A Monte Carlo study of dose distributions and energy imparted in computed tomography dosimetry phantoms

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Title:
A Monte Carlo study of dose distributions and energy imparted in computed tomography dosimetry phantoms
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xiv, 194 leaves : ill. ; 29 cm.
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Atherton, James Vincent, 1957-
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Nuclear Engineering Sciences thesis Ph.D
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Thesis:
Thesis (Ph. D.)--University of Florida, 1993.
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Includes bibliographical references (leaves 186-193).
Statement of Responsibility:
by James Vincent Atherton.
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Typescript.
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Vita.

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A MONTE CARLO STUDY OF DOSE DISTRIBUTIONS AND ENERGY
IMPARTED IN COMPUTED TOMOGRAPHY DOSIMETRY PHANTOMS












BY

JAMES VINCENT ATHERTON


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1993
















ACKNOWLEDGMENTS


The author wishes to extend copious and sincere thanks to his committee chairman,

Dr. William Properzio, for his many efforts on the author's behalf. The author also wishes

to extend his thanks to his advisor and cochair, Dr. Libby Brateman, for her hard work,

support and advice in the development of this project. Many thanks also go to the other

committee members, Drs. Hintenlang, Honeyman, and Roessler, for their encouragement

and assistance. Additional recognition is due Drs. Roessler and Brateman for rearranging

their travel schedules to accommodate the author. The author also wishes to extend his

sincere thanks to Dr. Walter Huda for his suggestions and advice. Also, thanks go to the

staff of The CancerCare Center and particularly Dr. Eric Rost for the use of the computers

and office equipment located there. The author would like to acknowledge the support he

received from the CDRH through the assistance of Drs. Orhan Suleiman and Marvin

Rosenstein.

The author wishes to extend his great thanks to his family for their continued love

and encouragement. Finally, the author wishes to extend his most sincere thanks to his

wife for her love and support during the course of this project.














TABLE OF CONTENTS


ACKN OW LED GM ENTS........................................................ ......................... ii

LIST O F TA B LE S.................................. ....... ... ....... ................. ..... vi

LIST OF FIGURES...................... .. ... ...... ...... ............. .. ..................... ix

A B STR A C T ................................ .. ..................... .................................. xiii

CHAPTERS

1. INTRODUCTION........................ .. ..... ..... ..................... 1

2. SURVEY OF COMMON CT SCANNER CHARACTERISTICS.................. 13

Introduction..................................................................... ......... ........... 13

Basic Operational Principles........................................... ................... 13

Differences Among CT Scanners from Various Manufacturers............. 15

Sum m ary............. ...... ............................... .................... .................. 27

3. THE MONTE CARLO TECHNIQUE.......................................... 30

Introduction............. ................................................ ................... .. 30

Sampling M ethods............ ......... ................................... ......... 32

Random Number Generation................................................ ............ 37

Data Analysis......................... ...... .. ... .. .......................... 38

4. THE EGS MONTE CARLO SYSTEM.................................. ....... 41

Introduction.................................................................................. ....... 41

EGS Components........................... ....... ... ........................ 43

Summary ......... .................. ............................. 65











5. VERIFICATION OF THE MONTE CARLO MODEL............................... 67

Conservation of Energy.................................................................... ..... 67

In Air Dose Calculation........ ..... ............... ............................. .......... 69

Primary X Ray Transmission Through a Cylinder................... ........... 71

6. DOSE DISTRIBUTIONS AND RELATED DOSIMETRIC QUANTITIES... 74

Introduction....................................................................................... 74

D ose Profiles.................... ................................ ........ ......... .......... 75

Integration of Dose Profiles............................ ................ 79

Photon Energy Variation............... .... .... .................. ...................... 89

Beam Filter V ariation................................................... ....................... 95

Slice Thickness V ariation...................... .......................................... 98

Phantom Size Variation................................ .. ............... 100

Phantom M material Variation............................................ ..................... 100

SA D V ariation.......................................... ...................... 102

Summary............ .... ... .......................... .................. 104

7. ENERGY IMPARTED RESULTS..................................... ........................ 106

Introduction........................................................................................ 106

Photon Energy V ariation.................................................. .................. 106

Beam Filter Variation....................... .... ......... .......... 112

Slice Thickness Variation............................... .. ............... 115

Phantom Size Variation.................................................... .................. 115

Phantom M material Variation................................................................ 115

SAD Variation ........................................................... ................... 119

Phantom Composition and Shape Variation................................. 120

Sum m ary............................................................................................... 128

8. DISCUSSION AND CONCLUSIONS.................................... ................... 130









APPENDICES

A. CT MANUFACTURER SURVEY FORM........... ........................................ 141

B. PRIMARY X RAY TRANSMISSION THROUGH A CYLINDER............. 145

C. DESCRIPTION OF MACRO $ELLCYL................................................... 148

D. LISTING OF PERMON1.MOR ................................................................. 156

REFEREN CE S.............................................. ..... .............. ....... .................. 186

BIOGRAPHICAL SKETCH..................................... .. .. .................... 194














LIST OF TABLES


Table 1-1 Comparison of effective dose equivalents for standard exams for
three common CT scanners ............................................................................. 6

Table 1-2: Ratios of the effective dose equivalent to the energy imparted for
different exams and scanners......... ...... .... .... .. .......... ............ ........ ............... 9

Table 1-3. The range of parameters examined in this work............................... 12

Table 2-1: Half value layers for the GE 9800 scanner as function of detector
and kV p................. ............. ................. ...... ........... ......... .. ................. 19

Table 2-2: Comparison of head CTDI values [mrad./mAs]............................ 28

Table 2-3: Comparison of body CTDI values [mrad/mAs].......................... 29

Table 5-1: Illustration of conservation of energy for four separate Monte Carlo
runs with an acrylic phantom and no filter with a 65 cm SAD......................... 68

Table 5-2: Comparison of CTDI values per incident fluence for various energies
and acrylic rod diameters............................. .................. ....................... 70

Table 5-3: Comparison of percentage transmission of primary radiation through
acrylic cylinders of various diameters as calculated analytically and with the EGS
code..................... ... ................................................ ............ .................. 72

Table 6-1: C(r,e) as a function of integration limits e and radial position r in a
head phantom at 50 keV ......................................................... ........... .... 83

Table 6-2: Summary of spectra data.......... .................. ................................... 92

Table 6-3: Comparison of CTDIa. values for the three beam filters used............ 92

Table 7-1: Energy dispositions for an 8 cm acrylic phantom with no beam filter
and a 5 mm slice thicknesses........ ............... ............................. 107

Table 7-2: Energy dispositions for a 16 cm acrylic phantom with no beam filter
and a 5 mm slice thickness.............................................. 108








Table 7-3: Energy dispositions for an 8 cm acrylic phantom with no beam filter
and 5 mm slice thickness for four spectra............................................................ 109

Table 7-4: Percentage energy dispositions for an 8 cm acrylic phantom with no
beam filter and 5 mm slice thickness for four spectra..................................... 109

Table 7-5: Energy dispositions in an 16 cm acrylic phantom with no beam filter
and 5 mm slice thickness for four spectra....................................... ...... ....... 110

Table 7-6: Percentage energy disposition for four spectra in a 16 cm radius
acrylic phantom with no beam filter................................................................... 110

Table 7-7: Energy dispositions at 50 keV for 8 cm and 16 cm radii acrylic
phantoms for a 5 mm slice thickness, for three filter configurations............................ 113

Table 7-8: Percentage energy dispositions at 50 keV for 8 cm and 16 cm radii
acrylic phantoms for a 5 mm slice thickness, for three filter configurations .......... 113

Table 7-9: Comparison of energy disposition categories of monoenergetic
beams with two typical CT spectra on a 16 cm acrylic phantom with the GE
beam filter.......................... ....................... ... ................................ 114

Table 7-10: Percentage comparison of energy disposition categories of
monoenergetic beams with two typical CT spectra on a 16 cm acrylic phantom
w ith the G E beam filter....................................................... ......................... 114

Table 7-11: Effect of slice thickness on energy dispositions for 50 keV incident
photons on an 8 cm radius acrylic phantom with no filter................................ 116

Table 7-12: Energy dispositions for acrylic phantoms with indicated radii for 50
keV incident photons and 5 mm slice thickness.................................... ........ 116

Table 7-13: Variations of the energy dispositions with phantom material in an 8
cm radius phantom ...... .............. .. ...... ................. ................................... 117

Table 7-14: Variations of energy dispositions with phantom material in a 16 cm
radius phantom ......................................... ...................................................... 118

Table 7-15: Variation of energy dispositions with change in the SAD for 50 keV
incident photons and a 5 mm slice thickness ............ ........... ...................... 120

Table 7-16: Energy dispositions in the anthropomorphic head phantom for the
indicated incident energies with no beam filter............................. .................. 122

Table 7-17: Energy dispositions in the anthropomorphic head phantom with the
GE filter for the indicated incident energies............................ ................ 123








Table 7-18: Energy dispositions in the anthropomorphic body phantom for the
indicated incident energies and no beam filter.................................................... 124

Table 7-19: Energy dispositions in the anthropomorphic body phantom for the
indicated incident energies with the GE filter................................................... 125

Table 7-20: Energy dispositions for four x-ray spectra on anthropomorphic-type
phantoms with no filter..................................................................... .................. 127

Table 7-21: Percentage energy disposition for four x-ray spectra incident on
anthropomorphic-type phantoms................................... .............................. ... 127

Table 8-1: Comparison of the arithmetic average of the central value and 1 cm
depth value of the normalized C(r,L/2) ("average") and the normalized mean
CTDI (C) for monoenergies and spectra, calculated with no beam filter.......... 132














LIST OF FIGURES


Figure 1-1: Schematic drawing of CT scanner geometry..................................... 1

Figure 3-1: Discrete probability density function............................................ 34

Figure 3-2: Discrete cumulative probability distribution............................... 34

Figure 3-3: top: Continuous probability distribution; bottom: cumulative
distribution functions..................... ............... ...................... 35

Figure 3-4: An illustration of the Rejection Method. Only the random pairs that
fall under the curve are accepted................................................ ................... 37

Figure 4-1: Flow control diagram of the EGS system....................................... 54

Figure 4-2 Illustration of the geometry described by subroutine HOWFAR in the
user code CTMONO with five radial zones and five planar zones...................... 58

Figure 4-3. Source and phantom configuration for cylindrical phantom............. 60

Figure 4-4(a): Head Phantom..................... .......................... 64

Figure 4-4(b): Body Phantom..................................................... ...................... 65

Figure 5-1: CTDI in air per incident fluence for acrylic rods of various sizes........ 71

Figure 6-1: Central portion of dose profiles in a 5 mm diameter acrylic rod for a
3 mm slice thickness for the indicated incident photon energies......................... 76

Figure 6-2: Dose profiles along the central 5 mm in a 5 mm diameter acrylic rod
for 80 keV photons and the indicated slice thicknesses.................................. 77

Figure 6-3: Normalized dose profiles at the center and at 1 cm depth in an 8 cm
radius acrylic phantom for 80 keV incident photons and a 5 mm slice thickness... 77

Figure 6-4: Normalized dose profiles at the center and at 1 cm depth in a 16 cm
radius acrylic phantom for 80 keV incident photons and a 5 mm slice thickness... 78

Figure 6-5: Portion of the dose profile along the positive z-axis at the center of a
16 cm radius acrylic phantom for 80 keV photons and 5 mm slice thickness........ 78








Figure 6-6: C(r,t), normalized to the CTDIair value, as a function of the
integration limit I in an 8 cm radius acrylic phantom at 50 keV for a 10 mm slice
thickness..................................................... ........................................ 82

Figure 6-7: C(r,), normalized to the CTDIair value, as a function of the
integration limit e in an 8 cm radius acrylic phantom at 50 keV for a 5 mm slice
thickness..................................... ....................... ....................... ................... 82

Figure 6-8: C(r,e), normalized to the CTDIair value, as a function of the
integration limit e in an 8 cm radius acrylic phantom at 50 keV for a 1 mm slice
thickness.......................... ................ ....................... .. ............................ 83

Figure 6-9: C(r,e) values, normalized to the CTDIair value, as a function of the
integration limit e in an 8 cm radius acrylic phantom at 50 keV for a 5 mm slice
thickness in a 140 cm long phantom................................................................ 86

Figure 6-10: C(r,e), normalized to the CTDIair value, as a function of the
integration limit e in an 8 cm radius acrylic phantom for a 120 kVp spectrum
with 5.4 mm HVL, for a 10 mm slice thickness................................................ 86

Figure 6-11: C(r,t), normalized to the CTDIair value, as a function of the
integration limit t in an 16 cm radius acrylic phantom at 50 keV for a 5 mm slice
thickness .......................... .............. ............................ .... ...... 88

Figure 6-12: C(r,e), normalized to the CTDIair value, as a function of the
integration limit e in a 16 cm radius acrylic phantom for a 120 kVp spectrum
with 5.4 mm HVL, for a 10 mm slice thickness....................................................... 88

Figure 6-13: C(rL/2) as a function of radial location in the 8 cm radius
phantom, normalized to the CTDIair at the same energy, for a 5 mm slice
thickness and various incident energies.................................................... 90

Figure 6-14: C(r,L/2) as a function of radial location in the 16 cm radius
phantom, normalized to the CTDIair at the same energy, for a 5 mm slice
thickness and various incident energies....................... ............................... 90

Figure 6-15: Plots of the four spectra used in this work..................................... 92

Figure 6-16: C(r,L/2) as a function of radial location for four spectra for 5 mm
slices in an 8 cm radius acrylic phantom........................ ... ...... ........ 94

Figure 6-17: C(r,L/2) as a function of radial location for four spectra for 5 mm
slices in a 16 cm radius acrylic phantom................................ ... ............ 94








Figure 6-18: C(r,L/2) as a function of radial location in the acrylic 8 cm radius
phantom at 50 keV for a 5 mm slice thickness, and the indicated beam filter........ 96

Figure 6-19: C(r,L/2) as a function of radial location in the acrylic 16 cm radius
phantom at 50 keV for a 5 mm slice thickness and the indicated beam filter....... 96

Figure 6-20: C(r,L/2) as a function of radial location for the GE filter in a 16 cm
acrylic phantom at 63 cm SAD for the 5 mm slice thickness, and various incident
beam s........................................................................... ....... ................. ........ 99

Figure 6-21: C(r,L/2) as a function of slice thickness in a 8 cm radius acrylic
phantom at 80 keV ........................................................ .............................. 99

Figure 6-22: C(rL/2) plotted against relative distance from the center of the
phantom for various values of phantom radius R ................................................ 101

Figure 6-23: C(r,L/2) plotted against radial location for various phantom
materials in a 8 cm radius phantom for 50 keV photons and 5 mm slice thickness 101

Figure 6-24: C(rL/2) plotted against radial location for various phantom
material in an 16 cm radius phantom for 50 keV photons and 5 mm slice
thickness...................... .. .. ...... ............... .. .. ................ ........................... 103

Figure 6-25: C(r,L/2) plotted against radial location for various SAD values for
a 5 mm slice thickness and 50 keV photons........................................ 103

Figure 7-1: Percentage energy dispositions for an 8 cm acrylic phantom with no
beam filter and a 5 mm slice thickness........................................................... 107

Figure 7-2: Percentage energy dispositions for an 16 cm acrylic phantom with
no beam filter and a 5 mm slice thickness ......................................................... 108

Figure 7-3: Percentage energy dispositions acrylic phantoms with indicated radii
for 50 keV incident photons and 5 mm slice thickness....................................... 116

Figure 7-4: Percentage variations with phantom material. Data are for an 8 cm
radius phantom and 5 mm slice thickness at 50 keV........................................... 117

Figure 7-5: Percentage variations with phantom material. Data are for a 16 cm
radius phantom and 5 mm slice thickness at 50 keV........................................ 118

Figure 7-6: Percentage energy disposition for the anthropomorphic head
phantom for a 5 mm slice and no beam filter........................... ................ 122

Figure 7-7: Percentage energy disposition for the anthropomorphic body
phantom for a 5 mm slice and the GE beam filter............................................... 123








Figure 7-8: Percentage energy disposition for the anthropomorphic body
phantom for a 5 mm slice and no beam filter...................................................... 124

Figure 7-9: Percentage energy disposition for the anthropomorphic body
phantom for a 5 mm slice and the GE beam filter............................................... 125

Figure B : Source-phantom geometry.............................................................. 145

Figure C1: Geometry of intersection of a vector and an elliptical cylinder........... 148

Figure C2: Geometric possibilities of particle intersecting an elliptical cylinder.... 150

Figure C3: Geometry used to pick incident particle parameters.......................... 151








Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

A MONTE CARLO STUDY OF DOSE DISTRIBUTIONS AND ENERGY
IMPARTED IN COMPUTED TOMOGRAPHY DOSIMETRY PHANTOMS

by James Vincent Atherton

August 1993

Chairman: William Properzio, Ph.D.
Cochairman: Libby Brateman, Ph.D.
Major Department: Nuclear Engineering Sciences

The EGS4 Monte Carlo system was used to study the factors that affect dose

distributions and the energy imparted to dosimetry phantoms in computed tomography

(CT). The energy imparted is a useful quantity since it is directly correlated with the

stochastic risk to the patient. Also, since the energy imparted per slice is independent of

both the precise anatomic location of the scanned slice and the total number of slices

imaged, the total energy imparted from an exam can be easily found.

The parameters that were investigated were incident energy, beam shaping filter,

slice thickness, phantom size, phantom composition, and source-to-axis-distance (SAD).

These parameters were evaluated for irradiation by both monoenergetic x-ray beams and

typical x-ray spectra incident upon acrylic cylinders modeled on standardized Computed

Tomography Dose Index (CTDI) phantoms. In addition, two phantoms similar to the

anthropomorphic phantoms used in previous Monte Carlo investigations of patient

dosimetry were studied, and the results obtained from these phantoms were compared to

those obtained from the CTDI phantoms. In all cases the results obtained were

normalized to the dose to acrylic determined "in-air" at the isocenter of the scanner in the

absence of any other phantom. The results quantitatively demonstrate the effects of

modifying key CT parameters on energy deposition patterns in CT dosimetry phantoms.








Also, a method is presented that allows estimation of the total energy imparted per slice to

a CTDI phantom.

In general, the factors most affecting the quantitative values of energy deposited in

CT dosimetry phantoms are beam filtration, the size and composition of the object being

scanned, and the incident energy. The results also suggest that the energy imparted is only

moderately dependent on the characteristics of the incident x-ray spectrum.

The results of this work may be used as a basis for estimating the energy imparted to

a phantom or a patient for non-standard or newly developed CT exams or scanners. The

values of the energy imparted may also be used to estimate other quantities such as the

effective dose. These estimates may then be used as indices of stochastic risk for

comparisons between scanners or imaging modalities, exam optimization, or for

population-based studies of radiation detriment.














