Optimal delivery techniques for intracranial stereotactic radiosurgery using circular and multileaf collimators


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Optimal delivery techniques for intracranial stereotactic radiosurgery using circular and multileaf collimators
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ix, 306 leaves : ill. ; 29 cm.
Wagner, Thomas H., 1966-
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Radiosurgery   ( lcsh )
Nuclear and Radiological Engineering thesis, Ph.D   ( lcsh )
Dissertations, Academic -- Nuclear and Radiological Engineering -- UF   ( lcsh )
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Thesis (Ph.D.)--University of Florida, 2000.
Includes bibliographical references (leaves 297-304).
Statement of Responsibility:
by Thomas H.Wagner.
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University of Florida
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Copyright 2000


Thomas H. Wagner

This work is dedicated to my loving wife and friend, Nanette P. Parratto-Wagner.


I would like to express my sincere appreciation for the guidance provided me by

the members of my supervisory committee. I would especially like to thank my

committee chairman, Dr. Frank J. Bova, from whom I have learned much about medical

physics and whose mentoring has been a key component of the successes I have enjoyed

during my doctoral work. I would also like to thank and acknowledge the contributions

of Dr. Sanford L. Meeks, who has always encouraged me to go beyond simply the

academic requirements and to strive to publish my work, and of Dr. Beverly L. Brechner,

who invested much of her personal time in teaching and coaching me through elementary

topology and set theory, as well as contributing key ideas for our joint sphere packing

project. I am grateful to Dr. Willaim A. Friedman and Dr. John M. Buatti for all of the

valuable clinical feedback, mentoring, and support I have received from them during my

doctoral work and association with the University of Florida stereotactic radiosurgery

program. I also owe special thanks to Dr. Taeil Yi for contributing his ideas and for his

initial computer programming efforts in our joint sphere packing project, and to Dr.

Yunmei Chen, whose contributions played a key role towards the success of our joint

sphere-packing project. Drs. Brechner, Yi, and Chen are all from the University of

Florida Department of Mathematics. I would like to give my most sincere thanks to Dr.

Lionel G. Bouchet for many hours of insightful technical discussions about numerous

aspects of my research, and especially for his computer programming efforts towards

transferring image and anatomical structure data between the several treatment planning

systems in our lab. I am grateful to Russell D. Moore for his invaluable aid in helping me

navigate and use the myriad of Unix computer systems necessary to perform my research

work, and to Lisa Mandell for her assistance in gathering radiosurgery patient data from

the University of Florida SRS-patient database. I would like to thank Dr. Wesley E.

Bolch for his support and encouragement for the entire time I have been a graduate

student at the University of Florida, and to Dr. Kelly D. Foote for many hours of

insightful conversations about radiosurgery and neurological surgery, and for his patient

tutoring and assistance in contouring brain lesions and other intracranial structures.

Finally, I am deeply indebted to my friend and wife, Nanette, without whose support and

loving encouragement I would not have had the strength to begin, let alone complete, the

last several years of my life in graduate school.

ACKNOWLEDGMENTS ......................................................................................... iv

A B STR A CT .................................................................................... ...viii


1 IN TRO D U CTION ......................................................................................................

Megavoltage Photon Radiotherapy And Radiosurgery .................................... ...... 1
Technical Evolution and Improvements Stereotactic Radiotherapy.......................... 5
Linear Accelerator Radiosurgery and Radiotherapy Treatment Techniques.................. 6
Technical Evolution and Improvements Linear Accelerator Radiation Delivery ....... 9
Research Problem: Comparison of SRS Treatment Methods................................... 13

2 EVALUATION OF TREATMENT PLANS ...............................................................17

D ose C alculation...................................................................................................... 18
Isodoses and Dose-volume Histograms ................................................................... 30
Physical Dose-volume Figures of Merit .............................................................. 37
Biological M odels................................................................................................... 49

CO LLIM A TO RS .........................................................................................................57

Circular Collimator SRS Dosimetry.................................................................... 58
Single Isocenter Treatment Planning.................................................................... 60
Multiple Isocenter Radiosurgery Planning Tools .................................... ............ 70
Multiple Isocenter Radiosurgery Planning via Sphere Packing ............................... 80
Converting Sphere-Packing Arrangements to Radiosurgery Plans .......................... 94
Application to Phantom and Clinical Targets......................................... ........... ... 95
Results Phantom Targets ...................................................................................... 97
Results Clinical Targets ........................................................................................ 98
Sphere Packing as a Mulitple-Isocenter Radiosurgery Planning Tool ..................... 103
Sphere Packing Algorithm: Potential Developments ................................................. 105
C conclusion ............................................................................................................ 108

4 SHAPED BEAM SRS ............................................................................................ 109

Introduction ...................................................................... ........................................ 109
Generation of Isotropic Beam Bouquets............................................... .......... .... 111

Rotation of Beam Bouquets ........................................................................................ 127
Generation of Beam's Eye Views (BEVs).................................................................. 129
Field Shaping with Multileaf Collimators ............................................................. 134
Shaped Field D osim etry.............................................................................................. 139
Optimization of Isotropic Beam Bouquet Orientation................................................ 140
Limits on Adjusting Beam Positions from the Initial Isotropic Beam Bouquet......... 147
Appropriate Number of Beams for Use in Shaped Beam SRS Planning ................... 157
Application of Isotropic Beam Bouquets Nine Beam Plan for Meningioma .......... 162
Dynamic Arcs with MLC...................................................................................... 169
C conclusion ............................................................................................................ 172

5 INTENSITY MODULATED SRS WITH FIXED BEAMS .....................................174

Introduction........................................................................................................... 174
Intensity Modulated Radiotherapy (IMRT).............................................................. 175
IMRT Treatment Planning with CadPlan/Helios.................................................. 182
Example Nine Beam and Nine Intensity-Modulated Beams for Meningioma........ 187
Multiple Isocenters as a Special Case of IMRT.......................................................... 196

6 SRS METHODS COMPARISON ....................................................................199

Introduction........................................................................................................... 199
Clinical Example Case Data....................................................................................... 202
Comparison of Alternative SRS Treatment Delivery Methods to Multiple Isocenter
SRS with Circular Collimators ................................................................................... 268
Strengths and Weaknesses of Multiple Isocenters and IMRT.................................. 277
Applying the Results of this Research to New SRS Cases....................................... 287
C onclusions............................................................................................................ .. 289

7 CONCLUSION.........................................................................................................291

LIST OF REFERENCES.......................................................................................... 297

BIOGRAPHICAL SKETCH ...................................................................................... 305

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



Thomas H. Wagner

August 2000

Chairman: Francis J. Bova
Major Department: Nuclear and Radiological Engineering

The University of Florida stereotactic radiosurgery (SRS) system is a well-

established system of single fraction, highly conformal linear accelerator based radiation

therapy for intracranial lesions. As originally implemented, the system is characterized

by the delivery of circular radiation beams with multiple arcs of megavoltage photon

beams all impinging on the target. The introduction of multileaf collimators (MLCs) and

computer-controlled treatment machinery offers the opportunity to plan and deliver more

complex radiation treatments that may apply dose to the target while applying less dose

to non-target tissues. There are numerous treatment techniques that may be employed

with such equipment, including multiple static fields, dynamic conformal arcing

treatments, and treatment with intensity modulated radiation therapy (IMRT) fields. This

expanded list of patient treatment options poses a new problem to the treatment planner:

determining the optimum treatment method for a given patient. Although many centers


report the expanding use of these and other treatment techniques, there are few, if any,

reports that offer definitive comparisons of the major treatment techniques against one


The purpose of this research was to compare radiosurgery plans using MLC static

fields, dynamic conformal arcs, and IMRT, with radiosurgery plans using circular

collimators and multiple isocenters, and to determine which of the MLC-based treatment

methods provides the best results. Representative clinical example cases from the

University of Florida radiosurgery patient database were used to examine the dosimetric

performance of each type of treatment delivery. Analysis of these clinical example cases

shows that circular collimators with multiple isocenters deliver dose distributions equal to

or better than MLC-based techniques for every example case studied. IMRT

radiosurgery treatments can yield results comparable to multiple isocenters, for cases

involving larger sized targets that would require more than about fifteen isocenters, and

for targets with relatively smooth-surfaced three-dimensional shapes. For most target

shapes, multiple isocenter dose distributions are more conformal and provide a steeper

dose falloff outside of the target than the MLC-based treatment methods. The software

tools developed for this research can also be employed in a clinically useful timeframe to

develop patient-specific optimized treatment plans to assist in this determination.


Megavoltage Photon Radiotherapy And Radiosurgery
Conventional external beam radiotherapy, or teletherapy, involves the

administration of radiation absorbed dose to cure disease. The general teletherapy

paradigm is to irradiate the gross lesion plus an additional volume suspected of

containing microscopic disease not visible through physical examination or imaging, to a

uniform dose level. External photon beams with peak photon energy in excess of 1 MeV

are targeted upon the lesion site by registering external anatomy and internal radiographic

anatomy to the radiation (beam) source. Due to uncertainty and errors in positioning the

patient, the radiation beam, which is directed at the lesion, may need to be enlarged to

ensure that errors and uncertainty in patient positioning do not cause the radiation beam

to miss some or all of the target. Unfortunately, enlarging the radiation beam results in a

relatively large volume of non-diseased tissue receiving a significant radiation dose in

addition to the target. For instance, adding only a 2mm rim to a 24mm diameter spherical

volume to ensure that the 24mm diameter target is covered even with a 2mm positional

error will increase the irradiated volume from 7.2 cm3 to 11.5 cm3, an increase of 60%

(Bova 1998a).

It has long been known that for many cancerous diseases, a radiobiological

advantage is realized by administering the total radiation dose in small doses (fractions)

over an extended period of time. Bergonie and Tribondeau discovered this principle in

the early twentieth century by experimenting to determine the doses of x-rays to the testes

required to sterilize male goats (Hall 1994a). They discovered that they could not

administer a high enough dose of x-rays to the testicles to cause sterilization without also

causing a severe skin reaction in tissue adjacent to the testicles. However, when they

administered the x-ray dose--in small doses, given once a day over several weeks,

sterilization without an adverse skin reaction was possible. They postulated that the

testes were a model of tumor tissue, while the adjacent skin served as a model of dose-

limiting, normal tissue. Although these assumptions are now known to be false, the

conclusion was valid, that in most cases, for a given level of normal tissue toxicity, better

tumor control may be achieved with multiple dose fractions over an extended time.

In contrast to conventional, fractionated radiotherapy, stereotactic radiosurgery

(SRS) involves the administration of a relatively large, single dose of radiation (10 Gy to

20 Gy) to a small volume of disease, thereby abandoning the advantages provided by

fractionation. Lars Leksell conceived of the idea of radiosurgery in 1951 (Leksell 1951).

His original idea involved using many beams of orthovoltage x-rays converging on an

intra-cranial target to create a lesion. Leksell's idea, known as the Gamma Knife or

gamma unit, was practically implemented in 1967 using an array of 170 Co-60 sources,

each of which emitted megavoltage gamma rays through radioactive decay (Leksell

1983; Colombo 1998). The decay gamma rays were collimated with holes bored in a

large radiation shield to converge to a single point inside the patient's head. The use of

secondary collimator helmets, each with different size holes (4, 8, 14, and 18 mm

diameter), allowed for the creation of several sizes of a spherical, high dose region (Maitz

1998). Irregularly shaped lesions could be produced by "stacking" spherical regions

together to build up complex shapes. The patient's skull was positioned with sub-

millimetric precision using a minimally invasive hearing that attached to each size of

collimator helmet (Wasserman 1996).

Linear accelerators linacss) were first used for radiosurgical use in the 1980s.

Betti and Derichinsky reported using a linear accelerator with multiple fixed, isocentric

beams in 1983 (Betti 1983). Several investigators reported using multiple converging

arcs with a linac by 1985 (Colombo 1985; Hartmann 1985). By fitting an isocentric

linear accelerator with circular collimators, multiple beams and/or arcs of radiation could

be made to converge upon the machine's center of rotation, where the patient's tumor had

been positioned. By such a means, dose distributions very similar to those of the gamma

knife could be produced with megavoltage photons from a linac, rather than from the

decay of radioactive sources. A significant practical difficulty of using an isocentric

linear accelerator, however, lies in overcoming the mechanical inaccuracy of rotation

inherent in heavy rotating equipment. A common upper limit on allowable mechanical

error ("wobble") of rotational center of linear accelerator gantrys is 2mm, plus an

additional 2mm error in the treatment couch rotation accuracy. Added in quadrature, the

resultant total possible error between the expected and actual radiation isocenter can be as

high as 2.8 mm, about an order of magnitude larger than the mechanical error associated

with gamma unit treatments. If the mechanical inaccuracy of the treatment machine

cannot be resolved, the radiosurgeon would need to increase the size of the radiation

therapy beam in order to ensure that the target being treated is completely covered.

Addition of even one or two millimeters of extra margin to a radiosurgical treatment

beam has a markedly undesirable effect, however, by drastically increasing the volume

treated to the target (prescription) dose. It is therefore strongly to the radiosurgeon's

advantage to improve the mechanical and overall system accuracy in positioning the

patient, in order to allow using treatment beams of the minimum necessary size (Meeks

1998a). By reducing the volume of non-target tissue irradiated to high levels, the

likelihood of incurring a complication (adverse reaction to the radiation treatment) may

be minimized.

