spectra (primary and scattered) at dierent depths (1 cm, 5 cm, 10 cm and 20 cm) in a 10 cm x 10 cm eld at CAX are shown in Figure

4.2. Note that in Figure 4.2 it is the total primary spectrum (Ψp

E = Ψ

[0,Emax],p

E ) that

is shown and not the partial spectrum as used by the response model (Ψ[0,EA],p

E ).

Scattered photon spectrum at 10 cm depth for dierent eld sizes and o axis (OAX) distances were also extracted and are shown in Figure 4.3.

Calculated (with use of the spectrum calculated from the uence kernels)

Spencer-Attix stopping power ratio for seven dierent TPR20,10:s are shown in Figure 4.4,

the cut-o energy used was 10 keV (∆ = 10 keV, see equation (2.5)). Figure 4.4 also shows published Spencer-Attix stopping power ratios determined using Monte Carlo simulations by Andreo (1994). The values determined by Andreo (1994) were chosen since these are used in the used in IAEA:s calibration protocol, TRS 398 (Andreo et al., 2000).

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Figure 4.1: The incident photon energy uence spectra created by the software for the beam qualities used in the experiments (6 MV and 15 MV).

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Figure 4.3: Calculated scatter photon spectra for three square elds (10 cm x 10 cm, 20 cm x 20 cm and 30 cm x 30 cm) at 10 cm depth at CAX (left) and four OAX po-sitions, 0 cm, 5 cm, 17 cm and 25 cm from CAX at 10 cm depth in a 30 cm x 30 cm eld (right). The incident spectrum was a 6 MV photon beam (see Figure 4.1). The amount of low energetic scatter photons at CAX is increasing with increasing eld size and the amount of scatter photons is decreasing as the distance from the CAX is increasing. 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 1.085 1.09 1.095 1.1 1.105 1.11 1.115 1.12 1.125 1.13 TPR 20,10 sw,air Andreo (1994) Fluence kernels

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0 5 10 15 20 25 30 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 Depth [cm]

Normalized correction factor

10 cm x 10 cm 20 cm x 20 cm 30 cm x 30 cm −20 −15 −10 −5 0 5 10 15 20 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

Lateral distance from CAX [cm]

Normalized correction factor

Figure 4.5: The calculated normalized correction factor for three square eld sizes as a function of depth (left) and OAX distance (right). The blue solid line represent a 10 cm x 10 cm eld, the red dashed line represent a 20 cm x 20 cm eld and the purple dash-dotted line represent a 30 cm x 30 cm eld. The beam quality was 6 MV (see Figure 4.1).

4.2 Experiments

Relative depth doses at CAX and dose proles at 10 cm depth were measured for three square elds 10 cm x 10 cm, 20 cm x 20 cm, 30 cm x 30 cm at the source to isocenter distance (SID) 100 cm, the source to surface distance (SSD) was 90 cm. Relative dose proles for three irregular elds were also measured (see Figure 4.6) at 10 cm depth. In the measurements of the square elds two beam qualities were

used, 6 MV (TPR20,10 = 0.681) and 15 MV (TPR20,10 = 0.760), the measurements

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−20 −15 −10 −5 0 5 10 15 20 −20 −15 −10 −5 0 5 10 15 20 [cm] [cm] −20 −15 −10 −5 0 5 10 15 20 −20 −15 −10 −5 0 5 10 15 20 [cm] [cm] 1 2 −20 −15 −10 −5 0 5 10 15 20 −20 −15 −10 −5 0 5 10 15 20 [cm] [cm]

Figure 4.6: The three irregular elds used to evaluate the software. The blue solid lines indicate the leafs of the MLC, the black solid lines indicate the position of the jaws and the dashed black lines indicate where dose proles were scanned. Field A (left) and eld B (center) was intended to investigate the eect of MLC leakage. Field B was also intended to investigate the eect of scatter since there is a large variation of scatter in this eld. A more realistic eld that could be used in conformal radiotherapy is eld C (right).

The measured response factors at 10 cm depth at CAX are presented in Table 4.1 and 4.2.

