Assessing the effect of electron density in photon dose calculations ...

This is an author produced version of an article that appears in: MEDICAL PHYSICS The internet address for this paper is: https://publications.icr.ac.uk/2642/ Copyright information: http://www.aip.org/pubservs/web_posting_guidelines.html Published text: J Seco, P M Evans (2006) Assessing the effect of electron density in photon dose calculations, Medical Physics, Vol. 33 (2), 540-552

Institute of Cancer Research Repository https://publications.icr.ac.uk Please direct all emails to: [email protected]

Assessing the effect of electron density in photon dose calculations J. Secoa兲 Francis H. Burr Proton Therapy Center, Massachusetts Hospital, Harvard Medical School, 30 Fruit Street, Boston, Massachusetts 02114

P. M. Evans Joint Department of Physics, Institute of Cancer Research and The Royal Marsden NHS Foundation Trust, Downs Road, Sutton, Surrey SM2 5PT, United Kingdom

共Received 6 July 2005; revised 28 November 2005; accepted for publication 29 November 2005; published 31 January 2006兲 Photon dose calculation algorithms 共such as the pencil beam and collapsed cone, CC兲 model the attenuation of a primary photon beam in media other than water, by using pathlength scaling based on the relative mass density of the media to water. In this study, we assess if differences in the electron density between the water and media, with different atomic composition, can influence the accuracy of conventional photon dose calculations algorithms. A comparison is performed between an electron-density scaling method and the standard mass-density scaling method for 共i兲 tissues present in the human body 共such as bone, muscle, etc.兲, and for 共ii兲 water-equivalent plastics, used in radiotherapy dosimetry and quality assurance. We demonstrate that the important material property that should be taken into account by photon dose algorithms is the electron density, and not the mass density. The mass-density scaling method is shown to overestimate, relative to electrondensity predictions, the primary photon fluence for tissues in the human body and water-equivalent plastics, where 6%–7% and 10% differences were observed respectively for bone and air. However, in the case of patients, differences are expected to be smaller due to the large complexity of a treatment plan and of the patient anatomy and atomic composition and of the smaller thickness of bone/air that incident photon beams of a treatment plan may have to traverse. Differences have also been observed for conventional dose algorithms, such as CC, where an overestimate of the lung dose occurs, when irradiating lung tumors. The incorrect lung dose can be attributed to the incorrect modeling of the photon beam attenuation through the rib cage 共thickness of 2 – 3 cm in bone upstream of the lung tumor兲 and through the lung and the oversimplified modeling of electron transport in convolution algorithms. In the present study, the overestimation of the primary photon fluence, using the mass-density scaling method, was shown to be a consequence of the differences in the hydrogen content between the various media studied and water. On the other hand, the electron-density scaling method was shown to predict primary photon fluence in media other than water to within 1 % – 2% for all the materials studied and for energies up to 5 MeV. For energies above 5 MeV, the accuracy of the electron-density scaling method was shown to depend on the photon energy, where for materials with a high content of calcium 共such as bone, cortical bone兲 or for primary photon energies above 10 MeV, the pair-production process could no longer be neglected. The electron-density scaling method was extended to account for pair-production attenuation of the primary photons. Therefore the scaling of the dose distributions in media other than water became dependent on the photon energy. The extended electron-scaling method was shown to estimate the photon range to within 1% for all materials studied and for energies from 100 keV to 20 MeV, allowing it to be used to scale dose distributions to media other than water and generated by clinical radiotherapy photon beams with accelerator energies from 4 to 20 MV. © 2006 American Association of Physicists in Medicine. 关DOI: 10.1118/1.2161407兴 Key words: photon dose calculation algorithm, mass-density heterogeneity correction, electrondensity heterogeneity correction I. INTRODUCTION Photon beam dose algorithms, used in radiotherapy treatment planning, are based on Monte Carlo calculations of primary photon beam attenuation and dose-deposition kernels in water.1 When applying these dose algorithms to media other than water, a heterogeneity correction is used. Heterogeneity corrections performed to nonwater materials have evolved with computing speed and the understanding and modeling 540

Med. Phys. 33 „2…, February 2006

of the radiation transport and interaction through regions of varying density. Algorithms such as a pencil beam 共PB兲, a collapsed cone 共CC兲, Batho, etc., all have a heterogeneity pathlength correction based on the mass-density value 共␳兲 per voxel obtained from CT data, which is known as an equivalent pathlength 共EPL兲.1 The EPL method scales the beam dose distribution to take into account changes with depth of the primary fluence in a medium other than water.

0094-2405/2006/33„2…/540/13/$23.00

© 2006 Am. Assoc. Phys. Med.

540

541

J. Seco and P. M. Evans: Effect of electron density in photon dose calculations

O’Connor2 and Sontag and Cunningham3 suggested the basis of the EPL method, which relates the dose in two media of different density but equal atomic composition, irradiated by the same external photon beam. O’Connor2 proposed that the ratio of scattered photon fluence to that of primary fluence at any point in two media are equal provided all geometric distances 共including field sizes兲 were inversely scaled by the mass density. The O’Connor theorem relates the dose in two media of different density and equal atomic composition, both irradiated by the same external beam. Monte Carlo has been used to assess how conventional dose calculation algorithms such as PB and CC perform in or around heterogeneities,1,4–8 where heterogeneity corrections use the EPL method. Ahnesjö8 showed that the CC algorithm overestimated the dose in the lung for small fields due to an overestimation of the primary dose and oversimplified modeling of electron transport in convolution algorithms. In regions of electronic disequilibrium, the CC was also shown to differ from MC predictions. These differences occurred at interfaces between heterogeneities and for small field sizes, where CC again overestimated the dose from primary photons. Although the CC algorithm calculates a dose reduction in the region of lateral electronic disequilibrium, it does not calculate the dose from primary photons accurately. Woo and Cunningham9 also observed this when evaluating CC in lowdensity media like cork. They showed that CC overestimated the dose at the proximal distal interfaces and within lung tissue and that the differences increased with increasing energy. The purpose of this paper is to 共a兲 assess if differences in the electron density between water and other media, with a different atomic composition, can influence the accuracy of conventional photon dose calculations algorithms; 共b兲 determine a universal heterogeneity correction, based on electron density that can be applied to conventional photon dose algorithms to scale the dose distributions correctly for media other than water; 共c兲 compare this new electron-density heterogeneity correction with the standard EPL mass-density heterogeneity correction in 共i兲 tissues present in the human body 共such as bone, muscle, etc.兲 and in 共ii兲 water-equivalent plastic materials, used in radiotherapy quality assurance phantoms; and 共d兲 assess the accuracy of this electrondensity heterogeneity correction as a function of the photon energy.

