(

PERCENT DEPTH DOSES FOR

DIAGNOSTIC RADIOLOGY

by

Jean-Pierre Bissonnette

Medical Physics Unit

McGill University, Montréal

September 1991

A Thesis submitted to the

Faculty of Graduate Studies and Research

in partial fulfillm ~n t of the

requirements for the degree of

Master of Science

in

Medical Physics

@ Jean-Pierre Bissonnette 1991

0'

ABSTRACf

A new model is proposed for the calculation of relative depth doses for di?ignQstic radiology using a direct photon transport/ray tracing technique which incorporates both primary and first scatter dose. The x-ra y spectra are generated from computer algorithms based on the 8irch and Marshall semi­empirical model; the spectra are established by matching caIculated and mea­sured transmission data. The algorithms for the generation of x-ray spectra and for the calculation of depth doses are described. Re!ative depth doses are determined for a number of radiographie techniques. The calculations are compared with measured and published depth doses; the agreement is very good for tube voltages below 90 kVp. It is suggested that relative integral doses obtained from relative depth doses give an accurate representation of risk reductions obtained with different radiological techniques. The integral dose reductions predicted by the model are within 8.5 % of those from mea­sured data.

11

1 RÉSUMÉ

L'auteur présente un nouveau modèle pour le calcul par ordinateur de la dose en pro{',:mdeur en radiologie diagnostique à l'aide d'intégrations sur les trajectoires empruntées par les rayons X, ce qui permet l'addition de la dose produite par les photons n'ayant subi qu'une seule diffusion à la dose primaire. On calcule les spectres de rayons X par l'utilisation d'algorithmes informatiques basés sur le modèle semi-empirique de 8irch et Marshall; les spectres utilis~s sont déterminés par ajustement des transmissions calculées aux transmissions mesurées. L'auteur décrit les algorithmes menant au cal­cul de spectres de rayons X ainsi qu'au calcul de la dose en profondeur le long de l'axe central. Nous avons obtenu les doses en profondeur pour rlusieurs

techniques radiographiques; le modèle proposé prédit adéquatement les doses en profondeur pour un éventail de tensions de tubes entre 10 et 90 kV. On propose que les doses intégrales relatives obtenues par intégration sur les doses en profondeur sont une représentation adéquate des réductions du

risque encouru lors de l'utilisation de diverses techniques radiographiques. Les doses intégrales relatives prédites sont à moins de 8.5 % de celles obtenues

avec les mesures.

, III

ORIGINAL CONTRIBUflON

Ray tracing integrations have recently been used to calrulate the fjrst scatter dose to optimize the design of tissue compensa tors and the effect of lead attenuators in radiotherapy. They also have been used in diagnostie ra­diology to compute scatter to primary ratios in order to quantify the contribu­tion of scattered radiation to radiographie images. Computer code has been written to use this technique to calcula te primary beam and first scatter doses along the central beam axis for diagnostic radiology applications involving simple geometries. Application of ray tracmg integrations for diagnostic radi­ology dosimetry has never been attempted before. This enables the simula­tion of irradiations involving various tube voltages, field sizes and filter ma­terials on a much shorter time scale than model calcula tians based on Monte Carlo computations.

Comparison of calculated and measured depth doses is pel'formed. It is found that calculated depth doses agree very weIl with measured ones for tube voltages below 90 kV. Integral dose ratios obtained from calculated and measun d depth doses also agree very weIl. AlI integral dose ratios obtained from calculated depth dose agree within 8.5 % of the integral dose ratios ob­

tained from measured depth doses.

IV

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ACKNOWLEDGEMENTS

The research leading to the completion of this thesis was carried out at

the Medical Physics, Radiation On col ogy and Diagnostic Radiology depart­

ments of the Montreal General Hospital. 1 would like to thank the members

of each department for their help on this project, but 1 would aIso hke to ac­

knowledge individu aIs who provided significant help.

1 am sincerely grateful towards Dr. L. John Schreiner, who provided

this research topie. His encouragement, support, direction, patience and

friendship were all essential to the completion of this work. 1 thank him for

giving a genuine effort to the establishment of this diagnostic radiology

physics laboratory, and 1 sincerely hope that he will continue to support stim­

ulating research in this field.

1 also thank individual members of the Department of Medical Physics

for their help and support. In particuIar, 1 wouid like to thank Dr. Noël Blais

for his patience and experience in diagnostic radiology. 1 aiso thank Dr.

Conrado Pla and Dr. Brenda Clark for their Macintosh computing expertise,

Mr. Lajos Palotay for constructing the phantom used in the measurement of

depth doses, Mr. Marc Bussière for performing sorne of the measurements,

and Professor Ervin B. Podgorsak for making my stay with the Medical

Physics Department a very stimulating and enjoyable experience. 1 also

would like to thank Dr. Geoffrey W. Dean for providing a working experience

in radiation safety to help support myself financially.

Je tiens aussi à remercier les membres de ma famille immédiate, ma

filleule Janika, ainsi que tous ceux qui me sont chers. Ce travail n'aurait pas

eu lieu sans eux.

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1

TABLE OF CONTENTS

Page

ABSTRACT ii

RÉSUMÉ iii

ORIGINAL CONTRIBUTION iv

ACKNOWLEDGEMENTS v

TABLE OF CONTENTS vi

LIST OF FIGURES

LIST OF TABLES

CHAPTER 1

CHAPTER 2

ix

xii

INTRODUCTION 1

CALCULATION OF X-RA Y SPECTRA 5 2.1 Basic review of radiation quantities 6 2.2 X-ray production processes in the x-ray tube 9

2.2.1 Review of the thin and thick target theories Il

2.3 The Birch and Marshall model for the generation of x-ray spectra 14

2.3.1 The bremsstrahlung continuum 15 2.3.2 The characteristic Unes 19

2.4 The Tucker, Barnes and Chakraborty model for x-ray spectra generation

2.4.1 Bremsstrahlung theory 2.4.2 Characteristic radiation theory

21 22

24

VI

\' 1 i

,. 1 CHAPTER 3 DETERMINA TION Of CENTRAL-AXIS DOSES IN •

DIAGNOSTIC RADIOLOGY 26 3.1 Measurement of percent depth dose 26 3.2 Calculation 32

3.2.1 Primary radiation 35 3.2.2 Scattered radiation 36

3.2.2.1 Rayleigh scattering 36 3.2.2.2 Compton scattering 38

3.2.3 Calculation of doses 40

CHAPTER 4 EXPERIMENTAL METHODS AND MATERIALS 46

4.1 Computer calculations 46

4.1.1 Mass attenuation coefficients 46

4.1.2 X-ray spectra 48 4.1.3 Percent depth doses 51 4.1.4 Determination of the integral dose 54

4.2 Apparatus and experimental technique 57

CHAPTER 5 RESULTS AND DISCUSSION 65 5.1 Comparison of the spectral models 65

5.2 Percent depth doses 72

5.2.1 Comparison with published data 76 5.2.2 Effect of the peak kilovol tage on

depth dose 79 5.2.3 Effect of added filtration on dose 81 5.2.4 Effect of the field size on dose 81 5.2.5 Off-axis doses 83

5.3 Ftisk assessment 87 5.4 Concluding remarks 92

APPENDIX A PROGRAMME FLOWCHARTS 94

APPENDIX B SELECTED SUBROUTINES 96

APPENDIX C PERCENT DEPTH DOSES - 1 97

," r

~ 'j

Vll1

APPENDIX D PERCENT DEPTH OOSES - 2 102

APPENDIX E DETERMINATION OF THE PRIMARY INTEGRAL

DOSE FROM PHOTON SPECTRA 104

REFERENCES 106

r

1\

l LIST OF FIGURES

Page

Figure 2.1 : Schema tic geometry for target attenuation. 19

Figure 2.2: Cûmparison of calculated spectra obtained with 12° and 20°

target angles. 20

Figure 2.3: Schematic geometry for target attenuation proposed by Tucker et al. (1991). 23

Figure 3.1: Energy dependance of various radiation detectors. 28

Figure 3.2: Normalized glow curves from LiF:Mg and Li2B407:Mn. 32

Figure 3.3: Primary and first scatter dose contribu tions to depth dose calcula tions. 34

Figure 3.4: DifferentiaI molecular cross section for 80 ke V photons undergoing Rayleigh scattering in polystyrene. 38

Figure 3.5: Compton differential cross section per molecule for 80 keV

photons in pol ystyrene. 41

Figure 3.6: Geometry used for calculation dose at a point in phan tom taking primary and first scattered radiation into account. 44

Figure 4.1: Calculated and measured attenuation data measured for

various x-ray beams. 51

Figure 4.2: Calculated 80 kV spectra from a constant potential and a

~ single phase generator. 52

.'

x

, 1 Figure 4.3: Determination of R(() to calculate the doses in phan tom '"

from scattered radiation. 55

figure 4.4: Percent depth doses measured wi th ion chambers and thermolummescent dosimeters. 61

Figure 4.5: Geometry used for depth dose measurements in pol ys tyr-ene. 62

Figure 5.1: Ratio of the differential energy suggested by Birch and Marshall to that of Tucker et al. 67

Figure 5.2: Comparison of 100 kV p bremsstrahlung spectra calculated from the Kramers, the Birch and Marshall, and the Tucker et al. models. 68

Figure 5.3: Transmission curves calculated fronl the spectra described in Fig. 5.2. 69

Figure 5.4: Matched transmission curves from spectra ca1culated with the three spectral models. 71

Figure 5.5: Measured depth doses versus calculated depth doses with primaryand with primary + first scattered radiation. 75

Figure 5.6: Calculated depth doses versus published doses. 78

Figure 5.7: Calculated percent depth doses for various kVp. 79

Figure 5.8: Calculated vs measured percent depth doses for various kVp . 80

Figure 5.9: Calculated vs measured relative depth doses for various beam filtration. 82

., ~

l \ 1

1 Figure 5.10: Calculated vs measured percent derth doses for two field sizes. 84

Figure 5.11: Variation of percent depth dose with field area. 85

Figure 5.12: Measured beam profiles. 86

Figure A.l: Flowchart of the Spectrum 1.1 programme. 94

Figure A.2: Flowchart of the depthDose programme. 95

xii

LIST OF TABLES

Page

Table 2.1: Values of the Thomson-Whiddington constant. 18

Table 2.2: Principal K-emission lines for tungsten. 21

Table 4.1: X-ray beam filtration used in dose measurements and

calculations. 63

Table 4.2: Combinations of peak kilovoltages and filtrations used for

measurement of depth doses in polystyrene. 64

Table 5.1: Shifts in inherent filtration needed to match spectra from Tucker et al. (1991a) and I<ramers (1923) with referenee spectra from Birch and Marshall (1979). 70

Table 5.2: Shifts in kVp and target angle needed to mateh spectra from

Tucker et al. (1991a) and I<ramers (1923) with referenee

spectra from Birch and Marshall (1979). 70

Table 5.3: Relative importance of calculated primary and scatter contribu-

tions to the depth dose for sorne radiographie techniques. 73

Table 5.4: The contribution to seatter dose from air scatter and phantom

scatter for sorne radiographie techniques. 73

Table 5.5: Risk estimates relative to standard radiographie techniques:

Calculated primary integral dose (total energy imparted) and

integraI doses determined from depth dose measurements and calculations. 91

~

TableC: Percent depth doses for diagnostic radiology. 98

Table D: Variation of the percent depth dose of a 79 kV beam using

1.5 mm Al additional filtration with field sizes.

\111

103

CHAmRI

INTRODUcnON

Wilhelm Conrad Roentgen discovered x-rays in 1895. X-rays were used

immediately after their discovery for diagnostic imaging and for therapy.

Detrimental effects of radiation were observed soon after.

The main goal of diagnostic radiology is to obtain maximum diagnosis

information with application of as Iowa quantity of radiation as reasonably

achievable (ALARA principle). It is an established fact that the risk involved

in an exposure to radiation increases with absorbed dose. If an effective dose

equivalent of 1 mSv per examination is assumed, the probability of occur­

rence of an adverse effect resulting from a radiological examination is 10-5 per

year. This figure results from an estimated 10-2 total mortality risk per Sv per

year from radiation-induced cancers (ICRP, 1977). Since the doses involved

are very small, the risk associated with diagnostic radiology exposures for an

individual is almost negligible. However, if the whole population of Canada

(about 26 million) is estimated to undergo, on the average, one radiological

examination per year, there will he 260 new incidences of fatal radiation-in­

duced cancers per year in the whole population. The detrimental effect of di­

agnostic radiology exposures on the population is therefore the main motiva­

tion for the reduction of doses in diagnostie radiology.

Risk reductions can be obtained by the selection of an appropria te ra­

diographie technique. For example, projections can he chosen to minimize

dose to specifie organs (Le. choose the AP projection rather than the PA pro­

jection for a chest x-ray) and attenuators of a known material can he placed in

the pa th of the x-ray beam to greatly reduce the intensity of certain x-ray ener­

gies present in the beam. This last technique is called x-ray filtration. The

goal of filtration is to remove certain unwanted photon energies from the

photon spectrum directIy emitted by the x-ray tube and thus obtain as close to

a monochromatic x-ray beam as possible: it is hoped that x-ray filters will re-

move low energy photons whieh do not reach the detector and high energy photons which are likely to undergo Compton interactions in tissue and de­grade the image. To minimize the radiation risk involved in a certain radio­logical examination, the radiographie techniques, which are determined by the selection of the tube kilovoltage (kV p), the photon output (mAs) and the added filtration, must necessarily be compared from bath the dosimetric and imaging point of view. One way to approach this problem is the determina­tion of radiographie techniques producing equivalent images. The dose or risk involved with those techniques would be determined a posteriori. It is then a simple matter to select which of the techniques producing equivalent images involves minimal risk.

The risk involved in radiation exposures can be assessed by a number of parameters. One of the parame ter used for risk assessment is the entrance exposure or surface dose, which describes the dose measured at the surface of the patient. This parameter has been extensively used in the past to assess pa­tient dose savings relative to a standard radiographie technique (Burgess, 1981;

Burgess, 1985; Horner et. al., 1988; Villagran et. al., 1978; Wang et. al., 1984;

Wesenberg et. al., 1987; Kohn et. al., 1988; Rossi et. al., 1982), assuming that the savings indicated were reflecting the actual risk reduction. The exposure al the surface of the patient with the patient removed (i. e. without backscatter) is sometimes called the entrance exposure. This last definition will be applied throughout this work. Both parameters are easy to measure, but almost no consideration is made about the distribution of dose in the phant<.'m. Furthermore, the entrance exposure, as used in the present context, disregards the contributions to dose from radiation backscattered in the patient (Nagel, 1989). Thus, the entrance exposure and the surface dose depend on the radio­graphie technique used, but they are independent of the organ distribution within the patient. Risk is ideally evaluated with the effective dose equiva­lent, HE (ICRP, 1977). This parameter, adopted by the United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) in 1982

(UNSCEAR 1982), is a quantitative estimate of tht. risk involved in a specifie exposure to radiation. It consist of a sum of the effective dose H, involved with a number of critical organs, which includes the gonads, the female breast, the red bone marrow, the lungs, the thyroid, the bone surface and the remainder organs. Ht can be calculated from a volume integral of the dose

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distribution of a concerned critical organ (Huda and Sandison, 1984). Risk has also been evaluated in terms of calculated or measured integral doses (Carlst;on, 1963; Gustafsson, 1979; Richards et al., 1970; Chakera et aL, 1982;

Yamaguchi et aL, 1983; Koedooder and Venema, 1986; Nagel, 1989). Carlsson (1963), Gustafsson (1979) and Wall and Shrimpton (1981) justified the use of the integral dose to assess risk reduction. A numher of these studies have also reported that the risk reductions indieated by the integral dose ratios are Jess than the reductions specified by the entrance exposure or the surface dose ra­tios applied to the same technique. The determination of the integral dose re­quires, however, knowledge of the incoming photon fluence, of the percent depth dose and of the beam geometry (Carlsson, 1963; Johns and Cunningham, 1984).

The easiest parameter to measure is the entrance exposure, where an ion chamber positioned in the beam can he used to measure the exposure re­sulting from various radiographie techniques. The evaluation of the effective dose equivalent, on the other hand, requires the determination of dost! distri­butions in the patient. A knowledge of the volume and the position of the eritieal organs in the irradiated volume, and of the geometry of the radio­graphie technique used for the radiological examination are aIso needed. The evaluation of the effective dose equivalent therefore involves an intense ef­fort in measurement and ea1culation. The integral dose seems to he an ade­quate compromise to assess the risk involved with various radiographie techniques applied to the same radiographie examination. Its determination is more complex than the entrance exposure, but easier to perform than the effective dose equivalent since it only requires percent depth dose curves and the geometry involved in the examination (Johns and Cunningham, 1984).

The determination of the integral dose can be made with minimal ef­fort by the use of reliable computer algorithms. Dose calculation techniques in use in radiotherapy can be adapted for the needs of diagnostic radiology to evaluate percent depth dose in phantom, from which one can extract integral doses. The time involved in risk estimation can then he reduced, and the op­timization of the radiographie technique needed for a certain examination eased.

3

1 This thesis presents the theoretical considerations and the computer al­gorithms involved in the calculation of x-ray intensities per energy interval (x-ray spectrum) and of doses in phantom for simple gecmetries. Calculated x-ray spectra are verified with the EQSPEC method (Boone 1986; Boone 1989) while calculated relative depth doses are verified by direct measurements in phantom. The theories involved in the determination of x-ray spectra and doses in phantom and the techniques involved in the computation and the verification of these theoretical consideration are addressed. Finally, the inte­gral doses in phan tom established from calculated and measured relative

depth doses are compared.

, .

CHArrERII

CALCULATION OF X-RA Y SPECfRA

Many of the quantitative parameters of radiation dosimetry are deter­

mined by the incident photon intensities per energy interval, i. e. the x-ra y

spectrum. Knowledge of a photon spectrum can he used directIy to calculate, for example, photon and energy fluences of the photon œam, exposure, the

primary dose to a point in phantom, transmission data O. e. exposure calcu­

lated after various thicknesses of attenuating material which can, in turn, yield the half-value layer) and the effects of primary radiation on image con­trast (Barton, Frank and Pfaff, 1983; Johns and Yaffe, 1985; Koedooder and

Venema, 1986; Boone, 1986; and Nagel, 1989). The measurement of man y of

these parameters is a labor-intensive process. Accurate x-ray spectra are also required for reliable Monte Carlo simulations of diagnostic radiology parame­

ters (see, for example, Chan and Doi, 1983; Johns and Yaffe, 1983; Persliden,

1986; Carrier and Blais, 1987). Direct experimental determination of x-ray

spectra, which yields the most reliable information, necessitates specialized

equipment not often encountered in radiology departments. Ge (Li) and

Si (Li) detectors have been successfully used for the determination of diagnos­

tic x-ray spectra (see, for example, Marshall et al., 1975; FeweU and Shuping,

1977). It is however necessary to collimate the x-ray beam to decrease the

count rate and to cool the detector to reduce the leakage current. Small silicon

photodiodes have recently been suggested to measure diagnostic x-ray spectra

(Aoki and Koyama, 1989), without collimation, at room temperature. This

technique is however subject to distortions because of pulse pileup (Aoki and

Koyama, 1989). Furthermore, statistical errors are larger for high energy pho­

tons and relatively long measurement times are necessary for high voltage

x-ra ys (Aoki and Koyama, 1989). This detector also reqlùres a light-tight envi­ronment. For these reasons, measurement of spectra is not practical for rou­

tine clinical spectral characterization. Therefore, a model which would char­

acterize the photon spectrum for diagnostic x-ray beams could potentially ac-

celerate the evaluation of different x-ray imaging techniques, for which knowledge of the aforementioned parameters is crucial.

In this chapter, three models for the generation of x-ray spectra, includ­ing the model from Birch and Marshall (1979) which has bren selected to per­form this work, will he presented. A review of the definitions of various ra­diation parameters is aiso given in order to emphasize their relationship with

the x-ray spectra. A discussion of x-ray production and of spectral rnodels is th en undertaken. The comparison of the implementation of these spectral models is left to Chapter V.

2.1 Basic review of radiation quantities

Radiation fields can he described using either quantitative or qualitative approaches. The quantitative approach makes use of physical quantities such as exposure, photon fluence and dose, where one's interest lies in the number of photons present at sorne reference point or in the total energy irnparted to

some mass. On the other hand, the qualitative aspect of a radiation field is

typically described by the beam quality, which specifies the bearn's ability to penetrate sorne material of known composition. The following discussion on x-ray beam quality is based primarily the development proposed in Attix (1986) and in Johns and Cunningham (1984).

X-ray bearn quality is usually expressed in terms of the half-value layer.

The half-value layer is the thickness of specified material which attenuates a

beam of radiation such that the exposure rate after the attenuator is one half of its original value (Johns and Cunningham, 1984). The measurernent of the half-value layer is performed sa that the contribution of aU scattered radia­

tion,other than any which rnay be initially present in the beam, is exc1uded. The half-value layer is usually expressed in mm of aluminum for x-ray tube potentials below 120 kVp while for potentials hetween 120 ta 400 kVp copper thicknesses are reported. The x-ray beam quality is an essential parameter for the calclliation of dose distribution in phantom: two beams producing identi­cal exposure in air may not produce identical doses (energy deposited by ioniz­

ing radiation per unit mass) at sorne depth in phantorn. Half-value layer is

.. l

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quickly determined from a few transmission mcasurements or by the use of

special tooIs, such as the University of Wisconsin X-Ray Test Cassette (Jacobson et al., 1976). The haU-value layer on its own, however, is a crude specification of beam quality since it tells very little about the composition of the radiation beam. The number and energy of the photons of the beam can­not be established from the haU-value layer itself. Other quantities, such as the second half-value layer and the homogeneity coefficient, can be used to complement the haU-value layer, but they still do not reveal the detailed en­

ergy distribution of photons. In addition, a given haIf-value layer and homo­

geneity coefficient combination does not necessarily result from a unique spectral distribution: one can obtain identical half-~':due layers and homo­

geneity coefficients from two x-ray beams with different kVp's by using approu

priate attenuating materials (x-ray filters) in the path of the ryeam. Thus, as pointed out by Hale (1966), the determination of the haH-value layer and of the homogeneity coefficient is not a complete description of diagnostic beam

quality. Cowdrey et al. (1970) have also shown that dose calculations involv­

ing specification of the effective beam energy from hali-value layer measure­ments (Johns and Cunningham, 1984) differed significantly from doses calcu­lated from full photon spectra. A thorough specification of the quality of the

x-ray beam th us necessitates the knowledge of the number of photons per en­ergy interval in the beam, that is the spectral distribution of the photons emit­

tOO from an x-ray tube excited to the spedfied peak kilovoltage (kVp). Spectral

reconstruction can be achieved by Laplace transformation of more extensive

transmission data measurements (Archer and Wagner, 1988; Dance, 1987;

Rubio and Mainardi, 1984; and Baird, 1981). However, such transformations of transmission data are ill-posed problems very sensitive to measurement error (Baird, 1981; Chu and Fenster, 1983). AIso, Laplace transform spectra

which are consistent with the measured transmission data may not be physi­

caUy realistic (Dance, 1987). The EQSPEC approach proposed by Boone (1986,

1988) and described in §§ 4.1.2 however does quantify the beam appropriately.

