Improving Ion Computed Tomography

PhD dissertation

David Christoffer Hansen

Health Aarhus University

2014

Improving Ion Computed Tomography

PhD dissertation

David Christoffer Hansen

Health Aarhus University

Experimental Clinical Oncology

P U B L I C AT I O N S

The work in this dissertation is based on the following four publica-tions:

Optimizing SHIELD-HIT for carbon ion treatment - DC Hansen, ALühr, N Sobolevsky, N Bassler 2012, Physics in Medicine and Biology,57 2393

Improved proton computed tomography by dual modality imagereconstruction - DC Hansen, JBB Petersen, N Bassler, TS Sørensen2014, Medical Physics, 41 031904

The image quality of ion computed tomography at clinical imagingdose levels - DC Hansen, N Bassler, TS Sørensen, J Seco 2014, submit-ted to Medical Physics

A comparison of dual energy CT and proton CT for stopping powerestimation - DC Hansen, J Seco, TS Sørensen, JBB Petersen, JW Wild-berger, F Verhaegen, G Landry 2014, submitted to Medical Physics

iii

The trouble with having an open mind, of course,is that people will insist on coming along

and trying to put things in it.- Terry Pratchett

—

A C K N O W L E D G M E N T S

I would like to thank main supervisor Thomas Sangild Sørensen forhis great help and support during my time as a Ph.D student. Ithas been a pleasure working with you and I wish you all the best inCopenhagen.

My co-supervisors, Jørgen Breede Baltzer Petersen who got me star-ted in the field of ion CT and Niels Bassler, who took me under hiswing and got me into research. A special thanks to Armin Lühr forteaching me how to write a paper and how to wonder.

The exchange-students at MGH in Boston with whom I had a great6 months, particularly Nora Hünemohr and Guillaume Landry. Iwould also like to thank Joao Seco, for having me there in the firstplace.

Simon Rit without whose help in implementing the path-filteredbackprojection I would have come to some entirely wrong conclu-sions.

Jens Overgaard for helping me get funding for the last three years,and Lene Skovlund Jensen for her aid in my battles with bureaucracy.

My parents, for supporting and believing in me.Finally, I would like to thank my girlfriend Mette for her love and

support during my studies and her patience during my periods ofwriting and particularly my periods of not writing.

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C O N T E N T S

english summary xidanish summary / dansk resumé xiii1 introduction 1

1.1 Ion CT 31.1.1 Ion CT in the third millennium 41.1.2 Limitations of ion CT 5

1.2 Aim 6

i theory and methods 92 the interactions of ions with matter 11

2.1 The range of an ion 112.1.1 Range straggling 12

2.2 Scattering 123 tomographic reconstruction 15

3.1 The system transform of ion CT 163.2 Iterative reconstructions 17

3.2.1 Nonlinear reconstruction 173.2.2 Compressed Sensing 18

3.3 Direct reconstruction 194 simulation and preparation of data 21

4.1 Simulating an Ion CT scanner 214.2 Preprocessing 224.3 Reconstruction on the GPU 22

ii results and discussion 255 ion ct reconstruction from noisy data 276 summary of article results 317 discussion 39

7.1 Validity of the Monte Carlo simulations 397.2 The articles 40

8 future directions 43

iii articles 451 optimizing shield-hit for carbon ion treatment 472 improved proton computed tomography by dual

modality image reconstruction 673 the image quality of ion computed tomography

at clinical imaging dose levels 854 a comparison of dual energy ct and proton ct

for stopping power estimation 107

bibliography 122

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L I S T O F F I G U R E S

Figure 1 Illustration of the problem of the scattering oflight ions in matter. The black path demon-strates the ion trajectory through the red phantom.The blue line shows where the ion would havehit on the detector if scattering could be ig-nored. Scattering has been exaggerated to il-lustrate the point. 4

Figure 2 Setup of modern day proton computed tomo-graphy (CT), consisting of two position detect-ors (usually silicon strip) in front of the targetand two after, in order to record both positionand direction. This is then followed by a calor-imeter measuring the energy (usually a scintil-lating crystal). 6

Figure 3 Simplified detector setup used in the Geant4Monte Carlo simulations of article 2 and 3. Con-sists of two ideal detectors, one before and oneafter the phantom, both measuring position,kinetic energy and direction of each individualparticle. 22

Figure 4 Resolution as defined by the modulation trans-fer function (MTF) 10% point as a function ofthe number of iterations for three different it-erative solvers: weighted least squares (WLS),nonlinear conjugate gradient total variation (NCG-TV) and DROP with total variation superior-ization (DROP-TV). This is compared with theRit-filtered backprojection (FBP) method. 28

Figure 5 Reconstruction error as a function of the num-ber of iterations for three different iterative solv-ers: weighted least squares (WLS), nonlinearconjugate gradient total variation (NCG-TV) andDROP with total variation superiorization (DROP-TV). This is compared with the Rit-FBP method.28

Figure 6 Article 1: partial charge-changing cross sec-tion of 12C on water for SHIELD-HIT08 andSHIELD-HIT10A compared with the experimentaldata (Toshito et al., 2007; Golovchenko et al.,2002) used for adjusting SHIELD-HIT10A. 32

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Figure 7 Article 1: relative yield of fragments emittedwithin a 10◦ forward angle from a 400MeV u−1

12C beam in water, compared with experimentaldata (Haettner et al., 2006). Dashed line: SHIELD-HIT08 simulation. Solid line: SHIELD-HIT10Asimulation. The markers are experimental data.Where error bars are not visible, they are smal-ler than the markers. 32

Figure 8 Article 2: boxplot showing the root mean square(RMS) error of the DMR reconstruction meth-ods in the inserts of the density slice of the Cat-phan phantom used for evaluation. The valuesbelow the material names are the ground truthrelative stopping powers. 34

Figure 9 Article 2: the MTF of the images reconstructedwith the dual modality reconstruction (DMR)method, as well as the MTF of the cone-beamCT prior. The area below the cone-beam CTMTF is shaded for ease of comparison. 34

Figure 10 Article 3: modulation transfer function for fourdifferent combinations of ion species and en-ergy. 36

Figure 11 Article 3: mean systematic error as a functionof dose, plotted for different ion species andenergies. The larger markers indicate the pointof 10mSv calculated using the ICRP weightingfactors (ICRP-60, 1990). 36

Figure 12 Article 4: residual errors in stopping power es-timates based on dual energy CT (DECT), singleenergy CT (SECT) and proton CT in each of theinserts of the Gammex phantom. Note that theDECT, and to some degree the SECT data werefitted to the inserts in this phantom. Errorbarsshow the 1σ confidence interval. 38

Figure 13 Article 4: residual errors in stopping power es-timates based on DECT, SECT and proton CT ineach of the inserts of the CIRS phantom. Error-bars show the 1σ confidence interval. 38

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A C R O N Y M S

CT computed tomography

CTDI CT dose index

CTDEI CT dose equivalent index

DECT dual energy CT

DMR dual modality reconstruction

FBP filtered backprojection

I-value mean excitation potential

MLP most likely path

MTF modulation transfer function

LET linear energy transfer

RMS root mean square

SECT single energy CT

WEPL water equivalent path length

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E N G L I S H S U M M A RY

Radiotherapy using ionized particles, known as particle therapy, providesa more accurate and gentle treatment than conventional radiotherapyusing x-rays. This is mainly due to the finite range of the particlesinside the patient, resulting in the patient receiving no, or very little,dose downstream of the tumour. The accuracy of particle therapy ishowever limited by the precision with which the particle range canbe predicted inside patients. Currently this is done based on x-raycomputed tomography (CT) scans, but the inherent physics of theproblem limit the precision to about 3.5%.

One of the alternatives that have been proposed to solve this, is theuse of ion computed tomography (ion CT). Ion CT uses the treatmentbeam in particle therapy to do imaging through measurement of theenergy loss, and is able to achieve sub-percent precision. It doeshowever have several limitations, including the limited range of theparticles, restricting the scanned objects to a diameter of 30− 37cm,depending on the facility.

In this PhD dissertation, several of these limitations are addressed.A new algorithm, combining ion CT with x-ray cone-beam CT hasbeen developed, which helps compensate for the limited range. Withthe new algorithm, high quality images could be reconstructed evenin cases where ion CT projections could only be obtained in 50% ofthe angular interval.

Accurate simulation of ion CT is highly dependent on the modelsof nuclear fragmentation. These were studied in the Monte Carlocode SHIELD-HIT, and the models were adjusted to experimentalpartial cross sections. When compared with experiments measuringnuclear fragmentation in water, the models were found to be accuratefor carbon ions, but with some deviations for neon.

For ions heavier than protons, the doses for ion CT reported in liter-ature are too high to be of clinical use and in several cases larger thanan actual treatment dose. It was unclear whether this was an inherentlimitation of using these heavier ions, or simply the result of the ex-perimental procedure used. A methodology for comparing the dosefrom ion CT with standard CT was developed and used to comparedifferent ion at clinical dose levels. It was found that protons allowedfor the lowest imaging dose, but at higher doses, similar to x-ray CT,helium and carbon ions gave images with less noise and better res-olution. It was concluded that helium is a good candidate for ionCT, as it gave the best compromise between dose and image quality.Another proposed solution to the problem of range estimation in lit-erature is dual energy x-ray CT, which allows a more accurate range

xi

calculation than conventional x-ray CT. A detailed comparison wasmade between ion CT using protons and dual energy CT. While dualenergy CT showed a significant improvement over standard x-ray CT,it still showed deviations in the order of 1.0%. Proton CT on the otherhand showed maximum deviations of 0.3%.

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D A N I S H S U M M A RY / D A N S K R E S U M É

Strålebehandling med brug af ioner, kendt som partikelterapi, giveren mere præcis og skånsom behandling end traditionel stråleterapimed hårde røntgenstråler. Dette skyldes, at partiklerne bremses nedog stopper inden i patienten. Dermed bliver området på den anden si-de af tumoren ikke, eller kun meget lidt, bestrålet. For at dette kan ud-nyttes til fulde, kræves det, at man med stor nøjagtighed kan beregne,hvor partiklerne stopper i patienten. I partikelterapi gøres dette nor-malt ud fra røntgen CT skanninger, men her er præcisionen begræn-set til omkring 3.5% af den underliggende fysik. En mulig løsninger i stedet sat bruge ion CT, hvor patienten skannes med de sammepartikler, der også bruges til selve strålebehandlingen. Med ion CTkan man beregne partiklernes rækkevidde med en præcision, der erbedre end 1.0%, men ion CT har også visse begrænsninger. Blandtandet gør den begrænsede rækkevidde af ionerne, at man ikke kanskanne områder tykkere end 30− 37cm, alt afhængig af acceleratorenpå behandlingsstedet.

I denne ph.d afhandling undersøges flere af disse begrænsninger.Her præsenteres en ny algoritme, som kombinere ion CT med derøntgenbilleder, der laves i behandlingsrummet. Dermed kan manrekonstruere ion CT billeder, selvom partiklerne kun kan nå igennempatienten i halvdelen af de vinkler, man normalt ville skanne fra.

Simulationer af ion CT er i høj grad afhængige af de fysiske model-ler, der bruges til kerne-spaltning. Modellerne blev i denne afhand-ling undersøgt i Monte Carlo koden SHIELD-HIT, og de blev tilpasseteksperimentelle tværsnit. Modellerne passede godt med eksperimen-telle målinger af kernefragmentation af kulstof i vand, hvorimod dervar større afvigelser for neon. I tidligere undersøgelser af ion CT medioner tungere end brint har dosis altid været meget høj, i flere tilfæl-de højere end den dosis man normalt ville bruge til behandling afkræft. Det var uklart i hvor høj grad dette skyldes de metoder, derblev anvendt, og sammenligningen med de doser, der normalt givesunder røntgen CT blev yderligere besværliggjort af, at stråling medioner normalt anses for mere skadelig end røntgenstråling, og derforikke er direkte sammenlignelig. For at undersøge dette nærmere blevder udviklet en model, der gjorde det muligt at sammenligne dosisfra ion CT og røntgen CT direkte. Med protoner kunne der opnåsgod billedkvalitet ved selv meget lave stråledoser, men ved højeredoser, svarende til dem der normalt bruges ved røntgen CT, gav bå-de helium og kulstof CT billeder med højere opløsning og mindrestøj. Et alternativ til ion CT er "dual energy CT", dvs røntgen CT vedto forskellige bølgelængder. Dette giver også mulighed for en bedre

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bestemmelse af partiklernes rækkevidde, og der blev derfor lavet endetaljeret sammenligning med proton CT. Dual energy CT gav væ-sentligt bedre resultater en normal røntgen CT, men viste stadig fejlstørre end 1.0%. Til sammenligning var den maksimale afvigelse forproton CT mindre end 0.3%.

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1I N T R O D U C T I O N

In the last two decades radiotherapy using ions, commonly referredto as particle therapy, has become increasingly popular. Over 60000patients have been treated since 2007, more than in the 50 years priorcombined (Particle Therapy Co-Operative Group (PTCOG), 2014). Themost common form of particle therapy is performed with hydrogenions (protons) and has several advantages over traditional radiother-apy using mega-voltage (MV) photons (Durante and Loeffler, 2009).

The primary advantage is a superior dose distribution, as particletherapy irradiates a smaller volume of healthy tissue during treat-ment. This is due to the finite range of ions in matter. By carefullyselecting the energy of the incoming ions, such that they stop insidethe area to be irradiated, no dose is delivered to the tissue down-stream from the target. This has two implications. First of all, a lowerdose to the healthy tissues leads to fewer side effects of the treatment.Secondly, since the dose to tumour is often limited by what is toler-ated by the normal tissue, the target dose can be increased leading tobetter tumour control for some tumours.

However, the high accuracy also makes particle therapy less robustto uncertainties in the treatment planning. If the estimated density ofa patient is off by 5% in conventional radiotherapy the resulting errorin the actual dose will be of a similar magnitude. On the other hand,if the estimated density of a patient is off by 5% in particle therapythe range of the particles will be off by 5%. This can result in part ofthe tumour receiving no dose at all or in very high dose being givento surrounding radiosensitive organs. Lomax (2008) studied the ef-fects of treatment uncertainties in proton therapy and found a ±3%range uncertainty to lead to a 5% underdosage of the target volumeand a 6% overdosage of a radiosensitive organ in the studied patient.The impact of range uncertainty is however specific to each anatom-ical site. In a lung tumour for instance, a 1% error in the estimatedstopping power of the ribs can lead to a much larger dose error in thelung and internal organs due to the high density gradients (Schue-mann et al., 2014). Clinically, all such errors are handled by irradiat-ing an extended volume around the tumour to ensure target coverageand by avoiding beam angles where radiosensitive organs could behit. Such increased margins implicate an increased dose to nearbyhealthy organs. If only a few beams are used for the treatment, theresulting dose will be close to the full prescription dose (Seco et al.,2012) which can lead to severe side-effects in the patient.

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The current state-of-the-art for range calculation is known as thestoichiometric method (Schneider et al., 1996), in which x-ray CTscans of a phantom with known composition are made and a cal-ibration curve is calculated based on a set of standard human tissues.The uncertainty of this method is estimated to be about 3.5% (Yanget al., 2012). A recent study (Moyers, 2014) revealed that the stoppingpower predicted from x-ray CT varies significantly (3%± 25%) fromone treatment facility to another however, indicating that the clinicaluncertainties may be even higher. Part of the problem is that fun-damental quantities, such as the mean excitation potential (I-value) ofhuman tissue, are simply not known to high precision. Because ofthis, even if the calibration itself was perfect, the uncertainty wouldstill be on the order of 2% (Besemer et al., 2013). Another problemis that the stoichiometric method only works on the human-like ma-terials on which it was calibrated. Surgical implants in and on thepatient will therefore have significantly larger errors for the range cal-culations. This effectively prevents treatment through such implantsunless their water equivalent thickness is known from other sources.The problem is most severe in the case of metal implants, which affectthe entire CT image due to metal artefacts. This can introduce errorsas large as 18% in the range calculation for particle therapy (Jäkel andReiss, 2007).

Many techniques have been suggested to improve the range estim-ation (Knopf and Lomax, 2013). They can generally be put in one oftwo categories: in vivo dose measurement and in vivo stopping powermeasurement. In the former methods, the main goal is to measurethe delivered dose in the patient. Any systematic difference betweenthe delivered and the expected dose would result in a change of thetreatment plan for the next dose fraction. While this is naturally lessdesirable than getting the dose exactly right from the first fraction,this strategy has the advantage that it takes into account all sources oferror including positioning and delivery. The two most promising ap-proaches are measurement of the PET isotopes produced during treat-ment (Bennett et al., 1978) and measurement of the prompt gammaemissions (Min et al., 2006). Both methods face the issue that the cor-relation between the measured quantity and dose is non-trivial, andhigh spatial accuracy is difficult to obtain with current detectors.

This work focuses on the second category of range estimation, stop-ping power measurements, for which two promising techniques exist.The first is based on DECT. By obtaining DECT scans at two differ-ent spectra, electron density and mean ionization potential of thetissue can be obtained through a few approximations (Yang et al.,2010; Hünemohr et al., 2014). Stopping power can then be calculatedanalytically via the Bethe-Bloch equation. The second method is ioncomputed tomography (ion CT), which is the main topic of this thesis.

2

1.1 ion ct

Ion CT using protons was originally proposed by Cormack (1963), inthe article which also laid the mathematical foundation for x-ray CT.Cormack proposed that proton CT would be useful for the otherwisequite complicated problem of estimating the range of radiotherapyions in patients. Half a century later, this remains the primary motiv-ation for the research in proton and ion CT. Ion CT is based on thesame basic principle that is the foundation of x-ray CT: the measure-ment of a linear quantity integrated along a line. In x-ray CT, x-raysare sent through the patient, the surviving fraction measured and log-transformed to give the integrated x-ray attenuation. For ion CT, ionsare sent through the patient. Their energy-loss is measured and theirwater equivalent path length (WEPL) is calculated. A 3D image canthen be obtained through tomographic image reconstruction.

A few years after Cormack proposed proton CT, the first protonradiograph was performed by Koehler (1968) in which a sensitiv-ity to density changes as low as 0.2% was demonstrated. This wasfollowed by two papers demonstrating the clinical benefit of protonradiography; one on the detection of strokes (Steward and Koehler,1973b) and one on the detection of tumours (Steward and Koehler,1973a). Crowe et al. (1975) was the first to demonstrate ion CT, us-ing 900MeV helium ions to scan the brain of a patient. Cormackand Koehler (1976) then demonstrated proton CT experimentally, al-beit with a radially symmetrical phantom and relatively low resolu-tion. Hanson (1979) created the first fully 3d proton CT, taking 45hto scan a 30cm diameter phantom. He was also the first to suggestthat proton CT could be made feasible by using the spare capacity ofa radiotherapy accelerator. The spatial resolution in all these caseswas quite poor, due to the multiple scattering of the protons (see fig1). Hanson (1979) reported a resolution of 3.8mm and Takada et al.(1988) reporting one of 3.2mm, using different strategies to improveresolution.

Huesman et al. (1975) was the first to propose measuring the anglesof the protons, in addition to the energy and position, which wouldallow the use of path modelling in the reconstruction and signific-antly improve the spatial resolution. It did however require very fastand accurate detectors that did not exist at the time. A second prob-lem of this approach is that of reconstruction time. The only way todeal with non-straight trajectories (at the time) would have been it-erative reconstruction which would have taken several days with thehardware available then.

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Figure 1: Illustration of the problem of the scattering of light ions in matter.The black path demonstrates the ion trajectory through the redphantom. The blue line shows where the ion would have hit onthe detector if scattering could be ignored. Scattering has beenexaggerated to illustrate the point.

1.1.1 Ion CT in the third millennium

With the advent of modern silicon detectors, used in particle physics,and high performance computers path modelling became a realisticprospect and for this purpose Williams (2004) published a methodfor calculating the most likely path (MLP) of a charged particle frommeasurements of the energy, angle and position. This was tested byLi et al. (2006) using Monte Carlo simulations, and compared with acomputationally simpler approach based on cubic splines interpola-tion. While the MLP yielded a higher resolution than the cubic splinethe difference was small and both methods yielded significantly bet-ter resolution than the conventional straight line approach.

Further research into efficient reconstruction was performed bythe same group. They introduced a more efficient MLP formalism(Schulte et al., 2008), optimized the calculations of the system matrix(Penfold et al., 2009) and investigated more efficient solvers (Penfoldet al., 2010a).

Schneider (1995) proposed that a single proton radiograph couldbe used for calibrating the piece-wise linear fit that is the basis forthe stoichiometric method for estimating stopping power from x-rayCT. This was later realised when the first in vivo proton radiographywas performed on a dog (Schneider et al., 2004) and used for doseplanning (Schneider et al., 2005). Using this method, the differencebetween the WEPL of the proton radiograph and the digitally recon-structed radiograph of the planning CT was reduced from 3.6mm to0.4mm.

Two groups have produced prototype proton CT scanners that al-low for path modelling. One is in California, USA (Hurley et al., 2012)

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and the other in Firenze, Italy (Sadrozinski et al., 2013). Both usingsimilar set-ups as shown on figure 2. Both prototypes are based ontwo pairs of silicon strip detectors providing position and direction ofthe protons as they enter and leave the patient. The second pair of po-sition detectors is followed by a calorimeter made from scintillatingcrystal: a thallium doped caesium iodide crystal in the US prototypeand a cerium doped yttrium aluminium garnet in the Italian.

Ion CT using heavier ions is less affected by scattering due to thelarger momentum of the ions, making a straight line approximationmore reasonable. This allows for the standard reconstruction meth-ods from x-ray CT to be used, and Ohno et al. (2004) demonstratedthis with helium, carbon and neon ions, achieving an electron dens-ity resolution better than 2.0%. This was done using two positiondetectors made of scintillating fibres and a NE102A plastic scintil-lator. The same group subsequently used this setup to verify thex-ray CT based range calculation used in the clinic (Shinoda et al.,2006), finding an agreement of better than 1.6% for fat, muscle andbone. The decreased scattering also allows for the effective use of in-tegrating detectors, rather than particle counting ones, and Muraishiet al. (2009) demonstrated the spatial resolution of 0.6mm for carbonions and 1.0mm for helium ions using a scintillating screen. Telse-meyer et al. (2012) used a standard commercially available flat-paneldetector to do carbon CT, and achieved an agreement of better than0.7% when compared with other experimental stopping power meas-urements. Both of these works reported excessively large doses, withTelsemeyer et al. (2012) reporting a scan dose of 8Gy and Muraishiet al. (2009) a dose of 40Gy, loosely 3 orders of magnitude larger thanthe dose from a standard x-ray CT. In both cases integrating detectorswere used, which require a larger fluence of particles when comparedwith particle counting detectors and thus result in a higher dose.