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Table 4.1: Experimental response factors for dierent detectors, EFD (unshielded diode, uncor-rected and coruncor-rected), PFD 1990 and PFD 2730 (shielded diodes) in a 6 MV photon beam at 10 cm depth. Ionization chamber (RK chamber) measurements, corrected with the Spencer-Attix stopping power ratio was used as reference.

Response factor

Field EFD EFDcorr PFD 1990 PFD 2730

10 cm x 10 cm 1 1 1 1

20 cm x 20 cm 1.018 ± 0.002 1.003 ± 0.002 1.002 ± 0.002 1.005 ± 0.002 30 cm x 30 cm 1.035 ± 0.003 1.007 ± 0.003 1.003 ± 0.002 1.011 ± 0.003

Table 4.2: Experimental response factors for dierent detectors, EFD (unshielded diode, uncor-rected and coruncor-rected), PFD 1990 and PFD 2730 (shielded diodes) in a 15 MV photon beam at 10 cm depth. Ionization chamber (RK chamber) measurements, corrected with the Spencer-Attix stopping power ratio was used as reference.

Response factor

Field EFD EFDcorr PFD 1990 PFD 2730

10 cm x 10 cm 1 1 1 1

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5 10 15 20 25 30 0 20 40 60 80 100 120 140 160 Depth [cm] Relative Readout Uncorrected Diode Shielded Diode 1990 Shielded Diode 2730 Corrected Ionization Chamber Corrected Diode 5 10 15 20 25 30 −4 −2 0 2 4 Residual −8 −6 −4 −2 0 2 4 6 8 10 0 20 40 60 80 100

Lateral Distance from CAX [cm]

Relative Readout −8 −6 −4 −2 0 2 4 6 8 10 −4 −2 0 2 4 Residual

Figure 4.7: Depth dose (left) and dose prole (right) measurements for a 10 cm x 10 cm eld at 6 MV. The data has been normalized to the dose for a 10 cm x 10 cm eld at 10 cm depth (6 MV) using the output factors. The reference data was measured using a pin point chamber and the reference output factors was measured using a RK chamber. Measurements were made with shielded (2730 and 1990), unshielded and corrected unshielded diode. 5 10 15 20 25 30 0 20 40 60 80 100 120 140 Depth [cm] Relative Readout Uncorrected Diode Shielded Diode 1990 Shielded Diode 2730 Corrected Ionization Chamber Corrected Diode 5 10 15 20 25 30 −4 −2 0 2 4 Residual −100 −8 −6 −4 −2 0 2 4 6 8 20 40 60 80 100

Relative Readout −10 −8 −6 −4 −2 0 2 4 6 8 −4 −2 0 2 4 Residual

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5 10 15 20 25 30 0 20 40 60 80 100 120 140 160 Depth [cm] Relative Readout Uncorrected Diode Shielded Diode 1990 Shielded Diode 2730 Corrected Ionization Chamber Corrected Diode 5 10 15 20 25 30 −4 −2 0 2 4 Residual −10 −5 0 5 10 15 0 20 40 60 80 100 120

Relative Readout −10 −5 0 5 10 15 −4 −2 0 2 4 Residual

Figure 4.9: Depth dose (left) and dose prole (right) measurements for a 20 cm x 20 cm eld at 6 MV. The data has been normalized to the dose for a 10 cm x 10 cm eld at 10 cm depth (6 MV) using the output factors. The reference data was measured using a pin point chamber and the reference output factors was measured using a RK chamber. Measurements were made with shielded (2730 and 1990), unshielded and corrected unshielded diode. 5 10 15 20 25 30 0 50 100 150 Depth [cm] Relative Readout Uncorrected Diode Shielded Diode 1990 Shielded Diode 2730 Corrected Ionization Chamber Corrected Diode 5 10 15 20 25 30 −4 −2 0 2 4 Residual −10 −5 0 5 10 15 0 20 40 60 80 100

Relative Readout −10 −5 0 5 10 15 −4 −2 0 2 4 Residual

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5 10 15 20 25 30 0 20 40 60 80 100 120 140 160 Depth [cm] Relative Readout Uncorrected Diode Shielded Diode 1990 Shielded Diode 2730 Corrected Ionization Chamber Corrected Diode 5 10 15 20 25 30 −4 −2 0 2 4 Residual −15 −10 −5 0 5 10 15 0 20 40 60 80 100 120

