Automatic beam angle selection in IMRT planning using genetic algorithm.

Abstract

The selection of suitable beam angles in external beam radiotherapy is at present generally based upon the experience of the human planner. The requirement to automatically select beam angles is particularly highlighted in intensity-modulated radiation therapy (IMRT), in which a smaller number of modulated beams is hoped to be used, in comparison with conformal radiotherapy. It has been proved by many researchers that the selection of suitable beam angles is most valuable for a plan with a small number of beams (< or = 5). In this paper an efficient method is presented to investigate how to improve the dose distributions by selecting suitable coplanar beam angles. In our automatic beam angle selection (ABAS) algorithm, the optimal coplanar beam angles correspond to the lowest objective function value of the dose distributions calculated using the intensity-modulated maps of this group of candidate beams. Due to the complexity of the problem and the large search space involved, the selection of beam angles and the optimization of intensity maps are treated as two separate processes and implemented iteratively. A genetic algorithm (GA) incorporated with an immunity operation is used to select suitable beam angles, and a conjugate gradient (CG) method is used to quickly optimize intensity maps for each selected beam combination based on a dose-based objective function. A pencil-beam-based three-dimensional (3D) full scatter convolution (FSC) algorithm is employed for the dose calculation. Two simulated cases with obvious optimal beam angles are used to verify the validity of the presented technique, and a more complicated case simulating a prostate tumour and two clinical cases are employed to test the efficiency of ABAS. The results show that ABAS is valid and efficient and can improve the dose distributions within a clinically acceptable computation time.

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INSTITUTE OF PHYSICS PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 49 (2004) 1915–1932PII: S0031-9155(04)65223-7
Automatic beam angle selection in IMRT planning
using genetic algorithm
Yongjie Li1,2, Jonathan Yao2and Dezhong Yao1
1School of Life Science and Technology, University of Electronic Science and Technology
of China, Chengdu 610054, People’s Republic of China
2Topslane Inc., Pleasant Hill, CA 94523, USA
E-mail: dyao@uestc.edu.cn
Received 25 June 2003
Published 29 April 2004
Online at stacks.iop.org/PMB/49/1915
DOI: 10.1088/0031-9155/49/10/007
Abstract
Theselectionofsuitablebeamanglesinexternalbeamradiotherapyisatpresent
generally based upon the experience of the human planner. The requirement
to automatically select beam angles is particularly highlighted in intensity-
modulated radiation therapy (IMRT), in which a smaller number of modulated
beams is hoped to be used, in comparison with conformal radiotherapy. It has
been proved by many researchers that the selection of suitable beam angles
is most valuable for a plan with a small number of beams (?5). In this
paper an efficient method is presented to investigate how to improve the dose
distributions by selecting suitable coplanar beam angles. In our automatic
beam angle selection (ABAS) algorithm, the optimal coplanar beam angles
correspond to the lowest objective function value of the dose distributions
calculated using the intensity-modulated maps of this group of candidate
beams. Due to the complexity of the problem and the large search space
involved, the selection of beam angles and the optimization of intensity maps
are treated as two separate processes and implemented iteratively. A genetic
algorithm (GA) incorporated with an immunity operation is used to select
suitable beam angles, and a conjugate gradient (CG) method is used to quickly
optimize intensity maps for each selected beam combination based on a dose-
based objective function. A pencil-beam-based three-dimensional (3D) full
scatter convolution (FSC) algorithm is employed for the dose calculation. Two
simulatedcaseswithobviousoptimalbeamanglesareusedtoverifythevalidity
of the presented technique, and a more complicated case simulating a prostate
tumour and two clinical cases are employed to test the efficiency of ABAS.
The results show that ABAS is valid and efficient and can improve the dose
distributions within a clinically acceptable computation time.
0031-9155/04/101915+18$30.00© 2004 IOP Publishing LtdPrinted in the UK1915