1.1.2 Limitations of ion CT

Ion CT requires an accelerator and a gantry to deliver the beam forthe tomography. As these are still very expensive, even in the contextof medical scanners, the only feasible way to implement this is to usethe equipment already in the hospitals for particle therapy. This iseven more true for ions heavier than protons, where only a singlegantry exists for carbon ion therapy(Haberer et al., 2004), though asecond is under construction(Shirai et al., 2011). This imposes severallimitations. First of all, accelerators for proton therapy typically havea maximum energy of about 230MeV (Particle Therapy Co-OperativeGroup (PTCOG), 2014), corresponding to a range of roughly 32cm.While this would be sufficient for scanning most paediatric patients,as well as scans of the head, many other treatment sites could notbe scanned from all angles at this energy. Newer facilities tend to

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Figure 2: Setup of modern day proton CT, consisting of two position detect-ors (usually silicon strip) in front of the target and two after, inorder to record both position and direction. This is then followedby a calorimeter measuring the energy (usually a scintillating crys-tal).

have a higher maximal energy of about 250MeV and a few as high as330MeV. For carbon ion facilities, similar circumstances apply, with amaximum range of about 31cm. A second limitation is the lateral fieldsize, which may not be large enough to cover the patient, particularlyin compact facilities.

1.2 aim

While the theoretical advantages of ion CT were well established,many questions remained unanswered and motivated this thesis. Theaim of this thesis is thus to investigate ion CT for dose calculation inparticle therapy, and to develop methods to help quantify and over-come the challenges baring clinical use.

The main hypothesis is that ion CT can be made suitable for clinicaluse when considering both dose and the limited range of ions inmatter. Additionally, it is hypothesised that it compares favourablewith other imaging modalities that could be used for stopping powerestimation.

The aim of the first article is to investigate the nuclear models em-ployed in Monte Carlo transport codes, specifically in the SHIELD-HIT code. While SHIELD-HIT was ultimately not used in the follow-ing articles, the issues studied remain important for the accuracy ofthe Monte Carlo simulations subsequently performed.

The aim of the second article is to develop a method to overcomethe limited range of ion CT by allowing ion CT to be made froma limited span of angles. This was achieved by including an x-raycone-beam CT prior in the reconstruction.

The aim of the third article is to develop a method for evaluatingthe dose of ion CT, such that it can directly be compared to x-rayCT with respect to the risk of carcinogenesis. This was subsequently

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used to compare the image quality of ion CT with different ions, atthe same dose equivalent.

The aim of the fourth article is to compare ion CT with dual energyx-ray CT for the use in particle therapy dose planning. Here realisticdetector models were used for the ion CT in order to allow for a faircomparison with the experimental DECT.

The thesis starts with a chapter on the theory and methods used,described in greater detail than in the articles themselves. This isfollowed by a section on the key results, and finally a discussion andfuture works.

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Part I

T H E O RY A N D M E T H O D S

2T H E I N T E R A C T I O N S O F I O N S W I T H M AT T E R

2.1 the range of an ion

The mean energy (E) lost per waylength of an ion moving throughmatter is described by the stopping power S(E) ≡ −dEdx . At the en-ergies commonly used for radiotherapy, the stopping power is welldescribed by the Bethe equation (Bethe, 1930, 1932). It is commonlydescribed in the formulation by Fermi (1950):

S(E) = ρ4πe4

mev21

u

Z

Az2[

ln2mev

2

I+ ln

1

1−β2−β2 −

C

Z−δ

2

], (1)

where

4πe4

mev21

u

Z

Az2 = 0.307075

z2

β2Z

A, (2)

in units of MeVcm/g. e is the electron charge, u the atomic mass unit,z the charge of the ion, me the mass of the electron, v the velocity ofthe ion and β the relativistic beta. Z and A are the charge and atomicmass of the element of the target material. I is the mean excitationenergy of the target medium, C/Z and δ/2 are the shell correctionsand density-effect corrections respectively. The mean energy lost bya charged particle moving a distance ∆x through a specific mediumcan then be found by solving the differential equation given by (1).

Similarly, ignoring the variations in the stopping power due to thestatistical nature of the process, the range of an ion in a particular me-dium can be calculated by simply integrating over the inverse stop-ping power

R(E0) =

E0∫0

S(E)−1dE. (3)

This is generally referred to as the continuous slowing down ap-proximation. Similarly, if the energy of an ion travelling through amedium is known before (E0) and after (E1), the thickness of the me-dium can be calculated by

t(E0,E1) = R(E0) − R(E1) =

E1∫E0

(dE

dx

)−1

dE. (4)

This is the basic principle behind ion CT, where the scanned objectof patient is assumed to be made entirely of a single medium, typ-ically water. The relationship between stopping power and the en-ergy lost after travelling a distance ∆x is non-linear, as the stopping

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power itself depends on the energy at a particular point. If the energymeasurements are instead converted into medium equivalent rangethrough equation (4) the relation becomes linear, assuming the ratiobetween the stopping power of the scanned material and the stoppingpower of water remains constant. This is accurate only if the terms in-side the parenthesis of (1) are equal for the reference material and theactually scanned material. In the energy range used, both the shellcorrection and the density-effect corrections only play a minor rolehowever, so the term is dominated by the I-value. For human tissues,the introduced error is on the order of 0.3%.

2.1.1 Range straggling

As previously mentioned, the energy loss of an ion in matter is astatistical rather than a deterministic process and equation (1) onlyyields the average energy loss. The variation was first described byBohr (1948), who introduced the concept of energy straggling

dσ2E(x)

dx≈ 1

4πε20e4ρe (5)

where ρe is the electron density and ε0 is the vacuum permittivity.This is valid for energy losses large enough that the Gaussian approx-imation holds, yet small enough that the energy of the ion can beassumed constant. A more accurate methodology was establishedby Tschalär (1968b,a). Following Schulte et al. (2005), this gives thedifferential equation

dσ2E(x)

dx= κ(x) − 2

(dS(E(x))

dx

)σ2E (6)

where κ(x) is given by

κ(x) = z2ρeK1− 1/2 ·β21−β2

. (7)

The range straggling as a function of energy is then given by thesolution of the equation

dσ2RdE

=1

S(E)

dσ2E(x)

dx(8)

2.2 scattering

Ions do not move in straight lines through a medium, but are scatteredby many soft collisions with the nuclei of the material. This processis known as multiple Coulomb scattering. Following the theory of

12

Fermi and Eyges (Eyges, 1948; Rossi and Greisen, 1941), scatteringpower of ions in matter is defined as

T ≡ d〈θ2〉

dx(9)

where 〈θ2〉 is the RMS projected multiple Coulomb scattering angle.This is analogue to the definition of stopping power. However T isonly approximately local, in the sense that it not only depends on theenergy of the particle and the composition of the medium, but alsoon the length travelled. A particle which has travelled a length x thenhas the RMS angle

〈θ2〉(x) =∫x0

(x− x ′)T(x, x ′)dx ′ (10)

which is the first Fermi-Eyges moment. Equivalently, a particle withstarting energy E0 has the following RMS angle when reaching energyE1:

〈θ2〉(E0,E1) =∫E1E0

T(E0,E)S(E)

dE. (11)

Gottschalk (2010) contains a thorough comparison of a number ofdifferent formulations for the scattering power T and presents thedifferential Molière scattering power, which is used in this work. It isgiven by

TdM = fdM(pv,p0v0) ·(Esz

pv

)21

Xs(12)

where Xs is the scattering length which is 46.88 cm for water, and

Es =(2πα

)1/2mec

2 with α being the fine structure constant.

fdM = 0.5244+ 0.1975 log

(1−

(pv

p0v0

)2)+ 0.2320 log(pv)

(13)

− 0.0098 log(pv) log

(1−

(pv

p0v0

)2)

with p0,v0, p and v, being the momentum and velocity at the initialand current point respectively in units of MeV. The constants hold noimmediate physical meaning and are simply the results of a fit madefor protons with an energy E0 6 300MeV.

13

3T O M O G R A P H I C R E C O N S T R U C T I O N

The word tomography originates from the greek words tomos, mean-ing section or slice, and graphy, meaning to write or record (Gross-mank, 1935). It refers to the imaging of the interior of an object bysubjecting it to some form of radiation and measuring the response.This includes, but is not limited to, x-ray computed tomography (CT),positron emission tomography (PET) and magnetic resonance ima-ging (MRI). In general, tomographic reconstruction is the process ofrecovering the image that caused a specific response to be observed.Making the assumption that the process involved can be described asa linear transformation A, the problem is then to solve

A(u) = b. (14)

where u is the image to be recovered and b is the measured response.However, this problem only has a unique solution in the case where A

perfectly describes the physical process and b is sufficiently sampledand does not contain any noise.

More generally, a weighted least-squares problem is solved:

minu‖W(A(u) − b)‖2 (15)

where W is the co-variance matrix of the data. Letting f(u) be the in-ner part of the equation ( f(u) = W(A(g)−b) ), it becomes equivalentto solving the problem

∇f(u) = 0 (16)

which yields the normal equation

A†W†WAu = A†W†b (17)

Cormack (1963) was the first to suggest the possibility recoveringa physical object by measurement of a complete set of line integralsthrough the object. Cormack proposed to use either x-ray attenuationor the energy loss of protons as means of obtaining the line integrals.This corresponds to defining A as

A(u) =∫

Luds (18)

where L is the set of lines along which measurement has been done.In the simple case where the line integrals are straight and form a

15

complete basis, A is the Radon transform (Radon, 1917, 1986) andequation (14) can be solved via the inverse Radon transform.

More generally, when the line integrals are not straight, no generalmethod of inversion exists. By writing A as a matrix, it is of coursepossible to solve equation (15) through simple Gaussian elimination.As both u and b can be quite large (106−1010 elements) and Gaussianelimination scales with the number of elements cubed, this wouldquickly become impractical. Instead, the most practical way to solvethis is via iterative methods, as will be described in section 3.2.

3.1 the system transform of ion ct

Due to the scattering, the assumption that light ions move in straightlines through a dense medium is a poor one. Based on the measure-ment of the direction and position of each individual ion before andafter the scanned object, an estimate of the ion path through the objectcan be made. By assuming the scanned object consists of a uniformmedium, the MLP can be calculated, as was first done by Williams(2004) for ion CT. In theory, it would be possible to gain a more ac-curate estimate of the MLP by incorporating different materials withdifferent densities into the calculation. While the problem has an ana-lytical solution (Øverås, 1960), it would involve a number of nestedintegrals equal to the number of different materials and densities. Aseven the single material approach is quite computationally expensive(Li et al., 2006; Karonis et al., 2013), such calculations would effect-ively prohibit ion CT reconstruction on a clinically relevant time-scale.Li et al. (2006) proposed the use of a simple cubic spline as the pathestimate, which was investigated in more detail by Wang et al. (2011).While the MLP does yield a more accurate path estimate and a slightlyhigher resolution, the cubic spline path is consistently within the 99%probability interval and much faster to calculate. For that reason, thecubic spline path is used exclusively in this work.

Using any given path estimate, the i’th entry of A(u) is then givenby

A(u)i =∫10

u(s(t))dt (19)

where u(x) is the medium equivalent stopping power at location xand s(t) is the function defining the path such that s(0) is the startand s(1) the end of the curve. For the cubic spline, this is given by

s(t) = (2t3−3t2+1)x0+(t3−2t2+ t)p0−(2t3−3t2)x1+(t3− t2)p1(20)

where x0, x1, p0 and p1 are the position and directions before andafter the object respectively. The length of p0 and p1 are determined

16

from the fact that equation (20) should be consistent with the straightline case, so ‖p0‖ = ‖p1‖ = ‖x1 − x0‖.

3.2 iterative reconstructions

While an exhaustive summary of iterative methods for inverse prob-lems is beyond the scope of this work, a brief description is in orderdue to the practical importance. Iterative methods for image recon-struction can be split into two groups: unsymmetrical methods andstatistical methods. The unsymmetrical methods attempt to solveequation (14), ignoring measurement error. This includes methodssuch as the algebraic reconstruction technique (ART), the simultan-eous algebraic reconstruction technique (SART) and variations of theiterative filtered backprojection. The statistical methods (in the caseof CT) solve minimization problems such as equation (15). This workprimarily deals with the latter kind.

The simplest way to solve eq. (15) is the gradient descent method,also known as steepest descent, given by

uk = uk−1 −αkgk (21)

where uk is the image at the k’th iteration, gk = ∇f(uk−1) and αk isthe step size. In theory, the step size should be the solution of

minα‖A(uk +αgk) − b‖2 (22)

which has an analytical solution and is known as the Cauchy step size.This converges in the sense that ‖∇f(uk−1)‖ → 0 when k → ∞. Animproved version can be made by changing the search direction gk

in such a way that it is conjugate with all previous search directions.This is known as the conjugate gradient method (Hestenes and Stiefel,1952), and is guaranteed to converge in n iterations, assuming thereare n elements in the image to be reconstructed. In practice, goodresults can be obtained in far fewer iterations. An ad-hoc explana-tion for this is that the frequency of the information in each iterationincreases with the number of iterations. Therefore, later iterationsprimarily give the noise component of the image, which holds littleinterest.

3.2.1 Nonlinear reconstruction

For ion CT, an added constraint preventing negative values should beused, so the problem becomes

minu‖W(A(u) − b)‖2 subject to u > 0 (23)

as negative stopping power is unphysical in patients. This makesthe minimization problem non-linear so no analytical solution to the

17

step-size αk is available. The step-size could instead be found bysolving (22) (subject to the non-negativity constraint) numerically tohigh precision. This is not required however, as convergence for theconjugate gradient method has been proven under much weaker con-ditions, known as the Wolfe conditions:

f(uk−1 +αkgk) 6 f(uk−1) + c1αkgk · ∇f(uk−1) (24)

αkgk · ∇f(uk−1 +αkgk) 6 c2gk · ∇f(uk−1) (25)

where c1 < c2 and both constants are positive and less than 1. To-gether these rules ensure that both the function and the curvaturedecrease sufficiently in each step. The inexact line-search further re-moves the analytical conjugency, and the search direction must beconstructed via

gk = βkgk−1 −∇f(uk). (26)

Here βk is an implementation dependent parameter that reduces tothe standard conjugate gradient method in the linear case. The non-negativity constraint can now be imposed by first doing an iterationignoring the constraint, clamping the negative values to 0 and us-ing the difference between this and the previous iteration as the newsearch direction. A detailed implementation can be found in the ap-pendix of article 2.

3.2.2 Compressed Sensing

The non-linear conjugate gradient can solve any convex optimizationproblem for which the gradient can be evaluated, which means thereconstruction can easily be extended to include things such as com-pressed sensing (Donoho, 2006):

minu‖W(A(u) − b)‖2 + λ‖T(u)‖1 subject to u > 0 (27)

where ‖ ‖1 is the L1-norm defined as the sum of absolute values. T(u)is a transform under which the true, noise-free, image is sparse suchas the wavelet transform or the image gradient (total variation). Thisreduces noise and allows for accurate reconstruction from fewer data.In the case where T(u) is a linear transformation, the gradient is thengiven by

∇‖T(u)‖1 = T†sgnT(u) (28)

where sgn is the elementwise signum function. For functions such asthe image gradient where T(u) is the elementwise norm of n of linearoperators Bk such that

T(u)i =

√√√√n∑k=0

Bk(u)2i (29)

18

the i’th element of the gradient of the function is then given by

∇‖T(u)‖1 =n∑k=0

B†k (Bk(u)� T(u)) (30)

where � indicates elementwise division.

3.3 direct reconstruction

In the simple case of parallel-beam x-ray CT, the image can be recon-structed via the formula

u = R−1(p) = R†(k1D ∗ p) (31)

where R† is the adjoint of the Radon transform, more commonlyknown as the back-projection. k1D ∗ p represents the convolution ofthe projection data p with the 1D kernel k1D. For the exact recon-struction, k1D is the so called ramp-filter, defined in Fourier space assimply the absolute value of the frequency. The combined method isgenerally known as the filtered back-projection. This has been exten-ded to many other geometries, including fan-beam, spiral and cone-beam scans and non-straight integration paths, such as geodesics. Fora full review, see (Helgason, 1999). No general inversion formulaexists for the so called generalized Radon-transform, but Rit et al.(2013) presented an approximate method for use in proton CT. Foreach scanned angle θ, a 3D projection is made via the equation.

pθ = A†θ(bθ)�A

†θ(1) (32)

where A†θ is the path-backprojection from angle θ and bθ refers to the

protons measured from this angle. � denotes elementwise division,and 1 is the vector with 1 in every entry and the same length as b. Thesecond term can then be seen as a normalization and the entire termsimply as the average WEPL through each voxel. If the protons movedon straight trajectories, no variation would be observed along thebeam direction and equation (32) would simply produce standard 2Dtomographic projections. The final reconstruction from t projectionsis then performed via

u =1

t

θt∑θ=0

kθ ∗ pθ (33)

where kθ ∗ pθ is the convolution of the ramp filter orthogonal to thebeam direction and the axis of rotation. In this work, this algorithmis referred to as the Rit-FBP.

19

4S I M U L AT I O N A N D P R E PA R AT I O N O F D ATA

4.1 simulating an ion ct scanner

While the idea to measure the position and angle of every ion wasfirst proposed by Huesman et al. (1975), the design typically usedis based on the proton radiography setup of PSI (Schneider, 1994;Schneider et al., 2004). In this setup, a detector is placed in front ofthe patient and another behind the patient. These measure positionand direction of each individual particle. This is then followed by acalorimeter. In this work, all ion CT data are simulated, using MonteCarlo transport codes. In article 2 and 3, a simplified setup is used(fig. 3), in which detectors are assumed to be perfect, and capableof registering position, direction and energy simultaneously. Detect-ors for proton CT have improved dramatically over the last decade,and is receiving significant amount of attention from many researchgroups (Johnson et al., 2003; Menichelli et al., 2010; Sipala et al., 2011a;Civinini et al., 2013; Poludniowski et al., 2014). By restricting the re-search to simulating any specific detector setup, the results would re-flect the limitations of that particular detector, and potentially not begenerally applicable. In article 4, comparison between experimentaldual energy CT and simulated proton CT are presented, and in or-der to make comparison fair, a more realistic setup is used (fig. 2).This was modelled after the setup in Sadrozinski et al. (2013), andconsist of 4 detectors, two in front and two after, to obtain positionand direction and a calorimeter. While article 1 deals with the nuc-lear models in the Monte Carlo code SHIELD-HIT(Gudowska et al.,2004; Geithner et al., 2006; Hansen et al., 2012) and their accuracy, thiscode was not used in the subsequent articles and instead the MonteCarlo code Geant4(Allison et al., 2006) was used. The reason for thechange was due to the fact that the somewhat complex scoring waseasier to set up in Geant4. This consideration outweighed the fasterruntime and easier geometry setup of SHIELD-HIT. In Geant4, theelectro magnetic physics list 3 was used. Elastic and inelastic nuc-lear collisions were enabled, and the Quantum Molecular Dynamics(QMD)(Niita et al., 1999) model was used for the nuclear fragment-ations of ions heavier than protons. It was shown by Böhlen et al.(2010) that this model was the most accurate for particle therapy ap-plications. As specified in the articles, version 4.96 and 4.10 of Geant4were used.

21

Figure 3: Simplified detector setup used in the Geant4 Monte Carlo simula-tions of article 2 and 3. Consists of two ideal detectors, one beforeand one after the phantom, both measuring position, kinetic en-ergy and direction of each individual particle.

4.2 preprocessing

The preprocessing of the data between the Monte Carlo simulationand the reconstruction was done using the SciPy stack1 in the pro-gramming language Python2.

The energies of the particles were converted to water equivalentrange using the ICRU range tables (International Commision on Ra-diation Units and Measurements, 1993, 2005; Sigmund et al., 2009).A tabulation of the RMS scattering angle in water was calculated bynumerically integrating equation (11). The expected RMS angle forall particles was then calculated via cubic interpolation and particleswhich deviate more than 3σθ are discarded, following the approachof Schulte et al. (2008). The co-variance matrix W was calculated asa diagonal matrix with the range straggling 1

σRin the diagonal. This

was done by solving equation (8) numerically from E0 (the mean en-ergy of the ions) to a number of tabulated values and then evaluatedusing cubic interpolation.

4.3 reconstruction on the gpu

All image reconstruction was done on the GPU, using the Gadget-ron (Hansen and Sørensen, 2013) framework for image reconstruc-tion, which besides MRI has also previusly been used for cone-beamx-ray CT reconstruction. The basis for the implementation of the re-construction is that the ion CT system transform A is never explicitlyformed as a matrix, sparse or otherwise. Instead, only the applicationof A and its adjoint were calculated, or equivalently, the matrix-vectorproduct.

1 http://www.scipy.org

2 https://www.python.org

22

Reconstruction was done on Nvidia GPUs, using the CUDA pro-gramming language. Each ion in the ion CT was mapped to singlethread on the GPU. When calculating the ion CT transform A(u), thepath integral (equation (19)) was calculated discretely in ntot steps,as the sum

A(u)i =ntot∑n=0

‖si(n+1ntot) − si( nntot

)‖+ ‖si( nntot) − si(n−1ntot

)‖2

·u[

si(n

ntot)

]

(34)

where u [x] refers to the value of the image u at the physical coordin-ate x, as calculated by the nearest neighbour approach and si(t) isthe function defining the path of the i’th ion, as defined in equation20. The adjoint transform A†(b) was implemented in the same man-ner, but all reads from the image u were replaced with atomic adds,so that the thread responsible for the i’th ion adds the appropriatelength to each image voxel it passes.

As the number of non-zero elements in a sparse matrix represent-ation of A would be on the order of the number of ions times thewidth of the image (∼ 1010 for a 2D image), the matrix-less approachhas a significant saving in the amount of memory needed. Addition-ally, the main bottleneck for most GPU computation is the memorytransfer on the GPU itself, suggesting that the advantage of a matrix-less approach is greater on the GPU. As this work does not includea matrix-variant of the reconstruction code, it is difficult to estim-ate the advantage, but Karonis et al. (2013) report a reconstructiontime (not including construction of the sparse matrix) of 21.47 s foran optimized sparse matrix algorithm using a cluster with 120 TeslaM2070 Nvidia GPUs. Using the same number of proton histories (131million), the same image resolution (256× 256× 36) and the same al-gorithm (DROP (Penfold et al., 2010b)), the code used in this worktakes 330.29 s on an Nvidia Titan Black GPU. The Titan Black GPUis roughly 5 times faster than the Tesla M2070. Accounting for thedifference in number of GPUs and GPU speed, the code used in thiswork is estimated to be 35% faster than the matrix-version. Using thefiltered-backprojection method of Rit et al. (2013), described in sec-tion 3.3, the reconstruction takes 91.52 s. If the number of GPUs andGPU speed are again taken into account, this is more than 5 timesfaster than the matrix method. Just comparing the raw times, filtered-backprojection on a single GPU is only about 4 times slower than thematrix implementation on a medium-sized cluster, essentially allow-ing clinical reconstruction times on a single computer.

