A REPETITIVE INTEGRAL MEASUREMENT TECIHIQUE FOR X AND
GANIlA RADIATION DOSI:ETRY AND DEPTH DOSE SCANNING
By
],LAroENCE 11EKPRELL FITZGEPAL1D
A DISSERTATION P1REZSENTED TO THF GRADUATE COUNCIL OF
THE LTNIVESITY OF FLORIDA
IN PARIAL. FULFILLMENT OF IHE REQUlRE:MEt.TS FOR THE
DEGREE OF DOCTOR OF PilLOSOPHY
UNIVERSITY OF FLORIDA
1974
To my Mother and late Father who worked so hard and made
many sacrifices that I might have the
opportunity of an education.
ACKNOWLEDGE [ENTS
I would like to express my appreciation to the mer.,iers of my super
visory conmrittee for their assistance in this investitiation anJ their
support during miy graduate study. Special appreciation is given to Dr.
Walter Mauderli, my committee chairman, for his guidance in all phases
of this vxork. My committee cochairnan, Dr. Emirett Bolch, is greatfully
acknowledged for his guidance and advise throughout my graduate study.
Sincere thanks go to M r. John Preisler for his many skills and sug
gestions in fabrication of mechanical apparatus used in this investi
gation. The assistance of the staff of the North East Regional Data
Center is acknowledged. My sincere appreciation goes to Mrs. Hughlene
Sadler for her assistance in typing working copies and to Mrs. Juanita
Bradley for her excellent job of typing the final manuscript.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ...................................................
ABSTRACT ...........................................................
CHAPTER
1 INTRODUCTION ..............................................
2 RADIATION DTTECTiON AND DE'TH DOSE ........................
Phantoms and Depth Dose .................................
Chemical Change .........................................
Film ..................................................
Chemical Dosimeters ...................................
Heat Production .........................................
Thermolumincsccnce ......................................
Light Production ........................................
Electrical ..............................................
Solid State ...........................................
Gas Detectors .........................................
Summary .................................................
3 THE 10NIZATION MrET OD .....................................
Ionization Chambers and Radiation Measurement ...........
Measurement of Small Currents ...........................
Measurement Methods .....................................
Depth Dose Scanning .....................................
Summary .................................................
Page
iv
ix
3
4
6
6
9
11
12
15
18
18
19
22
24
24
27
29
36
38
CHAPTER Pagg
4 THE RATE MEASURE',MENT AND THE REPETITIVE INTEGRAL
MEASUREMENT METHODS ...................................... 39
The Lateral Dose Function ............................... 39
Depth Dose Scanning and the Rate Measurement ........... 41
Limitations of the Rate Measurement Method ............. 46
The Scanning Speed ................................... 46
The Current Magniitude .................................. 47
The Chamber Volume ..................................... 47
The Time Constant ...................................... 48
Correction Technique ..................................... 49
The Integral Method .............................. ...... 51
The Repetitive Integral :easurerent rMthod .............. 54
Constant and Linear Current ........................... 57
Summary ...................................... ......... 58
5 COMPARISON OF THE RATE AND REPETITIVE INTEGRAL,
IMEASUI'E FJ'N"T METHODS ...................................... 60
The Step Function Response .............................. 60
Point Detector Response ............................... 62
Cylindrical Detector Response .......................... 66
The Size of the Measuring Interval .................... 71
Lateral Dose Function Response .......................... 73
The Effect of Scanning Speed ............................ 79
Advantages of the Repetitive Integral
Measurement cM thod .................................... 81
Elimination of the Effect of the Time Constant ........ 81
Application to Pulsed Radiation Sources .............. 82
Elimination of Amplifier Zero Drift .................. 82
Reduction of Signal Noise ............ .................. 83
Independence of Scanning Speed ........................ 83
Summary ............................. ...................... 84
6 THE REPETITIVE INTEGRAL ELECTROMETER AMPLIFIER ............ 85
The Integration Circuit ................................ 86
CHAPTER Page
The Switching Electrometer .............................. 88
The Switching Method .................................. 88
Reset Network Analysis ................................ 92
The Repetitive Integral Amplifier Circuit ............... 96
The Operational Arplifier ............................. 96
The Resetting FieldEffect Transistors ............... 98
The Compensating Pulse Level Buffer ................... 98
The Inverting Operational Amplifier ................... 98
The Input Protection Diodes ........................... 99
The VoltageDivider Gain Adj'isLnent ................... 99
Amplifier Construction .................................. 105
Amplifier Control Logic ................................. 108
The Digital Clock ..................................... 109
The Switching Logic ................................... 110
Amplificr Testing .................................... ... 319
Linearity Testing ..................................... 119
Stability Testing ..................................... 20
Electrical Noise ........................................ 122
Power Hum ............................................. 123
iicrophonics .......................................... 123
Amplifier Drift ....................................... 124
Input Leakage ......................................... 125
Ground Loops .......................................... 125
Random Noise .......................................... 126
Amplifier Noise ....................................... 128
Summary ................................................. 133
7 THE DEPTH DOSE SCANNING SYSTEM ............................ 135
The Mechanical Scanning System .......................... 136
Stepping Motors ........................................ 136
Drive Assembly ....................................... 137
The Scanning Logic .................................... 138
The Water Phantom ..................................... 149
System Testin ......................................... 153
Adaptive Scanning Rate................................. 154
The Data Processing and Recording System ................ 155
The AnalogtoDigital Converters ..................... 156
CHAPTER Page
The Shift Registers ................................... 156
The Optical Couplers .................................. 158
The Arithmetic Processing Units ....................... 159
BinarytoBinaryCodedDecinal Converters ............. 159
The Data Processing Circuit ........................... 159
The Magnetic Tape Recorder ............................ 162
System Testing ........................................ 163
The Detector and Amplifier System ....................... 163
Detector Construction ........... ......... ...... ....... 16',
Design Considerations ................................. 167
System Testing ........................................ 167
The Total System ........................................ 168
Computer Processing .................................... 171
Summiry .................................................. 176
8 RESULTS AND CO'CLUSIO:IS ................................... 178
Depth Dose Scanning S stemi Results ...................... 178
Conclusions ............ ................................ 184
BIBLIOGRAPHY ....................................................... 190
BIOGRAPHICAL SKETCH ...................................... .......... 195
viii
Abstract of Dissertation Presented to the Graduate
Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
A REPETITIVE INTEGRAL MEASUREMENTT TECHNIQUE FOR X AND
GAMMA RADIATION DOSIMETRY AND DEPTH DOSE SCANNING
by
Lawrence Terrell Fitzgerald
June, 1974
Chairman: Walter Mauderli, D. Sc.
CoChairman: W. Ermett Bolch, Ph. D.
Major Department: Eiviroimnental Engineering
The purpose of this study was to iL.prove measuring techniques in
depth dose scanning of a radiation treatment field with regard to in
creasing accuracy, speed of measurement and the design of an easily
transportable system. A new measurement technique and a new detector
transport system have been developed to fulfill these objectives.
The new measurement technique, called the repetitive integral Method,
estimates the input current flowing from a radiation detector by repeti
tive cycles of integrating and resetting over short time intervals and
has been shown theoretically to be superior to the standard rate
measurement method for the realtime dynamic current measurements re
quired in depth dose scanning. Electronic instrumentation which
functions in the repetitive integral mode has been developed and experi
mental results follow the theory. The repetitive integral measurement
method has the following advantages:
1. The effect of the amplifier time constant is eliminated.
2. The method is better suited for measurement of pulsed radiation
sources.
3. Longterm amplifier drift may be eliminated.
4. The integrating circuit acts to attenuate higherfrequency noise.
5. Amplifier thermalnoise output is negligible.
6. The method is independent of scanning speed.
The new detector transport system which uses stepping motors and
digital logic guarantees the accuracy and reproducibility of the new
depth dose scanning system. Digital techniques permit precise detector
positioning to within a small fraction of a millimeter and ensure
perfect correlation between detector position and amplifier measurement
cycles.
Application of the repetitive integral measurement method in con
junction with the detector transport system enables accurate and repro
ducible measurements to be made at greater scanning speeds than previ
ously possible. The depth dose scanning system developed is completely
selfcontained, is housed in two units which are easily transported in
a small van and provides a means of rapidly determining complete depth
dose distributions for a variety of radiation therapy equipment.
CHAPTER 1
INTRODUCTION
Shortly after the earliest therapeutic applications of xrays,
it became evident that some measure of absorbed energy, or dose, was
required to prevent overtreatment and to enable the compilation of
treatment results which would be transferable to other patients. As
the energies of the radiation used in xray therapy increased the need
developed for dose data at various depths below the skin surface.
These doses below the skin have come to be called depth doses.
Complete depth dose determination for a radiation treatment
field requires measurement at many points throughout the field.
Scanning devices which move a radiation detector through a radiation
field have been designed for obtaining depth dose data automatically.
These scanning devices have been slow in operation, limited in
accuracy and so large that transport from one location to another
has been difficult.
The development of computer techniques in xray treatment
planning during the 1960s has made possible the computation of
absorbed doses from very complicated treatment plans and has provided
a means of computation for individual patients. It is generally
agreed among radiation therapists that computerized treatment planning
has improved radiation therapy. Overtreatment complications may be
avoided and new treatment plans, not previously used because of
calculation complexity, may be computed. The computer techniques
now available make use of all the known physical parameters concerning
radiation absorption in tissue, with the limiting factor in dose
determination being the accuracy of the basic depth dose data for
the radiation fields used in treatment.
The aim of this work has been the improvement of measuring
techniques used in depth dose scanning. Primary consideration has
been given to increasing the accuracy and speed of measurement and to
designing a system easily transportable. This has been accomplished
with the use of digital design techniques and a new measuring method.
All testing has been limited to cobalt60 gamma radiation, but
the results also apply to the :rays produced by other types of
therapy equipment. The radiation detection system has been designed
to be compatible with the radiation intensities encountered in
radiation therapy equipment. All references to scanning in later
chapters pertain to depth dose scanning and the term radiation always
refers to x or gamma radiation or to both collectively.
Reviews of the development of radiation therapy and the various
methods of depth dose measurement are given in Chapter 2. Chapter 3
discusses the ionization measurement method and its limitations in
depth dose scanning. A new measurement technique is developed and
described in Chapters 4 and 5. A new electrometer amplifier is dis
cussed in Chapter 6. Chapter 7 discusses the mechanical scanning
design and the integration of mechanical and electronic systems into
the complete depth dose scanning system. Measurement results, system
evaluation and conclusions are given in Chapter 8.
CHAPTER 2
RADIATION DETECTION AND DEPTH DOSE
Professor Wilhelm Conrad Roentgen made his historic discovery of
xrays on Novenber 8, 1895 and published his results soon thereafter.1
Roentgen had used his hand in some of his experiments to test the pene
trating power of xrays; a fact which suggested a possible application
in medicine. Just 23 days following the public announcement of
Roentgen's discovery, Grubbe,2 a businessmaninventor, performed the
first xray therapy on a breast carcinoma in Chicago. The treatment
was suggested by the physician treating Grubbe for xray dermatitis
which he had contracted in duplicating Roentgen's experiments. Grubbe
had the dubious distinction of being the first to use xrays thera
peutically and the first to experience the harmful effects of excessive
xray exposure.
In the early days of radiation therapy, skin reactions were ob
served in both physician and patient, making it evident that some unit
of dose for specification of quantity was necessary. The dosage was at
first judged by the effect on the skin, with erythema and epilation dose
becoming the mainstay of clinical radiotherapy. No information concern
ing the dose received at any point under the skin was available. The
skin dose was of primary importance as it was the limiting factor in
therapy. The early xray apparatus used no beam filtration and
operated at low voltage; so the amount of soft radiation which was
readily absorbed by the skin was quite large.
Radiation therapy is generally considered to have come of age
around 1920 when xra: tubes had been improved so that they could
tolerate 200 kv and the value of beam filtration had been recognized.
Beyond this point, progress was rapid in the development of higher
voltage machines which produced radiation with greater penetration,
and the uses of xradiation in therapy continued to grow. Already in
1936 Elward3 reported that there were over 400 different diseases
which were treated with xray therapy. As the uses of xray therapy
have expanded and as higherenergy beams have been developed, there
have been demands for more and better dose determination. The skin
dose is no longer adequate as it was in the early days because with
higherenergy beams the doses below the skin may be the limiting
factor in therapy. Doses at points throughout the treatment volume
are thus required. A most interesting review of the development of
radiation dosimetry may be found in papers by Glasser4 and Victoreen.5,6
Phantoms and Depth Dose
As no direct measurement of the radiation received at the tumor
is possible it must be determined by measurement in simulated tissue.