II. BACKGROUND CONVERTING CT NUMBERS TO MASS DENSITY FOR RADIOTHERAPY DOSE CALCULATION Computerized tomography 共CT兲 measures the spatial distribution of the linear attenuation coefficient ␮共x , y兲.10–12 However, the physical quantity ␮共x , y兲 is strongly dependent on the spectral energy used during the CT scan. Therefore, the attenuation coefficient is displayed as a CT number relative to the attenuation of water and defined by the following expression: Medical Physics, Vol. 33, No. 2, February 2006

NCT =

␮mat − ␮H2O ␮ H2O

541

* 1000,

共1兲

where ␮mat共x , y兲 and ␮H2O共x , y兲 are the linear attenuation coefficients for a material mat and water, respectively, and the CT number, NCT, comes in units of Hounsfield Units 共HU兲, with each HU equivalent to 0.1% of the linear attenuation of water.12 The NCT definition given in Eq. 共1兲 is such that water with a density of 1.0 g / cm3 has a CT number of 0. Since Hounsfield numbers are proportional to x-ray attenuation, they depend primarily on the electron density and, especially for heavy atoms, on the atomic number, because of the dominant interaction at the CT scanning energies in Compton scattering. However, the electron density is a quantity that is not used in commercial planning systems like Helax-TMS,13 Pinnacle,14 etc. The derivation of interaction parameters needed by the dose engines is instead based on a mapping from Hounsfield numbers to mass density. For every CT scanner used to obtain patient images for radiotherapy treatment planning, a CT calibration phantom with inserts with known mass densities is usually scanned to obtain the CT numbers associated with those mass densities.13,14 In Fig. 1, we present an example CT calibration curve to mass density that was obtained for materials present in the human body such as, air, lung, water, muscle, and bone. In addition, the CT calibration curve to electron density is also presented for a comparison. Both curves were normalized respectively to the mass density and electron density of water. During planning, the treatment planning system 共TPS兲 reads in the CT numbers and then determines the mass density of each voxel of the patient using the selected CT to the mass-density calibration curve 共see Fig. 1兲, while the electron-density curve is not used in the commercial dose calculation algorithms. The mass density is used to look up mass attenuation coefficients and is used for massdensity EPL scaling during superposition, based on the O’Connor theorem.1,2 The mass density is also used to scale the dose deposition kernel during the superposition to account for the effects of heterogeneities on scattered radiation.1,2 This is accomplished by tracing a ray line between a TERMA 共total energy released per unit mass兲 interaction site and the dose computation point and accumulating the radiological EPL distance along the ray. In the present work, we assess if differences observed in Fig. 1 between mass density and electron density can influence the accuracy of conventional photon dose calculations algorithms.

III. METHODS AND MATERIALS A. Mass-density heterogeneity correction used in commercial photon beam dose algorithms

The idea of the EPL heterogeneity correction methods is to scale the dose distribution, in a medium other than water and to take into account changes with depth in the primary fluence in this media relative to water. The heterogeneity corrections are usually obtained using a ray-tracing procedure, where increments along a ray line from the source ac-

542


542

FIG. 1. Example CT calibration curves are presented, which allow the conversion of CT numbers 共in Hounsfield units, HU兲 to either mass density or electron density. Both curves are normalized, respectively, to the mass density and electron density of water.

count for mass-density variations along its trajectory. The increments are usually found using mass densities. The EPL is given by the following expression: EPL = 兺 li i

␳i ␳ H2O

,

共2兲

where li is the physical ray line length through medium i , ␳i is the mass density of i, and ␳H2O is the mass density of water. The EPL heterogeneity correction method is based on the O’Connor theorem 共O’Connor2兲, which states that the ratio of the secondary scattered photon fluence to that of primary photon fluence is constant in two media, provided all geometric distances are scaled inversely with mass density. The photon dose algorithms in many commercial treatment planning systems are based on convolution models calculated in water and adapted to media other than water using EPL scaling methods. The energy deposition by secondary particles around a primary photon interaction site is, in homogeneous media, independent of the location of the site and suitable for description by a precalculated distribution. Full three-dimensional 共3-D兲 superposition/convolution methods such as PB and CC are based, respectively, on pencil and point kernels. For the PB case, the dose distribution in water is generated for monoenergetic photon beams using Monte Carlo and then scaled using the EPL method to account for heterogeMedical Physics, Vol. 33, No. 2, February 2006

neities only in the direction of the incident photon fluence. For the CC case, the energy deposition kernel is obtained using a Monte Carlo simulation of photons forced to interact at a certain point in water. The dose calculation is a two-step procedure. First, the total energy released in the material T共rជ兲共also known as TERMA兲, is calculated for the applied radiotherapy field, where heterogeneities are accounted for in the direction of the incident photon fluence using the EPL method. Then, this is convolved with the energy deposition kernels, which are also scaled to take into account heterogeneities. Ahnesjö and Aspradakis1 describe the PB and CC dose algorithms in detail.

B. Electron-density heterogeneity correction for photon beam dose algorithms

In the present study, we define the electron-density heterogeneity correction, eEPL, for photon beams by the following expression: eEPL = 兺 lei i

␳ ei ␳ eH O

,

共3兲

2

where lei is the physical ray line length though medium i, ␳ei is the electron density of i, and ␳eH O is the electron density 2 of water. The electron density of a medium med, ␳emed, is given by

543


␳emed = ␳med ⫻

冓冔 Z A

,

共4兲

med

where ␳med is the medium density and 具Z / A典med = 兺i f i ⫻ 共Z / A兲i is the effective atomic number to atomic weight ratio of a compound or mixture, with f i the fraction by weight of the chemical element i and 共Z / A兲i the atomic number to atomic weight ratio of chemical element i. In the following paragraphs, a brief discussion is presented on the assumptions made in the electron-density heterogeneity correction. The heterogeneity correction relates the photon fluence in a medium med with the photon fluence in water, such that the ratio of secondary scattered photon fluence to that of primary photon fluence is equal in both media 共O’Connor2兲. The scattered photon fluence appears as a consequence of the attenuation of the primary photon beam in a medium via photon interaction processes such as Rayleigh scattering, Compton scattering, etc. Primary photon beam attenuation in any media is characterised by the linear attenuation coefficient, ␮med, in units of cm−1, representing the fraction of the primary photons in the beam that will interact per unit pathlength. Therefore, in order for the ratio of secondary scattered photon fluence to primary photon fluence for medium med to be equal to that of water, the linear attenuation coefficient of med must be proportional to that of water, ␮H2O. In the case of the O’Connor theorem for media with an identical atomic composition but different mass densities, the proportionality constant is the ratio of mass densities:

␮med =

␳med ␳med ␮H O ⇒ EPL共lH2O兲 = lmed , ␳ H2O 2 ␳ H2O

共5兲

where the linear attenuation coefficient is related to the average length the photon will travel in a media med before interacting, which is defined by lmed = 1 / ␮med and lH2O = 1 / ␮H2O for water. The linear attenuation coefficient of any medium med, ␮med, is defined by the sum of the various photon interaction processes that may occur at any set energy for radiotherapy photon beams, PE Rayleigh Compton PP + ␮med + ␮med , ␮med = ␮med + ␮med

共6兲

where PE and PP are, respectively photo-electric and pairproduction. The electron-density heterogeneity correction proposed in Eq. 共3兲 is obtained by assuming that the primary photon beam is attenuated in med only by Compton interactions. This is a very good approximation for radiotherapy beams with photon energies between 100 keV and 10 MeV, where the Compton interaction is the dominant interaction process. Compton , for a The Compton linear attenuation coefficient, ␮med generic medium med is Compton ␮med ⬀

冓冔 Z A

␳med ⫻ ␴KN ,

共7兲

543

then obtained by dividing the linear attenuation coefficient of med by that of water: Compton eEPL共lH2O兲 ␳emed ␮med ␮med , ⬇ Compton ⇒ = ␮ H2O ␮ H O lmed ␳ eH O 2

共8兲

2

where lmed = 1 / ␮med and lH2O = 1 / ␮H2O. Note that because Z / A is approximately 21 for most chemical elements in the body 共except hydrogen兲, the Compton linear attenuation coefficient of a compound will depend significantly on the hydrogen content of the material. A comparison has been performed between the massdensity EPL and the electron-density eEPL correction methods for human tissues and water-equivalent plastic substitutes, which have very different electron-density values and that are used in radiotherapy. C. Mass-density and electron-density characteristics of human tissues

A set of tissues taken from ICRP 44,15 ICRP 23,16 and the HELAX-TMS planning system 共Nucletron B.V., Veenendaal, The Netherlands兲 was studied. The tissues and their atomic composition are given in Table I. The values presented in Table I are in fraction by weight of each elemental component. The tissues indicated in Table I are those usually used in radiotherapy dosimetry with Monte Carlo, PB, or CC dose calculations. The mass density and 具Z / A典 values are presented at the bottom of the table, where the electron density can be obtained for each tissue using Eq. 共3兲. D. Water-equivalent plastics used in dosimetry and quality assurance

It is common practice to use water-equivalent plastic phantoms for basic quality assurance of high-energy photon beams. A plastic phantom is considered water equivalent if it has the same absorption and scattering characteristics for photons and electrons as water. However, since radiation interactions depend on the density and atomic composition of the material, as well as the energy of the photons and electrons, it is impossible for the two materials to be strictly equivalent over the whole spectrum of energies used in radiation therapy. Many materials have been investigated as substitutes for water.17–21 We have compared the mass-density and electrondensity heterogeneity correction methods in several waterequivalent plastics, given in Table II. These water-equivalent plastics are used in many clinics for quality assurance and calibration at depths of either 5 or 10 cm. The waterequivalent plastics have approximately the same mass density but very different electron density 共cf. Table II兲. E. Homogeneous multilayer phantom used to evaluate photon beam attenuation in a media

med

where 具Z / A典med␳med is the electron density of med and ␴KN is the Klein–Nishina cross section for an interaction between a photon and a single static and free electron. Equation 共3兲 is Medical Physics, Vol. 33, No. 2, February 2006

The FLURZnrc Monte Carlo user code was used to model a homogeneous cylindrical phantom, in order to simulate the photon beam attenuation in a homogenous medium made of

544


544

TABLE I. Atomic composition of various tissues in the human body 共Refs. 15 and 16兲, ordered from lowest mass-density value, ␳, to the highest value. The effective atomic number to atomic weight ratio of the compound is 具Z / A典 = 兺i共Z / A兲i f i, where 共Z / A兲i is the atomic number to atomic weight ratio of chemical element i and 共f i兲 is the fraction of mass of the element in the compound presented in the table. Chemical elements presented in the table are identified by their chemical symbol.

H C N O Na Mg P S Cl Ar K Ca

Z

Z/A

1 6 7 8 11 12 15 16 17 18 19 20

0.992 0.500 0.500 0.500 0.478 0.494 0.484 0.499 0.479 0.451 0.486 0.499 ␳共g / cm3兲具Z / A典

Water 共H2O兲

Air

11.19

88.81

75.50 23.20

Lung

Adipose

Tissue

Muscle

Cartilage

Bone

10.30 10.50 3.10 74.90 0.20

11.40 59.80 0.70 27.80 0.10

10.12 11.10 2.60 76.18

10.20 14.30 3.40 71.00 0.10

9.60 9.90 2.20 74.40 0.50

0.20 0.30 0.30

0.10 0.10

0.20 0.30 0.10

2.20 0.90 0.30

3.40 15.50 4.20 43.50 0.10 0.20 10.30 0.30

1.30 0.20 1.000 0.555

1.203⫻ 10−3 0.499

0.40

0.260 0.550

the materials presented in Tables I and II. FLURZnrc calculated the photon fluence in the cylindrical 共RZ兲 geometry and was used to assess which heterogeneity correction method: 共a兲 EPL or 共b兲 eEPL correctly predicts the scattered photon fluence as the photon beam is attenuated in a media other than water. The RZ phantom was made up of 60 slabs with a waterequivalent thickness of 0.5 cm and two concentric cylinders of radius 0.01 and 100 cm, to model the attenuation of a pencil beam. Therefore, in media other than water, the thickness of each slab was calculated using either 共a兲 the standard