23

Part II

R E S U LT S A N D D I S C U S S I O N

5I O N C T R E C O N S T R U C T I O N F R O M N O I S Y D ATA

Many algorithms have been used for the reconstruction of ion CT,from the filtered backprojection used by Hanson (1979) to the iterativestatistical methods used in this work. In this chapter, a comparison ismade between the Rit-FBP method described in section 3.3 (Rit et al.,2013) and three iterative methods: the total variation algorithm (NCG-TV) used in article 2, the statistical weighted least squares (WLS) ap-proach used in article 3, and the total variation superiorization al-gorithm (DROP-TV) used by Penfold et al. (2010b). These algorithmshave all proven to produce high resolution proton CT images and theiterative methods were shown to accurately reconstruct data free ofdetector-noise. No study of reconstruction accuracy was made for theRit-FBP however and the doses used in the original paper correspon-ded to a CT dose index (CTDI) of about 40mGy which is somewhathigher than what has been used in other papers. Additionally, itmight be expected that this method, being based on the filtered back-projection, would be less tolerant of noise in raw data.

The study is based on the simulated proton scan of the Gammexphantom used in article 4. Here, the detector is modelled after Sip-ala et al. (2011a,b) and an energy resolution of σE(E) = max(3% ·E, 24.15MeV2

E + 1.76MeV) was used. The scan was done at a CT doseequivalent index (CTDEI) of 5.0mSv.

In all cases were the algorithm settings of the respective papersused and the reconstruction was done on a 34 cm × 34 cm grid of1024× 1024 pixels. For the iterative reconstructions, 120 full iterationswere done.

Resolution is measured via the task based MTF (Richard et al., 2012),using the cortical bone insert of the Gammex phantom. It is importantto emphasize that this is not directly comparable with the standardMTF as used in article 2 and 3.

The resolution as a function of the number of iterations can be seenon figure 4 and the reconstruction error on figure 5. None of theiterative methods reach the resolution of the FBP method until afterthe 45th iteration. The spike seen for the NCG-TV method is a resultof an artefact overenhancing the edges in the early iterations and isnot a true measure of the resolution. The Rit-FBP method reaches aresolution 4.23 cm/linepair. All three iterative methods reach theirmaximum resolution after about 60 iterations and all are within 5%of the FBP method.

The FBP method has an error of 0.38± 0.19%. This is only matchedby the NCG-TV method, which has a minimal error of 0.30± 0.39%

27

0 20 40 60 80 100 120

Iterations

0

1

2

3

4

5

6

MT

F=

10%

WLS

DROP-TV

NCG-TV

Rit-FBP

Figure 4: Resolution as defined by the MTF 10% point as a function of thenumber of iterations for three different iterative solvers: weightedleast squares (WLS), nonlinear conjugate gradient total variation(NCG-TV) and DROP with total variation superiorization (DROP-TV). This is compared with the Rit-FBP method.

0 20 40 60 80 100 120

Iterations

10−1

100

101

102

Max

imum

syst

emat

icer

ror

(%)

WLS

DROP-TV

NCG-TV

Rit-FBP

Figure 5: Reconstruction error as a function of the number of iterationsfor three different iterative solvers: weighted least squares (WLS),nonlinear conjugate gradient total variation (NCG-TV) and DROPwith total variation superiorization (DROP-TV). This is comparedwith the Rit-FBP method.

28

after 49 iterations. After 120 iterations, none of the methods are dif-ferent at the 1σ confidence level however.

To conclude, all the methods tested were able to accurately recon-struct ion CT images from noisy data. Of these algorithms, the Rit-FBP

method reaches the same resolution and image quality more than 30times faster than any of the iterative methods, as the computationalburden of this method is loosely that of two iterations. In the casewhere a full set of data can be acquired, this algorithm should thusbe favoured.

29

6S U M M A RY O F A RT I C L E R E S U LT S

article 1 - improving nuclear models in shield-hit

In this paper, the accuracy of the nuclear models in the Monte Carlocode SHIELD-HIT were investigated and the model parameters tunedto experimental measurements of total and partial charge-changingcross sections of carbon ions. This was partly a continuation of thework done in (Hansen, 2011), but significantly expanded and com-pared with additional experimental data.

A marked overestimation of about 20% was observed in the totalcross section of carbon ions on carbon, neon and on aluminium tar-gets in the range of 100MeV u−1 to 500MeV u−1 in the old versionof SHIELD-HIT (08). This was fixed in the new version (SHIELD-HIT10A), by adjusting the parametrization of the ion-ion inelasticcross section. A general overproduction of ∆Z = 2 and ∆Z = 3

fragments in the partial-charge changing cross section was correctedby changing the interaction volumes in the Fermi-breakup model ofnuclear fragmentation (figure 6). These changes were tested againstmeasurements of the fragmentation yield of 430MeV u−1 carbon ionsand 670MeV u−1 neon ions in water. The general fragmentationyield for carbon on water agreed within 2σ, and showed good agree-ment with the experimental data in general (figure 7). For neon ionson water, larger deviations were observed, and the fluorine chan-nel in particular was overestimated by a factor of 2. In conclusion,SHIELD-HIT10A showed good agreement with experiments and thedeviations observed were no larger than those seen using other MonteCarlo codes (Böhlen et al., 2010).

31

0 100 200 300 400 500E (MeV/u)

0

100

200

300

400

500

600

σ (m

b)

SHIELD-HIT08SHIELD-HIT10A∆Z=1

∆Z=2

∆Z=3

Figure 6: Article 1: partial charge-changing cross section of 12C on water forSHIELD-HIT08 and SHIELD-HIT10A compared with the experi-mental data (Toshito et al., 2007; Golovchenko et al., 2002) used foradjusting SHIELD-HIT10A.

0 5 10 15 20 25 30 350.00.20.40.60.81.0

N/N

0

Bragg peak

HHeC

0 5 10 15 20 25 30 35Depth (cm H2O)

0.000.020.040.060.080.10

N/N

0

Bragg peakLiBeB

Figure 7: Article 1: relative yield of fragments emitted within a 10◦ forwardangle from a 400MeV u−1 12C beam in water, compared withexperimental data (Haettner et al., 2006). Dashed line: SHIELD-HIT08 simulation. Solid line: SHIELD-HIT10A simulation. Themarkers are experimental data. Where error bars are not visible,they are smaller than the markers.

32

article 2 - combined ion and conebeam ct

In the second article, a method for partially overcoming the limitedrange of proton CT is presented, called dual modality reconstruction(DMR). This is done by applying prior knowledge based on a differ-ent imaging modality, in this case cone-beam x-ray CT. The basic ideabehind the proposed algorithm is that there are ranges of the x-raybased stopping power estimation which are more accurate than other.A weighted difference between the reconstructed image and the x-rayprior can thus be used as a regularization term in the reconstructionto allow for imaging even when only part of the full 360◦ of viewscan be obtained. The method was tested in the case where half theangles were used (90◦ DMR) and in the case where a quarter of theangles were used (45◦ DMR). The accuracy of the reconstruction isshown on figure 8 and the resolution on 9. In all cases, the DMRreconstructions yielded more accurate stopping power than the x-rayestimate. For the 90◦ DMR, a mean root-mean-square error of 0.01was obtained, but it was generally worse than the standard protonCT. The 45◦ DMR had comparable accuracy to the 90◦ DMR, but yiel-ded a significantly lower resolution. In conclusion, the DMR methodwas able to significantly improve stopping power estimation whencompared with cone-beam x-ray CT.

33

Delrin

(1.36) Teflon

(1.79) Air1

(0.00107) PMP

(0.882) LDPE

(0.978)Polysty

rene

(1.03) Air2

(0.00107) Acrylic

(1.16)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

RM

SE

rror

(rel

ativ

est

oppi

ngp

ower

)

Proton CT

DMR

45◦ DMR

90◦ DMR

Other 90◦ DMR

Converted CBCT prior

Figure 8: Article 2: boxplot showing the RMS error of the DMR reconstruc-tion methods in the inserts of the density slice of the Catphanphantom used for evaluation. The values below the materialnames are the ground truth relative stopping powers.

0 2 4 6 8 10 12

Linepairs / cm

0.0

0.2

0.4

0.6

0.8

1.0

MT

F

CBCT prior

Proton CT

Full DMR CT

90◦ DMR CT

45◦ DMR CT

Figure 9: Article 2: the MTF of the images reconstructed with the DMR

method, as well as the MTF of the cone-beam CT prior. The areabelow the cone-beam CT MTF is shaded for ease of comparison.

34

article 3 - the image quality of ion computed tomographyat clinical imaging dose levels

In this article, the merits and flaws of different ion species for ion CTwere investigated, specifically protons, helium and carbon ions. Inorder for a fair comparison to be made, the different ions needed tobe evaluated at the same biological impact, preferably in a way thatmade it possible to also compare with standard x-ray CT dose. Tothis point, the standard CTDI was extended with the ICRP conceptof radiation quality factor (ICRP-103, 2008). This is effectively a scal-ing factor for biological impact, and combined gave the CTDEI. AtCTDEI = 10mSv, helium ions showed to yield the highest spatial res-olution (figure 10, followed by carbon, 330MeV protons and 230MeVprotons. As carbon ions are heavier than helium ions, it might havebeen expected that they would yield a higher resolution. With theconstrained dose however, there were simply not enough carbon ionsto reach the maximal resolution. The mean systematic error as a func-tion of CTDEI is shown on figure 11. Assuming that clinically use-ful ion CT should provide measurements better than 0.5%, all stud-ied ions were able to provide accurate stopping power at doses ofCTDEI = 10mSv and above. With this criteria, carbon fails at a doseof CTDEI = 5mSv and helium at dose of CTDEI = 1mSv. Protonsat both 330MeV and 230MeV are able to produce accurate images atCTDEI = 1mSv however. In conclusion, helium ions deliver the bestcompromise between high image quality and low dose.

35

0 5 10 15 20

Frequency (linepairs/cm)

0.0

0.2

0.4

0.6

0.8

1.0

Modu

lation

Tra

nsfe

rFun

ctio

n

230 MeV 1H

330 MeV 1H

230 MeV/u 4He

430 MeV/u 12C

Figure 10: Article 3: modulation transfer function for four different combin-ations of ion species and energy.

0 5 10 15 20

CT Dose Equivalent Index (mSv)

0

2

4

6

8

10

Mea

nsy

stem

atic

erro

r(%

)

230 MeV 1H

330 MeV 1H

230 MeV/u 4He

430 MeV/u 12C

Figure 11: Article 3: mean systematic error as a function of dose, plotted fordifferent ion species and energies. The larger markers indicatethe point of 10mSv calculated using the ICRP weighting factors(ICRP-60, 1990).

36

article 4 - a comparison of dual energy ct and protonct for stopping power estimation

A comparison of proton CT and DECT was performed, and comparedagainst the SECT methods used clinically. This was done in twophantoms: a Gammex RMI 467 electron density phantom and a CIRSModel 002H5 IMRT Phantom. Two SECT methods were considered,one using the stoichiometric composition of human tissue as the basefor the conversion from Hounsfield units to stopping power, and onein which the calibration was done directly on the Gammex phantom.

Maximal errors of 0.63%, 1.71% and 7.44% for proton CT, DECT andSECT respectively were reported, consistent with what can be found inliterature (Hünemohr et al., 2013; Sadrozinski et al., 2013; Yang et al.,2010, 2012). With the exception of a single insert in the Gammexphantom, the error of proton CT was within 2σ. This was also thecase for DECT in the Gammex phantom on which the method wascalibrated, but the errors were significantly larger in CIRS phantom.In general the SECT methods perform worse than DECT and protonCT. This is true even for the Gammex phantom with the SECT methoddirectly calibrated to this data, indicating that even with a perfectcalibration SECT cannot achieve an accuracy better than 3.5%.

37

Lung300

Liver

CB2-50

Mineral

Bone

CorticalBone

Lung450

Brain

Adipose

Inner

Bone

CB2-30

Water

Breast

Solid

Water

−6

−4

−2

0

2

4

6

8

Rel

ativ

ere

sidu

aler

ror

(%)

Insert error

DE CT

Proton CT

SE CT - Gammex

SE CT - Stoichiometric

Figure 12: Article 4: residual errors in stopping power estimates based onDECT, SECT and proton CT in each of the inserts of the Gammexphantom. Note that the DECT, and to some degree the SECT datawere fitted to the inserts in this phantom. Errorbars show the 1σconfidence interval.

Bone

Adipose

Muscle

Lung

PlasticWater

−12

−10

−8

−6

−4

−2

0

2

4

Rel

ativ

ere

sidu

aler

ror

(%)

Insert error

DE CT

Proton CT

SE CT - Gammex

SE CT - Stoichiometric

Figure 13: Article 4: residual errors in stopping power estimates based onDECT, SECT and proton CT in each of the inserts of the CIRSphantom. Errorbars show the 1σ confidence interval.

38

7D I S C U S S I O N

The main contributions of this thesis are: An evaluation of the recentimprovement of the nuclear models in the Monte Carlo code SHIELD-HIT. A method for reconstructing ion CT from limited angles. Amethod for evaluating the delivered dose of an ion CT scan in a waythat is directly comparable with what is currently used for x-ray CT,and accounts for differing biological impact. An evaluation of themerits and inherent problems in relation three different ion specieswhen used for ion CT. A comparison of the image quality of protonCT and DECT.

Consistent with past literature (Schulte et al., 2005; Li et al., 2006)it has been demonstrated that ion CT images can be made with highstopping power accuracy and low dose. In particular, it is demon-strated in article 3 that, given sufficiently accurate detectors, protonCT could image as low as CTDEI = 1mSv. This is slightly lower thanthe doses reported in low-dose CT thorax scans and much lower thanhead scans which can easily reach CTDI = 40mGy even when usingiterative reconstruction (Siemens-Healthcare, 2014). In article 4, it isdemonstrated that proton CT compares favourably with both SECT

and DECT for stopping power estimation. As article 3 demonstratesthat helium and carbon ion CT yield as good a stopping power estim-ation as proton CT, it can be inferred that these would also comparefavourably with SECT and DECT.

It is important to emphasise that the iterative reconstruction em-ployed in new commercial x-ray CT scanners is fundamentally differ-ent from the iterative reconstructions used in article 3 and 4 and inseveral other works on ion CT (Penfold et al., 2009, 2010a). In x-rayCT, the dose advantage reported (Hara et al., 2009) stems largely fromthe use of non-linear regularization which strongly reduces noise(Panin et al., 1999; Tang et al., 2009) thereby allowing reconstructionfrom lower dose projections. Similar regularisation is used in article2 and was previously used by Penfold et al. (2010b). In neither casewas it used to de-escalate the dose used however. It thus seems likelythat the dose advantage of ion CT may be larger than the factor of 2reported by Schulte et al. (2005).

7.1 validity of the monte carlo simulations

A general concern present in article 2, 3 and 4 is how comparableMonte Carlo simulations are with real ion CT scans. Range/energystraggling is of course part of the Monte Carlo simulation in all cases,

39

and is the major source of noise in article 2 and 3. Even in the case ofa 200MeV proton loosing almost all its energy, the range uncertaintywould only amount to about 1% (Sadrozinski et al., 2013). This is sig-nificantly less than the < 2% reported (Hurley et al., 2012; Sipala et al.,2011b) from calorimeters for proton CT. While the relation betweendetector noise and image noise is non-trivial, it is clear that greaternoise in raw data would result in greater noise in the reconstructedimage. The minimal doses for acceptable reconstruction results es-tablished in article 3 might thus not reflect what could be achievedexperimentally for protons today. For heavier ions, the noise levels re-ported in calorimeters are lower (0.8% (Ohno et al., 2004)) and the lackof detector modelling therefore less of a concern. In article 4 a morerealistic noise-model is used. This model assumes a Gaussian distri-bution, and a minimal noise level of 1.76MeV. The model is thereforeunrealistic for particles that have lost almost all their energy, but theseparticles would be exceedingly rare in a safe ion CT scan. For the res-olution studies in article 2 and 3, the Monte Carlo simulations aremore realistic. The silicon strip detectors used in (Sadrozinski et al.,2013) have a pitch of 228µm, corresponding to a spatial resolution ofσ ' 65µm. This is much smaller than the reported resolutions, andthus unlikely to change the validity of the results dramatically.

7.2 the articles

While the results in article 1 are not directly related to ion CT, thework was of great importance for studying the impact of the model-ling of nuclear fragmentation on radiobiological models which wasdone in Lühr et al. (2012). Here it was established that the cross sec-tions for nuclear reactions had the greatest effect on dose distributionand radiobiological models, whereas the specific fragments producedonly resulted in minor deviations. This was important for judgingthe accuracy of the mean quality factors calculated in article 3. Asthe nuclear cross sections in Geant4 are well tested (Grichine, 2009;Böhlen et al., 2012; Dudouet et al., 2014), the calculations can thusbe considered reliable. It should be noted that the biological modelshave considerable uncertainties (ICRP-103, 2008), which may causethe actual biological impact to be quite different from what could becalculated using the mean quality factor.

The DMR method presented in article 2 allowed for full resolutionand a stopping power accuracy better than 0.02 in units of water equi-valent stopping power, even when removing the data from two 90◦

spans. This effectively extends ion CT to be used for sites where theenergy of the ions means they can only penetrate the patient fromhalf the scan interval. The main advantage of the DMR method, whencompared with alternative methods relying on spiral CT scans (Wanget al., 2012), is the use of a cone-beam image prior. This allows for

40

the proton CT and cone-beam prior to be acquired simultaneously,thereby bypassing the need for some sort of registration to accountfor the differing anatomy of the patient.

The disadvantage is the long reconstruction time of the iterativemethod compared with the filtered-backprojection method. Combin-ing the projection-space prior of Wang et al. (2012) into the Rit-FBP

would be straight forward, potentially allowing for very fast recon-struction even from limited data. This relies on the projection priorto be almost perfect however, and removes the flexibility present inthe statistical framework of DMR. While the results in article 2 arepromising, the method is tested on a phantom with materials that arenot human-like with respect to stopping power. While there is noreason to believe the method would not work in patients, it shouldbe validated on more human-like materials beforehand.

The third article introduces a concise way of evaluating dose sothat it is directly comparable with the CTDI which is standard for re-porting x-ray CT dose. This allows for the first direct comparison ofdifferent ions for ion CT at clinically relevant dose levels. The weak-ness of the method largely hinges on the use of the ICRP linear en-ergy transfer (LET) dependent quality factor Q (ICRP-103, 2008). Thisis based on in vitro experiments and cannot be expected to correlateexactly with the risk of carcinogenesis in humans. It should how-ever provide a correct order of magnitude. It also has the advantageover a fixed weighting-factor that it can better differentiate betweenthe biological impact of high-energy (low LET) and low-energy (highLET) ions, which is particularly important for carbon ions. For pro-tons, the high LET part of the radiation stems exclusively from theheavier fragments produced in inelastic nuclear collisions. As thesehave both a greater mass and a greater charge, they have high LET,but also very short range (� 1mm). This means that the calculatedQ factor is highly dependent on the cross-section of the material inwhich it is calculated. As the macroscopic cross section per mass ismore than 10% higher for air than for human tissues, this means theQ factor should not be calculated in air-filled cavities, such as ioniz-ation chambers. One possible concern with the study is the use ofideal detectors, which means that the dose/quality correlation on fig-ure 11 possibly shows a smaller error than what could be obtainedclinically. This would particularly impact carbon and helium, as thereare fewer particles to average out detector noise. It is therefore likelythat carbon CT would require near ideal detectors to be feasible at aclinically reasonable dose level.

The fourth article compares proton CT with another method forstopping power estimation: DECT. While proton CT generally providesa more accurate estimation of stopping power, DECT has the consider-able advantage that it is available from all major vendors and couldbe directly adopted in clinical practice today. This is particularly at-

41

tractive when considering that the standard x-ray CT method yieldslarge deviations when presented with material only slightly differentfrom that to which it was calibrated. DECT also has the advantageof being available for all treatment sites and for larger patients thanwhat could be scanned with the proton energies used clinically today.Therefore it might then have been relevant to compare DECT with theDMR method from article 2, but this would have required cone-beamCT scans of the phantoms, which was not logistically possible.

42

8F U T U R E D I R E C T I O N S

The future of ion CT is limited mainly by two aspects, namely highquality detectors and available beam time. When ion CT uses radio-therapy beamlines, it competes with radiotherapy treatment for beamtime, which is always at a premium. Ion CT therefore needs to be fastenough that it will not significantly slow down the current radiother-apy workflow, or it will need dedicated beamlines. The requirementsfor beam quality and beam current are however much more lenientfor ion CT. This allows for the use of compact accelerators, such asdielectric wall accelerators (Caporaso et al., 2007) or laser accelerators(Bulanov and Khoroshkov, 2002), which could conceivably bring thecost of an ion CT machine down to the cost of other medical imagingscanners. For detectors, there are two problems. First of all, to makethem large enough to cover the entire field needed for a clinical scanand second of all to make them fast enough that a scan can be madein a matter of minutes (Sadrozinski et al., 2013).

From the reconstruction point of view, ion CT can be considered asolved problem. While reconstructions may still be two orders of mag-nitude slower than x-ray reconstructions, a full 3D ion CT reconstruc-tion would still be available in less than 5min, making it clinicallyfeasible. Where iterative reconstruction could make an impact is byreducing the number of scanning angles needed, decreasing the scantime and potentially by further reducing the dose required. Finallythe problem of limited angles is by no means completely solved. Themethod presented in this work requires experimental validation, andfurther improvements could likely be made to the statistical model.

If these challenges can be overcome, ion CT could substantially im-prove particle therapy. Even if it will not be possible to scan everypatient with ion CT, it would allow in vivo verification of other meth-ods such as DECT. These methods could also be calibrated directlyon patient scans, eliminating the errors associated with using tissuesubstitutes in phantoms.