The tissuesimulating material, called a phantom, should be a material
which absorbs and scatters radiation as does human tissue. Spiers7
pointed out that a good simulator of tissue should have the same
density, same atomic number, and contain the same number of electrons
per gram as tissue. Dry pressed wood of Masonite or hardboard have
been used as phantoms, but these are not reproducible from one therapy
center to another and do not absorb exactly as tissue. Several other
reports8'9 have described solid phantom materials which are tissue
equivalent. Pressed wood and other solid materials are very limited
in their usefulness since the measuring probe may not be moved from
one point to another within the medium. Water and soft tissue have
the same effective atomic number10 of 7.42, the same density and 3.34
x 1023 and 3.36 x 1023 electrons per gram respectively. The radiation
absorption of water and soft tissue is thus essentially the same and
water is usually the phantom choice since it is readily available, is
of constant consistency and allows movement of the measurement probe
within the mediur.. Water phantoms, however, require the use of probes
sealed against moisture.
Water phantoms are generally used as simulators of tissue when
the radiation dose is derived at some depth below the skin. Measure
ments made in water are usually specified as a ratio of the recorded
value at the depth of interest to the value at some other fixed point.
Johns and Cunningham11 define the percentage depth dose at a depth d
as follows:
dose at depth d
Percentage depth dose at depth d = 100
dose at maximum buildup
At energies less than 400 kv the depth of maximum buildup is practically
zero and the surface dose is used in the definition.
Although many different instruments have been used in radiation
detection and in depth dose determination, the operating principles
are few. The basic requirement of any radiation detector is that it
interact with radiation so that its response is proportional to the
radiation dose. Table 1 classifies the primary types of radiation
detectors according to the detector medium and the effect produced
in the detector. Each of the basic types of instrumentation will be
discussed briefly, particularly for therapy applications in depth dose
determination with x and gamna radiation.
Chemical Change
Film
Photographic emulsions have been used for radiation dosimetry
from the very discovery of xrays. Film is a rather unique dosireter
as it allows dosimetry of ionizing radiation over dose ranges of
millirads to m.cgarads, over tine periods of microseconds to months and
over areas of square microns to square meters. The small thickness of
film makes it particularly suited to radiation dosimetry as it can be
introduced into a radiation field with a minimum of spatial distortion.
Film has a high resolution, an advantage which is used in alpha and
beta particle autoradiography.
In the dosimetry of x and gamma radiation by photographic film,
the film is exposed by secondary electrons produced by one or more of
the primary processes (photoelectric effect, Compton effect, pair
production) whereby x and gamma radiation interact with matter. Johns
and Cunningham11 and Hendee12 describe the interactions of x and gamma
TABLE 1
PRIMARY EFFECTS IN RADIATION DETECTORS AND DETECTOR MEDIA
Instrumentation
Detector
Medium
Radiation
Effect
Film
Chemical Dosimeter
Calorimeter
Thermoluminescent Dosimeter
Scintillation Counter
Solid State
Ionization Chamber
Geiger Counter
Proportional Counter
Photographic Emulsion
Solid or Liquid
Solid or Liquid
Crystal
Crystal or Liquid
Semiconductor
Gas
Gas
Gas
Chemical Change
Heat Production
Thermoluminescence
Light Production
Electrical
(Pulse or Current)
radiation in detail. The important fact concerning the interaction of
x and gamma radiation with the 'ilm emulsion is that the mass attenu
ation coefficient for the photoelectric effect varies with the cube of
the atomic number. The film emulsion is made up largely of silver
bromide which has a high atomic number. Thus a large amount of the
energy deposited in the silver emulsion is from lowenergy photons.
The primary defect in film dosimetry of x and gamma radiation is the
dependence of sensitivity on photon energy.
Photographic methods with plates and paper were used in the early
1900s for dose determination. Strips of photographic paper were placed
on the patient's skin in the treatment area during the time of treatment
with xrays. The degree of blackening on the paper was determined by
comparison with a standard chart. Paper strips covered with different
thicknesses of aluminum were used to estimate depth dose. These
methods were unsatisfactory since standard paver, standard developer,
standard temperature and a standard scale were required. Also the
method gave results only after the radiation dose had been given. The
energy dependence of film was especially important at that time, as no
filtration was used, and the beam was composed largely of lowenergy
photons.
The energy dependence of film has been overcome to some extent by
selective filtering and the method has again been used for depth dose
determinations. Granke et al.13 have used film dosimetry for determi
nation of tissue doses with 2 Mev xrays by placing sheets of film
between different sections of an Alderson14 RANDO (RAdiation ANalog
DOsimetry) phantom, a tissue equivalent phantom containing a human
skeleton. Isodose curves of depth dose values for different radiation
fields were also produced with film dosimetry. Loevinger and SpiralS
used film dosimetry in radium therapy and developed a method combining
multiple films to determine the dose for a total treatment.
Even under the best of conditions, film dosimetry presents
significant problems. Irregular emulsion thickness contributes to
varying sensitivity over different portions of the film. Variations in
film processing and storage conditions from one laboratory to another
make it difficult to exchange results with other researchers. For
these reasons and the problem, of energy dependence film dosimetry is
not the method of choice when accurate results are required. Filn
dosimetry has been most useful in personnel monitoring where highly
accurate results are not required. Film dosimetry as a method of depth
dose determination for x and gamma radiation has been replaced with
more reliable and easier techniques.
Chemical Dosimeters
With chemical dosimeters the radiation dose is determined by the
chemical change in a particular medium. Roentgen1 in his first report
of the discovery of xrays called attention to chemical reactions
created by the rays. Very early dosimetry used crude chemical dosime
ters which changed color on exposure to the soft xrays then used in
therapy and diagnosis. In 1902 small discs of fused potassium chloride
and sodium carbonate which became discolored on radiation exposure were
used and various colorations were related to erythema doses. Barium
platinocyanide formed the basis of a dosimeter developed in 1904 and
was used extensively for many years. Again a color change indicated
the degree of xray exposure. This system was highly energy dependent
because of the high atomic number of the dosimeter material. These
early chemical systems were rather insensitive and required exposures
of 1,000 roentgens or more to produce visible changes. Due to these
problems and the rapid growth of the ionization method, interest in
chemical dosimetry waned, but was rekindled with the advent of atomic
weapons, nuclear reactors and the use of highintensity radiation
sources. These developments created a need for dosimeters for measuring
large doses of x and gamma radiation and have provided an impetus to
further research in the area of chemical dosimetry.
The Fricke dosimeter, described by Fricke and Morse,16 is a
ferrousferric sulfate systtn whose principal drawbacks are a lack of
sensitivity and a dependency on linear energy transfer of the radiation.
The Fricke dosimeter is generally considered the best for 4 40 kilorad
doses of x and gamma radiation. It has been accepted as a standard in
radiation chemistry and it may be used to measure dose in absolute units
to within 1 2 percent. Methods of preparing a Fricke dosimeter are
given in the National Bureau of Standards Handbook 85.17 The Fricke
dosimeter has been recommended for calibration of highenergy electron
beams used in radiation therapy.18
Chemical dosimetry is now used principally for measuring higher
doses than those encountered in radiation therapy but has found some
clinical application in determining the doses received by interior body
structures, since the dosimeter may be inserted in body cavities. An
aqueous solution enclosed in a suitable container is excellent for
biological dosimetry since it closely approximates the density and
atomic composition of biological tissues. Chemical dosimeters have
been particularly useful for x and gamma ray dosimetry in animals
exposed in atomic field tests.
Stenstrom and Lohmannl9,20 used a chemical dosimeter derived from
methylene blue for determination of depth doses but, in general,
chemical dosimeters have found very little use for depth dose measure
ments in radiation therapy primarily due to .their insensitivity.
However, some researchers have used the ferrousferric system.21
Presently chemical dosimeters, which may be used for doses as low as
400 rads,22 are available and thus are sensitive in the range of
therapy doses.
Hleat Production
Calorimetry has been employed to measure the heat produced from
the dissipation of energy by radiation. A calorimeter contains an
absorber specifically designed to convert all or part of the incident
radiation into heat. The absorber contains a temperaturesensitive
element and is thermally insulated. Calorimetry permits measurements
of radiation energy in terms of fundamental energy units thus providing
an absolute basis for evaluation and comparison of experiments or
treatments.
Calorimeters have for many years been applied to many different
problems but their use with ionizing radiation has been limited due
to the small magnitude of energy involved. The calorimetric method
has a primary advantage over other dosimetric methods as it measures
the energy deposited directly whereas other methods use secondary
processes. Calorimeters also have some disadvantages. Their sensitivity
is adequate for radiation intensities and dose rates used in diagnostic
and therapy but is insufficient for use in radiation protection. Also
in some types of radiation processes energy, which should be measured,
may be lost and in some processes energy is gained from sources other
than the radiation beam. Scattering of radiation out of the calorimeter,
photonuclear reaction or the emission of a penetrating neutron,
bremsstrahlung production by secondary electrons and endothermic or
exothermic chemical reactions are such processes.
Calorimetry was used by early investigators in radiation measure
ment. Curie and Laborde23 used the method in the first determination
of the rate of energy release by radium disintegration. Stahel24
employed the calorimetric method in measuring the energy absorbed by
water on exposure to x and gamma radiation. The skin erythema dose in
humans was related to these measurements, given in ergs per cubic
centimeter. Genna and Laughlin25 demonstrated the determination of
absorbed doses in radiation therapy with cobalt by the calorimetric
method. Rollo et al.26 have used calorimetry for measurement of total
energy deposition of Grenz rays. The calorimetric method has not played
a large role in depth dose measurement primarily because the ionization
method provides a simpler way of measurement. As more interest is
expressed in absolute units the calorimeter will surely play a role as
a primary standard for these units.
Thermoluminescence
Thermoluminescent dosimeters have been classified by Fowler and
Attix27 as solid state integrating dosimeters. This broad classification
includes photographic film, solid scintillators and plastic films
containing dye. Semiconductor devices could also be included in this
classification but they shall be discussed in a separate section.
Nonelectrical solid state dosimetry systems may be further divided into
two categories: destructive readout and nondestructive readout. The
nondestructive readout systems are those in which the effect produced
by the radiation is not destroyed by measurement of the effect.
Nondestructive systems may have their radiation effect measured any
number of times. Destructive readout systems may have their radiation
effect measured only once as the method required in reading destroys
the effect.
Nondestructive systems include glasses, plastics and dyes which
change color upon radiation exposure. Other nondestructive effects
are radiophotoluminescence of phosphate glass, degradation of
luminescence of anthracene and other crystals and a change in electron
spin resonance in alanine. Most of the nondestructive system materials
are useful only for doses greater than 104 rads but some dyes, phosphate
glass and alanine are suitable for doses in the range given in radiation
therapy (100 to 400 rads). A dyed polymethyl methacrylate, called
Perspex Red 400,* which becomes black on irradiation is commercially
available. It has been found useful for xray and electron beam dose
for depth dose measurements in xray contact therapy where the dose
rate is very high. Phosphate glass has a high effective atomic number
and is thus quite energy dependent. Bradshaw et al.28 used powdered
*May be obtained from Imperial Chemical Industries Ltd., Plastics
Division, Welwyn Garden City, Herts, England.
alanine, an amino acid, for depth dose measurements in animals and
satellites. This method is difficult due to the very small electron
spin resonance signals produced. Nondestructive readout systems for
radiation dosimetry are in general not suitable for routine depth
dose measurements in the range of therapy doses.
Destructive readout systems include the thermoluminescent
materials and infraredstimulated luminescence. Many of these systems
are suitable for measurement from below 10 rads to 104 rads. The
thermoluminescent materials have had the greatest impact of the two
systems on radiation dosimetry methods.
Many materials are thermoluminescent but only a few have been
found useful for therapeutic dosimetry applications. The ideal thermo
luminescent phosphor must have a strong light output and be able to
hold trapped electrons for some period of time at a temperature suit
able for the particular dosimetry application. The three most impor
tant phosphors used in radiation dosimetry in the range of therapy
doses and lower are calcium sulfate, calcium fluoride and lithium
fluoride.