The University of Florida radiosurgery system was developed in the mid-1980s as

a solution to the above mentioned problem of linear accelerator radiosurgery (Friedman

1989; Friedman 1992; Meeks 1998b). By using an isocentric system, as indicated by the

arrows in Figure 1-1, to position the patient and to provide tertiary, circular collimation of

the x-ray beam, the system allows a linear accelerator to be used to deliver radiosurgical

treatments (Figure 1-2) with mechanical accuracy comparable to a gamma unit.

Figure 1-1: University of Florida radiosurgery system : isocentric subsystem in place
under the gantry of a linear accelerator linacc).

Figure 1-2: Time lapse photograph showing arc rotation of the linac gantry about the
patient during treatment. The patient's lesion has been positioned at the radiation
isocenter with a stereotactic hearing and the isocentric subsystem.

Technical Evolution and Improvements Stereotactic Radiotherapy
Although minimally invasive, the hearings associated with gamma unit and

linear accelerator radiosurgery make the administration of multiple radiation doses

infeasible. Some physicians have experimented with leaving the stereotactic hearing on

the patient for extended periods to allow a series of treatments over this time. Although it

is possible to overcome problems with local infection and patient discomfort, these

difficulties have generally caused other practitioners to avoid using this method of

stereotactic radiotherapy (Schwade 1990). The application of non-invasive stereotactic

localizing techniques allows easier delivery of repeated radiation treatments to a

stereotactically-located lesion. The Gil-Thomas-Cosman headframe is one example of

how this may be done, relying on a headframe which locates to the patient's head by

means of a dental mold (a "biteplate", or "biteblock"), occipital headrest, and a strap to

tightly hold the frame to the head (Reinstein 1998). Systems such as these, which

combine the functions of immobilization and positioning, tend to suffer reduced accuracy

because of immobilization forces that are inevitably applied to the reference positioning

system. Optically guided systems have been developed recently, however, which de-

couple the positioning function from the immobilization function. Such systems have

been demonstrated to provide patient positioning with smaller errors than previous

systems. One such optically guided system has demonstrated the ability to position the

patient within about 1.1 mm error at isocenter, which while not as small a positioning

error as attainable with an invasive stereotactic heading, is still significantly better than

previous systems (Bova 1997; Bova 1998b). Because of such new, non-invasive

stereotactic techniques, radiosurgery treatments can be administered in multiple fractions

(Figure 1-3). Such fractionatedd radiosurgery" is generally known as stereotactic

radiotherapy (SRT).

Linear Accelerator Radiosurgery and Radiotherapy Treatment Techniques
The major treatment techniques used to deliver linac radiosurgery treatments are

circular collimators with arcs, conformally shaped beams, and intensity modulated

radiotherapy (IMRT). Circular collimators can be used to create spherical regions of high

dose. When used with linear accelerators, the circularly collimated beam is rotated

around the target at isocenter by moving the gantry in arc mode while the patient and

treatment couch are stationary, producing a para-sagittal beam path around the target.

Betti and Derichinsky developed their linac radiosurgery system with a special chair, the

"Betti chair," which moved the patient in a side to side arc motion under a stationary

linac beam, and which produced a set of para-coronal arcs. With modem, computer

controlled linear accelerators, more complex motions other than these simple arcs are

possible. The Montreal technique, which involves synchronized motion of the patient

couch and the gantry while the radiation beam is on, is an example of this, producing a

"baseball seam" type of beam path (Wasserman 1996). The rationale of using arcs with

circular collimators is to concentrate radiation dose upon the target, while spreading the

beam entrance and exit doses over a larger volume of non-target tissue, theoretically

reducing the overall dose and toxicity to non-target tissue.


Figure 1-3: Patient positioned under the linear accelerator with biteblock optically guided
system. A system of stereo cameras out of the picture's field of view senses the position
of the reflective spheres attached to the biteblock in the patient's mouth. This system
allows precise and repeatable patient positioning without the need for an invasive
stereotactic hearing (shown attached to the patient in Figure 1-2). The white mask is an
immobilization aid to assist the patient in remaining motionless during the treatment.

The technique of multiple converging arcs delivered with circular collimators

produces a spherical region of high dose with a steep dose gradient, or falloff. This dose

distribution is adequate for treating a sphere or round target, but will treat a large volume

of non-target tissue to high dose if the sphere encompasses an irregularly shaped target.

Multiple spheres of varying sizes may be "stacked" together to produce a high dose

region which conforms closely to the shape of the target while still maintaining a sharp

dose gradient, as shown in Figure 1-4 (Meeks 1998b).

Figure 1-4: Conformal dose distribution produced by circular collimators and multiple
isocenters. Several isocenters, each with a set of converging arcs, have been placed near
one another to conform the composite dose distribution to the target's shape.

An alternative means of delivering a linac radiosurgery treatment is to employ

beams which are shaped to conform to the target's shape as seen from the direction of the

beam, or "beam's eye view." Conventional radiotherapy practice is to use diagnostic x-

ray radiographic or flouroscopic images of the patient obtained in a simulator session to

determine the beam's shape. In recent years, the three dimensional image sets from

computed tomography or magnetic resonance image scans have been used to construct

three-dimensional models of the patient and internal structures, such as the target. These

models of each patient structure may be used to determine the placement of radiation

beams, and to design each beam's shape.

Technical Evolution and Improvements Linear Accelerator Radiation Delivery
Both the gamma unit and early linear accelerator radiosurgery systems produced

similar sphere-like dose distributions using either multiple static, circular beams (gamma

unit) or multiple circular beams swept though several arcs (linear accelerator). However,

the linear accelerator offers additional flexibility over the gamma unit in that multiple

collimation devices may be used to produce non-circular beams, and beams with non-

uniform intensity profiles across the beam. This additional flexibility in radiation

delivery potentially offers the ability to more closely tailor the dose distribution to the

target volume with a linear accelerator than with a gamma unit. The beam shaping and

modulation devices used with linear accelerators for these purposes include custom beam

shaping blocks, wedge beam filters, custom beam compensating filters, and multi-leaf

collimators. Also, linear accelerators typically can deliver a much larger range of

radiation beam sizes, upwards to a 40 cm x 40 cm square field at 100 cm from the

radiation source.

The simplest beam-shaping device used with linear accelerators (other than the

machine's secondary collimators, which typically produce rectangular fields up to a 40

cm x 40 cm square field at the machine's isocenter) is the custom block. Such blocks are

individually constructed by pouring low melting point alloy (cerrobend) into a mold,

which is attached to a mounting tray. The edges of the apertures defined in the hardened

metal block are designed to match the divergence of the radiation beam emanating from

the treatment machine. A separate block must be manufactured for each beam that will

be used to treat the patient. Although offering the best possible match between the shape

of the target and the shape of the beam-defining aperture, the time and cost of

manufacturing such blocks limits the number of blocks and radiation beams which can be

used to treat a patient.

Wedge beam filters may be used with or without the presence of beam shaping

devices such as the custom blocks mentioned above. Wedge filters are placed in the path

of the photon beam in order to tilt the shape of the isodose distribution. This provides a

simple one-dimensional intensity modulation across the treatment field, which is often

advantageous to the treatment planner in obtaining a more homogeneous dose

distribution in the target volume. This is desirable in certain cases, for example where

the patient's anatomy changes significantly over the extent of the field. Proper placement

of a wedge filter in this case can effectively compensate for missing tissue on one side of

a treatment field. Wedges are also commonly used to reduce dose heterogeneity

("hotspots") in regions of beam overlap inside the target (Khan 1994, Ch. 11).

The idea of using a filter to modulate the beam intensity across the treatment

beam is extendable to a two-dimensional intensity modulation. A typical such 2D

compensating filter is generally used to adjust the beam intensity over a grid of small

square regions, with the goal of obtaining a uniform dose distribution in a plane near the

target. Such devices are designed for each patient, and are typically constructed by

placing differing thicknesses of dense, radiation absorbing material such as brass in a

checkerboard type pattern on a tray that is placed in the beam path. This can be done by

hand, or by use of a computer-controlled milling machine which custom machines a

single piece of radiation absorbing material into the desired shape (Purdy 1996).

Although the dose distribution around the target may be made more homogenous by such

devices, they share a disadvantage of custom blocks in excessive labor costs.

Additionally, compensators requiring manual construction can remain a source of

potential treatment errors despite quality assurance. These factors limit the usefulness

and number of fields to which such beam modifying devices may used (Hall 1961;

Sundbom 1964; Grijn 1965).

Multileaf collimators (MLCs) are mechanical beam collimating devices that can

combine some or all of the functions of beam shaping blocks, wedge filters, and custom

compensating filters discussed above (Figure 1-4). The most common type of MLC

consists of two banks of opposed leaves of radiation absorbing metal that can be moved

in a plane perpendicular to the beam's direction. The MLC can be rotated with the

treatment machine's collimator in order to align the leaves for the best fit to the target's

projected shape. The simplest use of an MLC is simply as a functional replacement for

custom made beam shaping blocks, in which the rectangular MLC edges are used to

approximate a continuous target outline shape (Figure 1-5) (Brewster 1995). However,

the MLC may be used in a more sophisticated fashion to form many different beam

shapes of arbitrary size and intensity (by varying the amount of radiation applied through

each beam aperture). In this manner, radiation fields with a similar dose profile as a

shaped, wedged field may be delivered using only the computer-controlled MLC. MLCs

can also deliver intensity modulated dose profiles similar to those achievable using

custom beam compensators, but without the disadvantages of fabrication time or of

needing to manually change a physically mounted beam filter between each treatment

field (Stemick 1998). Thus, a.computer-controlled MLC and treatment machine offer the

potential to deliver more sophisticated radiation treatments to each patient with the same

time and cost resources available. Moss investigated the efficacy of performing

radiosurgery treatments with a dynamically conforming MLC in arc mode, and concluded

that dynamic arc MLC treatments offered target coverage and normal tissue sparing

comparable to that offered by single and multiple isocenter radiosurgery (Moss 1992).

Nedzi (Nedzi 1993) showed that even crude beam shaping devices offered some

conformal benefit over single isocenter treatments with circular collimators.


Figure 1-5: Multileaf collimator (arrows) attached to the gantry of a linear accelerator.
The MLC leaves define a small square aperture in this picture.

Figure 1-6: The narrow, rectangular MLC leaves conform the radiation field's shape to
approximate an irregularly shaped target's shape (solid line), as seen in this beam's eye
view (BEV).

Research Problem: Comparison of SRS Treatment Methods
The potential for improvement presented by some of these newer and more

sophisticated treatment delivery methods has spurred interest in their evaluation relative

to the more traditional linac SRS methods of multiple intersecting arcs and circular

collimators. These comparisons generally show that for small to medium (up to about 20

cm3) intracranial targets, multiple static beams offer acceptable conformity and target

dose homogeneity while offering a straightforward treatment planning process. Static

beam IMRT techniques generally performed comparably to or better than static beam

plans. A common conclusion by many of these investigators is that the use of multiple

isocenters with circular collimators results in a poor quality treatment plan, as evidenced

by the performance of the multiple isocenter plans they used to compare with the static

beam and IMRT plans. Even in reports more favorable to multiple isocenter linac SRS,

the investigators frequently note difficulty in achieving conformal and homogeneous

plans, and also note needing a large amount of time to plan and deliver these treatments.

Based on a number of recent comparisons of radiosurgery methods in the literature, one

could roughly expect to obtain a reasonably conformal (exposing up to about the same

volume of non-target brain tissue as target tissue to the target dose level) and

homogeneous (maximum dose not more than about twice the minimum target dose) dose

distribution for various IMRT- and mMLC treatment techniques, and moderate-to-large

sized, irregularly shaped intracranial lesions. Typical multiple isocenter plans presented

fare considerably worse, though, in terms of dose conformity, homogeneity, and in

treatment planning and delivery times (Laing 1993; Hamilton 1995; Woo 1996; Shiu

1997; Cardinale 1998; Kramer 1998; Verhey 1998).

A potential problem with these comparison studies is that they do not equitably

compare the full potential of multiple isocenter radiosurgery with circular collimators. A

qualitative inspection of the multiple isocenter dosimetric results shown in these

comparisons leads one to suspect that in many cases, sub-optimal multiple isocenter plans

are being compared with reasonably optimized static beam and dynamic MLC arcs/IMRT

plans. Although the multiple isocenter treatment plans in these comparisons in the

literature may represent a level of plan quality achievable by an average or unfamiliar

user, they do not represent the average level of plan quality in the University of Florida

experience. Unlike other evaluations readily available in the literature, an evaluation of

the best employment of an MLC in radiosurgery treatments at the University of Florida

must consider the typical quality of treatment plan that is readily achievable in the

University of Florida clinical experience.