43

Part III

A RT I C L E S

O P T I M I Z I N G S H I E L D - H I T F O R C A R B O N I O NT R E AT M E N T

David C Hansen, Armin Lühr, Nikolai Sobolevskyand Niels Bassler

Published in 2012 inPhysics in Medicine and Biology

Volume 57, page 2393 to 2409

47

Optimizing SHIELD-HIT for carbon ion treatment

David C Hansen1,2, Armin Luhr2,3, Nikolai Sobolevsky4 and NielsBassler2,3

1Department of Clinical Medicine, Aarhus University, Brendstrupgardsvej 100,DK-8200 Aarhus N, Denmark2 Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120,DK-8000 Aarhus C, Denmark3 Department of Experimental Clinical Oncology, Aarhus University Hospital,Nørrebrogade 44 Bldg. 5, DK-8000 Aarhus C, Denmark4 Department of Neutron Research, Institute for Nuclear Research of the RussianAcademy of Sciences, 7a, 60th October Anniversary prospect, Moscow 117312,Russia

E-mail: dch@ki.au.dk

Abstract. The SHIELD-HIT Monte Carlo Transport code has been widely used inparticle therapy, but has previously shown some discrepancies, when compared withexperimental data. In this work, the inelastic nuclear cross sections of SHIELD-HIT are calibrated to experimental data for carbon ions. In addition, the modelsfor nuclear fragmentation were adjusted to experiments, for the partial chargechanging cross section of carbon ions in water. Comparison with fragmentation yieldexperiments for carbon and neon primaries were made for validation. For carbonprimaries, excellent agreement between simulation and experiment was observed,with only minor discrepancies. For neon primaries, the agreement was alsogood, but larger discrepancies were observed, which require further investigation.In conclusion, the current version SHIELD-HIT10A is well suited for simulatingproblems arising in particle therapy for clinical ion beams.

PACS numbers: 87.10.Rt, 87.55.K-, 87.55.Qr, 24.10.Lx

Keywords: Carbon ion therapy, Monte Carlo Transport, Nuclear fragmentation,SHIELD-HIT

Submitted to: Phys. Med. Biol.

1. Introduction

The application of Monte Carlo (MC) transport codes in particle therapy researchcovers a broad range of topics. Common for all topics is that they require

Optimizing SHIELD-HIT for carbon ion treatment 2

detailed knowledge about the particle beams used clinically. Such knowledge canbe obtained by solving the Boltzmann transport equation. However, doing thisanalytically is not possible for anything but the simplest geometries and beams.Therefore, the only way to obtain knowledge about the particle-energy spectrumin a given point is either by experiment or by solving the transport equation bythe Monte Carlo method. Most ion treatment planning systems work with pencilbeam like algorithms where pre-calculated pencil beam dose kernels are appliedon a voxel-based structure. Such dose kernels can be calculated with MC codeswhere the full beam line is modelled. However, pencil beam algorithms with pre-calculated dose kernels are known to be inaccurate at media boundaries with strongdensity gradients, such as bone/tissue interfaces, lung or even if metal artifacts arein the treatment field (Schaffner et al. 1999). As computer power becomes cheaper,a full MC simulation can be realized which will give a more realistic treatmentplan. Some developers of commercial treatment planning software offer this asan option for photon/electron treatment planning. Moreover, MC codes allowdetailed theoretical studies of new types of particle beams for radiotherapy. Thisis in particular important for exotic particles where high current beams may notbe available for experiments and clinical facilities are far from being realized, as inthe case of antiproton radiotherapy (Bassler et al. 2010b, Bassler et al. 2008b, Bassleret al. 2008a, Bassler & Holzscheiter 2009).

Apart from dose planning, dose assessment is also relying on MC calculations.In a recent study, the stopping power ratios used for ionization chamber dosimetrywere investigated. Using the SHIELD-HIT MC code it was found that nuclearfragmentation only had a minor influence for the mixed particle field created bycarbon ion beams (Luhr et al. 2011a), when following the formalism of IAEA TRS-398 protocol (IAEA TRS-398 2001).

For calculating the radiobiological properties of the applied heavy ion beam,the Local Effect Model (LEM) (Kramer et al. 2000, Kramer & Scholz 2000) canbe used during the optimization of the treatment plan. The LEM, however,needs the particle-energy spectrum as an input parameter. These purely physicalinput parameters can either be derived experimentally or calculated by MCcodes. Furthermore, dose delivery verification using either PET or prompt gammarays (Parodi et al. 2007) relies on MC codes that can accurately simulate thedistribution of radioactive nuclei after irradiation of the patient.

Finally, the generation of secondary neutrons which increases the risk ofdeveloping a secondary radiation-induced cancer has been subject of muchdiscussion. Here MC codes are used to model the neutron fluence emerging frombeam nozzles and the patient itself (Hall 2006, Xu et al. 2008), and again for researchpurposes of more exotic beam environments (Bassler et al. 2010a)

For ion beams Monte Carlo transport codes such as Geant4 (Agostinelli

Optimizing SHIELD-HIT for carbon ion treatment 3

et al. 2003, Allison et al. 2006), FLUKA (Fasso et al. 2005, Battistoni et al. 2007),PHITS (Niita et al. 2006, Iwase et al. 2002), MCNPX (Pelowitz 2005) and SHIELD-HIT (Gudowska et al. 2004, Geithner et al. 2006) are frequently used. These codesall rely on sophisticated models for nuclear interactions, which have to balancespeed and accuracy, while remaining as general in terms of energy and particletype as possible. SHIELD-HIT is a Monte Carlo code specialized for ion therapy.However, previous studies have suggested that there were some inaccuracies inthe nuclear models implemented in SHIELD-HIT when comparing to experimentalfragmentation yields (Gudowska et al. 2004). Since then several updates were madeto SHIELD-HIT, and the more recent changes will be treated in the next section.The purpose of this paper is thus to give a presentation of the recent improvementsof the nuclear models in SHIELD-HIT10A and a new benchmark of the underlyingnuclear models against available experimental data.

The SHIELD-HIT Monte Carlo transport code (Gudowska et al. 2004) is derivedfrom the SHIELD code (Dementyev & Sobolevsky 1999), originally developed at theJoint Institute for Nuclear Research (JINR), in Dubna, Russia. Later, developmentwas moved to the Institute for Nuclear Research (INR) at the Russian Academyof Sciences in Moscow. The SHIELD code includes transport of arbitrary nuclei,nucleons, anti-nucleons, pions and kaons up to 1 TeV/u and down to 1 MeV/u.In SHIELD-HIT, the same particles can be transported, but here the upper andlower limit ranges from 2 GeV/u to 25 keV/u. Previous to this work, the latestversion of SHIELD-HIT was 08. In SHIELD-HIT10A several parameters regardingthe nuclear reactions were changed along with several updates to the user interfaceand performance, which were partially discussed in (Hansen et al. 2012).

SHIELD-HIT has earlier been used for various tasks within particle therapy,calculation of stopping power ratios (Henkner et al. 2009, Luhr et al. 2011a, Luhret al. 2012), fluence correction factors (Luhr et al. 2011b), and even antiprotonbeams (Bassler et al. 2008b).

In this article, we focus on the changes to the nuclear models used in SHIELD-HIT. The next section will give a more detailed description of the parameters of thenuclear interaction models as well as how SHIELD-HIT works in general.

2. The SHIELD-HIT Monte Carlo code

Before starting the Monte Carlo transport in SHIELD-HIT the stopping power,cross sections, ranges and optical depths are calculated for all the materials in thesimulation at various energy intervals. These tables are then used for interpolationof the values during the transport phase, using linear or quadratic interpolation.The energy grids used in the tables were refitted from SHIELD to SHIELD-HIT toyield the accuracy required in medical applications.

Optimizing SHIELD-HIT for carbon ion treatment 4

2.1. Electronic interactions

Stopping powers in SHIELD-HIT are calculated using the Bethe-Bloch formula(Fano 1963), with effective charge correction (Hubert et al. 1989) but without theshell-correction terms. For energies below the stopping power maximum, theLindhard-Scharff equation (Lindhard & Scharff 1961) is used. The exact energywhere the transition from the Bethe-Bloch to the Lindhard-Scharff descriptiontakes place is determined by the requirement that the two functions and theirderivatives equal in this point resulting in a continuously differentiable descriptionof the stopping power in SHIELD-HIT. Further details on the internal calculation ofstopping powers were earlier discussed by (Geithner et al. 2006, Luhr et al. 2011a).Alternatively, the stopping power can be interpolated from external tables, followingthe ICRU format, on an individual material basis. The software library libdEdx(Toftegaard et al. 2010, Luhr et al. 2012) allows direct use of the PSTAR, ASTAR(Berger et al. 2005, Paul & Schinner 2001, Paul & Schinner 2002), and ICRU stoppingpower tables in SHIELD-HIT, as it directly exports in the required text format. Inthis work, the stopping power tables published in the ICRU report 49 and 73 (witherratum) (International Commision on Radiation Units and Measurements 1993,International Commision on Radiation Units and Measurements 2005) are used forwater and air, and the Bethe-Bloch equation for any other material. SHIELD-HITincludes multiple Coulomb scattering through two dimensional Fermi (Gaussian) orMoliere distributions (Bethe 1953) and fluctuations of ionization energy losses in theform of Gaussian or Vavilov energy straggling. In this work, Moliere and Vavilovdistributions were used for all calculations where applicable.

2.2. Nuclear interactions

It is common in Monte Carlo transport codes to handle nuclear interactions in twoseparate steps. First, the probability that a nuclear event happens is sampled,based on the total inelastic nuclear cross section. Then, if an event happens, aseparate model for nuclear fragmentation is used to sample the outcome and getthe fragmentation yields.

2.2.1. Total inelastic cross sections In SHIELD-HIT up to version 08, the totalinelastic cross sections for nuclear interactions are taken from (Barashenkov 1993).The nucleon-ion collision cross-sections are interpolated from tables based onexperiments and theoretical calculations. These tables exist for 18 referencenuclei and non-tabulated isotopes are then interpolated linearly weighted with thegeometrical nucleus cross section. For an element with nucleon number A this isassumed to be a factor A2/3. For ion-ion collisions, the cross sections are given by a

Optimizing SHIELD-HIT for carbon ion treatment 5

strong absorption model type equation:

σR(E) = πr20

(1− B(E)

Ecm

)

(A1/3

P + A1/3T + 1.85

A1/3P A1/3

T

A1/3P + A1/3

T

+ λ(A, E)− δB(E)

)2

(1)

where r0 is the nucleon radius set to = 1.3 fm, E is the kinetic energy in thelaboratory frame, Ecm is kinetic energy in the centre of mass frame, λ(A, E) is the deBroglie wavelength of the projectile in the centre of mass system and AP and AT arethe projectile and target nucleon numbers, respectively. B(E) is the Coulomb barriergiven by

B(E) =1

4πε0

ZPZT

r0(A1/3P + A1/3

T ) + λ(A, E)(2)

where ZP and ZT are the atomic numbers of the projectile and target, respectively,and

δB(E) =

C1 + C2 log( EAT

) for EAT

< E1,

C3 + C4 log( EAT

) for E1 ≤ EAT

< E2,

C5 for E2 ≤ EAT

,

(3)

where C1, C2, C3, C4, C5, E1, and E2 are model parameters. For E/AT ≤ 10 MeV/u,the cross section for E/AT = 10 MeV/u is used, but scaled with the correct Coulombbarrier given by equation 2.

2.2.2. Nuclear fragmentation Nuclear fragmentation in SHIELD-HIT is handled bythe Many Stage Dynamical Model (MSDM) (Botvina et al. 1997). This model consistsof 3 steps: fast-cascade, pre-equilibrium and de-excitation/equilibrium. In the initialfast-cascade, it is assumed that all long range interactions can be neglected and thetime evolution of the system is dominated by short range interactions (i.e. collisions).For energies below 1 GeV/u it is done using the Dubna-cascade model (Toneev& Gudima 1983) and for higher energies it is handled by the Quark-gluon stringmodel (Amelin et al. 1990).

The second step is called the pre-equilibrium step, which is an exciton model.Here the excited nucleus is treated as a particle-hole system and solved using theBoltzmann Master Equation (Gudima et al. 1983) (BME). This takes the long rangeinteractions into account but neglects the collision geometry.

The last step is called the de-excitation or equilibrium step, where one of 3different models is used. For fragments with nucleon number A ≤ 16, the Fermi-breakup model (Fermi 1950, Botvina et al. 1987) is applied. For heavier fragmentsA > 16 which are excited to less than 2 MeV/u an evaporation model (Botvina et al.1987) is used and otherwise the statistical model of multifragmentation (Botvina

Optimizing SHIELD-HIT for carbon ion treatment 6

et al. 1987, Bondorf et al. 1995). For ion therapy the energies currently used aremainly below 400 MeV/u. At these energies only a negligible small fraction of thesimulated secondary particles reach an energy above 1 GeV/u, and thus only theinfluence of the Dubna cascade was considered in this work. Since carbon ions andpossibly oxygen ions are the heaviest projectiles used for treatment, and the humanbody primarily consists of relatively light elements, the Fermi-breakup model isdominating the de-excitation step. For heavier ions, which have been proposed fortreatment, such as oxygen and neon, the evaporation model becomes relevant.

3. Changes in the nuclear models of SHIELD-HIT

3.1. Corrections of the total cross sections

The tabulated proton-ion cross sections implemented in SHIELD-HIT08 agree wellwith experiments (most within 2σ), and can be considered sufficiently accurate.Thus, these cross-sections were not changed in SHIELD-HIT10A.

In contrast, the agreement for the ion-ion cross sections based on the analyticalparametrization was less convincing, and adjustments were needed. The totalinelastic cross sections were changed by simply calibrating the parameters of theBarashenkov cross section formula (equation 1) to experimental data for carbonions (Takechi et al. 2009, Fang et al. 2000, Kox et al. 1987, Sihver et al. 1993, Zhanget al. 2002). This was done using a coarse brute-force optimization and then findingthe local minimum with the Simplex-Downhill algorithm (Powell 1973). Carbon wasused for the calibration for several reasons: First of all due to its role as the dominantion in heavy ion therapy. Second, carbon is relatively easy to use in both as a primarybeam and as a target in experiments, so relatively many data sets are available. TheTripathi (Tripathi et al. 1996, Tripathi et al. 1997, Tripathi et al. 1999) cross sectionsused in Geant4 were also considered for comparison, as the experimental data usedin the calibration cannot also serve as validation.

3.1.1. Proton-nucleus cross sections As can be seen on figures 1 and 2, the totaland inelastic cross sections for protons fit well with available experimental data. Anotable exception is the inelastic cross section for boron, where some discrepanciescan be observed below 40 MeV/u. This is most likely because the boron crosssections are interpolated from the tables of carbon and beryllium which is notsufficiently accurate at low energies. Some disagreement can also be observedfor the inelastic proton-carbon cross section, but here the experimental data is lessconclusive.

Optimizing SHIELD-HIT for carbon ion treatment 7

100 200 300 400 500E (MeV)

0

200

400

600

800

1000

1200

1400

1600

1800

σtot(m

b)

H

He

Be

C

O

Figure 1: Total nuclear cross sections inSHIELD-HIT for protons on varioustargets, compared with experimental

data (Schwaller et al. 1979).

101 102

E (MeV)

0

200

400

600

800

σinel

(mb)

H

Be

C

B

Al

O

Figure 2: Inelastic nuclear cross sectionsin SHIELD-HIT for protons on varioustargets, compared with experimental

data (Sihver et al. 1993).

101 102 103

E (MeV/u)

600

800

1000

1200

1400

1600

1800

2000

σinel

(mb)

SHIELD-HIT10A

SHIELD-HIT08

Geant4 (Tripathi)

Be

Al

Figure 3: Inelastic nuclear cross sectionsfrom Geant4 (Tripathi), SHIELD-HIT08

and SHIELD-HIT10A for 12C onberyllium and aluminium targetscompared with experimental data

(Takechi et al. 2009, Kox et al. 1984, Koxet al. 1985).

101 102 103

E (MeV/u)

600

800

1000

1200

1400

1600

1800

2000

σinel

(mb)

SHIELD-HIT10A

SHIELD-HIT08

Geant4 (Tripathi)

C

Ne

Figure 4: Inelastic nuclear cross sectionsfrom Geant4 (Tripathi), SHIELD-HIT08and SHIELD-HIT10A for 12C on carbon

and neon targets, compared withexperimental data (Takechi

et al. 2009, Fang et al. 2000, Koxet al. 1984, Kox et al. 1985, Kox

et al. 1987, Sihver et al. 1993, Zhanget al. 2002).

3.1.2. Ion-nucleus cross sections When comparing SHIELD-HIT08 with theexperimental data for carbon on various targets (figures 3 and 4), a markedoverestimation of the cross section is observed in the region of about 100 MeV/u

Optimizing SHIELD-HIT for carbon ion treatment 8

SHIELD-HIT08C1 C2 E1 C3 C4 E2 C5

Z > 2 −2.05 1.9 100 MeV/u 0.8 0.426 560 MeV/u 2.07

SHIELD-HIT10Z > 4 −1.112 1.427 195MeV/u 1.624 0.2336 560 MeV/u 1.98

2 < Z ≤ 4 −1.768 1.652 195MeV/u 1.585 0.178 560 MeV/u 1.98

Table 1: Parameters used in the Barashenkov cross section model (equation 1) forinelastic nucleus-nucleus collisions in SHIELD-HIT08 and SHIELD-HIT10A.

to 500 MeV/u for C, Ne and Al targets and to a minor degree for Be targets aswell. A similar, but less pronounced overestimation is seen in the Tripathi (Geant4)data set. Optimizing the constants in the Barashenkov formula to the availableexperimental data yielded the values in table 1. It was not possible to make thedata for beryllium fit using the same constants as for the heavier materials, so aseparate set of constants were used for beryllium and lithium. This way, excellentagreement between SHIELD-HIT10A and the experimental cross sections could beachieved. The parametrization of σR(E) in equation 1 with the parameter in table1 produces an unphysical jump in the cross section at 560 MeV/u, and the crosssections from 450 MeV/u to 700 MeV/u were computed using spline interpolationto ensure a smooth transition. It was not possible to verify the applied smoothing asno data was available in this region and the Tripathi cross section shows no similartransition in this region. A possible way to compensate for the lack of experimentaldata would be to compare the data with non-empirical theoretical models such asoptical limit Glauber calculations (Takechi et al. 2009) being, however, beyond thescope of the current work.

3.2. Corrections to the models for nuclear fragmentation

The steps in the MSDM model for nuclear fragmentation are highly interdependent,and therefore they are difficult to validate against experimental data separately.Conversely, it is difficult to isolate any discrepancies to a single step. The Fermi-breakup model is the dominating model in the final step and therefore deservescloser attention. This model requires calibration of two parameters known as the

relative free volume (Vf rV0

) and the relative free Coulomb volume (Vc

f rV0

). The orderof magnitude of these parameters is physically motivated (Fermi 1950) and theyare usually set to 1.0. The Fermi-breakup model was also recently validated inGeant4 (Pshenichnov et al. 2010).

In this work, the model was calibrated to experimental partial charge-changingcross sections from a 12C-beam on water (Toshito et al. 2007, Golovchenko et al.2002). The partial charge-changing cross section is the cross section for a process,in which a fragment is produced with a specific charge. A program was created tosimulate these two experiments for a given projectile on a pure element with the

Optimizing SHIELD-HIT for carbon ion treatment 9

naturally occurring isotope content. This was done by simulating 100 000 collisionsfor every energy and then multiplying the obtained partial fragmentation yieldswith the corresponding inelastic cross section. For composite targets such as water,the results for the basic elements were combined by simply adding the cross sectionsfor the elements contained in the molecule. In order to compare with what could bedetected in experiments, fragments with an energy below 25 MeV/u were filteredout. For carbon this corresponds to a residual range of about 2 mm in water or 2 min air. Performing the calibration with computer optimization was impractical dueto the time required to calculate the partial charge-changing cross sections. Instead,

the optimization ofVc

f rV0

andVf rV0

was done manually, against partial charge-changingcross sections of 12C on water. Water was chosen for two reasons. First, it is thestandard reference material for radiotherapy and second, most of the experimentaldata was available for this material.

0 100 200 300 400 500E (MeV/u)

0

100

200

300

400

500

600

σ (m

b)

SHIELD-HIT08SHIELD-HIT10A∆Z=1

∆Z=2

∆Z=3

Figure 5: Partial charge-changingcross section of 12C on water for

SHIELD-HIT08 and SHIELD-HIT10Acompared with the experimental

data (Toshitoet al. 2007, Golovchenko et al. 2002)used for adjusting SHIELD-HIT10A.

0 100 200 300 400 500E (MeV/u)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

σ (m

b)

SHIELD-HIT08SHIELD-HIT10A∆Z=1

∆Z=2

∆Z=3

Figure 6: Partial charge-changingcross section of 12C on PMMA for

SHIELD-HIT08 and SHIELD-HIT10Acompared with experimental data

(Toshito et al. 2007).

3.2.1. Comparison with partial charge-changing cross section experiments The Fermi-breakup volumes best fitting the experimental data and therefore used in SHIELD-

HIT10A were found to beVc

f rV0

= 0.65 andVf rV0

= 18.0. Comparison of thepartial charge-changing cross sections of 12C between SHIELD-HIT10A SHIELD-HIT08 and experimental data can be seen in figures 5 and 6. SHIELD-HIT08 isgreatly overestimating the ∆Z = 2 and ∆Z = 3 channels for both PMMA andwater when compared with experimental data, but the ∆Z = 1 channel is in

Optimizing SHIELD-HIT for carbon ion treatment 10

relatively good agreement. For SHIELD-HIT10A good agreement could be achievedwith the data to which it was optimized (figure 5). The lower cross sectionsseen experimentally below 250 MeV/u for ∆Z = 2 are not reproduced by thefragmentation model, however, it is possible that they are artifacts coming froma systematic difference between the different experiments included. In order tovalidate the adjustments of the charge-changing cross sections, SHIELD-HIT10Awas compared with experimental data for PMMA (figure 6). It can be seen thatSHIELD-HIT10A is in good agreement and within 2σ of the data everywhere.

4. Comparison with fragmentation experiments

To validate the changes made in SHIELD-HIT10A an extensive comparison withexperimental data is required. A detailed experimental investigation of nuclearfragmentation of 400 MeV/u carbon ions has been made at GSI (Haettner 2006,Haettner et al. 2006). This data is particularly well suited for benchmarking SHIELD-HIT as it focuses on the fragmentation occurring in carbon ion therapy.

In the experiment, a 400 MeV/u carbon ion beam was shot at a water phantomof variable length. The energies and time of flight of the individual fragments werethen measured at angles from 0◦ to 10◦ relative to the direction of the primarybeam. From this information, the fragmentation yields, differential in angle andenergy, could be obtained for every element.

For this carbon experiment, the SHIELD-HIT simulations were split up intotwo parts. First, a simplified simulation to find the total fragmentation yields. Here,everything in the beamline, up to and including the water phantom, was simulated.But, instead of simulating the detector geometry explicitly a filter was added tothe detection code, ensuring that only particles in the forward 10◦ were scoredin the phantom. To compensate for the lacking exit window, the simulation datawas shifted 8 mm backwards when comparing with the experimental data. Forthe simulation of the angular and energy differential data, the full experimentalgeometry was simulated. The detector was modelled as cylinder shells, to exploitthe experimental cylindrical symmetry.