CHAPTER 1

INTRODUCTION



Computerized tomography (CT) is unlike any other radiological imaging method in

that an x-ray beam, tightly collimated in one dimension, is revolved about an object and

produces a complex three-dimensional pattern of energy deposition. The energy

distribution is a function of the x-ray spectrum, the x-ray beam filtration, collimation, and

the nature of the object scanned. A schematic drawing of the basic geometry of CT

scanners is shown in Fig. 1-1.

Since the clinical introduction of CT in the early 1970s, dosimetric investigations

have generally followed one of two tracks. One track has been the specification of

radiation doses to exact locations in patients and phantoms. The other track has been the

evaluation of total radiation risk to the patient from CT exams. The risk has generally

been assessed in one of two ways: organ-by-organ dose determination; or by estimation of

the total energy imparted to the patient.



x-ray source direction of source motion


Figure 1-1: Schematic drawing of CT geometry.








Initial investigations of radiation dose in CT followed the first track, reporting doses

to specific points or groups of points in phantoms. Horsley and Peters (1976) reported on

the dose to the skull from multiple scans from an EMI scanner as measured in a Rando

phantom. Dixon and Eckstrand (1978) used a film dosimetry system to measure surface

dose in a 19 cm diameter water phantom for both single and multiple slices. McCullough

and Payne (1978) used thermoluminescent dosimeters (TLD) to measure surface dose for

single and multiple scans in circular and elliptical acrylic phantoms. Shope et al. (1982)

used TLD to measure point doses for single and multiple scans in both cylindrical and

anthropomorphic phantoms in scanners from ten manufacturers. They presented surface

and center doses for the cylindrical phantom and the maximum dose measured in the

anthropomorphic phantom. Fearon and Vucich (1985) measured surface doses in

pediatric patients for several standard exams and compared the results to in-phantom

measured values. Storrs and Byrd (1988) measured the dose to the lens of the eye for

pediatric patients. There have been several studies of fetal or intrauterine dose from CT

(Wagner, et al. 1986, Guidozzi et al. 1987, Felmlee et al. 1990).

In 1981 Shope et al. proposed the Computed Tomography Dose Index (CTDI) as a

dose descriptor in CT. The CTDI is measured in cylindrical acrylic phantoms that are in-

tended to approximate the size and shape of a patient's head and body. The CTDI, as

defined by the United States Food and Drug Administration (FDA), is given by:

+7T
CTDI = T (z)dz (1-1)
7T

where n is the number of scans per slice, D(z) is the dose profile for a single scan along a

line perpendicular to the scanned slice, and T is the nominal slice thickness (FDA 1985).

The CTDI is given in units of absorbed dose to acrylic for a specific depth in one of the

two cylindrical phantoms. The CTDI is meant to approximate the dose at the depth of

measurement for a point in middle of fourteen contiguous scans. The CTDI is measured

the center of the phantom and at one centimeter depth from the surface. The FDA has








mandated that CT manufacturers supply users with CTDI values for a particular CT

scanner model for the range of tube potentials, tube currents, scan times, slice thicknesses,

and gantry angles likely to be used clinically. The FDA requirement has had the effect of

making the CTDI a standardized dose descriptor. Apart from its use as a dose estimator,

the CTDI can be employed for intercomparisons among scanners, and as an acceptance

testing and quality control check.

The CTDI is specified by the FDA to be given for two depths and with integration

limits that are a function of the nominal slice thickness.' It should be noted that the FDA

specifies only a minimum phantom length of 14 cm for CTDI measurements, and the

phantoms commercially available are 16 cm long.2 Note that the line integral of the dose

can be calculated at any depth in the phantom and with any integration limits (up to the

size of the phantom, of course). Viewed in this light, the CTDI as specified by the FDA is

a special case of a "family" of CTDI's. Part of this work investigates these different

classes of CTDI's and their relationship to the FDA's definition. A new, more generalized

dose index is defined, and its relation to the total energy deposited is demonstrated.

The second main track of dosimetric investigations in CT have endeavored to

evaluate the total radiation risk to the patient undergoing an examination. For doses of

less than 1 Gy, the total risk to the patient is considered to be the sum of the risks

associated with the individual irradiated organs (ICRP 1977). In diagnostic radiology the

main concern is with stochastic effects, e.g., carcinogenesis, genetic injury, and concepts

injury (ICRP 1986). Although doses from CT scans are greater than in conventional

radiography, the doses are below the threshold for deterministic effects such as lens

opacification or skin erythema (ICRP 1990). The method of estimating risk by using the

weighted sum of the doses to individual organs is the basis of the Effective Dose

Equivalent HE (ICRP 1977) and the Effective Dose E (ICRP 1990) quantities. The

1 Spokas (1982) has pointed out a weakness with the definition of the CTDI in that it combines a dose
measurement with an imaging parameter that is not measured. He suggested an alternate descriptor of CT
performance, the effective width w, but the CTDI remains to be the most widely used CT dose descriptor.
2 Nuclear Associates, Carle Place, NY 11514








weighting values used in calculation of E were intended to take into account the risk of

fatal cancer induction, the risk and the severity of non-fatal cancer induction, the

probability of severe hereditary effects, and the relative length of life lost for general

populations. The effective dose E is given by

E =wTHT
T

where HT is the equivalent dose in organ or tissue T and wT is the weighting factor for

tissue T. The tissue weighting factor represents the relative contribution from tissue or

organ T to the overall stochastic effects from uniform irradiation of the whole body. To

ensure that a uniform equivalent dose is numerically equal to a uniform equivalent dose

over the whole body, the sum of tissue weighting factors is one (ICRP 1990). The values

ofE and HE are given in units of Sv and are numerically equal to the absorbed dose in Gy,

because the radiation weighting factor wR (formerly the quality factor Q) is unity for

photons.3 The calculation of HE is similar to that for E, but different risk estimates (hence

different weighting factors) and fewer organs are used in the calculation. The risk

estimates used in the determination of the weighting factors for the effective dose

equivalent are based on an occupationally exposed population.

The effective dose equivalent and its successor the effective dose are both intended

as a method to assess non-uniform irradiations for use in radiation protection. The

weighting factors used in the calculation of HE are based on radiation risk estimates for

radiation workers, but the factors used in the calculation of E are based on risk estimates

for whole populations (ICRP 1990). The effective dose equivalent (and to some extent

the effective dose) have been used as a gauge in establishing radiation risks to patients in

diagnostic radiology. In the case of the effective dose equivalent, the patient population

has different demographics and presumably different risk factors than do radiation


3 As given in ICRP 60, HT = wRDTR where DTR is the absorbed dose averaged over organ or tissue
R
T from radiation R, and wR is the radiation weighting factor.








workers. In spite of this limitation, the effective dose equivalent continues to be used as a

yardstick in comparing patient doses in diagnostic radiology (Shrimpton et al. 1991, Jones

et al. 1991, Le Heron 1992). The use of the effective dose equivalent as an absolute

indicator of patient risk may not be as appropriate as the use of the effective dose, but it

does provide an objective method of comparison of differing irradiation conditions. One

aim of this project is the development of a method allowing estimation ofHE or E in CT,

based on the estimation of energy imparted values.

Organ dose assessment in CT has generally followed two approaches: measurement

in anthropomorphic phantoms, and Monte Carlo-based calculations in idealized math-

ematical phantoms. In general, physical measurement in an anthropomorphic phantom has

the advantage using of a realistic phantom but tends to be a very labor-intensive process.

Monte Carlo simulation can be used to study a wide range of irradiation configurations,

but it has the disadvantage of being computationally demanding and time intensive.

Huda and Sandison (1985) used TLD to measure organ dose from an EMI scanner

in a Rando phantom and presented risk estimates for standard exams, based on the ICRP

risk estimates for radiation workers. Huda and Sandison (1986) also published effective

dose equivalents based on the EMI scanner measurements. Fearon and Vucich (1985)

measured organ doses in a pediatric anthropomorphic phantom, although they presented

neither HE values nor risk estimates. In 1989 Huda et al. extrapolated the EMI

measurements to other models of CT scanners and presented HE values for standard

exams. Nishizawa et al. (1991) measured organ doses in a Rando phantom for a wide

range of CT scanners used in Japan. They similarly determined the effective dose

equivalents for standard exams for selected CT scanners.

There have been two significant studies using mathematical modeling for organ dose

assessment. In 1985 Drexler et al. published organ dose tables calculated by using Monte

Carlo simulation for standardized CT exams. These tables provide doses to a number of

organs in mathematically modeled male and female phantoms. The CT geometry was








based on that of a Siemens Somatom scanner. Faulkner and Moores (1987) used the

Drexler data to provide organ dose estimates for three different types of CT scanners.

Panzer et al. (1989) have also used the Drexler data to estimate organ doses, based on a

survey of 122 facilities in the former West Germany. Neither Faulkner and Moores nor

Panzer et al. presented effective dose equivalent values for standard exams. Recently,

Shrimpton et al. (1991) and Jones et al. (1991) of the National Radiation Protection

Board (NRPB) in the United Kingdom published organ dose tables for a range of CT

scanners and standard exams. The NRPB results were obtained using Monte Carlo

simulation on a mathematical anthropomorphic phantom. Values of HE and E were

included with the organ dose values presented.

Comparison of results from the organ-dose studies listed above shows a wide

variation in the results, even when comparing the same type of scanner for the same type

of exam. A comparison of findings from various authors is given in Table 1-1, which lists

effective dose equivalent values for four standard exams for the scanners which the studies

had in common. It is seen that the values of the effective dose equivalent for the same

exam and the same machine can vary by more than a factor of two. These variations seem

to be mainly due to three reasons: differences in the clinical protocol of "standard"

exams; differences in the treatment of the non-specified "remainder" tissues; and the


Table 1-1: Comparison of effective dose equivalents for standard exams for three common
CT scanners.
Effective dose equivalent in mSv by scanner
Scanner: GE 9800 Siemens DRH Siemens DRG
_study*-
Exam A B C A C B-(male) -(female) C
Head 1.9 4.3 2.3 2.5 2.6
Chest 15.7 7.7 9.0 16.0 5.8 6.5 6.5 5.3
Abdomen 6.3 10.3 9.5 8.1 2.6 2.7 7.0
Pelvis 6.7 11.0 13.4 7.9 6.2 12.5 5.5
* Data are from Huda et al. (1989) abbreviated A; Nishizawa et al. (1991) abbreviated B;
and Shrimpton et al. (1991) abbreviated C.








differences between TLD measurements in physical phantoms and mathematical modeling.

Thus, even with standardized measurement techniques and standard phantoms, estimation

of organ dose and patient risk provides inconsistent and widely varying results. Further-

more, organ dose assessment is difficult to perform. One must either use a large number

of small dosimeters placed in an anthropomorphic phantom, or one must employ a

mathematical model which must be constructed and tested for accuracy. Moreover, in

order to study different examinations and exposure geometries, multiple measurements

must be performed. Organ dose studies therefore tend to be labor and time intensive and

usually provide results for only a limited number of examination types or exposure

geometries.

Another method of assessing the radiation risk to the patient has been through

estimation of the total energy deposited or imparted to the patient from an exam. As the

studies cited below indicate, the energy imparted to the patient (formerly "integral dose")

has been shown to be a useful estimator of stochastic risk in diagnostic radiology. The use

of the energy imparted to a patient as a risk estimator assumes a homogeneous mix of

radiosensitive and radioresistant tissues within the body (Wall et al. 1988). This method is

both simpler and cruder than organ dose estimation, and several arguments can be made

for its use. The energy imparted is a physical quantity and is independent of organ-

weighting factors. These weighting factors are based on risk estimates which in turn are

subjectively-judged and subject to change following developments in the knowledge of

radiobiology (Greening 1986). They are subjective values and are not a physical quantity.

Also, organ dose determinations, as demonstrated above, vary widely in their results and

ignore inter-patient variations in organ dimensions and positioning (Harrison 1983).

Concerning the efficacy of using energy imparted as a risk indicator, Wall et al. (1988,

p. 8) state:

...the distribution of sensitive organs within the trunk may in many
circumstances be sufficiently uniform for the errors involved in this
approach to be no greater than those associated with using a suitable








combination of specific organ doses and weighting factors to express
the overall risk.

They conclude (p.55):

The reasonable degree of correlation that is evident between energy
imparted and health detriment for a wide range of x-ray examinations
suggests that in appropriate circumstances it [the energy imparted]
can represent a useful practical quantity for estimating the risk to
patients.

The studies below show that the energy imparted can be used outright as an indicator of

patient risk, or it can be used to estimate other risk indicators.

Bengtsson et al. (1978) used the energy imparted as a basis for risk estimates for

conventional radiographic examinations. They based their estimates on the risk coefficient

1.65x 10-2 Sv' for radiation workers (ICRP 1977) and identified a risk of 2.0x 104 J-1,

which corresponds to a ratio of 12.1 mSv/J. Southon (1980) reported values for energy

imparted in comparisons of six different CT scanners but did not present any explicit risk

estimates corresponding to the energy imparted values. Shrimpton (1985) studied the

relationship between the energy imparted and the effective dose equivalent in common

radiographic examinations and found a linear relationship (within a factor of two) between

the two of 13.8 mSv/J. Le Heron (1992) claimed "good agreement" between the energy

imparted and the effective dose in conventional radiography, although he did not present

any values. He based his work on the NRPB organ dose data (Jones and Wall 1986).
Alm Carlsson and Carlsson (1986) discussed the merits of using a quantity called the

mean absorbed dose D, given as D = s/M, where e is the total energy imparted and Mis

the mass of the patient. In a homogeneous whole-body exposure, the ratio of HE/D

would be 1.0 Sv/Gy, which corresponds to a HE/8 ratio of 14.3 mSv/J for a 70 kg patient

or phantom. The authors analyzed studies of patient dose in radiology (Laws and

Rosenstein 1978, Rosenstein 1982, Huda 1984, and Stentstrom et al. 1986) and

concluded that in all cases, the ratios HE/D were in the range of 0.44 Sv/Gy to 2.8 Sv/Gy

with a mean near 1.0 Sv/Gy. For a 70 kg patient or phantom, these values represent HE/E








Table 1-2: Ratios of the effective dose equivalent to the energy imparted for different
exams and scanners.
H/ __ratios in mSv/J
Exam type EMI 5005 GE 9800 Siemens DRH
Head 10.5 16.9 16.5
Chest 28.1 19.5 21.5
Abdomen 14.2 23.5 21.5
Pelvis 17.8 29.4 24.1
The values shown for the EMI scanner are based on data from Huda (1984) and Huda and
Sandison (1986). The data for the GE and Siemens scanners are from a private
communication from Dr. P.C. Shrimpton of the NRPB.



ratios of 6.3 mSv/Jto 40 mSv/J with a mean near 14.3 mSv/J. The radiological studies

included in this assessment all involved examinations of the head and trunk.

In the specific area of CT imaging, Huda (1984) investigated the relationship

between energy imparted and risk for four standard exams in CT. The risk estimates were

based on those of the ICRP (ICRP 1977). He based his conclusions on measurements

made in an anthropomorphic phantom scanned in an EMI 5005 scanner. The results were

also used to calculate HE values for the exams (Huda and Sandison 1986). The ratios of

HEf/ based on these two works are shown in Table 1-2. Also included in Table 1-2 are

values of HE/e for the GE9800 and Siemens DRH, obtained from Dr. P.C. Shrimpton of

the NRPB Medical Dosimetry Group in the United Kingdom. The GE and Siemens data

are based on the NRPB CT organ dose study (Shrimpton et al. 1991 and Jones et al.

1991). The values for the GE and Siemens units are based on the same scan protocols

(mAs, slice width, number of slices) as used in the EMI exams.

The values of HE/ shown in Table 1-2 are in the range of HE/e values derived from

the Aim Carlsson and Carlsson data, although somewhat larger. The data for the two

more modem scanners are quite consistent. These data show how energy imparted values

may be employed to estimate the effective dose equivalent for given irradiation conditions.

By combining the data in Table 1-2 with values of the energy imparted in the different








anatomic regions, it should be possible to make estimates of the effective dose equivalent

for a wide variety of possible exposure conditions.

The overall goal of this project is provision of the fundamental knowledge necessary

for the development of a method allowing a simple and convenient way of estimating the

energy imparted per CT slice. Values of the energy imparted may then be used as an

index of stochastic risk. The organ dose studies cited above provide data for only a

limited number of exam types and scanner types. A more generalized method, employable

by clinical medical physicists without access to specialized and costly dosimetry systems,

would provide a way of estimating a risk-based index for non-standard or newly

established exams or CT scanners. The energy imparted per CT slice has the advantage of

being independent of both the number of slices and the exact anatomic location of each

slice. Energy imparted values and subsequent risk estimates can be used for exam

optimization, intercomparisons between different imaging modalities, and population-

based studies of detriment.

The basis of this project is the assumption that the energy imparted can be used as

an index of the stochastic risk to a patient. This assumption is clearly justified on the basis

of the works cited above. Furthermore, Carlsson and Aim Carlsson (1990) have stated

that, for risk estimates, only crude dosimetry is necessary. The wide availability and use of

CTDI phantoms make them an attractive object to study. However, the question of how

closely CTDI phantoms model actual patients or anthropomorphic phantoms has not yet

been determined. This work therefore studies the factors that determine how much energy

is deposited in both CTDI and anthropomorphic phantoms and the patterns of that energy

deposition. Quantified estimates of the energy imparted to anthropomorphic phantoms are

compared to the energy deposited in CTDI phantoms.

This study uses the EGS Monte Carlo system to investigate patterns of energy

deposition in phantoms (Nelson et al. 1985). The influence of beam energy, beam








filtration, phantom size, phantom composition, source-to-axis distance (SAD), and slice

thickness on the pattern of energy distribution is systematically determined.

The variables studied are listed in Table 1-3. Each of the parameters was varied

over the stated range with the remaining variables held constant at their default value. For

each run (i.e., a series of Monte Carlo simulations with identical initial parameters), the

energy deposition pattern is found as a function of radial and longitudinal position within

the phantom. The dose to each position is calculated, as are line integrals of the dose.

The doses as well as the energies deposited in regions of the phantom are normalized to

the dose measured in a small acrylic rod placed at the isocenter of the hypothetical

scanner. In addition, the Monte Carlo method allows labeling of photons so as to

differentiate between scattered and primary contributions to any quantity of interest,

therefore allowing identification of scattered and primary components of the energy

deposited and of the energy exiting the phantom. Calculations performed in this manner

allow the study of the nature of the dose distributions in a way that is not practical by

physical measurement. The CTDI is investigated as a function of radius and as a function

of the integration limits. The relationship between the CTDI and the energy imparted is

studied for the range of irradiation conditions considered.