Calcium sulfate was used in depth dose measurement in body cavi
ties of experimental animals as early as 1954. This phosphor has the
remarkable ability to measure exposures in the microroentgen range.
Its main disadvantage is significant "fading" at room temperature, a
factor which apparently is dependent upon the method used in phosphor
preparation.
Calcium fluoride has a linear response to gamma radiation from
a few milliroentgens to 500 roentgens. This phosphor does not have
significant problems of "fading" but its response is energy dependent
at energies below 0.1 Mev.
Lithium fluoride dosimeters may be found in a number of shapes
and sizes and are widely used in clinical dosimetry. Lithium fluoride
dosimeters are approximately tissue equivalent and are of sizes that
may be placed in body cavities or in phantoms for depth dose studies.
Vacirca ct al.29 have reported a filmthermoluminescence system for
determining body doses in diagnostic radiography. Thermoluminescent
dosimeters have been used by Puite et al.30 for the purpose of
intercalibration of xray units at many different institutions so that
experimental results may be transferrable. The dosimeters were ideal
for this purpose as they could be sent from one institution to another
through the postal service.
The search for other useful thermnolurminpscent materials continues
with the hope that dosimeter "fading," which is currently a problem,
may be overcome.
Dixon and Watts31 have reported on barium fluoride as a thermo
luminescent dosimeter which exhibits good characteristics, but "fading"
is still a serious problem. Jayachandran32 reports that lithium
borate is more tissue equivalent than other thermoluminescent dosime
ters and should be evaluated for clinical use. Scarpa33 has suggested
that beryllium oxide may be a useful thermoluminescent material.
Light Production
One of the earliest means of radiation detection was by scin
tillation counting. Rutherford in his experiments with alpha particles
used a zinc sulfide crystal as a primary detector and visibly observed
the light flashes produced by the alpha particles striking the crystal.
This was very tedious and with the development of gasfilled detectors
the scintillation method fell into disuse. Interest in the method was
revived in the late 1940s and phenomenal development has taken place.
Many solid and liquid substances emit light when exposed to x,
alpha, beta or gamma radiation. Photomultiplier tubes have made it
possible to detect light flashes that correspond to the absorption of
a single quantum or a single charged particle and stimulated the search
for better scintillators. Scintillators may be classified as organic
crystals of the hydrocarbon type, inorganic crystals and powders of
the alkali halide.and zinc sulfide types, liquid and plastic solutions
in hydrocarbon solutes, noble gas types and glass types. Murray3 has
listed the various scintillators available and describes the important
characteristics of each.
Two methods of measurement are used with scintillation detectors.
With the first method individual particles or quanta are counted. In
the second method the average DC current is measured as the total
light output of the scintillator. The DC output of the photomultiplier
tube is proportional to the rate at which radiation energy is absorbed
in the scintillator if the light output of the scintillator is propor
tional to the absorbed energy. The inorganic crystals are of partic
ular importance for measuring x and gamma radiation. The alkali
halides are most often used since they can be grown into large crystals
of good transparency, a requirement for efficient detection of x and
gamma radiation. The inorganic crystals are activated by addition of
certain metals to make them highly efficient scintillators. Sodium
iodide activated with thallium is the most useful scintillator for
x and gamma radiation detection.
For purposes of x and gamma ray dosimetry it is desirable to have
a scintillator whose output per roentgen is constant over the entire
range of energies for which the detector is to be used. The output
per roentgen is a comparison between the scintillator measurement and
the results obtained from the standard freeair ionization chamber.
All inorganic scintillators contain elements with atomic numbers
higher than that of air and thus have a greater output per roentgen
at low energies due to the photoelectric effect. At high energies
a larger output is expected due to pair production. With organic
crystals the output per roentgen at low energies is determined by the
photoelectric absorption of carbon which is less than with air. The
output per roentgen of a scintillator wil1 be constant only if it has
the same effective atomic number as air, that is, the scintillator must
be air equivalent. The most efficient scintillator with the closest
effective atomic number to air is anthracene with an effective atomic
number of 5.8. The effective atomic number for air is 7.64. Plastic
and liquid scintillators have atomic numbers similar to air and thus
should be suited for biological dosimetry. The inorganic scintillators
such as sodium iodide may be used to determine depth dose for highener
gy radiation but the effects of lowenergy scatter must be considered.
Simple pulsecounting methods cannot provide a useful measure of x
or gamma ray exposure unless pulseheight discrimination is used.
Scintillation detectors may be used for direct measurement of depth
dose in body cavities but calibration is required with radiation of
the same energy.
Electrical
Solid State
Electrical solid state detectors or semiconductor detectors are
solid state analogs of the ionization chamber. The ionizing particle
or x or gamma ray reacts with the sensitive volume of the detector to
produce ionization. The ionization produces a conductivity through
the solid which is a function of absorbed dose rate in the material.
In some materials'the induced conductivity change is permanent due to
radiation damage and is thus a measure of total absorbed dose.
A semiconductor consists of a pn junction where the pregion is
deficient in electrons and the nregion has an excess of electrons.
When an electric field is connected across the semiconductor the
region in the vicinity of the junction becomes depleted of ions. The
presence of ionizing radiation in this region causes a current to flow
which is a measure of the amount of radiation. The size of the voltage
pulse produced is proportional to the energy lost in the detector by
the incident radiation.
The energy required to produce an ion pair in most gases is 3040
ev whereas it is only 3.5 ev for a silicon semiconductor. Thus many
more ions are produced in a semiconductor detector than in a gas
detector of the same mass for the same amount of energy absorbed. This
means that the pulse from a semiconductor detector is about ten times
larger than that from a gas detector. Semiconductor detectors have
a linear response, excellent resolution, a fast rise time and are
small in size. They are practically 100 percent efficient for partic
ulate radiation but are much less sensitive to x and gamma radiation
due to the small depletion region. Thicker depletion layers may be
obtained by applying a higher bias voltage or by the use of silicon
with higher specific resistance. Increased bias voltage increases
noise and thus methods of increasing the resistivity of silicon have
been developed. In the "lithium drifted" silicon detector lithium is
diffused into the silicon; a process which increases the resistivity
about 1,000 times. Thick junctions may be obtained with this technique
which improves sensitivity to x and gamma radiation.
Semiconductor detectors have been used in clinical dosimetry for
depth dose measurements.35 Silicon detectors are particularly useful
for measuring depth dose distribution in bone since the atomic number
of silicon (14) is near that of bone (13.8).
Gas Detectors
Roentgen reported in his second report36 that positive and negative
charged bodies in air were discharged when placed in an xray beam and
that the rate of discharge was proportional to the intensity of the
beam. This remarkable observation has provided the basis for gasfilled
radiation detectors used today.
All gasfilled detectors operate on the principle of collection of
the ions produced in the gas by particulate, x or gamma radiation. The
detectors consist of a gasfilled tube with a center electrode. The
chamber wall forms a second electrode. With the proper potential
difference electrons are collected on the center electrode. The
operation of the gas detector depends on the voltage difference. As
the voltage difference increases, the current collected increases
until it approaches asymptotically the saturation current for the
radiation intensity being measured. All ions produced in the chamber
are collected at the saturation voltage and thus the current is pro
portional to the total ionization produced, which is proportional to
the amount of radiation striking the chamber. If the voltage is
increased much beyond the saturation voltage, secondary ionization,
caused by collision of the original ions formed by the radiation with
other gas molecules, begins. As soon as secondary ionization occurs
a rapid multiplication of ions in the chaber takes place and the total
current depends very strongly on the applied voltage. A detector
operating in the region of the saturation voltage in which secondary
ionization is not taking place is called an ionization chamber, or
simply an ion chamber. Such a chamber is said to be working in the
ionization chamber region, indicating operation at saturation voltage
with no secondary electron emission. The current produced in an ion
chamber is very small and requires amplification before measurement.
The Geiger counter, a gasfilled counter operating in the ion
multiplication region, is not a precise instrument for measurement
of absorbed dose because its response is not directly proportional
to the energy absorbed in its sensitive volume. It has a detection
efficiency of nearly 100 percent for alpha and beta particles and is
sensitive to x and gamma radiation, but to a much smaller degree.
Geiger counters have been used for the measurement of lowdose rate
radiation fields. In this case the recorded counts are related
indirectly to dose by considering the efficiency with which secondary
electrons are ejected from the cathode walls as a function of x or
gamma ray flux. These results are then related to the dose rate
measured with an airwall ion chamber. Counter response in terms of
dose rate is a function of the energy of the x or gamma radiation.
Geiger counters are not suitable for measurement of lowenergy radia
tion due to the photoelectric interaction in the chamber wall. Neither
are they suitable for depth dose measurement in a scattering medium
because of the mixture of energies present at different depths.
Sinclair37 has discussed the use of Geiger counters in the measurement
of radiation fields and the problems involved.
Proportional counters have not been used for measurement of
radiation fields due to the complex instrumentation required to ensure
higherstability highvoltage power supplies and amplifiers of great
sensitivity.
Because of the many difficulties in making both direct and indirect
measurement of radiation as indicated in previous sections, the prin
cipal method of dosimetry has been based on the ionization of gases,
particularly ionization of air. Although air ionization is only an
indirect measure of energy absorption in other media it has proven to
be of the greatest value in medical radiology. The effective atomic
number of both soft tissue and water is 7.42 which is quite close to
that of air (7.64). Thus the absorption of ionizing radiation per gram
of air is almost the same as the absorption in tissue or water, and
ionization in air offers an accurate and reproducible method of depth
dose dosimetry in water phantoms. The unit of radiation exposure, the
roentgen, was defined in 1928 at the Second International Congress of
Radiology in terms of ionization in air. The definition of the unit
of exposure was modified slightly in 193738 but is still essentially
as originally defined. The acceptance of air ionization as the means
of defining and measuring x or gamma ray exposure was due largely to
the simplicity and reproducibility of the method. Another important
consideration was the fact that the measured energy absorption, al
though not in tissue, is in an approximate tissueequivalent material.
Air is a particularly suitable gas to use in ionization chambers
because of its effective atomic number, its availability and its
constancy of composition.
Air ionization chambers operate in the saturation voltage region,
where the output signal is proportional to the energy deposited in the
chamber by the impinging radiation. Air ionization chambers are used
routinely in equipment calibration and in depth dose measurements with
water phantoms.
Summary
Many methods of radiation detection are available and all have
been used at one time or another for determination of depth dose. The
principal method of depth dose determination has been based on ion
ization of air. This method is simple to implement and is easily
reproducible from one therapy center to another. The method is partic
ularly suited to measurement of data to be used in treatment planning
for xray therapy patients since the output of an ionization chamber
is proportional to the energy absorbed in the air medium. Air has
essentially the same effective atomic number as soft tissue and water
and thus absorbs radiation as these two media. It is because of
these many advantages that air ionization is the method applied in
depth dose measurement in this research.
Further details of the air ionization method and its limitations
as applied in depth dose scanning are presented in the following
chapter.
CHAPTER 3
THE IONIZATION METHOD
The physical phenomenon of air ionization has formed the basis
of the most widely used and reliable method of measurement of x and
gamma radiation. The ionization of air by xrays was observed by
Roentgen immediately after the discovery of the rays and was discussed
in his early publications. The Curies also observed air ionization by
gamma rays from radium and noted that the time required to discharge a
goldleaf electroscope was proportional to the intensity of the radia
tion incident upon it. The electroscope may be considered as the first
ionization chamber. Many refinements have been made to ionization
chambers over the years, but the principle of operation remains the
same.
Ionization Chambers and Radiation Measurement
By 1925 ionization measurements had become standardized to the
extent that agreement had been reached as to what quantity should be
measured. Serious problems of wavelength dependence due to wall effects
were, however, still present. The early ionization chambers designed
for dose measurements in radiation therapy were made of various materi
als, usually metal, and were quickly discovered to be unsatisfactory
24
because of photoelectrons emitted from the walls. The wall effects
were a particular problem because ionization chambers fabricated at
different therapy centers did not produce the same results due to
different materials and wall thicknesses. Most radiation physicists
were already at this time using the unit which was later defined as
the roentgen as a means of comparing results. This unit called for
measurement of "the associated corpuscular emission per cubic centi
meter of air at standard temperature and pressure," which meant that
the measurement must depend only on air ionization with no ionization
resulting from wall interactions included.