The experiment has previously been used in benchmark studies of Monte Carlotransport codes (Bohlen et al. 2010, Pshenichnov et al. 2010). Here, FLUKA andGeant4 were compared with the experimental data. For Geant4, two differentmodels of nuclear fragmentation were tested. First, the Binary Cascade light ion(BIC), which is a cascade model similar in concept to the Dubna Cascade Model usedin SHIELD-HIT. The second is a quantum molecular dynamics model (QMD), whichis similar to the model used by FLUKA. Like SHIELD-HIT, FLUKA and Geant4 usea de-excitation step with Fermi-breakup, though with different implementations.

Earlier, a similar experiment was done at Berkley (Schimmerling et al. 1989).

Optimizing SHIELD-HIT for carbon ion treatment 11

Figure 7: Setup of experiment(Haettner 2006) used for validation ofSHIELD-HIT10A. The 12C ions hit a

water target, and fragments are measuredat an angle θ from the particle entry.

Water phantom

d ≃ 30cm d ≃ 90cm

T1

Detector

T2

Detector

Figure 8: Setup of experiment(Schimmerling et al. 1989) used for

validation of SHIELD-HIT10A. The 20Neions hit a water target, and fragments are

measured on the beam-axis.

Here, a 670 MeV/u 20Ne-beam was shot at a variable size water phantom. Thevelocity and type of fragment were measured on the beam axis, using two timeof flight detectors with radius 0.95 cm. The exact placement of the objects in thebeamline were not given in (Schimmerling et al. 1989). However, a distance ofabout 120 cm was reported in an earlier work of the same authors (Schimmerlinget al. 1986) and assumed here in the simulation. For the neon experiment, only thewater phantom itself was simulated, as the positions of the rest of the beamlineelements were not described in (Schimmerling et al. 1989). The beam energywhen hitting the phantom was 626.25 MeV/u with an full width at half maximum(FWHM) of 5 cm, as per (Schimmerling et al. 1989). Like the simulation of thecarbon experiment, a filter was added to the detection code so only particles hittingboth parts of the time of flight detector (T1 and T2 on figure 8) were scored.

For all simulations 4 · 107 primary particles were used.

4.1. Comparison with fragmentation yields

The total fragmentation yields from a 400 MeV/u 12C beam in water are shownin figure 9. For the carbon ions SHIELD-HIT10A shows agreement within 2σ

of the experimental data, while SHIELD-HIT08 shows in general a lower fluence,consistent with the overestimated inelastic cross section seen in figure 4. The slightdisagreement between SHIELD-HIT10A and the experimental data is probably dueto boron fragments being misidentified as carbon in the experiment (Haettneret al. 2006). SHIELD-HIT10A shows excellent agreement with the hydrogenfragments before the Bragg peak, but a slight overestimation after. SHIELD-HIT08 generally underestimates the number of hydrogen fragments. Both codesunderestimate the amount of helium produced, on the order of 15%. For beryllium

Optimizing SHIELD-HIT for carbon ion treatment 12

0 5 10 15 20 25 30 350.00.20.40.60.81.0

N/N

0

Bragg peak

HHeC

0 5 10 15 20 25 30 35Depth (cm H2O)

0.000.020.040.060.080.10

N/N

0

Bragg peakLiBeB

Figure 9: Relative yield of fragments emitted within a 10◦ forward angle from a400 MeV/u 12C beam in water, compared with experimental data (Haettner

et al. 2006). Dashed line: SHIELD-HIT08simulation. Solid line:SHIELD-HIT10Asimulation. The markers are experimental data. Where error bars

are not visible, they are smaller than the markers.

and lithium, SHIELD-HIT08 is greatly overestimating the number of producedfragments being consistent with the partial charge-changing cross sections in section3.2.1. SHIELD-HIT10A is within 2σ for the lithium fragments, but is overestimatingthe beryllium fragments somewhat, in particular after the Bragg peak. A similarscenario is seen for both SHIELD-HIT10A and SHIELD-HIT08 with the boronfragments, where the estimation is generally too high. For SHIELD-HIT10A thisis particularly surprising, as the partial charge-changing cross sections for the boronchannel fitted the experimental data very well. It has however been suggested thatsome underestimation is done by the experiment for this channel, as it was partlymasked by the primary carbon ions (Haettner et al. 2006). This cannot be the solesource of the problem, however, as the problem continues beyond the Bragg peak,where all the primaries are gone. This could in part be caused by an overestimationof the boron channel below 100 MeV/u, where the partial-charge changing crosssection predicted by SHIELD-HIT10A is much larger than at higher energies.

Only some of the data compared with the experiment is explicitly discussedhere. However, a comparison with all the data from the carbon experiment (Haettneret al. 2006) is available in the online version of this article.

Looking at the angular resolved data (figures 10 and 11), SHIELD-HIT08 and

Optimizing SHIELD-HIT for carbon ion treatment 13

−2 0 2 4 6 8 10θ (deg)

0

5

10

15

20

25

N/N

0(1 sr)

SHIELD-HIT10A

SHIELD-HIT08

Experimental data

Figure 10: Angle resolved relative yieldof hydrogen fragments at a depth of25.8 cm water from a 400 MeV/u 12C

beam, compared with experimental data(Haettner et al. 2006)

−2 0 2 4 6 8 10θ (deg)

0

5

10

15

20

25

30

35

40

N/N

0(1 sr)

SHIELD-HIT10ASHIELD-HIT08Experimental data

Figure 11: Angle resolved relative yieldof helium fragments at a depth of 34.7 cm

water from a 400 MeV/u 12C beam,compared with experimental data

(Haettner et al. 2006)

−2 0 2 4 6 8 10

θ (deg)

0

5

10

15

20

25

30

35

N/N

0(1 sr)

SHIELD-HIT10A

SHIELD-HIT08

Experimental data

Figure 12: Angle resolved relative yield of boron fragments at a depth of 25.8 cmwater from a 400 MeV/u 12C beam, compared with experimental data (Haettner

et al. 2006)

to some extent SHIELD-HIT10A are underestimating the hydrogen fragments inthe forward angles, while overestimating it at larger angles. The opposite effectis seen for helium, where both codes overestimate the small angles, but slightlyunderestimate the larger ones. Looking at the boron spectrum on figure 12, it isclear that the overestimation is mainly from the central angles, which is where theexperimental boron yield might be masked by the primary carbon fragments.

For the fragmentation yield of neon, seen on figure 13, agreement withexperimental data is in general not as good as that seen for carbon. As the

Optimizing SHIELD-HIT for carbon ion treatment 14

0 5 10 15 20 25 30 35 400.0000

0.0004

0.0008

0.0012

0.0016

N/N

0

Be

B

C

0 5 10 15 20 25 30 35 400.0000

0.0008

0.0016

0.0024

0.0032

N/N

0

N

O

F

0 5 10 15 20 25 30 35 40Depth (cm H2O)

0.00

0.02

0.04

0.06

0.08

0.10

N/N

0

Ne

Figure 13: Relative yield of fragments from a 670 MeV/u 20Ne beam in watercompared with experimental data (Schimmerling et al. 1989). The solid line is the

SHIELD-HIT10A simulation. The markers are experimental data. Whereexperimental error bars are not visible, they are smaller than the markers.

neon experiment was simulated in less detail and detects only a smaller angularfraction of the fragments, this was to be expected. However, the discrepanciesseen for neon in the lowest panel of figure 13 are surprising where the SHIELD-HIT10A simulation is consistently higher than the experiment. As neon is theprimary particle, it would have relatively little angular divergence, and thus be lesssensitive to setup errors. The difference might be caused by an overestimation ofneon isotopes in SHIELD-HIT, as the other fragmentation yields would consistentlybe too low if the inelastic cross section was underestimated in general. For thelighter elements (Be, B and C), the agreement is generally good, though with someoverestimation of the beryllium fragments far from the Bragg peak. For nitrogen,the yield is generally overestimated by SHIELD-HIT10A, though agreement is goodfor the oxygen yield. The largest overestimation is seen for the fluorine yield,which could be the result of an inaccurate angular distribution in SHIELD-HIT10A.While SHIELD-HIT10A presents in general the right order of magnitude for allfragments, further investigations should be made before SHIELD-HIT10A is usedfor simulations of neon beams which require very high accuracy.

Optimizing SHIELD-HIT for carbon ion treatment 15

0 5 10 15 20 25 30 350.00.20.40.60.81.0

N/N

0 HHe

0 5 10 15 20 25 30 35Depth (cm H2O)

0.000.010.020.030.040.050.060.070.08

N/N

0

Bragg peakLiBe

0 5 10 15 20 25 30 350.000.020.040.060.080.10

B

Figure 14: FLUKA, Geant4-QMD, Geant4-BIC LI (Bohlen et al. 2010) andSHIELD-HIT10A simulations of the relative yield of fragments emitted within a 10◦

forward angle from a 400 MeV/u 12C beam in water, compared with experimentaldata (Haettner et al. 2006). Dashed line: FLUKA simulation. Dashed-dotted line:

Geant4-QMD simulation. Dotted line: Geant4-BIC simulation. Solid line:SHIELD-HIT10A simulation. The markers are experimental data. Where error bars

are not visible, they are smaller than the markers.

4.2. Comparison with other Monte Carlo Codes

When comparing SHIELD-HIT10A with other Monte Carlo codes (figure 14), sometendencies can be observed. For hydrogen, helium and lithium, SHIELD-HIT10Ashows similar agreement to FLUKA and Geant4-QMD, but without the tendency tounderestimate the hydrogen fragments after the Bragg-peak. In addition, SHIELD-HIT10A does not have the underestimation of helium seen with Geant4-QMD,nor the underestimation of lithium seen with FLUKA. For the heavier boronand beryllium fragments, however, FLUKA and Geant4-QMD show significantlybetter agreement with experimental data than SHIELD-HIT10A. For boron, it isparticularly interesting to note how similar the Geant4-BIC model is to SHIELD-HIT10A. This suggest that the overestimation may be general to the cascade models,and it would be interesting to compare this with other codes using cascade models,such as MCNPX. None of the Monte Carlo codes show the best agreement with

Optimizing SHIELD-HIT for carbon ion treatment 16

experimental data everywhere, though Geant4-BIC shows the largest discrepancies,as previously reported in (Bohlen et al. 2010). SHIELD-HIT10A shows similar orbetter agreement than FLUKA and Geant4-QMD for the light fragment yields, butalso presents some discrepancies for the heavier fragments.

5. Conclusion

In this work, the improved nuclear models in SHIELD-HIT10A were presented andcompared with a wide range of experimental data for nuclear interactions, as wellas to the previous version of SHIELD-HIT (SHIELD-HIT08) and other Monte Carlocodes. SHIELD-HIT08 showed some overestimation for the total nuclear reactioncross section above 100 MeV/u, which was corrected in SHIELD-HIT10A. The newcalibration of the models for nuclear fragmentation in SHIELD-HIT10A showedsignificantly improved agreement with experiments for the partial charge-changingcross sections for carbon on various materials, with an agreement that was in generalwithin 2σ. When compared with fragmentation yield experiments of carbon ions ona water phantom, SHIELD-HIT10A showed good agreement with the data. Theagreement was comparable to that obtain in (Bohlen et al. 2010) with the MonteCarlo codes FLUKA and Geant4. While some discrepancies were observed forthe SHIELD-HIT10A boron and beryllium yields the code showed similar or betteragreement than FLUKA and Geant4 for the light fragments. When compared withfragmentation yield experiments of neon ions on water, larger discrepancies wereobserved, although SHIELD-HIT10A was in general in the right order of magnitude.In general, SHIELD-HIT10A is well suited for the simulation of problems arising inparticle therapy, providing fast and accurate data. In order to further improve thenuclear models of SHIELD-HIT, more experimental data is required, in particular forenergies below 100 MeV/u. Here the cross sections are the largest and the particleshave the highest linear energy transfer, which is directly related to the biologicaleffect of particle therapy.

Acknowledgments

The authors wish to thank T. T. Bohlen for fruitful discussions on Monte Carlosimulations of fragmentation yield experiments and making data available in digitalform. In addition we wish to thank M. Takechi for providing experimental data andM. H. Mortensen for digitizing data only available in print form.

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I M P R O V E D P R O T O N C O M P U T E D T O M O G R A P H YB Y D U A L M O D A L I T Y I M A G E R E C O N S T R U C T I O N

David C. Hansen, Jørgen Breede Baltzer Petersen,Niels Bassler, and Thomas Sangild Sørensen

Published in 2014 inMedical Physics

Volume 41, Page 031904

67

Improved proton computed tomography by dual modality image reconstruction

David C. Hansen,1, a) Jørgen Breede Baltzer Petersen,2 Niels Bassler,1 and ThomasSangild Sørensen31)Experimental Clinical Oncology, Aarhus University2)Medical Physics, Aarhus University Hospital3)Computer Science, Aarhus University; Clinical Medicine,Aarhus University

(Dated: 30 August 2014)

Purpose: Proton CT is a promising image modality for improving the stoppingpower estimates and dose calculations for particle therapy. However, the finiterange of about 33 cm of water of most commercial proton therapy systems limitsthe sites that can be scanned from a full 360◦ rotation. In this paper we proposea method to overcome the problem using a dual modality reconstruction (DMR)combining the proton data with a cone-beam x-ray prior.Methods: A Catphan 600 phantom was scanned using a cone beam x-ray CTscanner. A digital replica of the phantom was created in the Monte Carlo codeGeant4 and a 360◦ proton CT scan was simulated, storing the entrance and exitposition and momentum vector of every proton. Proton CT images were recon-structed using a varying number of angles from the scan. The proton CT imageswere reconstructed using a constrained non-linear conjugate gradient algorithm,minimizing total variation and the x-ray CT prior while remaining consistent withthe proton projection data. The proton histories were reconstructed along curvedcubic-spline paths.Result: The spatial resolution of the cone beam CT prior was retained for the fullysampled case and the 90◦ interval case, with the MTF = 0.5 (Modulation TransferFunction) ranging from 5.22 to 5.65 linepairs/cm. In the 45◦ interval case, theMTF = 0.5 dropped to 3.91 linepairs/cm For the fully sampled DMR, the max-imal RMS error was 0.006 in units of relative stopping power. For the limitedangle cases the maximal RMS error was 0.18, an almost five-fold improvementover the CBCT estimate.Conclusion: Dual modality reconstruction yields the high spatial resolution ofcone beam x-ray CT while maintaining the improved stopping power estimationof proton CT. In the case of limited angles, the use of prior image proton CTgreatly improves the resolution and stopping power estimate, but does not fullyachieve the quality of a 360◦ proton CT scan.

a)dch@ki.au.dk

1

I. INTRODUCTION

The primary clinical advantage of particle therapy over conventional radiotherapy isthe inherent sparing of healthy tissue.

Utilizing the full potential of particle therapy is however limited by the ability toaccurately predict the range of the ions at a given energy. The uncertainty in the rangecalculations lead to increased margins in the prescribed dose distribution to ensuretumor control, which results in increased dose to the healthy tissue.

The calculation of stopping power in patients is currently the single largest contrib-utor to the range uncertainty in particle therapy1. This stems from the fact that thestopping power inside the patient is calculated based on kilovoltage x-ray computedtomography (CT) scans. The Hounsfield units from these scans are heavily effected bythe average atomic number of the material, whereas stopping power is primarily de-pendent the electron density. A calibration for patient-like materials to convert betweenHounsfield units and stopping power is therefore used, but can lead to systematic un-certainties as the physics involved do not provide a unique mapping.

Several methods have been proposed to reduce the range uncertainty of particle ther-apy either by verifying the range in vivo (see reference2 for an extensive review) or byimproving the stopping power map used for the dose planning. In the latter category,kV-KV and kV-MV dual energy CT promises a theoretical improvement3, however thishas not yet been reproduced clinically4. Proton CT has been proposed as a possible so-lution to reduce the range uncertainty5. For proton CT the energetic protons penetratethrough the patient and their energy loss is measured. A reconstruction of the measureddata then yields a 3D map of the relative stopping power with respect to water of thepatient. This has a number of advantages, such as a significantly lower noise level anddose to the patient6. A problem in relation to proton CT is however the limited rangeof most commercial proton accelerators, which are typically on the order of 230 MeV,corresponding to a range of less than 33 cm of water. For larger patients and certainanatomical regions this is insufficient to penetrate the patient from all angles. Full angu-lar coverage is required for the tomographic reconstruction and such missing data leadsto severe artifacts in the reconstructed image.

It was previously proposed by Wang et al7 to overcome this by using prior knowledgein the form of x-ray CT. They used projections from megavoltage x-ray CT (MVCT) inplace of the missing data. Their method did not take into account proton scatteringhowever, which is known to severely limit the resolution8. Due to the reliance on pro-jection images rather than particle-per-particle reconstruction, incorporating scatteringis a non-trivial extension. The method is also susceptible to changes in the anatomy ofthe of patient as well as misalignment between the x-ray and proton scans. In addition,they only exclude the exact beam spots that cannot penetrate the patient, and not theentire angle. This puts a much higher demand on beam delivery for the proton CT scanand prevents the use of passively scattered beams.

In this work the dual modality reconstruction (DMR) method for proton CT is pre-sented, combining kilovoltage cone beam CT (CBCT) and proton CT. By using CBCT,which could be acquired simultaneously with the proton CT, the DMR method bypassesthe geometrical issues associated with using priors obtained at a different time and/or

2

location from the proton scan. Cone-beam CT is commonly available in commercialx-ray radiotherapy systems and is also being incorporated into proton therapy, mak-ing a combined CBCT/proton CT reconstruction particularly attractive. The method ofproton CT used in this work is a particle-per-particle reconstruction, which requires so-phisticated detectors to implement in practice9, but yields a better spatial resolution thanthe simpler straight-line projection technique8. The DMR method presented is howeverextremely general, and could be used with almost any iterative method of proton CTreconstruction.

II. THEORY

A. Proton CT Reconstruction

Image reconstruction of proton CT can be formulated as a linear least squares prob-lem:

minu‖Au− f‖2

2 (1)

where u is the relative stopping power map to be reconstructed, f is the projectiondata, and A is a linear transformation between the image and the projection data. ‖.‖2

2indicates the squared l2 norm, given by ‖u‖2

2 = ∑i x2i . A is also often denoted the

projection transform or system matrix. In addition to solving eq. 1 a non-negativitycondition is often imposed

ui ≥ 0 for all i (2)

to prevent negative stopping powers, which would be unphysical in a clinical setting.For proton CT, f is a vector of the water equivalent range traveled by every single proton.This is calculated according to

fi =∫ E0

Ei

(dEdx

)−1

dE (3)

where dEdx is the stopping power of the reference material, usually water, and E0 and Ei

are the energies before and after traversing the patient respectively.

1. The projection transform

The system transform A can be viewed as a simulation of the physical process thatcaused the data f to be observed. More explicitly, we could write A as a sparse matrix,where the element aij of the matrix equals the intersection length of the i’th protonwith the j’th voxel. We model this following the cubic spline approach of8. Here, everyproton trajectory is modeled as a third degree polynomial, fitted to the entrance andexit coordinates. If the entrance and exit points are x0 and x1 respectively, and themomentum directions are p0 and p1, we then define p0 = p0 · ‖x0 − x1‖ and p1 =p1 · ‖x0 − x1‖. The proton trajectory inside the reconstruction volume is then given by

s(t) = (2t3 − 3t2 + 1)x0 + (t3 − 2t2 + t)p0 − (2t3 − 3t2)x1 + (t3 − t2)p1 (4)

3

where 0 ≤ t ≤ 1 and the coefficients stem from the boundary conditions s(0) = x0,s(1) = x1, ∂s

∂t (0) = p0 and ∂s∂t (1) = p1. Outside the reconstruction volume, the protons

can be assumed to move in straight lines. This is motivated by the fact that the protonsscatter far more per unit length in water-like materials such as patients than they do inair. Most iterative reconstruction methods, including the one used in this paper, onlyrequire the computation of Au (the forward projection) and ATf (the back projection).Explicitly storing A, which would have on the order of 1010 non-zero elements, is thusnot required.

B. Dual Modality Reconstruction

In order to combine x-ray and proton CT, two assumptions are made about the stop-ping power map from the x-ray CT (referred to as the prior, up). 1) The error in theprior is primarily systematic, rather than random and 2) the error in the Hounsfieldunit to stopping power conversion is proportional to the Hounsfield values. For lowHounsfield materials, the attenuation scales with electron density, like stopping power,but becomes dominated by a Z3.62 effect for larger values.

In addition to the systematic uncertainties present in the stopping power maps fromstandard x-ray CT, cone beam CT has a number of additional artifacts. First of all thereis a higher level of noise. More importantly however, x-ray scattering leads to cuppingartifacts. This causes a change in Hounsfield units as a function of depth in the patient.

To compensate for this, total variation regularization was employed. Minimizing thetotal variation (TV) norm has succesfully been used as a noise reduction technique forconventional x-ray CT and in10 TV superiorization was used for proton CT. Total varia-tion minimization enforces that the image is sparse under the gradient transform given

by TV(u)i,j,k =√(ui,j,k − ui,j−1,k)2 + (ui,j,k − ui−1,j,k)2 + (ui,j,k − ui,j,k−1)2. This corre-

sponds to assuming that the reconstructed image is piece-wise constant. Total variationminimized reconstruction is achieved by solving the problem

minu‖TV(u)‖1 subject to ‖Au− f‖2

2 < ε (5)

where ‖.‖1 is the l1-norm given by ‖u‖1 = ∑i ‖ui‖. The prior image is then incorporatedinto the reconstruction by adding an additional term

minu‖W(u− up)‖2

2 + ‖TV(u)‖1 subject to u‖Au− f‖22 < ε (6)

where W is the covariance error matrix of the prior. In this work, W was assumed to bea diagonal weighting matrix given by Wi,i = up

−1i , but if a more detailed model of the

image covariance matrix was available it could be used instead.

III. METHOD

A. The Phantom

A cone beam x-ray CT scan was acquired at 125 kV of a Catphan 600 phantom11

using the on-board imager (OBI) of a Varian Trilogy linear accelerator12. This phantom

4

FIG. 1. Stopping power maps of the Geant4 models of the CPT528 resolution section (left) andCPT404 density section (right).

consists of five sections, two of which are evaluated in this study: the CTP404 section(referred to as the density section) and the CTP528 (referred to as the resolution section)as seen on FIG. 1. The CPT404 section consists of 8 12mm diameter rods, of differentmaterials (see table I), as well as 4 smaller rods, 3 of which contain air and one containsteflon. The center of the phantom contains 5 acrylic spheres, of diameters from 10 mmto 2 mm. The CPT528 section contains 21 aluminium line pair/cm gauges.