The research problem posed is to evaluate the major SRS treatment delivery

methods that could be implemented clinically at the University of Florida, and other

institutions using the University of Florida radiosurgery system. Many claims are being

circulated about some of the newer methods mentioned earlier of employing teletherapy

beams for SRS treatment. The University of Florida radiosurgery system has

demonstrated the ability to plan and deliver tightly conformal dose distributions to

irregularly shaped targets near radiosensitive structures, while maintaining a sharp dose

gradient away from the target towards radiosensitive structures (Meeks 1998a; Meeks

1998b; Meeks 1998c; Foote 1999; Wagner 2000). While it may be attractive to

contemplate the replacement of the current circular collimator system with more

advanced and elaborate treatment delivery methods, such a decision should be based on a

reliable study.

The purpose of this research is to investigate the optimal implementation

of a multileaf collimator (MLC) system for SRS at the University of Florida. An MLC

could be employed in several different ways: 1) dynamic conformal arc treatments with

templated arc sets, 2) multiple fixed, conformal beams, and 3) multiple fixed, intensity

modulated beams. These treatment delivery options are to be compared against multiple

isocenters with circular collimators. In order to ensure a proper comparison, a reasonable

optimization strategy is employed for each treatment delivery technique to guard against

inadvertently biasing the comparison against one or the other treatment methods. To this

end, automatic planning and optimization tools were developed for multiple isocenter

SRS and for multiple static beam SRS. Due to the fewer number of variables involved,

treatment planning for dynamic MLC arc treatments will be based primarily on standard

arc templates or sets. Chapter two provides a discussion of treatment plan evaluation

techniques and tools. Chapter three is devoted to optimal treatment planning methods

with circular collimators, chapter four to shaped beam radiosurgery planning, and chapter


five to fixed beam IMRT planning. Chapter six is devoted to the actual comparisons of

each technique to an array of example cases, on which the guidelines and

recommendations for optimal employment of an MLC at the University of Florida are



Evaluating the suitability of a stereotactic radiosurgery or radiotherapy treatment

plan requires the human planner to assimilate and analyze a vast quantity of three

dimensional dose information. Given the distribution of radiation dose in three

dimensions in the vicinity of the target, the planner must assess how well the prospective

plan accomplishes the treatment goals of uniformly irradiating the target to a high dose

level while sparing nearby radiosensitive structures from the effects of a large radiation

dose. This chapter presents currently accepted methods and tools for analyzing

stereotactic radiosurgery and radiotherapy dose distributions.

In two dimensional radiotherapy planning, doses are calculated on a two

dimensional slice in a single plane through the target, assuming that the slice chosen is

representative of the entire target region, and that the slice is semi-infinite in extent

(extends infinitely in both directions perpendicular to the plane of interest). Dose

distributions are generally displayed as isodose curves superimposed upon either the

patient contour or a single CT image slice though the region of interest. Plan evaluation

is based upon inspection of the isodose curves overlaid upon this single slice or image.

In three dimensional radiation therapy planning, evaluation of the three dimensional dose

distribution involves the processing of considerably greater amounts of information.

The calculation of radiation absorbed dose is fundamental to radiation therapy, in

order to predict and control the radiation dose delivered to the lesion, and to non-target

regions inside the patient. This section provides a discussion of general methods of dose

calculation for radiotherapy and radiosurgery situations, followed by a presentation of

methods to evaluate the efficacy of a radiation dose distribution. The general aims of

radiotherapy and radiosurgery are simple: to deliver a high, uniform dose to the target

while minimizing the radiation dose to non-target structures. There are several tools

available to the human treatment planner to quantify the degree to which these goals are

accomplished: 1) isodose curves and distributions, 2) dose-volume histograms, 3)

physical dose-volume figures of merit, and 4) biological models of tissue response to

radiation. The following sections explain the use of each of these tools in radiation

therapy and radiosurgery treatment planning, after a discussion of methods for calculating

radiation dose distributions.

Dose Calculation
The purpose of dose calculation in radiotherapy is to be able to accurately

determine the dose to target and non-target structures inside the patient. An ideal

calculation of absorbed dose to matter in all regions of interest in megavoltage external

beam radiotherapy would correctly account for all of the interactions between the

megavoltage photons in the therapy beam and the matter in the patient. The most

accurate current methods of computing the spatial distribution of the deposition of

radiation dose involve probabilistically simulating the transport of many individual

radiation beam particles from their point of emission in the radiation source, using

random number processes (hence the name "Monte Carlo" to describe this calculation

method). Enough particles must be simulated to provide a statistically significant tally of

radiation particle interactions in each region of interest, often requiring lengthy

computing times to simulate the radiation transport of many (millions of) particles. This

dose calculation method is attractive because it is based on first principles of radiation

physics, and can therefore correctly account for any specific patient situation. However,

the amount of computation time generally required by present day computers limits its

usefulness in clinical situations.

Because of these difficulties in calculating absorbed dose distributions from first

principles of physics, the most common approach taken in radiation therapy has been to

use simpler models relying on direct measurements of dose. Typically, these models

involve using various radiation detectors to directly measure the dose distribution in a

water phantom, and applying corrections to the measured dose distributions to account

for differences between the water phantom and each actual patient situation. The dose

calculations in this report rely on such models of dose distributions. A brief discussion of

the dose calculation procedure for a rectangular solid water phantom follows, in order to

facilitate the explanation of the dose calculation process for clinical radiosurgery


The dose profile as a function of depth in a water phantom (setup shown in Figure

2-1) from a normally incident radiation beam is shown in Figure 2-2. This plot shows the

absorbed dose measured in water with a stationary radiation detector placed at the

isocenter of the linear accelerator. As the detector's depth to the water surface is

increased by adding water to the water phantom (which is a tank of water), there is a

greater thickness of water interposed between the radiation source and the detector, so

that the water absorbs more of the radiation beam. The curve is approximately

exponential in shape, but is not a pure exponential due to the non-linear variation in

scattered radiation dose to the detector with changes in water depth, and due to beam

hardening effects at greater depths. The radiation absorbed dose data measured in this

manner is commonly referred to-as "Tissue phantom ratio" (TPR) when the dose is

normalized to the dose at a particular depth (Khan 1994). TPR data is measured for each

circular radiosurgery collimator (Duggan 1998), or may be interpolated for a given

collimator from data tables of several collimators spanning a range of sizes (Surgical

Navigation Technologies 1996).

Linear accelerator
(radiation source)

\ -Colli
Source to axis
distance (SAD)

Water surface

Central axis
of beam

Radiation detector at isocenter
of the linear accelerator

Figure 2-1: Schematic of setup for measuring radiation dose as a function of depth in a
water phantom




E 0.600


0 5 10 15 20 25 30
Depth to water phantom surface (cm)

Figure 2-2: Tissue-phantom ratio (TPR) curve in water phantom for a 6 MV photon beam
shaped with a 30 mm diameter circular collimator.

The dose profile in a plane perpendicular to the central axis (along the "cross

beam direction" in Figure 2-1) varies with distance from the central axis, and is thus

measured with a radiation detector as well in order to allow calculation of the radiation

dose at off-axis points. A plot of the radiation field intensity as a function of off-axis

distance, in a plane 100 cm from the radiation source, is shown in Figure 2-3 for a 30 mm

diameter circular radiosurgery collimator. This off-axis dose data is frequently

normalized to either its maximum value, or to the dose at the central axis, and is also

known as "off-axis ratio", or OAR. Like TPR data, OAR data may either be measured

for each individual radiosurgery collimator, or may be interpolated from a table of

measured OAR values for selected collimators. Due to changes in the relative dose

profile with depth (due to changes in scattered dose and beam hardening effects), OAR

profiles are usually measured at several depths in order to provide measured data under

conditions close to those for which dose is being computed.


-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Off axis distance (cm)

Figure 2-3: Off-axis ratio (OAR) curve for a single circular collimator

The last measured quantity needed to calculate dose to any point inside the water

phantom is output factor, also referred to as total scatter factor, Sc,p (Khan 1994). Sc,p

accounts for changes in the dose due to changes in the radiation field size, and to changes

from the scattered dose due to changes in the volume of phantom irradiated. Sc,p is the

ratio of dose to a reference point on the central axis of the beam, to the dose to the same

point but under standard conditions. In linear accelerator radiosurgery treatments, Sc,p is

a function only of the collimator size (diameter) used, since the linear accelerator

secondary collimators are normally placed in a constant position when using the

radiosurgery circular collimators. A plot of output factor as a function of radiosurgery

collimator size for a 6 MV linear accelerator at the University of Florida is shown in

Figure 2-4.

--a a ao M!



a 0.95

4 /
^ 0.85

o 0.80

5 10 15 20 25 30 35
Collimator diameter (mm)

Figure 2-4: Output factor as a function of circular radiosurgery collimator diameter for a
6MV linear accelerator at the University of Florida.

The above measured dosimetric data may be used to compute the dose to a point

in the water phantom of Figure 2-5, as given in Eq. (2-1):

r is
(2-1) Dose(P) = k -MU TPR(coll, depth(P)) OAR(P) Sc,co dists-P

In this equation, k is the treatment machine's calibration constant, normally 0.01 Gy/MU,

MU is the number of monitor units delivered by the treatment machine, dists-cp is the

source to calibration point distance (nominally 100 cm for an isocentric machine

calibration), dists.p is the distance from the source to point P, TPR(coll, depth(P)) is the

TPR for the circular collimator being used at the depth of point P, Sc,p-coll is the total

scatter factor, or output factor, for the circular collimator being used, and OAR(P) is an

off-axis ratio representing the variation of dose away from the field's central axis (Khan

1994; Surgical Navigation Technologies 1996; Duggan 1998). Equation (2-1) provides

corrections to the measured dose data (TPR, OAR, and Scp) to account for changes in the

dose at point P from the dose to the reference point at the linear accelerator's isocenter.

If point P is moved away from the isocenter in any direction, these corrections are needed

due to changes in 1) the distance-of the dose point P from the source, 2) the distance of

dose point P from the central axis of the beam, 3) of changing collimator sizes, and 4) of

the changing water depth (attenuation) above point P. This dose model is simple and

accurate under conditions similar to the conditions under which TPR, OAR, and Sc,p were

measured, i.e.- a flat surfaced, homogeneous mass of water.


I i

Collimated /

beam !

Surface I



Figure 2-5: Parameters for calculation of dose to point P in a water phantom from a single
radiation beam. OAD is the off-axis distance.

The simple dose model of equation (2-1) is used as the dose engine of the

University of Florida radiosurgery treatment planning system, and of most other

I depthP)
I OAD(P) 1 \
--- Isooenter
/ Isooenter


radiosurgery dose planning systems that utilize circular collimators. In applying equation

(2-1) to a radiosurgery treatment, one must assume that the contents of the patient's head

are homogeneous, water equivalent, and that the geometry is similar to the reference

measurement geometry shown in Figure 2-5. Since linear accelerator radiosurgery

treatments with circular collimators are delivered with arcs of radiation rather than static

beams, each arc is simulated as a set of static beams, spaced along the path of the arc.

Typically, an arc of 100 degrees is simulated as eleven static beams, spaced 10 degrees

apart (Figure 2-6). The radiation dose to any point is the sum of the doses to that point

from all of the individual beams in the arc. Likewise, the dose to any point due to

multiple arcs and isocenters is the superposition (sum) of the doses of all of the arcs

associated with each isocenter. For any arrangement of radiosurgery beams or arcs, the

dose is calculated to a grid of points spaced closely together. Isodose curves

corresponding to the locus of isodose points may be constructed by interpolation amongst

the points in the viewing plane for which dose has been computed. Dose grid point

spacing should be no further apart than 2 mm, in order to properly sample the rapidly

changing dose distributions characteristic of radiosurgery dose distributions (Schell



Figure 2-6: A 100 degree arc of radiation produced by a continuously moving beam,
approximated by 11 beams spaced 10 degrees apart. Crosshairs indicate the center of
rotation (isocenter).

There are several departures from the idealized geometries shown in Figures 2-1

and 2-5 that can occur in clinical patient treatment situations, and which can potentially

lead to errors between the actual dose to a point and the dose calculated with the dose

model in equation (2-1). Inhomogeneities inside the patient or at the patient's surface can

cause significant dose errors under certain circumstances, such as for large field sizes.

An irregular (not flat) patient surface is one such example of inhomogeneity, shown in

Figure 2-7. The curvature of the patient surface causes points 1 and 2 in Figure 2-7 to lie

at different depths from the surface with respect to the beam. The resulting tissue deficit

shown will cause the photon beam to undergo less attenuation from the source to point 2

than from the source to point 1. If the TPR for central axis point 1 is used to calculate the

dose at points 1 and 2, then the dose to point 2 will be underestimated by equation (2-1),

since point 2 is not as deep as point 1 (point 2 has a numerically larger TPR than point 1).

For a typical adult patient with a cranial radius of curvature of about 7.5 cm, the tissue

deficit from the center of a 10x10 cm2 square field is about 1.5 cm, which corresponds to

a dose error of about 6% (for 6 MV photons attenuated at about 4% per cm of depth) at

point 2, if the TPR for central axis point 1 is used instead. The magnitude of the tissue

deficit increases as the field size increases, and decreases for smaller field sizes. For a 40

mm diameter circular field, the tissue deficit for the same radius of curvature is only

about 2.6 mm, corresponding to slightly less than a 1% dose error. The tissue deficit and

dose error for a 20 mm diameter field are only 0.6mm and 0.2%, respectively. Thus, for

small (< 40 mm diameter) radiosurgery fields, the effect of surface irregularity

(inhomogeneity) can be neglected without introducing undue error (several percent) into

the dose calculation (Ahnesjo 1999).