Duane39 realized that in order to measure the ionization in 1
cubic centimeter of air correctly it should be surrounded by a large
volume of air. Duane constructed a large freeair chamber but Glasser"0
pointed out that the chamber was still too small for correct measure
ment. Failla~1 and several European physicists independently developed
several different types of freeair chambers which were large enough.
Taylor,42 after studying these chambers, developed the American stand
ard freeair chamber. The development of the standard freeair chamber
provided a means of standardized air ionization measurement and set the
stage for the definition of the unit of radiation exposure in 1928.
In 1932 Taylor43 reported agreement between the standard freeair
chambers of several national laboratories to within 1 percent. This
meant a standard of radiation exposure was defined on an international
basis and it then became possible to exchange, internationally, results
of radiation therapy. A vast body of knowledge of doseeffect re
lationships has been built up over the years since the establishment
of the standard ionization measurement.
LZ
Standard freeair ionization chambers are large, bulky and very
sensitive instruments and are not suited for routine radiation measure
ment. The purpose of a freeair ionization chamber is to provide a
primary standard which can be used to calibrate other instruments which
may be used routinely. A dosimeter used in dose determination in
radiation therapy must respond like tissue to radiation exosure and
must be small enough to *determine the dose in a small volume. First
attempts to produce small ionization chambers, called "thimble chambers"
because of their size and shape, were not very successful. The
chambers were highly wavelength dependent and thus did not respond as
tissue to the radiation. Fricke and GlasserL concluded that a cham
ber having a wall with effective atomic number of 7.69 would have the
same effective atomic number as air and would thus produce the same
ionization per cubic centimeter as a standard freeair chamber. This
conclusion was however not borne out in practice. The first successful
thimble chamber was developed by Victoreen in 1927 and is discussed in
his review5 of the development of thimble chambers. His chamber was
fabricated from carbon with an aluminum electrode and was designed to
give the same output per cubic centimeter as a freeair chamber. Each
chamber was balanced to the proper response by calibration with a
freeair chamber. The Victoreen chamber was attached to a condenser
which was charged prior to the radiation measurement. The loss of
charge during irradiation was proportional to the dose. This type of
cavity or thimble chamber has been made available commercially* and
*This instrument is available from Victoreen Instrument Company,
Cleveland, Ohio.
has been the primary instrument for dose determination in radiation
therapy for many years. The instrument is commonly called today the
"condenser rmeter" or simply the "rmeter."
The design requirements of ionization chambers used in depth
dose scanning are not quite as rigid as those discussed above. Thimble
chambers are used to measure absolute values of radiation exposure and
thus must provide the same result as would a standard freeair chamber.
The definition of depth dose given earlier involves the ratio of the
measured result at the depth in question to the measured result at the
depth of maximum buildup. Since a ratio is involved it is not essential
that the ionization chamber used in depth dose scanning produce the
same result as would a standard freeair ionization chamber. In depth
dose scanning both primary and scattered radiation are measured and
so the ionization chamber walls should be made of a material which will
not produce significant wall effects.
The Victorecn rmeter used the voltage drop on a condenser as a
measure of radiation exposure. Later, ionization chambers were devel
oped which operated at constant potential supplied by a power supply
and the ionization current was measured directly by an electrometer
during the time of irradiation. The constant potential ionization
chamber is the type used in depth dose scanning.
Measurement of Small Currents
Ions produced by radiation interactions are collected in ionization
chambers under the influence of a polarizing voltage. The collection
of charge over a period of time constitutes a current flow. The
currents of interest in radiation measurement are very small, ranging
from 10~ to 1016 amperes with l010 to 10!2 amperes typical of
those encountered in depth dose scanning. These small currents place
severe restrictions upon circuit design and make electrometers manda
tory. An electrometer is an instrument with very high input impedance
and is used to measure small currents.
One of the earliest electrometers to be used in radiation measure
ment was the quartz fiber electrometer, whose operation depends upon
the attractive or repulsive force existing between two conductors, one
of which is charged by the signal to be measured and the other by a
polarizing voltage. The original Victoreen rmeter used this type of
electrometer. These electrometers are sensitive to shock and vibration
and have been largely replaced by vacuum tube electroneters which are
more rugged and less sensitive to damage. Vacuum tube electrometers
are however prone to drift. In an effort to overcome the drift prob
lem, the vibratingcapacitor electrometer was developed. This electrom
eter is extremely stable and highly sensitive so that currents down to
1015 to 1016 amperes may be measured. These excellent instruments
are however rather bulky and expensive.
Solid state electrometer devices were developed slowly because
solid state components with high input resistances were not available.
The fieldeffect transistor has now solved this problem. Mauderli and
Bruno45 developed the first solid state electrometer amplifier using
fieldeffect transistors. This amplifier has good sensitivity and is
extremely stable. It is small and may be placed in small probes in
close proximity to the ionization chamber.
The small currents produced in ionization chambers present some
measurement problems not encountered with larger currents. The sensi
tivity of an electrometer makes it particularly susceptible to insu
lation difficulties not present in less sensitive current measurement
devices. The input to the electrometer must be highly insulated from
its support. The insulation must be stable with time and should be
impervious to water. The electrometer is placed as near the ionization
chamber as possible and thus must be ionizationproof. No air spaces
should surround the electrometer or a second source of ionization and
collection may occur. The electrometer must be surrounded by some
highly insulating material to exclude all air spaces. Ceresin wax is
often used due to.its high insulating properties and low melting point.
Measurement Methods
Electrical current measurements are performed by one of two
methods, the IRdrop method or the lossofcharge method, also called
the rateofdrift method. Figure 1 illustrates a circuit typical of
small current measurement. The ionization chamber operates at constant
potential supplied by a power supply, a battery in this case. The
collection of ions at the center electrode causes a current flow which
is measured by the electrometer. The resistance R and the capacitance
C include the total resistance and capacitance present in the circuit.
If all circuit capacitance is ignored, that is, C = 0, the essential
characteristics of the IRdrop method are easily seen. The resistor R,
in this case, is the load resistor and is an added resistance used only
E
4)
Cl)
cc
0
rJ
U
r4
CI
*U
in the IRdrop circuit. When a polarizing voltage sufficient to cause
saturation in the chamber is applied, the magnitude of the ionization
current is proportional to the rate of ion production which in turn
is proportional to the radiation intensity. If I is the constant
ionization current, then the voltage drop, as a function of time,
apparent at the electrometer is, by Ohm's law,
V(t) = I R (1)
and the origin of the name of the method becomes quite clear.
The lossofcharge method of current measurement may be demon
strated from Figure 1 if the total circuit resistance is made infinite.
The definition of capacity C, is
C = (2)
V
where C is the capacity in farads, q is the charge, in coulombs, stored
in the capacitor and V is the potential, in volts, at the terminals of
the capacitor. The current I is defined as
I = dq (3)
dt
that is, the rate of change of charge with time. Differentiating
equation (2) with respect to time t, and substituting in equation (3)
yields
dV(t) I (4)
dt C
Integrating equation (4) the voltage at time t is
i It
V(t) = dt = C (5)
assuming the capacitor has no charge at time zero and the ionization
current I is constant. The term lossofcharge is used because the
current flow is in the direction to neutralize the charge placed on
the chamber by the polarizing voltage.
The actual current measurement circuit is neither of the two
idealized circuits just discussed but is a combination of the two
as shown in Figure 1. In the real case C represents the total
capacitance of the circuit including that of the ionization chamber
and R is the total resistance including any load resistance. Applying
Kirchhoff's law to the circuit in Figure 1
C dV(t) + V(t) I=0 (6)
dt R
and a differential equation relating voltage and current results.
For constant current I the solution is easily shown to be
V(t) = I R [1 exp (t/RC)] (7)
if the capacitor has no initial charge. If the current is not constant
but a function of time the solution of equation (6) is
ft
V(t) = [exp (t/RC)/C) exp (x/RC) J(x) dx (8)
J 0
if the capacitor has no initial charge. Equations (7) and (8) express
the voltage change at the electrometer.
Both the IRdrop method and lossofcharge method are used
routinely in radiation measurement, depending on the particular
application. With the idealized IRdrop circuit, it is clear from
equation (1) that the voltage output of the electrometer is pro
portional to the ionization current at any instant of time. The
voltage produced at a particular time is in no way affected by any
changes in the ionization current, which occur before that point in
time. If the method is used to sample the ionization current at
prescribed points in time, the measured values when plotted may not
come even close to approximating the true current function.
The lossofcharge circuit in its ideal realization will result
in a measurement value which is the area under a curve describing
the input signal. The measurement is made over some time interval as
opposed to an instantaneous measurement with the IRdrop method. In
this case, the signal change at every instant of time, during the
measuring interval, is considered in the total measurement and hence
this method is inherently the most sensitive and contains the most
information about the signal. It will later be shown that noise is
negligible compared to the signal with the lossofcharge method.
The method cannot give exact information concerning the signal at any
instant of time but the measurement value is proportional to the
average of the input signal over the measurement interval. The IRdrop
method and the lossofcharge method are often referred to as the rate
method and integral method, respectively.
The two measurement methods have been discussed up to this point
in terms of the idealized circuits. However, no matter which of the
two methods is chosen for a radiation measurement, the actual electrom
eter circuit is that shown in Figure 1 where both a capacitance and
resistance are present.
The capacity of the ionization chamber and other components is
always present and the circuit can never have infinite resistance.
When a rate measurement (IRdrop) is desired a load resistor is added
and the capacity is that inherent to the circuit. When an integral
measurement (lossofcharge) is desired additional capacity is added
and the load resistance is removed, although some circuit resistance
is still present. In either case both the R and the C components of
Figure 1 are present and thus the circuit has an associated RC time
constant. Equation (7) clearly shows the effect of this time constant
for constant current. The output from the actual current measuring
circuit reaches the value of the idealized IRdrop circuit only after
a certain time period depending on the value of the time constant. The
output voltage reaches 99 percent of its final value after a time of
4.6 time constants. The measured value differs from the final value by
less than 105 only after a time of more than 12 time constants has
elapsed. If the input current varies with time, equation (8) expresses
the measurement result for the circuit in Figure 1.
It would be desirable to have C in Figure 1 equal to zero when
making rate measurements, in which case equation (8) reduces to
equation (1). Likewise it would be desirable to have R in Figure 1
equal to infinity when making integral measurements, in which case
equation (8) reduces to equation (5). These ideal conditions are
impossible to achieve. When a rate measurement is desired the value
of the time constant is made as small as possible but is limited by
the size of the load resistor which must be used and the circuit stray
capacitance. When an integral measurement is desired the value of
the time constant is made as large as possible but this is again
limited by the size of the capacitor which must be used and the leakage
resistance.
The rate method has always been used in depth dose scanning
because the value of the ionization current must be approximated at
many points in time. The integral method has never been used in depth
dose scanning due to the nature of the measurement. The presence of
a finite nonzero time constant makes it impossible to make measurements
in a radiation field instantaneously. If the radiation field is constant
during the time of measurement the proper measured value may easily be
obtained by waiting an appropriate length of time until the instrument
has reached its final value. However, if the radiation field is
changing, as is the case when depth dose scanning is performed, the use
of the rate method may lead to significant errors particularly in the
region where the radiation field is changing most rapidly.
Depth Dose Scanning
In the 1920s and early 1930s xray equipment had reached a stage
of development such that the equipment was reliable and capable of
delivering doses which were reproducible. The value of beam filtration
had been recognized and xray equipment was able to operate at higher
voltages. With beam filtration and higherenergy operations, skin and
air doses did not correlate with observed tumor reactions and thus the
need for dose measurements beneath the skin became apparent.
The first depth dose measurements were made using Victoreen
condenser ionization chambers embedded within various tissuesimulating
phantom materials. This method was very cumbersome and extremely time
consuming. With solid phantom materials it was difficult to position
the chamber precisely at the desired depth making it necessary to use
water phantoms. The use of water presented still another problem, that
of waterproofing the condenser ionization chamber. Depth dose measure
ments performed in this manner were limited to the central beam axis
because of the great amount of time required in charging the chamber,
waterproofing it, placing it in the phantom and reading the chamber.