The first step in this project was to determine the ranges of variation of the

important characteristics of CT scanners. Although this study uses a simplified model of

CT scanners, it is desirable to base the mathematical model on realistic geometries and

characteristics. Results from the study are then more pertinent to clinical situations. To

accomplish this goal, a survey form was mailed to several CT manufacturers that asked for

descriptions of their scanners. The results provided a good overview of the characteristics

of common CT scanners in clinical use. The results from the survey were used in

determining both the range of values and the default values of the important scanner

characteristics in the Monte Carlo calculations.









Table 1-3: The range of parameters examined in this work.
Parameter default Range investigated
Beam Energy 50 keV 40, 50, 60, 80, 100, 120, 140 keV,
typical spectra
Beam Filter none none, perfect4, GE9800
Slice Thickness 5 mm 1, 2, 3, 5, 10 mm
Phantom Material acrylic acrylic, water, bone, fat, lung,
muscle5
Phantom Radius 8, 16 cm 6, 8, 10, 12, 14, 16, 20 cm
SAD 65 cm 50, 65, 80 cm
Phantom Type CTDI CTDI, anthropomorphic-analog


This dissertation is organized in the following way. The results from the CT

manufacturer survey are presented in Chapter 2. Chapter 3 provides a background of the

past uses of Monte Carlo simulation in diagnostic radiological physics and describes the

basic techniques of Monte Carlo simulation. The EGS system, along with the details of

the models used in this work, are presented in Chapter 4. The accuracy of the Monte

Carlo model is verified in Chapter 5. Chapter 6 introduces a new quantity, the

Generalized CTDI (GCTDI), and presents the results concerning dose deposition patterns,

the relationship of the energy imparted to the GCTDI, and the variation of the GCTDI as

a function of the variables in Table 1-2. The results concerning the energy imparted to

phantoms are presented in Chapter 7. Chapter 8 discusses the results and presents

conclusions concerning the entire study.







4 A perfect filter is constructed such that the total path length for any ray emerging from the x-ray source
within the fan beam is constant. For a right cylindrical cylinder of radius R with long axis normal to the
scan plane, the filter thickness t is found from 2R = t + L where L is the path length through the
cylinder.

5 In this category the phantoms are assumed to be of the same dimensions as the CTDI phantoms but
comprised of the different materials listed. The material data were obtained from ICRU Report No. 44
(ICRU 1989).













CHAPTER 2

SURVEY OF COMMON CT SCANNER CHARACTERISTICS



Introduction


The first step in this project was to determine the ranges of variation of the

characteristics of CT scanners. This study uses a simplified mathematical model of CT

scanners which is based on realistic geometries and characteristics. A survey form was

mailed to several CT manufacturers which asked for descriptions of their scanners. The

results provided a good overview of the characteristics of common CT scanners in clinical

use. The results from the survey were used in determining the both the range of values

studied and the default values of the scanner characteristics.

Basic Operational Principles


Computed tomography was introduced to the radiology community in the early

1970s. The modality's explosive growth is illustrated by the fact that approximately 2.2

million CT studies were performed in hospitals in 1980, only seven to eight years after the

first commercial CT units became available (Bunge and Herman 1987). CT is now

considered an essential element in radiological imaging.

In a computed tomography examination, the patient is exposed to a highly

collimated beam of x rays at a number of angular increments. The transmitted x rays are

absorbed by a series of detectors, and the data from these projections are used to create a

two-dimensional map of linear attenuation coefficients, i.e., an image.








The configuration of x-ray tube, x-ray detectors, and the manner in which these

elements are moved, has changed through several so-called "generations." The first CT

scanner, introduced by Hounsfield (1973), had a single detector and a pencil x-ray beam,

which moved in a translate-rotate manner, i.e., the detector and x-ray tube translated along

a line parallel to the image plane acquiring data. The tube and detector were then rotated

one degree, and the process repeated. To overcome the problems of slow scan times and

poor resolution, a second generation was devised with multiple detectors and a small fan-

shaped x-ray beam. The second generation retained the original translate-rotate motion.

In the third generation of CT scanners, the translate motion of the tube and

detectors was replaced with a rotate-only motion. A broad fan-shaped x-ray beam is used

with several hundred detectors, and the entire tube-detector system rotates around the

machine's isocenter. The fourth generation CT scanner has stationary detectors, and only

the x-ray tube rotates around the patient.

Recently a new type of CT scanner has been developed by Imatron.' In these

scanners, the patient lies on a large evacuated chamber. An electron beam is scanned

across a series of small tungsten targets lining the interior of the evacuated chamber.

These targets act like the anode in a standard x-ray tube and produce x rays which are

collimated, transverse the patient, and are detected by an array of detectors. Both the

tungsten targets and the array of detectors cover an arc of 2100. These machines have the

advantage of fast (50 100 ms) scan times but have slightly poorer image quality than

conventional CT scanners.

The following manufacturers provided information regarding their scanners:

Elscint2, Picker3, Siemens4, and General Electric5 (GE). Elscint supplied data concerning

four of their scanners, all third generation. Picker provided information on four of their


1 Imatron, Inc., San Francisco CA 94080.
2 Elscint Ltd., Haifa, Israel
3 Picker International Inc., Highland Heights, OH 44143
4 Siemens Medical Systems Inc., Iselin, NJ 08830
5 GE Medical Systems, Milwaukee, WI 53201








fourth generation models. Siemens supplied data for three of their scanners, which are all

third generation. GE provided data for two versions of the same model scanner.

Differences Among CT Scanners From Various Manufacturers


This section outlines the differences and similarities among scanners from different

manufacturers. A questionnaire was designed to allow comparisons among scanners in

physical construction, radiologic technique factors available, and reported CTDI. A copy

of this questionnaire is presented in Appendix A. The questionnaire was sent to the major

manufacturers of CT scanners. This questionnaire is the primary source of the information

presented in this chapter; however, not all manufacturers complied with the request for

complete information. The data provided are given in this section. One manufacturer did

not return the questionnaire; and one supplied the data requested, with the exception of a

description of their beam-shaping filter. These missing data are described below.

The sections below describe the features of the various CT scanners provided by the

manufacturers. Two tables summarize the CTDI data at the end of the section. (Note

that the electrical potential across the x-ray tube is abbreviated as "kVp", the current

through the x-ray tube is represented by "mA", and the product of the exposure time and

the tube current is abbreviated as "mAs".)


ELSCINT EXEL 2400 Elite/2400E

Radiologic technique factors available: Four peak tube potentials are available:

(100, 120, 130, and 140 kVp); however, Elscint recommends that only 120 and 140 kVp

be used routinely. The tube currents available are 50, 100, 120, 150, and 200 mA, and the

scan times available are 0.5, 1, 2, 4, and 8 s, with 1 to 4 s routinely used. The generator is

3-phase with continuously-supplied voltage.

Tube characteristics: There are two focal spots of nominal sizes 1.5 mm x 1.4 mm

and 0.8 mm x 1.4 mm. The manufacturer specified neither the criteria used for focal spot








size selection nor the orientation of focal spot sizes with respect to the anode-cathode

axis. The anode angle is 70, and the anode is constructed of tungsten-on-graphite

composite.

Geometry: The four slice thicknesses available (1.2, 2.5, 5, and 10 mm) are

specified at the isocenter. The fan beam arc is 220. The source-to-isocenter distance is 63

cm, and the isocenter-to-detector distance is 46 cm. The scanner is third-generation with

three scan angles (2020, 3600, 3820) available, although 3600 and 3820 are routinely used.

The beam is oriented such that heel effect intensity variation is not present in the scan

plane. The gantry tilt angles were not supplied.

Filtration: The inherent filtration is equivalent to 1.1 mm Al equivalent at 100 kVp.
The unit has both a flat, fixed, added filter and a shaped compensating filter, the details of

which are proprietary. The couch attenuation is equivalent to 1.4 mm Al (the kVp was

not specified).

CTDI: The CTDI for the head phantom using the recommended head technique

(120 kVp, 300 mAs, 10 mm) is 3.9 cGy (13.0 mrad/mAs) in the center and 4.6 cGy (15.3

mrad/mAs) in the 12 o'clock (maximum value) position. For the body phantom and

technique, the CTDI is 1.7 cGy (5.4 mrad/mAs) in the center and 3.6 cGy (11.4

mrad/mAs) at 12 o'clock (140 kVp, 315 mAs, 10 mm). Elscint has provided CTDI slice-

width and voltage dependence data. The maximum deviation from the specified CTDI

values is 20%.


ELSCINT 2400

Radiologic technique factors available: Four peak tube potentials are available (100,
120, 130, and 140 kVp); however, Elscint recommends that only 120 and 140 kVp be

used routinely. The tube currents available are 50, 100, 120, 150, and 200 mA, and the

scan times available are 0.5, 1, 2, 4, and 8 seconds, with 1 to 4 s routinely used. The

generator is 3-phase with continuously-supplied voltage.








Tube characteristics: There are two focal spots of nominal sizes 1.5 mm x 1.7 mm

and 0.8 mm x 1.7 mm. The anode angle is presently 100 (although it will be changed to 70

in the near future, according to the manufacturer), and the anode is constructed of

tungsten on graphite composite. Again, neither the orientation of the focal spots nor the

criteria used for focal spot size selection were specified.

Geometry: The four slice thicknesses available (1.2, 2.5, 5 and 10 mm) are specified
at the isocenter. The fan beam arc is 220. The source-to-isocenter distance is 63 cm, and

the isocenter-to-detector distance is 46 cm. The scanner is third-generation with three

scan angles (2020, 3600, 3820) available, although 360 and 3820 are routinely used. The

beam is oriented such that heel effect intensity variation is not present in the scan plane.

The gantry tilt angles were not supplied.

Filtration: The inherent filtration is 1.1 mm Al equivalent at 100 kVp. The unit has

both a flat, fixed, added filter and a shaped compensating filter, the details of which are

proprietary. The couch attenuation is 1.4 mm Al equivalent (at an unspecified kVp).

CTDI: The CTDI for the head phantom using the recommended head technique
(120 kVp, 300 mAs, 10 mm) is 3.8 cGy (12.7 mrad/mAs) in the center and 4.5 cGy (15

mrad/mAs) in the 12 o'clock (maximum value) position. For the body phantom and

technique (140 kVp, 310 mAs, 10 mm), the CTDI is 1.6 cGy (5.2 mrad/mAs) in the center

and 3.5 cGy (11.3 mrad/mAs) at 12 o'clock. Elscint has provided CTDI slice-width and

voltage dependence data. The maximum deviation from the specified CTDI values is
20%.


ELSCINT 1800

Radiologic technique factors available: Four peak tube potentials are available (100,
120, 130, and 140 kVp); however, Elscint recommends that only 120 and 140 kVp be

used routinely. The tube currents available are 50, 100, 120, 150, and 200 mA, and the








scan times available are 0.5, 1, 2, 4, and 8 s, with 1 to 4 s routinely used. The generator is

3-phase with continuously-supplied voltage.

Tube characteristics: The nominal focal spot size is 1.5 mm x 17 mm. The anode

angle is 100, and the anode is constructed of tungsten-on-graphite composite.

Geometry: The three slice thicknesses available (2, 5, and 10 mm) are all specified

at the isocenter. The fan beam arc is 220. The source-to-isocenter distance is 63 cm, and

the isocenter to-detector-distance is 46 cm. The scanner is third-generation, with three

scan angles (2020, 3600, 3820) available, although 3600 and 3820 are routinely used. The

beam is oriented such that heel effect intensity variation is not present in the scan plane.

The gantry tilt angles were not supplied.

Filtration: The inherent filtration is equivalent to 1.1 mm Al at 100 kVp. The unit

has both a flat, fixed, added filter and a shaped compensating filter, the details of which

are proprietary. The couch attenuation is 1.4 mm Al equivalent at an unspecified kVp.

CTDI: The CTDI for the head phantom using the recommended head technique

(120 kVp, 300 mAs, 10 mm) is 3.1 cGy (10.3 mrad/mAs) in the center and 3.6 cGy (12.0

mrad/mAs) in the 12 o'clock (maximum value) position. For the body phantom and

technique (140 kVp, 310 mAs, 10 mm), the CTDI is 1.3 (4.2 mrad/mAs) cGy in the center

and 3.1 cGy (10 mrad/mAs) at 12 o'clock. Elscint has provided CTDI slice-width and

voltage dependence data. The maximum deviation from the specified CTDI values is

20%.


GENERAL ELECTRIC 9800

The GE 9800 CT scanner is available with two different detector systems. The

older version, with xenon detectors, has several different characteristics from the newer

version, with "HiLight" (solid state) detectors. These differences are noted where

appropriate.








Three peak tube potentials available (80, 120, and 140 kVp); GE recommends 120
kVp for routine use. The tube currents available are 10, 20, 40, 70, 100, 120, 170, 200,

240, and 300 mA, and the scan times available are 2, 3, 4, and 8 s. The generator is 3-

phase with continuously-supplied voltage.

Tube characteristics: The nominal focal spot size is 0.9 mm x 0.7 mm. The anode
angle is 70, and the anode is constructed of tungsten and rhenium on a titanium-zirconium-

molybdenum substrate.

Geometry: The 9800 series are third generation scanners. The four slice
thicknesses available are 1.5, 3, 5, and 10 mm, all specified at the isocenter. The fan beam

arc is 45. The beam is oriented such that heel effect intensity variation is not present in

the scan plane. The source-to-isocenter distance is 63 cm, and the isocenter-to-detector

distance is 47 cm. The gantry can tilt 200.

Filtration: The inherent filtration of the x ray tubes were not explicitly listed.
However, the values of the half-value layers of the central rays are listed in Table 2-1
below.

CTDI: The CTDI of the 9800 varies with the type of detector. Solid State: The
CTDI for the head phantom using the recommended head technique (120 kVp, 340 mAs,

10 mm) is 4.0 cGy (11.8 mrad/mAs) in the center and also 4.0 cGy (11.8 mrad/mAs) at 1

cm depth. For the body phantom and technique (120 kVp, 340 mAs, 10 mm), the CTDI

is 1.1 cGy (3.2 mrad/mAs) in the center and 2.0 cGy (5.9 mrad/mAs) at 1 cm depth.

Xenon: The CTDI for the head phantom using the recommended head technique (120
kVp, 340 mAs, 10 mm) is 5.0 cGy (14.7 mrad/mAs) in the center and 4.8 cGy (14.1

mrad/mAs) at 1 cm depth. For the body phantom and technique (120 kVp, 340 mAs, 10


Table 2-1: Half value layers for the GE 9800 scanner as function of detector and kVp.
Detectors 80 kVp 120 kVp 140 kVp
_mm All [mm All mm All
Solid State 3.9 6.2 7.1
Xenon 3.6 5.0 6.0








mm), the CTDI is 1.4 cGy (4.1 mrad/mAs) in the center and 2.5 cGy (7.4 mrad/mAs) at 1

cm depth. GE provided CTDI slice-width and voltage dependence data, although the
maximum variation of the CTDI values were not provided.


PICKER 1200 SX

Radiologic technique factors available: The peak tube potentials available cover the

range from 100 to 140 kVp in 5 kVp steps. The tube currents range from 5 to 200 mA in

15 mA increments, and the scan times available are 1 to 20 s. The generator waveform is

high-frequency; the frequency was not provided.

Tube characteristics: The two nominal focal spot sizes are 0.5 mm x 1.65 mm and
0.9 mm x 2.4 mm. The orientations of the focal spots with respect to the scan plane were

not indicated. The anode angle is 120, and the anode is constructed of a tungsten-rhenium

track on molybdenum.

Geometry: The slice thicknesses available range from 1 to 10 mm in 1 mm

increments at the isocenter. The fan beam arc is 240 for head scans and 48 for body

scans. The source-to-isocenter distance is 64 cm, and the isocenter-to-detector distance is

85 cm. The scanner is fourth-generation with three scan angles (2200, 360, 3980)

available, although only 3600 is routinely used. The beam is oriented such that heel effect

intensity variation is not present in the scan plane. The gantry can tilt 200.

Filtration: The inherent filtration is 3 mm Al equivalent at 100 kVp. The unit has

both a flat, fixed, added filter and a shaped compensating filter, the details of which are

proprietary and not disclosed. The couch attenuation is 0.8 mm Al equivalent at an
unspecified kVp.

CTDI: The CTDI for the head phantom using the recommended head technique
(130 kVp, 240 mAs, 10 mm) is 3.7 cGy (15.4 mrad/mAs) in the center position and 4.1

cGy (17.1 mrad/mAs) at 1 cm depth. For the body phantom and technique (130 kVp, 240

mAs, 10 mm), the CTDI is 1.5 cGy (6.3 mrad/mAs) in the center and 4.2 cGy (17.5








mrad/mAs) at 1 cm. Picker has provided CTDI slice-width, voltage dependence, slice

thickness, added filtration and scan angle dependence data. The maximum deviation from

the specified CTDI values is 15%.


PICKER 10 Premier

Radiologic technique factors available: The only peak tube potential available is 130

kVp. The tube currents available are 20, 45, 65, 85, 105, and 125 mA, and the scan times

available are 2 and 4 s. The solid-state, high-frequency generator is mounted on a rotating

frame.

Tube characteristics: The nominal focal spot size is 0.9 mm x 2.0 mm. The anode

angle is 120, and the anode is constructed with a tungsten-rhenium track on molybdenum.

Geometry: The slice thicknesses available are: 2, 5, and 10 mm, all specified at the

isocenter. The fan beam arc is 480. The source-to-isocenter distance is 64 cm, and the

isocenter-to-detector distance is 85 cm. The scanner is fourth-generation with two scan

angles (3600 and 3790) available, although 3600 is routinely used. The beam is oriented

such that heel effect intensity variation is not present in the scan plane. The gantry can tilt

300.

Filtration: The inherent filtration was specified to be 3 to 4 mm Al equivalent at 130

kVp. The unit has both a flat, fixed, added filter and a shaped compensating filter, the

details of which are proprietary and not disclosed. The couch attenuation is 0.8 mm Al

equivalent at an unspecified kVp.

CTDI: The CTDI for the head phantom using the recommended head technique

(130 kVp, 260 mAs, 10 mm) is 4.2 cGy (16.2 mrad/mAs) in the center position and 3.9

cGy (15.0 mrad/mAs) in the 12 o'clock position. For the body phantom and technique

(130 kVp, 260 mAs, 10 mm) the CTDI is 1.6 cGy (6.2 mrad/mAs) in the center and 3.5

cGy (13.5 mrad/mAs) at 12 o'clock. Picker has provided CTDI slice-width, voltage








dependence, slice thickness, added filtration and scan angle dependence data. The

maximum deviation from the specified CTDI values is 15%.