I '

I i
1 ii

Assumed flat surface
for dose model

Tissue deficit due to
surface irregularity

Figure 2-7: Tissue deficit due to an irregularity (inhomogeneity) in patient surface.

The assumption that the interior of the patient is a homogeneous, water equivalent

material can also lead to errors between the calculated (equation (2-1)) and actual dose to

a point in some cases. For instance, the dose model in equation (2-1) does not account

for the changes in beam attenuation due to differences in electron density from that of

water, such as those encountered near air cavities (e.g.- sinuses) and bone. Although

methods such as the Batho power-law correction (Khan 1994; Ahnesjo 1999) exist to

correct calculated doses for these effects, such corrections are generally not needed to

obtain sufficient calculation accuracy in stereotactic radiosurgery or radiotherapy

situations with many beams. A study by Ayyangar on two typical radiosurgery cases

compared a simple dose model similar to that of equation (2-1), with a Monte Carlo dose

model with and without inhomogeneity corrections. Not correcting the simple dose

model for the passage of the beam through the cranium caused the uncorrected dose

model to overestimate the dose by 1.5% to 2.5%, an acceptable amount of error.

Applying a TAR ratio method correction, similar to the Batho power-law correction,

reduced the dose calculation error further (Ayyangar 1998).

The small size of the beams typical of stereotactic radiosurgery and radiotherapy

allow the use of a relatively simple dose calculation model without sacrificing accuracy

of dose calculation. This simplicity is important in that it allows much faster dose

computations throughout the volume of interest, which is especially important given the

large typical numbers of beams for which dose must be calculated in radiosurgery.

Isodoses and Dose-volume Histograms
Isodose curves overlaid upon the patient's three-dimensional image set are an

important plan evaluation tool, just as in two-dimensional planning. To evaluate a three-

dimensional dose distribution by this method, however, the planner must examine the

isodose distributions in a number of planes through the target region, which can be

cumbersome for large targets occupying many planes in an image set. It is possible to

display three-dimensional renderings of three dimensional dose distributions on a flat

computer display screen, but these are also very difficult to analyze. The problem with

evaluating a three-dimensional radiosurgery dose distribution, with it's sharp dose

gradients, lies in discerning the dose received by many possibly overlapping structures

around the target. Although a number of commercially available treatment planning

systems can render three-dimensional views of arbitrary isodose volumes in various

shades of translucency, along with any structures that the user has identified, it is very

difficult to determine precise (sub-millimetric) spatial relations between the target

volume, particular isodose volumes, and radiosensitive structures. For this reason, it is

usually necessary to evaluate a large number of two dimensional isodose plots through

the region of interest to determine plan suitability. In a single two dimensional isodose

plot, one may readily determine whether a particular isodose surface coincides with the

intersection of any particular volume with the image plane, to an accuracy of within one

image pixel in the plane of interest. Even this level (within an image pixel) of visual

inspection precision can still lead to significant errors in assessing the volume of dose

coverage for small intracranial targets. Consider a 20 mm diameter spherical

radiosurgery target, for which we wish to evaluate dose coverage by inspection of an

isodose line overlaid on the image set. A 10% volume error results if the isodose line is

shifted half of one image pixel (one image pixel is 0.67 mm x 0.67 mm in a transaxial

plane for a 512 x 512 CT image acquired with a 35 cm diameter field of view) inward or

outward, which is the spatial resolution limit of our ability to discern positional shifts in

the image set. The 20 mm sphere, 4.2 cm3 volume, would apparently be equally well

covered by an isodose surface ranging in volume from 3.8 cm3 to 4.6 cm3. The volume

error problem worsens as target size decreases, and results in a 20% volume error for a 10

mm diameter target. One can imagine then the difficulty in evaluating a large number of

these isodose plots to within submillimetric image pixel resolution: on each image slice,

one must examine and remember the isodose surface which encloses the target, and

which isodose surfaces (and to which extent) intersect nearby radiosensitive structures.

This is straightforward if somewhat tedious to do on one image slice, but the difficulty is

magnified tremendously when each slice in a large region must be examined, and the

dose area information from each slice integrated with the information from all the other

image slices. A method of comparing the volumes of dose coverage that is less error-

prone is desirable.

One commonly used solution to this problem is to use dose-volume histograms

(DVHs). DVHs are a method of condensing large quantities of three dimensional dose

information into a more manageable form for analysis. The simplest type of DVH is a

"direct" histogram of volume versus dose (Lawrence 1996), as shown in Figure 2-8. This

is simply a histogram showing the number of occurrences of each dose value within a

three dimensional volume. Unfortunately, the spatial information of which specific

volumes are exposed to each dose level is lost in the process of constructing a DVH. For

this reason, DVHs are generally used clinically in conjunction with the evaluation of

multiple isodose plots as mentioned earlier.

The ideal treatment planning situation is one in which the target volume receives

a uniform dose equal to the maximum dose, and the non-target volume receives zero

dose. This would be represented in a direct DVH by having a target histogram with only

one non-zero bin at 100% dose (normalized to maximum dose), and to have a direct DVH

of the non-target volume with all dose bins receiving zero dose. Plots of ideal direct

DVHs for target and non-target volumes are shown in Figure 2-8. Figure 2-9 shows

direct DVHs for target and non-target volumes for a more typical (non-ideal)

radiosurgery dose distribution. Figure 2-10 shows direct DVHs from two hypothetical

radiosurgery plans for a radiosensitive structure.



10 20 30 40 50 60 70 80 90 100
Relative dose

10 20 30 40 50

Relative dose

Figure 2-8: Ideal target (A) and non-target volume (B) direct DVHs. Note that in the
ideal direct DVH of the non-target volume (right side), the plot is empty, since there is no
non-target volume receiving any dose in the ideal case.








0 10 20 30 40 50 60 70 80 90 100
Dose (% of max)

FIB 0.120



S 0.040



10 20 30 40 50 60 70 80 90 100
Dose (% of max)

Figure 2-9: Typical (non-ideal) radiosurgery direct DVHs for target volume (A) and non-
target volume (B).

A 1.0


1 0.6

o 0.4



60 70 80 90 100


n- 2.5




| 0.5

> 0.0

10.0 20.0 30.0 40.0


Dose (% max)

Figure 2-10: Direct DVHs for a radiosensitive non-target structure in two hypothetical
treatment plans

10 20 30 40 50

Dose (% max)

Figure 2-11: Cumulative DVH plot of the direct DVH data shown in Figure 2-10.

Ideal cumulative target and non-target DVHs


1.0 < .0 4 0
2 0.8 -
S--4--Target volume
S0.6 Non-target volume -----
It 0 .4 - -
0.2 -------------- -- --------

0.0 20.0 40.0 60.0 80.0 100
Dose (percent of maximum)



Figure 2-12: Ideal cumulative DVH curve for target and non-target volumes.

The purpose of plotting two DVHs on a single set of axes is to allow a direct comparison

between two or more dose distributions. In the example of Figure 2-10, "Plan 1" and

"Plan 2" are being compared with respect to the radiation dose distribution delivered to

the radiosensitive structure. As can be seen in the figure, it can difficult to evaluate

competing plans using such direct histograms (Drzymala 1991). Above about 40 units of

dose, both plans appear to be identical, but the two plans expose differing volumes of

brainstem at doses less than about 40 units. It is difficult to determine whether one plan

is better than the other from Figure 2-10. Plotting the dose-volume information in the

form of a cumulative DVH makes it simpler to evaluate two similar DVHs against one

another. A cumulative DVH is a plot of the same dose volume information as before, but

with the modification that the y value displayed for each bin x is the volume receiving >

dose x. Peaks in a direct DVH correspond to inflection points on a cumulative DVH

curve. A cumulative DVH plot of the data in Figure 2-10 is shown in Figure 2-11.




Cumulative DVHs will be used throughout the remainder of this report, unless otherwise


The information in DVHs can be used to compare rival treatment plans in many

situations. Optimal DVH curves-for target structures will be as far towards the upper

right hand corner of the plot as possible, while the optimal DVH curves for non-target

structures will be as close as possible to the lower left hand comer of the plot axes as

possible, as shown in Figure 2-12. Thus, one may readily evaluate two rival treatment

plans based upon their DVHs, if the DVH curves for each plan do not intersect, since the

more desirable curve will lie either above and to the right of the other (if it is a DVH

curve of a target volume) or below and to the left if it is a non-target volume DVH curve.

With these rules for evaluating cumulative DVHs, we can use Figure 2-11 to evaluate

Plan 1 and Plan 2 for the radiosensitive structure. Since the curve for Plan 1 always lies

below and to the left of the Plan 2 curve, and the brainstem is a non-target tissue, we can

conclude that Plan 1 is the preferred plan in order to minimize the radiation effects to the

brainstem. The relative ease of this comparison underscores the general utility of

cumulative DVHs (Figure 2-11) over direct DVHs (Figure 2-10) (Lawrence 1996;

Kutcher 1998).

Unfortunately, it is rare for the cumulative DVH curves of rival treatment plans to

separate themselves from one another so cleanly. A more general and common

occurrence in comparing treatment plans is shown in Figure 2-13, in which the DVH

curves cross one another, perhaps more than once. The general rules above for

evaluating DVHs cannot resolve this situation, in which case we must use other means to

evaluate the treatment plans. The next section discusses physical dose volume metrics

for treatment plan comparison.


10 20 30 40
Dose (relative units)

Figure 2-13: Crossing cumulative DVH curves

Physical Dose-volume Figures of Merit
The three properties of radiosurgery and radiotherapy dose distributions which

have been correlated with clinical outcome are dose conformity, dose gradient, and dose

homogeneity (Meeks 1998a). The conformity of the dose distribution to the target

volume may be simply expressed as the ratio of the prescription isodose volume to the

target volume, frequently referred to as the PITV ratio (Shaw 1993):

(2-2) PITV = Prescription isodose volume / target volume.

Perfect conformity of a dose distribution to the target, i.e. PITV = 1.00, implies that the

prescription isodose volume exactly covers the target volume while covering no non-

target tissues. Typically, perfect conformity of the prescription isodose surface to the

target volume is not achievable, and some volume of non-target tissue must be irradiated

to the same dose level as the target, resulting in PITV ratios greater than unity. The most

conformal treatment plans are those with the lowest PITVs, if all of the plans under

comparison provide equivalent target coverage. This stipulation is necessary because the

definition of PITV does not specify how the prescription isodose is determined. It is

possible (but undesirable) to lower, and thus improve, the PITV by selecting an isodose

level which incompletely covers the target as the prescription isodose, and therefore

reduces the numerator of Eq. (2-2). Unless otherwise stated, prescription isodose levels

in the remainder of this report are selected to ensure that > 95% of the target volume

receives the prescription isodose. This ensures a more consistent basis of comparisons

for all treatment plans.

A sharp dose gradient (fall off in dose with respect to distance away from the

target volume) is an important characteristic of radiosurgery and stereotactic radiotherapy

dose distributions. Dose gradient may be characterized by the distance required for the

dose to decrease from a therapeutic (prescription) dose level to one at which no ill effects

are expected (half prescription dose).

Figure 2-14: Transaxial, sagittal, and coronal isodose distributions for five arcs of 100
degrees each delivered with a 30 mm collimator. Isodose lines in each plane increase
from 10% to 90% in 10% increments, as indicated. The :socenter is marked with

'-*--- -- -----------------~-_ -- ----------
Figure 2-15: Dose crossplots through the isocenter, corresponding to the isodose
distributions shown in Figure 2-14. The sharpest dose fall-off, from dose D to half-dose
0.5D, occurs between dose D of 80% to 0.5D = 40%, which occurs in a distance of 4.6
mm. The D to 0.5D fall-off distance is larger for 90-45% (5.1mm) and for 70-35%
(4.9mm) doses.

For illustrative purposes, a typical radiosurgical dose distribution, delivered with

five converging arcs and a 30 mm collimator to a hemispherical water phantom, is

depicted in Figures 2-14 and 2-15. The isodose surfaces in Figure 2-14 are normalized to

the point of maximum dose, such that 100% corresponds to the maximum dose. The

close proximity of the higher (50-90%) isodose lines to one another is a qualitative

measure of the steep dose gradient. A quantitative measure of gradient is obtained from

examining the dose profiles along orthogonal directions in the principal anatomical

planes (transaxial, sagittal, and coronal), as shown by cross-plots in Figure 2-15. This

figure shows the gradient between several dose levels, D, and half of D, in several

directions. The data show that for this single isocenter dose distribution, the steepest

dose gradient (distance between isodose shells of dose D to and 50% of D) occurs

between the 80% and 40% isodose shells, and is 4.6 mm for the 30 mm collimator and

five arcs. The steepest dose gradient is generally achieved between the 80% and 40%

isodose levels, and for this reason single isocenter dose distributions are prescribed to the

80% isodose shell (Meeks 1998c). The dose gradient is relatively independent of

direction (AP, Lateral, and Axial) between about the 90% and 40% isodose shells, since

the dose distribution is almost spherically symmetric between these isodose shells. Table

2-1 lists dose gradient information between the 80% and 40% isodose shells for single

isocenter dose distributions with 10 to 50 mm diameter collimators.