0.950 0.556

1.000 0.550

1.050 0.550

1.100 0.547

22.50 1.850 0.515

mass density EPL method defined in Eq. 共2兲, where EPL = 0.5 cm, or 共b兲 the electron-density eEPL method defined in Eq. 共3兲, where eEPL= 0.5 cm. For example, in the case of bone, the mass density of bone relative to water is ␳BONE / ␳H2O = 1.85, while the electron density relative to water is approximately ␳eBONE / ␳eH O = 1.72. Thus, for the stan2

dard mass-density EPL method the slab thickness is approximately lmed = 0.27 cm, while for the electron-density eEPL method the slab thickness is lmed = 0.29 cm. A 1 MeV photon pencil beam 共of width 0.01 cm兲 was used to irradiate the

TABLE II. Atomic composition of various water-equivalent materials 共Refs. 17–21兲, which are used for radiotherapy phantoms, ordered from lowest mass-density value, ␳, to the highest value. The effective atomic number to atomic weight ratio of the compound is 具Z / A典 = 兺i共Z / A兲i f i, where 共Z / A兲i is the atomic number to atomic weight ratio of chemical element i and 共f i兲 is the fraction of mass of the element in the compound presented in the table. Chemical elements presented in the table are identified by their chemical symbol.

H C N O F Mg Cl Ca Ti Br

Z

Z/A

1 6 7 8 9 12 17 20 22 35

0.992 0.500 0.500 0.500 0.474 0.494 0.479 0.499 0.459 0.438 ␳共g / cm3兲具Z / A典

Water 共H2O兲 11.19

88.81

WE210

Plastic H 2O

WT1

Solid H 2O

8.21 66.33 2.21 20.65

9.25 62.82 1.00 17.94

8.10 67.20 2.40 19.90

8.09 67.22 2.40 19.84

RW3

Polystyrene

A-150

PMMA

7.59 90.41

7.74 92.26

10.13 77.55 3.51 5.23 1.40

8.05 59.98

0.80 0.40 2.20

0.96 7.95

0.10 2.30

0.13 2.32

31.96

1.84 1.20

1.000 0.555

Medical Physics, Vol. 33, No. 2, February 2006

1.006 0.540

0.03 1.013 0.545

1.020 0.540

1.035 0.539

1.045 0.536

1.060 0.538

1.127 0.547

1.190 0.539

545


545

FIG. 2. The ratio of primary photon fluence in human tissues to fluence for a 1 MeV beam in water is presented as a function of depth in water, using, respectively, the mass density EPL and the electron-density eEPL estimates of the water-equivalent depth for tissues in the human body.

cylindrical phantom. This energy was selected because it is indicative of the average energy of a 6 MV treatment beam. IV. RESULTS A. Mass-density versus electron-density heterogeneity correction for a monoenergetic photon beam

The results were analyzed by plotting the ratio of the primary photon fluence in a material to that in water, corrected for differences in either the mass density or the electron density of the material relative to water. This ratio indicates how good the EPL or the eEPL methods are at estimating the fluence in a medium, if the fluence in water is known. If the heterogeneity correction is a good estimate of the photon attenuation in a media other than water, then this ratio of primary photon fluence in a material to that of water should be 1.00. 1. A 1 MeV photon beam irradiating human body tissues In Figs. 2共a兲 and 2共b兲, the ratio of primary photon fluence in human tissues to fluence in water for a 1 MeV beam is presented using, respectively, the EPL and the eEPL estimates of the water-equivalent depth. As an example, 5 cm of water in Fig. 2共a兲 corresponds in the EPL approximation to 2.70 cm of bone and to 19.00 cm of lung. On the other hand, 5 cm of water in Fig. 2共b兲 corresponds in the eEPL approximation to 2.91 cm of bone and to 19.40 cm of lung. It is possible to observe from this figure that performing the standard mass-density EPL scaling leads to an overestimation of the primary photon fluence in the medium relative to water, Medical Physics, Vol. 33, No. 2, February 2006

while the electron-density eEPL scaling method correctly predicts the primary photon fluence of the human tissues relative to water. In the case of bone and air, the EPL method predicts significantly higher primary photon fluence than in water because their atomic compositions are very different from that of water. Both air and bone have a lower hydrogen content than water; therefore their effective atomic number to atomic weight ratio is very different from that of water, where 具Z / A典bone / 具Z / A典H2O = 0.928 and 具Z / A典air / 具Z / A典H2O = 0.899. For the remaining materials presented in Fig. 2, i.e., lung, adipose, tissue, muscle, and cartilage, the hydrogen content is of the order of that in water 共cf. Table I兲. Therefore, smaller differences are observed for the primary photon fluence prediction in these materials relative to that of water. In the case of the eEPL scaling method, since it takes into account the electron density of the media, it therefore accounts for differences in the hydrogen content between the tissues in the human body and water. Thus eEPL accurately predicts the primary photon beam attenuation in media like bone and air, which have a significantly different atomic composition than water. The eEPL scaling method predicts to within 0.5% the photon fluence in tissues of the human body relative water, for depths in water up to 20 cm. 2. A 1 MeV photon beam irradiating waterequivalent plastics The primary photon fluence in water-equivalent plastics is presented in Figs. 3共a兲 and 3共b兲, using, respectively, the EPL and the eEPL estimates of the water-equivalent slab thickness for that material. It is possible to observe that at the depths of 5 and 10 cm in water, the EPL method predicts a

546


546

FIG. 3. The ratio of primary photon fluence in water-equivalent plastics to fluence for a 1 MeV beam in water is presented as function of depth in water, using, respectively, the mass-density EPL and the electron-density eEPL estimates of the water-equivalent depth for the plastics.

systematically higher fluence of 1%–2%, in the medium relative to water, while the eEPL accurately predicts the medium fluence relative to water at the same depths. This difference is largest for the plastics: RW3, polystyrene, and PMMA, which have a lower content of hydrogen than water. This lower content of hydrogen leads to a lower effective atomic number to atomic weight ratio to that of water, where the values are 具Z / A典RW3 / 具Z / A典H2O = 0.966, 具Z / A典Polystyrene / 具Z / A典H2O = 0.969, and 具Z / A典PMMA / 具Z / A典H2O = 0.971. Since the eEPL scaling method correctly accounts for differences in the electron density of the water-equivalent plastics relative to water; it therefore accurately predicts the primary photon fluence in these water-equivalent materials. The eEPL scaling method predicts to within 0.5% the photon fluence in water-equivalent plastics relative water, for depths in water up 20 cm.