PICKER IQ and IQ T/C

Radiologic technique factors available: The only peak tube potential available is 130

kVp. The tube currents are 20, 45, 65, 85, 105, and 125 mA, and the scan times available

are 2 and 4 s. The solid-state, high-frequency generator is mounted on a rotating frame.

Tube characteristics: The nominal focal spot size is 0.9 mm x 2.0 mm. The anode

angle is 13.5, and the anode is constructed with a tungsten-rhenium track on

molybdenum.

Geometry: The slice thicknesses available are 2, 5, and 10 mm, all specified at the

isocenter. The fan beam arc is 480. The source-to-isocenter distance is 64 cm, and the

isocenter-to-detector distance is 85 cm. The scanner is fourth-generation with two scan

angles (3600 and 3790) available, although 3600 is routinely used. The beam is oriented

such that heel effect intensity variation is not present in the scan plane. The gantry can tilt

300.

Filtration: The inherent filtration was specified as 3 to 4 mm Al equivalent at 130

kVp. The unit has both a flat, fixed, added filter and a shaped compensating filter, the

details of which are proprietary and were not disclosed. The couch attenuation is 0.8 mm

Al equivalent at an unspecified kVp.

CTDI: The CTDI for the head phantom using the recommended head technique

(130 kVp, 260 mAs, 10 mm) is 4.7 cGy (18.1 mrad/mAs) in the center position and 4.4

cGy (16.9 mrad/mAs) in the 12 o'clock position. For the body phantom and technique

(130 kVp, 260 mAs, 10 mm), the CTDI is 1.7 cGy (6.5 mrad/mAs) in the center and 3.9

cGy (15.0 mrad/mAs) at 9 o'clock. Picker provides CTDI slice-width, voltage

dependence, slice thickness, added filtration and scan angle dependence data. The

maximum deviation from the specified CTDI values is 15%.








PICKER PQ-2000

Radiologic technique factors available: The peak tube potentials available are 80,

100, 120, 130, and 140 kVp. The tube currents are 30, 50, 65, 100, 125, 150, 175 and

200 mA, and the scan times available are 1, 1.5, 2, 3, and 4 s. The solid-state, high-

frequency generator is mounted on a rotating frame.

Tube characteristics: There two nominal focal spot sizes are 0.9 mm x 2.4 mm and

0.6 mm x 1.65 mm. The anode angle is 12.50, and the anode is constructed with a

tungsten-rhenium track on molybdenum.

Geometry: The slice thicknesses available are 1.5, 2, 3, 4, 5, 8, and 10 mm, at the

isocenter. The fan beam arc is 44. The source-to-isocenter distance is 64 cm, and the

isocenter-to-detector distance is 85 cm. The scanner is fourth-generation with a scan

angle of 3600. The beam is oriented such that heel effect intensity variation is not present

in the scan plane. The gantry can tilt 30.

Filtration: The inherent filtration is 3 mm Al equivalent at 100 kVp. The unit has

both a flat, fixed, added filter and a shaped compensating filter, the details of which are

proprietary and were not disclosed. The couch attenuation is 0.8 mm Al equivalent at an

unspecified kVp.

CTDI: Picker has indicated that the CTDI report for the PQ-2000 is not complete

but that the CTDI values can be expected to be similar to those for the IQ. The CTDI for

the head phantom using the recommended head technique (130 kVp, 250 mAs, 10 mm) is

4.7 cGy (18.8 mrad/mAs) in the center position and 4.4 cGy (17.6 mrad/mAs) in the 12

o'clock position. For the body phantom and technique (130 kVp, 250 mAs, 10 mm), the

CTDI is 1.7 cGy (6.8 mrad/mAs) in the center and 3.9 cGy (15.6 mrad/mAs) at 9 o'clock.

Picker has provided CTDI slice-width, voltage dependence, slice thickness, added

filtration and scan angle dependence data. The maximum deviation from the specified

CTDI values is 15%.








SIEMENS Somatom PLUS

Radiologic technique factors available: The peak tube potentials available are 80,

120, and 137 kVp. The tube currents available are 70 to 300 mA (the increments were

not specified), and the scan times available are 0.7, 1, 2, 3, 4, 5, and 6 s. The generator

has a high-frequency waveform.

Tube characteristics: The nominal focal spot size is 1.1 mm x 1.8 mm. The anode

angle is 100, and the anode is tungsten-on-graphite.

Geometry: The slice thicknesses available are 1 through 10 mm in 1 mm

increments, specified at the isocenter. The fan beam arc is 420. The source-to-isocenter

distance is 70 cm, and the isocenter-to-detector distance was not supplied. The scanner is

third-generation with two scan angles (3600 and 2400) available, although 3600 is

routinely used. The beam is oriented such that heel effect intensity variation is not present

in the scan plane. The gantry can tilt 250.

Filtration: The inherent filtration is 2.5 mm Al equivalent at 120 kVp. The unit has

only a flat, fixed, added filter, which is 0.2 mm Cu. The couch attenuation is 1.2 mm Al

equivalent at an unspecified kVp.

CTDI: The CTDI for the head phantom using the recommended head technique

(120 kVp, 500 mAs, 10 mm) is 3.9 cGy (7.7 mrad/mAs) in the center position and 5.0

cGy (9.9 mrad/mAs) at the 1 cm-depth position. For the body phantom and technique

(120 kVp, 290 mAs, 10 mm), the CTDI is 1.2 cGy (4.1 mrad/mAs) in the center and 2.8

cGy (9.6 mrad/mAs) at 1 cm depth. Siemens provides CTDI slice-width, voltage

dependence, slice thickness, and scan angle dependence data. The expected variations in

CTDI values are given as 15%.


SIEMENS Somatom HiO

Radiologic technique factors available: The peak tube potentials available are 85,

and 133 kVp. The tube currents are 70 to 225 mA (the increment was not specified), and








the scan times available are 1.3, 2, 2.7, 4, and 8 s. The generator has a high-frequency

waveform.

Tube characteristics: The nominal focal spot size is 1.8 mm x 1.8 mm. The anode

angle is 100, and the anode is tungsten-on-graphite.

Geometry: The slice thicknesses available are 1 through 10 mm (the increment was

not specified) at the isocenter. The fan beam arc is 420. The source-to-isocenter distance

is 70 cm, and the isocenter-to-detector distance was not supplied. The scanner is third-

generation with two scan angles (3600 and 2400) available, although 3600 is routinely

used. The beam is oriented such that heel effect intensity variation is not present in the

scan plane. The gantry can tilt 250.

Filtration: inherent filtration is 2.5 mm Al equivalent at 120 kVp. The unit has only

a flat, fixed, added filter, which is 0.1 mm Cu. The couch attenuation is 1.2 mm Al

equivalent at an unspecified kVp.

CTDI: The CTDI for the head phantom using the recommended head technique

(133 kVp, 350 mAs, 5 mm) is 4.1 cGy (11.7 mrad/mAs) in the center position and 5.3

cGy (15.1 mrad/mAs) at the 1 cm-depth position. For the body phantom and technique

(133 kVp, 225 mAs, 10 mm), the CTDI is 1.2 cGy (5.2 mrad/mAs) in the center and 2.4

cGy (10.7 mrad/mAs) at 1 cm depth. Siemens provides CTDI slice-width, voltage

dependence, slice thickness, and scan angle dependence data. The expected variations in

CTDI values are given as 15%.


SIEMENS Somatom CRX

Radiologic technique factors available: The only peak tube potential available is 125

kVp. Neither the tube currents nor the scan times were provided. The high-frequency

generator pulses the tube potential at a frequency of 360 pulses per second.

Tube characteristics: The nominal focal spot size is: 1.6 mm x 1.6 mm. The anode

angle is 10, and the anode is tungsten-on-graphite.








Geometry: The slice thicknesses available are 2, 4, and 8 mm, specified at the

isocenter. The fan beam arc is 32. The source-to-isocenter distance is 70 cm, and the

isocenter-to-detector distance was not supplied. The scanner is third-generation with a

scan angle of 3600. The beam is oriented such that heel effect intensity variation is not

present in the scan plane. The gantry can tilt 250.

Filtration: The inherent filtration is 2.2 mm Al equivalent at 125 kVp. The unit has

only a flat, fixed, added filter, which is 0.2 mm Cu. The couch attenuation is 1.2 mm Al

equivalent at an unspecified kVp.

CTDI: Only the surface CTDI values were supplied by Siemens. The CTDI for the

standard head technique (125 kVp, 550 mAs, 8 mm) is 4.7 cGy (8.6 mrad/mAs). The

CTDI for body scans is 2.6 cGy (8.0 mrad/mAs) using the standard technique (125 kVp,

330 mAs, 8 mm).




Data Not Obtained

Below is the list of data not provided by the manufacturers.

PHILIPS: All Models

Philips has not returned the questionnaire.


PICKER: All Models

Picker has returned the questionnaire but has refused to provide the construction

details of their shaped compensating filter(s).


SIEMENS: Somatom CRX

Neither the available tube currents nor scan times were supplied.








Summary


The CT system information above may be summarized as follows. According to the

manufacturers' recommendations, most scanners use 3600 rotation of the tube, continuous

voltage supply, and very similar peak tube voltages. In addition, the source-to-isocenter

distances are similar (between 63 and 70 cm) as well; this indicates the x-ray output for

different scanners (specified at a given distance) would depend primarily on the filtration

present. The major difference among the scanners is the type of x-ray beam filtration, i.e.,

the presence or absence of a beam-shaping filter. The presence of a shaped compensating

filter is expected to produce a more homogeneous depth-dose distribution pattern in the

phantom than the use of a flat filter because of the preferential filtering of the beam near

the periphery of the phantom. The variation in depth dose distributions can be seen from

inspection of the ratios of the CTDI at the central position to the CTDI at the 1 cm depth

position. Table 2-2 below lists the CTDI for typical exams as a function ofmrad per mAs

for the head exam in the center of the phantom and at 1 cm depth.6 Table 2-3 lists the

CTDI for body exams as function ofmrad per mAs as measured in the center and at 1 cm

depth. Note that the mrad/mAs values are listed for the manufacturer-recommended

technique.

The head-CTDI data in Table 2-2 show that the Picker units, all fourth generation

scanners, produce larger CTDI values than do the other units which are all third

generation scanners. Also, the effects of the shapes and materials of the beam-shaping

filters are readily observed. The Siemens units have only a fixed, flat filter, and the ratios

of the central-to-peripheral CTDI values are smaller than for any other manufacturer's

scanner. This situation is contrasted with the GE 9800-series the ratio of the CTDI

values is almost unity. The ratios for the Picker units are also close to one with the

exception of the 1200SX. The 1200SX presumably has a beam-shaping filter of a

6 The unit mrad/mAs, while a non-SI unit, is commonly used in clinical medical physics and for
convenience will be used here.








different design than do the other Picker units. The Elscint units have ratios between the

values of the Siemens units and the GE and Picker units, demonstrating the effects of

differences in the filter designs.

The body-CTDI data in Table 2-3 show that the fourth generation Picker scanners

also produce larger CTDI values than do the third generation scanners. In general, the

effect of the beam-shaping filter is reduced. All of the units, including the Siemens

scanners with no beam-shaping filter, have central-to-peripheral CTDI ratios of

approximately 0.35 to 0.55. The Picker 1200SX has smaller ratio than the other Picker

units do, again illustrating a probable difference in beam-filtration.


Table 2-2: Comparison of head CTDI values [mrad/mAs].
manufacturer's recommended technique.


Manufacturer/Model Center 1 cm Ratiot
Elscint
Exel 2400 Elite/E 13.0 15.3 0.85
2400 12.7 15.0 0.85
1800 10.3 12.0 0.86
General Electric
9800- Xenon 14.7 14.1 1.04
9800 Hilight 11.8 11.8 1.00
Picker
1200SX 15.4 17.1 0.90
IQ Premier 16.2 15.0 1.08
IQ, IQ T/C 18.1 16.9 1.07
PQ2000 18.8 17.6 1.07
Siemens
Somatom Plus 7.7 9.9 0.78 *
Somatom HiQ 11.7 15.1 0.78*
Somatom CRX ** 8.6
$ CTDI at center + CTDI at 1 cm depth
5 mm slice thickness
** not supplied


Values are derived from the









Table 2-3: Comparison of body CTDI values [mrad/mAs]. Values are derived from the
manufacturer's recommended technique.
Manufacturer/Model Center 1 cm Ratiol
Elscint
Exel 2400 Elite/E 5.4 11.4 0.47
2400 5.2 11.3 0.46
1800 4.2 10.0 0.42
General Electric
9800- Xenon 4.1 7.4 0.55
9800 Hiliht 3.2 5.9 0.54
Picker
1200SX 6.3 17.5 0.36
IQ Premier 6.2 13.5 0.46
IQ, IQ T/C 6.5 15.0 0.43
PQ2000 6.8 15.0 0.45
Siemens
Somatom Plus 4.1 9.6 0.43
Somatom HiQ 5.2 10.7 0.49
Somatom CRX ** 8.0

$ CTDI at center + CTDI at 1 cm depth
** not supplied













CHAPTER 3

THE MONTE CARLO TECHNIQUE



Introduction


The Monte Carlo method is a calculational technique that utilizes random sampling

to solve problems numerically that may be difficult or impossible to solve analytically.

Radiation transport is simulated in the Monte Carlo technique by following individual

particles on a "random walk" through given geometries and media. The term particle, as

used here, refers to either photons or electrons. Random numbers are used to sample the

probability distributions that describe particle behavior. By recording events of interest

that occur during a large number of these random histories, one may accumulate average

values for the events or quantities of interest (Cashwell and Everett 1959). In contrast to

analytic methods, Monte Carlo simulation can use realistic cross sections, model realistic

beam conditions, and model complex geometries (Chilton et al. 1984). The price paid for

using the method is lengthy calculational times. Monte Carlo simulation can also assess

quantities that cannot be physically measured, e.g., the percentage of dose contributed by

Compton interaction recoil electrons (Ito 1988).

The Monte Carlo method has been used in the studies of nuclear medicine physics,

radiation oncology physics, diagnostic radiological physics, and in radiation protection.

Use of the method has increased greatly with the advent of small, powerful computers and

the availability of several general-purpose Monte Carlo codes to the medical physics

community (Andreo 1991). As an illustration of the explosive growth the method has

seen in the field of radiological physics, consider the following. A review of uses of the








method by Raeside in 1976 had 86 references. In 1991 Andreo reviewed work in the field
since the Raeside paper and had 299 references in a list ten pages long (!).

In the field of diagnostic radiological physics the Monte Carlo method has been used

to study both dosimetry and imaging. Dosimetry studies have investigated dose
deposition in individual organs from specific radiographic examinations and in anthro-

pomorphic phantoms. The absorbed dose from chest radiography was estimated using

Monte Carlo methods by Koblinger and Zarind (1973). Doi and Chan (1980) used the

method to evaluate absorbed dose in film/screen mammography. Morin produced a
Monte Carlo model of a CT scanner in 1980 to investigate artifact removal but did not

consider patient dose. Dance (1984) used the method to calculate integral dose (energy

imparted) in xeromammography. Patient dose from dental radiography was estimated by
Gibbs et al. (1984) using a method described by Pujol and Gibbs (1982). In 1983 Beck et
al. published a work describing a Monte Carlo model for estimating integral dose in CT

examinations. The method used by Beck et al. was based on a simple elliptical,

homogeneous phantom and estimated integral dose to large volumes of the phantom.

They did not, however, do any direct measurements in a phantom to verify their

calculation, and did not take into account fan beam intensity variations.

Several investigations have used Monte Carlo methods to estimate organ doses in

anthropomorphic phantoms from a wide range of diagnostic radiological procedures.

Rosenstein (1976) reported on the use of the Medical Internal Radiation Dose (MIRD)

phantom to calculate organ doses from simple radiographic exams. The phantom used

simple mathematical expressions to model the shapes and sizes of various internal organs.
In 1985 Jones and Wall of the NRPB in the United Kingdom published a study that was
similar to Rosenstein's in method. They used the MIRD phantom with modifications
suggested by Christy (1980), e.g., breasts were incorporated. The results of the study

were similar to Rosenstein's, although some differences occurred and were attributed to

differences in x-ray spectra and phantom geometry. Kramer et al. (1982) at the








Gesellschaft fir Strahlen-und Umweltforschung (GSF) in Germany produced patient

organ dose data based on Monte Carlo simulations with male and female anthropomorphic

phantoms (ADAM and EVA), similar to the MIRD phantom. In 1985 Drexler et al.

produced a set of tables which allow estimation of organ dose based on calculations with

the ADAM and EVA phantoms. The results for simple radiographic exams are similar to

those of Rosenstein's and Jones and Wall. In addition, organ dose data for CT were

included for pelvis, liver, lung, and head exams. These data were described in Chapter 1.

The NRPB study (Jones et al. 1990, Shrimpton et al. 1990) of CT organ doses lists organ

doses for a range of standard examinations and CT scanners.

Sampling Methods


In the Monte Carlo method of numerical analysis, statistical results are obtained by
sampling appropriate probability distributions. The general methods with which
probability distributions are sampled are described in this chapter. The next chapter

describes the specific probability distributions and the sampling methods used in the EGS

Monte Carlo system, which is a general purpose, public domain code. The following

treatment is based on that of Carter and Cashwell (1975) and Chan (1981).

Monte Carlo simulation constructs a set of random samples {xi} based on a set of

random numbers (r,} that are uniformly distributed on the unit interval. The samples {xi)

are distributed according to a probability density function, or PDF, p(x) such that:

p(x) dx = the probability that any x, will lie between x and x+dr.
We may also define the cumulative distribution function, or CDF, P(x) in terms of p(x):


P(x)= p(y)dy (3-1)
--O
which is the sum of probabilities of x, falling inside each infinitesimal interval between -oo

and x. Because the probabilities of mutually exclusive events are additive, P(x) is

interpreted as the probability that any given x, is less than x. The function P(x) is non-








decreasing in x, because p(x)>O for all x. The probability integrated along all possible

outcomes is unity.

The properties of probability distribution functions and cumulative distribution

functions are used in the three sampling methods described below.