Relative Readout −15 −10 −5 0 5 10 15 −4 −2 0 2 4 Residual

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5 10 15 20 25 30 0 50 100 150 Depth [cm] Relative Readout Uncorrected Diode Shielded Diode 1990 Shielded Diode 2730 Corrected Ionization Chamber Corrected Diode 5 10 15 20 25 30 −4 −2 0 2 4 Residual −15 −10 −5 0 5 10 15 0 20 40 60 80 100 120

Relative Readout −15 −10 −5 0 5 10 15 −4 −2 0 2 4 Residual

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−100 −8 −6 −4 −2 0 2 4 6 8 10 20 30 40 50 60 70 80 90 100

Relative Readout −10 −8 −6 −4 −2 0 2 4 6 8 −4 −2 0 2 4 Residual −100 −5 0 5 10 10 20 30 40 50 60 70 80 90 100

Relative Readout −10 −5 0 5 10 −4 −2 0 2 4 Residual

Figure 4.13: Dose prole measurements for eld A (left) and C (right) at 6 MV, the elds are shown i Figure 4.6. The data has been normalized to the dose for a 10 cm x 10 cm eld at 10 cm depth (6 MV) using the output factors. The reference data was measured using a pin point chamber (black) and the reference output factors was measured using a RK chamber. Measurements were made with shielded diode 2730 (blue), unshielded (red) and corrected unshielded diode (green).

−150 −10 −5 0 5 10 10 20 30 40 50 60 70 80 90 100

Relative Readout −15 −10 −5 0 5 10 −4 −2 0 2 4 Residual −150 −10 −5 0 5 10 20 40 60 80 100

Relative Readout −15 −10 −5 0 5 10 −4 −2 0 2 4 Residual

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5 Discussion

The primary spectra calculated at dierent depths (shown in Figure 4.2) proves that the primary uence decreases with increasing depth, due to attenuation in water. The primary spectrum is also shifted towards higher energies as the depth increases, due to the beam hardening eect caused by that low energy photons are more attenuated than photons with higher energies. It can be seen that the proportion of scattered photons (see Figure 4.2) is increasing as the depth increases, and that the spectrum of scattered photons is shifted towards higher energies as the depth increases. In the spectra of the scattered photons, a large peak can been seen at 0.511 MeV, from annihilation photons. Both the primary and scattered photon spectra appears wiggly when they are expected to be smooth. The wiggles are caused by dierent energy resolutions in the kernels and the incident photons. The secondary particles in the kernels had an energy resolution of 10 keV and incident photons had an energy resolution of 100 keV, in the primary spectra (shown in Figure 4.2) this can be seen, only every tenth bin contain photons. If a larger number of incident photon energies had been used in the calculations these kind of eects would been expected to be smaller, i.e. the spikes would been closer together and the magnitude less varying.

The spectrum of scattered photons at CAX becomes shifted towards lower energies as the eld size increases (see Figure 4.3) as a consequence of more scattered photons being produced in larger elds, and also due to the larger scattering angle required to reach CAX from the eld periphery. Figure 4.3 also shows that the low energy part of the scattered photon spectrum at CAX is mainly caused by photons incident in the outer regions of the eld. Figure 4.3 shows that the amount of scattered photons is decreasing as the OAX distance is increasing and a large decrease of scattered photons can be seen outside the eld due to absence of primary photons. Due to the lack of primary photons, the average photon energy decreases substantially outside of the primary beam.

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kernel calculated spectra (see Figure 4.4) agrees within 0.2% with data published by Andreo (1994) using Monte Carlo simulations. It can be noticed that the val-ues obtained from the uence kernels calculations are in general greater than the published values, possible reasons for this could be dierent incident photon spec-trum, dierences in stopping-power data or dierent methods for determination of

TPR20,10. Since only ratios of stopping power ratios are used for the corrections

calculated in this work, the agrement with published work is judged to be sucient. The calculated correction factor curves (see Figure 4.5) shows that the correction becomes larger as the depth and eld size increases, this was expected since the amount of scattered photon increases with depth and eld size. At depths in the build-up region CPE is not established, which the response model requires, and therefore correction factors in this region should not be trusted. The OAX distance variation shows that the correction becomes smaller when the OAX distance becomes larger until the penumbra region is reached, then the correction factor shows a very steep gradient. This steep gradient arises from the lack of primary photons outside the eld, here only scattered photons are present, this causes the correction factor to change rapidly.