B. Accuracy of mass-density and electron-density heterogeneity corrections with photon energy

The accuracy of the mass-density and the electron-density heterogeneity corrections will depend on the photon energy at which these energies are performed. We evaluate for 共a兲 tissues in the human body and 共b兲 water-equivalent plastics the accuracy of the mass-density and the electron-density heterogeneity corrections as a function of the photon energy. The results were analyzed by plotting the ratio of the photon attenuation lengths in water relative to a material med, corrected for differences in either the mass density or the electron density of the water relative to the material. This ratio indicates how good the EPL or the eEPL method is at estimating the photon pathlength in water, if the pathlength in the medium is known. If either the EPL or the eEPL heteroMedical Physics, Vol. 33, No. 2, February 2006

geneity correction is a good estimate of the photon attenuation length in a media other than water, then this ratio should be 1.00.

1. Tissues in the human body In Figs. 4共a兲 and 4共b兲, the ratio of photon attenuation H2O lengths in water relative to a material, ␳med 共lH2O / lmed兲 for H2O is either tissues defined in Table I, is presented, where ␳med the ratio of the mass density or the electron density of water relative to the medium, respectively, for the EPL and the eEPL methods. The photon attenuation lengths lH2O and lmed, were calculated using attenuation coefficients from the XCOM photon cross-section database,22 where the values were divided, respectively, by the mass density and electron density of water and med in order to allow a comparison with the EPL and eEPL models. For energies between 100 keV and 7 MeV, the EPL model tends to systematically predict larger photon attenuation lengths in water for the tissues defined in Table I. For lung, tissue, muscle, and cartilage the EPL method predicts a photon attenuation length that is, on average, 2% larger than H2O 共lH2O / lmed兲 value. In the case of bone and air, a the ␳med larger discrepancy is observed between the EPL estimate and H2O 共lH2O / lmed兲 value 共maximum differences are, respecthe ␳med tively, 7% and 10% and occurring for photon energies close to 1 MeV兲, with the EPL model providing a poor estimate. H2O 共lH2O / lmed兲 and the The differences observed between ␳med EPL estimate of the photon attenuation length for all materials in Table I are a consequence of the lower content of hydrogen in these media relative to water. In the case of bone and air, where the discrepancies are larger, the hydrogen content is significantly lower than in water.

547


547

H2O FIG. 4. The ratio of photon attenuation lengths in water relative to a material, ␳med 共lH2O / lmed兲 is presented for tissues in the human body. Photon attenuation lengths in water, lH2O and med, lmed were calculated using attenuation coefficients from the XCOM photon cross-section database 共Ref. 22兲, where the values were divided by the 共a兲 mass density and 共b兲 electron density of water and med in order to allow a comparison, respectively, with the EPL and eEPL models for tissues defined in Table I. If either the EPL or the eEPL heterogeneity correction is a good estimate of the photon attenuation length in a media other than water, then this ratio should be 1.00, as represented by a dotted line. In 共c兲 is presented the photon fluence spectrum of 10⫻ 10 cm2 6 MV photon beam, obtained at 100 cm source to surface distance.

For the same energy range 共100 keV to 7 MeV兲, the eEPL model provides a significantly better estimate of the photon attenuation length in water than EPL, for nearly all materials defined in Table I, except bone. For this energy interval, the H2O 共lH2O / lmed兲 for all mateeEPL estimate is within 2% of ␳med rials in Table I, except bone. In the case of bone, the eEPL method provides a good estimate of the photon attenuation length to within 3%, for photon energies between 100 keV and 4.0 MeV. However, for photon energies above 4.0 MeV, the eEPL method incorrectly estimates the photon attenuaH2O 共lH2O / lmed兲 is a tion length. The increase with energy of ␳med consequence of the increase in the probability of pair proMedical Physics, Vol. 33, No. 2, February 2006

duction occurring in bone, which is not accounted for in the eEPL model. The high probability of pair production in bone is a consequence of the high content of calcium 共Ca兲 and phosphor 共P兲, which have a significantly higher Z value than chemical elements present in other media in Table I and because the probability of pair production increases with Z2. The photon fluence spectrum of a 10⫻ 10 cm2 field is presented in Fig. 4共c兲 for a clinical 6 MV photon beam, obtained from Seco et al.,23 where it is possible to observe that approximately 95% of the photon fluence is below 3 MeV. This photon fluence spectrum was used to assess the accuracy of performing the EPL or the eEPL heterogeneity cor-

548


TABLE III. The percentage difference between the EPL or the eEPL methods and the photon attenuation length in water, lH2O, is presented for tissues in H2O the human body, and which are defined respectively by ␳med 关共EPL H2O 关共eEPL− lH2O兲 / lmed兴100. This percentage differ− lH2O兲 / lmed兴100 and ␳med ence is calculated at each energy bin in Figs. 4共a兲 and 4共b兲, and then summed over the whole energy spectrum, where a weighting is performed per energy bin with the aid of Fig. 4共c兲. H2O ␳med

Air Lung Adipose Tissue Muscle Cartilage Bone

共EPL− lH2O兲 lmed 共%兲

100

H2O ␳med

10.0 0.8 0.0 1.0 0.9 1.4 6.5

共eEPL− lH2O兲 lmed 共%兲

100

−0.1 0.1 0.2 0.2 0.1 0.0 −0.9

rection on a clinical 6 MV photon beam for tissues in the human body. The energy spectrum was normalized to unity such that it could be used to perform weighted calculations of the percentage difference between the EPL or eEPL method and the photon attenuation length lH2O in water. The results obtained are presented in Table III, where the values represent how accurately primary beam attenuation is modeled by either the EPL or the eEPL methods for the 10 ⫻ 10 cm2 energy spectrum. The eEPL model accurately predicts, to within 1%, the primary photon beam attenuation for tissues in the human body, where the maximum difference was −0.9% for bone. The EPL model also provides a good