In general, however, radiosurgery dose distributions are not spherically

symmetric, and are tailored to fit the target's shape through manipulation of arc

parameters (Meeks 1998c), multiple isocenters, beam shaping, or intensity-modulation.

Additionally, dose distributions are often manipulated to steepen the dose gradient in the

direction of adjacent radiosensitive structures. This additional complexity makes it

necessary to complement the dose cross-plot with other methods to evaluate the dose


Figure 2-16 illustrates this point with a hypothetical radiosurgery target

shaped like a three-dimensional letter "F", which is covered by a multiple isocenter dose

distribution using 10 mm collimators and five converging arcs at each of eight isocenters.

The 70% isodose shell, which covers the hypothetical target, represents the prescription

isodose and is shown along with the half of prescription isodose (35%), and twenty

percent of prescription isodose (14% = 0.2 x 70%). Unlike the single isocenter,

spherically symmetric dose distribution of Figures 2-14 and 2-15, the multiple isocenter

dose distribution is asymmetric, and the prescription isodose to half-prescription isodose

gradient therefore has a directional and spatial dependence. Depending on where the

dose cross-plot is centered and the direction, the distance between the prescription (70%)

and half-prescription (35%) isodose shells varies from 2 to 7 mm. In order to obtain a

representative sample average gradient distance, it would be necessary to take a large

number of gradient measurements at many points at the target's edge. However, a

method has been proposed which uses easily obtainable DVH information to generate a

numerical measure of the overall dose gradient, and which may be used with arbitrary

dose distributions.


~1- *

Wireframe representation
of 3D target

-4' *-4' --

-- 4- 4
7^ 0,*

Figure 2-16: Irregular "F" shaped target and multiple isocenter dose distribution in
hemispherical water phantom.

The LF Index (gradient) score, or UFIg, has been proposed as a metric for

quantifying dose gradient of a stereotactic treatment plan. From treatment planning

experience at the University of Florida, it has been observed that it is possible to achieve

a dose distribution which decreases from the prescription dose level to half of

prescription dose in a distance of 3 to 4 mm away from the target. Taking this as a guide,

a gradient score UFIg may be computed as

(2-3) UFIg = 100- 00 [(REff,50%Rx REff,Rx) 0.3cm]}

where Refso%Rx is the effective radius of the half-prescription isodose volume, and RefRx

is the effective radius of the prescription isodose volume. The "effective radius" of a

volume is the radius of a sphere of the same volume, so that Reff for a volume V is given


(2-4) R.,

The volumes of the prescription isodose shell and the half prescription isodose

shell are obtained from a DVH of the total volume (or a sizeable volume which

completely encompasses the target volume and a volume which includes all of the half

prescription isodose shell) within the patient image dataset. The UFIg score is a

dimensionless number that exceeds 100 for dose gradients less than 3mm (steeper falloff

from prescription to half-prescription dose level), and which decreases below 100 as a

linear function of the effective distance between the prescription and half-prescription

isodose shells.

Table 2-1 summarizes dose-volume and gradient information for single isocenter

dose distributions delivered with five converging arcs and 10, 20, 30, and 50 mm circular

collimators. UFIg is calculated for each dose distribution using DVH information as

described above. Since the dosegradient for single isocenter arcing dose distributions

(with circular collimators) is achieved between the 80% and 40% isodose shells, the

volumes and effective radii of the 80% and 40% isodose shells are listed, as well as the

difference between these radii. The dose gradient is steepest for the smallest collimators

(about 10 mm diameter) with an effective distance between the 80% and 40% isodose

shells of 2.4 mm and a corresponding UFIg of 106. Dose gradient gradually worsens as

the field size collimatorr size) increases. At a 30 mm diameter field, what many consider

to be the upper limit on radiosurgery target size, the effective dose gradient is about 4.5

mm (UFIg -85).

Table 2-1: Single isocenter (five converging arcs) dose-volume and gradient information
for 10-50 mm circular collimators.
(cm3) (mm) (cm3) (mm) (mm)
Coll. Vsoo. Reff8o% V40% Reff40% Eff. Gradient UFIg
10 0.3 4.2 1.2 6.7 2.4 106
20 3.9 9.8 9.7 13.2 3.5 95
30 13.9 14.9 30.8 19.4 4.5 85
50 67.4 25.2 111.6 29.9 4.6 84

g 0.800 -
j 0.700
c 0.600
z" 0.500 -
c 0.400
| 0.300 -
0.000 -

- Total volume


-_--~ -:

10 20 30 40 50 60 70 80 90 1C
Dose (% of maximum)

Figure 2-17: Target and total volume DVHs for F-shaped target in Figure 2-15.

This methodology can be applied to the dose distribution shown in Figure 2-16.

Figure 2-17 shows the DVHs for the F-shaped target volume and for a large 352 cm3

cubic volume enclosing the region of interest. From this DVH it can be seen that >95%

of the target volume receives >70% of maximum dose, which is necessary to support

selection of 70% as a prescription isodose for this target. The volume receiving 709 o of

the maximum dose is 5.2 cm3 (Rff = 10.8mm), with 22.6 cm3 (R ff= 17.5mm) receiving

35% of maximum dose. The effective dose gradient is therefore 17.5 mm 10.8 mm=

6.7 mm, corresponding to a UFIg = 62.

There is another key piece of dose-volume information contained in this DVH

(Figure 2-17), which bears on the practice of multiple isocenter radiosurgery. Table 2-2

shows the resulting prescription to half-prescription dose gradient resulting from using

various isodose shells as the prescription isodose. The important information in Table 2-



2 is that the steepest dose gradient for most (properly planned) multiple isocenter dose

distributions lies between the 70% and 35% isodose shells, with an effective distance

between the prescription and half-prescription isodose shells of 6.8 mm, corresponding to

a UFIg score of 62. Therefore, in multiple isocenter radiosurgery planning, the planner

should attempt to fit the 70% isodose shell to the target (as opposed to the 80%, 60%, or

other isodose shells) in order to maximize the dose gradient and non-target tissue sparing

(Meeks 1998c).

Table 2-2: Dose gradient variation with selection of prescription isodose shell for
multiple isocenter "F"-shaped dose distribution
Rx isodose Gradient (mm) UFIg
90 8.6 44
80 7.1 59
70 6.8 62
60 7.3 57
50 8.3 47
40 9.7 33
30 10.7 23
20 9.3 37

Dose conformity is another important characteristic of a radiosurgery treatment

plan which should be considered in plan evaluation. A means of con erting dose

conformity in terms of PITV into a conformal index score on a common scale with UFIg

has been proposed, the UFIc score. The UF Index conformall), or UFIc, is defined as

(2-5) UFIc =IO0( Target volume = (PITV)- l00.
Prescription isodose volume

The UFIc converts PITV into a numerical score expressing the degree of conformity of a

dose distribution to the target volume. UFIg score increases as the dose gradient

improves, and the UFIc score increases as dose conformity improves. Perfect conformity

(assuming the target is adequately covered) of the prescription isodose volume to the

target is indicated by a PITV = 1.00 and a UFIc = 100.

As both dose gradient and dose conformity are both important parameters in

judging a stereotactic radiosurgery or radiotherapy plan, an overall figure of merit for

judging radiosurgery plans should incorporate both of these characteristics. Since clinical

data to indicate the relative importance of conformity versus gradient is currently lacking.

an index, the UF Index (UFI) is proposed which assigns equal importance to both of these

factors. The overall UF Index score, or UFI, for a radiosurgery or radiotherapy plan is

the average of the UFIc and UFIg scores (Bova 1999).

Dose homogeneity is considered by some to be an important factor in evaluating

treatment plans. A homogeneous dose distribution throughout the target volume (target

dose within +7% and -5% of the prescribed dose to the target's periphery) is desirable for

conventional, fractionated radiotherapy (Landberg 1993). In radiosurgery, however, the

importance of a homogeneous target dose distribution is less clear. Several studies have

associated large radiosurgical dose heterogeneity (maximum dose to peripheral dose

ratio, or MDPD, > 2.0) with an increased risk of complications (Nedzi 1991; Shaw

1996). However, some radiosurgeons have hypothesized that the statistically significant

correlation between large dose inhomogeneities and complication risk may be associated

with the relatively non-conformal multiple isocenter dose distributions with which some

patients in these studies were treated, and not with dose inhomogeneity alone. One

theory is that the extreme "hot spots" associated with large dose heterogenities may be

acceptable, if the dose distribution is very conformal to the target volume and the hot spot

is contained within the target volume. Non-conformal dose distributions could easily

cause the hot spots to occur outside of the target, greatly increasing the risk of a treatment

complication. The extensive successful experience of gamma unit treatments

administered worldwide (almost all treatments with MDPD > 2.0) lends support to this

hypothesis (Flickenger 1997). Therefore, as a general principle, one strives for a

homogeoneous radiosurgery dose distribution, but this is likely not as important a factor

as conformity of the high dose region to the target volume, or the dose gradient outside of

the target.

As was shown in Tables 2-1 and 2-2, in order to maintain as steep a dose gradient

as possible, the 70% (of maximum dose ) isodose shell is generally used for planning

multiple isocenter treatments, while the 80% isodose shell is used for single isocenter

treatments. An additional benefit of selecting the 70% to 80% isodose shell, rather than

the 50% isodose shell commonly used in gamma unit radiosurgery, as the prescription

isodose is an improvement in treatment efficiency, in terms of the total number of

monitor units which must be delivered. Setting the 50% isodose shell as the prescription

isodose surface rather than 70% would require 1.4 times as many monitor units to be

given to deliver the prescription dose to the target. Also, this would impart a larger

integral dose to the patient in order to deliver the same peripheral target dose. Although

the 70% and 80% prescription isodose levels were chosen based primarily on maintaining

the steepest possible dose gradient, they represent a guideline for acceptable dose

inhomogeneity in linear accelerator radiosurgery dose planning (Meeks 1998c; Meeks


Biological Models
In planning stereotactic radiosurgery (SRS) or stereotactic radiotherapy (SRT)

treatments, the object is to minimize the dose to radiosensitive non-target structures while

covering the target with a conformal and homogenous dose distribution. In multiple

isocenter SRS planning, non-target structures are protected primarily by the steep dose

gradient inherent in stereotactic irradiation. In single isocenter SRS plans, several

techniques (arc start and stop angles, couch angles, and differential collimators) are

generally used to enhance dose conformity and to steepen the dose gradient in the

direction of especially radiosensitive structures, such as the brainstem (Meeks 1998a;

Meeks 1998; Foote 1999). Such treatment plans can be evaluated on the basis of dose

gradient and conformity, which can be determined from dose-volume histograms of the

target and surrounding volumes (Shaw 1993; Bova 1999). When multiple critical

structures are to be spared as part of the optimization process, such as in the problem of

deciding beam orientations in conformal beam SRS and SRT, the treatment plan

evaluation problem can shift away from determining obvious differences in the

conformity and gradient of competing plans. In such cases, biological indices, such as

the normal tissue complication probability (NTCP), may be used to evaluate rival

treatment plans, each of which demonstrates comparable dose gradient and conformity to

the target.

An example of this occurrence is shown in Figures 2-18(a) and 2-18(b), which

depict DVHs for the total intracranial volume ("cubic") and several radiosensitive

structures for two hypothetical radiosurgery plans. An analysis of both sets of target

DVHs (not shown) and total volume DVHs would indicate that both plans cover the

target with similar dose homogeneity (at least 95% of the target receives >69% of

maximum dose for the first plan, and >72% of maximum dose for the second plan) with

very similar dose conformity (PITVs of 1.42 and 1.40) and gradient (UFIg of 76 and 82).

However, the two plans are not equivalent, due to the doses received by the radiosensitive

structures (e.g.-brainstem, and left and right optic nerves). One can see qualitatively that

plan 2 improves (reduces) the overall dose received by these radiosensitive structures,

since the DVH curves for the left and right optic nerve are shifted downward and to the

left in Figure 2-18(b) relative to Figure 2-18(a). However, a quantitative measure of this

effect is desirable. Normal tissue complication probability (NTCP) models have been

proposed as one such quantitative measure.

- unopt-cubic
- unopt-brainstem
..----- unopt-r-optnerve
- unopt-1-optnerve

0 5 10 15 20 25 30 35 40

Dose (% of maximum)

Figure 2-18(a): DVHs for hypothetical radiosurgery plan (Plan 1).

0.9 ... ...... ._ -- opt-cubic
0.8 -*- opt-brainstem
0.7 --.... opt-r-optnerve
0.6--- .. .. -- opt-l-optnerve

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40 0

Dose (o of maximum)

Figure 2-18(b): DVHs for hypothetical radiosurgery plan (Plan 2).