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In the following text the percentage refers to the percentage of the relative readout of a 10 cm x 10 cm eld at 10 cm depth, i.e. the same units used in the residuals in Figure 4.7−4.14. The depth dose curves for the square elds measured at 6 MV (Figure 4.7, 4.9 and 4.11) shows that the maximum deviation for the uncorrected diode is about 6.5%. One of the shielded diodes (1990) show a maximum deviation of approximately 1.0% and the other shielded diode (2730) deviates about 2.8%. The corrected diode shows an agreement with the corrected ionization chamber within 0.7%. The build-up region was excluded from this discussion because of lack of CPE and presence of electron contamination with dierent stopping power.

In the same cases for 15 MV (Figure 4.8, 4.10 and 4.12) the uncorrected diode deviates about 4.5% from the corrected ionization chamber, the corrected diode deviates approximately 1%. For 15 MV one of the shielded diodes (1990) shows a slightly better agreement with the corrected ionization chamber than the corrected diode, the other shielded diode (2730) shows as much as 2% deviation from the corrected ionization chamber.

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This deviation is most likely not an eect of the response model, the eect may arise from an approximation made in the uence calculation algorithm. As can be seen from the proles the attening lter has a considerable eect. The attening lter was implemented by assuming a constant incident spectrum and intensity within the eld. In reality the incident spectrum varies as a consequence of the variable thickness of the attening lter, this will cause the incident spectrum in center of the eld to be more shifted towards higher energies (Mohan et al., 1985; Lee, 1997), i.e. the beam quality is higher in the center than o axis (Hanson et al., 1980). This was believed to cause an underestimation of the low energy photons in the spectrum. Tailor et al. (1998) introduced an empirical relation between the o axis

beam quality (TPR20,10) and the o axis angle, this relation was used to estimate

the eect of the attening lter. Correction factors at a few measurement points of the dose proles for the 30 cm x 30 cm eld were recalculated using the beam qual-ity by Tailor et al. (1998), the diode readouts (shown in Figure 4.11 and 4.12) with

corrections calculated with varying TPR20,10 were now in better agreement with the

corrected ionization chamber, approximately within 1% for both 6 MV and 15 MV. This conrms that the attening lter eects the o axis correction factors. Another possible reasons for this o axis deviation could be the diodes angular dependent response.

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6 Conclusions

This work has shown an alternative way of removing the energy dependence of silicon diodes, how this can be implemented in a user friendly software and also validation of the software, when used in large photon elds. For relative dosimetry in the center of large square elds (e.g. relative depth dose curves) it was found that corrected unshielded diode measurements work well, there was a agreement within 1% with the ionization chamber. In o axis measurements of the large square elds the corrected unshielded diode deviated more with increased distance from CAX, as much as 2% was observed, it was found that some of this deviation originated from o axis beam softening since this eect was not considered in the spectrum calculation algorithm. Taking the eect of the attening lter in consideration an agreement of 1% (within the eld) of corrected unshielded diode and the corrected ionization chamber was estimated. For o axis measurements of irregular elds the maximum dierence between the corrected unshielded diode and the corrected ionization chamber was 1%, it was believed that better agreement of the corrected unshielded diode and the ionization chamber could be achieved by a more detailed model of the treatment head in the spectrum calculation algorithm, e.g. modeling the MLC leakage. This work has shown that this new approach is an alternative to shielded diodes and that corrected diodes will in some cases provide more reliable results. A natural next step to develop this method further is to solve some of the issues in the uence calculation algorithm and also to validate this correction method in IMRT elds, where segments of dierent sizes are mixed.