B. Monte Carlo simulated proton scan

All simulations were done using the Monte Carlo framework Geant413,14, version 9.6.We used the Standard 3 physics list for electromagnetic interactions and enabled thehadron elastic and binary ion models, for elastic and inelastic collisions respectively.Cuts were set to 0.1 mm. Digital replicas were made of the CPT404 and CPT528 sectionsof the Catphan 600 phantom. For the materials of the rods in the phantom, stochiometricdata is provided by the manufacturer. The casing and background are made of twodifferent kinds of proprietary plastics without any data provided by the vendor. Thesewere respectively assumed to be made of water, and PMMA . Both the CTP404 andCTP528 inserts contain a small air cavity at the edge to allow determination of rotationof the phantom. This cavity was not included in the Geant4 model. Neither were thepositioning wires in the CTP404 insert, as material and thickness of the wire was notspecified by the manufacturer.

In this work, the proton CT follows a design similar to the one originally suggestedin6. A detector is placed before the phantom and another after the phantom. For ev-ery individual proton, at both detectors, the position, direction and momentum of theproton is measured. For a real scanner, this information could be achieved using four

5

Name Composition Density (g/cm3)

Air 0.755N, 0.232O, 0.0128Ar, 1.24 · 10−4C 0.0012Teflon C2F4 2.16PMP H14C7 0.83LDPE H4C2 0.92

Polystyrene H8C8 1.05Delrin H2C1O1 1.41Acrylic H8C5O2 1.18

TABLE I. Material and composition used in the digital version of the rods in the CPT404 sectionof the Catphan 600 phantom.

6

strip-silicon detectors followed by a calorimeter and a prototype of such a device iscurrently being tested9. In this work virtual, perfect detectors were used, placed 1cmbefore and after the phantom. The two sections were scanned separately, using a flatparallel beam pattern for 360 distinct angles, with a target dose of 5 mGy (assuming1 mm slice thickness), corresponding to about 40 million protons for these particularphantoms. The energy used was 230MeV and an ideal beam source was used, meaningno modeling was done of the beam head.

C. Pre-processing of the data

The length traveled in the reference medium (water) was calculated based on the en-trance and exit energies. This was done according to ICRU recommendations15, usingthe stopping power library libdEdx16,17. Note that this differs from the Geant4 definitionof water due to the difference in I-value. While it would have been straightforward touse the stopping power values provided by Geant4, this would have given unrealisti-cally good estimates of the actual stopping power. In a clinical scenario, it would beimportant to use the same stopping power table in the reconstruction and in the subse-quent treatment planning. All events in which more than 0.5% of the energy had beenlost and the angle between entrance and exit were more than 3σ were removed, whichis common for proton CT18. Here σ was calculated for protons in water using the scat-tering power by Gottschalk19. This had two purposes. First of all to remove all protonsinvolved in elastic or inelastic nuclear interactions, as these would not fit into the linearreconstruction model. The second is that the cubic spline approach is a better fit forrelatively small angles. About 20% of the proton histories were removed this way.

D. X-ray conversion

Commonly Hounsfield to stopping power conversion is done by calibrating a piece-wise linear function to a stochiometric calibration of human materials20 and a similarapproach was used in this work. However, some of the materials in the Catphan phan-tom are poorly estimated by a calibration to human tissue. In particular delrin andteflon, which have a much higher stopping power than human tissue with a similarHounsfield value. Therefore, the conversion was done using a two-step piece-wise lin-ear function, splitting at 0 Hu. This was fitted to the stopping-power and Hounsfieldvalues of the materials present in the density region of the phantom. The stoppingpower was calculated at an energy of 230 MeV and the Hounsfield units at a photonenergy of 60 keV, which was estimated to be the mean energy of the cone-beam spec-trum. Both were calculated with the Penelope electromagnetic physics-list in Geant4.While exact knowledge of the materials inside a patient will not be available clinically,the CBCT scanner would be calibrated to detailed measurements which would accountfor the scanner spectrum. The conversion used here can therefore be considered to beof similar quality to what could be obtained in a clinical setting.

The sections not included in the Geant4 model in section III B were removed from theCBCT images and replaced with data from constant regions of the scan. The ground

7

truth used to compare the results is calculated as a voxel map with Geant4 using thestopping power of 230 MeV protons. In each voxel, the geometry was sampled 100 timesat random, the stopping power of the material in that point calculated and finally thevalues were averaged.

E. Reconstruction algorithm

Computationally it is very expensive to solve the constrained reconstruction problemin equation 5 and 6 directly. However these can be recast to an equivalent problem

minu

α(‖W(u− up)‖2

2 + ‖TV(u)‖1

)+ ‖Au− f‖2

2 (7)

which was solved using a non-linear conjugate gradient algorithm21. The prior imagewas used as a starting estimate. Here α should be chosen in such a way that the condition‖Au− f‖2

2 < ε still holds. For all reconstructions, 100 iterations of the solver were usedand the non-negativity constraint was imposed. For the fully sampled reconstructionα = 20. In the other reconstructions, this was scaled with the percentage of projectionsused.

F. Calculation of the Modulation transfer function

The spatial resolution of CT images is commonly estimated through the use of themodulation transfer function. This is usually calculated by measuring the point spreadfunction (PSF) from a small high density bead in the resolution phantom. However, thenon-linear nature of total variation makes this a poor choice22. Instead the modulationmethod23 is employed in this work. Here the MTF is calculated via the equation

MTF( f ) =π√

24

M( f )M0

(8)

where M( f ) is the noise-adjusted variance the line pairs at frequency f and M0 is themean value of the line pair and background stopping power. The disadvantage of thismethod is that it becomes less accurate for frequencies smaller than fc/3, where fc is thecutoff frequency. However, this is less of an issue as the high frequency components aretypically of larger interest.

G. GPU Implementation of the projection transform

Calculation of the forward and backward projections (Au and ATf respectively), aswell as all vector operations needed for the reconstruction were performed on an NvidiaGeforce Titan GPU using the Gadgetron image reconstruction framework? . The imple-mentations of both the forward and the backward transform assign one GPU thread toevery proton history. For the forward projection, the integral of the proton path throughthe material was calculated by evaluating the path at n = 3 · Nu equally spaced values

8

of t, where Nu is the maximum number of voxels along a single dimension. The integralwas then calculated by

(Au)k =n

∑i=1

uk(si)‖si − si−1‖+ ‖si − si+1‖

2(9)

where si = s(δt · i) and u(s) is the value of the image at the physical point s. The numberof evaluations was chosen to ensure that almost no image voxels were missed by theevaluation. Increasing this number did not improve image quality. The backprojectionATf was calculated simlarly to the forwards, but rather than reading at each point, everythread would write to the image voxel using the atomic add operation.

H. Cases studied

For both the spatial resolution insert and the density insert, four different cases werestudied in this work: A standard proton CT, a fully sampled dual modality CT and twodual modality CTs with limited angles. For the latter, a set of data was created wheredata from the angles in the interval [90◦, 180◦] and [270◦, 360◦] were removed (referredto as the 90◦ case), and another set where the intervals [45◦, 180◦] and [225◦, 360◦] wereremoved (referred to as the 45◦ case). In order to investigate the geometrical effects,the 90◦ case was repeated with the other half of the data for the density phantom. Theestimated dose of the CBCT is 6mGy24, making the total dose of the fully sampled DMRCT 11mGy.

IV. RESULTS

A. Spatial Resolution

The modulation transfer functions of the reconstructed images as well as the priorare shown in figure 3. The CBCT MTF and proton CT MTF are very similar, with theCBCT being slightly better above 4 linepairs/cm. The fully sampled DMR MTF followsthe standard proton MTF closely. In the 90◦ cases, the MTF generally fall betweenthe standard proton CT and CBCT MTFs above 4 linepairs/cm. At 45◦, the resolutionis significantly worse and the MTF drops off faster than in the other cases. This isquantified in table II, which shows the MTF( f ) = 0.5 points of the reconstructions. Herethe 45◦ value is more than 30% lower than the CBCT value. This may in part be dueto minor mismatches between the Monte Carlo model and the actual phantom, whichhas a larger effect in this case due to the lack of geometrical information. In the othercases the MTF( f ) = 0.5 values are very similar, deviating no more than 8%. It shouldbe noted that the MTF of the CBCT is similar to what has been reported elsewhere forthe same scanner type, using other methods for estimating the MTF25.

9

FIG. 2. Fully sampled DMR image of the CPT528 resolution section (left) and CPT404) densitysection (right).

0 2 4 6 8 10 12Linepairs / cm

0.0

0.2

0.4

0.6

0.8

1.0

MTF

CBCT prior

Proton CT

Full PIP CT

90 ◦ PIP CT45 ◦ PIP CT

FIG. 3. Plot of the Modulation transfer function (MTF) of the reconstructed images. The areabelow the CBCT MTF is shaded for ease of comparison.

B. Density

The accuracy of the reconstructed stopping powers are shown on figure 4. For allinserts, the converted CBCT shows the highest RMS error, ranging from 0.008 for air to0.088 for teflon. The large relative error (compared with the stopping power) in the airinserts most likely stem from image noise in the CBCT resulting from scattering arte-facts. For higher density materials, the errors are most likely a result of the Hounsfieldto stopping power conversion. The standard proton CT and the DMR consistently havevery similar RMS error, ranging from 0.002 to 0.006 (mean 0.003) and have the low-

10

Delrin

(1.36) Teflo

n

(1.79)

Air1

(0.001

07)PMP

(0.882

) LDPE

(0.978

)Poly

styren

e

(1.03)

Air2

(0.001

07) Acrylic

(1.16)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09RM

SErr

or(r

elat

ive

stop

ping

pow

er)

Proton CT

DMR

45◦ DMR

90◦ DMR

Other 90◦ DMR

Converted CBCT prior

FIG. 4. Boxplot showing the root mean square (RMS) error of the inserts in the CT404 slice. Thevalues below the material names are the ground truth relative stopping powers.

est error in all cases except that of teflon. The two 90◦ cases yield similar results formost materials with an RMS error ranging from 0.003 to 0.018 and a mean of 0.01. Forpolystyrene and teflon, large difference can however be observed. As the two cases onlydiffer in the geometry, the reason for this must be geometric interplay. Due to the lackof information perpendicular to the beam directions, errors in one part of the imagemay be compensated by errors of the opposite sign in other parts of the image, and thisis not fully corrected by the DMR method. Due to the high weighting of low stoppingpower materials in the prior, the large relative error in the air inserts are also seen in the90◦ cases. Such errors would presumably be smaller for larger cavities (ie. lung), butcould be problematic if present in patient scans for dose planning. This could likely bereduced using a more sophisticated covariance matrix for the prior. In the 45◦ case, theRMS error ranges from 0.003 to 0.017 (mean 0.01), similar to the 90◦ case.

While not shown visually, the cupping artifacts present on cone-beam CT images dueto scattering were not discernible in any of the DMR images.

V. DISCUSSION

The use of prior information has successfully been used in most parts of tomographicreconstruction, including MRI? and x-ray CT26. While using prior information fromother modalities is relatively common for PET imaging27, it is a relatively new concept inCT and MRI where combination is typically done after the image reconstruction, using

11

MTF( f ) = 0.5 (linepairs/cm)

Proton CT 5.22DMR 5.37

90◦ DMR 5.5645◦ DMR 3.91

Prior 5.65

TABLE II. Resolution where MTF( f ) = 50%.

12

image fusion. In x-ray CT, prior information has been used successfully via the priorimage constrained compressed sensing (PICCS) algorithm26. This method is similar toDMR in so far as both methods employ a prior image and use total variation. PICCShowever relies on the difference between the prior and the final image to be piecewiseconstant. If a conventional CT was used as the prior, this would be a good assumptionand the PICCS algorithm could be competitive with DMR. Cone-beam CT images arehowever, considerably more noisy and subject to scattering artifacts, which would bereflected in the final result. The use total variation term is often controversial, as it leadsto a visually displeasing ’staircase’ effect28. As proton CT is primarily intended for doseplanning and not for target delineation or diagnostics, this should not be a significantissue.

The use of cone-beam CT as the prior in DMR does have several advantages. First ofall, cone-beam CT is already present in many treatment rooms and therefore it wouldbe possible to perform the proton and x-ray scans simultaneously. This would effec-tively bypass any issue of alignment between the prior and the final image which wouldotherwise be present, even using in-room CT. Such errors would be even more severeif the planning CT was used, in which the patient could not be expected to have theexact same anatomy Provided that this was solved, for instance by using CBCT to doa deformable registration to the patient anatomy,the method presented is very general.Conventional MV or kV x-ray CT could just as easily be used as priors, or even MRI, ifa suitable error-weighting W was provided. The results would largely be dependent onthe quality of the error weighting as well as the quality of the conversion to stoppingpower.

The primary limitation of this work is the choice of the phantom used in the study.As mentioned previously, the materials of the rod inserts are not good tissue substitutes.Other phantoms were considered but these were either too large, having a radius closeto the range of a 230 MeV proton or were so intricate that they could not be replicatedinside the Monte Carlo code with sufficient precision.

This study was only done in 2d, and all results are assumed to be directly applicablefor a full 3d reconstruction. 3d does however give a greater potential for compressedsensing to sparsity in the data, which would further improve the results. Most phan-toms currently used for CT calibration and validation are however inadequate for suchstudies, as they typically have very little variation along the axis of rotation and wouldtherefore give unrealistically good results. A more human like phantom should be em-ployed in such studies. It would also be interesting to investigate the use of a differentsparsifying transform, such as framelets, for the compressed sensing. Using more ad-vanced methods of compressed sensing is also likely to improve quality in the case of alimited angle acquisition.

VI. CONCLUSION

In this work, the dual modality reconstruction (DMR) method for combining x-raycone-beam CT prior with proton CT imaging was presented. Using DMR with the fullproton data did not significantly alter the resulting image. With 50% of the data removedin 90◦ intervals, DMR retained the high resolution of the cone-beam CT. Removing 75%

13

of the data in 135◦ intervals noticeable decreased image resolution. In both cases, thestopping power estimates were greatly improved over the prior image however. Themaximal RMS error of 0.018, compared with that of 0.088 for the CBCT and the meanerror of the inserts was 0.01 compared to 0.03.

VII. APPENDIX - THE NONLINEAR CONJUGATE GRADIENT

The Hybrid Conjugate Gradient of Dai & Yuan21

Input: convex cost function f (x), cost function gradient ∇ f (x), starting image u

ρ← 0.5δ← 0.001g0 ← ∇ f (u0)d0 ← −g0i← 0while Not converged do

αi ← 1while f (ui + αdi) > f (ui) + α · δ · gi · di do

αi ← αi · ρend whileu∗i+1 ← ui + αidiui+1k ← max(u∗i+1, 0)d∗i ←

ui+1−uiα

gi+1 ← ∇ f (ui+1)βDY ← gi+1·gi+1

d∗i ·gi

βHS ← gi+1·gid∗i ·gi

β← max(min(βDY, βHS), 0)di+1 ← βd∗i − gi+1i← i + 1

end while

In the present case, the cost function is given by eq. 7. The gradient for the linear termscan easily be calculated via the normal-equation ∇ fA(u) = 1

2 AT(Au− f).

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28T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restora-tion,” SIAM Journal on Scientific Computing 22, 503–516 (2000).

16

T H E I M A G E Q U A L I T Y O F I O N C O M P U T E DT O M O G R A P H Y AT C L I N I C A L I M A G I N G D O S EL E V E L S

David C. Hansen, Niels Bassler,Thomas Sangild Sørensen and Joao Seco

Published in 2014 inMedical Physics

Volume 41, Page 111908

85

The image quality of ion computed tomography at clinical imaging dose levels

David C. Hansen,1, a) Niels Bassler,2 Thomas Sangild Sørensen,3 and Joao Seco41)Deptartment of Experimental Clinical Oncology, Aarhus University Hospital2)Department of Physics and Astronomy, Aarhus University3)Department of Computer Science, Aarhus University; Department of Clinical Medicine,Aarhus University4)Department of Radiation Oncology, Massachusetts General Hospital and HarvardMedical School, Boston, MA, USA

(Dated: November 2, 2014)

1

Purpose: Accurately predicting the range of radiotherapy ions in vivo is impor-tant for the precise delivery of dose in particle therapy. Range uncertainty iscurrently the single largest contribution to the dose margins used in planning,and leads to a higher dose to normal tissue. The use of ion CT has been proposedas a method to improve the range uncertainty, and thereby reduce dose to normaltissue of the patient. A wide variety of ions have been proposed and studied forthis purpose, but no studies evaluate the image quality obtained with differentions in a consistent manner. However, imaging doses ion CT are a concern whichmay limit the obtainable image quality. In addition, the imaging doses reportedhave not been directly comparable with x-ray CT doses due to the different bio-logical impact of ion radiation. The purpose of this work is to develop a robustmethodology for comparing the image quality of ion CT with respect to particletherapy, taking into account different reconstruction methods and ion species.Methods: A comparison of different ions and energies was made. Ion CT projec-tions were simulated for 5 different scenarios: Protons at 230 and 330 MeV, heliumions at 230 MeV/u and carbon ions at 430 MeV/u. Maps of the water equivalentstopping power were reconstructed using a weighted least squares method. Thedose was evaluated via a quality factor weighted CT dose index (CTDI) called theCT dose equivalent index (CTDEI). Spatial resolution was measured by the mod-ulation transfer function. This was done by a noise-robust fit to the edge spreadfunction. Secondly, the image quality as a function of the number of scanningangles was evaluated for protons at 230 MeV. In the resolution study, the CTDEIwas fixed to 10 mSv, similar to a typical x-ray CT scan. Finally, scans at a rangeof CTDEI’s were done, to evaluate dose influence on reconstruction error.Result: All ions yielded accurate stopping power estimates, none of which werestatistically different from the ground truth image. Resolution (as defined by theMTF = 10% point) was the best for the helium ions (18.21 line pairs/cm) andworst for the lower energy protons (9.37 line pairs/cm). The weighted qualityfactor for the different ions ranged from 1.23 for helium to 2.35 for carbon ions.For the angle study, a sharp increase in absolute error was observed below 45distinct angles, giving the impression of a threshold, rather than smooth, limit tothe number of angles.Conclusion: The method presented for comparing various ion CT modalities isfeasible for practical use. While all studied ions would improve upon x-ray CTfor particle range estimation, helium appears to give the best results and deservesfurther study for imaging.

a)dch@oncology.au.dk

2

I. INTRODUCTION

Particle therapy using light ions can greatly spare the radiation dose given to thenormal tissue of the patient, when compared with traditional radiotherapy using MVx-rays. The light ions slow down through the patient, ideally stopping in or aroundthe target volume and thus not depositing further dose downstream. The range of theparticles depend on their energy and the composition of the patient, the latter of whichis currently the largest source of uncertainty in particle therapy dose planning1. Thepatient composition is estimated from x-ray CT, which results in systematic errors in therange calculation, resulting in a larger dose to the normal tissue of the patient due toincreased dose margins.

In recent years several papers have been published on the subject of ion radiographyand tomography, which is a promising new image modality2–12. Ion CT is similar to themore traditional x-ray CT, but uses the energy loss of protons and other light ions as themethod of generating medical images, rather than the attenuation of x-rays. This hasseveral advantages, the most important of which is the fact that it produces an in vivomap of the relative stopping power of the patient. This is of particular importance toparticle therapy, where the accuracy of current stopping power estimates is the largestcontributor to the dose margin1. A better estimate would therefore allow for a moreaccurate treatment of the patients, sparing more normal tissue and thus reducing sideeffects. Other advantages of ion CT, at least for some ions, include a lower dose neededfor imaging and a better soft tissue contrast13. The most widely studied ions for ion CTare protons, due to their wide availability.

The main disadvantage of protons is that they are relatively light, causing multiplescattering to limit spatial resolution14 and energy straggling to increase the number ofparticles required. The spatial resolution can be greatly improved by modeling the pathof every individual proton in the reconstruction15, but requires quite sophisticated de-tectors as energy, as well as exit and entry angles of the individual protons are neededon a particle by particle basis. The particle by particle approach also results in a sig-nificantly longer reconstruction time for proton CT when compared with regular x-rayCT, taking several minutes on large computer clusters16, where iterative x-ray CT canbe done in less than a minute on a single GPU17. Both of these problems could, at leastpartly, be remedied by the use of heavier ions. These would scatter less and experi-ence less straggling, which is the largest source of noise for proton CT8. Fewer particleswould be therefore be required, leading to shorter reconstruction times.

The first experimental helium ion tomography scans18 were done at doses of 0.3 mGy,but were performed at a low resolution of 110× 110 pixels. Recent experimental studieson carbon ion tomography were done at higher resolution and report good results, butalso very high doses to the patient, orders of magnitude above what would be clinicallyfeasible2,7. This is in contrast to previous studies on carbon and proton radiography19,20,which reported significantly smaller doses than x-ray radiography. It can thus be in-ferred that the high doses in these studies are not necessarily an inherent limitationof carbon CT. Only few articles have compared different ions for the use in imaging.In Seco et al. 11 , protons and carbon ions were compared for the use in radiography,concluding that a higher spatial resolution was obtained with carbon ions. In Muraishi

3

et al. 2 , helium and carbon ions were compared for ion CT, again concluding that a higherresolution could be obtained using carbon ions. In neither article, dose was consideredas a limiting factor and in the latter work no attempt was made to model the ion trajec-tories. Both of these issues are important in a fair comparison as the spatial resolutionof proton and helium CT benefits strongly from modeling the particle trajectory15, whilethis has less of an impact when using heavier particles. Additionally, the biological im-pact of different ions is known to differ widely21, suggesting that physical dose may notbe a meaningful quantity in comparing different ions for CT or when comparing withx-ray CT.

In this article we build upon the reconstruction framework Gadgetron22, which waspreviously used for proton CT12, and allows for non-linear iterative reconstruction in-cluding the use of path modeling. A method for weighting the dose in a way that makesit directly comparable with x-ray CT scanners is presented and used to make a fair com-parison between hydrogen, helium, and carbon ions for the use in ion CT at clinicallyfeasible dose levels. Finally, we investigate the impact of the image quality as a functionof the number of beam angles.

II. THEORY

A. Reconstruction

Iterative CT and MR image reconstruction is typically treated as a weighted least-squares problem

minu‖W(Au− f)‖2

2 (1)

where A is the so called system matrix or system transform, u is a vector containingthe desired image to be reconstructed, f is a vector of the measured data and W is thecovariance matrix of the image noise. For parallel-beam CT, A corresponds to the Radontransform, and W is a diagonal matrix containing the inverse variance of the detectormeasurements. Following our previous work12, ion CT can also be modeled as a linearreconstruction problem but with the additional constraint that the resulting stoppingpower map u cannot contain negative values:

minu‖W(Au− f)‖2

2 s.t. u ≥ 0. (2)

In this work we modeled ion CT particle by particle in the reconstruction, similar to whathas previously been done for proton CT23. The linear transformation A thus correspondsto the integral of each particle track through the stopping power map. The approach ofLi et al. 15 was used, modeling each track via acubic spline path. The measured quantityf is the water equivalent length traveled by each track. Similarly to CT, W is the diagonalcovariance matrix containing the variance of each individual particle, calculated via thestraggling theory of Tschalar24,25. The weighted least squares approach corresponds tothe assumption that the range-straggling has a Gaussian distribution. This is obviouslyan approximation, but useful as it results in a linear optimization problem which is muchfaster to solve. As the noise contribution from a real detector is to good approximationGaussian8, the assumption that measured data is normally distributed is reasonable.