Inversion Method of Sampling

Assume that a given series of events E,, E2,..., E, are mutually exclusive and have

probabilities pl, p2,..., p, such that Ep, = 1. Such a series of events may be the type of

interaction a photon undergoes the occurrence of a photoelectric absorption, for

instance, precludes the occurrence of any other type of interaction. If a random number C,

uniformly distributed on the unit interval, is selected such that

Pl + P2+..+Pi-1 <+P, +..+Pi
then the random number C uniquely determines the event E,. Next, a probability density
function p(x) can be constructed such that p(x) = p,, with the understanding that x is

mapped on the interval 0 < x < n to the events E, ... E,, and that event E, is determined
from (i-1) < x
If the PDF is normalized, the sum of the rectangular areas must be unity. We next define a

cumulative distribution function P(x) which is written as


P(x)- fp(x')dx' (0 0

and shown in Fig. 3-2. Obviously P(0) = 0 and P(n) = 1. Also, since P(i) =p, + ... + i,
the term P(x) can be interpreted as Pr{( < x}, i.e., the probability that any randomly

selected value ofx is less than a given x. Furthermore, if a random number CE [0,1] is

selected and the equation


= P(x)= p(y)dy (3-3)
0























1 2

Figure 3-1: Discrete probability density function.


1 2 n-1

Figure 3-2: Discrete cumulative probability distribution.


p(x)


0










E p =1
n




P1 P,



P,

o


E,


E,







is solved for x, x will fall on the interval (i- 1) < x < i with frequency p,, thereby
determining a value for i and thus determining event E,.
The discrete case can be extended to the continuous case (see Figure 3-3) by making
the step size ofp(x) arbitrarily small. If Ax is an infinitesimally small increment of x, and x
is given in the range [x,x+Ax], then the CDF is given by definition. That is, as Ax
approaches zero, the probability that x falls between x and x+Ax is approximated by


P(x + A)- P(x) = dP(x) A
dr (3-4)
= p(x)Ax.
If C is a random number and ifp(x) is defined over the interval a < x
= P(x)= p(y)dy (3-5)
a
determines x as a function of x representing a random sample of the function p(x).





P (x)



I I I I
x
a x x+Ax b





P (x + x) -
P (x)
- -
I I
o x
a x x+Ax b
Figure 3-3: top: Continuous probability distribution; bottom: cumulative distribution
functions.








Rejection Method of Sampling

The rejection method was first described by Kahn (1950). The method is used in
situations in which the probability distribution function p(x) is bounded and calculable. If
p(x) is defined on an interval [a,b] and Mis the maximum value offx) on the interval
[a,b], then the function p*(x) can be defined such that

p*(x)=p(x)/M. (3-6)
Next, two random numbers, C, and C2 are generated, and x*, a possible value for x, is
computed from x* = a + Cl(b a). If 2 the process is repeated. A rigorous proof of the rejection method is given by Spanier and
Gelbard (1969).
The efficiency of the method is defined as the probability that a given set of random
numbers is not rejected. The efficiency E is simply the ratio of the area under the curve

p(x) to the area under the circumscribing rectangle and is given by

E- p(x)
M(b a) (3-7)
Clearly PDFs that are "spiked" are not efficiently sampled using the rejection method.
Such PDFs may be sampled using the composition method, described below. The
rejection method is illustrated graphically in Figure 3-4.

Composition Method of Sampling

The composition method has been described by Carter and Cashwell (1975), Nelson
et al. (1985), and Williamson (1988). If a PDFp(x) can be put in the form of

p(x)-= a,(xg,(x)
'' (3-8)
wheref(x) and g,(x) are PDFs, g,(x)E[0,1], and ai are positive real numbers, thenp(x) can
be sampled as follows. A random number C, is chosen and i is found such that









i-1 n n
la; j=i =i j=i (3-9)

Next, a sampled value x* fromf(x) is selected. For example, iff(x) is sampled using the

direct method described above, then x* is found from

X*
C2= f,{x) dr
= ((3-10)

where C2 is a random number. Finally, another random number C3 is chosen. If

C3 < g,(x *), then x* is accepted; if not, the process is started again.


Random Number Generation


The basis of all Monte Carlo simulations is the availability of a long sequence of

numbers such that the occurrence of each number in the sequence is unpredictable and the




Uniformly distributed Region of rejection
Uniformly distributed
ordered pairs





( .r2) .. .



*


0

a t
Region of acceptance
Figure 3-4: An illustration of the Rejection Method. Only the random pairs that fall under
the curve are accepted.








sequence of numbers will pass statistical tests designed to detect departures from

randomness (Hammersley And Handscomb 1964). Series of random numbers are

available from published tables (Tippett 1927, Rand Corporation 1955), physical sources

(James And Arason 1973), and algorithms (Knuth 1969).

Sequences of numbers obtained from an algorithm are called pseudorandom

numbers to reflect their deterministic origin and are routinely used in modern Monte Carlo

simulation (Morin et al. 1988). The algorithm used to produce random numbers in the

EGS system is called the multiplicative-linear-congruential method (Lehmer 1951).

Random integers I are generated according to

I, = (ai_, +c)mod 2k (3-11)

where I, is the ith random integer in the sequence, a is multiplier, c is the increment, and k

is the number of bits in the integers of the computer. In the EGS system, and on a 32-bit

computer, the following values are used:

a = 663608941,

c = 0, And

k =32.

The actual random number used in sampling any given distribution is

Ii (3-12)

The first number Io used to start the sequence is called the "seed number". In the EGS

system the default seed number is 987654321, and Ce [0,1]. The cycle length, which is

the length of the random number sequence before it starts repeating itself is approximately

109 (Ehrman 1981).


Data Analysis


Monte Carlo simulation is a statistical method; therefore, an estimate of statistical

uncertainty must be determined. The following method of uncertainty estimation is








described by Rogers and Bielajew (1990). The total number of histories for a particular

run is broken up into NB equal batches, and an estimate of the quantity of interest x,(e.g.,

energy deposited in a region) is obtained for each batch. The final estimate of the quantity

x is (x), the mean of the x, values. The estimate of the variance of the mean, s2, is then

1 t
s) 1 N(x, (x)) (3-13)

Because x, values are assumed to be drawn from a normal distribution, it can be assumed
that the interval ((x)- s,(x)+ s) contains the true mean in about 68% of all cases, and the

interval ((x) 2s, (x) + 2s) contains the true mean in about 95% of the cases.' In this work,

results are presented +ls, and NB is 10 for all cases.

For each run the following data are recorded:

kinetic energy of the incident beam

phantom material

source-to-axis-distance (SAD)

phantom radius (or semi-axis lengths)

slice width in cm

number of histories.

The following data categories are scored for each run:

total energy incident on the phantom

scattered energy absorbed

primary energy absorbed


1 Given n independent, identically distributed, random variables xl, x2, ... x, with mean m and variance
a2, the Central Limit Theorem is written as

P X -mn 1 b -t2/2
Pa n b --je dt


1 "
where X, = -1 xi from which it follows that the distribution of the sum of n independent, identically
distributed, random variables approaches a normal distribution.
distributed, random variables approaches a normal distribution.








total energy absorbed

scattered energy emerging from phantom

primary energy emerging from phantom

total energy emerging from phantom

GCTDI (for circular phantoms)2

dose (z,r) (for circular phantoms).

The mean value and the estimate of the variance s of each quantity are calculated in all

cases.


2 The Generalized Computed Tomography Dose Index, which is defined in Chapter 6.














CHAPTER 4

THE EGS MONTE CARLO SYSTEM



Introduction


The Electron-Gamma-Shower version-4 (EGS) Monte Carlo system is used in this

work. The EGS system is a general-purpose Monte Carlo code that simulates coupled

electron-photon transport over an energy range from several keV to several TeV (Nelson

et al. 1985). The simulation may be carried out in any material and in any geometry

specified by the user. EGS takes into account the predominant photon interactions in the

diagnostic energy range: coherent scattering, Compton scattering, and photoelectric

absorption. EGS can also simulate electron transport; however in this work all electrons

are assumed to be locally absorbed because of their short range and negligible

bremsstrahlung production. The continuous-slowing-down-approximation (CSDA) range

of electrons is <0.2 mm in acrylic at 100 keV (Berger and Seltzer 1983), and the radiation

yield (the fraction of energy radiated away) for a 100 keV electron is 0.0005 in acrylic and

0.001 in cortical bone (Johns and Cunningham 1983).

The original EGS system was designed to simulate and study the interaction of high

energy electromagnetic cascades with matter. In 1985, version 4 of the EGS system was

published (Nelson et al. 1985), the system having been expanded and the dynamic range

extended down to 1 keV for photons and tens of keV for charged particles.

The EGS system is written in the computer language Mortran3, which is a high-

level, structured language similar to FORTRAN (Cook, 1983). Mortran3 uses a macro-

processor facility for string replacement. Included in the EGS system is a translator that








converts the Mortran3 code into ANSI standard FORTRAN77. The problem-specific

code written by the user is combined with the standard EGS files to produce a large
program that is then translated into FORTRAN for compilation and execution. This mod-
ularity makes for ease in programming and debugging. (See Appendix D for an example

of a Mortran3 user code, which was written to simulate radiation transport in and around

a right circular cylinder irradiated by a fan beam with a perfectly compensating bow-tie

filter.)

The EGS system has been used by numerous investigators in the area of medical
physics. Rogers and Bielajew (1990) have provided an extensive list of the ways the
system has been benchmarked, and EGS has been shown to provide accurate results over
a wide range of medical physics problems. Examples include the work of Rogers (1982),

who modeled photon detector response functions with EGS and showed excellent (<1%)

agreement with measured values. The EGS system was used to calculate depth dose

curves in water for WCo beams and also showed excellent agreement with measurements,
including the buildup region (Rogers et al. 1985). Mohan et al. (1986) used the EGS
system to calculate dose distributions for 20 MV photon beams as well as for 6Co beams.

The calculated and measured data presented by Mohan et al. are virtually identical. The

system has also reproduced absolute dose measurements based on a monoenergetic 7-

MeV photon beam (Mach and Rogers 1984). The beam was produced in van de Graaff

accelerator using the 19F(p,acy)160 reaction, and the fluence was known to within 1.1%.

EGS calculated the dose per unit fluence to within the experimental measurement error of

1.6% as measured with a Baldwin-Farmer chamber and the AAPM TG21 protocol
(AAPM 1983). Herbold et al. (1988) used EGS to calculate buildup factors for photons
in the 15-100 keV range. Dose distributions around 125I seeds have been calculated using

EGS by Scarbrough et al. (1990) and by Cygler et al. (1990).








EGS Components


There are four major components of a Monte Carlo code (Rogers and Bielajew

1990):

(A) algorithms for sampling the physical processes,
(B) material data for the media of interest (e.g., cross section data),
(C) geometry description and methods for scoring quantities of interest, and,
(D) data analysis.

The manner by which EGS handles each of these components is described below, as

well as the specific code and subroutines written for this project. In general, the sampling

algorithms used and the material data are already programmed, and the EGS-user need not

be concerned with the inner details of the code. The user must however write subroutines

which describe the media to be used, the geometry in which the simulation is to be

conducted, the quantities to be scored or recorded, and the manner in which the resulting

data is to be handled. The method of data analysis used in this work was discussed in the

preceding chapter.

Algorithms

The EGS Monte Carlo system as used in this work is an analog Monte Carlo

system.' That is, one particle is followed at a time, and physical processes are simulated as

closely as possible. The particle is observed as it undergoes discrete events. The results

of these discrete events are accessible to the user and, depending on the problem at hand,

can be stored for later analysis. The code simulates these events by sampling the

probability distributions which govern the following physical processes:
1. the distance traveled by the particle between interaction sites,
2. the type of interaction that occurs at each site,


1 Several variance reduction methods are used in quantity scoring; however particle transport is strictly
analog. This is in contrast to another method, used by Rosenstein (1976) in which photoelectric cross
sections are excluded, and particles are assigned statistical weights based on their probability of having
survived photoelectric interactions at any point in their history. Energy deposition events are recorded as
the product of the energy of the particle and the weight of the particle.








3. the energy and/or direction of the particles) emerging from an
interaction site, and
4. determination of particle trajectories in the reference cartesian
coordinate system.

Particles being transported are described by a series of variables. The particle

properties stored are its current location in cartesian coordinates, direction cosines,

energy, charge, statistical weight, and the geometric region. (Geometric regions are

defined in the geometry subroutine HOWFAR, as explained below.) Particle properties

are stored on a last-in-first-out (LIFO) stack. The particle on the top of the stack is

transported until an interaction takes place, until its energy drops below a pre-set energy

threshold, or until the particle leaves the geometry of interest. If the particle leaves the

geometry or if the energy drops below the cutoff, the particle is discarded (after some

bookkeeping) and the simulation continues with the new top-most particle. The term

bookkeeping refers to the records kept and actions taken when a particle is discarded,

which in turn depend on the quantities of interest in the problem. Recording the location

and energy of a particle when it leaves the geometry of interest is an example of

bookkeeping. When an interaction occurs and more than one particle emerges (e.g., a

Compton-scattered photon and an electron), the particle with the lowest energy is put on

the top of the stack to prevent an overflow of particles. When the stack is empty, the

initial particle's history is complete and another history may commence.

Random Path Length Sampling

As listed above, the program must determine how far a particle travels from its

present location before it interacts with the medium. EGS uses inversion sampling to

determine the distance to the next interaction. In this case, the PDF is the probability that

a particle will not interact in the distance x and is given by

f ) exp(-x/,) (4-1)
X


where X is the mean free path. The CDF is found from










F(x,)= exp(-x'/ X)dr'= ,
0


(4-2)


where i is a random number on the unit interval. Solving for x,, (the randomly-sampled

path length), we obtain the expression

x, = -Xln(1- ,)= -Xln (4-3)

because if is uniformly distributed over the interval [0,1], then [1 Q] is also uniformly

distributed on the same interval. This result applies only if the medium is homogeneous,

of constant density, and infinitely large in size. In the more usual case for which the

photon is traveling through a finite-sized volume composed of different regions and

materials, a slightly more complicated sampling scheme is used. The procedure used in

EGS is as follows and is taken from SLAC-Report-265 (Nelson et al. 1985) which

describes the system.
1. Compute X at the present location.
2. Select a random number C and set t, = -Lx = X In C. (Nx is the
randomly-sampled number ofpathlengths.)
3. Calculate the distance d to the nearest boundary along the photon's
present direction.
4. Set t2 = the smaller of tI and d; transport t2.
5. Subtract t2/ from Nx; if the result is zero, then the photon has not left
the present region, and an interaction will occur: Leave this loop.
6. This step is reached if ti > d and a boundary is reached. If the new
region is the same material, go to step 2; otherwise, go to step 1.

Using this technique a photon may be transported through any type of heterogeneous

media.

Interaction Selection

Once the interaction point has been selected, the type of interaction that is to occur

must be sampled. EGS can account for the three main interactions that photons undergo

in the diagnostic range, i.e., Compton scatter, coherent scatter, and photoelectric

interactions. The probability of a given interaction occurring is proportional to its cross








section (Williamson 1988). If the possible interactions are numbered 1,2,...n, then the

branching ratio F(i) is defined to be



F(i)= (4-4)

where a0 is the cross section for thefth type interaction and a, is the total cross section,

given by

n
CY t oi, (4-5)
j=l

The number i of the interaction to occur is selected by choosing a random number C and

finding i such that

F(i 1) < C < F(i). (4-6)

As a simple example, suppose that there is a 80% chance of a photoelectric interaction and

a 20% chance of a Compton interaction for a given photon. If is less than 0.8, then the

photoelectric interaction is chosen; if not, the Compton interaction is chosen.

Once the point of interaction and the type of interaction has been sampled, the

interaction itself must be simulated in order to determine the energy and direction of the

particles) leaving the interaction point. The PDF of an interaction is the differential cross

section of that interaction (Williamson 1988).


Photoelectric Interactions. A photoelectric interaction occurs when a photon

interacts with an atomic electron. The photon is absorbed and an electron is ejected from

the atom with a kinetic energy equal to the difference between the photon energy and the

electron binding energy. Also, either a fluorescent x ray or an Auger electron is emitted

upon filling the vacancy in the inner shell (Evans 1955). If a photoelectric interaction has

been chosen, one of two possibilities occurs. If the photon energy k is less than the K-

edge energy EK, then the photon is discarded, and the energy k is simply deposited at the








location of the interaction. If k > E, a photoelectron is created with kinetic energy E

where
E=k- E, (4-6)

and a photon of energy EK is created and forcibly discarded (i.e., absorbed) at the point of

interaction. The photoelectron can then be transported until it drops below an energy

cutoff, at which point local deposition occurs. In this work, the electron energy cutoff is

500 keV, so all electrons are absorbed at the point of interaction. Thus in any case the
final result of a photoelectric interaction is the deposition of the total energy of the photon

at the point of the interaction.

Rayleigh Scattering. Coherent, or Rayleigh, scattering occurs when an incident

photon interacts with the electron cloud of an atom. The atom is neither ionized nor is it

excited; the only result of the interaction is a photon with the incident energy scattered

through a small angle from the initial trajectory (Johns and Cunningham 1983). For

Rayleigh scattering in a medium, EGS calculates total cross sections which are based on

the weighted sum of the cross sections of the atoms in the medium, the weight being the

proportion of atoms of a given type. The differential cross section for Rayleigh scattering

is given by (using the notation of Nelson et al., 1985):

daR (0) (1 + cos2 0)[F(q)]2 (4-7)

where re is the classical electron radius (2.8x10-13 cm2), 0 is the polar scattering angle,

Fr(q) is the atomic form factor, q is the momentum transfer parameter, q = 2ksin(0/2), and
k is the photon energy. The assumption of independent atoms is also used in calculating

the form factor for the medium, i.e., the atomic form factors are combined in a weighted
sum. This procedure is known to produce inaccuracies because molecular structure can

affect coherent scatter (Johns and Yaffe 1983). However, for photons in the diagnostic

energy range in water, the maximum discrepancy between cross sections is less than 10%.

The magnitude of the effect is not known for acrylic or other tissue-like materials;







however, the discrepancies are considered small enough to be ignored. The differential
cross section can be written, using the substitutions I = cosO, dna = 27?rd, and
q2 = 2k2(1-I),as


d Gq 2 2 22 2. 1

Next A(q) is defined as

A(q') [F(q)]2d(q2), (4-10)
0
and the cross section may be written as

daR(q2) 7rr2 2 (r1+2 )[F(q)]2 (4-11)
dq2 k- Aq ~( 2 jA(q2)

The term qn is the maximum value that q can take, i.e., when gt = -1. When expressed in
this form, [F,(q)]2/A(qq) can be used as a PDF (i.e., thef(x) function in the

composition method), and (1 + ,t2)/2 can be used as the rejection function (i.e., the g,(x)

function). The PDF is sampled by generating the random number C, and finding q2 from

A(q2) (4-12)
T= 2 '

The corresponding gI value is found from I* = 1- q2/2k2. A new random number C2 is
generated, and the value of 4* is accepted if

1+ 1 *2
2 < (4-13)
If p* is not accepted the procedure is repeated until a suitable value is found. To increase
calculational efficiency the EGS system sets up look-up tables for the form factors F(q)
which are based on the values given by Hubbell and Overbo (1979). Once the polar
scattering angle has been determined, the azimuthal angle is randomly sampled from the 2
7n distribution and is found from 27ix, where C is a random number on the unit interval.