Sorne radiation quantities used in dosimetry are defined in terms of the x-ray photon spectrum per cm2, (N v)

N-.JJlL v - d(hv) (2.1)

7

l which is the intensity of photons of energy h v, where h is Planck's constant (6.626176 x 1()-34 J s) and v the photon frequency, in a radiation field given in terms of the number N of photons of energy E per cm2• For instance, the pho­ton (<P) fluence for polyenergetic x-ray beams is defined as (Attix, 1986):

(2.2)

It should be pointed out that the x-ray spectrurn is usually defined as the dif­ferential form of Eqn. (2.2), that is, Nv = dtlJ(hv)/d(hv). This is why x-ray spectra are usually given in units of cm-2• The energy fluence ('11 of the beam is

1~"-

'1' = 0 N v hv d(hv). (2.3)

Many other radiation quantities such as the exposure (X), the kerma (K) and the dose absorbed (D) at a point in sorne phantom material and under elec­tronie equilibrium conditions are, in turn, derived from the energy or photon fluences (ICRU, 1980; Johns and Cunningham, 1984):

(2.4)

1"- - Ji.hv) K = Nv E"(-)dlhV), o P

(2.5)

(2.6)

where (Wle)~, is the inverse of the average ionization en erg y per ion pair for

air «Wle);;; == 1/33.97 eV lion pair; Rogers, 1991), Nv is the photon spectrum at the point of interest, (Jl(hv)/p) and (Jlab(hv)/p) are the mass attenuation and mass energy absorption coefficients of the medium of the phantom material, respectively, and E" is the average energy transferred to electrons of the medium at each interaction. From Eqn. (2.2) to (2.6), it is clear that the photon

(

spectrum emitted by the x-ray tube is required for the a priori ca1culation of

the quantities above.

2.2 X-ray production processes in the x-ray tube

X-rays are produced when electrons, which have been emitted by the cathode filament of an x-ray tube and accelerated by a high voltage towards the anode target, interact with the atoms of target. There are two ways an in­coming electron can interact with the atoms of the target and thereby lose sorne (or ail) of its kinetic energy as x-ra y photons. One process involves the interaction between the incident electron and the nucleus of a target atom. The electron is deflected from its course by the nuclear Coulomb field; the re­sulting deceleration produces the so-called bremsstrahlung photons. The other interaction occurs when the electron interacts with one of the bound electrons of a target atom. The collision may result in the ejection of the bound electron from the atom; the subsequent transitions of the remaining bound electrons belonging to higher energy levels produce x-ray photons. These x-ray photons have well defined energies characteristic of the atom that has been ionized, and thus the x-rays emitted by such interactions are called characteristic radiation. Electrons decelerated in the target of a diagnostic ra­diology x-ray tube lose about 1 % of their kinetic energy in bremsstrahlung production. Less than 1 % of the remaining energy emerges as characteristic radiation, and the rest is given up in heat producing collision interactions. Therefore, most of the initial electron kinetic energy is converted to heat. The generation of x-ra ys by deceleration of electrons in truck, high atomic number targets is thus very inefficient. The two x-ray production processes will now be discussed in greater detail, and a resulting model for x-ray spectral genera­tion proposed.

Bremsstrahlung, or braking radiation, is the process responsible for most of the x-rays produœd in an x-ray tube. In this process, an incident elec­tron of kinetic energy T approaches the nucleus of a target atom close enough to be influenced by the strong Coulomb field of the nucleus so that the elec­tron is deflected from its original course. Sorne of the electron's energy is lost in such a collision as a photon of energy hv while the electron continues with

9

kinetic energy T-hv. The mass radiative stopping power, which is the rate of

energy 10ss of electrons by bremsstrahlung radiation per unit material thick­

ness (dT/dt) divided by the density p of the material, can be obtained by quan­

tum mechanical reasoning (Heitler, 1984):

Srad = 1. (dL) = 4rÔ neZ [In ~T + ,Ll(J) _1]. p dx 137 lJ.o 3

(2.7)

Equation (2.7) shows that the energy lost to bremsstrahlung photons increases

with the atomic number (Z) of the material and with the electron kinetic en­

ergy (T). The constants ro and Jlo represent, respectively, the classical electron

radius (ro = 2.8179 x 10-15 m) and the electron rest mass energy (511 ke V)

The energy of the emitted bremsstrahlung x-ray photon depends on how close

the electron pas~es to the nucleus, the initial kinetic energy of the electron

and the charge of the nucleus. For exarnple, a head-on collision could bring

the electron to rest and the emitted photon's resulting energy would be equ'll

ta the initial kinetic energy of the electron. Such an event is unlikely. Elec­

trons usually lose their energy in stages and a single electron can be responsi­

hie for the production of any number of photons with energies varying con­

tinuously from 0 to the kinetic energy of the incident electron. Therefore, the

wide variation in the energy of photons produced by bremsstrahlung colli­

sions leads to the continuum of photons energies observed in x-ra y spectra.

Bremsstrahlung photons are produced in the target with aIl energies up to a

maximum specified by the applied potential difference (Duane and Hunt,

1915):

hvmax = eV 1 (2.8)

where h is Planck's constant, Vmax the frequency of the most energetic photon

produced, V the applied potential difference, and e the electronic charge

(1.60218 x 10-19 Cl.

Characteristic radiation results from the expulsion of an electron from

the inner shells of the target atom by an direct hit with an incident electron.

The resulting ionized atom is in an excited state: for the atom to return to its

ground state, an atomic electron from a higher energy outer shell sponta-

1 0

neously fills the vacancy created in the lower energy shen by the collision.

The transition of the electron is accompanied by the production of a photon of

energy equal to the change in binding energy experienced by the transition

electron. The energies of the photons produced are known with great preci­

sion and are characteristic of the atom that has been ionized. If the incident

electron has insufficient energy to eject a bound electron from a particular

shen, there will be no characteristic radiation from that shen. The production

of characteristic radiation is therefore dependant of the voltage applied be­

tween the anode and "he cathode of the x-ra y tube. Characteristic radiation can

contribute up to about 30 % of the diagnostic x-ray spectrum for a tungsten

anode (Johns and C1mnin~ham, 1984).

2.2. 1 Review of the thin and thick target theories

A detaHed model describing the production of x-rays in the target

would be prohibitively complex, since it would have to consider the many in­

teractions a single electron may undergo. Acceptable approximations to mea­

sured x-ray spectra have been obtained however with semi-empirical models.

The simplest model is for the case of thin targets (thin in the sense that

the electron undergoes at most one interaction). Early measurements of spec­

tra involving such thin targets suggested a fiat differential energy spectrum

(Kuhlenkampff, 1923), i. e., power radiated in increments of photon frequency

and target thickness per unit current incident on the target (the differential

energy spectrum is discussed in greater detail in §§ 2.3.1). The differential en­

ergy spectrum (Nv(hv) dv) can then be expressed as

Nv(hv)dvdx = nkZ;: dvdx (v< vmax)

Nv(hv)dvdx = 0 (v> vmax) (2.9)

(Kramers, 1923; Dyson, 1986) where N v defines the spectrum (see §§ 2.1), Z is

the atomic number of the target material, n is the number of atoms per unit

volume, T is the electron kinetic energy and k is a proportionality constant.

1 1

1 The model can be extended to thicker targets by integrating over the

target thickness or, equivalently, over the electron energy decrease dT as the electron travels deeper in the target (Oyson, 1986). The energy intensity per

energy interval or the energy spectrum Iv (I v = N v x h v) is then:

(2.10)

where To is the kinetic energy of the incident electron, which aiso determines

the maximum photon energy produced, hV"'aA (see Eqn. (2.8», and T = hv is the smallest electron energy capable of producing photons of energy hv within the target. The rate of electron kinetic energy loss per unit thickness of medium, designated the stopping power (dT / dx), can be evaluated using the

non-relativistic Bethe-Bloch equation (Bichsel, 1968):

dI.=_ -1 d.x (41tEof

21te4 L T n.Z log r (2.11)

In Eqn. (2.11), J is the mean excitation or ionization potential for the target

material. Since at low electron energies the logarithmic term of Eqn. (2.11)

variE's slowly with electron energy, the approximation that log Tf} is a constant

can be used and Eqn. (2.11) reduces to:

dT = _ k' nZ dx T'

(2.12)

where k' is a term collecting aIl the constants of Eqn. (2.11). Substitution of

Eqn. (2.12) in Eqn. (2.10) yield

/vdv = k Z2 n dvfT __ 1_ dT = .K.Zh (vmax - v) dv (2.13) k'nZ k'

ra

or (2.14)

1 2

(

(Dyson, 1986) where Ac is the constant kik.' Zh. This form of the brems­strahlung spectrum was first proposed by Kramers (1923).

Kramers' equation Œqn. (2.14» is the simplest model of the brems­strahlung spectrum; it is a linear function of the constant voltage applied to the tube (remember that Vmax is determined from the tube voltage, as stated by Eqn. (2.8». Note that characteristic radiation is usually neglected in the Kramers model (Barton, Frank, and pfaff, 1982; Boone, 1986). In practice, the increase of the mean x-ray beam energy as it goes through an absorber has to be taken into account and the contribution of the characteristic lines to the photon spectrum should be considered. Filtration effects produced on the photon spectrum by the x-ray tube housing (inherent filtration, usually ex­pressed in mm Al) or by deliberately placing absorbers in the x-ray beam (added filtration) can easily be calculated with a simple application of the ex­ponential attenuation law (Johns and Cunningham, 1984),

Nou,(hv) = Njn(hv) exp [-p(hv) t], (2.15)

where t is the thickness of the filter.

The agreement of spectra calculated with I<ramers' model with experi­mental data is acceptable only for certain applications, since there are five in­herent physical weaknesses in the model. The Kramers model assumes that: the contribution of characteristic radiation to the photon spectrum in negIigi­ble; the variation of the intensity of the x-ray beam with the angle at which x-ra ys are emitted from the focal spot (heel effect) is ignored; the electron stopping powers derived from Eqn. (2.12) are correct for aU incident electron energies; the differential energy spectrum is flat; and the absorption of the ra­diation produœd by the target itself is negligible. AlI of these assumptions in­duce errors in the Kramers spectra. Furthermore, Eqn. (2.12) is true only for non-relativistic electrons (e. g. the velocity for 30 keV electrons calculated classically and relatively differ, - 0.3 c and - 0.5 c, respectively; this indicates that the relativistic approach is required). Measured x-ray spectra have demonstrated that target attenuation is significant and that the tbin target en­ergy spectr\Jm is not flat, but has a tendency to dedine towards the rugh-en­ergy end (Dyson, 1986), in opposition with the predictions of Eqn. (2.9).

1 3

1

..­..

Because of these limitations, published spectra are usually assumed to be more reliable (Johns and Cunningham, 1984) for diagnostic radiology applica­tions and are preferred to the Kramers spectra. However, sin ce sorne of the errors inherent to the Kramers theory cancel each other for tube potentials above 80 kVp and target angles larger than 300 (Soole, 1972), the Kramers spec­

tra are often considered acceptable for radiotherapy conditions. More recent models (Birch and Marshall, 1979; Tucker et al., 1991a) eliminate a number of

aforementioned assumptions, and have been shown to be reliable for a much wider range of diagnostic radiology applications than the Kramers model (Birch and Marshall, 1979; Oyson, 1986; Tucker et al., 1991b).

2.3 The Birch and Marshall model for the generation of x-ray spectra

Birch and Marshall (1979) proposed an alternate semi-empirical model

to calculate photon or energy spectra for a wide range of circumstances in di­

agnostic radiology. The model was defined in terms of two input parameters:

the kinetic energy of incident electrons Ci. e. electrons emitted in a constant

potential x-ray tube) and the anode angle. The differential energy spectrum was determined by fits to experimental data. This differential energy spec­

trum was then incorporated into a theoretical bremsstrahlung model which took the attenuation of x-rays by the target itself into account. Th~ characteris­

tic radiation contribution to the spectrum was then added independently.

Characteristic and bremsstrahlung radiation intensities were subsequently corrected for the attenuation of photons in the target itself. Finally, the result­

ing spectrum can be attenuated with any desired filter by a simple application

of the exponential attenuation law given by Eqn. (2.15).

Spectra calculated from the Birch and Marshall model (Birch and Mar­

shall, 1979; Birch et al., 1979) have been used in a number of computer studies

conceming the effect of added filtration on image quality, beam quality, and dose. These studies involved either direct photon transport calculations (Harrison, 1981; Harrison, 1982; Johns and Yaffe, 1983; Johns et al., 1983; Koedooder and Venema, 1986; Boone, 1986; Boone, 1988; Nagel, 1988; Nagel,

1989, Schreiner et al., 1991) or Monte Carlo simulations (Shrimpton et al.,

1981; Persliden, 1986; Carrier and Blais, 1987).

1 4

(

(

2.3.1 The bremsstrahlung continuum

The energy intensity dl v of bremsstrahlung photons of energy E gener­ated by an electron traveling at a distance dx in an element of atomic number Z is given by (Birch and Marshall, 1979):

(2.16)

where N is Avogadro's number, p the density of the target material, A the atomic mass of the target material and Q the x-ray energy intensity of photons with energies between E and E + dE per incident electron flux (i. e. Q is the differential energy spectrum) in keV per unit energy interval (1 keV) per atom cm-2 pel' mA per 1 mm2 area at 0.75 m. The x-ray energy intensity func­tion Q is normalized in these units in order to reproduce the conditions used when spectra were determined experimentally (Birch and Marshall, 1979).

To obtain div for a soUd target, the 10ss of electron energy as the electron travels through deeper layers of the target must be considered. As noted ear­lier, photons of energy E = hv can only be emitted by electrons of kinetic ener­gies higher or equal to hv. The intensity of bremsstrahlung photons produced in the target is therefore the integral over the electron path from the surface of the targe t, where the electron has kinetic energy T 0 (i.e. surface of the tar­get), to the depth at Xv where the electron energy is reduced to T = E,

la,

pN lv=y 0 Qdx. (2.17)

The integral over x can be transformed to an integral over T by the following change in variables: d.x = (dx/dn dT. Introdudng a relativistic correction (recall that at 30 keV, the electron is already relativistic), the following is obtained:

(2.18)

1 5

.'

The relativistic total stopping power (dT/dt) can he obtained from either the relathistic Bethe-Bloch equation (Bichsel, 1968) or by interpolation of tabu­lated values (ICRU, 1984).

From Eqns. (2.16) to (2.18), it is apparent that the bremsstrahlung con­tinuum is strongly determined by the function Q. This function represents the differential energy spectrum, which is the energy distribution of brems­

strahlung photons produced by an electron of kinetic energy T colliding with a target atom of atomic number Z. Various attempts have been made to de­velop expressions for the differential energy spectrum Q, defined from the dif­ferential bremsstrahlung cross section (Evans, 19n):

dOr/Jd = _1_ (~)2 Q Z2 T + moc2 d(hv) . 137 moc 2 T hv

(2.19)

The quantum mechanical theory of bremsstrahlung was developed by Bethe and Heitler using the time-dependant perturbation theory and the Born ap­proximation, where the amplitude of the incident wave is much larger than that of the scattered wave. In the Born approximation, the bremsstrahlung cross section is obtained from the square of the modulus of the elastic scatter­ing amplitude brought by the weak perturbing interaction. A consequence of this approximation is the assumption that the emission of bremsstrahlung photons is considered as a scattering process in which the scattered particles are described by plane waves (Heitler, 1984; Marmier and Sheldon, 1969). The Bethe and HeitIer theory was thus valid for very high electron energies, i. e. when T» moc2 and (T - hv) »moc2. Sommerfeld (1931) developed a non­relativistic theory (T« moc2) lising exact wave functions. Not only is the Sommerfeld theory restricted to non-relativistic electron energies, but also to low Z targets (Heitler, 1984). The electron energies encountered in diagnostic

radiology range between 0 and the maximum tube voltage in the target (typically between 20 and 150 kV) and may he outside of the region of validity of both Sommerfeld and Bethe-Heitler approaches. Therefore, these theories are expected to fail for diagnostic radiology conditions. For example, a com­parison of x-ray spectra obtained from the bremsstrahlung cross section derived from the Sommerfeld theory with measured spectra indicates that the expression for Q obtained from the Sommerfeld theory overestimates the

1 6

mean energy of the x-ray beam (Birch and Marshall, 1979). For this reason, empirical expressions for the differential energy spectrum are often consid­ered for diagnostic radiology applications. Soole (1977) proposed a semi-em­pirical expression for Q which was determined by a series of trial and error computations performed until a good match of the differential energy spec­trum was obtained with published attenuation measurements. The spectra calculated with Soole's fit have heen found, however, to underestimate the mean energy of the x-ray beam (Birch and Marshall, 1979). Birch and Marshall (1979) fitted a fourth-order polynomial in terms of ç = hvlT, the ra­tio of the photon energy to the electron kinetic energy, to measured x-ray spec­tra. They established, for constant tube potentials between 30 and 150 keV and for any target material, that:

(JT = P(~) = 0.503 _ 0.94597 ; + 0.1553 ~2 + 1.1632 ~3 _ 0.6818 ;4. (2.20) Z2

Since x-ra ys are produced along ail of the electron pa th in the target, a correction must be made to the expression for the bremsstrahlung energy density (Eqn. (2.18» to indu de attenuation from the target materiallying be­tween the point of origin of the x-ray photon and the surface of the target. A relationship describing the residual energy of electrons at various depths within the target is necessary for the determination of the production of bremsstrahlung photons at these points. An empirical energy-loss relation for electrons penetrating a medium can be used to this end. Such a relation was obtained from measurement of the entrance and exit velocities of elec­trons incident upon thin foils (Whiddington, 1912). The electron kinetic en­ergy at depth x in the target has been determined as

T = (T6 _ pCx)1/2, (2.21)

where p is the density of the target material, To the kinetic energy of the inci­dent electron and C is a numerical parameter termed the Thomson­Whiddington constant. It should be noted that Eqn. (2.21) can be solved for x when T is zero to estimate the extrapolated range of incident electron with ki­

netic energy To. The parame ter C is not a constant but it is a slowly varying function of incident electron energy. Specifie values for C must he deter-

1 7

"

mined for different electron energies (Green and Crosslett, 1961; Tothill, 1968).

Birch and Marshall (1979) proposed values of C to calculate extrapolated

ranges in good agreement with electron ranges measured by Katz and Penfold (1952). These values are shown in Table 2.1. It should he noted that the

Thomson-Whiddington relationship is an oversimplification. It does not take into account the spread of electron energy that occurs at a given depth

nor the variation of electron population with depth. Furthermore, this rela­tionship assumes that the electrons travel in a straight line. However, it has

been derived from macroscopic measurements and il is, in the first approxi­mation, an adequate way to characterize the variation of the electron energy

as they penetrate a medium.

From the Thomson-Whiddington relationship (Eqn. (2.21», the depth x

at which the electrons have kinetic energy T is known. If a photon of any en­ergy is emitted from this point, it will have to travel through a certain thick­

ness y of material before it escapes the anode (see Fig. 2.1). This distance can he determined from x and the angle which the target makes in excess of 90° to the electron beam direction, i. e. the target angle. For a target angle 9, the dis­

tance y is:

(T2 T2) Y = x cot 9 = 0;,c cot 9,

Incident electron C energy (keV) (keV2 cm2 tr 1)

25 0.39 x 1()6

50 0.54 x 1()6

75 0.625 x 1()6

100 0.70 x 1()6

150 0.84 x 1()6

200 1.00 x 1()6

Table 2.1: Values of the Thomson-Whiddington con"tant, C, as calculated by Birch and Marshall (1979).

(2.22)

IS

(

(

cathode

[

Figure 2.1 : Schematic geometry showing the x-ray geometry used by Birch and Marshall (1979) to ca1culate target attenuation

where x, y, and 9 are defined in Fig 2.1. The bremsstrahlung radiation pro­duced within the anode is therefore reduced by an amount exp(-pv y), where Jlv is the linear attenuation coefficient of the target materiaI for photons of en­ergy h v. Insertion of this term and of Eqn. (2.20) in Eqn. (2.18) gives the equa­tion for the energy intensity of bremsstrahlung photons of energy hv:

Equation (2.23) can also be used to determine the bremsstrahlung photon spectrurn, i. e. the number of photons per energy interval (N v), since Nv = Ivlhv. Examples of spectra obtained with two different target angles at the same k V p and mAs are shown in Fig. 2.2.

2.3.2 The characteristic Unes

The characteristic radiation emitted by the target is added to the contin­uous bremsstrahlung spectrum using a fit to experimental data (Birch and

1 9

~ 1.2

1 1.0

a 0.8 c

~ 0.6 &:

Go

1 0.4 ~ i 02

0

0 20 40

--12° target

20° target

60 Photon en.rgy (k.v)

80

Figure 2.2: Comparison of calculated spectra obtained with 120 and 200

target angles. Both spectra are obtained with a 80 k V p'

Marshall, 1979). This empirical relationship for the energy intensity of thick target characteristic radiation, Ich, was suggested by Green and Cosslett (1968):

(T )1.63

{ch oc ~-l , (2.24)

where T" is the electron kinetic energy required to remove an electron from the K-shell of a target atom, i. e. the binding energy of the K-shell electron (69.5 keV for tungsten), and To is the electron kinetic energy.

There are six main characteristic tungsten K-Iines of varying intensity as shown on Table 2.2. The calculation of their respective intensities necessi­tates the use of appropriate weighting factors We which indicate the relative intensities of the characteristic lines. In practice, the K~l and K~3 lines and

the K~2/1 and K~2/2 lines are paired as single lines (with their weighting fac­tors summed) since they are not easily resolved.