A four-parameter model has been suggested by Lyman (Lyman 1985; Kutcher

1991) as the basis of such an NTCP figure of merit for evaluating subtle differences in

rival SRS/SRT plans. The basic four-parameter model is

1 -t
(2-6) NTCP = j Jexp 4t,
72=r, 2 t


(2-7) t= -TD ,

(2-8) v=-- ,and

(2-9) TD(1) = TD(v). Vn

Here, NTCP represents the probability of complication for an organ of volume Vref

resulting from uniform irradiation of partial volume to homogeneous dose D. TDso is

the tolerance dose for whole organ irradiation at which 50% of the patients receiving this

dose encounter a 50% risk of radiation induced complication within five years after

treatment. The tolerance dose for partial organ volume, v, and the entire organ volume is

given by Eq. (2-9) (Schultheiss 1983). The quantities n and m are fitting parameters

which govern the volume and dose dependence of the NTCP model. Quantity t is a

parameterization of the number of standard deviations separating the partial volume v at

dose D from TDso. TDso, m, and n model parameters used for comparisons were taken

from curve-fits (Burman 1991) to dose-volume tolerance data in the literature (Emami

1991). Since dose distributions in SRS/SRT are almost neN er perfectly homogeneous in

the region of interest, the four-parameter model cannot be used directly as presented.

Kutcher's method (Kutcher 1991) is used to reduce each non-uniform dose volume

histogram to an equivalent volume DVH receiving the maximum dose level. This

method involves treatment of each small dose bin in the differential dose volume

histogram as a volume receiving-a uniform dose Di, which is a reasonable assumption if

the dose bin size is small enough. "Small enough" means practically using dose bins no

larger than about 2 Gy each (Kutcher 1991). The effective volume, Veff, receiving the

maximum dose Di, is found by converting the volume in each differential DVH dose bin

to an effective volume Vefi, and summing them:

(2-10) Ve = V,. D--
\ Dm

The source data cited above is applicable for NTCP calculations involving

fractionated radiotherapy under typical regimes of about 1.8-2.0 Gy per fraction on a five

day per week treatment schedule. These model parameters must be adjusted in order to

use this four parameter model to calculate NTCP for single dose radiosurgery cases. The

biologically equivalent dose formalism (BED) may be applied to make this modification

(Fowler 1989; Smith 1998), in which a BED may be calculated from any particular

fractionation scheme delivering dose D in fractions of dose d by

(2-11) BED= D. .I
a P

The BED represents a biologically effective dose for tissues with an a/p ratio of

(o/p) when delivered in fractions of dose size d. In this relation, ct and P are the

coefficients in the linear-quadratic cell survival curve (Hall 1994a; Hall 1994b). To

gauge the biological effect of two-different doses, Di and D2, given in individual fraction

doses of di and d2, respectively, one would calculate and compare the BEDs calculated

for Dl and dl, and for D2 and d2 using equation (2-11). Unit analysis of Eq. (2-11) shows

that BED has units of Gray (Gy), although to indicate that the quantity BED is

biologically effective dose rather than a physical absorbed dose, BEDs are usually

subscripted with their a/P ratio, e.g. Gy2. For the purposes of comparing rival treatment

plans, 2.0 is an acceptable default a/p ratio for normal brain and nervous tissue (Smith

1998). To use the NTCP models in equations (2-6) through (2-9) with single fraction

SRS dose distributions, the volume element in each SRS DVH dose bin must be

transformed into a biologically equivalent dose using equation (2-11), and each organ's

tolerance dose (TDso, in units of Gy) must be transformed into a biologically equivalent

single fraction BETDso. Table 2-3 summarizes NTCP model data taken from Burman

(Burman 1991) and Emani (Emami 1991) for intracranial anatomy. The radiosurgery

BED for each organ's TDso is calculated for a fractionation schedule of 2 Gy per fraction

and an a/P ratio of 2.0, in units of Gy2.

Table 2-3: NTCP model data for intracranial sites
n m (Gy)
Brain (a)-_ 0.25 0.15 60
Brainstem ta) 0.16 0.14 65

Brainstem (b) 0.04 0.15 65
Lens (a) 0.3 0.27 18
Optic nerve (a) 0.25 0.14 65
Source:(a) (Burman 1991), (b) (Meeks 2000)

Although these models and data represent a commonly accepted method for

modeling the biological response of tissues to irradiation, the data used to fit the model

parameters remain sparse and somewhat uncertain (Zaider 1999). For the intracranial

anatomical sites listed in Table 2-3, the brain is the organ with the greatest amount of

clinical data, a total of six data points. The lens and optic nerve models are fitted for only

two data points corresponding to 5% and 50% complication probabilities for irradiation

of each entire organ. Thus, computing NTCP values with the four-parameter model is

possible, but even under the whole organ irradiation conditions under which the model

was created, significant interpolation between clinically observed data points is

necessary. For conditions of partial organ irradiation, calculation of NTCP values would

involve significant extrapolation beyond observed data (Burman 1991). For these

reasons, complication probabilities calculated by this means are intended to serve only as

a guide for evaluating rival treatment plans, and not for use as absolute probabilities of

complication (Kutcher 1996). A previous study of such simple biological models has


shown that it is possible to use them in the ranking of rival treatment plans, despite

uncertainties in the model parameters (Kutcher 1991).


This chapter presents the methods used in planning optimized single and multiple

isocenter radiosurgery treatments with circular collimators. Single isocenter LINAC

radiosurgery with evenly spaced arcs produces essentially spherical dose distributions at

high isodose levels (70-80%). To avoid covering excessive volumes of non-target tissue

to the prescription dose, however, the high dose region must be shaped to fit individual

targets. The high dose region near a single isocenter can be manipulated (within limits)

into a variety of ellipsoidal shapes that closely conform to a target's shape. For

irregularly shaped lesions, and those with an ellipsoidal shape in the transaxial plane,

however, multiple isocenters must be used to obtain a conformal dose distribution for

radiosurgery (Friedman 1998; Meeks 1998c).

The University of Florida radiosurgery planning algorithm, shown in Figure 3-1,

(Friedman 1998; Meeks 1998c) provides the basis for the treatment planning methods

outlined in this chapter. This algorithm organizes the tools (optimization variables)

available to the radiosurgery planner to efficiently generate conformal radiosurgery plans

that provide appropriate sparing of non-target tissues. The first step of the algorithm is to

determine whether the targeted lesion is adjacent to a radiosensitive structure. If so,

single isocenter arc parameters (presented in a later section) are adjusted to steepen the

dose gradient in the direction of the radiosensitive structure, if possible. If the lesion is

very irregular in shape or is an ellipsoid with the major axis aligned along the anterior-

posterior direction, multiple isocenters are used to conform the dose distribution to the

shape of the lesion. Due to the difficulty of standardizing this multiple isocenter planning

process, the major emphasis in this chapter is on a geometrically based and automated

method (sphere packing, developed in a joint project with the University of Florida

Department of Mathematics) which attempts to generate conformal multiple isocenter

dose distributions.

University of Florida
Treatment Planning Algorithm for Optimizaeton

IThe d' t a p cenP to a single, stationar ea tat is saed it ura

/ ,
circular collimator of diameter "coll" is computed at any point in the stereotactic space by,
\C3 l:cii i 3rucgture nrtte Cr sci nr^,r m^ i, ..*.:.-l ^R3CC *

(3-1) Dose(P) = k MU TPR(coll,depth(P)) OAR(P) Sc.p-col dists-P
ilie"3: r; 'L1'A 5. lirfb' ?nLI Ir '! '

/ .

S,1 3 ':*"C r i :C<;
j A;xi: C:i e t *' i"..:So

OosO S ieC', ion

Figure 3-1: University of Florida radiosurgery planning algorithm

Circular Collimator SRS Dosimetry
The dose at any point P due to a single, stationary beam that is shaped with a

circular collimator of diameter "coll" is computed at any point in the stereotactic space by

(3-1) Dose(P) = k MU. TPR(coll, depth(P)). OAR(P). S .p_*on. dists-p
C dists-cp,)

as sho\%n in Figure 3-2. In Eq. (3-1), k is the treatment machine's calibration constant,

normally 0.01 Gy MU, MU is the number of monitor units delivered with the beam, dists-

cp is the source to calibration point distance (nominally 100cm for an isocentric machine

calibration), dists.p is the distance from the source to point P, TPR(coll, depth(P)) is the

tissue-phantom ratio for the circular collimator being used at the depth of point P, S,p-co1l

is the total scatter factor, or output factor, for the circular collimator being used, and

OAR(P) is an off-axis ratio representing the variation of dose away from the field's

central axis (Khan 1994; Surgical Navigation Technologies 1996; Duggan 1998). Data

tables of measured TPR and Sc, values for all circular collimators in use are maintained

and directly used by the treatment planning system for dose calculation at each point.

Dose distributions can be determined by a) computing the dose to each point (from each

beam) in a grid of points in the viewing planes selected by the user, or b) to a grid of

points in a three dimensional region. Each arc of radiation, formed by rotation of the

gantry about the patient with the radiation beam on, is accurately modeled as a series of

stationary beams spaced approximately 10 degrees apart. Thus, a 100-degree arc is

approximated by spacing eleven beams 10 degrees apart (Figure 2-6). The dose

algorithm assumes the patient to be water equivalent, and each beam to be perpendicular

to the patient's surface. Determination of the patient's surface and the depths for each

central axis dose point is derived from the three-dimensional stereotactic computed

tomography (CT) image set of the patient. The dose calculation falls into the "three

dimensional imaging, one dimensional dose calculation" classification discussed in

Chapter 2. Although somewhat simple, this process is rapid and provides sufficient

accuracy for radiosurgery dose calculations (Schell 1995).



t \



Single Isocenter Treatment Planning
A single isocenter with multiple converging arcs may be used to create a spherical

dose distribution close in size to the diameter of the circular collimator. This type of

treatment produces a conformal dose distribution for spherical or near-spherically shaped

targets. The standard set of fourteen circular collimators used at the University of Florida

covers a range of sizes from 5mm to 40mm (5, 10, 12, 14, ..., 30, 35, and 40 mm )

diameter, projected at 100cm from the radiation source, allowing the planner to closely

match the diameter of the dose distribution to the target. As discussed in Chapter 2,

because the steepest dose gradient for single isocenter dose distributions lies between the

80% and 40% isodose shells, a collimator size should be chosen which covers the target

with the 80% isodose shell. This will ensure the steepest possible dose gradient between

the prescription and half-prescription isodose shells. A standard set of nine convenrent
t \
Surface / \

__ __OAP ....

Figure 3-2: Radiosurgery beam dose calculation for dose at point P.

Single Isocenter Treatment Planning
A single isocenter with multiple converging arcs may be used to create a spherical

dose distribution close in size to the diameter of the circular collimator. This type of

treatment produces a conformal dose distribution for spherical or near-spherically shaped

targets. The standard set of fourteen circular collimators used at the University of Florida

covers a range of sizes from 5mm to 40mm (5, 10, 12, 14, ..., 30, 35, and 40 mm )

diameter, projected at 100cm from the radiation source, allowing the planner to closely

match the diameter of the dose distribution to the target. As discussed in Chapter 2,

because the steepest dose gradient for single isocenter dose distributions lies between the

80% and 40% isodose shells, a collimator size should be chosen which covers the target

with the 80% isodose shell. This will ensure the steepest possible dose gradient between

the prescription and half-prescription isodose shells. A standard set of nine convergent

arcs (specific parameters listed in Table 3-1, in terms of International Electrotechnical

Commission (IEC) couch and gantry angles (IEC 1996)), is shown in a perspective view

in Figure 3-2, and is generally used as a basis for generating single isocenter dose

distributions. Couch and gantry angles are illustrated in Figures 3-3 through 3-6. An AP

xiew of a standard nine-arc set, delivered to an approximately spherical target with an

eighteen-millimeter collimator, is shown in Figure 3-7 along with the resultant isodose

distribution. Each arc is weighted equally with respect to dose to isocenter. The couch

angles are chosen to approximate an even and symmetrical beam distribution over the 2x

steradian solid angle above the patients head, while avoiding parallel opposed beams

which would adversely affect the dose gradient (Meeks 1998a; Meeks 1998b; Meeks


Table 3-1: Couch and gantry angles for standard University of Florida nine arc set.
(Angles are in accordance with IEC standards).
Gantry Gantry
Couch Start Stop
10 130 30
30 130 30
50 130 30
70 130 30
350 230 330
330 230 330
310 230 330
290 230 330
270 230 330

as a starting point for single isocenter radiosurgery plans. Each equally weighted arc

4- .4..-

spans 100 degrees of gantry rotation at one of nine couch angles.

Gantry 0 degrees

Couch rotation

Couch 0 degrees

Figure 3-4: Schematic depiction of couch and linac gantry angles. The linear accelerator
couch and gantry are positioned in the "home" position, couch at 0 degrees, and gantry at
0 degrees. Couch and gantry angles refer to the amount of clockwise rotation as shown
in the figure.

Gantry 30

Couch 55

Gantry 130

Figure 3-5: Linac couch (rotated clockwise) at 55 degrees, and gantry arcing between 30
and 130 degrees.

Gantry 330

s Couch

Figure 3-6: Linac couch at 305 degrees, and gantry arcing between 230 and 330 degrees.