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Acknowledgments

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Appendices

A Silicon Diode Correction Software

A prototype software was developed with MATLAB Version 7.8.0 (R2009a) by Robert Vorbau and Artur Omar to correct for the energy dependent silicon diode response. A screenshot of the software main window is shown in Figure B.2. The software is able to import les containing information about the eld specication ex-ported from MOSAIQ v1.6 (Oncology management system by IMPAC Medical Sys-tems) and les exported from the OmniPro-Accept 6.4A (Scanditronix-Wellhöfer) software, containing information from measurements made with the RFA 300 (Scan-ditronix) computerized water phantom. The software was based on uence pencil-beam kernel calculations and a response model proposed by Eklund and Ahnesjö (2009)

The developed software uses the algorithm illustrated as a diagram in Figure A.6. In the following subsections each block of the diagram will be discussed in detail.

A.1 Generating incident photon energy uence spectrum

Published Monte Carlo simulated incident photon energy spectra (Mohan et al., 1985), specied at the central axis, were converted to relative spectra, dierentiated in the direct photon beam in energy bins from 50 keV to 19950 keV in steps of 100

keV (∆ED).

The published spectra beam qualities were specied according to their nominal energy. This kind of specication may not accurately represent the actual beam

characteristics. Therefore, the beam quality was converted to TPR20,10 based on

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based on the same algorithm used by the software. 0 5 10 15 20 25 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 Energy [MeV]

Relative energy fluence

TPR = 0.627 TPR = 0.671 TPR = 0.730 TPR = 0.765 TPR = 0.805

Figure A.1: Relative incident photon energy uence spectra (Mohan et al., 1985) used by the

software with TPR20,10 as beam quality.

This block generates an incident relative photon energy uence spectrum, based on

any TPR20,10 value between 0.626 (≈ 4 MV) to 0.789 (≈ 20 MV). The spectrum is

linearly interpolated from ve published spectra, shown in Figure A.1.

A.2 Polyenergetic uence pencil beam kernels

The software is connected to a large database of monoenergetic uence pencil beam kernels, a concept introduced by Eklund and Ahnesjö (2008). These kernels were made from Monte Carlo simulated uence distributions by Eklund and Ahnesjö (2008).

The simulations were done with monoenergetic photon pencil beams incident per-pendicular to the surface of a semi-innite water slab, in discrete energies from

50 keV to 19950 keV (ED) in steps of 100 keV (∆ED). The simulated uences were

scored in ring voxels with a radial ring thickness of 0.1 cm (∆r) and a depth thickness of 0.1 cm (∆d), dierentiated with respect to the energy of the scattered particles in

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is illustrated in Figure A.2.

The monoenergetic uence pencil beam kernels in the database are related to re-spective simulated uence according to the following equations:

φp_E_D_,E_S(r, d) = Φ p ES(r, d) ED· n [cm−2· keV−2] (A.1) ψ_Ep_D_,E_S(r, d) = Ψ p ES(r, d) ED· n [cm−2_{· keV}−1] (A.2) ψ_Es_D_,E_S(r, d) = Ψ s ES(r, d) ED· n [cm−2_{· keV}−1] (A.3) where Φp

ES is the simulated primary electron particle uence, Ψ

p

ES is the simulated

primary photon energy uence, Ψs

ES is the simulated scattered photon energy uence,

r is the radial distance from the primary pencil beam, d is the depth in water, and

n is the number of simulated incident primary photons.

The algorithm takes advantage of the faster calculations provided by the use of polyenergetic kernels. The polyenergetic kernels are precalculated and stored in the database using the calculated incident relative photon energy uence spectrum

(ΨED/Ψ0) to weight the monoenergetic kernels,

φ[EA,Emax],p ES Ψ0 (r, d) = EXmax ED= EA ΨED Ψ0 φp_E_D_,E_S(r, d) [cm−2_{· keV}−2] (A.4) ψ[0,EA],p ES Ψ0 (r, d) = EA X ED= 0 ΨED Ψ0 ψ_Ep_D_,E_S(r, d) [cm−2· keV−1] (A.5) ψs ES Ψ0 (r, d) = EXmax ED= 0 ΨED Ψ0 ψ_Es_D_,E_S(r, d) [cm−2· keV−1] (A.6) where φ[EA,Emax],p