The expression yielding the energy intensity of characteristic lines, which includes the self-attenuation term exp(-J,Lv y) introduced earlier (with y

10

~

ï ... Transition Symbol Energy Weighting (keV) factor

K-Nm K~2/1 69.033 2.45

K-Nn K~2/2 69.101 4.77

K-Mm K~t 67.244 20.8

K-MII KP3 66.950 10.8

K-Lm Kat 59.321 100

K-Ln Ka2 57.984 57.6

Table 2.2: Principal I<-emission Unes for tungsten as taken from Storm and Israel (1970). Only the most intense lines are shown here.

as defined in Eqn. (2.22) and Fig. 2.1), is obtained by an integration over aU electron kinetic energies T which can produce characteristic x-rays. Thus,

where T 0 is the kinetic energy of the incoming electrons, and T" is the energy required to remove an electron from the K-shell. The constantk is an empiri­caUy determined normalization constant which incorpora tes the bremsstrahlung yield for tungsten; it is equal to 9.05 x 10-21 • k is also used to maintain the convention of Birch and Marshall that the intensity of charac­

teristic radiation is given per mA per 1 mm2 area at 0.75 m. In practice, Eqn. (2.25) is evaluated for aU of the characteristic Unes of interest, and the result­ing energy intensity is added to that calculated for the bremsstrahlung contin­uum to obtain the complete x-ra y spectrum.

2.4 The Tucker, Barnes and Chakraborty model for x-ray spectra generation

Tucker, Barnes and Chakraborty (1991a) have recently proposed a modi­fied semi-empirical model for the generation of x-ray spectra from tungsten targets based on the application of the results of the quantum mechanical the-

2 1

ory of bremsstrahlung. Similarly to the Birch and Marshall model, x-ray pro­duction is assumed to occur at varying depths in the target. The spectral model used by Tucker et al. (1991a), hereafter ca lIed the TBC model, is now presented. The optimal parameters for this model was determined by fitting the model to experimental spectral data from FeweU et al. (1981).

2.4.1 Bremsstrahlung theory

The quantum mechanical theory of bremsstrahlung production from an accelerated electron has been discussed above. The differential cross sec­tion for the emission of a bremsstrahlung photon with energy between h v and hv + d(h v) from an electron with kinetic energy T, (dO',ad), can be obtained from Eqn. (2.19). The average bremsstrahlung energy produced by an electron going through a distance dx along its incident direction in the target is (Tucker

et al., 1991a):

P dT,ad (h v) = A h V dU,ad dt (2.26)

where, again, p is the density of the target material and A is the atomic mass of

the target atoms. Assuming that the electron loses an energy dT from both ra­diative and collisional processes while penetrating a distance dx in the target material, the dT,ad(hv)/dT gives th~ fraction of electron kinetic energy radiated as bremsstrahlung photons of energy hv. Integrating this ratio over the elec­

tron energies present in the target will give the energy intensity of brems­

strahlung photons, Iv, of energy between hv and hv + d(hv):

(2.27)

where Nv is the x-ray spectrum and To is the initial electron kinetic energy.

Substitution of Eqns. (2.19) and (2.26) in Eqn. (2.27) yields:

.., ..,

(

(

The fraction of the x-ra ys transmitted by the anode as they escape is de­termined in a manner similar to that of 8irch and Marshall. The geometry used by the TBe model however differs from that shown in Fig. 2.1. In their model, Tucker et al. (1991a) suggest that the electron flow from the cathode is perpendicular to the surface of the target of angle 8, as depicted on Fig. 2.3. This assumption arises from the fact that the electric field near the edge of the anode is not parallel to the anode-cathode axis, but rather to the plane of the anode. Electrons are thus deflected from their course and enter the target with a direction normal to the surface of the anode l . The fraction of x-ra ys transmitted by the target, F(E,T,To), is then (Tucker et al., 1991b):

F(E T T) = [-JJfhV)(T6- T2

)] , ,10 exp () ,

pC sin 9+ ({J

anode cathode

X-rays

(2.29)

Figure 2.3: Schematic diagram showing the x-ray geometry used by Tucker et al. (1991b) to calculate attenuation of x-ray photons by the anode with target angle 9. The electron beam is effectively perpendicu­lar to the target surface. This results from distortions of the elec­tric field near the edge of the positively charged target. The for­mulation of the target attenuation allows the determination of the x-ray spectrum along the anode-cathode axis by choosing an angle'P different from 0, thus determining a different distance y travelled by x-rays in the anode.

ITucker, 1990 (private communication).

23

where C is the Thomson-Whiddington constant, JJ.(hv) is the linear attenua­tion coefficient of the anode and qJ is the angle off the central beam axis along which photons travel. Variation of fi' will therefore result in different off-axis spectra along the anode-cathode axis. These off-axis spectra thus enable the studyof the heel effect on image quality, beam quality and dose. The brems­strahiung spectrum is therefore given by (Tucker et aL, 1991b):

The function Q has the same meaning as above. However, Tucker et al. fitted their bremsstrahlung model to published spectra from Fewell et al. (1981). The expression of the function Q obtained by Tucker et al. (1991a) is:

Q = [3.685 X 10-2 + 2.900 X 10-5 To]

[ 2 3 4] X 1 - 5.049 ~ + 10.847 ~ -10.516 ~ + 3.842 ~ hvST

Q=O hv>T, (2.31)

where ç is hv/T, as above.

2.4.2 Characteristic radiation theory

Tucker et al. incorporate characteristic radiation in their model by com­bining the Green and Cossiett (1968) empirical relation, Eqn. (2.24), with the observations that the production of characteristic radiation increases to a max­imum with depth in the anode, and then decreases as the electrons lose more

energy as they penetrate the deeper layers of the target (Vignes and Dez, 1968). This buildup effect is attributed to backscattered electrons escaping the anode. This trend was modeled by Tucker et al. (1991a) with a parabolic probability

distribution of characteristic radiation production with depth x:

p(x/R) = (3/2)[1 -(x/Rf]

P(x/R) = 0

xSR

x>R. (2.32)

\ ~ This probability of characteristic x-ray production goes to zero at the depth (R)

at which the average electron energy equals the binding energy of electrons in the K-shell of target atoms (Tucker et al., 1991a). From these considerations, the intensity of a characteristic line with energy E" N(Ej), emitted from the tar­get atoms was modeled as

N( E,) = A. (~: - 1 )'"64' W. f p(xIR) exp [-IJ( Ei)xlsin (9 + '1')1 <Ix, (2.33)

where Tir. is the binding energy of the K-shell electron, Air. is a model parame­ter equal to 1.349 x 10.3 photons / electron, 8 and qJ are the angles defined on Fig. 2.3, and We is the weighting factor defined in §§ 2.3.2. The resulting inten­sity of characteristic photons is then added to the bremsstrahlung continuum to obtain the complete x-ray spectrum specified by the tube kilovoltage and the target angle. The exponential attenuation law can then he applied to the calculated spectra to yield filtered spectra. The comparison of the Kramers, the Birch and Marshall and the Tucker, Barnes, and Chakraborty models is presented in Chapter V.

25

CHAms III

DETERMINATION OF CENTRAL·AXlS DOSES

IN DIAGNOSTIC RADIOLOGY

In this chapter, the various dose measurement techniques aVdilable to

diagnostic radiology are described. The specifie problems associated with dose

measurements in diagnostic radiology will also be addressed. The theoretical

considerations needed for the numerical evaluation of dose contributions

from primal'y and first scattered radiation will be described. A specifie descrip­

tion of the computer programme used to calculate doses in phantom and the

comparison of calculated and measured depth doses is left to Chapters IV and

V.

3.1 Measurement of percent depth dose

Percent depth dose data are measured routinely in radiotherapy. Sorne

of the commonly used dosimeters for these measurements inc1ude ion cham­

bers, radiographie films and thermoluminescent dosimeters (TLD); these

could also be used to measure depth doses for diagnostic radiology conditions.

Unfortunately, it is not guaranteed that the dosimeters used in radiotherapy

will be practieal for dose measurement in p 'ntorn for diagnostic radiology

x-ray beams because of three limitations specifk to such beams. These limita­

tions are low output of the x-ray tube, variations in the tube output and rela­

tively low energy of the emitted photons.

Because the low output of x-ray tubes result in low doses in phan tom

(typieal surface doses are of the order of a few mGy), many exposures at large

tube outputs (mAs) are required to obtain reliable measurements for depths

larger than 5 cm. These repeated exposures quickly heat up the x-ray tube, and

sorne time must be spent between successive irradiations to prevent over­

heating and tube damage. These precautions increase the duration of the

depth dose measurements significantly compared to radiotherapy conditions. Also, unlike x-ray sources used in therapy, the output of a diagnostic radiol­ogy tube varies somewhat (a coefficient of variation of 0.05 is the accepted limit; Hendee and Rossi, 1976). If doses are measured by integration of a sig­

nal over many exposures, it become& difficult to compensa te for this variation over the exposures. Finally, the energies of photons produced in diagnostic radiology tubes range typically between 10 keV and the kVp. In the lower por­tion of this energy range, the photoelectric effect may have a significant role ir. the photon-matter interactions occuring inside some dosimeter materials and the dosimeter response may be energy dependant. The suitability of ion chambers, radiographie films and TLO's is now discussed in this light for the measurement of depth doses in diagnostic radiology. The dosimeter of choice will have the following properties: high sensitivity, small size and minimal energy dependance.

Ion chambers are the most commonly encountered dosimeters in ra­diotherapy. They are also extensively used for exposure measurements in di­agnostic radiology. In regard of the limitations of diagnostic radiology condi­tions, there are characteristics of ion chambers that have to he considered in viewof depth dose measurements. One problem involved with depth dose measurements with ion chambers in diagnostic radiology is the attenuation of low energy x-rays in the wall of the chamber which causes the response of the ion chamber to decrease as photon energy diminishes (see Fig. 3.1). An

ion chamber suitable for most of the energies present in diagnostic radiology beams must have very thin walls. Thin-walled ion chambers can be achieved for cylindrical chambers of small diameter (about 0.3 cm). However, thin­walled chambers of large diameter and sm ail volume are not mechanically stable. Small diameter end-window chambers produce very low signaIs when

used in phantom with diagnostic x-ray energies. Repeated irradiations at high tube loadings (in our experience, over 20 irradiations at 200 mAs at - 100 kVp for a - 40 cGy dose at the surface) are necessary and the measurement of dose to - 8 points in phan tom requîres a few hours if the cooling time determined from the x-ray tube rating charts is allowed between successive irradiations to

prevent tube overheating. Parallel plate chambers of relatively large volume and with air-equivalent walls are routinely used for exposure measurements.

27

.,. "

• 0 (,)

0 ut

0 -1 ;: oS e 1 1:

8. 1 >-a .. • 1: W

0 CJ

CI ut

0 -• .! i • .. 1 1:

1 • e >-e-l YI

1

0.1

0.001

1.5

1.0

.. , • • • •

o Lithium fluoride

• Silver halide in film

001 0.1 1 101

Photon .nergy (MeV)

• lonchamber

++ + + lithium fluoride +

+ 0 lithium tetrabOrate +

+

• • ~i· • 0

o 0 <>

0.5

0 50 100 Photon e"ergy (keV)

150

Figure 3.1: Relative energy dependance of various radiation detectors, inc1ud­ing two thermoluminescent dosimeters, the silver halides pre­sent in photographie film emulsions and a general use Farmer ion chamber. The energy response of the ion chamber, which shows the possible loss of sensitivity of ion chambers at low pho­ton energies, is taken from Johns and Cunningham (1984). Data for the other dosimeters is extracted from Hubbell (1982).

28

(

(

They also have been used in the past to de termine the energy imparted to an

irradiated volume from exposure measurements involving x-ray techniques with known exposure-to-integral-dose conversion factors (Alm Carlsson et al., 1984; Shrimpton et aL, 1981; Shrimpton and Wall, 1982; Shrimpton and Wall, 1983). The volume of the air cavityof these chambers is however large

enough to perturb dose measurements in phantom because of the volume of medium displaced by the chamber. Therefore, ionization chambers dû not ~onstitute an ideal choice for dose measurements in phantom under the con­ditions of diagnostic radiology. Ion chambers have been used by Harrison (1981) for depth dose measurements in water. It was noted that the ionization signaIs were very small, as discussed above, and, at large depths, the uncer­tainties in the measurement became significant. For this reason, the mea­surements were limited to depths smaller than 16 cm.

Film may seem at first thought a good practical dosimeter since the film

optical del1c;ity is related to dose through the characteristic curve (Johns and Cunningham, 1984). In radiotherapy, sheets of film may he placed vertically between layers of polystyrene to obtain a continuous depth dose curve (Almond, 1982; Evans and Schreiner, 1991). The result of the exposure is a permanent record of the experiment and the reading can he repeated. Film is

a good means to detect photons of energies varying between 10 keV and 3 MeV; it can record doses between 0.05 cGy and 10 Gy (Pitt and Scharmann,

1981), and therefore the use of film for dosimetry in diagnostic radiology

would drastically decrease the number of exposures needed to measure doses deep in the phantom. Film dosimetry depends significantly, however, on de­

veloping conditions; processing films separately (under identical conditions) or utilizing different film batches can result in variations of 5 % (Dutreix and Dutreix, 1969). This necessitates calibration of the film response for particular

conditions of use. More problematic is the physics of the photographie pro­cess itself. Film blackening by radiation depends primarily on the absorption

of photons by the silver halide crystals in the emulsion. The binding energy of the K shell electrons of sil ver occurs at 25.51 keV. Therefore, the silver

halide crystals absorb energy very strongly through the photoelectric process,

and the response is very energy dependant (Fig. 3.1). For these reasons, film should not he used for precise dosimetry if the radiation field used contains a

relatively large amount of photon energies in the diagnostie range.

29

"

TLO's are based on the thermoluminescent (TL) pro cess (Cameron et al., 1968; McI<inlay, 1981) which can be observed when certain crystalline ma­terials are heated after a previous irradiation at ambient temperatures. Some of the energy absorbed by the TLO crystal is recorded when electrons set in motion by ionizing radiation are trapped in lattice defects. Subsequent heat­ing of the crystal releases trapped electrons, and the stored cnergy is re-emitted in the form of opticallight. The variation of the light output with tempera­ture is called the glow curve, and the peaks in the glow curve, which corre­spond to different trap energy levels, are called glow peaks. If the heating rate of the crystal is constant, the light emitted is proportion al to the dose absorbed by the dosimeter (McI<inlay, 1981). The dose information is usually obtained from integration of the light output over sorne temperature limits or from the height of the most intense glow peak. Sorne of the general characteristics of TL dosimeters include their wide dose and energy ranges (10 J.1Gy-100 Gy and 10 keV-30 MeV, respectively; Pitt and Scharmann, 1981), low cost and their relatively slight energy dependance (see Fig. 3.1). Three well-known TL phosphors are magnesium and titanium-activated lithium fluoride (LiF:Mg,Ti, TLD-100), manganese-activated calcium fluoride (CaF2:Mn TLD-400), and manganese-activated lithium tetraborate (Li2 B4Ü7:Mn, TLO-BOO).

Lithium fluoride (TLD-I00) is the most commonly used thermolumi­

nescent material. Its effective atomic number is close to that of tissue

(ZeDtTLO.1OO) = 8.14. Zf!ffitis,ue) = 7.4). As seen in Fig. 3.1, the energy response of TLD-IOO is essentially constant compared to that of film. Another characteris­tic of nO-lOO is that the emission of light from aU traps is in the blue-green part of the visible spectrum (400 nm). Consequently, photomultiplier tubes found in commercial TLD readers are typically optimized for these wave­lengths. Six peaks are observed in the TLD-IOO glow curve. The dose infor­mation is usually obtained from either the height of the highest glow peak (- 225 OC) or from the area under the glow curve between - 140 oC and - 260 oC. The intrinsic sensitivity of LiF (the ratio of the energy emitted as light per unit crystal mass and absorbed dose) is 0.04 (Portal, 1981). This is a relatively low value compared with the majority of phosphors used. The sec­ond thermoluminescent dosimeter in common use is calcium fluoride (TLD-400). It has a single glow peak at 260 oC, the wavelength of the ernitted light is

30

500 nm, and it can then he used with commercial TLD reader without any re­strictions. The intrinsic sensiti vit y of this pho~phor is excellent: 0.44 com­pared with 0.04 for TLO-IOO and its dose response is linear up to 2000 Gy (Portal, 1981). On the other hand, CaF2:Mn is not soft tissue-equivalent (ZejJ{TLO-400) = 16.57) and has a large energy dependance at low photon ener­gies; for example, it is a factor of 15 more sensitive to 30 keV photons than to the radiation of C0-60. The dose measurements obtained with this phosphor are more suitable for dose measurements in bone-equivalent materials. The glow curve of lithium tetraborate (TLD-BOO) is composed of two groups of dis­tinct peaks (see Fig. 3.2). The first group, a series of lower temperature peaks lying between 50 oC and 90 oC, is uns table while the second corresponds ta a double peak at - 200 oC. The stable second group of peaks is suitable for dosimetry although 10 % of the trapped electrons are freed over a 2 months period when irradiated crystals are kept at ambient temperature (Portal, 1981). Unfortunately, exposure to light can result in the creation of a spurious signal (0.015 cGy) and in the loss of latent dose information (10 % after 3 hours of ex­posure to ambient levels of fluorescent light, i. e. 1000 lux). Humidity may also increase fading and produce sensitivity lasses (McI<inlay, 1981). The wavelength associated with the main peak is in the orange-red part of the vis­ible spectrum (about 600 nm). Since the photomultiplier tubes found in commercial TL readers are optimized for blue-green light, there is an effective 10ss of sensitivity and the practical sensitivity of TLD-Boo lies between 0.5 and 0.125 of the intrinsic sensitivity of LiF despite the higher intrinsic sensitivity of TLD-BOO, which is 0.073 (Portal, 1981).

Unlike TLD-IOO, the reading of TLD-8QO does not change the distribu­tion of traps in the crystal and, therefore, the phosphor does not necessitate pre-irradiation annealing. Il may however be necessary to anneal the crystal after high doses ~ave been measured in order to release any remaining trapped electrons or ta get rid of the measured background signal because of its light-sensitivity. As shown in Fig. 3.1, lithium tetraoorate is the dosimeter which presents the least variable response with energy. Furthermore, the ef­fective atomic number of TLO-800 is c10sest to that of soft tissue (Zf!.D{TLD-800) = 7.4). These characteristics make the material very attractive for dosimetry in diagnostic radiology despite its lower apparent sensitivity, its light-sensitivity and the deleterious effects of humidity on the material.

3 1

1.0

0.0 _________ -...1 _____ "--.......

o 100 200 300 400

Temperature (OC)

Figure 3.2: Normalized glow curves from LiF:Mg and Li284O,:Mn.

TLD-BOO crystals are thus the preferred dosimeter for the dose measurements presented in this work.

3.2 Calculation

As noted above, the determination of depth doses in diagnostic radiol­ogy requires laborious and time consuming measurements. An accurate computer calculation of doses can therefore accelerate the assessment of pa­

tient risk without having to chance possible tube damage. In this section, the basis for such calculations is presented.

Doses for narrow beam conditions (where scattered radiation is ex­cluded from the volume described by the radiation beam) can easily he calcu­

lated for any depth in phantom when the dose at the surface is known. The

dose D obtained at a point of interest in a homogeneous phantom from a par-

r •

aller (i. e. non-divergent), monoenergetic beam along the central beam axis is calculated from:

D =Doe-JU, (3.1)

where Do is the dose at the surface of the attenuating medium whose linear attenuation coefficient for the photon energy corresponding to that of the beam is Jl and where the thickness of the medium at the point of interest is t.

Equation (3.1) is never used in practice since monoenergetic x-ray beams have not been successfully achieved and narrow beam conditions are aImost never used in diagnostic radiology.

In general, the dose to a point in a patient or a phantom results from both primary and scatrer contributions. The relative amount of each depends on the geometry of the beam, on the beam quality and on the composition of the phantom. The primary radiation component behaves similarly to Eqn. (3.1); the scatter dose contribution, on the other hand, is hard to predict theo­retically because it is a complicated function of the physical properties of the beam and the phantom.

Two techniques, bath involving calculation, have been devised to de­termine the contribution of scattered radiation to the dose to a point in phan­tom: ray tracing integrations (Vaffe and Johns, 1983) and Monte Carlo simula­tions (Chan and Doi, 1983). Ray tracing is a numerical integration of the en­ergy fluence reaching a point Q in phan tom from all surrounding media; in the present case, these media include any x-ray filtration materials in the path of the x-ray beam, the air layer between the patient or phantom, and the phan­tom itself (see Fig. 3.3). For a single scattering event occuring in a single site in anyof the irradiated media (first scatter sites occur only in the volume de­scribed by the radiation beam), the probability of a photon emitted by the x-ray tube reaching that site and scattering at the angle 8 to arrive at Q can be calcu­lated. This is a relatively simple integration over the volume described by the radiation beam in aU of the irradiated media. A second integral, over photon energies, is necessary to acc, unt for the polyenergetic nature of the x-ray beam. For second and higher order scatter the calculation must consider, for

33

-------------------

X-ray tube

Added filtration -H+.--

Phan tom

Figure 3.3: Contributions to dose considered in the numerical evaluation of dose: (1) primary dose and first scalter dose due ta scatter in (2) the filter, (3) the overlying air and (4) the phantom itself.

each first scatter site, aU subsequent scatter sites in the phantom. Those sites -:an be either within or outside of the radiation field.

Monte Carlo techniques can be employed ta investigate the characteris­tics of the multiply-scattered photons for the purposes of both dosimetry and

imaging. The random trajectories of individual photons are simulated by the computer in Monte Carlo calculations. The probability for a particular inter­

action ta occur are calculated from x·ray cross section data while pseudo-ran­dom numbers are used to select which interaction, if any, takes place. A large number of photon "histories" are needed for the simulation to he statistically reliable. Physical quantities can then he estimated with sufficient precision. The time needed to generate a sufficient number of histories is of the arder of

sever aI minutes to a few hours, depending on the computer hardware: Monte Carlo studies typically generate -1()6 photon trajectories while direct photon

transport calculations involve less than 50 integrations per photon energy present in the x-ray beam.

( ..

The advantage of ray tracing integrations over Monte Carlo simula­tions is the computation time. Doses in phan tom may be obtained in a few minutes with ray tracing if the geometry is simple, which is the case for cen­tral-axis depth dose calculations involving primary and first scattered pho­tons. In this thesis, the ray tracing technique was preferred to calcula te doses in phantom. The comparison of calculated and measured relative depth doses is done in Chapter IV.