Figure 3-7: Standard nine-arc set delivered with 18mm collimator, and isodose
distribution in axial, sagittal, and coronal planes. The 80%, 40%, and 16% isodose lines
are shown in each plane. The inset at lower left shows an AP view of a patient's head,
with an overlay of the linac couch angles corresponding to each arc.

The standard nine arc set is well suited for conforming the high dose region to the

target, if the target is spherically shaped. However, in the case of an ellipsoidal shaped

target, or of an adjacent critical (radiosensitive) structure, it may be necessary to alter the

shape and gradient of the dose-distribution to improve dose conformity and gradient. The

University of Florida dose-planning algorithm (Figure 3-1) guides the selection of

appropriate isocenter arc parameters to manipulate to obtain optimal dose conformity and

gradient. The nine arcs in the standard set may be manipulated to change the shape of the

high (80%) isodose shell from a spherical shape to an ellipsoidal shape with the major

axis inclined in the sagittal or coronal planes.

The "arc elimination" tool or technique may be used to steepen the dose gradient

in a lateral or axial (along the cranial-caudal axis) direction. This is accomplished by

eliminating arcs that are aligned in the direction along which a steeper dose gradient is

needed. Figure 3-8 shows how elimination of the lateral arcs changes the overall dose

distribution, causing a steeper dose gradient laterally from the isocenter, and making the

dose gradient less steep in the inferior/superior direction. This technique is appropriate to

protect a radiosensitive structure that lies medial or lateral to the target at isocenter.

Figure 3-9 shows an application of this technique to steepen the dose gradient in an

oblique direction. Elimination of the most superior arcs would likewise steepen the

superior-inferior dose gradient, at the expense of a less steep lateral dose gradient. The

arc elimination tool allows the planner to selectively steepen the dose gradient in the

coronal plane.

The dose gradient may be altered in the sagittal plane by altering the start and stop

angles of each arc, and/or by altering the span of each nominal 100 degree arc. Figure 3-

10 shows the effect on the sagittal isodose distribution of shortening each arc by

removing the posterior 40-degree portion of each arc, which shortens each arc from 100

degrees to 60 degrees. The overall dose distribution tends to follow the directional

alignment of most of the beams in each arc. Use of this planning tool allows the dose

gradient to be steepened to protect critical structures lying anterior or posterior to the


In addition to protecting radiosensitive structures near the target, the arc

elimination and arc start/stop angle tools allow changing the shape of the high dose

region from a sphere to an ellipsoidal shape, which can improve the conformity of the

high dose region to the target if the target is an ellipsoid with the major axis in the sagittal

or coronal planes. However, if the target is an ellipsoid with the major axis aligned in the

transaxial plane, or if the target is irregularly shaped, multiple isocenters may be required

to achieve a dose distribution that conforms to the shape of the target.




Figure 3-8: Steepening the dose gradient in the lateral direction by elimination of the
most lateral four arcs from a standard nine-arc set. The 80%, 40%, and 16% isodose
lines are shown in each plane.


Figure 3-9: Rotating the distribution and steepening the dose gradient in an oblique
direction by elimination of four arcs entering from the patient's right side. The 80%,
40%, and 16% isodose lines are shown in each plane.

80%-40%-16% isodoses

Standard arcs: Zach arc 60 degrees
Zach arc 100 degr esI

Figure 3-10: Tilting the dose gradient in the sagittal plane by shortening each standard
100-degree arc (left side) to 60 degrees (right side).

lane views:

Multiple Isocenter Radiosurgery Planning Tools
Multiple spherical dose distributions may be placed adjacent to one another to

build up a composite dose distribution which conforms to the shape of an irregular target.

as was shown for the "F" shaped target in Chapter 2. When using multiple isocenters,

typically five arcs (Figures 3-11, 3-12) rather than nine arcs (Figures 3-3, 3-7) are used

with each isocenter, since the dose distribution from five arcs is very similar to that from

nine arcs, and less time is required to deliver five arcs than nine arcs. Couch and gantry

angles for the standard five arcs used in multiple isocenter planning at the University of

Florida are listed in Table 3-2, and are depicted in Figures 3-11 and 3-12. Figure 3-11

shows the resultant dose distribution from a standard five-arc set, which is very similar to

the standard nine arc dose distribution shown in Figure 3-7.



Figure 3-11: Standard five-arc set delivered with 18mm collimator, and isodose
distribution in axial, sagittal, and coronal planes. The 80%, 40%, and 16% isodose lines
are shown in each plane. The 70%, 35%, and 14% lines (not shown) are very close to the
80%, 40%, and 16% isodose lines.

Figure 3-12: AP superior-oblique view of the standard five arc set generally used for each
isocenter in multiple isocenter plans.

Table 3-2: Couch and gantry angles for standard University of Florida five arc set.
Couch Start Stop




Three factors strongly affect the dose distribution when using multiple isocenters:

1) collimator size, 2) inter-isocenter spacing, and 3) isocenter weighting. Collimator size

is chosen to match the region of the target which is being covered, and affects the

diameter of the spherical high dose region that is produced by each isocenter. Proper

selection of collimator size and isocenter location is a complex topic, which is addressed

in the next section, while the issues of isocenter spacing and weighting are discussed


The effects of isocenter spacing on the overall dose distribution may be seen in

Figure 3-13, which shows 50% and 70% isodose curves in an axial plane for two equally-

weighted isocenters at several inter-isocenter spacings, each with a standard five arc set

delivered with a 30mm collimator. For this discussion, it is helpful to consider each

isocenter as a solid, 30 mm sphere, corresponding approximately to the 70% isodose

surface of a five arc set. As a first approximation, one would expect a sphere separation

of about 30mm (the sum of the radii of each sphere) to be correct. As will be shown, this

is approximately correct, but slightly more separation is optimal.

The 70% volume in the dose distributions shown in Figure 3-13 correspond

approximately to the geometrical coverage of a 30 mm diameter sphere placed at each

isocenter. At an isocenter spacing of 40 mm, the 70% volume is slightly greater than the

sum of two 30 mm spheres, and the 50% volume (outer isodose line) is slightly larger.

The "waist" between the 70% isodose lines is so pronounced that the 70% isodose shells

are actually separated from one another. As the isocenters are moved closer together, the

70% isodose shell more strongly resembles two 30 mm diameter spheres. At about 32-33

mm inter-isocenter spacing, the isodose distributions are about ideal. As the isocenters

are moved closer to one another for distances less than about 32 mm, the 70% isodose

volume contracts dramatically. This is because each dose distribution is renormalized to

1000 o at the point of maximum dose, so that as the hotspot where the two spheres overlap

one another becomes more intense, the volume covered by 700o of this increasing

maximum dose becomes smaller and smaller. This can be seen by the rapid decrease in

size of the middle 70% isodose region, from 59 mm across with a spacing of 33mm,

down to a region only 17 mm across when the inter-isocenter spacing is reduced to 24

mm. The 70% isodose region shrinks to less than a third of its initial size, while the 509 o

isodose region shrinks much-more gradually from 66 mm to 53 mm (a 20% decrease).

For this example of two 30mm isocenters, the 70% isodose shell may be approximated

as two 30mm spheres, if an inter-isocenter spacing of at least 32mm is maintained.

To assist the human radiosurgery planner in maintaining this appropriate spacing

between isocenters, a table of empirically determined optimal inter-isocenter distances is

incorporated into the University of Florida radiosurgery treatment planning system (Foote

1999). The planner enters the table with the collimator sizes for two adjacent isocenters,

and the table returns the optimal inter-isocenter distance for these two collimator sizes.

This planning tool merely serves as an aid to provide recommended isocenter spacing,

and does not directly alter any treatment plan parameters. According to this isocenter

spacing table, the optimal spacing distance for two 30mm isocenters is 31mm. Inspection

of the isodose distributions for a variety of collimator sizes and spacings, as was done in

Figure 3-9 for two 30mm collimators, shows that for two isocenters with collimator

diameters of dl and d2, an approximate spacing of 0.52(dl + d2) to 0.60(dl + d2) will

yield an overall dose distribution similar in shape to two spheres of diameters dl and d2.

Isocenter weighting is another important aspect of multiple isocenter

treatment planning. When planning radiosurgery treatments with multiple isocenters and

when the isodose distribution is normalized to maximum dose, care must be taken to

consider the additive dose from all isocenters. Examples of this are shown in Figure 3-

14(a), 3-14(b), and 3-14(c). Figure 3-14(a) shows a dose profile (cross plot) through four

optimally spaced isocenters (each separated from the others by 13.8 mm), each with an

equally weighted five arc set and a 14 mm collimator (the relative dose weight of each

isocenter is 1:1:1:1). The individual dose profile for each isocenter is shown, along with

the total dose distribution along the crossplot from all four isocenters. The central

regions of the distribution near the two isocenters in the middle of the distribution receive

140o more dose than do the two isocenters at the edges. This is because each of the

middle isocenters receives the dose from its own five arcs, and also receives a substantial

contribution from each of its neighboring isocenters as well. The desired situation is to

have equal doses at each isocenter, rather than equal weighting of the arcs associated with

each isocenter. In order to compensate for the increased dose to the middle of the total

distribution, it is necessary to decrease the weight of the two isocenters in the middle.

Figure 3-14(b) shows the individual and total dose distribution after the isocenter dose

weighting has been adjusted to 1.17 : 0.94 : 0.94 : 1.17. Making this adjustment causes a

uniform dose to be received by each of the four isocenters. Although the total dose

distribution is still somewhat heterogeneous, it is actually more homogeneous than the

total dose distribution in Figure 3-14(a). This is shown in Figure 3-14(c), which shows

the total dose distribution for both situations. The overall dose distribution after adjusting

the weights is more homogeneous, in that the volume of the 70% isodose surface has

been increased, and the "hot" volume (hotspot) receiving more than 90% of maximum

dose has been reduced. Also, note that the prescription to half-prescription isodose

gradient is steeper for the adjusted weights distribution. This can be seen in Figure 3-15,

which shows the axial isodose distribution for both the equally weighted and the adjusted

weights plans. The 35% isodose shell is almost identical between the two plans, but

since the 70% isodose shell is larger for the adjusted weights plan, it is closer to the 35%

isodose shell, and offers a steeper dose fall off. In most radiosurgery planning situations,

the same advantage holds for adjusting the isocenter weights in order to improve the dose

homogeneity, and gradient, around the target volume. An automatic weighting tool to

perform this task has been implemented in the University of Florida treatment planning

system which iteratively adjusts the arc weights associated with each isocenter to achieve

a uniform dose to each isocenter (Foote 1999).

40 mm spacing

35 mm spacing

33 mm spacing

32 mm spacing

31 mm spacing

29 mm spacing

26 mm spacing

24 mm spacing

Figure 3-13: Effects of isocenter spacing on the multiple isocenter dose distribution. The
70% and 50% isodose lines are shown in a transaxial plane for two equally weighted
30mm isocenters, each with a five arc set.


Four 14mm isocenters spaced at 14mm, all weights equal

1.6 ------- --------. ----- ------ --- --- -------
1. Total dose
1.4 ---------------------- --- ---------------------

1.2 --------- -------- ---------------- -------- --------

M 1 ......... ........ .... ....- -... .......... ...

0.1 --------- --------
0.4 ------------ ---------- --
O H i I ,
I H I i I

0.2 ------------- ----- .--------------

0 25 50 75 100 125 150
Lateral distance (mm)

Figure 3-14(a): Dose profile through four optimally spaced 14mm isocenters,
each with an equally weighted five arc set. The dose profile for each
isocenter and the total combined dose profile are shown.

Four 14mm isocenters spaced at 14mm. isoc. weights adjusted 1.17 : 0.94 : 0.94 1.17

1.4 .--------- -------- .--. ------ ------- -------- ---------

1.4 --------- ---------- ------ -- -------- --------
0 -'-

1.2 -- --.. ---.--- +---

4 1 -

08 --------- --- -- ---------- ----- ------------ ---
Figure 3-14(b): Dose profile through four optimally spaced 14mm
0 ------------ -i-- ------- ----- ---- -------- -----------

0. ............. -- -........ .. ...........

0 25 50 75 100 125 150
Lateral distance (mm)

Figure 3-14(b): Dose profile through four optimally spaced 14mm

isocenters. The dose profile for each isocenter and the total combined
dose profile are shown.
i ----;--- .... -- ,

0.2 ......... ---- --- I -- I ...............

dose profile are shown.

Four 14mm isocenters spaced at 14mm, total normalized dose,adj(solid),unadj(dashed)
1 r I /-


Adjusted weights


cr 0.4

25 50 75 100
Lateral distance (mm)

Figure 3-14(c): Total

dose from four isocenters for the plans shown in Figures 3-9(a) and

Figure 3-15: Axial plane dose distribution (70%, 35%, and 14% isodose lines shown) for
four 14 mm isocenters. (A) all weights equal, (B) weights adjusted to obtain equal
isocenter doses.

44.7 mm \ Unadjusted (equal)
44.7mm V weights

- I

/I \

Multiple Isocenter Radiosurgery Planning via Sphere Packing
In a simple manner, multiple isocenter radiosurgery planning may be considered

as the problem of determining the positions and sizes of the multiple spherical high dose

regions or isocenters which will be used to fill up the target volume, or put another way,

of determining the sphere-packing arrangement with which to fill the target volume.