ES /Ψ0(r, d)is the polyenergetic primary electron particle uence

ker-nel generated by primary photons of energies between EAand Emax, ψE[0,ES A],p/Ψ0(r, d)

is the polyenergetic primary photon energy uence kernel generated by primary

photons of energies between 0 and EA, ψEsS/Ψ0(r, d) is the polyenergetic scattered

photon energy uence kernel, EA is the partitioning energy (denition in section

2.3.2), Emax is the upper energy limit of ΨED and Ψ0 is the total incident photon

energy uence (Ψ0 =

R

ΨEDdE).

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∆d

∆r

Figure A.2: The cylinder geometry of the pencil beams, where ∆r = 0.1 cm is the radial thickness and ∆d = 0.1 cm is the depth thickness of each ring voxel. Each voxel contains a uence spectrum.

only once, it will then be stored in the database and can be accessed at any time, thereby reducing the computation time.

A.3 Generating eld shape

The developed software can generate circular elds (specied according to radius), or may import RTP (RadioTherapy Plan) les exported from MOSAIQ v1.6 (Oncology management system by IMPAC Medical Systems), that contain the eld specica-tion informaspecica-tion. The software may only import StepNShoot IMRT elds and static elds. The exported .RTP les are written in the ASCII format that contain in-formation about an entire plan. For each eld and/or segment there is inin-formation about the position at isocenter of the jaws, the multileaf collimator (MLC) settings and the collimator angle.

This block returns a boolean matrix, that describes the eld shape according to the position at isocenter of the jaws and the collimator leafs. Each element in this eld

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A.4 Extracting measurement positions and readouts

ASCII les (.ASC) exported from OmniPro-Accept 6.4A (Scanditronix-Wellhöfer), containing information from measurements made with the RFA 300 water phantom (Scanditronix), may be imported by the software. The les contain information about the three dimensional measurement positions r = (x, y, d) and respective readouts.

A.5 Field projection

The imported eld shape, i.e., the generated eld matrix (section A.3) is specied at the source to isocenter distance (SID). However, the size of the eld will vary at dierent depths, due to beam divergence. This is taken into account by scaling the pixel area, Apixel(d) = Apixel,SID· SSD + d SID 2 (A.7)

where Apixel(d) is the pixel area at depth d, Apixel,SID is the pixel area at SID, and

SSD is the source to surface distance.

By scaling each pixel according to equation (A.7), the entire eld is projected from SID to the measurement depth d.

A.6 Superimposing kernel ring structure

The polyenergetic uence kernels with the dimensions described in section A.2 are superimposed on the projected eld, with the kernels centered at the measurement

position r = (x, y, d). The horizontal cross section area, ARi, of each superimposed

ring voxel, Ri, is calculated by assigning each eld matrix element area, i.e., pixel

area (Apixel) to a ring voxel. Each pixel area is assigned according to the position

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can be described in terms of sets, as Apixel(xp, yp, d)∈ ARi if      ri−1 ≤ rpixel < ri (xp, yp)∈ F (d) (A.8)

where rpixel is the radial distance from the measurement position to the central point

of the pixel, ri−1 and ri is the radial distance from the measurement position to the

lower boundary (i − 1) respectively upper boundary (i) of ring voxel Ri and F (d) is

the set that contains all points within the eld contour.

The ring area ARi is calculated by summing the area of all pixels assigned to ring

voxel Ri,

ARi ≈

X Apixel∈ARi

Apixel(xp, yp, d) [cm2] (A.9)

To make accurate calculations of the ring area, it is necessary that the constituent

pixels (Apixel) are made small enough to distinguish the ner details of the ring

contour. Since the rings are smaller close to the measuring position, and since these rings represent a large part of the total uence due to the large contribution from the primary particles, these rings are more sensitive to inaccurate area approximation. The algorithm uses two dierent pixel sizes (specied at isocenter), a smaller size of 0.001 cm in a 2 cm x 2 cm grid centered in the measurement position and a larger size of 0.01 cm outside the small grid. This approach enables a high accuracy while maintaining a reasonable computation time.