Centralaxis depth dose measurements did not adequately describe radia
tion fields used in therapy, particularly for isotope machines with
large penumbra. The lack of sharp beam edges due to large source size
in isotope machines made it imperative that entire radiation distribu
tions be determined in the form of isodose curves.
Kemp46 developed an automatic method of scanning a radiation field
with a directreading ionization chamber. The scanner plotted isodose
curves during the measurement through a selsyn motor arrangement and
a balancing circuit. His device though automatic was very slow, moving
at a speed of only 3 centimeters per minute. An entire set of 10
isodose curves for 1 radiation field required about 2 hours. Several
other researchers4750 later used depth dose scanners which were
essentially the same as that developed by Kemp. In 1954 Mauchel and
Johns51 developed an automatic scanner capable of measuring a set of
isodose curves in 10 minutes. Berman et al.52 reported a scanner using
a ratio circuit to eliminate variations in response due to variations
of output from an xray source.
In 1966 Fitzgerald et al.53 reported the first digital scanner
capable of recording data at discrete points during scanning. This
scanner, rather than plot isodose curves, measured data at 5 millimeter
intervals so that a dose matrix was formed representing the dose dis
tribution. The dose values were recorded automatically on punched
cards, a form suitable for use in radiation treatment planning computer
programs. This scanner was the first which recorded data suitable for
immediate computer processing. If required, standard isodose curves
could be produced from the measured data by a digital plotter attached
to a computer. A complete radiation field distribution could be re
corded in digital matrix form in about 15 minutes. This scanner used
a solid state electrometer amplifier45 and a small constantpotential
ionization chamber. In 1969 King54 reported a similar scanning device
which produced isodose curves during the scan. Glenn et al.55 developed
a computercontrolled scanner which plotted isodose curves during meas
urement and collected data in digital form for treatment planning.
A number of depth dose scanners* are now available commercially.
There are many different features available on these scanners but they
all have one thing in common, the measurement method. All use the rate
method of measurement, a distinct disadvantage.
Summary
The ionization method of radiation measurement had become the
method of choice by 1925. The development of the standard freeair
ionization chamber paved the way for the definition of the roentgen
as the unit of radiation exposure and established a means of standard
measurement so that transfer of radiation therapy results was possible
on an international basis.
The currents produced in an ionization chaber on exposure to
radiation are small and an electrometer is required to measure them.
Current measurements are nade by either the rate (IRdrop) method or
the integral (lossofcharge) method. The rate method has always been
used in depth dose scanning and the RC time constant associated with
the measuring circuit has posed a problem in this application.
The limitations of the rate method and the development of a new
measuring technique based on an integral measurement method is discussed
in the following chapter.
*Depth dose scanners are available from:
Victoreen Instrument Company, Cleveland, Ohio
Artronix Medical Systems, St. Louis, Missouri
SHM Nuclear Corporation, Sunnyvale, California
Scanditronix, Uppsala, Sweden
CHAPTER 4
THE RATE MEASUREMENT AND THE REPETITIVE
INTEGRAL MEASUREMENT METHODS
The Lateral Dose Function
A radiation detector moving across a typical symmetric teletherapy
radiation field should produce a response, which when plotted against
detector position, results in a curve of the type shown in Figure 2.
This curve is representative of all symmetric radiation fields and is
the idealized response of the detector, that is, no time constant error
is present.
Daniel and Wood,56 in a review of functions producing certain
basic shapes, include the function
1
h(x) =
A + BCx
For A and B positive, this function takes values between zero and a
maximum of 1/A. The range of x required to produce values from zero
to the maximum depends on the values of B and C. This function may be
used to approximate as closely as desired the radiation detector re
sponse as the detector moves into a radiation field. By properly
0 0 0 0 O
cO (D !
2S00 Hi d30 %
0C
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J
LU
0
0
choosing the values of A, B and C, the function h(x) can closely
approximate that portion of the curve shown in Figure 2 corresponding
to abscissa values ranging from 0 to 10 centimeters. The detector
response may be approximated only as it moves from outside the radi
ation field into the field since once h(x) reaches the maximum value
of 1/A it remains at this value for all larger values of x. The
other onehalf of the response curve, representing the change as the
detector moves from inside to outside the field, may be approximated
by subtracting a function of the same type as h(x) from h(x).
A new function was defined by combining two functions of the type
given by h(x). The new function, f(x), may be used to approximate the
entire detector response as it moves across the radiation field. This
function is given by
1 1
f(x) = for x > 0 (9)
A + BC(D) A C(xE)
The function f(x) has been defined as the lateral dose function and
was developed for use in mathematical analyses of detector response.
The curve in Figure 2 may be reproduced with this function with
A = 0.01, B = 2.0, C = 14.0, D = 1.0, and E = 6.0.
Depth Dose Scanning and the Rate Measurement
Depth dose scanners have greatly improved over the years. The
mechanical scanning apparatus has been improved in order to increase
the scanning speed, automate the scanning, and ensure the reproduc
ibility of positioning of the ionization chamber. It is generally
agreed that the present limitation on scanning speed is the amplifier
time constant. This limitation was recognized by the developers of
early scanners and was responsible for the long measuring times.
In the author's53 experience of depth dose scanning, the amplifier
time constant is normally a problem in that area of the radiation field
where a large intensity gradient exists. Figure 2 shows a plot repre
sentative of a single lateral scan across a radiation field and corre
sponds to a curve which would be determined by an idealized rate
measurement. It is clear that a rapid change of signal occurs as the
detector moves into the radiation field. The change from almost zero
signal to the maximum signal occurs within a few millimeters. A
reverse change occurs at the other field edge as the ionization chamber
moves out of the beam. It is in these two regions that the amplifier
time constant presents a problem.
The curve in Figure 2 is expressed in normalized units as a func
tion of detector (ionization chamber) position. The curve represents
the voltage output of an electrometer capable of making idealized rate
measurements or the ionization current at the electrometer input. When
the curve represents the ionization current, the response of a nonideal
ized ratemeasurement circuit to this input current, given by equation
(9), may be mathematically computed by evaluation of equation (8).
The dotted and dashed curves in Figure 3 are the computed responses
of a ratemeasuring circuit, with a finite time constant, to the input
current given by equation (9) and correspond to scans across a radiation
field in opposite directions at the same depth. Equation (9) is eval
uated with A = 0.01, B = 2.0, C = 14.0, D = 1.0, and E = 6.0. The
43
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dotted and dashed curves are obtained by numeric integration* of
equation (8) with the time constant equal to 300 milliseconds and a
scanning speed of 2.54 centimeters per second. Because of the nonzero
time constant, the curves show displacement toward the direction of
the scan in areas of rapid signal change. The curves correspond with
each other and with the solid curve, which represents the ideal
response, in the plateau areas since the input does not change and
the output reaches its final value in these regions. If no time
constant error were present the scans made in opposite directions
would be identical and would lie on the solid curve in Figure 3.
The deviations of the dotted and dashed curves from each other
and from the correct response, given by the solid curve, are greater
in Figure 3 than would be encountered in depth dose scanning due to
the long time constant and fast scanning speed chosen to more clearly
demonstrate the problem of the time constant. Figure 4 shows two
scans, in opposite directions, of a 10 x 10 centimeter cobalt treatment
field performed with the digital depth dose scanner53 previously
mentioned. The time constant in this case was about 100 milliseconds
and the scanning speed was just over 1 centimeter per second. The
shifting effect, indicative of time constant error, is still clearly
evident which shows that the rate measurement method, as applied in
depth dose scanning, has severe limitations.
*The method of numeric integration was threepoint quadrature,
generally known as Simpson's Rule and described in all texts of
integral calculus.
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Limitations of the Rate Measurerent Method
The accuracy of the rate measurement method, as applied in depth
dose scanning, is limited by the effect of the nonzero amplifier
time constant. The synergistic effect of several factors produces
this problem. The most important of these factors are the scanning
speed, the magnitude of the measured currents and the volume of the
ionization chamber used.
The Scanning Speed
The inaccuracy introduced by the nonzero time constant may be
completely eliminated by sufficiently reducing the scanning speed.
If scanning is so slow that the amplifier output reaches its final
value in a small enough time interval such that the ionization chamber
movement is insignificant, the nonzero time constant ceases to effect
the measuring accuracy. Slow scanning, however, defeats the very
purpose for which depth dose scanners were developed. If adequate
computer treatment planning is to be done, radiation depth dose data
of several hundred different treatment fields are required. A slow
scanner would be very impractical in view of the manhours required
for depth dose measurements. If a scanning ionization chamber is to
move no more than 0.5 millimeter during the period of amplifier output
buildup to 99 percent of its final value, the maximum scanner speed
is approximately 0.11 centimeter per second when the amplifier time
constant is 100 milliseconds. In this case the complete scan of one
radiation field would require almost 3 hours. Such a long scanning
time is impractical particularly with linear accelerators since the
generating tubes have limited operational life.
The Current Magnitude
The magnitude.of the current produced by the radiationinduced
ionization in an ionization chamber is directly related to the
intensity of the radiation and the volume of the detecting chamber.
A cobalt teletherapy unit typically has a radiation output of about
80 roentgens per minute at a treatment distance of 100 centimeters.
This radiation output would produce a charge of 80 electrostatic
units per minute per cubic centireter in an ionization chamber. The
collected ions result in a current I expressed by
I 80 esu/min/cc = 4.44 x 1010 C/sec/cc
[3 x ]09 esu/C] [b0 sec/min]
= 4.44 x 1010 A/cc
The ionization current may be made as large as desired, depending on
the chamber volume.
The Chamber Volume
The volume of an ionization chamber used in depth dose scanning
must be kept as small as possible to ensure good resolution. The
chamber response is a measure of the average ionization occurring in
the chamber in that portion of the radiation field covering the
sensitive volume and with a smaller chamber volume a smaller radiation
gradient is present across the chamber. Cylindrical ionization
chambers are most often used in depth dose scanning. The cylinder
axis is perpendicular to the direction of scanning and the radiation
beam axis so as to minimize the radiation gradient across the chamber.
The scanning chamber used in this study has an inside diameter of
3.8 millimeters and is 26.5 millimeters in length. The sensitive
volume is 0.21 cubic centimeter. The current delivered to the
electrometer from the ionization chamber is 9 x 1011 amperes with a
radiation output of 80 roentgens per minute.
The Time Constant
The amplifier time constant T is defined as
T = RC
where C is the total circuit capacitance and R is the feedback
resistance in the circuit. The value of C may be 1 picofarad in a
typical ratemeasuring circuit. With a rate circuit, the voltage
developed at the electrometer amplifier is the product of the input
current in amperes and the feedback resistance in ohms. This means,
that with the ionization current calculated above, a resistor of about
1011 ohms is required to produce a 10 volt output at the amplifier
output. The required selection of R completely determines the value
of T since C is fixed. The value of the time constant under these
conditions is
r = RC = [1 x 1011 ] [1 x 1012 f] = 0.1 sec = 100 msec
This value of T is fixed by the desired output voltage and may not be
reduced.
Correction Technique
Since the time constant limitation of the rate measurement method
cannot be eliminated in a satisfactory way, a means of correcting the
measured scan data was developed. Equation (6) is a differential
equation relating current and voltage in an electrometer amplifier and
has the form
C dV(t) V(t I(t) = 0
dt R
This may be rewritten as
I(t) R = V(t) + RC dV(t) = V(t) + T dV(t)
dt dt
This equation relates the final output voltage (I(t) R) to the
electrometer output voltage and its derivative. The derivative of
the output voltage may be approximated from output voltage values.
If Vi and Vi + 1 are two values of the electrometer output voltage
at two sampling points separated by a distance AX, the derivative of
the output voltage midway between the two measuring points may be
approximated by
Vi + 1 Vi
AX
v
where v is the scanner speed.
The value of the output voltage at the point midway between the two
sampled points may be approximated by
Vi + 1 + V.
2
The approximation to the correct output voltage is given by
I
Vi = I(t) R Vi + 1 + Vi + T Vi + 1 Vi (10)
2 AX
v
where T is the time constant of the amplifier.
This correction technique was applied to each sampled point in
each depth dose scan performed with the previously described digital
scanner. The correction greatly reduced the effect of the time constant
and produced scans which were essentially symmetric in both directions.