Compton Scattering. A Compton scattering event occurs when a photon strikes an
atomic electron. The electron emerges from the collision with an energy T and at an angle

by Evans (1955). As discussed above, the electron energy is assumed to be absorbed at
the site of the interaction; thus its direction following the interaction is not considered.
The relationship between the electron energy and the photon energy is found through
conservation of energy:
ko=k+ T, (4-14)
where ko is the incident photon energy, and k is the scattered photon energy. The
relationship of the scattered photon energy and direction to the incident photon is
described below.
Compton interactions are sampled using the composition method. For sampling
purposes, the Compton differential cross section can be written (Butcher and Messel
1960) as

d E. c I [1 E sin2 2
dc ko [+ 1-+ (4-15)


where C = k/ko, and 0 is the angle between the initial and scattered photon trajectories.

The relationship between k and 0 is given by

k = o (4-16)
1 +(1-cos)ko / m

where m is the electron rest energy. The differential cross section is thus in the form
f(8)g(S) whereA() = [C + 1/8] and

()[1 E sin2
g() 1 = sin2 (4-17)
1+82


The function f() may be factored as









2
f(S)=1-+E= -a,f,(8) (4-18)
8 i=1
where

a0{1 = In( 1d1 (4-19)

(l-0 ) ,n(1/0) 2S
2 (' 1 ) '(4-20)


and So = 1/(1 +2ko im). The function, is sampled by generating a random number C and
finding 8 = Eo exp(a1). Next the functionf2 is changed via the change of variable


61-t (4-21)

to the function
f '(') = a' fj"(') + a' f2'(') (4-22)

k' 1
where a = a =' 1 "(E') = 26'' and f "(E') = 1. The rejection function
where k1 2- +1 2
is calculated by finding t, where

m(l-8)
t= k (4-23)

and the required value of sin2 0 is found from
sin0= t(2- t). (4-24)
The algorithm that EGS uses for sampling the Compton differential cross section follows
(random numbers on the interval [0,1] are designated by ):
1. Calculate ko, Eg, a,, and a,.
2. Calculate 6 :
If a, 2 (a, + a)%&, then calculate 6 = Soexp( a1 2);
otherwise use = Eo +(1- O)E' ,








where C' is the larger of 3 and C if k" > (ko + 1)2 and
= 3 if k <(k +1)2 .


3. Calculate t and the function g(s) = 1 -I 0 2

If C (or ~5) < g(E ), then reject E, and go back to step 2 above.
4. If 8 is accepted, the scattered electron energy is found from
conservation of energy. The polar scattering angle 0 is then found
from

cos = (ko + m)k kom
cose9=
kok
and the azimuthal angle p is randomly selected.

Coordinate Rotation

This algorithm is used by EGS to determine particle trajectories following
interactions. A particle's spatial location is described a vector r = (x,y,z) located in a

cartesian coordinate system. The particle's direction is expressed by direction cosines.

Direction cosines are the cosines of the angles the trajectory makes with the x, y, and z

axes: a, P, and 0, respectively. The direction cosines may be expressed in terms spherical

coordinates:

u = cosa = cosq sin 0

v= cos3 = sinp sin (4-25)

w = coso
where 0 is the polar angle and p is the azimuthal angle (Williamson 1988). A vector
f = (u,v,w) may thus be defined. The advantage of using this system is that, under a

translation S, the new spatial coordinate r' = (x',y',z) may be found from
x' = x+uS

y'=y+vS (4-26)
z' =z+wS ,








or in vector notation,

r' = r + S (4-27)

Selection of a scattering interaction will return a polar angle of deflection. This

deflection is taken as that from the direction of the incident photon. When combined with

a randomly selected azimuthal angle, the new direction cosines may be determined.

Cashwell and Everett (1959) have described a method for determining the direction

cosines (with respect to laboratory coordinates) of photons after multiple collisions. For

systems for which 0 is the polar angle and w = cos 0 ,

uuw vv
U= -- --+w u
P P
0 (4-28)
S WV VU
v = -+-+w v
P P

w' = -u- +Ww

where u',v',w' are the direction cosines of the deflected photon's line of flight; u,v,w are

the direction cosines of the incident photon; u*,v*,w*, are the direction cosines of the

deflected photon's line of flight relative to u,v,w; and p = 1 for the unprimed,

primed, and starred frames of reference.


Material Data

The discussion of sampling methods in the previous section indicates the need for a

large amount of material data such as cross sections and branching ratios. Also, for

calculational efficiency the data must be in a form that can be accessed quickly. In the

EGS system a separate preprocessor code (PEGS Preprocessor for EGS) is furnished

to compute the physical quantities needed for the simulation by the EGS code. PEGS is

usually run only once, and it establishes data sets for the materials and energy ranges to be

used by the EGS user code.








PEGS constructs piecewise-linear fits over a large number of energy intervals of the

branching ratio and cross section data (Nelson et al. 1985). The data sets are optimized to

provide fast numerical manipulation. The materials used may be elements, compounds, or

mixtures. For elements, the user can either utilize the default values for physical density

and atomic weight or specify a value. This feature allows the use of different isotopes and

the simulation of exact experimental conditions. When compounds are specified, the user

must supply the number of elements in the compound, the density, the relative numbers of

each type of atom in the compound, and the state of the compound. The characteristics of

the compound (e.g., cross section) are based on the weighted sum of the component

atoms; the weighting is determined by relative number of each type of atom in the

compound. The information needed for the mixture option is the same as that for

compounds, except, instead of the relative numbers of atoms, the relative weight of each

type of atom present is specified.

It is also possible for the user to have Rayleigh scatter data included in the data set

prepared by PEGS for any material. The inclusion of Rayleigh scatter in the simulation

process is an option and must be specifically requested by the user, which is a feature that

is most probably a remnant of the high-energy origins of the EGS code. Because of the

nature of the narrow beams modeled, and because Rayleigh scatter usually only involves

small angles, it was felt that exclusion of Rayleigh scatter might introduce errors into the

results. Therefore, in order to make the EGS model as realistic as possible, Rayleigh

scatter is included for all of the simulations reported in this work.

Geometry and Scoring Method

The user of the EGS system must write three Mortran3 routines to describe the

problem at hand. These codes are called User Codes to emphasize their separate nature

from the main EGS routines. The user must write:

1. MAIN the driving program of the system,









2. HOWFAR the subroutine in which the geometry of the problem is
described, and,
3. AUSGAB the subroutine which scores quantities of interest.

The EGS system is basically a series of subroutines which share data by COMMON

blocks. Thus in writing MAIN, the user must follow a set of specific rules that dictates

the order in which each subroutine must be called, which Mortran3 macros are to be used,

and which subroutines are to be interconnected by which COMMON blocks. A simplified

flow control diagram for the EGS system is shown in Fig. 4-1.

The MAIN routine dimensions and initializes the data arrays to be used in the

simulation. It also sets COMMON variables which specify the cutoff energies, the units

(e.g., cm) and the media names. MAIN then calls the HATCH subroutine to read in the

media data created by PEGS. Once the preliminary initializations have been performed,

MAIN calls the subroutine SHOWER, which initiates one history for each time it is called.

The call to SHOWER specifies the parameters (e.g., location, energy, direction, region,

electric charge, and statistical weight) of the initial particle in each history. Some of these




UserControl formation





w MAN HOWFAR
Data ELECT EcHOTON


-~ -- -~ -

0
o HATCH SHOWER ELECTR PHOTON
(.I
,w


Figure 4-1: Flow control diagram of the EGS system








initial parameters may require sampling before the SHOWER call, such as finding a

random trajectory within a CT-type fan beam, or sampling the incident spectrum for a

photon energy. Finally, after the desired number of histories are simulated, the MAIN

routine analyzes the results and outputs the desired data.

The purpose of the subroutine HOWFAR is to provide information to EGS about

the nature of the geometry in which the Monte Carlo simulation is to be done. The user

describes the shapes, sizes, and materials of objects in the geometry. Included in the EGS

system are subprograms which are designed to aid the user in specifying geometric shapes

in the subroutine HOWFAR (Nelson and Jenkins 1987). These subprograms simplify the

specification of planes, spheres, cylinders, and cones; they determine the distance a particle

must travel to intersect the surface of a given shape. (An example of such a subprogram

that was written to define a right elliptical cylinder is listed in Appendix C.) Multiple

elements using simple geometric shapes may be combined to form complicated structures.

Different geometric regions within the geometry thus described are identified by region

numbers. The user must then systematically examine (via IF... THEN...ELSE statements)

all possible destinations of a particle exiting each region.

HOWFAR is called after a path length has been sampled from the exponential

attenuation distribution. Given a particle region, position, and trajectory, HOWFAR

calculates the distance to the nearest boundary of the region. If this distance is greater

than the sampled distance to the next interaction, HOWFAR puts the particle on the

boundary and sets the region number to that of the new region. The direction cosines are

unchanged, and a new path length is sampled. Note that regions may or may not be of

different materials.

The user can also tell HOWFAR to discard a particle if it enters into a region of no
interest. For example, if the user is interested in dose deposition locations within a

cylinder, particles that leave the cylinder can no longer deposit dose there. (In this work,

the assumption is made that the particles leaving the geometry of interest cannot be








scattered back in.) Thus when HOWFAR determines that a certain particle has left the

cylinder, it terminates the particle's history. Time and calculational effort are minimized,

because particles that can not contribute to the final results are eliminated.

Three different geometries are simulated in this work: a circular cylindrical CTDI-

type phantom, a small diameter acrylic rod, and an elliptical anthropomorphic-type
phantom. The details of each are described below.

The purpose of the subroutine AUSGAB is to record or score quantities of interest.

AUSGAB is called before a particle is discarded because it has fallen below the cutoff

energy. The user can also specify that AUSGAB be called in other situations, such as

before (or after) a specific interaction occurs. The user specifies the actions to take when

AUSGAB is called. For example, the user may wish to record energy deposition in

geometric regions. Energy deposition occurs when a photon or electron falls below the

energy cutoff level and is absorbed at its present location. If, for example, a Compton

electron is created, its kinetic energy is below the pre-set cutoff energy, and it will be

discarded. AUSGAB is called, and the kinetic energy of the electron is added to the total

energy deposited in the appropriate geometric location. The user can have AUSGAB

record any of the characteristics of a particle, i.e., its energy, spatial coordinates, direction

cosines, weight, charge, or current region.

Circular Cylindrical Phantom

The user code CTMONO was written by the author to simulate a CTDI-type

phantom. The subroutine HOWFAR used in CTMONO was based on the version in

DOSRZ, a code included in the EGS system. For this case, the subroutine HOWFAR

specifies the geometry as follows. The isocenter of the scanner is taken to be at the origin

of a cartesian coordinate system. A right cylindrical phantom is centered on the origin

with the long axis coincident with the z-axis. The default material is acrylic. The cylinder

is 16 cm long. The maximum radius is varied over a range of values, although most runs

use either 8 cm or 16 cm. For dose-scoring purposes the phantom is divided into annular-








shaped regions. These dose-scoring regions are bounded by planes normal to the z-axis,

and by concentric cylinders whose long axes are coincident with the z-axis. Figure 4-2

illustrates the general configuration of radial and planar zones. (Note that Fig. 4-2 is a

schematic drawing and not an actual representation of the sizes of the zones used in this

work.) Each annulus is defined by two planes and two cylinders. The regions between

planes are assigned identifying numbers from 1 to NZ, with NZ being the total number of

planar zones thus defined. Similarly, the zones or regions between cylinders are assigned

identifying numbers from 1 (the central core) to NR, with NR total radial zones. The

bounding-cylinders are also numbered 1 to NR, so each radial zone is associated with the

larger of its bounding cylinders. (Radial region 1 is bounded by cylinder number 1, radial

region 2 is bounded by cylinder number 2 on the outer surface and cylinder 1 on the inner

surface. The outer cylinder defining region NR is the radius of the phantom.) Any point

within the phantom is consequently located within a specific planar region (IZ), and a

specific radial region (IX). The annuli are assigned region numbers, IRL. The region

number of any location inside the phantom with known IX and IZ is found from

IRL = IZ + NZ(IX 1) + 1. (4-29)

Given a region number IRL, the corresponding IX and IZ values are found from the

expressions

IX = [(IRL 2)/NZ] + 1, and IZ = IRL 1 NZ(IX 1) (4-30)

The equations in (4-30) assume integer arithmetic and were obtained directly from the

DOSRZ code.

The cylinders defining the radial zones are assigned radii of:
0.25R 0.5 cm,
0.25R + 0.5 cm
0.50R 0.5 cm
0.50R + 0.5 cm
0.75R 0.5 cm
0.75R + 0.5 cm
R-1.0 cm, and
R, the radius of the phantom.






58

There are 60 planar zones. The thickness of the planar zones are as follows, with z given

in cm:
6 < IzI < 8 : 10 mm spacing
2 < zI < 6 : 5 mm spacing
1 < IzI 2 : 2.5 mm spacing
0.6 < zI 1 : 2 mm spacing
0.5 < kjz < 0.6 :1 mm spacing
0.1 < zl< 0.5 : 0.5 mm spacing
0 < z| < 0.1 : 0.2 mm spacing

The smaller spacing of the planes near the center of the phantom was initially selected to

provide good spatial resolution of the dose profile. It should be noted that integrals of the

dose profile, which are discussed in Chapter 6, are calculated with all the planar zones

collapsed into one large zone.

The geometry described in the subroutine HOWFAR was exploited for scoring

efficiency by utilizing the cylindrical symmetry inherent in the problem. AUSGAB scored

energy deposition in the annular regions described by a given IX and IZ. The total energy

sIL deposited in any given region IRL is used to calculate the absorbed dose DRL in that

region from the expression


R =R .- ..-- --
5= IX 5
4 IX = 4


2 IX=2
--- -





1Z=1 IZ=2 IZ=3 IZ=4 IZ=5
Side View Frnt View
Figure 4-2 Illustration of the geometry described by subroutine HOWFAR in the user
code CTMONO with five radial zones and five planar zones.










Dn = (4-31)
p VM
where p is the medium density and VIR is the volume of the region. VI is given by

VMR = (R4 4-1) T (4-32)
where Ri and R.x, are the radii of the indicated radial zones and Tz is the thickness of

planar zone IZ.

The x-ray beam incident on the phantom is simulated as follows. With the phantom

centered at the origin, a monoenergetic point source is located on the y-axis at a fixed

distance from the origin. This distance is 65 cm by default. To ensure that all the photons

emitted by the source strike the phantom, the initial direction for each photon is sampled

using the "SELECT-POINT-IN-RECTANGLE" algorithm in the DOSRZ series of

programs (see Fig. 4-3). This algorithm uses the fact that in the beam's-eye-view, the

projected area of the beam intercepting the phantom is a rectangle. For each incident

photon, the algorithm picks a random point within the rectangle, calculates the trajectory

from the source to that point, and finds the point of intersection of the trajectory with the

phantom. Once the location and trajectory of the incident photon is known, the radiation

transport simulation commences.

The energy of the incident photon is determined before the call to subroutine

SHOWER. For monoenergetic problems, the energy is simply a constant value. In cases

for which the source is polyenergetic, the x-ray spectrum is treated as a PDF. The EGS

system includes a source-handling subroutine, SRCRZ, that creates a CDF from a given

spectrum. For problems involving x-ray spectra, the CDF is sampled prior to each photon
history to assign the incident photon energy.

The CT-slice of thickness Tis assumed to be centered about the origin. For a

maximum phantom radius, R, the photons must be emitted so as to strike inside the

rectangular box of dimensions 2R in the x-direction and T in the z-direction, centered at

the origin and normal to they-axis. In order to determine the points within the rectangle,






60

two random numbers C, and C2 are selected, and the points in the box x, and zb are

selected by:

Xb = R(2C, 1)

Yb =0 (4-33)

zb = (T/2)(242 -1).

Once the two coordinates x, and zb are found, the direction cosines u, v, and w of the ray

connecting the source point to the point in the box are determined, and from the cosines,

the point of intersection with the phantom is found. The medium between the source

point and the phantom surface is assumed provide negligible attenuation (e.g., air). Note

that the cylindrical symmetry in the problem allows the use of only one source location.

Three beam filtration configurations were used in part of this investigation: no filter,

and two beam-shaping, or bowtie, filters. One filter is a perfect or ideal filter. That is, for

a given phantom size and material, the filter produces a beam for which the primary





Point source


Bowtie filter

S\ AD











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


Figure 4-3: Source and phantom configuration for cylindrical phantom.








fluence exiting the phantom is independent of angle within the fan beam because the total

path length is constant. The fan beam subtends angles over the range [-P1to2, OtJ2] with
respect to the y-axis in the x-y plane. The trajectory of any photon in the fan beam can
therefore be described by the absolute value of its angle with respect to they-axis, P. At
any angle within the fan beam the thickness of the filter is found from

tf(P) + L(P) = 2R (4-34)

where tX/) is the thickness of the filter at angle 1, L(3) is the thickness through the
phantom at angle 13 and R is the phantom radius. To ensure that the attenuation of the
beam, and not just the path length, is constant for any angle within the fan beam, the filter
is made from the same material as the phantom, i.e., acrylic.
The second beam-shaping filter is based on a real filter and modeled from data
supplied by General Electric. The geometric configuration in this problem is modeled on
the filter of a General Electric 9800-series CT scanner, the details of which were obtained

from the manufacturer and are proprietary. The shape of the filter was digitized, and a
fifth-order polynomial curve was fit to the data points. The distance through the filter as a
function of angle in the fan beam was then calculated and tabulated in a lookup table, as
was the linear attenuation coefficient for the filter material, Teflon (ICRU 1989).

For both filter types, a photon's initial trajectory angle within the fan beam is used to
find the corresponding thickness of the filter. In a totally analog Monte Carlo simulation,

the photon's random trajectory through the bowtie filter would be followed; however

considerable computing would be spent following photons that never enter the phantom

and thus are of no interest. In this study, a variance reduction method is used: a weight is
assigned to each photon. The weight is simply the probability that the photon was
transmitted through the filter, and is calculated by
W = exp[-pt(E)t(P)], (4-35)

where W is the weight assigned, p(E) is the linear attenuation coefficient of the filter
material at energy E, and t(P) is the thickness of the filter at angle 3 within the fan beam.








The weight of an incident photon is passed to all generations of its progeny. Jones and

Shrimpton (1991) have shown that there is negligible difference between actually

simulating transport in the filter and accounting only for attenuation effects as is done

here.