This block generates the ring area within the projected eld, ARi, of each uence

kernel ring voxel, Ri.

A.7 Fluence spectra calculations

The primary electron particle uence spectrum (Φ[EA,Emax],p

E ) generated by primary

photons of energies between [EA, Emax], the primary photon energy uence spectrum

(Ψ[0,EA],p

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R1 R2 R3 R4 R5 rpixel

r4

(xp, yp)

Figure A.3: Illustration of the kernel ring structure superimposed on the projected eld, centered at the measurement position with a pixel grid consisting of two dierent pixel sizes. See text for symbol denitions.

The ring area ARi is calculated by summing the area of all pixels assigned to ring

voxel Ri

ARi ≈

X Apixel∈A_Ri

Apixel(xp, yp, d) [cm2] (A.9)

To make accurate calculations of the ring area it is necessary that the constituent

pixels (Apixel) are made small enough to distinguish the ner details of the ring

contour. The rings are smaller close to the measuring position, therefore requiring a smaller pixel size. These rings also represent a large part of the total relative uence due to the contribution from the primary relative uence, therefore being more sensitive to inaccurate area approximation.

The algorithm uses two dierent pixel sizes, a smaller size (0.001 cm at isocenter) in a 2 cm x 2 cm grid (at isocenter) centered in the measurement position and a larger size (0.01 cm at isocenter) in the rest of the eld matrix. This approach enables a high accuracy while maintaining a reasonable computing time.

Figure A.3: Illustration of the kernel ring structure superimposed on the projected eld, centered at the measurement position with a pixel grid consisting of two dierent pixel sizes. See text for symbol denitions.

are calculated at the measurement position r = (x, y, d) by this block.

The uence spectra are calculated from the polyenergetic kernels and the calculated

ring area ARi, implementing the reciprocity theorem (Ahnesjö and Aspradakis, 1999)

and the inverse square law, as

Φ[EA,Emax],p E Ψ0 (r) = RXmax r = R1 φ[EA,Emax],p ES (r, d) Ψ0 SID SSD + d 2 · Ar, [keV−2] (A.10) Ψ[0,EA],p E Ψ0 (r) = RXmax r = R1 ψ[0,EA],p ES (r, d) Ψ0 SID SSD + d 2 · Ar, [keV−1] (A.11) and Ψs E Ψ0 (r) = RXmax r = R1 ψs ES(r, d) Ψ0 SID SSD + d 2 · Ar, [keV−1] (A.12)

where the uence spectra are dierentiated in energy E, corresponding to the same

dierentiation as that in energy ES for the scattered particles.

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due to the eects of direct source obscuring (Zhu et al., 2009), collimator leakage, attening lter, and other headscatter sources (Ahnesjö and Aspradakis, 1999). The uence calculation algorithm does not take these eects into consideration, therefore assuming a constant incident spectrum distribution over the eld. This assumption was expected to have a minor impact on the nal correction factor, due to the re-sponse factor being a silicon/water dose ratio where the doses are calculated from the same calculated uence spectra (equation (A.14)).

It was also assumed that all ring voxels are located at the same depth d, thereby

ignoring the dierence in depth (d0₎ _{at a distance from the central axis due to the}

beam divergence, illustrated in Figure A.4. This approximation was assumed to have a negligible eect on the result.

d d’

Figure A.4: Illustration of the beam

diver-gence, where depth d < d0_.

A.8 Response model

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The calculation of the K(E) factor, i.e., calculation of the primary silicon dose to silicon collision kerma ratio was performed by Eklund and Ahnesjö (2009). The sili-con dose was Monte Carlo simulated for a Scanditronix EFD silisili-con diode geometry, i.e., the correction factor is optimized for this specic silicon diode, as the dose to a silicon disk in water deposited by monoenergetic primary photons and the electrons released by them up to an incident photon energy of 13.45 MeV. The silicon colli-sion kerma was derived analytically by the use of the mass attenuation coecient on the monoenergetic primary photon energy uence and the energy absorption coe-cients. When the primary photon energy increases, the electron ranges also increases and the K(E) asymptotic approaches a small cavity. A function was tted to the calculated K(E) (Eklund and Ahnesjö, 2009),

K(E) = a0+ ea1·E+ a2ea3·E

2

(A.13)

a0 =−0.135, a1 =−0.0104 [MeV−1], a2 = 0.135, a3 =−0.0183 [MeV−2]

where E is the energy of the photons depositing dose in the silicon cavity. The tted function is shown in Figure A.5.