3.2.1 Primary radiation

The physical interactions possible in diagnostic radiology are the photo­electric, Compton or Rayleigh processes. Absorbed dose depends on the mass energy attenuation coefficient (Jlab/ p), which encompasses energy deposition for a photon undergoing a single interaction in the phantom. In the case of the primary beam, only photons which have not interacted with any of the surrounding media are considered. Since the radiation field reaching a point Q in phantom is usually attenuated by the x-ray filtration, the air and the overlying phantom, the primary beam originating at the x-ray tube reaching point Q in phan tom is reduced by an amount exp-I,J.lidi, where I.Jl;d; is the sum of the attenuations provided by the layers of materials of thickness d, and lin­ear altenuation coefficient p; a photon traverses before reaching the point of interest.

For a parallel beam of monoenergetic photons with an energy Eo inci­dent on an homogeneous phan tom and electronic equilibrium at point Q, the dose at point Q due ta primary radiation is given by

D"c.Q) = 4> Eo (~)p e-p"d, (3.2)

where tIJ the photon fluence (photons/ cm2) at the surface of the phan tom, (Jlab/ p)p the mass energy absorption coefficient (an2 / g) of the phantom mate­rial for the primary photons of energy Eo (in keV) and IIp the linear attenua­tion coefficient (cm-t ) of the phantom material at Eo and d is the depth in phantom. Equation (3.2) can he corrected for beam divergence using the in-

35

----------------- .- --- -- ---------_._--_._---------,

ver~,e square law. The primary dose from a monoenergetic x~ray beam then becomes

D,J.Q} = CIJ( (2 )Eo(lJab) e-lJrli, (j+d)2 P p

(3.3)

where f is the source~to-skin distance and dis the depth in phantom. The to~ tal primary dose to a point Q in phantom is therefore the integral over aU primary photon energies of Eqn. (3.3).

3.2.2 Scattered radiation

The first scatter dose at Q in an homogeneous phantom from aIl sur­

rounding elements in the radiation field is essentially produced by both the Compton (incoherent) and Rayleigh (coherent) interactions (Johns and Cunningham, 1984; Yaffe and Johns, 1983). Coherent scattering is the domi­nant scattering process for photons of energies less than 10 keV in water while Compton scattering prevails over ail processes in water for photon energies Iarger than 30 keV (Johns and Cunningham, 1984). 80th contribute equally to the first scatter dose for photons of primary energy of around 20 keV. Il is as­

sumed that the characteristic photons produced in water (or water-equivalent material) by the photoelectric effect, which have energies typically less than

1 keV, are totally absorbed at the interaction site.

3.2.2.1 Rayleigh scattering

Rayleigh scattering is an interaction between photons and bound elec­trons. The collision is regarded as taking place between the photon and the

whole atom: the transferred momentum is absorbed by the whole atom and the excited atom radiates electromagnetic waves of the same frequency and in

phase with the incident photon. For this reason, Rayleigh scattering is also known as coherent scattering. The atomic electrons absorb energy from the

incident wave and scatter it equally probably in all directions, without modify­

ing the photon energy. The atom is neither excited nor ionized at the end of

the interaction.

36

(

(

The probability that a photon of a given energy will interact with an atom of a material through which it passes is specified by the cross section «1)

of the particular interaction. The expression of this probability per unit soUd angle is called the differential scattering cross section ~"l1/dn). The Rayleigh cross section can be determined using the momentum transfer parameter (Hubbell et aL, 1975),

(3.4)

where 8 is the scattering angle and Â. the wavelength of the incident photon, Â. = hclE. The momentum transfer thus combines the dependance of the Rayleigh cross section on the scattering angle 9 and the photon energy. The Rayleigh differential atomic cross section is defined as (Hubbell et aL, 1975):

(3.5)

where F(q,Z) is the atomic form factor for coherent scattering, tabulated in Hubbell et al. (1975), ro is the classical electron radius and Z is the atomlC num­ber of the materia1. Squaring the atomic form factor yields the probability that the Z electrons of the atom take up a recoil momentum without absorbing any energy. The differential Rayleigh cross sections are strongly peaked in the forward direction. This is a result of the interference effect produced by a large number of point sources of coherent radiation. This effect is taken into ac­count by the form factor.

To obtain form factors for molecules, the sum of the contributions of aIl of the different atomic species must be performed. It is commonly assumed that the coherent cross sections of individual atoms combine independently. Under this assumption, form factors for molecules are calculated with (Chan and Doi, 1983; Morin and Berroir, 1983):

N

F!o, = 1: nj Ff(q ,Zj~ j = 1

(3.6)

where N is the number of atomic species present in the molecule and nj the

number of atoms of the same kind in the molecule. F!o, can then be substi-

37

--------------------------------------------------------------------------------------

tuted to F2(q,Z) in Eqn. (3.5). The differential cross section thus obtained for 80 keV photons in polystyrene is shown in Fig. 3.4. Il has however been shown that the differential cross section of water molerules for Rayleigh scat­tering does not peak at 0° as there are additional interference effects arising from inter-molecular interactions occuring in liquid water (Morin and Berroir, 1983; Yaffe and Johns, 1983). It is therefore preferable to use corrected form factors when they are available for the desired molecule. Correct form factors were tabulated for water molecules by Morin (1982); none have been compiled for polystyrene or lu cite.

3.2.2.2 Compton scattering

Compton (incoherent) scattering is one of the most important interac­tion mechanisms involving photons in tissue-like materials. Incoherent scat­tering occurs when a photon of energy hv eoUides with an atomic electron and

2.5 --c: c .! .2 ~ 2.0

1 ! !Il • 1 5 0 .. • '3 . .. Q U •

'0 .!! E 1.0 --c -.CIC .. E Je Q :::. 0.5 CI":

o ... - 0.0

o 5 10 15 20 25 30 Scatterlng angl. (degrees)

Figure 3.4: Differentiai molecular cross section for 80 keV photons undergoing Rayleigh scattering in polystyrene. The form factors for polysty­rene were obtained from adding the form factors for carbon and hydrogen under the assumption that each atom contributes inde­pendently to the interaction.

38

imparts sorne of its energy and momentum to the electron. The photon is scattered by an angle 8. Since only one electron is involved, photons are not scattered in phase. The energy of the scattered photon (h v1 can he determined from the scattering angle according to (Johns and Cunningham, 1984):

hv' - hv - [1 + a (1 - cos 8)] ,

(3.7)

where a is the ratio of the energy of the photon to the rest energy of the elec­tron (a = hvlmee2,where c is the speed of light and me is the electron rest mass). From conservation of energy, the kinetic energy of the recoil electron is h( v - v').

The differential cross section for the Compton interaction is usually de­termined from the product of the classical Thomson scattering cross section and the Klein-Nishina formula derived for a photon interacting with a free electron (Johns and Cunningham, 1984),

d O'jllCoh = r~ (1 + cos1 9) dD 2

{ 1 )2 /1 a2(1 - cos 8/ \ (3.8)

x 1 + a(l- cos 8) \ + [I + a(l - cos 8)] (l + cos2 8)/"

The cross section for Compton scattering is expressed in units of cm2 1 electron and it shows no dependance on the atomic numher of the target atom. The Klein-Nishina cross section has aiso been used to calcula te the Compton cross section arising from interaction with bound electrons. This approximation is valid in the cases where the energy of the incoming photon is much Iarger th an the electron binding energy. A correction is neœssary to take the binding energy of the electron into account for lower energy photons. The differential Compton cross section for atomic electrons is given by the product of Eqn. (3.8) with the incoherent scattering function S(q,z), where q is the momen­tum transfer as expressed by Eqn. (3.4) and Z is the atomic number of the atom. This scattering function represents the probability that an atom will he

raised to any excited or ionized state when a photon delivers a recoil momen­

tum to a bound electron. This probability becomes unit Y for photon energies

39

much larger than the electron binding energy. The Compton cross section now decreases for low energy photons because of the binding energy correc­tion. Values for the incoherent scattering function are tabulated in Hubbell et al. (1975). The incoherent scattering functions modify the Klein-Nishina cross section most prominently for small angle scattering of low energy photons in high atomic number elements.

In a manner similar to Rayleigh scattering form factors, the in coherent scattering function for a molecule S",(q) can be calculated with (Chan and Doi, 1983; Morin and Berroir, 1983)

N Sm(q) = L ni Sdq,Z;), (3.9)

i = 1

since, for inelastic scattering, each electron contributes independently to the total scattering probability (Eisberg and Resnick, 1985). The differential cross section for the Compton interaction for 80 keV photons in polystyrene is

shown in Fig. 3.5.

3.2.3 Calculation of doses

The photon transport/ray tracing calculation described here is an adap­tation of the technique proposed by EI-Khatib, Podgorsak and Pla (1986) for the determination of the broad beam attenuation in phantom obtained when lead compensators are used under radiotherapy conditions. A similar algorithm was also proposed by Robinson and Scrimger (1990) to optimize tissue com­

pensators. It consists of a volume integration of the photon fluences reaching a point of interest Q in phantom. These fluences can originate from primary

radiation and fr01n first scattered radiation due to the added filtration, air, and the phan tom itself. Only very simple geometries are considered here: the phan tom is homogeneous, and the technique assumes that the edges of the radiation beam are enclosed in the irradiated volume. A generic geometry is

depicted in Fig. 3.3.

4()

8.0 --c c • 0 ;; ;: • ~ :; 6.0 -• .. .. ~ ..

0 :s ~

~ 4.0 " i '0 - E c ::::" .1'1 .. E ! u 2.0 =.0 QW ,

0 --0.0

1 1

o

1 1 1

-- Klein-Nishina - - - Corrected

45 90 135 180 Seatterlng angle (degre •• )

Figure 3.5: Differentiai cross sections per molecule for 80 keV photons under­going Compton scattering in polystyrene. The solid line repre­sents the Klein-Nishina differential cross section for free elec­trons. The dashed line is the cross section obtained by multiplica­tion of the Klein-Nishina differential cross section with the in­coherent scatter function to correct for the electron binding en­ergy. The molecular incoherent scatter function for polystyrene was obtained under the assumption that each atom contributes independently to the interaction.

The primary dose at point Q along the central axis of a polyenergetic beam in phantom is obtained by integrating Eqn. (3.3) over aIl photon ener­gies present in the beam. Since the spectra (N v) calculated according to the Birch and Marshall semi-empirical theory (1979) described in Chapter II are al­ready normalized to one cm2, one can use them directly in the integration. In a typical situation, however, the radiation field is attenuated by the inherent filtration, the added filtration, air, and the phantom itself before it reaches the point of interest. Therefore,

Dp(Q)=( 12

211'-Nvxhvexp[-± JlP'fldfl](Jlab)d(hV1 (3.10) (jtt/) fI= 1 P

o

41

., where the Ny exp-I~p.n d" is the attenuated spectrum reaching the point of in­

terest (Q> at depth d in phantom (the input x-ray spectrum, N v, is already al­

tenuated by the inherent filtration). This attenuation can be provided by

three media the photons can go through before reaching Q: the added filtra­tion, air between the added filtration and the phan tom and the phan tom it­self.

For a monoenergetic beam, the first scatter dose at point Q at depth d in a homogeneous phantom from aU surrounding media in the radiation field

can be calculated from a volume integral. In spherical coordinates, the first scatter dose is given by (EI-Khatib, Podgorsak and Pla, 1986):

(3.11)

where da/dD is the differential cross section of the chosen scattering interac­tion (Rayleigh or Compton) per unit soUd angle (cm2/electron), (hv)" is the en­

ergy of the scattered photon in keV, ne is the electron density (electrons/ cm3 )

of the phantom material, JJs and (IlGb/ p)s are the linear and mass energy ab­

sorption coefficient for once-scattered photons in the phantom material and R(8) is the radial distance from Q to the edge of the of the media where the

scattered photon originates or the geometric edge of the beam, whichever cornes first. R(8) thus takes the beam divergence into account. The attenua­

tion probability specified by the path of the photon is then the product of the

two exponential terms found in Eqn. (3.11).

Integration of Eqn. (3.11) over r can be performed analytically if a paral­

leI beam is assumed. This assumption allows the differential cross section to

remain constant over aU r for a fixed 8. This integration yields (EI-Khatib,

Podgorsak and Pla, 1986):

D s (Q) = 2x ne Ny exp [ -Sipd]

x 1" (hv~ (#JGb) sin 8 J1a. exp [R(S) (pp cos 9 -Ils)] - 1 d8. P .r dD (Jip cos 8 - Sis)

o

(3.12)

Equations (3.10) and (3.12) can now be written to explicitly take into account the three materials the radiation field travels through.

Using the geometry shown in Fig. 3.6, the primary dose to a point Q in phan tom can he calculated from:

where XI, X2 and Xl are defined in Fig. 3.6, fis the source-to-skin distance, and

J.l~jlt, JJf/' and JJp represent the attenuation coefficients for the primary beam for the mter, air and the water-equivalent material. Here, N v is the x-ray spec­trum incident on the added filtration.

Equation (3.12) must also be modified to give the first scatter doses con­tributed to the point of interest in phantom by the three usuallayers. The first scatter dose contributed from the added x-ray filtration material is given

by:

1~ ...

og'l(Q) =21t nP" Nvexp[-iJU't (XI-X2 -X3)] (hV).r(J.lGb\W sin B.da. o P & dD

( lJai'X2 + IJWX3) exp [Rlilt(B) (Pf"tcos 9-- .utUt)]-1 xexp _ S S dB,

cos 9 (PP'ICOS 8 - IJtilt) (3.14)

that from the air layer is:

(3.15)

43

1

Figure 3.6: Geometry used for the calculation of the primary and first scatter doses to point Q from surrounding media (added filtration, air and the phan tom material itself) in a divergent beam.

while the expression for the dose from radiation scattered in the phantom is:

1- (J.lab)w. ,1 .... exp [RPilantoM( 8) (J.lpcos 8 - J.l;V)] - 1 x (hv)s - sm 8~ 1.. ) d8. (3.16)

P s dO \JlpCOS 8 - J.l't o

The first scatter dose al point Q in phan tom results from radiation scat­tered by the added filtration, air, and the phantom itself. Both the Rayleigh

and the Compton interactions are responsible for the production of scattered photons in diagnostic radiology. The first scatter dose can then he expressed as

(3.17)

where Ds.R and Ds,e, the scatter doses from photons scattered by the Rayleigh and the Compton interactions, respectively, result from the sum of their re­spective contributions from photons scattered from the added filtration, air and the phan tom material:

D ..... filter nIJÎr DW s.R = vi.R + U soR + s.R

D - ..... filttr Dai/' DW s,e - vi.e + s,e + s,e (3.18)

Equations (3.16) to (3.18) are evaluated for both the Rayleigh and the Compton

processes. The sum of aIl six integrals gives the first scatter dose at the point of interest. The relative importance of the different contributions to the dose

and the relevance of Eqns. (3.14), (3.15), and (3.16) in scatter dose calculations will he discussed in Chapter V.

45

1 CHAPTEBIY

EXPERIMENTAL METHODS AND MATERIALS

Models have been presented in Chapters n and III for the calculation of diagnostic x-ray spectra and for the calculation of doses in phan tom. The im­plementation of these algorithms and the use that has been made of them for the assessment of the risk reduction involved with various radiographie techniques is now presented. For programming simplicity, code has been written to calculate mass attenuation coefficients using the Ouellet and Schreiner parametrization (1991). Twenty-three tube voltage/filter combina­tions were tested. One such combination was tested for four field sizes and for off-axis doses, and all of the data resulting from measurements or calculations are found in the appendices. A discussion of the results is left to Chapter V.

4.1 Computer calculations

4.1.1 Mass attenuation coefficients

The effects of inherent or added filtration as weIl as the effects of in­coming x-rays on a phantom or screen-film system are considered by looking at the spectra obtained after transmission through a thickness t of a specifie material. Spectra are modified acC'ording to the exponential attenuation law (Eqn. (2.15» which requires known mass attenuation coefficients (~/ p). In this work, the mass attenuation coefficients were obtained using a set of paramet­

ric equations which generate ~/p for aU photon energies (hv) between the Ll

absorption edge (or a minimum of 1 keV) and 150 keV and for aU elements from hydrogen to uranium (1 S Z ~ 92) (Ouellet and Schreiner, 1991). These equations were determined by fits ta tabulated values of mass attenuation co­efficients (Hubbell, 1982; Storm and Israel, 1970):

Il (Z, hv) = A(Z) (hvf(Z) + C(Z) (hv~Z) + F(Z) cPz) la v. P

(4.1)

The constants A, B, C, D, F, and Gare defined in terms of the atomic number

of elements using polynomial equations in terms of l, A(Z) = ~ ai li, expo-

nentials of polynomial terms in l, ln A(l) = ~ Qi Z', or polynomials in ln (Z),

ln A(Z).: ~ ai (In Z)·. AlI of the mass attenuation coefficients can be generated

for the energy range and atomic numbers cited above with only 148 polyno­mial coefficients (Ouellet and Schreiner, 1991).

Another set of fits based on the equation proposed by Thinh and Leroux (1979):

(4.2)

was used to calculate mass attenuation coefficients between the Lt absorption edge and 1 keV for elements whose atomic number is above 27 (i.e. elements whose LI absorption edge is above 1 keV). Values for K(Z) and the values for n are aIso found in Ouellet and Schreiner (1991). An overall mean error of 1.4 % from tabulated values is observed (Ouellet and Schreiner, 1991).

Once the mass attenuation coefficients have been calculated for a11 of the desired elements, a mass-weighted sum can be used to calculate pl p for mole cule s,

P ..... tnO_le_c,u_e = I: W i (p.), p j P j

(4.3)

where Wj are the weight fractions of each of the constituent elements. This mixture rule ignores the variations in atomic wave functions resulting from changes in the state of aggregation of the atomic constituents of the molecule (Deslattes, 1969; McCullough, 1975; Johns and Yaffe, 1983). The induced errors are less than a few percent for photon energies above 10 keV except just above absorption edges (Deslattes, 1969; Hubbell, 1969). The sources of discrepancies of the p.1 p calculated from the mixture ruIe and those measured for the vari­ous phases of compounds are discussed by Deslattes (1969).

47

1

4.1.2 X-ray spectra

The programme Spectrum 1.1 generates x-ray spectra according to the Birch and Marshall semi-empirical model described in Chapter II. The pro­gramme was written on a Macintosh Plus computer (Apple Computer, lnc., Cupertino, California), which uses the Motorola 68000 rnicroprocessor.

Versions of the program were produced for the faster Macintosh models (Macintosh Il with a Motorola 68020 microprocessor and the Macintosh II FX and SE/30 models with the Motorola 68030 microprocessor) which use a mathematical co-processor (Motorola 68881). The programme was written in

the C programming language (THINK C, Symantec Corporation, Cupertino, California). A listing of sorne of the more important subroutines is included in Appendix C. The programme employs discrete means (summations) to calculate the integrals leading to a complete x-ray spectrum.

The Spectrum 1.1 programme evaluates x-ray spectra after specification of peak kilovoltage, target angle, inherent filtration (expressed as the equiva­

lent aluminum filtration thickness) and mAs according to the theory of Birch and Marshall for a tungsten target. Photon intensities are ca1culated for a

minimum bremsstrahlung photon energy of 5 keV and a maximum of

150 keV (in steps of 1 keV) for constant integer potentials varying between 30 and 150 kVp. K-characteristic photons are then added to the bremsstrahlung continuum. Contributions from photons of energies lower than 5 keV and of

characteristic radiation from the tungsten L-lines (- 8 to 12 keV; Birch;

Marshall and Peaple, 1982) are neglected since a realistic amount of inherent filtration (a minimum of about 0.6 mm Al) will effectively remove all of the

photons of energies below 10 keV. The programme includes algorithms for the cakulation of linear attenuation coefficients based on the Ouellet and

Schreiner parametrization (1991). The mass stopping powers are determined for integer electron kinetic energies varying between 1 keV and the peak kilo­

voltage (kV p) by a cubic splines interpolation (Press et al., 1986) of tabulated data (ICRU 1984). The Thomson-Whiddington constant is obtained from a linear fit to the data found in Table 2.1. The normalization constant for char­

acteristic radiation is taken from Boone (1988). The resulting spectra agree

weil with the spectra published in the HP A scientific report 30 (Birch,

(

(

Marshall and Ardran, 1979). The spectra can be modified further to account

for any added filtration.

The spectra obtained through Spectrum 1.1 are normalized for a dis­

tance of 0.75 cm from the source and for an area of 1 cm2. This normalization

originates from the fit to the differential energy spectrum proposed by Birch

and Marshall (1979) in order to obtain calculated spectra similar to the ones

they measured. Such geometries are not used in practice. It is a simple matter

to attenuate a spectrum to take into account a different source-to-skin distance

even though the modified spectrum would not appredatively differ from the

original one. The question of beam area arises when quantities linked with

irradiated volume are considered. This problem will he discussed in §§ 4.1.4.

The programme Spectrum 1.1 contains routines which permit the user

to optionally evaluate the exposure, the primary integral dose to a semi-infi­

nite water phantom of user-defined thickness and the transmission curve and

its interpolated first and second halE-value layers for either filtered or unfil­

tered spectra (see the flowchart in Appendix A).

Equation (2.3), defined by the ICRU (1980), is used to calculate the expo­

sure X. Transmission curves are calculated using the ratio of the exposure X

obtained for a chosen spectrum N" (which may or may not he filtered), to the

exposure X(l) obtained after attenuation through a thickness lof aluminum:

The transmission curve can then be expressed as a function of the thickness of

aluminum l,

1(1) = X (1) X

(4.5)

The x-ray model can be supported by comparison of calculated trans­

mission curves to measured ones using a chi-square minimization technique

similar to that of Boone (1986): the goodness of the fit of the calculated trans-

49

• c,

1 mission curves to measured transmission data was obtained by the determi­nation of

x2 = L (T fMtlSlITtd (t,) - TcalcMlllltd (t,))2, 1 Tealcwlattd (t,)

(4.6)

where T measwrtd (Iil and Tealculalttl (t,) represent the transmissions through thick­nesses li of aluminum obtained from measured data and from calculated spec­

tra, respectively. Of the three parameters which can be potentially he fitted to

calculate equivalent constant potential x-ray spectra, which Boeme designates

EQSPEC, (the kVp, the inherent filtration, and the anode angle), only the in­

herent filtration was varied in this implementation until a match of calcu­lated and measured transmission data was obtained. The target angle was fixed at 12°, as stipulated in the x-ray tube specifications. The kVp was set

from measured effective kV p'S to accurately de termine the equivalent con­

stant peak kilovoltage. These measurements will be discussed in §§ 4.2 .