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1916 Y Li et al
1. Introduction
Intensity-modulated radiation therapy (IMRT) has gained more and more attention because of
itsadvantageofproducingahighlythreedimensionalconformaldosedistributiontothetarget,
while sparing organs at risk (OARs) and normal tissues. Compared with traditional three-
dimensional conformal radiation therapy (3D-CRT) that conforms only two dimensionally to
thetargetprojectionintheplaneperpendiculartothebeamorientation, IMRTcansignificantly
improve the outcome of radiation therapy. Many efforts have been made to make IMRT an
easy implementation through increasing the automatization of beam set-up, shortening the
optimization time of inverse planning, enhancing the ability of dose verification, and so on
(Webb 2000).
Theselectionofsuitablebeamanglesinexternalbeamradiotherapyisatpresentgenerally
basedupontheexperience ofthehumanplanner. Normallyseveraltrial-and-errorattemptsare
neededinordertofindagroupofadequatebeamangles. Althoughthisleadstogoodtreatment
plans,theyarenotnecessarilyoptimal(Rowbottom et al1998a). Computeroptimizationofthe
beam directions has the potential to achieve optimal plans that are customized for each
individual patient.The requirement to automatically select beam angles is particularly
highlighted in IMRT, in which a smaller number of modulated beams is hoped to be used, in
comparison with conformal radiotherapy, in order to shorten the treatment time. It has been
provedbymanyresearchersthattheselectionofsuitablebeamanglesismostvaluableforaplan
with a small number of beams (?5) (Bortfeld and Schlegel 1993, Soderstrom and Brahme
1995, Stein et al 1997, Rowbottom et al 1998a, 1998b, 2001, Pugachev et al 2000). The
ultimate goal of the beam angle optimization is to achieve the best possible dose distribution
by using the minimum number of beams (Pugachev et al 2000).
The problem of selection of suitable beam directions in external beam radiation therapy
was studied as early as 1992 by Soderstrom and Brahme. They proposed entropy and Fourier
transform measures for the selection of suitable beam orientations. Bortfeld and Schlegel
(1993) used simulated annealing to optimize the beam directions in the frequency domain.
Their general conclusion is that the optimal beam configuration for multiple-beam irradiations
(with more than three beams) tends to be an even distribution over an angular range of 0 to 2π,
whereas itisveryimportanttooptimizethebeamdirectionswhenthenumber ofbeamsisonly
two or three. They also concluded that it is not in general useful to avoid beam orientations
through organs at risk for modulated beams. Similar conclusions have also been made by
Soderstrom and Brahme (1995), Stein et al (1997), Rowbottom et al (1998a, 1998b, 2001)
and Pugachev et al (2000).
The genetic algorithm (GA), as a powerful global optimization approach, has been
introduced by many researchers to solve radiation optimization problems. Ezzell (1996) used
GAtofindanoptimalcombinationofexternalbeams. Langeretal(1996)usedGAtooptimize
the beam weights for treatment of abdominal tumours. Yu and Schell (1996) chose GA for the
optimization of prostate implants. Wu and Zhu (2000) proposed a technique to optimize
the beam directions and weights using a mixed-encoding GA. Zhang et al (2001) used a
hybrid algorithm called guided evolutionary simulated annealing (GESA), which combined
the GA and SA together, to optimize the gamma knife treatment planning. Despite the
drawbackofextensivecomputation, similartosimulatedannealing, thepotentialofGAisvery
attractive.
In this paper, a new efficient technique is developed to implement automatic beam angle
selection(ABAS).InABASaspecifiednumberofanglecandidatesareselectedfromadiscrete
angle candidates pool using a GA, and then the intensity maps of these angles are optimized
usingaconjugategradient(CG)methodundertheguidanceofadose-basedobjectivefunction.

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Automatic beam angle selection in IMRT planning using genetic algorithm1917
After a great number of iterations (generations), the angle combination with the highest fitness
valueisregardedasgivingtheoptimalbeamanglesforthecase. Afinalrefinedbeamintensity
map optimization is performed using these optimal beam angles, under the guidance of a more
complicated dose–volume-based objective function. In section 3 two simulated test cases with
obvious optimal beam angles are used to verify the validity of the presented technique, and
a more complicated case simulating a prostate tumour and two clinical cases are employed
to test the efficiency of ABAS. The results show that ABAS is valid and efficient and can
improve the dose distributions within a clinically acceptable computation time.
2. Materials and methods
2.1. Objective function
A whole form of the objective function used in this paper can be written as
Fobj(? x(k)) = α · FOAR(? x(k)) + β · FPTV(? x(k))
NOAR
?
(1)
FOAR(? x(k)) =
i=1
NTi
?
j=1
δ · wj· (dj(? x(k)) − pj)2
(2)
FPTV(? x(k)) = γ ·
NTPTV
?
j=1
δ · wj·?dj(? x(k)) − pj
NTPTV
?
?2
−η ·
j=1
δ · wj·
?
dj(? x(k)) − pj− dj(? x(k)) · log
?dj(? x(k))
pj
??
(3)
dj(? x(k)) =
Nray
?
m=1
ajm· ? x(k)
m.
(4)
Here FOAR(? x(k)) is the part associated with all the OARs, FPTV(? x(k)) is the part associated
with the target, NOARis the total number of OARs, NTiis the point number in the ith OAR,
NTPTVis the point number in the target, δ = 1 when the point dose in the volume breaks the
constraints, elseδ =0, wjistheweightofthejthpoint, djisthecalculateddoseofthejthpoint
in the volume, pjis the prescribed dose of the jth point in the volume, a,β are the regularizing
factors that control the importance between target and OARs, γ,η are the importance factors
of the first and second parts in equation (1), Nrayis the total ray (also called pencil beam)
number, ajmis the dose deposited at the jth point from a unit weight of the mth ray, ? x(k)
intensity of the mth ray.
The second part in equation (3) is used to measure the homogeneity of dose distribution in
the target through the entropy information (Wu and Zhu 2002). According to the information
theory, maximum entropy is equivalent to minimizing information in the system, i.e. to
maximize the homogeneity of dose distribution.
In our objective function both dose and dose–volume constraints are embedded. A dose
constraint for the target is expressed as ‘all doses should be higher than DD maxand lower than
DD min’, and a dose–volume constraint for the target is expressed as ‘no more than Vmin% of
the target volume can absorb doses lower than DDV min’. For the OARs, a dose constraint is
denotedas‘alldosesshouldbelowerthanDD max’, andadose–volumeconstraintisdenotedas
‘no more than Vmax% of the volume can absorb doses higher than DDV max’. The dose–volume
mis the