4

B. Dose equivalent calculation

Ions generally have a higher biological impact than x-rays, with heavier ions havinga greater response than lighter ones21. Therefore, for the purpose of assessing safescanning doses for ion CT, physical dose cannot be used directly and another metric isneeded.

In x-ray CT, the CT dose index (CTDI) is often used for reporting the dose of a givenscanner26. This is defined as

CTDI =1

NT

∫ ∞

−∞D(x,y)(z) dz (3)

where N is the number of slices scanned, T is the slice thickness and D(x,y)(z) is thedose at a point (x, y, z). In practical applications, CTDIl is measured, is measured with apencil beam ionization chamber of length l, usually l = 100 mm, CTDI100, in which casethe infinite integral boundaries are replaced by −l/2 and l/2. In order to estimate themean dose across the phantom, the weighted CTDI is defined

CTDIw =13

CTDImid +23

CTDIedge (4)

where CTDImid and CTDIedge are the CTDI evaluated at the center and 1 cm from theedge respectively.

For experimental measurements of large fields, such as seen in cone-beam CT, it maybe infeasible to obtain ionization chambers of sufficient length to cover the scan field andscattering tail. In this case, the equilibrium dose in the center of the phantom should bemeasured27,28, which for standard CT is equivalent to equation 3, as originally defined inDixon and Boone 29 . In this work we propose to extend the CTDI to take the biologicalimpact of different radiation types into account, giving the CT dose equivalent index(CTDEI).

In principle, when assessing the radiation detriment of external radiation exposure,ICRP21 recommends to use the equivalent dose terminology which can be applied toindividual organs or total body with specified weighting factors. For instance, all dosefrom protons is multiplied by a weighting factor (wR) of 2 and all dose from heavierions by a weighting factor of 20. As noted by the ICRP this is a very conservativeestimate, and more realistic weighting factors should be used when exposed to e.g.cosmic radiation. The recent ICRP recommendations for low doses of ion beams aremotivated by occupational exposure of astronauts and aircraft crew to cosmic rays, butdo not consider exposure to low doses of ions from ion-CT. In particular, the ions usedin ion CT are relatively light and fast resulting in a low linear energy transfer (LET) asthe Bragg-peak falls outside the patient.

CTDI is however defined as a measurable quantity, and in order to retain thisparadigm we will follow the ICRP definitions on operational quantities as close aspossible, leading to the definition of CTDEI.

For operational exposure, ICRU recommends to use an LET-dependent quality factor

5

Q(L), which was originally defined in the earlier ICRP 60 report30:

Q(L) =

1 L < 10 keV/µm0.32L− 2.2 10 keV/µm ≤ L ≤ 100 keV/µm300/

√L L > 100 keV/µm

(5)

where L is the LET to water. It should also be noted that in ICRP 60 this relationship wasintended for the secondary ion fragments from neutron beams. Assuming equation 5 isaccurate, it should however be equally applicable to ion beams and in the more recentICRP 103, it was applied to calculate the mean quality factor of protons to demonstratetheir low-LET properties21.

It should be noted that the fact that equation 5 has a maximum larger than 20 whichis consistent with the equivalent dose weighting factor of wr = 20 being a reasonableaverage for certain spectra.

Following ICRP recommendations, any of these relationships are only valid in thelow-dose region, i.e. below 100 mGy.

From the quality factor, ICRP defines the dose equivalent H in a point as

H = D ·Q (6)

where D is the dose andQ =

1D

∫ ∞

0DL(L) ·Q(L)dL (7)

is the mean quality factor with DL(L) being the dose differential in LET to water. Ideally,this should be evaluated at the center of a 30 cm diameter sphere of tissue equivalentmaterial (an ICRU sphere). We suggest that any sufficiently tissue-like material couldbe used without altering the results significantly including the polymethylmethacrylate(PMMA) of which the standard CTDI phantoms are made. As the ratio of stoppingpowers for different materials is to good approximation constant31 for light ions at therelevant energies, the primary concern becomes that of the cross sections for inelasticnuclear collisions. Accounting for density the macroscopic cross sections of the humantissues (taken from Woodard and White 32) only show smaller variations and PMMAfalls well within this range. It should be noted that (dry) air has a higher cross sectionthan human tissue by about 10% for protons and is therefore not a good tissue substitute.For the purpose of calculating the Q value at dosimeter locations only it is thereforeimportant not to model the dosimeter air cavities, but instead have these filled with thephantom material, i.e. PMMA.

The CTDEI is then defined simply as

CTDEI = CTDI ·Q (8)

Note that for x-ray radiation, Q(L) = 1 in accordance with ICRP recommendations,so in this case eq. 8 transforms to eq. 3, i.e. CTDEI = CTDI.

Similarly, as we are interested in the average quality factor in the phantom, we intro-duce the weighted quality factor

QW =13CTDImid ·Qmid + 2

3CTDIedge ·Qedge

CTDIW(9)

6

Figure 1. Geant4 image of the 20 cm diameter phantom used in the study.

which is the dose average of the quality factor in the at the edge and in the center of thephantom.

In this work, most studies were performed at CTDEIW = 10 mSv which is a typicalexposure from a diagnostic or therapeutic cone beam x-ray CT scan28.

III. METHOD

A. Reconstruction

Equation 2 was solved using an ordered subset algorithm33 using GPU accelerationwith the Gadgetron Framework22 the implementation details of which are described ina previous work12. 100 iterations were run to ensure convergence, and 20 subsets wereused. In practice, many iterative algorithms yield information about the low frequenciesin the image in the early iterations and then proceed to give information about thehigher frequencies as the number of iterations increases. Due to numerical issues, thehigh frequencies are often dominated by measurement noise from the data and theapproximations in the system transform. For most applications, this is solved either bythe use of regularization, such as total variation3, or by simply using a solver with a slowconvergence such as SART (Simultaneous Algebraic Reconstruction Technique)34,35. Inthis work, neither was done, as we wished to asses the maximal resolution obtainable,without considering the impact of the regularization. For the same reason, no metricwhich heavily influenced by noise was used in evaluating the results. All images werereconstructed on a 1024× 1024 pixel grid of 21 cm× 21 cm.

B. Phantoms & simulation

In order to evaluate the resolution and accuracy of ion CT with the different ionspecies, a phantom was created in the Monte Carlo code Geant436,37 (version 9.6), shownon fig. 1. The phantom is made of water and consists of 12 cylindrical inserts (2 cm diam-eter) of tissue and bone like materials, taken from38. In addition, a box of cortical bone

7

was inserted, which allows for evaluation of the modulation transfer function (MTF) viameasurement of the edge spread. Replicas were made of the standard CTDI head andbody phantoms, which have a radius of 16 cm and 8 cm respectively. Both are madefrom PMMA and contain 4 holes of 1.3 cm diameter placed 1 cm from the rim as well asone hole in the center. When calculating the CTDI, these holes were filled with air. Thedose to water was then calculated by using the constant stopping power ratio term of1.2205 from Luhr et al. 31 . For the purpose of calculating Q, the holes were filled withPMMA. All phantoms were 15 cm in length.

The scan setup follows that of Schulte et al. 13 . A detector was placed in front ofand after the phantom, registering position and momentum of each ion. For a realscanner, such information could be achieved using four strip-silicon detectors followedby a calorimeter and a prototype of such a device is currently being tested39. A parallelbeam was employed, to scan a single slice, with a slice thickness of 1 mm. Except in thecase of the angle study, 360 discrete angles were used for the scans. In the simulation,while all particles were used for the CTDEI calculation, only the ions with the samecharge and mass as the primaries were registered by the virtual scanner detector.

Protons and helium ions which had lost more than 0.05% of the entrance energy andhad scattered more than 3σ were removed before the image reconstruction, as has beendone in previous works on proton CT12,15. This is done to filter out particles that arethe results of nuclear fragmentation, and therefore do not fit the reconstruction model.This was not done for carbon ions as it did not have the desired effect of increasingimage quality. The energy loss of the particles was converted into water equivalentlength using the ICRU recommended stopping powers40–42 in the computer programlibdEdx43,44.

C. Resolution measurement

The modulation transfer function is the predominant method for measuring resolu-tion in tomography. The MTF relates the spatial frequency of a scanned object, withhow large an intensity change it will induce in the reconstructed image.

Several methods exist for measuring the modulation transfer function for x-ray CT.The most common are via measurement of the pointspread function (PSF) or the edge-spread function (ESF). The point spread function is typically measured by scanning athin high density wire or small bead inside a plastic phantom45. The high density regionneeds to be small enough that it can be considered point-like, i.e. much smaller than thepixel size. This method was not used for the current work. The discrete nature of the ionCT tracks makes this problematic, particular at low ion fluxes where the probability ofan ion hitting the high density area would be low. This is thus particularly problematicfor heavier ions and low doses, both of which reduce the flux of ions. The edgespreadfunction, as the name implies, is the blurring of an edge of a large high density region.If the edge is aligned with either the y or the x axis of the image, each line of pixelsgives a measurement of the ESF. By slightly tilting the edge, the exact position of theedge with respect to the pixels is changed, allowing for sub-pixel measurement. Thismethod is better suited for ion-CT, as it can also be made quite robust against the noisein non-regularized iterative reconstruction. For further details, the reader is referred to

8

the work of Mori and Machida 46 .The MTF and the ESF are conceptually related by

MTF( f ) = F ∂ESF(x)∂x

(10)

where F indicates the Fourier transform. The differentiation makes this method verysusceptible to noise and correction for this has to be done. To avoid this, the methodof Mori and Machida 46 was used, following their recommendations for x-ray CT usinga hard filter, as it has the same Gaussian PSF that can be observed in ion CT. TheMTF = 10% point was used to summarize the MTF into a single number, as has forexample been done in a previous study on ion radiography11.

D. Number of scanning angles

Unless the ion gantry can be rotated while shooting, the speed of ion CT will dependheavily on the number of angles used for a scan as even a small overhead for stop-and-shoot could add several minutes to the scan time when using 360 or 720 angles as iscommon in x-ray CT. Reducing the number of angles is thus important for a clinicallyfeasible design. We investigated the number of angles required by performing a numberof different scans of the phantom with 230 MeV protons, each at the same dose, but witha differing number of scanning angles. This was done to separate the issues coming froma low number of scan angles, and from a low number of particles. In order to evaluatespatial resolution, the edge-based method could not be used, as it would be highlydependent on the exact choice of angles in the case of only a few scan directions. Dueto the 1 dimensional nature of the ESF, a beam running parallel with the edge wouldprovide more information than one which was offset by some degrees. This dependencecan be removed by evaluating an ESF from the edge of a circular insert instead, whichwe refer to as a pseudo-ESF. The distance to the center of the insert is evaluated from thecenter of each pixel. This leads to the task-based MTF47, rather than the standard MTF.While both give a measure of resolution, the two are not directly comparable. In thiswork, the task-based MTF was used only for the number of angles study. The standardMTF was used for everything else, as it allows direct comparison with other resolutionstudies in x-ray CT.

E. Ions

In this study, three ion species were evaluated. Protons at 230 MeV and 330 MeV,helium ions at 230 MeV/u and carbon ions at 430 MeV/u. These values were cho-sen as typical maximal energies for particle therapy facilities, either in current use orcommercially available. Note that only the 330 MeV protons have sufficient energy tofully penetrate the body CTDI phantom. Q values for other ions and energies are stillreported, but are not representative of what would be obtained in a safe ion CT scan.

9

Ion & Energy (MeV/u) ICRP wR Study Phantom CTDI Head Phantom CTDI Body PhantomQavg QW Qavg QW Qavg

H 230 2 1.30 1.32 1.29 1.29∗ 1.27∗

H 330 2 1.33 1.40 1.36 1.39 1.36He 230 20 1.23 1.23 1.22 1.39∗ 1.38∗

C 430 20 1.59 1.49 1.53 2.35∗ 2.92∗

Table I. Dose average quality factor over the entire phantom (Qavg) compared with the weightedquality factor (QW) from different ion/energy pairs in three phantoms as well as the ICRPweighting factors (wR). Asterisks indicate that the ion energy was insufficient to penetrate thephantom.

Ion & Energy (MeV/u) Study Phantom CTDI Head Phantom CTDI Body PhantomPrim. Frag Prim. Frag Prim. Frag

H 230 1.00 2.74 1.00 2.34 1.03∗ 2.30∗

H 330 1.00 2.66 1.00 2.33 1.00 2.15He 230 1.00 2.07 1.00 2.08 1.17∗ 2.02∗

C 430 1.58 1.64 1.51 1.63 3.07∗ 2.56∗

Table II. Dose average quality factor from the primary ions and from the produced fragmentsfrom different ion/energy pairs in three phantoms. Asterisks indicate that the ion energy wasinsufficient to penetrate the phantom.

F. Evaluation of accuracy

The primary motivator for ion CT is the accurate prediction of relative stoppingpower for dose planning. When calculating the ion range inside a patient for treat-ment purposes, the main problem is systematic, rather than random, errors. Randomerrors in the image would largely even out when doing dose calculation, and the ex-pected range of the ion beam would still be correct. Random errors are also related tothe number of ions used in the ion CT scan and can be reduced by increasing the scandose. A systematic error, on the other hand, results in a change in the expected range,which would not be reduced by increasing the scan dose48.

Therefore, systematic error was used as the metric for the precision of the reconstruc-tion. In practice, this was done by taking the mean of the pixels within a radius of 0.8 cmin each tissue insert and comparing it with the ground truth, as obtained from the MonteCarlo code itself. The smaller radius was used to eliminate errors due to blurring. Forall particles, the ground truth is the stopping power of the tissue computed at the initialenergy of the ion, divided by the corresponding stopping power of water.

IV. RESULTS

The weighted quality factor QW as well as the dose average Q (Qavg) for the entirephantom is reported in table I. For the two CTDI phantoms, the QW can be consideredconstant over the two different phantoms. For helium ions a slight increase in QW isseen from the head to the body phantom, while for carbon ions the increase is more

10

0 5 10 15 20

CTDEIW (mSv)

0

2

4

6

8

10

Mea

nsy

stem

atic

erro

r(%

)

230 MeV 1H

330 MeV 1H

230 MeV/u 4He

430 MeV/u 12C

Figure 2. Mean systematic error as a function of weighted CT dose equivalent index (CTDEIW),plotted for different ion species and energies. The larger markers with an X indicate the pointwhere the ICRP weighting factors would lead to a CT equivalent dose index of 10 mSv.

dramatic ( 40%). With the exception of carbon ions in the body phantom, Qavg and QWdiffer by less than 0.04 for all three ions in both CTDI phantoms. Qavg in the studyphantom agrees within 0.06 with that in the CTDI head phantom for all ions.

Qavg from primary ions and from the fragments alone is shown in table II. For pro-tons and helium ions in the study phantom as well as in the CTDI head phantom, thecontribution to Q stems exclusively from the fragments. For helium and 230 MeV pro-tons in the body phantom, a small contribution is also seen from the primary fragments.For carbon ions, the contribution from the fragments and the primaries are of similarsize for all phantoms.

The mean systematic error as a function of dose is shown for each phantom in fig-ure 2. At a CTDEI of 20 mSv all ions show a systematic error below 0.2%. Assumingthat clinical range calculations should be better than 0.5%, carbon ions show signifi-cant error at 5 mSv (1.22%) and helium ions at 1 mSv (1.37%). Protons at both energiesremain below 0.5% even at 1 mSv. For a dose of CTDI = 10 mSv/Q using the ICRP

11

0 5 10 15 20

Frequency (linepairs/cm)

0.0

0.2

0.4

0.6

0.8

1.0

Modu

lation

Tra

nsfe

rFun

ctio

n

230 MeV 1H

330 MeV 1H

230 MeV/u 4He

430 MeV/u 12C

Figure 3. Modulation transfer function estimated using the method from46 for three differentions at four energies at a CTDEI of 10 mSv.

Ion & Energy (MeV/u) CTDI Head Phantom CTDI Body PhantomQmid Qedge Qmid Qedge

H 230 1.22 1.24 1.47∗ 1.51∗

H 330 1.50 1.60 1.58 1.63He 230 1.68 1.66 1.74∗ 1.80∗

C 430 1.42 1.46 1.76∗ 1.77∗

Table III. Ion CT quality factor from different ion energy pairs in the two CTDI phantoms in thecentral and edge positions. Asterisks indicate that the ion energy was insufficient to penetratethe phantom.

factors of Q = 2 and Q = 20, for protons and heavier ions respectively (indicated bythe larger markers on the figure), both helium and carbon ions show significant errors(1.39% and 7.99% respectively) while protons at 230 MeV and 330 MeV are both belowthe 0.5% threshold (0.10% and 0.17% respectively). This is consistent with the fact thatthe dose for the carbon and helium ions are 10 times smaller than the proton dose.

The MTF at a CTDEI of 10 mSv is shown in figure 3. The 430 MeV/u C and230 MeV/u He have similar MTFs, with carbon ions reaching the 10% point at 17.27 linepairs/cm and heliums ions at 18.21 line pairs/cm. The protons have a consistently lowerMTF, with 330 MeV protons reaching the 10% point at 11.60 line pairs/cm and 230 MeVprotons at 9.37 line pairs/cm.

The mean systematic error of relative stopping power as a function of the number ofprojection angles used in the reconstruction is shown on figure 4. A significant drop inthe error is seen above 15 angles, from 8.20% at 10 angles, to 0.30% at 30 angles and toless than 0.1% at 90 angles.

12

0 50 100 150 200 250 300 350 400

# of angles used

0

1

2

3

4

5

6

7

8

9

Mea

nsy

stem

atic

erro

r(%

)

Figure 4. Mean systematic stopping power error of the tissue inserts as a function of the numberof angles employed.

The fits of the error function to the circular edge can be seen on figure 5. The corre-sponding 10%-point of the task-based MTF can be seen in figure 6. A general increasein the 10% point as a function of the number of angles can be observed. This follows thetheoretical prediction from x-ray CT49,50 up 60 angles, but the increase becomes smallerabove this limit.s The 95% confidence interval was calculated from the covariance matrixof the fit, but cannot be seen on the graph as the bars are smaller than the size of thesymbols.

V. DISCUSSION

It could be expected that the MTF would only depend on the amount of scatteringfrom each ion, in which case carbon ions would be expected to give the highest resolu-tion. This is only accurate when the number of particles hitting each pixel in the imagefrom each scan angle is much large than one, in which case the field of ions can be con-sidered continuous. For the carbon ions, and to some degree the helium ions, this is not

13

Figure 5. Relative stopping power as a function of distance from the center of the circular insertfor a varying number of angles used in the reconstruction. The dotted line is the error-functionfitted to the edge data. The solid red indicates the mean, and the shaded areas the 95% confidenceinterval, of the raw data binned in 40 bins for ease of view.

the case at clinical dose levels. The image resolution is instead limited by the numberof ion tracks, which is why carbon ions show a worse resolution than the helium ions,even though the opposite could be expected as carbon ions scatter less.

This can also be seen in the systematic error, where the carbon ions show clinicallysignificant errors below 10 mSv, earlier than both helium and protons. This is still wellwithin the range of the scan doses used clinically today, and all three ions can thus beconsidered suitable for ion CT.

In all cases where the ions can penetrate the phantom QW represents the averagequality factor well, whereas some deviations can be seen between the two in the bodyphantom, particularly for carbon ions. As the ions are stopping in the phantom, the LETdistribution is highly heterogeneous, causing the difference between the average qualityfactor and QW .

The weighted quality factor QW for carbon and helium is small when comparedwith the ICRP weighting factor of 20. For helium ions QW agrees much better withthe weighting factor of 2 for protons, as QW for protons and helium ions are almost

14

0 50 100 150 200 250 300 350 400

# of angles used

0

5

10

15

20

25

Tas

kba

sed

MT

F=

10%

(lin

epai

rs/c

m)

Figure 6. The 10% point of the task-based MTF as a function of the number of angles used andcalculated for 230MeV protons at a CTDEI of 10 mSv. Errors bars indicated the 95% confidenceinterval. Blue line indicates the theoretical streak free maximum resolution for x-ray CT49,50.

identical. While QW is slightly higher for carbon ions, it is still well below the factor of2. If QW is not used, a general factor of 2 on the dose from ion CT may thus reasonablefor estimating the safe scan dose.

For protons, where the ICRP weighting factor is 2, QW ranges from 1.27 to 1.40. It isinteresting to see that the QW actually rises slightly with particle energy for the protonsand that protons have a higher QW then helium ions. In both cases, the cause is thelarger fluence of ions causing more nuclear events and thus a higher dose average LET.

For protons, nuclear models are quite accurate, resulting in reliable QW estimate. Forheavier ions, the models are not as accurate51,52, but has been shown to only have aminor impact on the dose average LET53 and thereby the QW .

The results for the image quality as a function of beam angles indicate that accurateestimation of the stopping power might be obtained with as few as 15 angles, but isgreatly dependent on the required resolution. Comparing with the theoretical streak-free limit49,50 shown in figure 6, this is the limiting factor below 60 angles, whereas the

15

physics of ion CT appears to be the limitation above this limit. It may be possible tofurther reduce the number of angles by the use of compressed sensing reconstruction orby using prior information as was done in our previous work12.

Gantries that allow beam delivery while rotating would also reduce scan time andmay be more feasible for ion CT than for radiotherapy using ions. For ion CT, knowledgeof the exact gantry position is not required until after the scan, whereas this informationwould be needed online for radiotherapy.

VI. OUTLOOK

The CTDEI concept presented in this work can readily be used in future studies onproton and ion CT, both experimental and simulated. It would be difficult to measurethe CTDEI directly in a clinical setting, as it requires knowledge of the LET spectrum. Inpractice however, Qedge and Qmid could be taken from the data presented in this work, ornew ones could be calculated if different ions or energies were used. The CTDI would bethen measured experimentally, as is currently done for x-ray CT. Note that while Qedgeand Qmid are in principle independent of the dose, QW requires the measured CTDImidand CTDIedge to calculate the dose average. The CTDEI could then be evaluated throughequation 8. The authors hope that other groups will use this methodology, in order togive a simple and consistent basis for comparing doses from ion CT.