Twenty transmission curves were measured for a wide range of tube kilovolt­ages. The matching of calculated transmission curves indicated an inherent

filtration equivalent to 1.31 ± 0.03 mm Al in ail cases. Figure 4.1 shows the

agreement obtained between the transmission curves obtained from measure­

ments and from their calculated equivalent spectra.

While the Spectrum 1.1 programme generally assumes a constant po­

tential, another programme, SinglePhase, has been written to generate spectra for the voltage waveforms obtained with single phase generators using either

half or full wave rectification (i.e. V = kVp sin (rot), 0 S rot S 180°). The mean

x-ray spectrum for such situations is obtained by the summation of weighted

constant potential spectra: the sinusoidal waveform of the applied potential is

divided in ten equal steps between 0 S rot S 90°, the voltage at the midpoint

of each interval being V" = kVp sin (4.5 + 9n) 0 ~ n S 9. (4.7)

Multiplication by 2 gives the spectrum for a full wave rectified potential. The

weighting of each interval is thus given by:

1-'1

Wi = • sin 8d8 OS; i S 9, (4.8)

50

.(

(

1.0

110 kVp: R-filter c oR

J -1 0.1

1 c l! t-

61 kVp: samarium

0.01 0 5 10 15 20 25 30

Alumlnum thlckn ... (mm)

Figure 4.1: Attenuation data measured for various x-ray beams. The attenuation data obtained for the corresponding calcu­lated spectra are shown as salid Unes. The 110 kVp beam filtered by the commercial R-filter, the unfiltered 100 kV p beam and the 61 kVp beam filtered with 0.1 mm Sm are illustrated here. A 1.31 mm Al inherent filtra­tion is present in aU beams.

where Bi = 9 x i. The effect of the voltage waveform on the x-ray spectrum can be appreciated from Fig. 4.2.

4.1.3 Percent depth doses

A computer programme based on the ray tracing technique, depthDose, was written in the C programming language for a Macintosh SE/30 system. Using the spectra specified by Spectrum 1.1, depthDose calculates the primary and first scatter doses to a number of points in a polystyrene phan tom of user­defined thickness. The programme is relatively fast: it takes about 10 minutes to evaluate the dose to eleven points in a 30 cm polystyrene phantom for an 80 kVp beam. The user has to specify the source-to-skin distance (S50), the

5 1

't ~

.. ~ a. .. ~ 0

c 0 .. 0

oC a.

l N --• E .. 0 Z

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0

. . . . Constant potential

--Ha" rectified

\ 1 III t, Il 1 111

10 20 30 40 50 60 70 80 Photon energy (keV)

Figure 4.2: Comparison of an 80 kVp spectrum from a single phase generator and a 80 kV p constant potential spectrum generated using the Spectrum 1.1 and SinglePhase programmes.

source-t~filter distance (S1O), the radius of the circular field equivalent to the field of interest, the filter rnaterial, the filter thickness and the incident x-ray spectrum. Again, the linear attenuation coefficients are obtained using the Ouellet and Schreiner (1991) rnethod. The mass energy absorption coefficients for polystyrene were obtained from logarithmic interpolation of data found in Johns and Cunningham (1984) and Plechaty et al. (1975). The electron densi­

ties were obtained using Avogadro's number NA:

ne = P N ~ Z electrons/cm3, (4.9)

while those for air and polystyrene were taken from published tables (Plechaty et al., 1975; Johns and Cunningham, 1984). In the computation, it was as­sumed that each atom interacts independently from the other atoms making up the molecule and the atornic form factors used for Rayleigh scattering and the scattering functions used for the Compton interaction were divided by the

52

-, ...

r ., "

total number of electrons in the molecule in order to obtain du/dD in cm2/electron, as required by Eqns. (3.11), (3.12) and (3.14) to (3.16).

The computations for the primary dose contributions are standard, and the steps needed to calculate the first scatter dose are straightforward but nu­merous. The six integrals used to calculate the first scatter dose (three for the Compton interaction and three for the Rayleigh scattering) must be evaluated discretely for a number of angles between 0 and x, for all integer energies pre­sent in the x-ray spectrum, and for each point in the polystyrene phan tom at which the depth dose is required. The computer calculation assumes that:

(1) the incident beam is parallel (this assumption induces a small error, cos 6 ... 1 for small angles, but it resuIts in reduced compu­tation lime). Beam divergence is taken into account in the defi­nition of R(8) (see Chapter UI). Inverse square law corrections are made for the primary and the scatter dose.

(2) The phantom surface is flat, parallel to the added filtration and perpendicular to the central beam axis.

(3) The radiation field is completely covered by added filtration.

(4) The phantom is homogeneous.

(5) The beam profile is uniformly fiat over the area described by the radiation field.

Second and higher order scattering are neglected. This last assumption

can be vaUdated from published Monte Carlo studies. For example, Persliden's calculations (1983) have shown that the average number of scatter

events from a 70 kVp polyenergetic x-ray beam is likely to undergo in a 20 cm

water phantom is 2.18. In this example, the doses calculated for primary and first order scatter photons would account for a large fraction of the total dose (> 70 %). It is however expected that, for more penetrating beams and for

larger phantom thicknesses, the average number of scattering events per pho­ton will increase, and so will the second and higher order scatter doses.

53

Monte Carlo simulations are currently underway ta further examine the va­

lidity of this assumption.

Figure 4.3 shows the geometry of the integration steps involved in the

calculation of the scatter dose at a point Q in phan tom. As shown in Fig. 4.3c,

the incident photons are assumed ta he parallel. The integral in Eqn. (3.14) is evaluated for aU angles such that the distance from the central axis along the added filtration's upper surface increments by 0.1 cm until the beam's edge is

reached, as seen in Fig. 4.3a. Angles are scanned in a similar manner for Eqn.

(3.15) for 0 S 9 S; 6!i!; the integral is then evaluated in ten equal steps between

6!:!: and e::XI along the beam's edge, as shawn in Fig. 4.3b. 8t:~~ and 8::U are as defined in Fig. 3.6. The scanning of angles along the surface of the phan-

tom is performed similarly to Eqn. (3.14) for angles between 0 and 8::::u to

evaluate Eqn. (3.16). Twenty equal steps between 9::U and 1C/2 are subse­

quently used along the beam's edge. The contribution from backscattered

photons (K/2 S 6 S; TC) is evaluated in steps of 5 degrees along the beam or the

phan tom 's edge, whichever is closest at that angle.

4.1.4 Determination of the integral dose

Once the spectra equivalent ta the radiographie technique are deter­

mined from the mode l, integral dose calculations can he performed. A num­

ber of expressions are proposed to calculate the integraI dose (E) from spectral

data. The integral dose (or the energy imparted) was defined by the ICRU

(1980) as the difference between the incoming en erg y (R in) and the energy

transmitted through a volume of interest (Ro/ll) plus the total release of energy

from nuclear transformations (I Q),

(4.10)

The l Q term equals zero in the diagnostic energy range. Ri,. is easy enough to

calculate, but Ro", can only be calculated exactly by Monte Carlo techniques.

An analytic approximation ta X would however he easily achieved if only the primary radiation were ta ken into a ccoun t, i.e. the incoming x-ra ys are

a) b) c)

Figure 4.3: Steps involved for the determination of R(6) to calcula te the dose at point Q in phantom from radiation scattered by a) the added filtration, b) the air layer, and c) the phantom itself. When the scattering angle is determined, R( 8) is determined and used for the numerical evaluation of Eqns. (3.14) to (3.16). The incident photons are assumed to be parallel to the central beam axis at an Compton or Rayleigh scattering interaction sites.

assumed to interact only once in the phantom. A number of expressions to derive E from x-ray spectra (N v) have been proposed in the literature. Carls­son (1963) proposed an expression,

(4.11)

where both the coefficients AE for the subtraction of photons backscattered from the phantom (Berger and Raso, 1960), and Td, the transmitted fraction, are determined from Monte Carlo calculations and measurements. Since a rapid means of computation of the integral dose is desired and since both AE

and Td are available only for restricted applications, Eqn. (4.11) is impractical.

55

Koedooder and Venema (1986) formulated an expression based on the energy balance from the incoming and exiting energy spectra:

(4.12)

where x is the thickness of the phantom. This equation assumes however that any photon interacting with the phantom is totally absorbed. Equation (4.12) will consequently overestimate the integral dose.

Johns and Yaffe (1985) introduced an exact expression,

x = lltv

- N v hV[ 1 - e-J.'v,x] (JJab. v + x) d(hv), (4.13) o Jlv phanlom

where the ratio (#1ab.JJJv) is the fraction of energy removed from the primary beam which is deposited in an initial interaction and X is the fraction of the energy removed from the primary beam by a scattering process which is ab­sorbed on a subsequent interaction. X can only he evaluated by Monte Carlo ca1culations. Equation (4.13), however, can give the integral dose from pri­mary radiation if X is set to zero. This modification has been implemented in the Spectrum 1.1 programme to yield primary integrai dose. This quantity is expected to underestimate the absolute integral dose. If, however, the relative contribution of scattered radiation to the integral dose is constant, the relative integral doses obtained with Eqn. (4.13) may he satisfactory.

Since the integral dose is a quantity intimately linked with the irradi­ated volume, one needs to integrate over cos cp dA (Alm Carlsson, Carlsson and Persliden, 1984) to take beam divergence into account:

(4.14)

56

where qJ is the angle of incidence of photons impinging over an element of area dA (tp is uniquely determined by the position of dA relative to an assumed point focus). In most practical cases, however, the cosine of qJ is close to unit y and the integral over qJ would introduce only a very slight correction. Only strongly divergent beams should he treated this way. For most geometries used in diagnostic radiology, multiplication of the integral over photon ener­gies by the area covered by the field at the surface of the phan tom is sufficient.

The integral dose can also be extracted from depth dose data. It can be ca1culated using

E = p f D(x) A(x) dx, (4.15)

(Carlsson, 1963; Johns and Cunningham, 1984) where D(x) is the dose at a depth x in sorne phantom material of density p and A(x) is the area described by the beam at depth x. A derivation of Eqn. (4.13) with X set to zero from Eqn.

(4.15) is presented in Appendix E. A comparison of the three methods used to obtain the integral dose (i.e. measurements, the depthDose programme and Eqn. (4.13) with X set to 0 (see above» is presented in Chapter V.

4.2 Apparatus and experimental technique

The dose measurements were all performed on a c1inical x-ray system in the Department of Diagnostic Radiology at the Montreal General Hospital consisting of a Picker GX600 three phase twelve pulse generator (Picker International Canada Inc., St-Laurent, Québec) as weIl as a Machlett Dynamax 62U tube containing a 69B insert with a 120 tungsten target (Varian/Eimac, Georgetown, Ontario). The system's coefficient of variation and linearity of exposure coefficient were 0.02 and 0.03 respectively; these valu.2S are weIl be­low the limits suggested by Hendee and Rossi (1976) (0.05 and 0.1 respectively). The large focal spot of the tube 0.2 mm nominal, 2.2 mm effective) was used throughout the experiments to reduce the tube strain throughout the experi­

ments which involved measured tube voltages between 48 and 134 kV p'

57

TLD-BOO thermoluminescent crystals (Harshaw Chemical Co., Solon, Ohio), 3 mm x 3 mm x 1 mm in size, were used for measurements. Dose was read using a Harshaw mode12000-A TL detector and a Harshaw model 2000~B

automatic integrated picoammeter. The heating cycle of the crystals involvcd annealing for 20 minutes in a Thermolyne 200 furnace (Sybron corporation, Dubuque, Iowa), a preheat period of 5 seconds at 80 oC in the TLD reader fol­

lowed by readout of the TL light output by a photomultiplier tube using il

constant heating rate of 2.7 °C/s to a maximum temperature of 290 oC. The

dose information was obtained from the integration of the glow curve in the temperature interval between 90 and 270 oc. Each TLD-BOO chip was calibrated

monthly to determine the light output of the crystal from a known absorbed dose of radiation. The calibration was performed in the Department of Radiation Oncology at the Montreal General Hospital using a Theratron 780

Co-60 isocentric teletherapy unit (AECL, Ottawa, Ontario) with a dose rate of

80.2 cGy /min (on December l st, 1990) at depth of dose maximum in il

polystyrene phantom for a 10 cm x la cm field al Ba cm SSD. The TL sensitiv­ity calibration factor, C, for each particular TLD is given by

C= TL (Dw}60co'

(4.16)

where TL is the measured light output from the crystal which has received an absorbed dose «Dw)ecco' in Gy in water-equivalent material) of Co-60 radiation

The calibration factor for each chip was averaged over three irradiations. No

corrections were made for the variations of sensitivity in TLD-BOO at low x-ray

energies. The absolute absorbed doses measured by the chips in the determi­

nation of percent depth doses for diagnostic radiology were therefore obtained

by dividing their light output by their respective calibration factor C. Special

precautions for the handIing of lithium tetraborate crystal incIuded storage in

a Drierite filled de!"iccator (W. A. Hammond Drierite Co., Xenia, Ohio) and

shielding from ambient light using aluminum plates after the chips were ir~

radiated. Light in the x-ray room was reduced to a minimum during irradia­

tion in order to prevent the optical fading and spurious signaIs.

The depth doses were measured simultaneously at 10 depths with the

TLD-BOO crystals inserted in a single polystyrene sheet placed vertically be~

tween others to form a 30.5 cm x 30.5 cm x 20 cm phan tom. A 2 % standard deviation for the percent depth dose at ail depths was established from re­peated experiments. These variations are induced by variations in the tube output and by errors in the reading of the TLDs. The number of exposures at 200 mAs for the various radiographie techniques is such that surface doses of - 40 cGy were obtained for the aluminum standard techniques, i. e. - 35 R to­tal exposure (typically 20 to 50 irradiations, depending on the kV p; a complete depth dose curve was obtained within 45 minutes). The depth dose mea­surements involving the other fiIter materials were carried out at the same mAs and with the sa me number of exposures as the standard techniques. Doses were subsequently normalized to 100 % for the surface dose, and nor­malization to 100 % at the surface dose obtained with the aluminum added filtration was used to compare the depth doses relative to the standard tech­nique. In this work, the terms "relative depth dose" and "percent depth dose" are used interchangeably, even if the rigorous definition of the percent depth dose implies that aU doses are normalized to the maximal dose, which may not coïncide with the surface dose (a discussion of dose buildup in diagnostic radiology will be presented later). Two distributions of nos in polystyrene were used for the measurements. Since the density of the TL crystals is not identical to that of polystyrene, ca!"e has been taken to limit interference effects from the overlying dosimeters by ensuring that none of the TLO positions were directly below any other.

Percent depth doses were measured and calculated for a large number of peak kilovoltages between 45 and 145 kVp (nominal). Although a 3.5 %

ripple was established from observation of the waveform, the effective peak kilovoltage (effective kVp) of the waveform of the x-ray generator was consis­tentlyestablished using a NERO (Victoreen, Cleveland, Ohio) digital quality

assurance instrument. This quantity, equivalent to the kVp indicated by the modified Ardran-Crooks cassette (Victoreen, 1981; Jacobson et al., 1976), is closer to the average value of the whole waveform. This value determines the kVp of an equivalent constant potential spectrum generated by the Spectrum "1.1 programme. Transmission curves are calculated from computer spectra using Eqns. (4.4) and (4.5). The matching of the calculated transmis­sion curves to measured ones is done with a chi-square reduction test. The

59

agreement between some calculated and measured transmission curves is

shown in Fig. 4.1.

Preliminary depth dose measurements were performed with a Farmer­

type cylindrical ion chamber (6 mm diameter, 0.6 cm3 sensitive volume and

0.3 mm thick carbon wall), a fiat Markus ion chamber (6 mm diamelcr,

0.055 cm3 sensitive volume, and 2 mm height> and nD-IOO rods. Figure 44

shows the agreement obtained for percent depth doses measured with thesc

dosimeters with doses measured with TLD-SOO. The Farmer chamber mea­

surements involved 5 depths in phantom for a 62 kV p beam filtered with

1.5 mm Al and a 68 kVp beam filtered with 0.27 mm Zn while the Markus

chamber measurements were performed for 7 depths for a 45 kV p beam fil­

tered with 1.5 mm Al. With ionization chambers, a single percent depth dose

curve was obtained within three hours. This is approximately 5 limes the

measurement time involved with TLDs. Unfortunately, the ion chamber

measurements were limited to depths less than 10 cm in phantom because

the measurements performed below 10 cm were difficult to reproduce.

TLD-IOO crystals were also used for preliminary depth dose measure­

ments in phan tom for a 45 kV p beam filtered with 1.5 mm Al. The light out­

put obtained with the usual 550 (92.5 cm) were very small after 50 exposures,

and the measurements were repeated for a 45 cm SSD in order to reduce the

number of exposures n('eded to obtain high enough doses (20 exposures).

Maynoerd factors were used to obtain the percent depth dose for the standard

geometry. These SSD corrections necessitate the knowledge of backscatter fac­

tors (BSF), whose dependance on the x-ray spectral shape has been demon­

strated (Harrison, 1982). Unfortunately, BSF can only be roughly approxi­

mated when anyof the dosimeters mentioned in this section are used. It is

therefore preferable to use the correct 550 for the measurements. This is an­

other reason for using TLD-800 rather than TLD-IOO as the prime dosimeter in

this work.

The geometry described in Fig. 4.5 was maintained for both the mea·

surements and calculations. The 13.7 x 21.0 cm2 field size was chosen because

it described the largest irradiated volume which could be totally enclosed by

the phantom: maximal scatter was ensured.

60

120

100

: 0

80 ~

.c: -Q. • 60 ~ -c • 40 0 .. • Q.

20

a

120

100

: 0

80 'a

.c: -ca. • 60 'a -C • 40 u .. • IL

20

0

0

a

45 kVp, 1.5 mm AI added filtration

2

o Martus chambar ___ TLD-BOO

)( llO-100

4 6 Depth (cm)

8

62 kVp, 1.5 mm AI added filtration

2

o Farmer cham bar --- llO-aOO

o

4 6 Depth (cm)

a

10

10

Figure 4.4: Comparison of percent depth doses measured with ion chambers and TLD phosphors.

6 1

"

92.5 cm

123.0 cm

X·ray tube

Field size at the surface 1$

13.0cmX210cm

Figure 4.5: Geometry used for depth dose measurements in polystyrene. The field size chosen is the largest that could he totall y en­closed in the phan tom.

Six filter combinations, including the 1.5 mm Al standard filtration, were used to modify the incident x-ray spectra. The Niobix filter (0.05 mm Nb foil from North American Health Care Products Inc., Oakville, Ontario) and the R-filter (RSP Inc., San Carlos, California) were obtained commercially

while high purity holmium (55.56 keV K-edge), gadolinium (50.24 keV K­

edge) and ytterbium (61.33 keV K-edge) foils were obtained from chemical suppliers (Atomergic Chemetals Corporation, Plainview, New York). The thicknesses of the holmium, gadolinium and ytterbium foils were establishcd by dividing their masses by the product of the density of the material wi th the area of the foi!. These thicknesses were confirmed by direct measurement of the filter thicknesses with a micrometer. The thicknesses thus obtained were directly used in the Spectrum 1.1 and depthDose programmes. The filter combinations used throughout this work are listed in Table 4.1. Table 4.2 lists all of the kVp/filter combinations used. The kVp/filter combinations used for the standard, Niobix, R-filter and holmium mters are the same as those used in the work of Bissonnette, Blais, Schreiner and Podgorsak (I990) and Schreiner et al. (1991). These techniques have been established to give equiva­

lent image contrast. Percent depth dose were calculated and measured for ail

62

{ Desi nation Actual filler materials

Inherent filtration glass housing equivalent to 1.31 mm AI

Standard AI 1.5 mm AI + Inherent filtration

Niobix Niobix filter (0.05 mm Nb) + Standard AI

R-filter R-fUter (0.27 mm Zn) + Inherent filtration

Ho 0.112 mm Ho + Inherent filtration

Gd 0.116 mm Gd + Inherent filtration

Yb 0.101 mm Yb + Inherent filtration

Table 4.1: X-ray beam filtration used in dose measurements and calculations.

of the radiographie techniques listed in Table 4.2, and integral doses were ob­tained from these percent depth dose data. The calculation of the x-ray inten­sities involved with each of these techniques was made so that ail of the pa­rameters pertinent to each technique (i. e. mAs, kV p' target angle ... ) were re­flected in the calculation; the relative intensities of each techniques were ver­ified by comparison of calculated and measured exposures relative to the standard aluminum technique which gave an equivalent radiographie image. The dose calculations performed were expressed as quantities relative to the surface doses of the associated technique using aluminum filtration in order to directly assess the dose savings of a given technique with respect to these standard aluminum techniques.

The effect of the field size on the percent depth dose curves was studied for a nominal 80 kV p x-ray beam (effective kV p is 79) filtered with the stan­dard aluminum filtration (1.5 mm Al added to the 1.31 mm Al inherent fil­tration). 7 x 7 cm2, 10 x 10 cm2 and 15 x 15 cm2 fields were selected. The

model was also used for comparison with published percent depth doses in water from a single phase, full wave rectified x-ray generator (Harrison, 1981).

Off-axis depth doses were also measured for the 80 kVp beam, using a 13.7 x 21 cm2 field. Percent depth doses were measured at four off-axis posi­tions while the SSD and the field size remained constant; the surface crystal was moved in steps of 2 cm away from the central axis along the anode-cath­ode direction in order to determine if the fiat beam approximation is accept­able. The measured and ca1culated depth dose curves are presented and dis-

63

cussed in Chapter V, and the percent depth dose data are shown in Appendix C for all of the radiographie techniques listed in Table 4.2. The integration of measured and calculated percent depth doses over the irradiated volume yields the integral dose (Eqn. (4.15». The comparison of integral doses ob­tained from rneasured and calculated depth doses is reported in Chapter V.