548

estimate, to within 1.5%, of the primary photon beam attenuation in lung, adipose, tissue, muscle, and cartilage, where the EPL values for these materials are systematically higher than eEPL. However, for bone and air, EPL incorrectly estimates primary photon beam attenuation due to the significantly different atomic composition of these materials relative to water; therefore the EPL model will incorrectly predict the scattered photon fluence in this medium. 2. Water-equivalent plastics In Figs. 5共a兲 and 5共b兲, the ratio of photon attenuation H2O lengths in water relative to a material, ␳med 共lH2O / lmed兲, for water-equivalent plastics defined in Table II, are presented, H2O is either the ratio of the mass density or the where ␳med electron density of water relative to the medium, respectively, for the EPL and the eEPL methods. The photon attenuation lengths lH2O and lmed were calculated using attenuation coefficients from the XCOM photon cross-section database,22 where the values were divided, respectively, by the mass density and electron density of water and med in order to allow a comparison with the EPL and eEPL models. The water-equivalent plastics should have very similar absorption and scattering properties as water and therefore the EPL heterogeneity correction should provide a good estimate of the photon attenuation length lH2O in water of these materials. However, as may be seen in Fig. 5共a兲, this is not the case. All ratios are systematically lower than 1.00. These differences are a consequence of the difference in the hydrogen content in the plastics relative to water. Polystyrene and RW3 are the two plastic materials with a lower hydrogen

H2O FIG. 5. The ratio of photon attenuation lengths in water relative to a material, ␳med 共lH2O / lmed兲, is presented for water-equivalent plastics. Photon attenuation lengths in water, lH2O, and med, lmed were calculated using attenuation coefficients from the XCOM photon cross-section database 共Ref. 22兲, where the values were divided by the 共a兲 mass density and 共b兲 electron density of water and med in order to allow a comparison, respectively, with the EPL and eEPL models for water-equivalent plastics defined in Table II. If either the EPL or the eEPL heterogeneity correction is a good estimate of the photon attenuation length in a media other than water, then this ration should be 1.00, as represented by a dotted line.


549


TABLE IV. The percentage difference between the EPL or the eEPL methods and the photon attenuation length in water, lH2O, were calculated for waterH2O equivalent plastics, and that are defined, respectively, by ␳med 关共EPL H2O 关共eEPL− lH2O兲 / lmed兴100. This percentage differ− lH2O兲 / lmed兴100 and ␳med ence is calculated at each energy bin in Figs. 5共a兲 and 5共b兲, and then summed over the whole energy spectrum, where a weighting is performed per energy bin with the aid of Fig. 4共c兲. H2O ␳med

WE210 Plastic H2O WT1 Solid H2O RW3 Polystyrene A-150 PMMA

共EPL− lH2O兲 lmed 共%兲

100

H2O ␳med

2.8 1.7 2.9 2.9 3.5 3.3 1.2 2.9

共eEPL− lH2O兲 lmed 共%兲

100

0.0 −0.3 0.0 0.1 0.0 0.1 −0.3 0.0

549

C. Extension of the eEPL method to account for pair production

In the previous sections, the eEPL method was shown to be significantly more accurate than the EPL method at predicting the primary photon beam attenuation for all photon energies between 100 keV and 10 MeV and for all the materials defined in Tables I and II, except for bone, where for energies above 5 MeV, the EPL method was more accurate 共cf. Fig. 4共a兲兲. Since the eEPL model assumed that Compton interaction is the dominant process 共cf. Eqs. 共7兲 and 共8兲兲, its accuracy varied with the primary photon beam energy, depending on whether pair production could be neglected or not. In this section, the eEPL heterogeneity correction is extended to account for pair production. The eEPL heterogeneity correction can be extended to include pair production in the following extension to Eq. 共8兲: 1/2 eEPL ␳emed共1 + ␣fit兵1 + 具Z典med其ln共E兲E 兲 , = lmed ␳eH O共1 + ␣fit兵1 + 具Z典H2O其ln共E兲E1/2兲

共9兲

2

content than water, and, as may be seen in Fig. 5共a兲, are the H2O materials with the lowest value for the ratio ␳med 共lH2O / lmed兲 and therefore have different scattering properties to water. Since the scattering properties of the water-equivalent plastics are very different from that of water, the EPL heterogeneity correction will provide a poor estimate of the photon attenuation length lH2O for the media. In the case of the eEPL model, a significantly better estiH2O 共lH2O / lmed兲 than EPL. For energies mate is provided of ␳med between 100 keV and 6 MeV, eEPL provides a good estiH2O 共lH2O / lmed兲 to within 2% for all the watermate of the ␳med equivalent plastics. In this energy range, EPL systematically overestimates the photon attenuation lengths by up to 3%, where the largest differences are observed for polystyrene and RW3. For energies above 6 MeV, eEPL overestimates H2O ␳med 共lH2O / lmed兲 for nearly all the water-equivalent materials, except plastic H2O. This is a consequence of the increase of the photon attenuation in the plastic media with photon energy, due to the lower probability of pair production occurring in these plastics than in water. Since water is mainly composed of oxygen 共with Z = 8兲, and water-equivalent plastics are mainly composed of carbon 共with Z = 6兲; therefore the plastics have lower probability of pair production than water because they have a lower effective Z value than water, and the probability of pair production increases with Z2. In addition, the plastics have also a lower content in hydrogen than water and therefore have lower Compton and pairproduction interaction probabilities than water. The photon fluence spectrum of the 10⫻ 10 cm2 field, given in Fig. 4共c兲, was used to assess the accuracy of performing the EPL or the eEPL heterogeneity correction to a clinical 6 MV photon beam for the water-equivalent plastics given in Table II. The values obtained are presented in Table IV. The eEPL method correctly models the primary photon beam attenuation in the water-equivalent plastics to within 0.3%. While, the EPL heterogeneity correction overestimates by up to 3.5% 共the maximum difference occurring for RW3兲, the attenuation of the 10⫻ 10 cm2 6 MV photon beam. Medical Physics, Vol. 33, No. 2, February 2006

where the pair-production attenuation coefficient has been added to the numerator and denominator of Eq. 共8兲, in square brackets. The dependence of the attenuation coefficients on photon energy, E, and the medium atomic number, Z, are ␮Compton ⬀ Z / A * E−1/2 and ␮PP ⬀ 共Z / A given by 2 + Z / A兲 * ln共E兲, respectively, for the Compton and pairproduction interactions,24 with the Compton effect having a weak E−1/2 energy dependence. Pair production can occur in the electromagnetic field of the electron 共giving Z dependence兲 or the nucleus 共giving Z2 dependence兲; 具Z典 = 共1 / 具Z / A典兲共兺i f i共Z2 / A兲i兲, with f i the fraction by weight of the chemical element i and 共Z / A兲i the atomic number to the atomic weight of chemical element i; ␣fit = 1.775⫻ 10−3 is a fitting parameter that is independent of material and photon energy and was obtained by fitting Eq. 共9兲 to the photon attenuation lengths lH2O of all the materials defined in Tables I and II. In Figs. 6共a兲 and 6共b兲, the ratio of photon attenuation H2O 共lH2O / lmed兲, for the length in water, to that of a material, ␳med tissues defined in Table I and the water-equivalent plastics defined in Table II, lmed, are presented. The photon attenuation lengths lH2O and lmed were calculated using attenuation coefficients from the XCOM photon cross-section database,22 where the values were divided by the effective electron density of water and med, respectively, in order to allow a comparison with the eEPL estimate using Eq. 共9兲, where this effective electron density is defined by

␳eff = ␳e共1 + ␣fit兵1 + 具Z典其ln共E兲E1/2兲.