This correction technique was particularly attractive since the
measured data could be corrected after measurement with no increase
in scanning time required. Over 700 different radiation fields were
scanned and corrected as above. Many of the isodose curves plotted
from these data are included in the International Atomic Energy Agency
Dosimetry Catalog57 and have had worldwide distribution.
The correction technique provides only an approximation to the
correct chamber response since the differential equation is evaluated
using discrete data points. This correction technique then becomes
less satisfactory as the scanning speed increases. The fact that the
rate measurement method is unsuitable for the rapid sampling required
in depth dose scanning and the need for faster depth dose scanners
has promoted a search for a new measurement technique.
The Integral Method
An integral method of measurement is used in radiation therapy
when the total dose administered over some time period is desired.
The Victoreen rmeter, developed in the 1920s and still the primary
instrument for equipment calibration, is such an integrating device.
The loss of charge on the ionization chamber capacitor is the measure
of total accumulated dose during the time of exposure. The time of
exposure in this case is determined by a separate timing device.
Integrating dosimeters have been used primarily in dose monitor
devices which turn off the xray unit after a preset dose is delivered.
These monitors have used both electromechanical and electronic methods
of integration. Wheatly58 and Farr59 reported the use of a small motor
driven by an amplifier whose output voltage was proportional to the
ionization current. The motor speed varied linearly with applied
voltage and the number of motor revolutionsmade in a period of time
was proportional to the integral of the ionization current. After
a preset number of revolutions the xray unit was turned off. This
electromechanical means of digital integration has not been widely
used because the motor speed was not quite linear with voltage at low
voltage levels and primarily because better techniques became available.
When integration is performed electronically the integrating
capacitor is charged by the ionization current and the voltage present
after some time is proportional to the integral of the current. Inte
gration may also be accomplished by initially charging the capacitor to
some fixed voltage and letting the capacitor be discharged by the
ionization current. The charge lost during the measuring time is pro
portional to the integral of the current. Both methods have been used
in dose monitors to turn off an xray unit when the voltage on the
capacitor reached a preset voltage level. The dose monitor had to
work over a wide dose range from just a few roentgens to several hundred.
It was difficult to cover the entire required dose range with a single
integrating capacitor and thus a method of digital integration was
developed.
A single integrating capacitor may be used in a dose monitor,
operating over a wide dose range, if the capacitor can be recharged
when the voltage on the capacitor drops to a preset voltage or, with
the other method of integrating, if the capacitor can be reset to zero
when it reaches a preset voltage. The number of times the capacitor
goes through its voltage excursion and recharge or discharge cycles is
a digital measure of the integral of the ionization current. Farmer60
developed a measuring circuit of this type which used an electrostatic
relay to actuate a counter and at the same time recharge the integrating
capacitor. This device was limited to measuring dose rates of less than
40 roentgens per minute due to the recharge time of the capacitor.
Watson61 used a neon tube, which flashed after a particular voltage
built up on the capacitor, to discharge the capacitor. This method
was used in a Geigercounter type personnel monitor worn by persons
working in radiation therapy. This device could be adjusted to operate
over a dose range of 0.1 milliroentgen to 10 roentgens per minute.
Watson62 also developed a dose monitor for use in radiation therapy.
In his design, a movingcoil relay operated to reset the capacitor.
The device produced one count for every 10 roentgens and could operate
at up to 30 cycles per minute. Naylor,63 in a similar circuit, used
a cold cathode trigger tube to discharge the capacitor. Kemp et al..64
and Kemplay65 have used solid state electronics to reset the capacitor
with cycle repetition rates of 100 or more per minute. Lorenz and
Mauderli66 reported a similar solid state system for use with cobalt
teletherapy units.
Neither the electromechanical nor the electronic methods of
digital integration discussed above are applicable to depth dose scanning
for several reasons. Electromechanical methods are not sufficiently
accurate because of nonlinear motor response to voltage. The best of
the electronic methods has a repetition rate of just over a hundred
per minute which is too slow for measuring while scanning. The magni
tude of the ionization currents measured in the dose monitor devices
is of the order of 106 amperes, with 107 amperes minimum. These
currents are 1,000 to 10,000 times greater than those encountered in
depth dose scanning. The methods of recharging or discharging the
capacitor involve switching at the input tothe electrometer amplifier.
Switching transients and leakage currents become quite important when
currents below 107 amperes are measured and these problems have
limited the presently used digital integration techniques to measure
ment of large currents. Furthermore, the digital integration tech
niques discussed above are not sufficiently accurate for depth dose
scanning since the measurement may be in error by as much as one com
plete cycle.
The Repetitive Integral Measurement Method
A new method of measurement based on repetitive integratereset
cycles has been developed. Significant modification of the techniques
used previously in dose monitors allows the new method to be applied
to depth dose scanning. The method has been named the "Repetitive
Integral Measurement Method."
The details of the new method may best be described by reconsider
ation of some of the results developed in Chapter 3. The general
solution of the differential equation (6), relating ionization current
to the voltage output of the electrometer amplifier, is given by
equation (8) as
ft
V(t) = [exp(t/RC)/C] j exp(x/RC) I(x) dx
when RC becomes large this becomes
t
V(t) 1 I(x) dx (11)
C
showing that the voltage output of the electrometer is proportional
to the integral of the input current from time zero to time t. The
average current I(0, t) in the interval [0, t] is
t
1(0, t) = I(x) dx
t O
or
tI(O, t) = I(x) dx (12)
Substituting equation (12) into equation (11)
V(t) = tI(0, t)
C
or
I(0, t) = C V(t) (13)
t
and it is seen that V(t), the voltage present at the electrometer
output at time t, is a measure of the average current over the time
interval [0, t].
The integral measurement is a sum of all values of the input
current over the interval of integration and thus the value of the
integral can give no information concerning the variation of the input
current at any particular time. As nothing can be said concerning the
actual value of the input current at a specific time, it is reasonable
to estimate the value of the input current at the center of the interval
of integration by the average current 1(0, t). This may be expressed,
using equation (13), as
C
I(t/2) ~ I(0, t) = 0 V(t) (14)
Equation (14) shows that the voltage V(t) at the electrometer output
at time t is a measure of the input current at the center of the
measuring interval.
An input current may be approximated over a total time t by
estimating the current value at the center of each of m measuring
intervals by the average current in each of the intervals. The input
current function may be approximated by the average current in each of
m equal intervals as indicated by
p (n + 1) At
I[(n + 1/2) At] 1 I(x) dx = V[(n + 1) At]
At f At
nAt
n = 0, 1, 2, .. m 1
where At = t/m is the length of each of the m intervals. In the above
expression, the value of the input current at time (n + 1/2) At is
approximated by the average current in the interval [nAt, (n + 1) At]
which is, as shown by equation (14), proportional to the output voltage
of the electrometer at time (n + 1) At.
The voltage V[(n + 1) At] in the above equation is a measure of
the average current in the interval fnAt, (n + 1) At] only if the
integrating capacitor was in a reset state prior to start of the inte
gration at time nAt. The value of the integral over each of the m
intervals must be independent of the value in any other interval.
The relationship between output voltage and the average current
in an interval may be shown by combining equation (11) and equation
(12).
J (n + 1) At
V[(n + 1) At] = 1 I(x) dx = At I[nAt, (n + 1) At]
C nAt C
nAt
The output voltage at time (n + 1) At is proportional to the average
current in the interval [nAt, (n + 1) At] when the voltage at time
nAt is zero. If the capacitor is not reset at time (n + 1) At and
integration continues until (n + 2) At then
I (n + 2) At
V[(n + 2) At] = I(x) dx = t I[nAt, (n + 2) At]
C C
nAt
and V[(n + 2) At] is proportional to the average current over the
interval [nAt, (n + 2) At]. The necessity of resetting the capacitor
prior to integration in each interval is clear. Without resetting,
the output voltage reflects the average current from the time inte
gration began and not the average current in each separate time inter
val.
In summary, the repetitive integral measurement method approxi
mates an input current through repetitive cycles of integrating and
resetting. The output voltage generated in each integration cycle
is proportional to the average current over the interval and is used
to approximate the value of the input current at the center of the
measuring interval.
Constant and Linear Current
When the input current function is constant (I(x) = IO) the
average current in any interval [nAt, (n + 1) At] is given by
1 j (n + 1) At
I[nAt, (n + 1) At] = A I dx = IO
nAt
and in this case the current value in the middle of the measuring
interval is equal to the average current.
Likewise for a linear function (I(x) = kx, for k constant) the
average current in any interval [nAt, (n + 1) At] is given by
f(n + 1) At
I[nAt, (n + 1) At] kx dx = k(n + 1/2) At
and again the value of the average current over the measuring inter
val is equal to the current at the midpoint of the interval.
These results show that the repetitive integral measurement
method will give exact results with a constant or linear function of
current. This also implies that if it is possible to choose the inte
gration intervals so small that the function is essentially constant
or linear within the interval then the repetitive integral measurement
method will result in very good approximations to the actual values of
the input function at the center of the measuring interval.
Summary
The rate measurement method is limited in depth dose scanning
by the time constant of the amplifier. The time constant is determined
by the magnitude of the current to be measured and is of the order of
100 milliseconds with the currents encountered in depth dose scanning.
With a time constant of this magnitude, measurement values suffer
significant signal lag in regions where there is rapid change in the
input function. Correction techniques must be applied to measured
data, if scanning speeds exceeding a few millimeters per second are
used.
A new measurement technique based on integral measurement has
been developed for use in depth dose scanning. The method is called
the repetitive integral measurement method and approximates an input
current through repetitive cycles of integrating and resetting over
short time intervals. The output voltage at the end of each inte
gration cycle is proportional to the average current over the interval
and is used as an approximation of the input current at the center
of the interval of integration.
A comparison of both measurement methods is made in the following
chapter.
CHAPTER 5
COiMPARISON OF THE RATE AND REPETITIVE INTEGRAL
MEASUREMENT METHODS
The Step Function Response
The rate measurement method has been shown to present a problem
in those parts of the radiation field where rapid changes occur. When
the input signal increases or decreases rapidly the amplifier output
is late in increasing or decreasing due to the nonzero time constant.
The unit step function has the value zero up to a prescribed
point at which the function value becomes unity. It is an example of
a function with the most rapid change possible and is useful in
evaluating the response of an amplifying system to a changing signal.
The unit step function has been used to compare the responses of the
rate measurement and repetitive integral measurement methods.
The step function may be defined as a function of time in the
following way
f! t T
j(t) = t where T is a fixed time
iL t <
The responses of both methods of measurement to the above step function
are determined by evaluating the expressions
t
R(t) = [exp(t/T)/C] exp(x/T) j(x) dx (15)
for the rate measurement, where i = RC, and
(n + 1) At
S[(n + 1/2) At] ={exp[(n + 1) At/T]/C} exp(x/) j(x) dx (16)
nAt
n = 0, 1, 2,
for the repetitive integral method. The time unit in the above
equations is arbitrary. Equation (16) produces a value which is pro
portional to the amplitude of the step. For a meaningful comparison
with .the rate measurement this result must be normalized so that
equation (16) will produce a value equal to the step amplitude for
any measuring interval fully inside the step. This normalization is
performed by dividing the result of equation (16) obtained in each
measuring interval [nAt, (n + 1) At] by the result obtained in any
one interval contained entirely within the step.
Suppose the interval [kAt, (k + 1) At] lies entirely within the
step (kAt T) then the result of equation (16) in this interval is
(k + 1) At
S[(k + 1/2) At] = {exp[(k + 1) At/T]/C} exp(x/T) dx
kAt
= (l1 exp(At/T)]
C
The normalized values are given by
S[(n + 1/2) At]
N[(n + 1/2) At] = + 1/2) At]
Sj(k + 1/2) At]
(n + 1) At
exp[(n + 1) At/T] (n + 1
= T[1 exp(At/T)] exp(x/t) j(x) dx (17)
nAt
n = 0, 1, 2,
where the interval [kAt, (k + 1) At] lies entirely within the step.
Equation (17) gives values which may be compared directly with the
results produced by equation (15). The values resulting from evalu
ation of equation (17) in each of the intervals is interpreted as the
value of the input function j[(n + 1/2) At] in the middle of the
measuring interval. Equation (15) is evaluated at corresponding
points so that comparison is possible. It is to be noted that
equations (16) and (17) contain exponential terms not present in
equation (11), the equation on which the definition of the repetitive
integral measurement method is based. The exponential terms are
included so that the effect of the time constant T on the repetitive
integral method may be observed. When T becomes large the exponential
terms reduce to unity as in other discussions of the measurement method.