Conventional radiosurgery optimization schema are generally iteratively based,

dosimetrically driven algorithms. They require many computations in order to compute a

radiosurgical plan dose distribution, and then to evaluate the quality of the dose

distribution. Geometrically based radiosurgery optimization has been suggested as a

possible alternative means of optimizing radiosurgery treatment planning, since

geometrical solutions are generally much less computationally expensive than the large

iterative set of dosimetric calculations required for most other optimization strategies

(Wu 1996; Bourland 1997; Wu 1999). For instance, a high isodose region around a

single isocenter may be approximated by a sphere of a diameter approxirrately equal to

that of the circular collimator used. Given such a sphere's location and diameter in

stereotactic space, it is much easier to describe this sphere's spatial relationship to the

target volume than it is to compute a three dimensional dose distribution, and to then find

the relationship of this dose distribution to the target's volume.

Wu et al (Wu 1996; Wu 1999) proposed a geometrically-based sphere packing

optimization method for automated gamma unit radiosurgery, in which the shot

(isocenter) locations and sizes are selected according only to the target's three

dimensional shape. Grandjean (Grandjean 1997) et al report on their implementation of a

similar volume packing process for linac radiosurgery, but one in which ellipsoids as well

as spheres are graphically placed by a human user in a three dimensional representation

of the target. Both of these methods are similar, in that isocenter or shot placement is

based simply on obtaining the best geometrical agreement between the target's shape and

the shape of the high dose region characteristic of the treatment unit (i.e. a sphere or

ellipsoid). Due to non-geometric constraints imposed by the physics of radiation

dosimetry (e.g. due to dose interactions and contributions between neighboring

isocenters), multiple isocenter radiosurgery planning is not exactly a sphere-packing

problem. However, in many cases, a sphere-packing arrangement will translate into a

satisfactory radiosurgery plan, particularly if simple dosimetric adjustments are made to

the automatically generated plan (i.e. use of the isocenter weighting tool discussed in

the previous section).

An alternative sphere packing method is presented in this section that shows

potential to significantly aid the planning of complex, multiple isocenter cases. Based on

tests with irregularly shaped phantom targets and with a representative sampling of

clinical example cases, the method demonstrates the ability to generate radiosurgery

plans comparable to or of better quality than multiple isocenter linac radiosurgery plans

found in the literature.

The major steps of the sphere packing process are diagrammed in Figure 3-16. A

7.6 cm3 phantom target, similar in shape to a large acoustic neuroma, is shown in Figures

3-17 and 3-18 with its sphere packing arrangement, and will be used to illustrate the


Step 1: Read in target volume contours. Target volume information is obtained by

manually contouring the target on successive transaxial image slices in the University of

Florida radiosurgery treatment planning system. Code written in the MATLAB

language (Matlab v5.1, The Mathworks Inc., Natick, MA) processes the target contours

data, and computes the sphere packing arrangements.

Step 2: Map target points into 3D array. Each point identified from the target

contour data file is mapped into a three dimensional integer valued array. Each voxel, or

array element, corresponding to a target point is set to a value of 1. After mapping each

given target point into an array element, the program closes any gaps between "1" voxel

elements in the array, ensuring that the contour in each plane is a closed, continuous


Step 3: Build solid voxelized model. A fill routine assigns the voxels inside each

contour with values of 1, resulting in a solid, connected array of one-valued voxels

corresponding to the target volume, and zero-valued voxels outside the target (Figure 3-

17a). A default voxel size of 1 x 1 x 1 mm3 was used for this study, although this process

is general and may be applied to any voxel size.


T .

Build 30 voxel
model of target

-* procedure

Evaluate score for
-- maximum valued

Select voxel with
-- maximum score as
seed voxel

Locate max. score
in neighborhood of
seed voxel

Reset non-covered
target voxels to "1"


Place sphere


Figure 3-16: Block diagram of sphere packing process

a) b)

c) d) e) /

f) g) h)


Figure 3-17: Major steps of the grassfire and sphere packing process for a phantom target
(phantom target number three), shown in a coronal plane. a) Voxelized model of the
target, constructed from axial contours, b) Solid model after application of the grassfire
process in 3D. Voxel intensity (color) is a function of each voxel's value after the
grassfire process. c) First isocenter, 22 mm diameter, placed at best-scoring voxel, d)
The voxels inside the sphere are effectively removed for purposes of the grassfire
process. Note the change in the voxel values (color) near the target borders, at the arrow.
e) The situation after application of grassfire process. The deepest voxels are now
identified as candidate isocenter locations for the second isocenter. f) Placement of the
second isocenter. g) Voxels inside the second sphere are effectively removed. h)
Applying the grassfire process after the situation in g). Arrows indicate locations where
voxel values have changed.

Step 4: Grassfire. The outermost layer of voxels in the target model is then

identified and removed, and the process repeated until the deepest lying voxels have been

identified. This peeling and layering, or shelling, by a grassfiree" algorithm, is so called

due to the analogy of burning-off one outer layer of the target at a time, as in a fire (Blum

1973). This edge detection process identifies the outermost layer of voxels in the 3D

model, and adds the integer 1 to each outer layer voxel's value, converting all outer layer

voxels from 1 values to 2. Voxels lying one layer deeper inside the target are easily

identified as the set of "" valued voxels which are adjacent to "2" valued voxels. These

voxels one layer deeper than 2 are assigned a value of 2 + 1 = 3, with the algorithm

continuing application of this process until all 1-valued voxels have been assigned a layer

value, with the deepest lying voxels having the largest values (Figure 3-17b). Ideally, the

deepest lying voxel in the entire target volume would thus be quickly identified as the

best location for an isocenter. Such a maximum-valued voxel's value should also indicate

the size of the sphere to be placed there as well, since (layer number minus 1 ) should

indicate approximately the depth from the surface (in units of voxel size). For example, a

maximum-valued voxel with a value of 7 should lie (7-1) = 6 voxels from the surface of

the target, which suggests that a sphere 12 voxels in diameter would be required to cover

this volume.

a) b) c)



d) e) f)

Figure 3-18: Three-dimensional depiction of the example phantom target and spheres
placed by the sphere packing algorithm, a) Target volume b) Sphere packing
arrangement for five-isocenter plan c) Target volume superimposed on sphere packing
arrangement d) Prescription isodose surface (64% of maximum dose) superimposed on
sphere packing arrangement e) Prescription isodose surface superimposed over target
volume. The prescription isodose surface covers the target with the exception of isolated
'clipped" edge voxels (see arrow), f) Superposition of target volume, sphere packing
arrangement, and prescription isodose surface.

Step 5: Identify the optimal isocenter size and position (location). Ideally, a

shelling process would easily identify the deepest-lying region of the target volume as a

single voxel. However, the process described determines only an approximation to the

various layers of the target from the outside in, with each cubical voxel representing a

differential radial volume element. For the small intracranial target sizes and the 1 x 1 x

1 mm3 voxel sizes used, the discretization of the 3D target model generally does not

result in a unique identification of the deepest lying voxel. Hence, often more than one

voxel is identified as belonging to the deepest layer. In the example case we have been

following, the first application of grassfire process identifies 7 voxels with the maximum

value of 8, some of which are shown in Figure 3-17b. Interpreting the distribution of

these maximum valued voxels may be difficult, as these voxels do not always lie together

in one group. Even when the maximum valued voxels form a simple connected group,

simply taking the centroid of all such voxels will not necessarily yield the optimum

sphere location. This occurrence of multiple maximum valued voxels is especially a

problem after one or more spheres have been placed in the target volume, and many

target voxels lie at or near the surface of the target or another sphere. Using smaller

voxels does not necessarily result in unique identification of the deepest voxels, either,

but does result in much longer computation times by increasing the number of voxels

which must be processed. To resolve the ambiguity of multiple voxels apparently lying

in the same depth, a score function was used to further distinguish the maximum valued

voxels from one another.

An additional benefit of using a score function to rank candidate isocenter

locations was realized, in that a score function easily allows other factors to be

considered other than depth from the target's surface. For instance, it is often possible

and preferable to use a larger diameter sphere to cover a greater volume of target, at the

expense of covering a small volume of non-target tissue. This is particularly true of the

first isocenter, if multiple isorenters are to be used to conform the dose distribution to the

target. Renormalizing a multiple isocenter dose distribution to maximum dose causes

isodose constriction as the magnitude of the maximum dose is changed by the addition of

subsequent isocenters. The use of a score function allows the algorithm to take this factor

into consideration when attempting to optimize sphere placement. Other factors may be

considered as well, such as inter-isocenter distances.

The score function is computed at each maximum valued voxel location, for each

possible circular collimator (sphere) size, and the best-scoring (numerically largest score)

voxel from the list presented to the user. The score function is the product of several

independent factors: fl, fractional target coverage; f2, a penalty factor which is a function

of the volume of non-target tissue covered; and a third factor, f3, a function of all inter-

isocenter distances. In equation form, for a sphere k at a particular position, these

relations are

(3-2) Score = fi x f2 x f3,


Target volume covered by sphere
(3-3) f -= w, *
total target volume

volume of normal tissue covered by sphere
(3-4) f2 = e total target volume

(3-5) f; = f[dist(i,k)],


0, if dist(i,k) < 0.9dop,
(3-6) f[dist(i,k) i k] =- -- dist(i,k) 9, if 0.9dop < dist(i,k) < d ,
1.0, if dist(i,k) > dopt

and dist(i,k) is the distance between isocenters i and k.

Factor fi is the fraction of target coverage for a sphere of a specified size at the

voxel under consideration, which varies from 0.0 to 1.0. Factor f2 is a normal tissue

penalty function, so that f2 = 1.0 if no normal tissue volume is covered, and f2 0.0 as

increasing volumes of normal tissue are covered. Factor f3 is a function of the distance of

each isocenter to all other isocenters. Factor f3 serves to prevent placing spheres

(isocenters) too closely to one another, which results in excessive target dose

heterogeneity (Meeks 1998c). This isocenter to isocenter distance function is zero for

isocenter-isocenter distances less than 0.9dopt (dopt = empirically determined optimal

isocenter-isocenter distance for the two spheres under consideration, implemented in the

form of a look-up table accessible to the code (Foote 1999)), is unity for isocenter-

isocenter distances greater than or equal to the optimal distance, and varies linearly

between zero and unity for distances between 0.9 dopt and dopt. Terms w1 and w2 are

relative weighting factors, with which the user may control the behavior of the algorithm.

For example, an "aggressive" setting (relatively small penalty for normal tissue over-

coverage) can be chosen by decreasing the relative weight of w:. A conventional sphere

packing (the target volume is filled with non-intersecting spheres which do not extend

outside of the target volume) results when w2 -4 c, so that f2 = 0 if any normal tissue

voxels are covered by spheres, and f3 0 if any of the spheres are too close to each

other. For all of our work, wl =1 by default, with w2 being the only adjustable variable in

the score function. Although it could also be considered a variable, the inter-isocenter

distance factor f3 was left unchanged as it is accounted for in equations (3-5) and (3-6).

The optimal maximum-valued voxel is thus identified as the voxel with the

maximum score function value. However. this voxel is not necessarily the optimal

location at which to place a sphere. For this reason, an optimization loop is used to

search the neighborhood around the best scoring voxel found so far in the process. The

best scoring maximum valued voxel is input as a seed voxel, the score calculated (for all

14 circular collimator sizes) at that voxel and all 26 of its neighboring cubic I x xl mm3

voxels, and the largest score value of all these recorded. If the seed voxel is the best

scoring of all these 27 voxels, the optimization routine has converged, and the voxel is

used as the recommended isocenter (sphere center) location, with the collimator size

corresponding to the best score. If one of the 26 neighboring voxels yields a better

(higher) score, then it is made the seed voxel, and the process repeated for the new seed

voxel and its neighbors until no further improvement in score function is found. In our

example case, a best scoring voxel was soon located within one voxel of the best scoring

maximum valued voxel, and a 22 voxel (mm) diameter sphere was placed there, as shown

in Figure 3-17c.

Step 6: Place sphere corresponding to isocenter size and location. Spheres are

placed by setting voxels lying inside the sphere to a unique numerical value, such as the

maximum voxel value plus two. For instance, if the grassfire process identified the

deepest layer of voxels as those with a value of eight, voxels inside any placed spheres

would be set to a value of ten, which easily allows one to distinguish voxels inside a

sphere from other voxels.

If the user desires to place another sphere after the first sphere has been placed, all

target voxels not covered by-the first sphere are reset to values of 1, and the entire

grassfire process repeated (Figures 3-17d and 3-17e, steps 4-6). Note that target voxels

inside the sphere which was placed have been effectively removed for purposes of the

grassfire algorithm, causing the sphere's surface to be treated as an outer surface of the

target (see arrows in Figures 3-17d and 3-17e). Figures 3-17f through 3-17h similarly

depict selection of the second 14mm isocenter location after the first isocenter has been

placed. The algorithm continues to place three more isocenters, all 5mm in diameter,

before halting. Figure 3-14 depicts three-dimensional views of the sphere packing

arrangement, the target, and the prescription isodose cloud surrounding the target. Figure

3-19 shows this final dose distribution in three orthogonal planes through the center of

the target.

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