共10兲

The eEPL estimate provides an excellent estimate of the photon attenuation length, to within 1%, where only one single fitting parameter was required for all materials in Tables I and II and for energies between 100 keV and 20 MeV. The extended eEPL heterogeneity correction can now be used for materials with a high content of calcium 共such as cortical bone, bone-equivalent plastics, etc.兲 or higher Z elements and/or for primary photon energies above

550


550

H2O FIG. 6. The ratio of photon attenuation lengths in water relative to a material, ␳med 共lH2O / lmed兲, is presented for 共a兲 tissues in the human body and 共b兲 water-equivalent plastics. Photon attenuation lengths in water, lH2O and med lmed were calculated using attenuation coefficients from the XCOM photon cross-section database 共Ref. 22兲, where the values were divided by the effective electron-density of water, ␳eff H2O, and med, ␳eff med 共which account for pair production兲, respectively, in order to allow a comparison with the eEPL model for 共a兲 tissues in the human body and 共b兲 water-equivalent plastics as defined in Tables I and II. If the eEPL heterogeneity correction is a good estimated of the photon attenuation length in a media other than water, then this ratio should be 1.00, as represented by a dotted line.

10 MeV, where the pair-production process can no longer be neglected in calculating the photon attenuation lengths. V. DISCUSSION The systematically larger EPL ranges relative to lH2O observed in Figs. 4共a兲 and 5共a兲, lead to systematically higher estimates of the primary photon fluence in these media relative to water. This was also observed for the monoenergetic photon beam in Figs. 2共a兲 and 3共a兲. These higher estimates of the primary photon fluence are a consequence of the different hydrogen content between the media in Tables I and II and water. A higher estimate of the primary photon fluence will lead to a higher estimate of the primary dose in these media. This was also observed by Ahnesjö,8 who showed that the CC algorithm overestimated dose in lung. The incorrect lung dose can be attributed to the incorrect modeling of the photon beam attenuation through the rib cage 共thickness of 2 – 3 cm in bone兲 and through the lung and the oversimplified modeling of electron transport in convolution algorithms. The 2 – 3 cm of bone 共or 5 cm of water-equivalent thickness兲 can lead to differences between PB/CC and Monte Carlo dose predictions of approximately 2%–3% 关as may be seen from Fig. 2共a兲 by comparing the 5 cm value between water and bone兴. In addition to this, differences in the photon attenuation in lung can lead to differences of the order of 1%, as may be seen from Figs. 2 and 4. In the case of the eEPL method, a significantly better estimate is provided of the photon attenuation length for all materials, because differences in the hydrogen content of the various tissues and plasMedical Physics, Vol. 33, No. 2, February 2006

tics relative to water are accounted for by the ratio of the electron density of the media relative to water. The accuracy of eEPL heterogeneity correction was shown to be within 3% of photon attenuation length lH2O for nearly all materials studied and for primary photon energies between 100 keV and 10 MeV, where differences depended on whether pair-production could be neglected. The eEPL method can be used to scale the photon beam dose distribution to media other than water, for 4, 6, and 10 MV radiotherapy treatment photon beams. For radiotherapy treatment beams with photon energies higher than 10 MeV 共such as 15, 18, and 21 MV photon beams兲 and for tissues with high-Z chemical elements 共such as calcium in cortical bone兲, the scaling of the dose distributions must take into account pairproduction attenuation of the primary photons, and therefore the scaling of the dose distributions in media other than water becomes dependent on the primary photon energy. Attenuation of the primary photon beam by pair production can be accounted for by introducing a correction term to the electron density, as was done in Eq. 共9兲, where the extended eEPL method was shown to estimate the photon attenuation length lH2O to within 1% for all materials studied in Tables I and II and for energies between 100 keV and 20 MeV. The electron density was shown to influence photon dose calculations, where the largest differences relative to massdensity predictions were observed for air and bone. Therefore, the largest differences are expected between PB or CC predictions and a Monte Carlo dose prediction 共that accounts for electron-density variations in the patient兲, for treatment sites with bony structures or air cavities in the beam 共such as

551


head and neck tumors兲. For example, in the case of lung tumors, photon beams usually have approximately 2 – 3 cm of bone 共or 5 cm of water-equivalent thickness兲 in the beam due to the rib cage; this should lead to differences between PB/CC and Monte Carlo dose predictions. The electron-density effect on photon dose calculations was also shown to depend strongly on the energy of the photon beam used for treatment. Therefore, PB and CC dose estimates for energies equal to or higher than 10 MV become less accurate with increasing photon energy. This is a consequence of the increased probability of pair-production events occurring, which is not accounted for by the mass-density scaling methods used in the PB or CC algorithms, but is accounted for by the effective electron-density scaling method defined in the present work.

551

or for primary photon energies above 10 MeV, the pairproduction process can no longer be neglected in calculating the photon attenuation lengths. For these materials, the eEPL method was extended to account for pair-production attenuation of the primary photons, and therefore the scaling of the dose distributions in media other than water became dependent on the primary photon energy. The extended eEPL method was shown to estimate the photon attenuation length lH2O to within 1% for materials studied and for energies from 100 keV to 20 MeV. Thus, the eEPL method proposed in the present paper can be used to scale the photon beam dose distribution to media other than water accurately to within 1%, for clinical radiotherapy photon beams with energies from 4 to 20 MV. ACKNOWLEDGMENT