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1918Y Li et al
Figure 1. A chromosome with five genes (i.e. five beam angles): 0◦, 40◦, 100◦, 190◦and 300◦.
constraints are realized through the penalty of those points breaking Vmax% or Vmin% after
sorting the doses in ascending or descending order.
During the optimization of beam angles, the part associated with entropy measurement in
the objective function is not involved, and also those dose–volume constraints are skipped in
order to simplify the calculation. This is just a simple but popularly used dose-based quadratic
objective function. Whereas the whole form of the objective function is employed to optimize
the final intensity maps at those optimal angles using CG method.
2.2. Beam angle selection using the genetic algorithm
2.2.1. The genetic algorithm.
that simulates the natural process of evolution in which the fittest solutions survive after
numerousgenerationsofnaturalselectionandgenetics(Goldberg1989). Usually,anunknown
variable in a solution is represented as a gene in a chromosome, and a solution with many
variables is represented as a chromosome (also named individual). This process is called
coding. Fitnessisdefinedforeachindividualtoevaluateitsgoodness. Duringtheoptimization
process, GA keeps a group of candidate individuals (solutions) in each generation. Through
genetic operations, such as selection, crossover and mutation, those individuals with better
fitness will be propagated to the next generation with higher probability. After numerous
generations of genetics, the individual with the best fitness among the last population is
regarded as the optimal solution.
The genetic algorithm (GA) is a global optimization technique
2.2.2. Coding scheme of GA.
adopted, in which the combination of beam angles is represented by a chromosome (an
individual) with a length of user-specified beam number for the plan, and each gene in the
chromosome represents a trial beam angle. Figure 1 shows a chromosome with five genes
(i.e. five beam angles): 0◦, 40◦, 100◦, 190◦and 300◦. The genes in one chromosome are
different from each other, which means that there should be no two beams with the same
angles in one plan.
In this study a one-dimensional integer-coding scheme is
2.2.3. Determination of the angle search space.
coplanar ones. To reduce the search space covering the total 2π gantry rotational angle,
some beam angles that cannot be used because of the physical limitation or potential danger
for treatment are specified by planners or by a knowledge-based database. Then the angle
space excluding the discarded ranges is divided into equally spaced directions with a given
angle step, such as 5◦or 10◦. There are three main situations for beam angle restriction:
(1) beam orientations that would result in a collision between the treatment gantry and the
patient couch; (2) beam orientations passing through the OARs that have a extra low radiation
tolerance, e.g. the lens of the eye (Rowbottom et al 2001); and (3) beam orientations directly
passing the metal frames of the patient couch or immobilizing couch that can heavily attenuate
radiation.
Figure 2 shows an interactive tool for defining angle constraints in order to reduce the
search space of beam angle optimization. There are two orientation constraints defined so that
The beams in this study are restricted to

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Automatic beam angle selection in IMRT planning using genetic algorithm1919
Figure 2. An interactive tool for defining angle constraints.
no beam is allowed to pass through. Constraint A is defined to protect the spinal cord and to
avoid the right-hand side of the metal frame of the immobilizing couch, and constraint B is
defined to avoid the left-hand side of the metal frame of the immobilizing couch.
2.2.4. Initialization of the first generation.
in one generation), NPsize, is empirically set to double the total number of angle candidates
(Wu and Zhu 2000). For example, if there are six beam angles to be selected, NPsizecan be set
to 12 or a little more. Among these NPsizeindividuals, two strategies are used to implement
the initialization: (1) some are initialized with equispaced angles with different rotations, and
(2)therestarerandomly initialized. For example, ifthebeam number is6and theanglestepis
10◦forthediscretespaceofthewhole2π gantryangle,thenthegenesofthefirstindividualare
occupied by 0◦, 60◦, 120◦, 180◦, 240◦and 300◦, respectively, and 10◦, 70◦, 130◦, 190◦, 250◦
and 310◦, respectively, for those of the second individual. Analogous operations are applied
to the other four individuals, i.e. among the 12 individuals there are 6 individuals initialized
with equispaced angles. Then the remaining six individuals are initialized randomly.
The population size (i.e. the individual number
2.2.5. Definition of individual fitness.
value, and the purpose of optimization is to find the individual (i.e. a group of beam angles)
with maximum fitness. The fitness value is calculated by
The quality of each individual is evaluated by a fitness
Fitness(? s) = Fmax− Fobj(? s)
? s =?s1,s2,...,sNangle
?.
(5)
Here Fmax is a rough estimation of the maximum value of the objective function, ? s is a
group of angles to be selected, and Nangleis the number of beam angles to be selected. Both
Fmax and Fobj(? s) are calculated using equations (1)–(4). Especially, Fmax can be simply
determined by setting it to be a little greater than the maximum objective function value of the
individuals in the current generation, which ensures that all the fitness values are positive, a
requirement oftheselectionoperation tobementioned below. Itshould benotedthatthevalue