Added fil kVp HVL t HVLt Addcd fil kVp HVL 1

HVLt HVL2 HVL2

tration (nommaI) (mm Al) tration (nominal) (mm AI)

Standard 48 (45) 1.90 0.91 R-filter 68 (bS) 4.66 085

AI 57 (55) 2.07 0.73 85 (86) 5.92 0.83

62 (60) 2.22 0.69 121 (126) 8.02 n.83

70 (70) 2.50 0.67

74 (75) 2.67 0.66 Ho 66 (66) 3.74 0.82

79 (80) 2.90 0.66 93 (92) 4.94 0.79

97(00) 3.80 0.66 134 (145) 7.02 0.76

113 (20) 4.58 0.66

Gd 65 (65) 3.24 080

Niobix 66 (66) 3.66 0.78 89 (87) 4.27 0.77

85 (86) 4.96 0.77

105 (110) 6.18 0.77 Yb 62 (62) 3.32 0.79

120 (125) 6.99 0.78 79 (80) 4.23 0.78

113(20) 5.93 0.77

Table 4.2: Combinations of peak kilovoltages and filtrations for which depth doses in polystyrene were rneasured. The half-value layer, the homogeneity coefficient and the effective kV p define the beam quality of each radiographie technique. These techniques are identical to those in Bissonnette et al. (1990) and in Schreiner et al. (1991).

cussed in Chapter V, and the percent depth dose data are shown in Appendix C for aIl of the radiographie techniques listed in Table 4.2. The integration of measured and calculated percent depth doses over the irradiated volume yields the integral dose (Eqn. (4.15». The comparison of integral doses ob­tained from measured and calculated depth doses is reported in Chapter V.

Added fil kVp HVL) HVL) Added fil kVp HVL) HVL)

HVL2 HVL2 tration (nominal) (mm AD tration (nominal) (mm Al)

Standard 48 (45) 1.90 0.91 R-fiiter 68 (68) 4.66 0.85

AI 57 (55) 2.07 0.73 85 (86) 5.92 0.83

62 (60) 2.22 0.69 121 (126) 8.02 0.83

70 (70) 2.50 0.67

74 (75) 2.67 0.66 Ho 66 (66) 3.74 0.82

79 (BO) 2.90 0.66 93 (92) 4.94 0.79

97 ClOO) 3.80 0.66 134 (145) 7.02 0.76

113 (120) 4.58 0.66

Cd 65 (65) 3.24 0.80

Niobix 66 (66) 3.66 0.78 89 (87) 4.27 0.77

85 (86) 4.96 0.77

105 (110) 6.18 0.77 Yb 62 (62) 3.32 0.79

120 (125) 6.99 0.78 79 (SO) 4.23 0.78

113 (120) 5.93 0.77

Table 4.2: Combinations of peak kilovoltages and filtrations for whieh depth doses in polystyrene were measured. The half-value layer, the homogeneity coefficient and the effective kVp define the beam quality of each radiographie technique. These techniques are identical to those in Bissonnette et al. (1990) and in Schreiner et al. (1991).

64

CHAPTER V

RESUL TS AND DISCUSSION

In this chapter, a comparison of the spectral models presented in Chapter li (Kramers, 1923; Birch and Marshall, 1979; Tucker et aL, 1991a) is

performed. The percent depth doses obtained from these calculated spectra is compared with published (Harrison, 1981) and measured data, and the behav­

ior of percent depth dose curves when a number of parameters are varied is studied. A number of graphs will be shown to demonstrate the agreement he­

tween calculated and measured data. AlI measured and calculated percent

depth doses can he found in Appendices C and D. Finally, predicted integral dose ratios are compared with ratios obtained from measured depth dose data.

5.1 Comparison of the spectral models

The Birch and Marshall model (1979) was chosen to calcula te the x-ray spectra (which were subsequently used for depth dose ca1culations) because these spectra agree weIl with measurement and the model has been used ex­

tensively and successfully for both direct photon transport calcula tions and Monte Carlo simulations applied to diagnostic radiology (Harrison, 1981;

Shrimpton, Jones, and Wall, 1981; Harrison, 1982; Johns and Yaffe, 1983; Boone, 1986; Koedooder and Venema, 1986; Persliden, 1986; Boone, 1988; Nagel, 1989; Schreiner et aL, 1991). The models proposed by Kramers (1923)

and Tucker et al. (1991a) have, however, the advantage of being derived from

correct, albeit simplified, quantum mechanical considerations. As shown ear­lier, the Kramers model (1923), which is the easiest to implement, contains a

number of simplifying assumptions that are unsuited to diagnostic radiology

applications and it should be discarded for such use. The spectral models

proposed by Birch and Marshall (1979) and by Tucker et al. (1991a), hereafter referred to as the TBC model, have been supported by experimental evid:;mce.

Furthermore, the TOC model (1991 a) can he used to obtain off-axis spectra.

l The theoretical expressions from the 8irch and Marshall (1979) and the TBC (1991a) models for the bremsstrahlung energy spectrum (Eqns. (2.23) and

(2.30), respectively) can be compared by collecting terms common ta both ex­

pressions. The resulting expressions of these equations are:

Nv = k..fTV P(~) (1+ --L)(l dLrl exp (-l4hV) (T6 -T2) cot 8) dT, (5.1) E T moc2 p dx pC

lb

for the Birch and Marshall (1979) model, and

N v = k' llb

Q(T + moc2) (.1 dL)-1 exp [-l4hV) (T~ - T2 )] dT, (5.2)

E T P dx pC sin (8 + lP) Aov

for the TBC (1991 é:\) model.

The relativistic term proposed by 8irch and Marshall Œqn. (5.1»,

(I+T/moc2)/T, differs from the correct formulation from Tucker et al. (Eqn.

(5.2», (T+moc2)/T. These terms however differ only by a factor of moc2• Since

this factor is constant, it is expected that it will he of no importance for the de­

termination of relative quantities, as is the case in the present work.

Comparison of the exponential terms reveals that, if central axis

(li' = 0) spectra from small target angles are considered, these terms are nearly

identical since tan 8 ... sin 9. Furthermore, a simple geometrical correction

could he made to Eqn. (2.22) to calculate off-axis spectra in a manner similar to

the TBC model.

P(~) and Q are the fits ta the differential energy spectrum proposed for

each the Birch and Marshall (1979) and the TBC (1991a) spectral models, re­

spectively. The fit used by Birch and Marshall (1979) was normalized to re­

produce measured spectra for a definite geometry. Q was obtained by Tucker

et al. (1991a) by fitting their model to previously published spectra (Fe weIl et al., 1981). Since these function ultimately define the shape of the brems­

strahlung continuum, the ratio P(ç)/ Q indicates how the general spectral

66

.. shape differs from model to model. Figure 5.1 shows the P(~/ Q ratio for pho­tons produced by 50, 100 and 150 keV electrons. It can he seen from the figure that the TBC model generates more lower energy photons for a given kinetic electron energy (resulting in a softer spectrum) than the Birch and Marshall model. This observation is confirmed by the spectra (see Fig. 5.2).

Figure 5.2 shows central axis bremsstrahlung spectra (Le. x-ray spectra without characteristic radiation) ca1culated by the three models described in Chapter Il for the sa me radiographie technique (note that the target angle is unaccounted for in the I<ramers mode!). Tl,e transmission curves derived from these spectra, shown in Fig. 5.3, also indicate an appreciable variation be­tween the output spectra obtained from the three models with identical input parameters. For the technique shown in Fig. 5.2 (100 kVpl 10° target, 1.2 mm Al inherent filtration and 1.5 mm Al added filtration), the Kramers spectrum describes a less penetrating bearn than predicted by the other models. These particular bremsstrahlung spectra (i. e. x-ray spectra without the characteristic

'Ci ~ ! eu =ca - .. ~-u --0G. o· ->0 !f!t ~. eue .!:!-iD E .. 1

1.4

1.2

1.0

0.8

0.6

0.4

0.2 o

r· ,/'+ ..

,J)-/ \, . ~ . :' . " • • . : . .: • • . :

a .. : ~ .. . .. a : .. ' .: •.

20 40

•

150 keV electrons 100 keV electrons 50 keV electrons

60 80 100 1~0 140 160

Photon energy (keV)

Figure 5.1: Ratio of the differential energy spectra suggested by Birch and Marshall (1979) (P(Ç» to that proposed by Tucker et al. (1991a) (Q> for sorne electron kinetic energies; the data are normal­ized arbitrarily to 1 at the photon energy equal to half of the electron kinetic energy.

67

..

-(1) -'ë ~

>-~ ... ~ -e ... --~ A -~ 0

c 0 -0 ~ G.

1 0

0.8

0.6

0.4

0.2

0.0 0

Tucker,t al. -- Birch & Marshall

• Kramers

20 40 60 80 100 120 Photon energy (keV)

Figure 5.2: Comparison of 100 kV p bremsstrahlung spectra calculated from the Kramers (1923), the 8irch and Marshall (1979), and the Tucker et al. (1991a) models for a 10° tungsten target and with 1.2 mm Al inherent and 1.5 mm Al added filtration. AU spec­tra were normalized to equal number of photons, as in the convention of Tucker et al (1991a). The characteristic photons are not shown so that direct comparison with T~lcker et al. (J991a) is possible.

photons) were chosen in order to reproduce the comparison of the three spectral models performed in Tucker et al. (1991a). It should be noted that the difference between the 8irch and Marshall spectrum and the spectrum calcu­lated with the TBC model is less than that indicated by Tucker et al. (1991a). This may result from the difficulty Tucker et al. (1991a) had in implementing the Birch and Marshall model (1979). Whereas Tucker et al. (1991a) were un­

able to duplicate the spectra published in the Scientific Report 30 of the Hospital Physicists' Association (Birch et aL, 1979), our implementation of the 8irch and Marshall model (1979) reproduced these data very weIl.

Of more interest dre the EQSPEC's obtained with the three models which yield identical transmission curves. Since Figs. 5.2 and 5.3 have indi­cated that the spectra generated by the three implemented models for similar

68

1 1 .1

.,

102

-~ -c 101 0

;: u CIl .. -ou .. .. :: E 1 fit C ! ...

0.1 o

Birch & Marshall Tuckeret al. Kramers

10 20 30 40 50 60 70 Alumlnum thlckness (mm)

• • •

80

:Figure 5.3: Transmission curves ca1culated from the spectra described in Fig. 5.2.

conditions yielded different transmission data, identical transmission curves can only be obtained by modifying one or more of the three EQSPEC parame­ters (i. e. the kV p, the target angle, and the inherent filtration). An arbitrary reference transmission curve for a JOO kVp beam from a 10° tungsten anode

with 1.2 mm Al inherent and 1.5 mm Al added filtration was obtained using a bremsstrahlung spectrum ca1culated from the Birch and Marshall model (1979). This particular bremsstrahl ung spectrum is the reference spectrum and determines the reference transmission curvp. A number of transmission

curves were obtained from bremsstrahlung spectra ca1culated with the other two spectral models, and they were fitted to the reference transmission curve

similarly to the method proposed by Boone (1986) to establish spcctra equiva­lent to the reference spectrum. The results of this fit indicate that the bremsstrahlung EQSPEC obtained with the Birch and Man,hall and TBC models éll.'e essentially identical although the EQSPEC parameters are varied. In the present case (t00 kVp and 10° target), a shift in the inherent filtration was made to fit TBC and Kramers bremsstrahlung EQSPEC to the reference spectrum (+ 0.5 and + 2 mm Al, respectively). Other shifts in the EQSPEC pa­

rameters are listed in Tables 5.1 and 5.2. The transmission data calc.ulated

69

r

from spectra obtained with the Kramers (1923) and the TEe (I991a) spectral

models which produced the best fit to the transmission curve obtained from the reference spectrum are depicted in Fig. 5.4 with the EQSPEC associa ted

with these transmission curves. Il can readily be seen that, when the match between transmission curves is good, the corresponding EQSPEC calculated from either the Birch and Marshall (1979) or the TBC (1991a) model also match. The selection of the spectral model has therefore no consequence on the final EQSPEC result except for spectra from the Kramers model, which has

already been proved inadequate for diagnostic radiology purposes.

kVp Tucker et. al. Kramers brcms. + char. brcms.only brems. + char. brems.only

30 - 1.25 (0.0002) - 2.0 (0.080) 60 - 1.4 (0.0039) - 2.3 (0.87) 90 1.8 (0.084) 1.7 (0.041) 2.80.66) 2.6 (1.30>

120 2.2 (0.22) 2.3 (0.26) 3.3 (0.93) 3.0 (1.16)

Table 5.1: Inherent filtration (mm Al) needed to match EQSPEC from Tucker et al. (1991a) and Kramers (1923) with kVp and target angle (10°) identica' to a reference spectrum from Birch and Marshall with 1.2 mm Al inherent filtration. The goodness of fit parameter is indi-cated in the parenthesis <X2). Fits were performed for reference spectra with (brems. + char.) and without <brems. only) characteris­tic radiation.

(a) kVD (h) target angle kVp Tucker et al. Krarners Tucker et al.

30 30 (0.015) 34 (0.17) 8.5 (0.0005)

60 62 (0.090) 72 (0.73) 7.5 (0.0089) 90 98 (0.46) 113 (t.63) 6.0 (0.054)

120 138 (0.33) - 5.5 (0.13)

Table 5.2: (a) kVp correction needed to match spectra from Tuc1cer et al. and Kramers with inherent filtration and target angle identical to a reference spectrum from Birch and Marshall with a 10° target and 1.4 mm Al inherent filtration. (b) Target angle correction needed to match spectra from TBC with inherent filtration (1.4 mm Al) and kVp identical to the reference spectrum from a 10° target. The goodness of fit parameter <X2) ls in parenthesis.

70

-fi» = C = ~ I! -:a .. ca --= Q. -= o c: o ï CL

~ -c ..2 -u 1! -

1.2

1 0

08

0.6

0.4

0.2

0.0

100.0

100

1.0

0.1

o 20

.r'\. • • ". " "

-- Birch and Marshall Tucker et al. Kramers

•

40

• • • • "

60 80 100

Photon energy (kev)

-- Birch and Marshall Tucker et al.

• Kramers

120

•

o 10 20 30 40 50 60 70 80

Alumlnum thlckness (mm)

Figure 5.4: Best fits to a transmission curve obtained from a 100 kVp spectrum from a 10° target with 1.2 mm Al inherent and 1.5 mm Al added filtration generated with the Birch and Marshall model (1979). Transmission curves obtained from spectra generated from the Tucker et al. (1991a) and the Kramers (1923) models where only the inherent filtration was varied were fitted to the initial trans­mission curve. The EQSPEC generating the transmission curves are also shown, normalized to the same area.

7 1

The model proposed by Tucker et al. (1991a) is undoubtedly the most useful model for diagnostic radiology. Its theoretical grounds are solid, and it agrees with published spectral data. Nevertheless, results obtained previously with the 8irch and Marshall model (1979) should not be disregarded. As pointed out above, the origin of a calculated x-ray spectrum is unimportant if the derived transmission curve fits measured transmission data. Figure 5.4 clearly shows that, for matched transmission curves, the bremsstrahlung spectra calculated from the TBC model (1991a) is nearly indistinguishable from the Birch and Marshall (1979) spectrum. Furthermore, the ca1culation of off-axis spectra by both models can be used for a wider range of dose and image quality compLter experiments. Perhaps the TBC model (1991a) should be selected for further work because of its inherent qualities, but the matching of ca1culated to measured transmission data remains the selection criterion for calculated spectra.

5.2 Percent depth doses

It is customary to ignore scattered radiation when integral doses are cal­culated by direct photon transport (Harrison, 1981; Koedooder and Venema, 1986; Nagel, 1989). Unfortunately, the depth doses ca1culated for primary ra­diation significantly differ from ca1culations involving first scatter doses and

from measurement. Table 5.3 shows the percentage of calculated depth doses due to priI.,ary, first scatter coherent and first scatter incoherent radiation for

three radiographie techniques. As expected from calculated scatter cross sec­tions, the contributions to the total dose from coherently scattered photons decreases with increasing energy of the primary x-ray beam. This is because the proportion of low energy photons of the beam decreases with increasing kVp. Conversely, the dose contributions from incoherently scattered photons depends upon tht= spectrum of the selected radiographic technique since the Compton cross section reaches a maximUIll at photon energies between 35 and 45 keV for low Z materials (e. g. water, lucite, polystyrene). As seen in Table 5.3, increasing the energy of the primary x-ray beam from 30 kVp ta

68 kV p increases incoherent scalter doses while further increases in energy ta 113 kVp lead ta a reduction of the scalter dose. Table 5.4 displays the contribu­tion to scatter dose from air '3catter and phantom scatter, and also shows what

72

depth 30 kVp 0.5 mm Al> 68 kVp (0.27 mm Zn) 113 kVp (1.5 mm AI)

(cm) Plmary coherent tncoh Dnmarv coherent inrnh Dnmarv coherent mcoh 0.0 83.76 0.19 16.04 78.85 0.08 21.07 80.39 007 19.54 0.5 79.05 1.48 19.45 64.43 0.57 35.01 76.16 0.50 23.34 1.0 75.44 2.52 22.05 61.27 0.93 37.80 72.80 0.85 26.35 2.4 68.69 4.60 26.72 54.71 1.71 43.58 66.26 1.58 32.17 4.0 64.15 6.15 29.70 49.63 2.36 48.02 61.67 2.12 36.21 5.0 62.20 6.87 30.94 47.29 2.69 50.02 59.64 237 37.99 7.0 59.38 7.97 32.65 44.07 3.28 52.65 56.70 2.72 40.58 8.0 58.33 8.41 33.25 43.05 3.56 53.39 55.58 286 41.56 10.0 56.69 9.15 34.15 41.93 4.10 53.97 53.83 3.06 43.11 12.5 55.26 9.88 34.87 41.67 4.71 53.62 52.24 3.25 44.52 15.0 54.28 10.44 35.28 42.06 5.20 52.74 51.10 3.39 45.50 17.5 53.62 10.88 35.50 42.67 5.57 51.76 50.28 3.51 46.21 20.0 53.20 11.23 35.57 43.28 5.85 50.87 49.68 3.61 46.71 22.5 52.94 Il.52 35.54 43.84 6.05 50.11 49.24 3.70 47.()6 25.0 52.81 II.76 35.43 44.32 6.20 49.47 48.92 3.78 47.31 27.2 52.78 11.94 35.28 44.69 6.31 48.99 48.71 3.84 4745

Table 5.3: Relative importance of ealculated primary and sealter contributIOns to the dcpth doses for some radiographie techmques, dcfined by a kVp and addcd filtra t,on (an parenthesls) eombmation. The inherent filtration IS 1.31 mm AI and the tar­g~t angle is 120 in aU cases.

depth 30 kVp 0.5 mm AI) 68 kVp (0.27 mm Zn) 113 kVp (1.5 mrn AI) (cm) air phanlom back-

alf phantom tŒk-311' phantom back-

scatter scalter scallcr 0.0 1.77 98.23 100.00 1.06 98.94 100.00 0.93 99.07 100 (}() 0.5 1.20 98.79 55.86 0.40 99.60 42.57 0.67 99.34 69.12 1.0 0.91 99.09 41.38 0.33 99.67 35.04 0.53 99.47 54.85 2.4 0.55 99.47 27.20 0.21 99.79 24.99 0.32 99.68 3792 4.0 0.39 99.61 21.56 0.15 99.85 19.81 0.22 99.78 3028 5.0 0.33 99.68 19.64 0.12 99.88 17.83 0.18 99.82 27.53 7.0 0.25 99.75 17.22 0.09 99.91 15.32 0.14 99.87 24.01 8.0 0.22 99.78 16.39 0.08 99.92 14.52 0.12 99.88 22.79 10.0 0.18 99.82 15.16 0.07 99.93 13.49 0.09 99.91 20.94 12.5 0.15 99.85 14.08 0.06 99.94 12.86 0.07 99.93 19.33 15.0 0.12 99.88 13.28 0.05 99.95 12.58 0.06 9994 18 16 17.5 0.11 99.89 12.67 0.04 99.96 12.44 0.05 99.95 17.27 20.0 0.09 99.91 12.16 0.04 99.96 12.34 0.04 99.96 16.56 22.5 0.08 99.92 11.73 0.04 99.96 12.24 0.04 99.96 15.93 25.0 0.07 99.92 11.36 0.04 . ''.96 12.14 0.04 9996 15.48 27.2 0.07 99.93 11.07 0.03 99.97 12.05 0.03 99.97 15.10

'==z

Table 5.4: The contribution to scalter dose from aIr scatter and phantom scatter for sorne radIO­graphie techniques defined by a kVp and added filtratIon (in parenthesl!,) combl­nation (the inherent filtration IS 1.31 mm AI and the target angle is 12° ln ail cases). Dose contributions from radiation scattered in the added x-ray fIltratIOn 15

essentially non-existent. The fraction of the scalter dose (rom phantom scattcr which com~ from backscattered radiation is also mcluded.

73

fraction of the phantom scatter originates from backscattered radiation. In aH of the radiographie techniques studied, no significant contributions to the seatter dose from the added x-ray filtration were found. Also, in aU cases, the ealculated dose contribution from radiation scattered in air amounts to at most a couple of percent of the seatter dose, and thus to less than a percent of the total dose. Therefore, ealculations involving radiation scattered in air and in the x-ray filtration eould he neglected for they have Uttle effect on the over­aU dose (thereby allowing a-50 % reduction in the computing time). The programme was changed aecordingly.

Calculated and measured depth doses that were obtained for a 68 kV p

beam filtered with the R-filter and for a 113 kVp bearn used with the standard aluminum filter are shown in Fig. 5.5. Fig. 5.5 shows how agreement be­tween measured and ca1culated doses improves if first scatter is included in the ealculation. One area of improved agreement is in the buildup region of our 68 kVp beam used with the R-filter. Depth dose curves measured for low energy r~diation are usually considered to show no buildup since the range of the eleetrons set in motion by diagnostie energy photons is very small. Therefore, the depth dose is expeeted to decline with increasing depth as the primary photon beam is attenuated. The observed buildup doOes not result from the establishment of eharged partide equilibrium but from scattered ra­diation. Our caIculations show that, for sorne radiographie te.:hniques, the inerease in scatter in the phantom more than compensates for the attenua­tion of the primary beam, leading to a dose buildup below the surface (see Fig. 5.5). Therefore, buildup eurves similar to those found in radiation therapy can occur in diagnostic radiology. As seen in Table 5.3, such is thE' case for the 68 kVp beam since there is a large fraction of the photons whose energies eo­incide with the peak in the Compton cross section for polystyr,ene. Depth dose curves with such buildups have been reported previously for radio­graphie techniques using high kVp (Yamaguchi et al., 1983) and f()r orthovolt­age therapy units operating from 80 to 250 kV p, where the depth of dose max­imum is very shallow (Podgorsak et al., 1990). Harrison (1981) did not mea­sure depth dose at depths between the surface and 1 cm and so buildup was not observed. However, our results remain preliminary. More work is re­quired to explain this interesting observation.