VII. CONCLUSION

In this work, we have presented a method for evaluating ion CT scans in a manner tomake them comparable with x-ray CT and across ion species. The method was used tocompare the spatial resolution of ion CT with different ion species at the same dose andusing the same reconstruction method. It was concluded that, at clinical dose levels,helium ions gave the best resolution. At an equivalent dose of 10 mSv no significantdifference in the systematic error was observed between the images made with protons,carbon and helium respectively. The minimum equivalent dose shown to yield andacceptable image quality was 10 mSv for carbon ions, 5 mSv for helium ions and 1 mSvfor protons, all of which is within the range of what is currently given to patients fromx-ray CTs. The number of scanning angles needed for an ion CT image was studied andit was found that 60 distinct angles were sufficient for clinically acceptable range errors.

Based on the findings in this paper, we conclude that the ICRP weighting factor of 2for protons would be a realistic and conservative estimate for proton CT. Extending thisfactor to helium and carbon CT could also be considered safe, even relatively close tostopping.

ICRP acknowledges that only little knowledge exists for assessing the risk of low-dose ion beams. This underlines the need for further studies at dedicated experimentalradiobiology laboratories such as the proposed BioLEIR facility at CERN54.

16

ACKNOWLEDGMENTS

The authors would likely to thank the two anonymous reviewers, whose feedback sig-nificantly improved the manuscript. Bjarne Thomsen, Dept. of Physics and Astronomy,Aarhus University is thanked for providing cluster computing time. This project wassupported by CIRRO - The Lundbeck Foundation Center for Interventional Research inRadiation Oncology, The Danish Council for Strategic Research (http://www.cirro.dk).

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A C O M PA R I S O N O F D U A L E N E R G Y C T A N DP R O T O N C T F O R S T O P P I N G P O W E R E S T I M AT I O N

David C. Hansen, Joao Seco, Thomas Sangild Sørensen,Jørgen Breede Baltzer Petersen, Joachim E. Wildberger,

Frank Verhaegen and Guillaume Landry

Submitted in 2014 toMedical Physics

107

A comparison of dual energy CT and proton CT for stopping power estimation

David C. Hansen,1, a) Joao Seco,2 Thomas Sangild Sørensenn,3 Jørgen Breede BaltzerPetersen,4 Joachim E. Wildberger,5 Frank Verhaegen,6 and Guillaume Landry71)Department of Experimental Clinical Oncology, Aarhus University2)Department of Radiation Oncology, Massachussetts General Hospital and HarvardMedical School, Boston, USA3)Department of Clinical Medicine, Aarhus University, Aarhus ,Denmark4)Department of Medical Physics, Aarhus University Hospital,Denmark5)Department of Radiology, Maastricht University Medical Center (MUMC),The Netherlands6)Department of Radiation Oncology (MAASTRO), GROW School for Oncology andDevelopmental Biology, Maastricht University Medical Center (MUMC), Maastricht,the Netherlands;Medical Physics Unit, Department of Oncology, McGill University,Canada7)Faculty of Physics, Department of Medical Physics, Ludwig-Maximilians-University,Munich, Germany;Department of Radiation Oncology (MAASTRO), GROW School for Oncology andDevelopmental Biology, Maastricht University Medical Center (MUMC), Maastricht,the Netherlands

1

Purpose: Accurate stopping power estimation is crucial for high precision particletherapy and leads to improved outcome for the patient. Dual energy x-ray CT andproton CT have been both proposed as methods for obtaining patient stoppingpower maps. The purpose of this work is to give the first direct comparison ofthese two methods, compared with stopping power estimation using the state-of-the-art single energy CT method.

Methods: Two phantoms were scanned with dual energy and single energy CTwith a state-of-the-art dual energy CT scanner. Proton CT scans were simulatedusing Monte Carlo methods. The simulations followed the setup used in currentproton CT scanners and included modeling of detectors and the correspondingnoise characteristics.

Stopping power maps were calculated for all three scans, and compared withthe ground truth stopping power.

Results: Both dual energy CT and proton CT consistently gave better stoppingpower estimates than the standard CT method, reducing the maximal error to1.71% and 0.31% respectively from 2.61%.

Conclusion:In conclusion, while dual energy CT does give greatly improved stopping

power estimates and should be adopted clinically, proton CT can still providesignificant improvement to particle therapy.

a)dch@oncology.au.dk

2

I. INTRODUCTION

Accurate stopping power determination is required to calculate the ranges of chargedparticles in proton and light ion therapy and is critical for precise dose calculations andplanning. The primary advantage of particle therapy is the finite range of the particles,which leads to a significant dose sparing of the normal tissue in the patient. The pre-cision given by this finite range also makes particle therapy less robust to differencesbetween the dose planning models and reality. Inaccuracies in the estimated stoppingpower lead to inaccuracies in the predicted particle range, which again may lead to fail-ure of the treatment. The only clinical way to handle such inaccuracies is to increasethe dose margins used, with a larger dose to normal tissue as the result. State of the artfor estimating stopping power in vivo is the stoichiometric method based on x-ray CTscans1. This method is however limited by the fact that no direct relation between x-rayattenuation coefficient and stopping power exists. The result of this is systematic uncer-tainties of up to 3.4% in biological materials2 and potentially much higher in plastic andmetal inserts.

Several methods have been proposed to alleviate this problem, as recently reviewedby Knopf and Lomax 3 . One option is to measure the range of the particles in vivoduring treatment via PET or prompt gamma emissions. This would allow potentialover or under estimation of the dose to be adjusted interfractionally, but accuracy is stilllacking due to washout (PET), detector efficiency (prompt gamma) and the relativelylow number of photons produced (both)3. Another option is to use a different scanningtechnique to obtain better stopping power estimates directly. Proton CT has been widelytested for this purpose4–8 and a prototype head scanner is currently being test at theLoma Linda University Hospital9 as well as a smaller scanner in INFN-LNS, Catania,Italy10. These are however still limited by the fact that current scanning techniques arequite slow and take up valuable treatment beam time. In addition the maximum rangeof the protons could be a limiting factor for larger patients11. Another attractive option isdual energy x-ray CT (DECT), which is now clinically available from several vendors. Byobtaining photon attenuation coefficient values from two different x-ray spectra in eachvoxel, electron density and atomic number can be calculated. Yang et al. 12 demonstratedthat this could be used for stopping power calculations and recently Hunemohr et al. 13

compared the method with experiments and found an agreement better than 1.0%. AsDECT is somewhat simpler clinically, this raises the question whether proton CT holdsany clinical advantage for stopping power estimation. In this work we make a directcomparison between stopping power estimates obtained via the stoichiometric method,via DECT and via proton CT.

II. METHOD

Calculating the range of charged particles in proton and light ion therapy is donevia the concept of stopping power, defined as the energy lost per unit distance. For agiven medium and charged particle, this can be calculated theoretically via the Bethe

3

formula14

dEdx

= −4ρeπe4

mev21u

z2[

ln2mev2

I+ ln

11− β2 − β2

](1)

where ρe is the electron density, e is the electron charge, me the electron mass and uthe atomic mass unit, v the velocity and z the charge of the projectile. I is the meanexcitation energy of the medium and β = v/c.

For compounds, the mean excitation energy is typically calculated via the Bragg ad-ditivity ruleInternational Commision on Radiation Units and Measurements 15 , but thisis associated with some uncertaintySigmund, Schinner, and Paul 16 . In order to esti-mate the accuracy with which stopping power can be predicted, the data from13 is used.Here stopping power was measure experimentally, and the stoichiometric compositionof compounds reported. In this work, the theoretical stopping values were calculatedand compared with the measurements to calculate the standard deviation, taking intoaccount the experimental uncertaintyJakel et al. 17 .

A. Stopping power estimation from SECT

The Hounsfield units obtained in x-ray CT are a transformation of the attenuationcoefficients of the materials scanned. This can be calculated by18

µ(E) = ρNA

N

∑i=1

[wi/Ai

(Kph(E)Zn

i + Kcoh(E) · Zmi + KKN(E)

)](2)

where Kph(E), Kcoh(E) and KKN are the energy dependent photo-electric, Klein-Nishina,and coherent scattering coefficients respectively and Zi is the atomic number of the ithelement with relative contribution wi to the mass of the material made of N elementsand with mass density ρ. m and n are constants and are typically given the values 2.86,and 4.62 respectively18. Due to the large dependence on the atomic number, it is clearthat no direct relation can be established between x-ray attenuation units and stoppingpower. This is further complicated by the fact that the attenuation measured in x-rayCT is a weighted average from a spectrum of energies, rather than a mono-energeticcontribution.For SECT, an empirical relation must therefore be used. The simplest approach relies ona piece-wise linear relation based on measurements of Hounsfield units and stoppingpower of a phantom. The problem with this approach is that the stoichiometric compo-sition of the phantom materials is rarely a good substitute for human tissue. Instead,state of the art is the stoichiometric method1. For a real spectrum, the Hounsfield unitsare

HU = 1000 · ρ

ρH2O

N∑

i=1(wi/Ai)(Zi + Z2.86

i k1 + Z4.62i k2)

(wH/AH)(1 + k1 + k2) + (wO/AO)(8 + 82.86k1 + 84.63k2)− 1000 (3)

where k1 ≡ Kcoh

KKN and k2 ≡ Kph

KKN19. Here wH,AH, wO and AO indicate the fraction and

atomic weight of hydrogen and oxygen in water. x indicates the spectrum averaged

4

quantity, indicating that k1 and k2 are dependent on the x-ray spectrum and thereforethe specific scanner.

In the stoichiometric method, a phantom with inserts which need not be humantissue equivalent, but with known ρ, ρe and Zi is scanned. k1 and k2 are then found byfitting to equation 3. Theoretical Hounsfield units and stopping power relative to waterare calculated for a number of human like materials taken from20. A piece-wise linearrelation, consisting of 5 segments is then fitted to this. For human like materials, thisgives estimates accurate within 3.5% for tested materials12. For non-biological materials,such as plastics and metals, these errors can be significantly larger. As the stoppingpower comparison in this work is not done in biological tissue, we include both thestoichiometric and the simple piece-wise linear fit (referred to as stoichiometric SECTand Gammex SECT respectively.) in order to give a fair estimate.

B. Stopping power estimation from DECT

In DECT, attenuation coefficients for two x-ray spectra are obtained, either by scan-ning the patient with two different energies or by employing detectors with differentialenergy response21. This allows for the calculation of the electron density and meanexcitation potential needed to calculate stopping power from equation 1.

1. Electron density

Saito 22 introduced the dual energy subtracted quantity ∆HU

∆HU ≡ (1 + α)HU(H)− αHU(L) (4)

where HU(H) and HU(L) are the Hounsfield values obtained from the high and lowenergy spectra respectively and α is a scanner specific weighting. A linear relationbetween this quantity and electron density was then established

ρe = aρ∆HU1000

+ bρ. (5)

aρ and bρ can then be obtained via a calibration, while α is chosen to minimize the resid-uals in the calibration scan. The method was subsequently shown to have an accuracyof better than 1.0% for most materials23.

2. I-value

Equation 2 is valid for a single photon energy only, whereas the Hounsfield unitsmeasured in a CT scanner are the result of a full photon spectrum. In Landry et al. 24 , itwas established that the Hounsfield units could be approximated for human tissues by

HU(S) ≈ 1000 · ρ(

A(S) + B(S)Zn−1eff + C(S)Zm−1

eff

)− 1000 (6)

5

where ρ is the material density. A(S), B(S) and C(S) are energy response weightedaverages of the K factors from equation 2 and are dependent on the x-ray spectrum S.Zeff is the weighted average atomic number of the material in a given CT voxel. For aDECT scan, the ratio between the scans is then given by

HU(H)+10001000

HU(L)+10001000

≈ A(H) + B(H)Zn−1eff + C(H)Zm−1

eff

A(L) + B(L)Zn−1eff + C(L)Zm−1

eff

(7)

Yang et al. 12 established an empirical correlation for human tissues between the effectiveatomic number and the logarithm of the I-value through a piece-wise linear relation

ln(I) = aIZeff + bI (8)

where aI and bI are model parameters. In this work, we use two sections ((Zeff < 8.2)and (Zeff > 9)) fitted to the tissues from Woodard and White 20 . Only one material(thyroid) is present in the remaining region, for which a constant value is then used.

C. Phantom measurements

A dual source CT scanner (Siements SOMATOM Definition FLASH, Siemens Med-ical, Forchheim, Germany) was used for all x-ray scans. For dual energy scans, thelow spectrum was 80kVp and the high spectrum was 140kVp filtered with a Sn filter.For single energy scans, a standard 120kVp spectrum was used. A Gammex RMI 467(Gammex, Middleton, WI) electron density phantom was scanned with both dual andsingle energy. The phantom consists of 16 cylindrical inserts of different tissue-like ma-terials. The images were imported into MATLAB (MathWorks, Inc., Natick, MA) andthe average HU values were extracted for each insert using a region of interest cover-ing 90% of the insert, in a central slice of the phantom. These data were then usedto find the constants in equations 2, 5 and 7 (dual energy) as well as for the stoichio-metric and piece-wise linear fit (single energy). A second phantom, the CIRS Model002H5 IMRT Phantom (CIRS, Inc., Norfolk, VA) was also scanned and used for valida-tion. In all cases, the ground truth stopping power was calculated at 100 MeV based onthe stoichiometric information provided by the phantom manufacturers. The I-valueswere calculated based on the recommendations in International Commision on Radia-tion Units and Measurements 15 . All x-ray CT scans were performed with a computedtomography dose index (CTDI) of 30 mGy (for DECT this value was for the sum of thecontributions of the two scans). Phantom characteristics can be found in Landry et al. 24 .

D. Proton CT based stopping power estimation

In proton CT, high energy protons are shot through the patient and their energyrecorded on the other side. Based on the continuous slowing down approximation(CSDA), the water equivalent path-length (WEPL) can then be calculated via the equa-tion

WEPL(Ein, Eout) =∫ Eout

Ein

(dEdx

)−1

dE. (9)

6

Figure 1. Setup of the proton CT simulation, modelled after Sadrozinski et al. 30 and Sipalaet al. 31 .

From this water equivalent stopping power (WESP) can be reconstructed. In this work,we use an iterative reconstruction method previously described11,25. Here, the protontrajectories are modelled via cubic splines in order to account for scattering, as sug-gested in Li et al. 26 , and the reconstruction is done via weighted least squares usingthe Gadgetron framework27. Note that this is based on the assumption that the ratiobetween the stopping power of water and the stopping power of other materials is con-stant with energy. This would be accurate if all materials had the same I value. Whilethis is obviously an approximation, for clinically relevant energies and materials, thechange with energy is below 0.5%. The reconstructions were performed on a GeforceGTX Titan BLACK GPU (NVIDIA, Santa Clara, USA).

1. Monte Carlo Simulation

Digital replicas of the Gammex and CIRS phantoms, described in section II C, weremade in the Monte Carlo code Geant428,29. Proton CT scans were modelled after thedetectors used in Sadrozinski et al. 30 and Sipala et al. 31 (see figure 1). In order to reg-ister position and direction of the particles, four silicon strip detectors, each 200µmthick, were placed pair-wise on each side of the phantom. The distance between thedetectors in each pair was 5.0cm and the detectors were assumed to have a precision ofσxy = 200µm. A YAG:Ce calorimeter was placed after the last position detector. Thiswas modelled on the PRIMA calorimeter, and detector response and noise were donefollowing Sipala et al. 32 . Based on a fit of the experimental characterisation, an energyresolution of σE(E) = max(3% · E, 24.15MeV2

E + 1.76MeV) was used. All detectors wereassumed to have 100% efficiency, and only protons were considered in the scoring. Eachphantom was scanned using a uniform field of 250MeV protons at a CT equivalent doseindex (CTEDI)25 of 10mSv.

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1.0 1.1 1.2 1.3 1.4HU(H)HU(L)

6

7

8

9

10

11

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14

Zef

f

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−1000 −500 0 500 1000

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6 7 8 9 10 11 12 13 14Zeff

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−1000 −500 0 500 1000 1500

HU

0.0

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pow

er

SECT: Relative stopping power

Figure 2. Plots of the calibration curves for single and dual energy CT. Top left: Ratio between thehigh and low kVp HU values in the dual energy CT, fitted to effective atomic number. Top right:∆HU for the dual energy CT, fitted to electron density . Bottom left: Effective atomic numberfitted to the mean excitation energy (I-value) for tissues from Woodard and White 20 . Bottomright: HU value fitted to relative stopping power via the stoichiometric method for single energyCT. The blue dots are the measured data in the top row and calculated tissue properties in thebottom row, the red lines indicate the respective fits.

III. RESULTS

The uncertainty from the theoretical calculation of stopping power was found to beσ = 0.49%. The fits used for calculation of the relative stopping power for dual andsingle energy CT are shown in figure 2. Good agreement for electron density as afunction of ∆HU can be observed, while the effective atomic number as a function of theHU-ratio show a few outliers around a HU-ratio of 1.0. The two inserts having maximalresiduals are made of lung mimicking material.

The error for the inserts in the Gammex phantom can be seen on figure 3. Theaverage root mean square (RMS) errors are 2.69% for stoichiometric SECT, 1.57% for theGammex SECT, 0.54% for DECT and 0.21% for the proton CT. The maximum absolute

8

Lung300

Liver

CB2-50

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ativ

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ror

(%)

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DE CT

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SE CT - Gammex

SE CT - Stoichiometric

Figure 3. Residual errors in stopping power estimates based on DECT, SECT and proton CT ineach of the inserts of the Gammex phantom. Note that the DECT, and to some degree the SECTdata were fitted to the inserts in this phantom. Errorbars show the 1σ confidence interval.

errors are 7.44% for stoichiometric SECT, 3.6% for the Gammex SECT, 1.00% for DECTand 0.63% proton CT. For DECT, the results are consistently within 2σ of ground truth.This is also true for proton CT except in the case of cortical bone, which shows a minorunderestimation (0.38± 0.19%).

The corresponding errors for the CIRS phantom are shown on figure 4. Here theRMS errors are 1.61% for stoichiometric SECT, 3.56% for the Gammex SECT, 0.93% forDECT and 0.26% for proton CT. The maximum errors are 2.61%, 10.5%, 1.71%, and0.51% respectively. For proton CT, all results are within 2σ of the ground truth, but bothDECT and SECT show statistically significant deviations.

9

Bone

Adipose

Muscle

Lung

PlasticWater

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DE CT

Proton CT

SE CT - Gammex

SE CT - Stoichiometric

Figure 4. Residual errors in stopping power estimates based on DECT, SECT and proton CT ineach of the inserts of the CIRS phantom. Errorbars show the 1σ confidence interval.

IV. DISCUSSION

The good results of DECT for stopping power estimation in the Gammex phantomis not surprising, as this phantom was used for the calibration. Thus these data mainlyilluminate the errors from artifacts such as x-ray beam hardening and photon scattering.They agree well with the maximal error of 1.0% from Yang et al. 12 . For the single energyCT, the errors are significantly larger than the maximal error of 3.5% quoted, and indeedused clinically. This is most likely due to differences between the tissue substitutes inthe Gammex phantom and theoretical tissues used in the stoichiometric calibration. Asthe errors are largest in the two lung inserts, this suggest a particular difference for

10

low HU materials. Using the piece-wise linear SECT fit, the results are better, as couldbe expected when the same phantom was used for calibration. Even so, the maximalerror is large, which demonstrates that even if table values of human stoichiometriccomposition is completely accurate, SECT cannot perform better than 3.5%.

For the CIRS phantom, the errors for the SECT and DECT are more comparable. Theerrors for the DECT correspond more realistically to what could be achieved clinically.These errors are slightly larger than those reported in Hunemohr et al. 13 , where themaximal error were less than 1.0%. These were however based on measurements wherethe inserts were loaded centrally in a smaller diameter (16 cm) phantom. This meansthat artifacts from beam hardening and scattering were minimized. These artifacts couldintroduce additional errors. It should be noted that using statistical poly-energetic it-erative reconstruction can significantly reduce these effects33, at the cost of increasedreconstruction time. 1.0% is therefore a realistic estimate of the accuracy that can beachieved with the current generation of DECT. The phantoms used in this study are alsorelatively large, and for smaller regions such as the head and neck it is likely that DECTwould provide a greater advantage.

For the Gammex SECT, the results are clearly worse, as the Gammex phantom is nota good model for the CIRS composition.

As seen by the difference in SECT stopping power estimation between the two phan-toms, SECT is highly sensitive to stoichiometric variations and that the composition ofthe scanned material match that used for fitting. The primary advantage of DECT overSECT would thus mainly be the avoidance of the large deviations seen in a few materialsfor SECT. It can be hoped that the variations between the phantoms is larger than thatbetween individual patients.

A. Proton CT

In the studied phantoms, proton CT yielded the most accurate results for both phan-toms, as measured by the RMS error. However, the proton CT was Monte Carlo based,so some consideration must be given to how accurately this portrays what could be doneclinically. Specifically, while detector noise was modeled in this study, it was perfectlyGaussian and did not include the systematic uncertainties that could be present in areal detector. Comparing with literature, Hurley et al. 9 achieved 0.5% accuracy in anexperimental proton CT of a water phantom, which we consider to be a good estimateof what could be achieved in a clinical setting. This fits well with the accuracy achievedin this study.

In this light, proton CT compares favorably with DECT and in particular SECT. ProtonCT does not suffer from the artifacts present in x-ray CT, such as beam hardening. Fullspectral x-ray CT could potentially alleviate some of these artifacts, as beam hardeningcould be taken into account in the reconstruction. Proton also has the advantage thatit would image the patient directly in the treatment position, which could reduce setuperrors.

It does however suffer from longer reconstruction times; the reconstructions in thiswork took several hours on a top of the line GPU. This could partly be solved by theuse of non-iterative34 or non-statistical35 reconstruction methods which could be signifi-

11

cantly faster. No comparison has been made on the accuracy of these methods howeverand, based on the amount of data they process, they would still be at least an order ofmagnitude slower than their x-ray CT counterparts.

V. CONCLUSION

In this work, proton CT and dual energy CT were compared for the purpose of stop-ping power estimation and compare with single energy CT, as used in clinical practice.Maximal errors of 0.31%, 1.71% and 7.44% for proton CT, DECT and SECT were re-ported, consistent with results found in the literature. As DECT is clinically available,we suggest that it should become standard practice for particle therapy. Proton CTshowed a significant advantage in stopping power estimation over both the x-ray basedmethods. Provided the technical challenges can be overcome, proton CT would thereforebe the ideal method for in vivo stopping power estimation.

ACKNOWLEDGMENTS

This work was sponsored by the Lundbeck Centre for Interventional Research inRadiation Oncology (CIRRO). Bjarne Thomsen, Dept. of Physics and Astronomy, AarhusUniversity is thanked for providing cluster computing time.

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