The LATCH feature of the EGS system is used to differentiate between primary and

scattered photons. LATCH is a variable that can be associated with each particle, along

with energy, position, trajectory, charge, and weight. The value of LATCH can be passed

to all succeeding generations of particles. The initial value of LATCH is zero for all

photons and is changed if the photon undergoes an interaction. Subroutine AUSGAB is

called if a scattering interaction is about to occur, and the LATCH value is checked. If the

value was zero, it is then changed to ten, indicating a primary interaction. If the value was

ten, it is changed to one, indicating a previously scattered photon. If the value was one, it

is left as one. When a particle is to be deposited in a given region, the value of its LATCH

is checked to determine into which category the energy is to be placed. The four

possibilities are:
1. If the particle depositing the energy is a photon and LATCH is not
equal to zero, then clearly it had been scattered at least once and the
energy is deposited in the scatter category.
2. If the particle depositing has a LATCH of zero, then it must be either a
photoelectron from a primary interaction, and it is scored in the
primary bin, or a primary photon exiting the geometry.
3. If the particle depositing has a LATCH often, then it must be a
Compton electron from a primary interaction and is scored in the
primary bin.
4. If the particle has a LATCH of one, it must be is a multiply-scattered
particle and so is placed in the scatter bin.

Another variance reduction technique, discard-within-a-zone (DWAZ), was used
during the actual transport simulation. DWAZ dictates that when a particle does not have

enough energy to reach a boundary, the energy may be assumed to be deposited in the

zone in which it is being transported (Bielajew and Rogers 1988). The neglect of the

range of electron travel is justified on the basis of the extremely small distances (~0.1 mm)








traveled by electrons at diagnostic energies in the materials considered here (Berger and

Seltzer 1983), however the method also neglects any bremsstrahlung that may have been

created. Fortunately the radiative yield of these electrons is so small (<0.1 %) that it is

essentially zero in the materials used in this work (Johns and Cunningham 1983).

Small Acrylic Rod

The user code INAIR was written by the author to simulate an acrylic rod placed

alone, i.e., "in-air" at the isocenter of a CT scanner. In the subroutine HOWFAR, a small

right circular cylinder is described as centered at the origin of the coordinate system used,

i.e., the isocenter. In this case, the same planar geometry as the larger cylinder is used, but

only one radial zone is used because of the small sizes involved (from 1 mm to 10 mm in

diameter; see the following chapter). The rod is specified to be made of acrylic and 16 cm

long. This "in-air" rod was used for verification of the Monte Carlo model and also as a

method of normalization, as described in the next chapter. The other user-written

subroutines, MAIN and AUSGAB, used in INAIR were identical to those of CTMONO,

and were described in the previous section.

Anthropomorphic Phantoms

In order to compare the results from the cylindrical CTDI-type phantom to those

from an anthropomorphic-type phantom, a subprogram was written to simulate the

radiation transport in and around an elliptical cylinder. Anthropomorphic phantoms are

typically constructed (i.e., described mathematically) with elliptical bodies and heads

(Christy 1980). The energy deposited in and absorbed doses in such phantoms are

interpreted to be representative of these quantities in typical human subjects. The

algorithm used for the description of the elliptical cylinder, and a listing of the Mortran3

code, are given in Appendix C. This code was used in the user codes BODYPHAN and

HEADPHAN, which simulated anthropomorphic body and head phantoms, respectively.









+y

Soft tissue








skull



Figure 4-4(a): Head Phantom

Elliptical cylinders have at least two planes about which they are symmetric in
this case, three planes because the phantom is centered at the origin. Because of this
symmetry, it is necessary to use source locations in only one quadrant in the x,y-plane. In

the user codes BODYPHAN and HEADPHAN, the source location was randomly
sampled from the range 0 to 7i/2. The source angle 6 was defined as the angle between the

ray connecting the source and the origin and they-axis. The source angle was sampled by
finding 6 = (n / 2)C, where C is a random number on the unit interval.

The anthropomorphic head phantom was constructed of skeletal tissue and adult
soft tissue and was 24 cm long. A "skull" with semi-major and -minor axes of 10 cm and
8 cm, respectively, enclosed the soft tissue. The skull thickness was 9 mm throughout the

phantom (Christy and Eckerman 1987), thus the ellipse defining the inner boundary of the
skull had semi-major and semi minor axis of 9.1 cm and 7.1 cm, respectively. The head
phantom is illustrated in Fig. 4-4(a).
The anthropomorphic body phantom used in this work is 70 cm long and had semi-
major and semi-minor axes of 40 and 20 cm respectively. The phantom was constructed











spine


Figure 4-4b: Body phantom


of adult "soft tissue" as described by Christy and Eckerman (1987). A "spine" consisting

of skeletal material was located inside the phantom along the z-axis. The skeletal tissue

assumes a mix of cortical bone, trabecular bone, and red and yellow marrow (Christy and

Eckerman 1987). The spine is an elliptical cylinder of with semi-major and semi-minor

axes of 3.2 cm and 3.1 cm, respectively, as shown by Fig. 4-4(b) (Christy and Eckerman

1987).

The user codes BODYPHAN and HEADPHAN were tested by setting both semi-
major axes to the same value (making circular cylinders), setting the media to be acrylic,

and comparing values of the energy dispositions to previously computed results for the

same configuration. In all cases the results were within the statistical uncertainty of the

quantities examined.

Summary


This chapter has described the overall structure of the EGS system, as well as the
specific methods and algorithms used therein for radiation transport simulation. The








details of the individual user codes specifically written to investigate energy deposition in

phantoms from idealized CT scanners were also described.

The next chapter presents a comparison between data calculated with the EGS user

codes described above and data calculated by analytic means, and is intended to validate

the implementation of the EGS system. Chapters 6 and 7 present the results obtained with

the system and the user codes described above.

All user codes mentioned in this work were run on Intel2 80386-based micro-

computers equipped with Intel 80387 floating point co-processors. All these computers

were run under the MS3-DOS 5.0 operating system, and used the Lahey4 FORTRAN

F77L EM/32-OS386 environment for running the execution code.




























2 Intel Corporation, Hillsboro OR 97124
3 Microsoft Corporation, Redmond WA 98052
4 Lahey Computer Systems, Incline Village NV 89450













CHAPTER 5
VERIFICATION OF THE MONTE CARLO MODEL


The references cited in the previous chapter demonstrate that the simulation of
physical processes in the interaction of ionizing radiation with matter can be accurately

modeled by EGS. Rogers and Bielajew (1989) state that, although the system has been

shown capable of producing accurate results in a variety of situations, new code should

not be trusted until some appropriate experimental or previously calculated data can be

duplicated. The problem in comparing results is that high-quality experimental data are

scarce. Simple, well-specified experimental setups are needed. In this work, mono-

energetic photon beams in the diagnostic-energy range are incident on acrylic and other

tissue-like phantoms. Because of a dearth of experimental results for this situation,

comparisons were made between EGS results and theoretical relationships.


Conservation of Energy


The first method compares Monte Carlo results to those expected by applying the

principle of conservation of energy. The quantity of energy absorbed in a volume can be

written as
E=R. + Q, (5-1)

where E is energy absorbed, R, and TR are radiant energies incident upon and escaping

from the volume, and y Q is the change in rest mass (Carlsson and Aim Carlsson, 1990).

The change in rest mass is assumed to be zero for interactions in the diagnostic energy

range. Thus, the quantity of energy incident on a volume must be equal to the sum of that

deposited and that escaping from the volume.








Table 5-1: Illustration of conservation of energy for four separate Monte Carlo runs with
an acrylic phantom and no filter with a 65 cm SAD.
Energy [keV] 80 80 80 120
phantom radius [cm] 8 16 8 16
slice width [mm] 5 5 10 5
histories 2.0E6 5.0E6 5.0E6 5.0E6
Energy disposition Energy [keV]
category
1 Total incident 1.6000E+08 4.0000E+08 4.0000E+08 6.0000E+08
2 Scattered imparted 2.6312E+07 9.8485E+07 6.5857E+07 1.4813E+08
3 Primary imparted 2.3994E+07 7.5436E+07 5.9966E+07 1.1103E+08
4 Total imparted 5.0306E+07 1.7392E+08 1.2582E+08 2.5917E+08
5 Scattered out 9.3531E+07 2.1942E+08 2.3386E+08 3.2639E+08
6 Primary out 1.6163E+07 6.6584E+06 4.0315E+07 1.4437E+07
7 Total out 1.0969E+08 2.2608E+08 2.7418E+08 3.4083E+08
sum of rows 2 and 3 5.0306E+07 1.7392E+08 1.2582E+08 2.5917E+08
compare to row 4
sum of rows 5 and 6 1.0969E+08 2.2608E+08 2.7418E+08 3.4083E+08
compare to row 7
sum of rows 4 and 7 1.6000E+08 4.0000E+08 4.0000E+08 6.0000E+08
compare to row 1


All of the user codes written for this project record in separate bins the amount of

primary and scattered energy imparted to, and exiting from, the phantom. The total

energy imparted and total exiting the phantom are recorded separately. Table 5-1 shows

part of the raw EGS output from four different runs with different configurations. The

photon energy, radius, slice width and number of histories are as listed. The energy

disposition results have not been normalized and are in units ofkeV. The values in rows 1

through 7 are the raw, un-normalized data as output by EGS. The last three rows are the

sums of the primary and scattered energies for the imparted and exiting-phantom

categories. Note that the sum of these values is exactly the same as that calculated by

EGS. The sum of the scattered and primary energies equals the total energy for both the

energy imparted and energy exiting categories. The relationships between these quantities

were checked for all runs and found to be correctly accounted for in all cases. This

internal check shows that the user codes were written in a self-consistent manner.








In Air Dose Calculation


The next verification test compared the EGS results to an analytical calculation of

absorbed dose in a small volume of tissue-like material. In general, the absorbed dose D

to a small volume of material (assuming charged particle equilibrium) is given by:

D= E i (E)Sk, (5-2)
P
where E is beam energy, (p. /p)(E) is the mass energy absorption coefficient of the

medium for energy E, 0 is the photon fluence at the point of interest, and k is a constant

that transforms units (Attix 1986). If dose is given in Gy, energy in keV, the mass energy

absorption coefficient in cm2/g, and the fluence in photons/cm2, then k = 1.602x10-13

Gy-g/keV. The use of this formula assumes that only primary radiation produces the dose

in a small absorber and that any scattered radiation exits the volume of interest without

further interaction.

Next, consider the small volume of tissue-like material to be located in the middle of

a small (5 mm diameter) rod of such material, and the x-ray beam to be collimated so as to

strike only the middle of the rod. The maximum dose is approximated by (5-2) because

the small size of the rod produces negligible scatter and attenuation. Furthermore, if the

dose is integrated along the length of the rod, a value for the CTDIaf (i.e., the CTDI as

measured in air) can be calculated. The integration limits in this calculation are not

critical, because there is negligible dose deposition outside the primary-irradiated region.

This CTDIa value should be the same as the mean dose in the directly irradiated region.

Also, the CTDIr value should be independent of rod size (for sizes on the order of mm)

because of the small amount of attenuation occurring, as shown below.

The user code INAIR was written to simulate acrylic rods with various diameters (1

to 10 mm) at a distance of 65 cm from a point source ofmonoenergetic x rays collimated

to a narrow fan beam. In the interest of calculational efficiency, the size of the fan beam is








constrained to an angle that ensures that each photon strikes the rod, by using the

"SELECT-POINT-IN-RECTANGLE" method described in the previous chapter. The

photon fluence at the center of the rod is determined by the number of photon histories

run. If Nhistories are run, with a slice thickness oft and rod diameter of r, then the

incident fluence is NI(tr) photons/cm2, and the dose D produced by the beam is calculated

from (2) above. The dose was normalized to the incident fluence in units of photons per

steradian.1

For each run, the dose profile was found. The dose profile was integrated and a

value of CTDI, calculated. The results of the comparison between the analytic

calculation and the EGS simulation are listed in Table 5-2. These data are shown

graphically in Fig. 5-1. The values are listed for six energies and four rod diameters, and

all values are in units of Gy/(photon/sr). The analytical calculations were performed using

the attenuation data for acrylic from ICRU 44 (1989). The data in Table 5-2 show

excellent agreement between the two methods of calculation. The largest discrepancies

are seen at the lower energies: for the 5 mm rod, 2.6% at 40 keV, 2.5% at 60 keV, 1.6%

at 80 keV, and 0.8% at 100 keV. These values are well within the uncertainty of the mass

energy absorption coefficients, which are listed in ICRU 44 as in the range 3% to 10%.


Table 5-2: Comparison of CTDI values per incident fluence for various energies and
acrylic rod diameters.

CTDI per fluence
[Gy/(photon/sr)1
Photon energy [keV]
diameter
[mm] 40 60 80 100 120 140
analytic 6.82E-17 5.68E-17 6.94E-17 8.94E-17 1.13E-16 1.38E-16
1 6.94E-17 5.72E-17 6.97E-17 8.97E-17 1.13E-16 1.39E-16
3 6.92E-17 5.78E-17 7.06E-17 9.03E-17 1.14E-16 1.39E-16
5 7.00E-17 5.82E-17 7.05E-17 9.01E-17 1.14E-16 1.38E-16
10 7.07E-17 5.88E-17 7.11E-17 9.08E-17 1.14E-16 1.39E-16

SThe solid angle C subtended by such a fan beam is given by fR = tr/SAD 2 where SAD is the source-
to-axis-distance (Johns and Cunningham 1983).








The CTDI, values correspond to what would be measured in air if an appropriate

dosimeter (e.g., a stack of small TLD chips in a holder) were placed at a CT scanner

isocenter. This in-air measurement thus provides a convenient method of normalization

between different exposure conditions (e.g., different phantom sizes and slice thicknesses).

The 5 mm rod value is used for normalization because this size is the smallest one that can

realistically be used in a clinical scanner (Shrimpton et al. 1991).


Primary X Ray Transmission Through a Cylinder


The third method of verification is the comparison between EGS and analytical

results for transmission of primary photons in a tightly collimated fan beam through a

cylinder of acrylic. If a mono-planar fan beam ofx rays is incident on the cylinder in the

plane normal to the long axis of the cylinder, the mean transmission T of the fan-beam

through the cylinder is:


T= -1 exp[-p(E)L((p)]dqp (5-3)
qPmax 0




S14E-16 140 keV

1.2E-16 t k 120 keV

S-16 -100 keV

8E-17 80 keV

C 6E-17 -- n __ o

h 4E-17
40 keV
E-17 60 keV


1 2 3 4 5 6 7 8 9 10
diameter of acrylic rod [mm]

Figure 5-1: CTDI in air per incident fluence for acrylic rods of various sizes. Values from
the analytic calculation are shown as lines.








where p(E) is the linear attenuation coefficient of the phantom at energy E, p is the angle

within the fan beam, and L(p) is the distance through the phantom as a function of angle

within the fan beam. It is shown in Appendix B that L((p) is given by


L(c) [2 S2 tan2 11/2
1+tan9J (5-4)

where R is the radius of the cylinder, S is the distance from the source to the cylinder axis,

and 9.m = tan-(R/S). Table 5-3 shows the results obtained with the EGS simulation and
the analytical calculation.

There are minor discrepancies between the results: the analytic method gives
slightly larger values at the lower energies and smaller phantom sizes, while the EGS

values are slightly larger for the higher energies and larger phantoms. These differences

are on the order of tenths of a percent. The largest discrepancies are in the 80 mm
diameter phantom, over the 40 keV to 120 keV energy range, and are 1% to 3%. At 140

keV, the average discrepancy is approximately 5%, and this may be due to the interpolated
value of the mass attenuation coefficient used in the analytic calculation. The overall

uncertainties in the attenuation coefficients, as listed in ICRU Report 44 (ICRU 1989), are

greater than 10% in the range 10 keV to 40 keV, 3% to 10% in the range 50 keV to 100

keV, and less than 3% at energies greater than 150 keV.




Table 5-3: Comparison of percentage transmission of primary radiation through acrylic
cylinders of various diameters as calculated analytically and with the EGS code.
Percent transmission through acrylic cylinder
energy 8 cm 16 cm 24 cm 32 cm
[keV] analytic EGS analytic EGS analytic EGS analytic EGS
40 20.4 19.8 5.5 5.2 1.8 1.8 0.6 0.6
60 26.7 25.9 8.4 8.3 3.1 3.1 1.1 1.2
80 29.7 29.2 10.1 10.1 3.9 4.0 1.5 1.7
100 31.9 31.1 11.4 11.5 4.6 4.8 1.9 2.1
120 33.9 33.5 12.9 12.8 5.3 5.4 2.2 2.4
140 35.0 35.1 13.6 13.9 5.7 6.0 2.4 2.7








The discrepancies between the EGS and analytic results may be due to the

uncertainties in the attenuation coefficients and differences between the monoplanar x-ray

beam assumed in the analytic calculation and the finite-sized beam used in the Monte

Carlo model. As a result of the small discrepancies noted, the agreement between the two

methods may be considered to be excellent, indicating that the EGS user code was

correctly configured.

The comparisons made above show that EGS can closely duplicate results

calculated by the three analytic and independent means. The comparisons indicate that the

user codes produced for this project were written and implemented correctly.














CHAPTER 6

DOSE DISTRIBUTIONS AND RELATED DOSIMETRIC QUANTITIES




Introduction


The user code MONOCT was written by the author to determine the energy

deposition locations in a cylindrical phantom. The energy of the incident photons, the

SAD, the phantom radius, the slice thickness, and the phantom material can all be changed

over a range of values. In addition, two other user codes, PERFMON1 and GEMON1

were written to vary the type of beam filtration from none to a perfect filter to a GE9800

filter. A beam-shaping (or bow-tie) filter in a CT scanner is meant to reduce the dynamic

range of the fluence of primary photons exiting the patient. The perfect filter represents

an ideal situation that cannot be achieved in the clinical environment. The results using an

actual bow-tie filter (the GE filter) may thus be compared to the ideal filter.

The sections below demonstrate the effects of varying different parameters on the

dose distributions. These parameters are listed in Table 1-1 and include beam energy,

beam filter, slice thickness, phantom material, phantom diameter, source-to-axis distance

(SAD), and phantom type. First, dose profiles in air' are presented. Then, dose profiles

are shown as a function of radial location r in standard CTDI phantoms. A new quantity

the Generalized Computed Tomography Dose Index (GCTDI) is introduced, and the

relationships among the dose profiles, the standardized CTDI and the GCTDI are shown.

Next, a method of determining the energy imparted as a function of the GCTDI is

1 As described in Chapter 5, the dose to a small acrylic rod placed at the isocenter of the hypothetical CT
scanner.








outlined, and finally, the variations of GCTDI values as a function of both radial location

and beam parameters are shown.