74

68 kVp; A-tilter 120

)( Primary 100 -- Primary + firsl scalter • .,

0 Measured 0

" 80 z:. .. a. • 60 " )( .. c • 40 x u .. CIl D.

20

0 0 5 10 15 20 25 30

Depth (cm)

113 kVp; Standard AI filter 120

x Primary 100 -- Primary + first scatter

• [J Measured ., 0

" 80 z:. .. a. • 60 " x .. C • 40 u .. • D.

20

0 0 5 10 15 ::0 25 30

Depth (cm)

Figure 5.5: Comparison of ca1culated and measured depth doses for a 68 kV p beam filtered with the R-filter and for a 113 kVp beam with stan­dard aluminum filtration (the inherent filtration is 1.31 mm Al in both cases). Doses are calculated for primary and for primary and first scattered radiation.

75

( The results of the depth dose model will now he compared with pub­Iished and measured data. These data were taken for a wide selection of ra­diographie techniques involving various filtration schemes, field sizes and k V p'S in order to reproduce most of the conditions used in diagnostic radiol­ogy. AIso, off-axis depth dose measurements are presented.

5.2.1 Comparison with published data

Two of the more commonly cited compilations of percent depth doses for use in diagnostic radiology are Harrison (1981) and the British Journal of Radiology Supplement 17 (1983). Central axis depth doses calculated by the present model are now compared to these data.

To best reproduce published depth dose data with the programme depthDose, it is necessary to know the transmission data from the x-ray beams used in those measurements. The equivalent constant potential x-ray spec­trum ŒQSPEC) can then be determined, and dose calculations performed. Unfortunately, both of the above documents specify their beams by the half­value layer alone (i. e. only one point in the transmission curve; BJR sup­plement 17, 1983) or bya'1 half-value layer/homogeneity coefficient combina­tion (i. e. two points in the transmission curve; Harrison, 1981). As discussed in Chapter II, these parameters do not completely specify the x-ray beam (Hale, 1966).

In Harrison (1981), percent depth doses were measured from a full­wave rectified unit for a range of kV p'S, filters, and field sizes. The target an­gle, kVp and inherent and added filtration were specified for the radiographie techniques used. Since the kV p'S mentioned in Harrison (1981) were verified by direct measurement, the EQSPEC for these techniques were obtained by variation of only the inherent filtration until the stated haH-value layer /homogeneity coefficient combination was reproduced; the target angles were fixed as per Harrison (1981). The matching of calculated transmission curves to the beams spedfied by Harrison (1981) was difficult. Only an ap­proximate match of the calculated half-value layers and homogeneity coeffi­cients was obtained in most cases. The half-value layers were easy enough to

7 é'

.. reproduce (1.0, 2.0, 3.0, and 4.0 mm Al HVL). The homogeneity coefficients obtained with these half-value layers were however more difficult to match. Variations from the homogeneity coefficients quoted (between 0.54 and 0.75; Harrison, 1981) were typically in the 15 to 20 % range. Percent depth doses were calculated from these approximate EQSPEC's for a number of techniques with half-value layers between 1.0 and 4.0 mm Al (for example, see Fig. 5.6).

Since the EQSPECs could not be precisely determined, sorne discrepancies be­tween the current calculated and published depth doses were to be expected. Figure 5.6a depicts calculated and published depth doses for 60 kV p beams fil­

tered with different thicknesses of aluminum. The agreement between calcu­lated and measured depth doses is quite good (less than 5 % for ail depths),

espedally considering that the estimated standard deviations quoted by Harrison for his measurements lie between ± 3 and ± 6 % at all depths

(Harrison, 1981). Figure S.6b shows a departure of the calculated percent depth doses from published ones at large depths. This behavior is more ('vi­dent for the 100 kV p (4.0 mm Al HVL) data, where the variation between cal­culated (dotted Une) and measured data is of the order of 10 to 20 %. As noted

below, it is expected that second and perhaps third order scattered radiation contributions to the dose will become more important as the HVL of the tech­

nique increases.

It would have been of interest to reproduce the depth doses from

Yamaguchi et al. (1983), where dose buildup in diagnostic radiology is re­

ported. Unfol'tunately, only the kVp of the studied beams were indicated.

This information is insufficient to reproduce the experiment with the com­

puter modeI.

77

,-

1 0

" 10 ~ -G-• " -c • CI .. • a.

0.1

1 0

10 " .c: -Q. • " -c • CI .. • a.

0.1

o

o

• 60 kVp, 1.0 mm AI HVL • 60 kVp, 2.0 mm AI HVL

5 10 D.pth (cm)

• 90 kVp, 2.0 mm AI HVL • 100 kVp, 4.0 mm AI HVL

5 10

D.pth (cm)

15 20

15 20

Figure 5.6: Percent depth doses measured and ca1culated for identical radio­graphie techniques. (a): 60 kVp doses from Harrison (1981) (data points) versus predicted doses (soUd Unes). (b): 90 kV p and 100 k V p doses from Harrison (1981) versus predicted doses (soUd Une for 90 kVp and dashed line for 100 kVp)' The error bars only serve as an indication of a 3 % variation from ea1cu­lated data.

78

l 5.2.2 Effect of the peak kilovoltage on depth dose

It was seen in Chapter II that the kV p de termines the maximum energy of the x-rays produced. Increasing the kVp will produce a more penetrating x-ray beam. As a result, larger percent depth doses are observed at each depth. The effect of the kV p on normalized depth doses calculated with the ray trac­ing algorithm is illustrated on Fig. 5.7. Predicted depth doses are compared

with measured data in Fig. 5.8. It is apparent from Fig. 5.8 that the model seems ta break down at high kVp. This is not unexpected because high energy x-rays interact with the phantom material almost exclusively through the Compton effect. Consequently, the second and higher arder scatter dose be­

come more important at high kVp. Monte Carlo simulations are currently being performed to determine the kV p limit of the model, which accounts for

primary and first scatter doses only.

100

-- 48 kVp

CD - - 74 kVp

fil - - - 113 kVp 0

" 10 oC -a. CD

" -c CD 1 e CD a..

0.1 o 5 10 15 20 25 30

Depth (cm)

Figure 5.7: Calculated depth doses for the 48, 74 and 113 kVp x-ray beams (nominal) with 1.31 mm Al inherent and 1.5 mm Al added filtration.

~

~~"L'':"l"

~ -fit i a 3 ~ li ëii

100 ~o q

10

1

0.1

a::: 0.01

'i -fi ~ oC

i 'a fit .2 1i ëii a:

o 5 10 15 20 25 30 Depth (cm)

100 h" 12

10

1

0.1 "" • 1

o 5 10 15 20 25 30

Depth (cm)

-# -CD

i oC

t ~ i i

'i -a v oC

i ~

~ li Qi

100 ~~~O~~~~~~~~TT~~~~

10

1

0.1

o 5 10 15 20 25 30 Depth (cm)

100 ""~L:TI ïTïrrrTT1rrr"Tïïrrr"TïïrrrT"rrT-r"T"1

10

1

a: 0.1 o 5 10 15 20 25 30

Depth(cm)

Figure 5.8: Variation of percent depth dose with depth for four diagnostic x-ray beams using the standard 1.5 mm Al added and 1.31 mm Al inherent filtration. Measurements for 45,60,80 and 120 kVp beams are shown as dots while the calculated doses are shown as solid lines.

~

00 o

~ .......,.:

5.2.3 Eifect of added filtration on dose

The primary effect of added filtration on an x-ray beam is the harden­ing ot the x-ray beam, achieved by preferentiai attenuation of the incident low

energy photons by the fil ter. In addition, the K absorption energy of certain materials is high enough to produce considerable attenuation of high energy

photons. The reduction of the number of incident photons by additional fil­

tration therefore results in smaller doses for a given tube output (mAs) and

kV p' It is however expected that the attenuation of the incident spectrum by the added filtration will affect the dose distribution itself.

Some of the depth dose measurements performed by Schreiner et al.

(1991) have been repeated in this work using the TLD technique described

above. Calculations involving the equivalent theoretical spectra were aiso

achieved. Sorne of the relative depth doses obtained with both methods are

compared in Fig. 5.9. Again, a ± 2 % error is observed in all measurements.

The data shown in Fig. 5.9 are not percent depth doses since aU doses are

normalized to 100 % at the surface dose obtained with the technique u.,ing standard aluminum added filtration. If these data were presented as percent

depth doses (i. e. normalized ta 100 % at the "surface dose"), the increase of

the percent depth dose at the same depths with filtration would be clearly

seen. The data listed in Appendix C are all normalized to 100 % at the surface.

5.2.4 Effect of the field size on dose

The contribution of scattered radiation t'J the dose at a point Q in phan­

tom is a complicated function of the beam geometry and of the composition

of the media the radiation travels through. When a pend} beam is used, the

absorbed dose at Q is entirely due to primary radiation since the volume that

can scatter radiation is negligibly small. In a diverge. ,t beam, the volume

from which scattered photons susceptible to reach the point of interest in­

creases with depth in phantom. The number of photons scattered increases as

the size of the radiation field increases. Also, the dose due to scattered radia­

tion increases more rapidly for a deep point in phantom than for a shallow

100

-!! 80

1 o " ~ i " • > i • a:

60

40

20

o

100

80

60

40

20

o

o 5

o 5

o 66 kVp: Niobix filter

• 68 kVp: R-finer

10 15 20 25

Oepth (cm)

o 85 kVp: Niobix filter

• 85 kVp: R-finer

10 15 20 25

Oepth (cm)

30

30

Figure 5.9: Measured and calculated (solid lines) depth doses established for low and mid-range tube voltages. The data are normal­ized to 100 % at the surface dose obtained with the technique using the standard Al added filtration.

S2

· t

•

one. It is then expected that the percent depth dose at ci single depth will in­crease with field size.

Using a 79 kVp (effective) x-ray beam with the standard Al added filtra­tion, percent depth doses were measured for four different field sizes, 7 x 7 cm2, 10 x 10 cm2, 15 x 15 cm2, 13.7 x 21 cm2, whose radius of their equiva­lent circular field were used in the depthDose programme. Additional per­cent depth doses were calculated for 0 x 0 cm2 (i. e. prin\ary beam only) and 20 x 20 cm2 fields to show the behavior of depth dose for smaller and larger fields.

Figure 5.10 shows measured and calculated percent depth doses ob­tained for two of the field sizes studied while Fig. 5.11 shows the variation of the percent depth dose at depths of 10 and 15 cm with field size. The mea­sured and calculated percent depth doses obtained for aU of the above fields are shown in Appendix D.

5.2.5 Off-axis doses

Depth doses calculated with the current model are limited to the cen­tral-axis. If beam profiles were fiat at aU depths, this would imply that cen­trctl-axis depth dosps would he applicable anywhere in the field. It is known that this is not the case, mainly because of the heel effect. As stated earlier, the heel effect refers to spectral and intensity gradients along the cathode-an­ode axis: the x-ray heam towards the cathode is a more intense and a softer beam than in the direction of the anode. The increase of intensity of the x-ray beam along the anode-cathode axis is principally due to the differential atten­uation of the x-rays escaping the target, but non-negligible contributions to the heel effect may also arise from the angular anisotropy of the production of brpmsstrahlung photons and from the inverse square effects (Meredith and Massey, 1972; Fritz and Livingston, 1982). It is therefore expected that varia­tions in relative depth doses at different points in the field will arise, and it is desirable to quantify those variations .

83

l

100

: o 'a 10 ~

A • 'a -c ë 1 .. A.

• .. 0 'a

oC -ca. 1 -c • u .. • A.

0.1

100

10

0.1

o 5

o 5

i 1 1 1 Iii i 1 1 1 lit 1 1 1 1 1

Radius of equivalent -circular field is 2.95 cm.

10 15 20 25 30

10

Depth (cm)

Radius of equivalent circular field is 9.065 cm.

15 20 25 Depth (cm)

30

Figure 5.10: Percent depth doses for an 80 kVp (nominal) x-ray beam filtered with 1.5 mm Al versus depth for 7 x 7 cm2 and 13.7 x 21 cm2

fields at the surface. Measured data ( • ) are compared with cal­culated doses, shown as solid Unes. The error bars are not shown in the bottom graph since they are of the same scale as the upper plot.

16

14

• • 0 12 ~

1: - 10 a. • ~ - 8 1: • ~ 6 • Do

4

2

0 2 4 6 8 10 12

Equlval.nt radlui Of fl.ld (cm)

Figure 5.11: Variation of percent depth dose with field area for an 80 kVp (nominal) beam filtered with 1.5 mm Al. Doses at 10 cm (.) and 15 cm (.) are compareà with calculations (solid lines).

Dose profiles were measured at eight depths in phantom along the an­

ode-cathode axis to investigate the dose variations resulting from the heel ef­fect (for a 79 kVp beam with standard aluminum filtration). The results are

shown in Fig. 5.12. The profiles are relatively fiat at all depths except at the

surface. The surface dose varied from that at the central axis (from -15 %

towards the anode to + 5 % towards the cathode), indicating a shape of the

dose distribution similar to that in Meredith and Massey (1972). At depths

larger than 2.0 cm, however, the beam profiles are relatively fiat (± 3 %).

One of the objectives of the central-axis depth dose calculation is the

evaluation of the patient risk involved with radiological examinations by the

determination of the integral dose from either calculated or measured depth

doses (see §§ 5.3). When relative quantities are considered, it may be expected

that the variation in the beam profile will not produce drastic effects.

Therefore, given that relative risk is considered and that the beam profile is

relatively fiat at depths larger than 2.0 cm, the distortion of the beam profile

85

120

100 .. Il 0

" 80 .c: -c:a. .. 60 " -c .. 40 u .. .. D.

20

0

;;::=::::;: ..... ~ ~ >+- - - -)t- - - -x-- _ - -x- - _..:

~+------~----~----~~--~

b-"'-"6-- ... _ tr- "_"6-" -·-A

e-- - 0--- _ 0-- _0-- -0

- - - -- - - - - - --- - --5 o

Dlstanca .rom cantral axis (cm)

1Wnb +---- Surface

- -0- - 05 cm .J--G-- 1.0 cm

)( - 24 cm J--+-- 50em

·A·- 7.0cm - 10.0 cm

.. - 15.0cm

5

Figure 5.12: Variation of dose along the anode-cathode axis for various depths in phantom. Negative distances indicate points doser to the cathode. A 79 kVp beam with standard alu-rninurn filtration was used with a 13.7 x 21 cm2 field. An uncertainty of about ± 3 % is observed for most depths.

by the heel effect will not severely affect the determination of the integral dose. The fact that the heel effect is usually ignored in the calculation of inte­gral doses and effective dose equivâlents (Harrison, 1981; Alm Carlsson and Carlsson, 1986) support the validity of central-axis depth dose to obtain a bet­ter approximation to the integral dose (i. e. Eqn. 4.15) than if the entrance ex­posure or only the prirnary radiation are considered (Eqns. 4.11-4.13).

The results shown in Fig. 5.12 suggest that it rnay be interesting to modify the current model for the calculation of off-axis depth doses. In order to evaluate depth doses off the central axis, the heel effect would have to be

considered. Sorne of the assumptions used to simplify the calculation of the central-axis depth dose data would be invalidated: the incident beam would no longer be assumed to be parallel, and the symmetry assumptions which

86

simplify the current calculations would no longer hold. The modifications required to obtain doses out of the central axis would result in a significant increase of the computation time. Therefore, ca1culated off-axis depth doses would best be obtained from Monte Carlo simulations. The spectrum genera­tion model proposed by Tucker et al. (1991a) would also he required so that the heel effect could be incorporated into the calculation the x-ra y intensities.

5.3 Risk assessment

One of the possible applications of the programmes Spect'lJm 1.1 and depthDose is the evaluation of the relative risk involved by a patient under­going various radiological examinations. As noted in Chapter l, the entrance exposure, the integral Jose and the effective dose equivalent are currently used to esnmate radiological risk. The entrance exposure and the primary in­tegral dose can be obtained from the programme Spectrum 1.1 using Eqns. (2.4) and (E.6). Equation (4.15) can be used to obtain the integral dose directly from relative depth dose data, which can either be measured or calculated from the depthDose programme. Ideally, however, risk should be estimated from the determination of the effective equivalent, HE (Hu da et aL, 1989).

The effective dose equivalent was introduced by the ICRP (1977) and it has been accepted by the UNSCEAR as the most appropriate indicator of the patient risk in medical radiation dosimetry (UNSCEAR, 1982). In general, the probabilily of a stochastic effect (e. g. carcinogenesis and genetic defects) pro­duced by an exposure to radiations is assumed to he proportional to the ab­sorbed dose. The value of the proportionality factors however differs among the various tissues due to differences in radiosensitivity among the tissues. The effective dose equivalent is then evaluated by a weighted sum of the dose equivalent of each of the irradiated tissues:

(5.3)

where W, is the weighting factor for tissue t and H, is the mean dose equiva­lent for tissue (or organ) t. The weighting factors are established from the rel­ative sensitivities of tissues as appointed by the ICRP (1977), and H, is obtained

87

.. from the integration of the dose equivalent distribution DE(x,y,z) over the ir­radiated patient volume:

(5.4)

(Huda and Sandison, 1984), where pis the organ density, and m is the total or­gan mass. p is assumed to he constant. The use of Eqn. (5.4) is restricted to very low doses where only stochastic effects (i. e. carcinogenesis and genelic defects) may occur.

The evaluation of the effective dose equivalent th us necessitates the three dimensional dose distribution produced by the considered radiographic

examination and the determination of the volume occupied in the irradiated field by the various organs. The computation of the H, associated with aIl of the involved organs must then be done, and the weighted sum over aIl the organs concerned he performed. The evaluation of the effective dose equiva­lent thus involves more labour or computation time th an any of the other parameters used for relative risk assessment. Furthermore, the uncertainties in the weighting factors, caused by race, sex, or age variations in the occur­rence of stochastie radiation effects, induce errors up to factors of 2 or 3 in the determination of the effective dose equivalent (Wall and Shrimpton, 1981).

In theory, the effective dose equivalent is the best estimate of the risk in­volved in exposure to radiations. Unfortunately, the effective dose equiva­lent is difficult to calculate with precision.

The entrance exposure is extensively used to evaluate the relative risk

involved with various radiographie studies mainly hecause this parameter is the easiest to measure. Sorne conversion factors to obtain the effective dose equivalent from entrance exposure or surface dose measurements have been determined (Shrimpton et al., 1988; Huda et al., 1989). These conversion fac­tors depend on the radiological technique used. For example, if the half­value layer of the x-ray heam is increased from 1 to 7 mm Al (e. g. by adding filtration or by increasing the kVp), the conversion factor for PA chest exami­

nations changes by - 430 % (Schreiner et al., 1991).

88

-J' i{

(

Comparative risk can be successfully achieved from integral dose mea­surements, even if the specifie radiosensitivity of organs is ignoreà (Carlsson, 1963; Gustafsson, 1979; Wall and Shrimpton, 1981; Schreiner et al., 1991). Huda et al. (1989) and Schreiner et al. (1991) have also pointed out that the conversation factors for integral dose to effective dose equivalent c;how much less variation versus HVL « 10 %, over the same HVL range as in the previ­ous paragraph) th an the conversion factors for surface dose to effective dose equivalent.

Some of the radiological studies performed for this work were used to compare, in terms of risk, various radiographie techniques where a different tube kilovoltage was determined to obtain images similar to standard tech­niques when non-standard filters were used. The tube voltages of the stan­dard tflchniques were established at 62, 79 and 113 kVp (60, 80 and 120 kVp,

nominally) to cover the low, medium and high kVp ranges encountered in diagnostic radiology, and aIl used the standard 1.5 mm Al filtration. The de­termination of the kilovoltages used for the other filtration schemes is as de­scribed in Chapter IV. The EQSPEC spectra were determined for each of these radiographie techniques, and doses in phan tom were both calculated and measured.

Three methods have been used to estimate the integral dose relative to the established standard techniques. Integral doses were first estimated using the algorithm implemented in the Spectrum 1.1 programme, where Eqn. (E.6) is evaluated for a particular incident spectrum. As pointed out in Appendix E, integral doses calculated in this manner assume a parallel beam. Furthermore, only primary radiation is considered: the rising presence of scat­tered photons with increasing kVp and field size is likely to cause discrepan­cies with the real integral dose. The quantity obtained from Eqn. (E.6) will be interchangeably named total energy imparted or primary integral dose, since it is only an approximation of the integral dose: it represents the energy dif­ference between the incident and the exit radiation beams. The second inte­gral dose calculation used measured depth dose data to evaluate Eqn. (4.15) while the third method incorporated the calculated depth doses. The resuIt­ing risk relative to the standard techniques, as determined from the ratio of the integral dose obtained to the integral dose associated with the standard

89

.. technique, X/Istd, is presented in Table 5.5 for most of the studied techniques. The relatively large uncertainties (in general, ± 10 %) involved with mea­sured relative integral doses arise from the propagation of errors in the mea­

surement process. AlI of the integral dose ratios indicated from calct'lated depth doses are, in general, within the ± 10 % interval; the maximal devia­tion from measured integral doses, obtained from the comparison of calcu­lated and measured integral doses, is 8.5 %, ar.d the average deviation is

4.3 %. Unfortunately, sorne of the discrepancies between integral doses indi­

cated by calculated and measured depth doses are above the 5 % precision ex­pected by radiotherapy dosimetry (ICRU 1976). AlI of these values are how­

ever well below the 15 % uncertainty suggested by other workers using expo­sure-area products compared with Monte Carlo simulations (Shrimpton et al., 1981). Huda et al. (1989) consider that a 16 % difference between effective

dose equivalents obtained from Monte Carlo simulations and measurements

is good. These larger discrepancies are acceptable because the operating condi­tions used in diagnostic radiology are not as strict as in radiotherapy. Unlike

radiotherapy, the distribution of organs within a patient is not known a Pri­ori, the x-ray beam output is not monitored, and the irradiation geometry is not rigorously known. Thus, a ± 10 % variation in measured integral dose ra­tios is adequate and the observed 8.5 % discrepancy between calculated and measured integral dose ratios is very good.