Point Detector Response
Figure 5 shows a series of results for each measurement method
for several values of the time constant and several positions of
measurement points relative to the step location. The figure repre
sents the response of a point detector to a timedependent step input
63
POINT DETECTOR
0
7
r*TE A
RATE MEASUREMENT
0 6 0 10 12
T0 01
REPETITIVE INTEGR.. MIASUPEMENT
S T.O0 5
RATE fEAS#AEWENT
0 04 06 0
d
1.0 I 2
T T0 I
REPETITIVE INTEGRAL MEASUPENT
02 04 06 0 10 12
TT5 0
REPETITIVE INTEGRAL MEASUREMENT
Point Detector Response to Step Function Input.
ii
TIME 0
rTI 0
TIM4 0
TIME 0
TIME 0
TIME 0
0.2 0.4 0 6 0. 10 1.2
Figure 5.
when measurements are made at equally spaced time intervals. All
points are determined at time intervals separated by 0.2 units and
straight lines connect each series of points. For example, the
calculated values at time 0, 0.2, 0.4, 0.6, 0.8, 1.0 and 1.2 correspond
to the same series of measurements and are connected by a straight
line. Likewise the calculated values at time 0.04, 0.24, 0.44, 0.64,
0.84 and 1.04 correspond. It is emphasized that each value determined
with the repetitive integral measurement method is the normalized
result of an integration over a time interval. The value is plotted
at the center of the integration interval. In each interval of
measurement At = 0.2. The integral expressions in both equations (15)
and (17) were evaluated by threepoint quadrature. Equation (15) was
evaluated at points corresponding to the center of the interval of
integration for equation (17).
Figures 5a and 5b show the results of the two measurement methods
with a time constant T = 0.01. From equation (8) it is seen that the
basic integral expression evaluated for the two methods is identical
when T is the same. There is a significant difference however in the
way the integral expression is evaluated in each method. When a rate
evaluation is made the integral in equation (8) is determined by
integration from time zero to the time of sampling. When a repetitive
integral evaluation is made the integral is determined by integration
from time nAt to time (n + 1) At and the result is normalized as
discussed above. Figures 5a, 5b, 5c and 5d show that even with the
same value of the repetitive integral method reaches the step value
sooner than the rate method. The repetitive integral method overesti
mates the step in Figure 5b and rises too quickly due to integration
over an interval At. The rate method also overestimates the step
when measurements are made at the same interval At apart and the output
is represented by a straight line between points. This area where
overestimation of the step occurs is directly dependent upon the time
interval between sampling with the rate measurement and the time
interval over which integration is performed in the case of the
repetitive integral method. The repetitive integral method is clearly
superior to the rate method when T = 0.5, although again the step is
overestimated.
Figure 5e shows the response of the repetitive integral method
to the step input when T > 5.0. It was found that Figure 5e did not
change for values of T > 5.0, which means that the exponential term
in equation (16) is essentially unity and thus a simple integration of
the input function is performed. The response when T > 5.0 is then
that which is produced by the repetitive integral method as it was
originally defined, that is, a measure of the average of the input
function over the interval of measurement. The response to the step
input of the repetitive integral method when T < 5.0, as shown in
Figures 5b and 5d, is not a measure of the average of the input function
but an average of the product of the exponential function for the
appropriate T and the input function over the interval of measurement.
Only with T > 5.0 does the repetitive integral method produce an output
which is proportional to the average input current. As the average
input current is the only reasonable estimator of the input within the
interval, the repetitive integral method would in practice always use
a T > 5.0.
The repetitive integral method, as shown in Figure 5e, responds
most quickly to the step when the step occurs at the beginning of the
integrating interval in which case the step value is reached within
one measuring interval. The worst case in response is when the step
occurs in the middle of the integrating interval. In this case, two
intervals are required to reach the step value. Results which have
been obtained with values of T not shown in Figure 5 indicate that
the repetitive integral method with T > 5.0 always reaches the value
of the step before the rate method with T > 0.02. The repetitive
integral method with T > 5.0 reaches the step value at least two
intervals earlier than the rate method with T > 0.05.
All the results shown in Figure 5 and the above discussion of the
figure pertain to measurements made with a perfect point detector,
which is physically impossible to achieve. The dimensions of the
detector become important when evaluating its response to a changing
signal. Ionization chambers are most often cylindrical because of
certain electrical characteristics and because of simplicity of
fabrication. The response of the chamber is proportional to the volume
irradiated or if a twodimensional cross section is considered, the
response is proportional to the area irradiated. It is sufficient to
consider the twodimensional case since the entire length of the
sensitive volume is equally irradiated and the crosssectional area
irradiated is then proportional to the volume irradiated.
Cylindrical Detector Response
Figure 6 shows a crosssectional view of a cylindrical ionization
chamber of radius R. The shaded section represents that portion of the
chamber irradiated. The dividing line between the irradiated and the
67
LC
Ft
0
U
,4
0
u
0
C,
,,4
ox
unirradiated portions forms a chord of distance x from the center of
the cylinder. The response of the chamber is proportional to the area
irradiated and the fraction of full response is the ratio of the shaded
area to the total area. The area of the shaded section is given by the
expression7
x R2 X 2
A() = [x 2 x2 + R Arcsin (x/R)]
for x in the interval [R, R]. The ratio of the shaded area to the
total area is
A(x) 1
F(x) = = 0.5 [(x/R) /R)V ( R)2 + Arcsin (x/R)]
for x in the interval [R, R]. The response of the cylindrical detector
is described by F(x) as the chamber moves across an irradiation step.
If the step occurs at time ts and the chamber moves at a rate of k
distance units per time unit
0 t < ts R/k
g(t) = F(t) ts R/k < t < ts + R/k
1 t > ts + R/k
is the expression of the chamber response to a step input. The function
g(t) is used in equations (15) and (17) to compare the results of the
two measurement methods for the cylindrical detector.
Figure 7 shows the response of the cylindrical detector to a
step input. The chamber diameter in this example was taken to be the
CYLINDRICAL DETECTOR
ATE 0 01
RATE MEASUREMENT
.0O 01
REPETITIVE INTEGRAL MEASUREMENT
04 06 0 10 12
C
rT* 5
RATE MEASUREMENT
0 6 0 10 1 2
TIME 0
I
TIME 0
r
TIME 0
r
TIME 0
T> 0
REPETITIVE INTEGRAL MEASUREMENT
TIME 0 0.2 04 06 0 10 1.2
Figure 7. Cylindrical Detector Response to Step Function Input.
T0 5
REPETITIVE INTEGRAL MEASUREMENT
. . . v 
distance transversed during one measuring interval or, in time units,
it corresponds to the time between individual measurements. When
Figures 5a and 7a are compared, it is seen that the cylindrical
detector is delayed in reaching the step value and in the worst case
requires three intervals to reach the step amplitude whereas the point
detector requires only two intervals. The repetitive integral method
response shown in Figure 7b reaches the step value sooner than the rate
method, but overestimates the step more than the rate method. With
the large value of T = 0.5 in Figures 7c and 7d, the response is quite
similar to that with the point detector, but both methods anticipate
the step before it arrives due to the finite size of the detector. The
response of the repetitive integral method with T > 5.0 is shown in
Figure 7e. In the worst case, three intervals are required to reach
the step amplitude and in the best case, two intervals are needed.
As the cylindrical chamber is made smaller, the response will
become closer to that of the point detector. If a larger chamber is
used, the number of measuring intervals required to reach the step
value increases. The dimensions of the cylindrical ionization chamber
used in depth dose scanning are approximately the same as those used in
the example in Figure 7 and thus its response is well described by
Figure 7.
Figure 7e shows that the repetitive integral method, with T > 5.0,
is superior to the rate method in approximating the step input except
for the rate method with T = 0.01, and it is almost as good in that case
also.
The results shown in Figures 5 and 7 and some similar results,
which are not plotted, with other values of T indicate that the
repetitive integral method with T > 5.0, which is really a measure of
the average input, is far superior to the rate method with T > 0.02 in
responding to a rapidly changing signal.
The Size of the Measuring Interval
The closeness of approximation of the step by both methods depends
on two factors, the time interval between measurements and the diameter
of the detector. The effect of reducing the measuring interval is
easily seen with the point detector. Figure 8 shows the result of
reducing the measuring interval by ten times. Figures 8a and 8b show
the response of both methods of measurement for T = 0.1 for the rate
method and T > 5.0 for the repetitive integral method. Figure 8a
shows the best case when the step occurs at the beginning of the
interval. Figure 8b illustrates the worst case when the step occurs
in the middle of the interval of measurement. Figures 8c and 8d show
the same results for T = 0.01 for the rate method. Comparisons of
Figures 8 and 5 show the improvement in approximating the step. It
is possible to approximate the step as closely as desired with the
repetitive integral method and a point detector by choosing a suf
ficiently small interval of measurement.
In the case of the cylindrical detector, the closeness of approxi
mation of the step is limited by the detector diameter. Reducing the
measurement interval improves the approximation but no matter how small
the interval becomes, the step amplitude is first reached at a point
inside the step by a distance equal to the detector radius.
72
POINT DETECTOR
RATIO
0 .2 0.4 06
O 0.2 0.4 0.6
0.2 04
 RATE
T'0.1 RATE MEASUREMENT
T>5 0 REPETITIVE INTEGRAL
MEASUREMENT
I I
0.8 1.0
E
T*0 I RATE MEASUREMENT
T>5 0 REPETITIVE INTEGRAL
MEASUREMENT
0.6 08 1.0
/\RATE
T'0.01 RATE MEASUREMENT
T>5 0 REPETITIVE INTEGRAL
MEASUREMENT
0.6 0.8
0 02
0 0.2 0.4
\RATE
T'0.01 RATE MEASUREMENT
T>5.0 REPETITIVE INTEGRAL
MEASUREMENT
06 0.8 1.0
Point Detector Response to Step Function Input with
a Tenfold Reduction in the Size of the Measuring
Interval.
TIME
p~ I
TIME
TIME
0
TIME
Figure 8.
I  T i 1 
1~
0
73
w
Lateral Dose Function Response
The lateral dose function given by equation (9) represents the
input current expected at the electroaeter during a scan across a
radiation field. This function provides a means of evaluating both
measurement methods as they would be applied in depth dose scanning.
Equations (15) and (17) were evaluated, by numeric integration, using
the lateral dose function as the input function. The solid curve in
Figure 9 shows a plot of the lateral dose function. The dotted and
dashed curves are the result of evaluating equation (15) and correspond
to scans across the field in opposite directions. The amplifier time
constant was taken as 0.1 seconds, a value typical in depth dose
scanning, and the scanning speed was 2.54 centimeters per second. The
curves for the scans in opposite directions show the shifts from the
true value which are characteristic of the ratemeasurement method.
The repetitive integral measurement method was applied to the
lateral dose function under the same conditions as with the rate method
except that the time constant for the repetitive integral method was
chosen equal to 50 seconds so that the electrometer amplifier output
was proportional to the average current. Figure 10 shows a plot of
the lateral dose function and the results of the evaluation of equation
(17) for scans in opposite directions. The three curves superimposed
show no deviation greater than the line width.
Figure 10 shows that the repetitive integral measurement method
produces identical results for scans in opposite directions, completely
eliminating the shift introduced by the rate method. The results shown
in Figure 10 were determined with a scanning speed of 2.54 centimeters
per second and At = 0.125 seconds.
0 0o
o (0D
3SOO Hld30 %
z l
z
0 3
CO)
a
o W
I L
J
J
c)
24
O
o E
 u
The accuracy of the repetitive integral measurement method in
reproducing the lateral dose function was evaluated by consideration
of two types of errors, the function error and the displacement error.
Figure 11 shows a diagram of a function F, over interval [nAt, (n + 1)
At], the integral approximation of the midpoint and the two types of
errors. The repetitive integral method approximates the function value
at the midpoint of the interval by the integral of the function over
the interval. Only rarely does the function value at the midpoint
equal the integral value and, thus, the integral value lies above or
below the function value. The difference between the function value
and the integral value is denoted by AF and is called the function
error. The distance from the midpoint of the interval to the point
on the abscissa, where the function value and integral value are equal,
is denoted by AX and is called the displacement error.