VI. CONCLUSIONS Conventional photon dose calculation algorithms such as PB, CC, Batho, all perform EPL mass-density scaling of the dose distributions to media other than water. In this paper, we demonstrate that the important material property that should be taken into account by photon dose algorithms is the electron density, and not the mass density. We have shown that EPL mass-density scaling leads to an overestimation of the primary photon fluence for materials in the human body and water-equivalent plastic substitutes, where largest differences were observed for bone and air, respectively, 6%–7% and 10%. However, in the case of patients, differences are expected to be smaller due to the large complexity of a treatment plan and of the patient anatomy and atomic composition and of the smaller thickness of bone/air that incident photon beams of a treatment plan may have to traverse. Some incident beam directions may have 2 – 5 cm of bone 共corresponding to regions of rib cage, femur, skull, or jaws兲 upstream to the tumor site, while others may have no bone in the field of view. An overestimate of the dose to tissues in the human body, such as the lung, have also been observed by Ahnesjö1,8 and others,4,6,24 and were attributed to approximations made in both the mass-density scaling and the modeling of the electron transport in convolution algorithms. The overestimated lung dose can be attributed to the incorrect modeling of the photon beam attenuation through the rib cage 共thickness of 2 – 3 cm in bone兲 and through the lung and of the oversimplified modeling of the electron transport in bone and lung. In the present study, the overestimation of the primary photon fluence, by the EPL method, was shown to be a consequence of the differences in the hydrogen content between the various media studied and water. A new scaling method, eEPL, was proposed based on electron-density scaling 共instead of mass density兲 and was shown to accurately predict the primary photon fluence in media other than water to within 1%–2% for all the materials studied and for energies up to 5 MeV. For energies above 5 MeV, the accuracy of the electron-density scaling method was shown to depend on the photon energy, where for materials with a high content of calcium 共such as bone, cortical bone兲, or higher-Z elements, Medical Physics, Vol. 33, No. 2, February 2006

This work is supported by Cancer Research UK under Programme Grant No. SP2312/0201. a兲

Corresponding author. Electronic mail: [email protected] A. Ahnesjö and M. M. Aspradakis, “Topical Review: Dose calculations for external photon beams in radiotherapy,” Phys. Med. Biol. 44, R99– R155 共1999兲. 2 J. O’Connor, “The variation for scattered x-rays with density in an irradiated body,” Phys. Med. Biol. 1, 352–369 共1957兲. 3 M. Sontag and J. Cunningham, “The equivalent tissue-air ratio method for making absorbed dose calculations in a heterogeneous medium,” Radiology 129, 787–794 共1978兲. 4 A. O. Jones and I. J. Das, “Comparison of inhomogeneity algorithms in small photon fields,” Med. Phys. 32, 766–776 共2005兲. 5 C. Martens, N. Reynaert, C. De Wagter, P. Nilsson, M. Coghe, H. Palmans, H. Thierens, and W. De Neve, “Underdosage of the upper-airway mucosa for small fields as used in intensity-modulated radiation therapy: A comparison between radiochromic film measurements, Monte Carlo simulations, and collapsed cone convolution calculations,” Med. Phys. 29, 1528–1535 共2002兲. 6 M. R. Arnfield, C. H. Siantar, J. Siebers, P. Garmon, L. Cox, and R. Mohan, “The impact of electron transport on the accuracy of computed dose,” Med. Phys. 27, 1266–1274 共2000兲. 7 F. du Plessis, C. Willemse, and M. Lotter, “Comparison of the Batho, ETAR and Monte Carlo dose calculation methods in CT based patients models,” Med. Phys. 28, 582–589 共2001兲. 8 A. Ahnesjö, “Collapsed cone convolution of radiant energy for photon dose calculation in heterogeneous media,” Med. Phys. 16, 577–592 共1989兲. 9 M. Woo and J. Cunningham, “The validity of the density scaling method in primary electron transport for photon and electron beams,” Med. Phys. 17, 187–194 共1990兲. 10 R. A. Brooks and G. Di Chiro, “Review article: Principles of computer assisted tomography 共CAT兲 in radiographic and radioisotopic imaging,” Phys. Med. Biol. 21, 689–732 共1976兲. 11 H. H. Barrett and W. Swindell, Radiological imaging: The theory of image formation, detection and processing 共Academic, New York, 1981兲. 12 H. Huizenga and P. R. Storchi, “The use of computed tomography numbers in dose calculations for radiation therapy,” Acta Radiol.: Oncol. 24, 509–519 共1985兲. 13 HELAX-TMS, “Dose formalism and models in HELAX-TMS,” Manual accompanying the HELAX-TMS planning system, 1998. 14 ADAC-Pinnacle, “Pinnacle3 Physics Instructions for Use” Manual accompanying Pinnacle3 radiation therapy planning software obtainable from [email protected] 共document called PhysicsInstructions. pdf兲, 2004. 15 International Commission on Radiation Units and Measurements, “Tissue substitutes in radiation dosimetry and measurements,” ICRU 1989, Vol. 44. 16 International Commission on Radiological Protection, “Report of the task group on reference man,” ICRP 1975, Vol. 23. 1

552


17

D. R. White, “Tissue substitutes in experimental radiation physics” Med. Phys. 5, 467–479 共1978兲. 18 C. Constantinou, F. H. Attix, and B. R. Paliwal, “A solid water phantom material for radiotherapy x-ray and g-ray beam calibrations,” Med. Phys. 3, 436–441 共1982兲. 19 G. Christ, “White polystyrene as a substitute for water in high energy photon dosimetry,” Med. Phys. 22, 2097–2100 共1995兲. 20 W. Schneider, T. Bortfeld, and W. Schlegel, “Correlation between CT numbers and tissue parameters needed for Monte Carlo simulations of clinical dose distributions,” Phys. Med. Biol. 45, 459–478 共2000兲. 21 H. Saitoh, T. Tomaru, T. Fujisaki, S. Abe, A. Myojoyama, and K. Fukuda,


552

“A study on properties of water substitute solid phantom using EGS code,” Proceedings of the 10th EGS4 Users Meeting, Japan, 2002, pp. 55–64. 22 M. J. Berger, J. H. Hubell, S. M. Seltzer, J. S. Coursey, and D. S. Zucker, “XCOM: Photon Cross Sections Database,” National Institute of Standards and Technology, NIST. 23 J. Seco, E. Adams, M. Bidmead, M. Partridge, and F. Verhaegen, “Headand-neck IMTR treatments assessed with a Monte Carlo dose calculation engine,” Phys. Med. Biol. 50, 817–830 共2005兲. 24 P. Metcalfe, T. Kron, and P. Hoban, The Physics of Radiotherapy X-Rays from Linear Accelerators 共Medical Physics Publishing, Madison, WI, 1997兲.