The discrepancies between the measured integral doses and the calcu­

lated primary integral dost' ratios listed in Table 5.5 are in general larger than

those indicated by the comparison of risk reductions indicated by measured

and calculated depth doses. Furthermore, the primary integral dose usually underestimates the integral dose savings indicated by measured depth doses

since contributions to the dose in phantom from scattered radiation are not

accounted for. To verify the effect of this error on r~lative integral doses, the primary integral dose was extracted from the data compiled for the three

techniques listed in Table 5.3 and was compared with the integral doses ob­

tained from their respective calculated percent depth doses. The primary in­

tegral dose varies between 52.1 % (68 kVp beam filtered with the R-filter) and

70.4 % (30 kV p beam filtered with 1.5 mm Al) of the total integral dose. That

is, the contribution of primary radiation to the relative integral dose varies

with radiographie technique and, therefore, the contribution of once-scattered

90

(

radiation to the integral dose should he taken into account. It is of interest,

however, to note that aU but one of the integral dose savings indieated by the

total energy imparted lie helow the 15 % uncertainty limit dted from other

publications. If variations in the contribution of the primary integral dose to

the integral dose of the order of these cited above are considered acceptable,

the total energy imparted should be a sufCident parameter for the evaluation

of relative integral doses. Unfortunately, the direct evaluation of the integral

dose does not give any information on the dose distribution within the phan­

tom.

Technique 62 kVp (Al) 66 kVp (Nb) 68 kVp (Zn) 65 kVp (Ho) 62 kVp (Yb)

Technique 79 kVp (Al) 85 kVp (Nb) 85 kVp (Zn) 93 kVp (Ho) 89 kVp (Gd) 79 kVp (Yb)

Technique 113 kVp (Al) 113 kVp (Yb)

(a) LOW kVp Energy Imparted Measurements

1.00 1.00 0.59 (- 9 %) 0.65 0.50 (-14 %) 0.58 0.52 (- 4 %) 0.54

0.53 (-12 %) 0.60

(b) MEDIUM kVp Energy Imparted Measurements

1.00 1.00 0.72 (- 17 %) 0.86 0.61 (-16 %) 0.73 0.70 (-12 %) 0.79 0.64 (- 5 %) 0.67 0.61 (+ 5 %) 0.57

(c) HIGH kVp Energy Imparted Measurements

1.00 1.00 0.65 (- 6 %) 0.69

depthDose 1.00

0.65 (-) 0.60 (+ 3 %) 0.51 (- 6 %) 0.55 (- 8 %)

depthDose 1.00

0.82 (- 5 %) 0.75 (+ 3 %) 0.73 (- 8 %) 0.68 (+ 2 %) 0.60 (+ 5 %)

depthDose 1.00

0.67 (- 3 %)

Table 5.5: Risk estimates relative to standard radiographie techniques rep­resentative of low (a), medium (b) and high (c) tube kilovolt­ages. Relative risk is obtained from calculated primary integral dose (total energy imparted) and from integral doses de ter­mined from depth dose measurements and calculations. The results are normalized to 1.00 for the standard Al filtration. Deviation of calculated relative risks from measurements are given in %.

9 1

While the energy imparted calculations seem to give nearly identical results to the risk reduction as obtained with the more lengthy depth dose cal­culations, the latter approach is superior. The depth dose calculations reflect the physical situation more truly as scatter is considered. The calculated depth doses cal' be compared with measurement and, therefore, calculations can be verified directly. The error involved with the integration of depth dose data can be reduced by increasing the number of depths calculated.

Depth dose data can also be used to determine the effective dose equivalent if this risk parameter is required. This can be achieved in a manner analogous to the techniques used by treatment planning computers; one could con vert

the dose data into dose equivalent (by multiplication of dose with a quality factor of 1 for x-rays) and the use of this equivalent dose distribution in Eqn. (5.4) for each of the "irradiated organs".

5.4 Concluding remarks

The motivation for performing computer experiments in diagnostic

radiology is to reduce the work involved in the assessment of a large number of quantities useful for both dose and image quality studies. While no de­tailed theory of x-ray production in a target has been successfully derived for

diagnostic radiology, a number of semi-empirical models have b€en shown to

give acceptable approximations to measured x-ray spectra. These calculated x-ray spectra can then be used to evaluate a large number of quantitIes related

to dose or beam quality. The percent depth dose is such a quantity Doses cal­

culated with the depthDose programme described in this thesis has been shown to give a quick, reasonable approximation to measured depth doses for

tube potentials commonly used in diagnostic radiology by considering only

primary and first-scattered radiation. The calculated integral dose savings rel­ative to standard techniques were also shown to agree with measurements.

Calculations and measurements have been made for various radiologi­

cal conditions. Sorne of the results have been interesting and unexpected (e. g. observation of dose buildup for some techniques). Further work involv­

ing the comparison of percent depth doses calculated with the present model

with those from other simulations and measurements are required to more

•

clearly investigate those results. This will be achieved by measurement of

depth doses for a number of other techniques, leading to the expansion of

Table 5.5. Comparison of depth doses calculated with the present model with

those obtained from Monte Carlo simulations will enable the assessment of

the conditions where second and higher order scattered radiation contribu­

tions ta the dose become significant. This work is currently underway.

Finally, the model will he adapted to imaging and contrast problems since risk

reduction alone without imaging considerations may be misleading for the

determination of an optimal radiographie technique .

93

APPENDIXA

PROGRAMME FLOWCHARTS

1 Start : ...

Use saved spectrum ? Calculate new spectrum

yas no - Enter kVp. target angle.

Use 'iltered spectrum ? Inherent filtratIOn

Bremsstrahlung()

no Charact9f1stld) -yas

'1 , --Display 'iltered spectrum. 1 ~ yas Display unfilît;red

Modify spectrum wlth spectrum. Adj tliiratlon ?

exponential attenuation law.

1 no , Calculate quantities derived no --'rom displayed spectra ?

yas

Calculate transmission curve wrth -- - Calculate exposure and --tirst and second hait-value layers. primary Integr'll dose. Display transmission data.

AttenuatlonCurv"J. HVL() IntOosa.) -,

1 --Figure A.1: Flowchart of the SpectI'um 1.1 programme for the computation

of diagnostic x-ray spectra from constant potential tubes. Spectra and transmission data can be saved as SPCT (format de­fined by the programme) or as ASCII files when displayed. Sorne subroutines (in italics) are listed in Appendix B.

;( ""

1 Start 1

-550 (cm) -510 (cm)

. - thickness of filter (cm) Enter geometry. _ phantom thickness (cm)

- radius of equivalent circular field (cm).

, Select files containing the anenuation coefficients for

Enter data: the fiRer and the unfinered spectrum.

for every depth do:,

r-----------1-~-: Calculale primary dose , Calculate seatter dose:

(Rayleigh and COrf1)ton scattering)

Next photon energy

InitLoop.), doJadia(Ioop(), do_basic_IooP(), geLab,_ value.)

no ....... ----.... : Is photon energy • tMntu?

i yas

1 End 1

t--~'" Radiation scattered in fiRer layer1()

1-1

Radiation scanered in air lay.~)

Radiation scattered in phantom laye~), do_bac/cscatteOoo,,)

1

Figure A.2: Flowchart of the depthDose programme for the calculation of depth doses in polystyrene. The output doses are saved as ASCn files. The selection of the form factors and scattering functions needed to obtain the differential cross sections for the scattering interactions is done by modifying the code. Some subroutines (in italics) are listed in Appendix B.

95

APPENDIXB

SELECTED SUBROUTINES

Computer code written for the Spectrum 1.1, the SinglePhase and the depthDose programmes is available upon request. The code for each of these programmes was written in the C programming language (see for example, Kernighan and Ritchie, 1988).

Inquiries should be directed to:

L. John Schreiner, Ph.D. Montreal General Hospital 1650 avenue des Cèdres Montréal, Québec H3GIA4

(514) 934-8052

Jean.,Pierre Bissonnette London Regional Cancer Center 790 Commissioners Road East London, Ontario N6A 4L6

(519) 685-8600 ext. 3146.

96

APPENDIXC

PERCENT DEPTH DOSES -1

Percent depth doses are presented for ail of the radiographie techniques listed in Table 4.2, covering a wide range of potentials used in diagnostic radi­ology and a number of filtration schemes. Tables C- a to C- f were obtained from measurements involving the technique described in Chapter 4 using the geometry shown in Fig. 4.5 (radius of equivalent circular field is 9.065 cm) and from the depthDose programme. The source-fil ter distance was 5 cm in ail cases except for tables C- band C- c, whieh involved the commercial R-filter and Niobix filters. In these cases, the source-filter distance was 27 cm.

The standard deviation of a single depth-dose measurement in the range of depths between 0 and 5.0 cm is less than 4 % and it is less th an 3 % for depths greater than 5.0 cm. The estimated overall uncertainty, however, is about ± 2 % for depths between 0 and 5.0 cm and ± 1.5 % below 5.0 cm These figures were obtained from repeated measurements.

Surface doses were of the order of 40 cGy for the standard techniques.

97

98

~

~. Table C: Percent depth doses for diagnostic radiologyl.

(a) 1.5 mm Al added filtration

kVp

depth 48 57 70 79

(cm) eXE calc exe cale eXE cale exe cale 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0

0.45 83.5 91.26 91.8 92.08 95.8 92.92 80.7 93.41 1.0 80.84 82.48 84.18 85.17

2.35 57.7 58.45 64.9 61.26 76.8 64.27 64.0 66.06 4.0 38.30 41.45 44.92 47.03 5.0 29.5 29.46 36.7 32.49 47.0 35.89 39.3 37.99 7.0 17.34 19.85 22.74 24.58 8.0 13.30 15.49 18.07 19.73 10.0 8.0 7.83 10.2 9.44 20.5 11.41 13.6 12.70 11.9 3.5 4.74 6.0 5.91 14.6 7.37 8.5 8.35 15.15 1.1 2.02 1.5 2.66 7.2 3.50 4.3 4.()9 17.5 1.10 1.50 2.05 2.44 20.0 0.7 0.58 0.6 0.82 3.3 1.16 1.5 1.42 22.5 0.30 0.45 0.66 0.82 24.95 0.8 0.16 0.4 0.00 1.4 0.38 0.5 0.49 27.2 0.7 0.09 0.2 0.00 0.7 0.23 0.1 0.30

kVp

depth 62 74 97 113

(cm) exe calc exe calc eXE calc cXE cale 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 0.5 91.4 91.59 97.0 92.38 105.8 93.48 103.0 93.99 1.0 85.5 83.20 88.2 84.64 102.3 86.68 100.1 87.62 2.4 65.1 61.84 69.8 64.43 83.3 68.2C 83.5 69.98 4.0 42.90 45.89 50.36 52.53 5.0 35.1 33.90 41.0 36.85 52.4 41.33 55.2 43.52 7.0 21.6 21.04 24.4 23.58 36.3 27.56 38.5 29.56 8.0 16.55 18.83 22.45 24.29

10.0 9.4 10.24 12.0 11.99 20.8 14.86 23.0 16.35 12.5 5.63 6.82 8.85 9.!t3 15.0 2.1 3.10 1.3 3.89 8.4 5.27 10.5 6.03 17.5 1.72 2.22 3.14 3.66 20.0 0.7 0.95 0.0 1.28 1.9 1.87 4.4 2.22 22.5 0.53 0.73 1.12 1.35 25.0 1.8 0.30 0.1 0.42 0.7 0.67 1.0 0.82 27.2 0.18 0.26 0.43 0.53

..,. "'" 1 exp:: percent depth dose measured with TLD-800

calc !! percent depth dose calculated with the depthDosc programme

1 J

99

(b) 1.5 mm Al + 0.05 mm Nb added filtration

kVp kVp

depth 85 120 depth 66 105

(cm) ex cale ex cale (cm) ex cale ex cale 0.0 100.0 100.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0

0.45 98.4 95.07 109.0 95.77 0.5 96.7 93.63 100.8 95.00 1.0 88.37 89.85 1.0 94.9 86.75 98.4 89.36

2.35 71.5 71.43 102.1 74.34 2.4 79.6 67.73 88.9 72.77 4.0 53.10 56.84 4.0 49.32 55.54 5.0 49.7 43.89 73.7 47.77 5.0 46.0 40.06 58.9 46.42 7.0 29.60 33.28 7.0 30.4 26.11 41.2 31.98 8.0 24.21 27.65 8.0 21.00 26.43 10.0 21.5 16.12 37.0 18.99 10.0 13.0 13.53 26.9 17.95 11.9 15.0 10.91 27.5 13.23 12.5 7.78 11.01 15.15 6.3 5.57 14.4 7.09 15.0 6.1 4.46 21.8 6.72 17.5 3.42 4.51 17.5 2.56 4.10 20.0 1.7 2.04 5.7 2.78 20.0 1.9 1.47 9.2 2.50 22.5 1.22 1.72 22.5 0.84 1.52 24.95 0.5 0.73 2.1 1.07 25.0 1.5 0.49 1.1 0.93 27.2 0.7 0.46 0.6 0.70 27.2 0.30 0.60

(c) 0.27 mm Zn (R-filter) added filtration

kVp kVp depth kVp 121 depth 68 85

(cm) ex cale (cm) ex cale ex cale 0.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0

0.45 93.5 107.7 0.5 105.8 106.6 104.1 107.0 1.0 81.5 101.7 1.0 103.3 99.75 102.6 100.8

2.35 60.7 85.89 2.4 93.9 80.39 89.0 82.80 4.0 67.46 4.0 60.69 63.90 5.0 33.7 57.53 5.0 53.5 50.27 56.3 53.66 7.0 21.5 40.86 7.0 37.0 33.71 37.2 36.98 8.0 34.10 8.0 27.34 30.39 10.0 11.4 23.39 10.0 17.7 17.74 18.6 20.24 11.9 16.15 12.5 10.17 11.98 15.15 4.9 8.48 15.0 5.4 5.80 4.8 7.05 17.5 5.31 17.5 3.30 4.14 20.0 0.6 3.24 20.0 0.4 1.89 0.2 2.44 22.5 1.98 22.5 1.08 1.44 24.95 0.2 1.23 25.0 0.6 0.62 0.1 0.86 27.2 0.79 27.2 0.39 0.54

(

100

(d) 0.11195 mm Ho added filtration

kVp kVp

depth 134 dcpth 65 93

(cm) ex cale (cm) ex cale ex cale 0.0 100.0 100.0 0.0 100.0 100.0 100.0 100.0

0.45 90.6 95.50 0.5 88.5 93.30 98.4 94.13 1.0 89.46 1.0 83.7 86.29 96.5 87.85

2.35 80.8 74.03 2.4 69.7 67.21 82.5 70.22 4.0 56.78 4.0 48.86 52.60 5.0 57.7 47.85 5.0 39.5 39.64 49.7 43.47 7.0 33.54 7.0 25.9 25.76 33.7 29.31 8.0 27.96 8.0 20.68 23.97 10.0 30.3 19.34 10.0 10.6 13.27 18.2 15.96 11.9 19.6 13.57 12.5 7.59 9.54 15.15 10.8 7.37 15.0 1.89 4.33 5.46 5.69 17.5 4.73 17.5 2.46 3.39 20.0 4.4 2.95 20.0 1.12 1.40 0.16 2.02 22.5 1.85 22.5 0.80 1.20 24.95 0.9 1.16 25.0 2.16 0.46 0.99 0.72 27.2 1.1 0.76 27.2 0.28 0.46

(f) 0.11605 mm Gd added filtration

kVp kVp

depth 65 depth 88

(cm) ex cale (cm) ex cale

0.0 100.0 100.0 0.0 100.0 100.0 0.45 87.8 93.65 0.5 98.8 93.75 1.0 85.60 1.0 94.3 87.14

2.35 69.4 66.60 2.4 78.2 68.90 4.0 47.38 4.0 51.00 5.0 41.9 38.16 5.0 48.4 41.85 7.0 24.47 7.0 30.5 27.84 8.0 19.53 8.0 22.63

10.0 13.3 12.39 10.0 16.5 14.89 11.9 8.7 8.01 12.5 8.76 15.15 1.9 3.80 15.0 2.6 5.17 17.5 2.21 17.5 3.05 20.0 0.8 1.25 20.0 0.1 1.80 22.5 0.70 22.5 1.06 24.95 0.0 0.40 25.0 0.4 0.63 27.2 1.2 0.24 27.2 0.40

.",.

1.'

101

" (e) 0.10145 mm Yb added filtration '.

kVp

depth 62 79 113

(cm) exe calc exe calc exe cale 0.0 100.0 100.0 100.0 100.0 100.0 100.0 0.5 93.9 93.01 100.0 93.70 104.8 94.55 1.0 95.2 85.76 93.1 87.05 96.0 88.67 2.4 73.1 66.25 79.2 68.70 91.0 71.87 4.0 47.75 50.73 54.71 5.0 43.9 38.55 49.5 41.57 62.3 45.68 7.0 24.6 24.83 32.5 27.56 45.6 31.42 8.0 19.86 22.36 25.96

10.0 9.8 12.65 16.5 14.66 27.0 17.63 12.5 7.17 8.60 10.81 15.0 0.1 4.06 2.5 5.03 10.3 6.60 17.5 2.30 2.95 4.03 20.0 0.9 1.30 0.3 1.72 2.9 2.46 22.5 0.74 1.01 1.50 25.0 0.0 0.42 0.3 0.59 0.0 0.92 27.2 0.26 0.37 0.59

f '.

=

APPENDIXD

PERCENT DEPTH DOSES - 2

Percent depth doses are presented for a range of field sizes. The input x-ray spectrum was a 79 kV beam from a 12° anode with an 1.31 mm Al in­herent filtration corresponding to an 80 kV beam from the x-ray unit de­scribed in section 4.2. The added filtration consisted of the standard alu­minum filtration (1.5 mm Al). Measurements were performed for four field

sizes of the radiographie techniques listed in Table 4.2 while calculations were

performed for seven fields sizes, inc1uding zero-area. Table D was obtained ei­ther from measurements involving the technique described in chapter threc

using the geometry shown in Fig. 4.5 or from the depthDose programme. The source-filter distance was 5 cm in ail cases.

The standard deviation of a single depth-dose measurement in the range of depths between a and 5.0 cm is less than 4 % and it is less than 3 % for depths below 5.0 cm. ~he average overall standard deviation, however, is

about ± 2 % for depths between a and 5.0 cm and ± 1.5 % below 5.0 cm.

102

{ 1

Table D: Variation of the percent depth dose of a 79 kV beam using 1.5 mm Al additional filtration with field sizes1 •

dcpth 7x7 10x 10 13.7x 21

(cm) cale exp cale exp cale exp 0.0 100.0 100.0 100.0 100.0 100.0 100.0

0.45 92.75 93.05 91.8 93.41 95.8 1.0 83.92 84.48 85.17

2.35 64.15 57.7 64.96 64.9 66.06 64.0 4.0 45.12 45.93 47.03 5.0 36.20 29.5 36.98 36.7 37.99 39.3 7.0 23.10 23.79 24.58 8.0 18.41 19.04 19.73

10.0 11.69 8.1) 12.18 10.2 12.70 13.6 11.9 7.60 't5 7.96 6.0 8.35 8.5 15.15 3.66 .'.1 3.,36 1.5 4.09 4.3 17.5 2.16 2.29 2.44 20.0 125 0.1 1.32 0.6 1.42 1.5 22.5 o.n 0.77 0.82 24.95 0.42 0.8 0.45 0.4 0.49 0.5 27.2 0.26 0.1 0.28 0.2 0.30 0.1

dcpth OxO 20x20 depth 15x 15

(cm) cale cale (cm) cale ex 0.0 100.0 100.0 0.0 100.0 100.0

0.45 88.83 93.56 0.5 92.61 91.4 1.0 76.96 85.43 1.0 85.08 85.5

2.35 54.42 66.46 2.4 65.26 65.1 4.0 35.95 47.47 4.0 46.88 5.0 28.08 38.40 5.0 37.84 35.1 7.0 17.27 24.90 7.0 24.47 21.6 8.0 13.59 20.01 8.0 19.64

10.0 8.47 12.89 10.0 12.63 9.4 11.9 5.44 8.49 12.5 1.27 15.15 2.59 4.17 15.0 4.19 2.1 17.5 1.52 2.50 17.5 2.42 20.0 0.87 1.45 20.0 1.40 0.7 22.5 0.50 0.84 22.5 0.82 24.95 0.30 0.50

1 25.0 0.48 1.8

27.2 0.18 0.31 27.2 0.30

exp lE percent depth dose measured with TLO-BOO cale !il percent depth dose calculated with the depthOose programme

103

APPENDIXE

DETERMINATION OF THE PRIMARY INTEGRAL

DOSE FROM PHOTON SPECTRA

The def!nition of the integral dose (E) usually encountered in textbooks is the one expressed by Eqn. (4.10):

d

X = p 1. D(x) A(x) dx, (E.l)

where x is the depth along the central beam axis in phantom, D(x) is the dose at a depth x, A(x) is the area of the radiation field at depth x, and p is the density of the phantom material (Johns and Cunningham, 1984). In diagnostic radiology, the range of electrons set in motion by the x-ray photons is 50 sm aIl that the photon beam is not appreciatively attenuated by the thickness of phan tom described by the electron range, and a state of electronic equilibrium can be assumed. Under electronic equilibrium conditions, the absorbed dose at a depth x in phan tom for a monoenergetic beam can be

calculated with:

/Jab Pœ D(x) = '1'- = N v exp( -11...x) h v-, P P

(E.2)

where '1' is the energy fluence at depth x, whkh is equal to the number of

photons per cm2 at depth x (Nv exp(-I1...x» times the photon energy, and l1ab/P is the mass energy absorption coefficient of the material of interest. Substitution of (E.2) in (E.}) yields

I= plt1

Nvexp (-I1...x) hv/Jab A(x) dx. o P

(E.3)

In the special case of a parallel beam, A(x) == A and integration of Eqn. (E.3)

yields

\04

J.lab I= Nvexp (-Jl~) hv-A. Jlv

(E.4)

The integral dose due to primary radiation from a polyenergetic beam is then obtained from integration of Eqn. (E.4) over ail photon energies present in the beam:

f.

...... I= Nvexp (-Jl~) hvJ.lab A d(hv),

Jlv o

(E.5)

Since Nv is normalized to a 1 cm2 field, A = 1 and Eqn. (E.5) thus becomes

(E.6)

which is the same as Eqn. (4.8) w:th X set to zero, the expression used in the Spectrum 1.1 program to evaluate the integral dose due to primary radiation.

105

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