Both AF and AX were determined for both measurement methods, as
applied to the lateral dose function. These errors were determined
from the same data used in plotting Figures 9 and 10. The time constant
was taken as 0.1 second for the rate method, as this corresponds to
the minimum value used in depth dose scanning. These errors were
determined for each measuring interval, as the lateral dose function
increased from 0 to 100 percent. In the case of the repetitive
integral method, the value of the integral in each interval [nAt, (n + 1)
At] was normalized to a 100 percent maximum by dividing each integral
value by the integral value obtained when the input function remained
at its maximum during the entire measuring interval. This corresponds
to normalization to the value obtained at the central axis.
Table 2 gives the values of AF and AX for both the repetitive
integral method and rate method, as applied to the lateral dose
function. All errors are the difference between the actual value
and the experimental value. With the rate method, the function error
AF exceeds 21 percent in the region where the function is increasing
rapidly. The displacement error AX continually increases, a fact
observed in Figure 9. Both the function errors and the displacement
errors are large with the rate method showing the unsuitability of
the method in this application. Both the function errors and dis
placement errors with the repetitive integral method are quite small
showing that the method very closely approximates the lateral dose
function.
The Effect of Scanning Speed
It has been noted, from experience in depth dose scanning, that
the rate measurement results are directly dependent upon the speed
of scanning. The effect of scanning speed may be observed by consider
ation of equation (8), which may be rewritten as
V(t) = exp[(x t)/T] I(x) dx (18)
0
where T = RC.
The above equation gives the results of measuring the current I(x)
under the influence of a time constant T. If the scanning speed is
increased m times, the current function in equation (18) becomes I(mx)
and a larger range of function values is included in the integral. If
the measurement of the current is performed at time t/m, then
t/m
V(t/m) = C exp[(x t/m)/] I(mx) dx (19)
0
In order to compare the results of equations (18) and (19), it is
convenient to make a simple change of variable in equation (19) by
letting y = mx, then
V(t/m) = exp[(y t)/mT] I(y) dy (20)
f 0
Equation (20) shows that the expression under the integral sign is
identical with that in equation (18), except that the time constant is
effectively increased by m times. When the rate measurement method is
used, the effect of increasing the scanning speed m times is an increase
in the time constant by m times, relative to V(t), as given by equation
(18).
With the repetitive integral measurement method, the value pro
portional to the average current over a single measuring interval is
given by
I (n + 1) At
V[(n + 1) At] = C I(x) dx (21)
nAt
If the scanning speed is increased m times, the current function
becomes I(mx), as above. If the measuring interval At is divided by m
then
(n + 1) At
V[(n + 1) t] = I (mx) dx (22)
m C t
nAt
m
Again changing variables by letting y = mx and substituting into
equation (22)
(n + 1) At
V[(n + 1) At= I I(y) dy (23)
m mC 
m nAt
Comparing equations (23) and (21), it is seen that the only effect of
increased scanning speed on the repetitive integral method is a
reduction of the magnitude of the output voltage by 1/m. Theoretically,
this is of no consequence, since all the results of the repetitive
integral method are normalized and, thus, this factor disappears. It
is of practical importance, however, since the magnitude must be large
enough to be distinguishable from noise. It may be said that the
repetitive integral method is independent of scanning speed as only
the signal magnitude changes.
Equations (18) through (23) apply equally well in describing
the effect of reduced scanning speed on both measurement methods.
Advantages of the Repetitive Integral Measurement Method
The previous results in this chapter indicate certain advantages
of the repetitive integral measurement method, when compared to the
rate measurement method. These advantages, which are summarized below,
are of particular importance in depth dose scanning and provide the
basis for the development of a totally new approach in scanning.
Elimination of the Effect of the Time Constant
When the rate method is applied in depth dose scanning, the lag in
output response, a result of the nonzero time constant, causes the
plot of output voltage versus detector position to be shifted away
from the correct response toward the direction of the scan. The
magnitude of the shift is directly dependent upon the value of the
time constant. Figure 9 shows results typical of those obtained in
depth dose scanning with the smallest possible time constant employed.
The results of scans in opposite directions show the lack of symmetry
about the central axis, which leads to a zigzag effect when the
recorded data are plotted. Figure 10 and Table 2 show that the
repetitive integral method applied to the same input data produces
much smaller displacement and function errors and reproduces the plot
of the input function so well that no deviations are evident. Figures
5 and 7 show the superior response of the repetitive integral method
applied to a step input.
Application to Pulsed Radiation Sources
Pulsed radiation sources such as betatrons and linear accelerators
pose a particular problem in radiation measurement because the radiation
output is not constant, as with cobalt, but appears in short pulses of
microsecond duration and, thus, determination of the dose rate is
uncertain. The most reasonable measure of the dose rate would be the
average dose rate over some interval. The repetitive integral method
is ideally suited for this type measurement, since the integral result
is proportional to the average input current.
Elimination of Amplifier Zero Drift
This advantage of the repetitive integral method does not result
from the theoretical development of the method but becomes evident
when the electrical analog of the repetitive integral procedure is
developed. All DC amplifiers have a certain amount of drift from
zero, which is included as part of the value measured by the rate
method. The repetitive integral method can completely eliminate all
longterm zero drift (drift over a period much longer than the measuring
interval). Amplifier zero drift may be eliminated by making two
measurements during integration, the first soon after the start of
integration and the second at the end of the measuring interval, and
using the difference of the two measurements as the integral value over
the interval. Any zero drift, which has occurred prior to the interval
of measurement, is eliminated. Drift occurring during the interval of
measurement is not eliminated by this method.
Reduction of Signal Noise
It is well known that the RC circuit shown in Figure 1 has a
certain filtering effect on the input signal. When the value of T is
large the circuit functions as an integrator and produces a filtering
effect which severely attenuates highfrequency signals while passing
lowfrequency signals. The integrator forms what is called a lowpass
filter. The filtering effect attenuates highfrequency noise components
superimposed on the input and reduces total signal noise. A further
discussion of noise is included in the next chapter.
Independence of Scanning Speed
The repetitive integral method is theoretically independent of
scanning speed, as only the magnitude of the output is reduced. The
practical realization of the method by electronic means, however, is
limited by the minimum signal which may be distinguished from noise.
The method is, then, practically limited by the magnitude of the output
which is dependent upon speed. The rate method is quite dependent
upon speed, as the time constant is effectively increased. The effective
increase in time constant is proportional to the increase in speed.
Summary
The repetitive integral measurement method has been shown to be
superior to the rate method in responding to a step input and tothe
lateral dose function, which simulates the input current expected
during depth dose scanning.
The repetitive integral method, when applied in depth dose scanning,
has the following advantages:
1. The effect of the time constant is eliminated.
2. It is better suited for measurement of pulsed
radiation sources because the result produced is
proportional to the average current over the
measuring interval.
3. Amplifier longterm zero drift may be eliminated
by taking the difference of two measurements during
integration.
4. The integrating circuit acts as a lowpass filter,
thus attenuating highfrequency noise components.
5. It is theoretically independent of scanning speed
as only the magnitude of the signal is reduced.
The actual implementation of the repetitive integral method by
electronic means is presented in the following chapter.
CHAPTER 6
THE REPETITIVE INTEGRAL ELECTROMETER AMPLIFIER
The magnitudes of the currents produced in small ionization
chambers such as those used in depth dose scanning are of the order
of 1012 to 1010 amperes, and thus sensitive amplifying devices are
required for their measurement. Amplifiers used for current measure
ment in this range are usually called electrometer amplifiers or simply
electrometers. The voltage output of the electronic electrometer
amplifier is usually proportional to the input current.
Frequently, operational amplifiers are used in the construction
of electrometers. An operational amplifier is simply a highgain,
directcoupled amplifier. The operational amplifier often has a
differential input and is used with external feedback networks. The
ideal operational amplifier is represented by the symbol
el
+ K eo
e2
where el and e2 are the input voltages, e0 is the output voltage and K
is the gain of the amplifier. The idealized amplifier properties
usually assumed are
Gain (K) =
e0 = 0, when el = e2
Input impedance =
Output impedance = 0
Bandwidth =
No temperature drift.
These properties can never be realized in practice but the assumption
of idealness is useful in the analysis of feedback circuits.
The Integration Circuit
The electrical analog of the mathematical operation of integration
can be easily obtained with the use of the inverting input of an
operational amplifier with the addition of a feedback capacitor. The
integration circuit is diagrammed below
R ei
+ >i 'o 0
Since the input impedance is assumed infinite, no current flows into
the amplifier. Applying Kirchhoff's law at the junction at the amplifier
input,
el ei d(e0 ei)
+ C =
R dt
e0
The outputvoltage may be expressed as e0 = Kel. Hence, el K
As K in the ideal case is infinite, ei = 0 and
de0 el
dt RC
e0 = el dt + e0(t = 0) (24)
0
where e0(t = 0) is the voltage across the capacitor at the start of
integration. The output voltage is seen to be proportional to the
integral of the input. In this discussion, the noninverting amplifier
input is connected to voltage common so the amplifier acts as a single
input device. This technique of integration has been used for many
years with analog computers for the solution of differential equations.
The repetitive integral measurement method described in Chapters
4 and 5 approximates the input function by a series of integration
performed over short time intervals. Each of the individual inte
grations is independent of any other integration, which means the
feedback capacitor is discharged to zero voltage before integration
begins in each interval. In the case of depth dose scanning, the
ionization current is approximated at n points across the radiation
field by repetitively integrating in n time intervals whose length is
is At with the integrating capacitor being reset to zero before inte
gration begins in each interval. An operational amplifier with feedback
capacity is utilized to perform the integration with the repetitive
integral electrometer.
The Switching Electrometer
The repetitive integral measurement method demands that the
integrating capacitor be reset to zero and held at zero until inte
gration of the input is to begin, at which time the capacitor must
be allowed to charge. Therefore, the electrometer must be switched
at the appropriate time in order that the amplifier output and input
are shorted and the integrating capacitor discharged. This procedure
is simple to describe but poses quite a problem to accomplish with an
electrometer. Switching at the input of such a sensitive electrometer
is quite difficult since transients accompanying both making and 
breaking of the switch may send the operational amplifier into immediate
saturation. Leakage currents through the switch during integration
are also a problem with a sensitive electrometer.
The methods6065 of digital integration discussed in Chapter 4
employed switching at the input of the electrometer, but the currents
involved were 1,000 to 10,000 times larger than those encountered in
depth dose scanning. Switching at the electrometer input is easily
accomplished with these large currents since leakage currents through
the switching apparatus, even though present, cause no problems due to
the low sensitivity of the amplifier.
The Switching Method
The connection of an electrometer output to input for the purpose
of resetting may be accomplished by either mechanical or electronic
means. Mechanical resetting using relays was tried and immediately
abandoned because the switching noise drove the amplifier into saturation.
A method of resetting by electronic switching was successful and has
been used in the construction of the repetitive integral electrometer.
The reset network is illustrated in Figure 12.
The switching of the integrator to the reset mode is accomplished
with a pair of pchannel enhancementtype metal oxide silicon field
effect transistors (MOSFET). These transistors, indicated as A and B
in Figure 12, are normally nonconducting with zero gate voltage and
become conducting when a negative voltage is applied at the gate. The
integrating capacitor is reset during the conducting phase when the
electrometer output is connected to the input through the transistors.
The first requirement of such a switching arrangement is that the
leakage through the switch be so small as to produce a negligible
effect on the integration. It was shown in a previous chapter that
an ionization current of 1010 amperes, which is typical in depth dose
scanning, will produce 10 volts at the output of the electrometer
when a feedback resistor of 1011 ohms is used. A leakage of just 1
picoampere into the input will result in an output of 100 millivolts
which is 1 percent of the value measured.
Two transistors are used together as shown to reduce leakage
problems. The drain of one transistor is connected to the source of
the other and this junction goes to voltage common over a small
resistance keeping the junction point of the two transistors essentially
at zero volts during integration. The source of transistor A in
Figure 12 goes to the input of electrometer F. The input voltage to
F never exceeds a fraction of a millivolt so the voltage across
transistor A is very small during integration. Furthermore during
the integration cycle the gate voltage on both transistors is zero,