Dose Profiles


Energy deposition in each radial and planar location was found for each run of the

user code, and the dose was calculated for each region. This dose calculation allowed the

dose profiles along the longitudinal and radial axes to be examined. Also, the doses along

longitudinal axes were integrated to find the CTDI as function of radius in the phantom.

The dose profiles D(z) calculated with a 5 mm diameter rod in air and for a 3 mm

slice thickness are shown in Fig. 6-1 for the range of energies from 40 to 140 keV. The

doses are normalized to the incident fluence in units of Gy/(photons/sr). The size of the

symbol at each data point approximates the size of the error bars, indicating 1 standard

deviation. Figure 6-1 illustrates the small amount of dose deposition outside the directly

irradiated region. The values of the doses in the directly irradiated region show the close

agreement with the analytically derived doses shown in Table 5-2. Figure 6-1 shows how

the mean value of the dose in the directly irradiated region is approximately equal to the

CTDI,. (The small variations in the dose as function ofz in the directly irradiated region

are due to the statistical fluctuations of the number of dose-deposition events occurring in

each planar region.) The doses immediately adjacent to the central region (i.e., <1 mm)

are approximately 2% of the mean dose in the central region. This percentage drops to

tenths of a percent within several centimeters and to hundredths of a percent near the ends

of the rod.

Figure 6-2 shows the comparison among the dose profiles for four slice thicknesses

(1, 3, 5, and 10 mm) at 80 keV in the 5 mm diameter acrylic rod. This figure illustrates
the independence of CTDIj values on the slice thickness, as anticipated. The area under

the four curves are directly proportional to the slice thickness; when divided by the slice

thickness, the integrated doses are the same.









Illustrative examples of dose profiles found in-phantom are shown in Figs. 6-3

through 6-5. The dose profiles for a 5 mm slice thickness in an 8 cm radius (the FDA

"head" phantom) and 16 cm radius (the FDA "body" phantom) acrylic phantom are

illustrated in Figs. 6-3 and 6-4, respectively. In both figures, the dose profile is shown for

80 keV photons and for two radial locations: at the center of the phantom and at 1 cm

depth from the periphery of the phantom, which are the locations specified by the FDA for

CTDI measurements. All dose values have been normalized to the CTDI, for the

appropriate incident energy. In Fig. 6-3, the error bars for each data point are

approximately the same size of the symbol at each point. This is also the case for the 1 cm

depth dose data shown in Fig. 6-4. Figure 6-5 illustrates the uncertainties in the dose

profile in the center of the CTDI body phantom for the 5 mm slice thickness and 80 keV

1.4E-16- -
--4 40keV-6.82E-17
1.2E-16- 60 keV 5.68E-17

-1- 80 kcV 6.94E-17
1E-16- r-
)D(z) | -- 100 keV 8.94E-17
[Gy/(photon/sr)l
8E-17- l____ -- 120 keV-1.13E-17

-0-- 140keV-1.38E-17
6E-17-


4E-17-


2E-17-- --


0- .. .-e s 1r- -
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
z [cm]
Figure 6-1: Central portion of dose profiles in a 5 mm diameter acrylic rod for a 3 mm
slice thickness for the indicated incident photon energies. The analytically-derived values
of CTDIr are shown in the legend for comparison. Error bars are approximately the size
of the symbol at each data point.

































-0.6 -0.4 -0.2 0 0.2 0.4 0.6
z [cm]
Figure 6-2: Dose profiles along the central 5 mm in a 5 mm diameter acrylic rod for 80
keV photons and the indicated slice thicknesses. Error bars are approximately the same
size as the symbol at each data point.


0.60 -


0.50 1---------------


S0.40


0.30


Z 0.20


center ofphantom
-- - -


0.10 --------


-8 -6


-2 0 2
z [cm]


1 cm depth


4 6


Figure 6-3: Normalized dose profiles at the center and at 1 cm depth in an 8 cm radius
acrylic phantom for 80 keV incident photons and a 5 mm slice thickness. Error bars are
approximately the size of the symbol at each data point.


- ------- -- _- -


-- i


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








0.8 ---
0.7 -....------- -- -- --.. --

0.6 ------------------- ----- ---- ------ -V"",

S0.5 ------------------ ---- -------- ------- --------------

c0.4 o----- f--------- ----------

0 center of phantom --

0.1 ---------------- -- ---------


-8 -6 -4 -2 0 2 4 6 8
z [cm]

Figure 6-4: Normalized dose profiles at the center and at 1 cm depth in a 16 cm radius
acrylic phantom for 80 keV incident photons and a 5 mm slice thickness. Error bars are
approximately the size of the symbol at each data point for the 1 cm depth data. (Figure
6-5 shows error bars for the center of phantom data.)


4E-18


3.5E-18


3E-18


D(z) 2.5E-18
jGy/(photon/sr)l

2E-18


1.5E-18-


1E-18


5E-18I



0 1 2 3 4 5 6 7 8
z [cm]
Figure 6-5: Portion of the dose profile along the positive z-axis at the center of a 16
cm radius acrylic phantom for 80 keV photons and 5 mm slice thickness. The error
bars indicate 1 standard deviation.








incident photon energy. The error bars show 1 standard deviation, and the doses are

normalized to the incident photon fluence. The variability in the plot is a consequence of

the small slab thicknesses used. Slab thicknesses two tenths of a millimeter were used in

the center of the phantom to provide good spatial resolution of the dose. Not many dose

deposition events occurred in these small volumes, producing larger uncertainties

associated with the doses in those regions. These results are in good qualitative

agreement with published dose profiles (Dixon and Ekstand 1978, McCullough and Payne,

1978, Southon 1981, Shope et al., 1982, Fearon and Vucich, 1985, McCrohan et al.,

1987).

It is important to note that the data displayed in the following sections that concern

line integrals of dose profiles are found by measuring the dose only in the cylindrical

regions, i.e., the planar zones are collapsed into one large region. By accumulating the

dose in the cylindrical shells, the uncertainties associated with the line integrals are

reduced to less than 1% in all cases.


Integration of Dose Profiles


Dose distributions may be used to determine the total energy imparted to the

phantom. For a homogeneous medium, total energy imparted e can be found using the

expression

f= DdM=pDdV (6-1)
M V
where p is the physical density of the medium (Carlsson and Aim Carlsson 1990). In a

cylindrical phantom of length L and radius R centered on the origin of a cylindrical

coordinate system, (6-1) can be written as

+L/2 2xR
= p J fJD(z,O,r)rdrdOdz. (6-2)
-L/2 0 0
The result after the angular integration is










R +L/2
e= f D(z,r)27prdrdz. (6-3)
0 -L/2

The definition of the CTDI for a single slice as specified by the FDA, presented in (1-1), is

I +7T
CTDI =- D(z)dz,
7T

where D(z) is the dose profile for a single scan along a line perpendicular to the scanned

slice, and T is the nominal slice thickness. Note that dose profiles can be integrated along

any path length and at any radial location in the CTDI phantom. We can therefore define

a Generalized Computed Tomography Dose Index, (GCTDI), with variable integration

limits, and at any radial location within the phantom:

1+
C(r,e) = D(z,r)dz. (6-4)


Here D(z,r,) is the dose profile parallel to the z-axis for a single scan along a line

perpendicular to the scanned slice, at radius r from the phantom center, and C(r,e)

represents the GCTDI at that radial location r and with integration limits f. In the FDA

definition, e is 7Tand r is defined at two places: at the center of the phantom and at a 1

cm depth from the phantom periphery.2 If the integration in (6-4) is carried out along the

total length of the phantom, the integration limits are L/2. If(6-4) is substituted into

(6-3), we obtain

R
= 27CpT C(r,L12)rdr. (6-5)
0

Because the integration limits have been fixed in the GCTDI (i.e., L/2), it can be assumed

that the GCTDI is exclusively a function of radial location within the phantom.3 With the


2 The investigators who first proposed the quantity CTDI, (Shope et al., 1981), stated that the CTDI is a
good approximation of the dose at a location for a series consisting of more than 8 or 10 scans. The
reason that the FDA specifies 14 scans is not in the literature to the author's knowledge.

3 With the tacit assumption of fixed incident x-ray beam characteristics.








definition of a mean or effective value of the GCTDI, C, of constant value and

independent of r, (6-5) can be rewritten as

R
S= 27pT7 r rd =xR'pT (6-6)
0

which shows that the energy imparted is simply the product of the mass of the directly

irradiated slice pxR2T and the mean GCTDI.4 Substituting (6-6) into (6-5) the explicit

expression for the mean GCTDI is found:

2
~ f C(r,L/2)rdr (6-7)
0

Note that the use of the GCTDI implies integration over the entire length of the phantom.

The analysis above shows that the GCTDI is a useful parameter in calculating the

total energy imparted to a CTDI phantom. Energy imparted can be used as an index of

machine performance or of patient risk, as discussed in Chapter 1. A comparison between

the energy deposition characteristics in CTDI phantoms and anthropomorphic-type

phantoms is provided in the next chapter. The next two sections will show the

relationships between the integration limits used, the slice thickness, and the incident x-ray

beam energy for the CTDI head phantom and the CTDI body phantom.


CTDI Head Phantom


In order to investigate the differences between the GCTDI and the CTDI, data from

several MONOCT runs with different slice thicknesses were analyzed. The resulting dose

profiles were integrated over a range of limits from 0.2 mm to 8 cm for the 16 cm-long

phantom. Figures 6-6 through 6-8 demonstrate the effect of changing the integration

limits e on the calculated value of the CTDI in a 8 cm radius acrylic phantom at 50 keV


4 Here the term directly irradiated slice is taken to mean the volume xR2T. While this definition is not
strictly true because of beam divergence, it is used here because the beam divergence is very small (<10)
for a 10 mm slice, and because of its use in calculating the energy imparted in CTDI phantoms.









for three slice thicknesses. Figure 6-6 shows the data for a 10 mm slice thickness, 6-7 for

a 5 mm slice thickness, and 6-8 for a 1 mm slice thickness.

Data for fixed integration limits for the 8 cm radius phantom are summarized in

Table 6-1. The integration limits chosen represent the total possible integration length for





0.9 ----. -- ------. ---- ----.- ----- ----- --------- ----------- .---------. .





0. ------
0. ---------- -- --------- -- ~~~i ~------ ----- ----~--------i ~ ~ ~~ -----------------




S03








Figure 6-6: t(r,e), normalized to the CTDIa value, as a function of the integration limit f
in an 8 cm radius acrylic phantom at 50 keV for a 10 mm slice thickness.



0 ..- ...- .... .-- ..- ---. -- ....----......--..---. ..... ... .......- -.... .
r an
0.5 -






- ---- ^ -^ r - ---* -- r = 0 cI
r=7cm






















0.1 -- ..---------------- .------------ .-----------. -----------.-----------r
0.1














0 1 2 3 4 5 6 7 8
Integration limits [cm]

Figure 6-7: C(r,e), normalized to the CTDIj value, as a function of the integration limit e
in an 8 cm radius acrylic phantom at 50 keV for a 5 mm slice thickness.
0.9 -


0.7 ------------- -
06 ----r=O-m -
0.6 ------ ------ --- ------ ------- ----------- r=2cm --
0.4 ------------ ------ ------ ------ r ----------- r=4cm
Z 0.3 ---------- ----- ------ --- ------- -------- ------------ -- r=6cm -
0.2 -- -------- -------- -- -- -------- I- ----------7c
0.1


0 1 2 3 4 5 6 7 8
Integration limits [cm]

Figure 6-7: C(r,t), normalized to the CTDI, value, as a function of the integration limit t
in an 8 cm radius acrylic phantom at 50 keV for a 5 mm slice thickness.






83



---- ---...-----------. --------. ----------- ---------------------------... .--
0.9 ---------- ---------------------------------- --- ---- --- ----
0.8 ..------------j.------. ----.----- ----.--.----- ------------------.... ..
0.8 ---------- -----
U -0.7 -- ---- ---- --- ---- -- ---- --- -- --
0.6 ---- ---------.------ ------ --- ------- ------------- r=-0 -



0.3 ----------- ------ --------- ----- -- ----------- r=2an
0 .2 - --. I ..- -. ..-- - - - ---- r = c m .





0 1 2 3 4 5 6 7 8
Integration limits [cm]

Figure 6-8: QCr,), normalized to the CTDIa, value, as a function of the integration limit e
in an 8 cm radius acrylic phantom at 50 keV for a 1 mm slice thickness.




Table 6-1: (r,) as a function of integration limits t and radial position r in a head
phantom at 50 keV. The results shown are the percentages of the values at each (
normalized to the value at M=8.0 cm.

cm| r= 0 cm r=2cm r=4cm I r=6cm r=7cm
10 mm slice thickness
8.0 100.0 100.0 100.0 100.0 100.0
7.0 96.9 96.5 97.0 97.5 97.8
5.0 87.1 87.9 89.7 92.1 93.7
5 mm slice thickness
8.0 100.0 100.0 100.0 100.0 100.0
5.0 88.1 87.9 89.6 92.1 93.6
3.5 76.8 76.8 79.9 85.0 87.9
1 mm slice thickness
8.0 100.0 100.0 100.0 100.0 100.0
5.0 87.8 88.0 89.6 92.1 93.6
0.7 35.0 36.9 42.1 53.1 62.0








the 16 cm phantom (8 cm), the integration length obtained with a typical pencil chamber

used for CTDI measurements5 (5 cm), and the 7T definition of the FDA.

The results indicate that, in the 1 cm slice thickness, integrating over the 7 slice

thicknesses produces CTDI and GCTDI values within 3% of each other at all radial

positions. Measurement of the CTDI with a 10 cm pencil chamber underestimates

C(r,L/2) by approximately 6% to 13%, depending on measurement location.

The 5 mm slice results show that the FDA definition of the CTDI produces dose

estimates that are 12% to 23% less than the corresponding C(r,L/2) values, depending on

measurement location. Use of a 10 cm pencil chamber underestimates C(rL/2) by 7% to

12%, very similar to the results for the 1 cm slice.

The truncation effect is most obvious in the 1 mm slice data, where the FDA

definition produces dose estimates that are 38% to 65% less than the corresponding

C(r,L/2) values. The 5 cm integration limit (i.e., the pencil chamber) underestimates

C(r,L/2) by 6% to 12%.

The FDA-defined CTDI value can underestimate the actual point dose from a series

of contiguous slices anywhere from 6% (at the 1 cm depth for 10 mm slices) to 65% (at

the center position for 1 mm slices) at 50 keV. This finding is significant because CTDI

values are routinely used as estimates of actual patient dose. The patient doses estimated

from CTDI values are underestimates in any case because the CTDI is defined as dose to

acrylic, not to tissue, and thef-factor is smaller for acrylic than for tissue.6 Furthermore, if

conventional CTDI measurements were used in place of the C(r,L/2) in the integration

method described above, underestimation of the energy imparted to the phantom would

5 Nuclear Associates, Carle Place NY 11514
6 Thef-factor is the ratio of absorbed dose in a medium to exposure in air, and is defined as

In =0.(873 -en / P) med
fm = 0.873
(een / P)air
(Attix, 1986). At energies less than 150 keV, the values of the mass energy absorption coefficients for
water and for soft tissue are several percent larger than those for acrylic (ICRU 1989). Thef-factors and
absorbed doses for these materials are accordingly larger than that for acrylic.








result. The data in Table 6-1 can be used to calculate the energy imparted within the

integration limits shown. When the 7T integration limit data are used to calculate C with

(6-7), the energy imparted is underestimated by 50% for 1 mm slices, 18% for 5 mm
slices, and by 4% for 10 mm slices in the CTDI head phantom.

Another factor which can lead to underestimation of energy imparted to a patient is

the physical size of the phantom a phantom of a more realistic length would absorb

more energy. (The anthropomorphic phantom described by Christy and Eckerman (1987)
is 24 cm long.) The data shown in Figs. 6-6, 6-7 and 6-8 indicate that not all the scattered
radiation is accounted for in the dose profile integration. One would expect an

asymptotic relationship between the GCTDI and the integration limits used. The user

code LONGCT was written to determine the length of the phantom required to absorb all

the radiation scattered out of the primary beam. LONGCT simulated 50 keV photons
incident on a cylindrical acrylic phantom 140 cm in length, and of 8 cm radius. A 5 mm

slice thickness was used. The results are shown in Fig. 6-9. The results indicate that, for
circular phantoms 8 cm in radius, integration limits greater than approximately 20 cm

produce no significant (<1%) change in the GCTDI. Although not many patients would

be expected to have 40 cm-long heads, these results indicate that CT exams of the head

can deposit energy at locations relatively remote from the exam site. The data also

indicate that if FDA's minimum phantom length of 14 cm was used, the energy imparted

underestimate would be even greater.

The next investigation of the effects of varying integration limits in CTDI and

GCTDI calculations uses spectra in place of monoenergetic incident x-ray beams. Figure
6-10 shows results obtained when a 120 kVp spectrum of 5.4 mm Al half-value layer

(HVL) is incident on an 8 cm radius acrylic cylinder. The spectrum was obtained from
Birch et al. (1979), and the data are shown for a 10 mm slice. For integration limits larger

than the slice thickness, the effect of limiting the integration length is very similar to that
exhibited by the monoenergetic x rays, which allows the conclusion that the CTDI/GCTDI






86

truncation effect in an 8 cm radius phantom is essentially energy independent in this

model.



1.2 .-- .------ ---------------------- 7 -----------------.-------------- ----------------------------
1.2





,
1 --- - -

0.8 -- ----- -- ---- -- -------
r = 0 cr
0.6 -- -- ----- ---- r= 2cm




0.2 -------- ---------------------------------------------------
0.2 --------------:---------------- -t----------- ---i------------ -: -7, ~ ----------------


0 5 10 15 20 25 30
Integration limits [cm]

Figure 6-9: C(r,e) values, normalized to the CTDI, value, as a function of the integration
limit t in an 8 cm radius acrylic phantom at 50 keV for a 5 mm slice thickness in a 140 cm
long phantom.'



0.9 ..--------- -- --.-- --- .---....------------ .----. -- o- --



0.8s ----- ----'-- ---- ----- 1 ----- ----- ----------- 2 .




S 0.73 ---- ------- -- ------- ----- ---------- ---------- -- --r=6cm
0.6 --- ------- -- ----- -- ----- -- ------- ------- ----------- 0-_ r = 07cm
0.7

0.4 -- :::::---- ----------- ----:::: :::::::::: r- -







0 1 2 3 4 5 6 7 8
Integration limits [cm]


Figure 6-10: C(r,), normalized to the CTDIr value, as a function of the integration limit
t in an 8 cm radius acrylic phantom for a 120 kVp spectrum with 5.4 mm Al HVL, for a
10 mm slice thickness.