0 5 10 15 0.5 0.6 0.7 0.8 0.9 1 Energy [MeV] K(E) Fitted K(E) (µ_en)_w,Si / (L_∆)_w,Si

Student Version of MATLAB

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The response factor (RFcalc₎ _{(according to equation (2.14)) is implemented in the} software as RFcalc = PEmax ∆ Φ[EA,Emax],p_E Ψ0 L∆ ρ Si∆E + (T.E.)Si + PEmax 0 K(E) Ψ[0,EA],p_E Ψ0 + Ψs E Ψ0 µen ρ Si∆E PEmax ∆ Φ[EA,Emax],p_E Ψ0 L∆ ρ w∆E + (T.E.)w + PEmax 0 Ψ[0,EA],p_E Ψ0 + Ψs E Ψ0 µen ρ w∆E (A.14) where T.E. is the tracks end term with mass collision stopping power for the respec-tive material; (T.E.) = Φ [EA,Emax],p E Ψ0 (∆) Scol(∆) ρ ∆, (A.15)

where ∆ = 189 keV is the cut-o energy and EA= 5050 keV as proposed by Eklund

and Ahnesjö (2009). The EA-value was determined by experiments and the

∆-value correspond to the energy of electrons with range about 0.5 mm in silicon (the thickness of the chip).

This block generates a correction factor (CFcalc_{), dened as}

CFcalc = 1

RFcalc. (A.16)

A.9 Corrected measurement readout

The correction factor CFcalc_{(A, r), is normalized to the same specied reference}

condition as the measured shielded silicon diode readout, of reference eld Aref, and

reference depth at the central axis rref, as

CFcalc_norm(A, r) = CF

calc_{(A, r)}

CFcalc_(A

ref, rref) (A.17)

where A is the generated eld (section A.3). The readout is then corrected as

Ddiode_norm,corr(A, r) = Ddiode_norm(A, r)· CFcalc

norm(A, r) (A.18)

where Ddiode

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B Spencer-Attix Water/Air Stopping Power

Ratio Calculation

The diode correction software was extended to include a correction factor for change in Spencer-Attix water/air stopping power ratio, for measurements made with an ionization chamber. The Stopping Power correction factor algorithm (Figure B.1) is a slight modication of the silicon diode energy dependence correction factor algorithm (Figure A.6). Therefore, in this section only the dierences between these two algorithms will be discussed, for detailed information about each block in Figure B.1 refer to Appendix A.

The Spencer-Attix stopping power block in Figure B.1 is implemented according to Spencer-Attix cavity theory (dened in section 2.2.2), as

sw,air= PEmax ∆ ΦE Ψ0 L∆ ρ w∆E + ΦE Ψ0(∆) Scol(∆) ρ w∆ PEmax ∆ ΦE Ψ0 L∆ ρ air∆E + ΦE Ψ0(∆) Scol(∆) ρ air∆ (B.1)

where ∆ = 10 keV is the cut-o energy and ΦE is the total electron particle uence

spectrum, dened according to the notations and denitions detailed in section A.7, as ΦE(r) Ψ0 = Φ [0,Emax],p E (r) Ψ0 + Φ [0,Emax],s E (r) Ψ0 [keV−2] (B.2)

The readout is then corrected as

D_norm,corrIC (A, r) = D_normIC (A, r)· sw,air(A, r)

sw,air(Aref, rref) (B.3)

where DIC

norm,corr(A, r)is the corrected normalized ionization chamber readout, DICnorm(A, r)

is the measured normalized readout, Aref = is the specied reference eld, and rref

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C Acronyms

CAX central axis CF correction factor

CPE charged particle equilibrium

IMRT intensity modulated radiation therapy LEE lateral electron equilibrium

MLC multi leaf collimator MU monitor unit

OAX o axis RF response factor

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