BEAM ANGLE OPTIMIZATION IN IMRT USING PATTERN

SEARCH METHODS: INITIAL MESH-SIZE CONSIDERATIONS

Humberto Rocha

, Joana Matos Dias

1,2

, Br´ıgida Ferreira

3,4

and Maria do Carmo Lopes

3,4

INESCC, Rua Antero de Quental 199, 3000-033 Coimbra, Portugal

FEUC, Av. Dias da Silva 165, 3004–512 Coimbra, Portugal

Servic¸o de F´ısica M´edica, IPOC-FG, EPE, Coimbra, Portugal

I3N, Departamento de F´ısica, Universidade de Aveiro, Aveiro, Portugal

Keywords:

Beam angle optimization, Derivative-free optimization, Radiation therapy.

Abstract:

In radiotherapy treatments, the selection of appropriate radiation incidence directions is decisive for the quality

of the treatment, both for appropriate tumor coverage and for enhance better organs sparing. However, the

beam angle optimization (BAO) problem is still an open problem and, in clinical practice, beam directions

continue to be manually selected by the treatment planner in a time-consuming trial and error iterative process.

The goal of BAO is to improve the quality of the radiation incidence directions used and, at the same time,

release the treatment planner for other tasks. The objective of this paper is to discuss the beneﬁts of using

pattern search methods in the optimization of the BAO problem. Pattern search methods are derivative-free

optimization methods that require few function value evaluations to progress and converge and have the ability

to avoid local entrapment. These two characteristics gathered together make pattern search methods suited to

address the BAO problem. Considerations about the initial mesh-size importance and other strategies for a

better coverage and exploration of the BAO problem search space will be debated.

1 INTRODUCTION

The purpose of radiation therapy is to deliver a dose

of radiation to the tumor volume to sterilize all can-

cer cells minimizing the collateral effects on the sur-

rounding healthy organs and tissues. An important

type of radiation therapy is intensity modulated ra-

diation therapy (IMRT), where the radiation beam is

modulated by a multileaf collimator. Multileaf colli-

mators enable the transformation of the beam into a

grid of smaller beamlets of independent intensities. A

common way to solve the inverse planning in IMRT

optimization problems is to use a beamlet-based ap-

proach leading to a large-scale programming prob-

lem. Due to the complexity of the whole optimiza-

tion problem, the treatment planning is usually di-

vided into three smaller problemswhich can be solved

sequentially: beam angle optimization (BAO) prob-

lem, ﬂuence map optimization (FMO) problem, and

leaf sequencing problem. In clinical practice, most of

the time, the number of beam angles is assumed to be

deﬁned a priori by the treatment planner and the beam

directions are still manually selected by the treatment

planner that relies mostly on his experience, despite

the evidence presented in the literature that appropri-

ate radiation beam incidence directions can lead to a

plan’s quality improvement (Das and Marks, 1997).

Here we will focus our attention in the BAO problem,

using coplanar angles, and will assume that the num-

ber of beam angles is deﬁned a priori by the treatment

planner. Many attempts to address the BAO problem

can be found in the literature including mixed integer

programmingapproaches (Lee et al., 2006), neighbor-

hood search approaches (Aleman et al., 2008), hybrid

multiobjective evolutionary optimization approaches

(Schreibmann et al., 2004), and gradient search ap-

proaches (Craft, 2007). The BAO problem is quite

difﬁcult since it is a highly non-convex optimiza-

tion problem with many local minima (Craft, 2007).

Therefore, methods that avoid being easily trapped in

local minima should be used. Pattern search methods

are suited to addressthe BAO problemsince they have

the ability to avoid local entrapment. Here, we will

discuss the beneﬁts of using pattern search methods

in the optimization of the BAO problem. Considera-

tions about the initial mesh-size importance and other

strategies for a better coverage and exploration of the

BAO problem search space will be debated.

355

Rocha H., Matos Dias J., Ferreira B. and do Carmo Lopes M..

BEAM ANGLE OPTIMIZATION IN IMRT USING PATTERN SEARCH METHODS: INITIAL MESH-SIZE CONSIDERATIONS.

DOI: 10.5220/0003716803550360

In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 355-360

ISBN: 978-989-8425-97-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

2 BEAM ANGLE OPTIMIZATION

PROBLEM

In order to model the BAO problem as a mathemati-

cal programming problem, a quantitative measure to

compare the quality of different sets of beam angles is

required. Most of the previous BAO studies are based

on a variety of scoring methods or approximations of

the FMO to gauge the quality of the beam angle set.

When the BAO problem is not based on the optimal

FMO solutions, the resulting beam angle set has no

guarantee of optimality and has questionable reliabil-

ity since it has been extensively reported that optimal

beam angles for IMRT are often non-intuitive. Here,

for modelling the BAO problem, we will use the op-

timal solution value of the FMO problem as measure

of the quality of a given beam angle set (Aleman et

al., 2008; Craft, 2007). Thus, we will present the for-

mulation of the BAO problem followed by the formu-

lation of the FMO problem we used.

2.1 BAO Model

Let us consider k to be the ﬁxed number of (coplanar)

beam directions, i.e., k beam angles are chosen on a

circle around the CT-slice of the body that contains

the isocenter (usually the center of mass of the tu-

mor). Here we will consider all continuous [0

◦

, 360

◦

]

gantry angles instead of a discretized sample. Since

the angle −5

◦

is equivalent to the angle 355

◦

and the

angle 365

◦

is the same as the angle 5

◦

, we can avoid

a bounded formulation. A basic formulation for the

BAO problem is obtained by selecting an objective

function such that the best set of beam angles is ob-

tained for the function’s minimum:

min f(θ

, . . . , θ

)

s.t. θ

, . . . , θ

∈ R

(1)

Here, the objective f(θ

, . . . , θ

) that measures the

quality of the set of beam directions θ

, . . . , θ

is the

optimal value of the FMO problem for each ﬁxed set

of beam directions. Such functions havenumerouslo-

cal optima, which increases the difﬁculty of obtaining

a good global solution. Thus, the choice of the solu-

tion method becomes a critical aspect for obtaining a

good solution. Our formulationwas mainly motivated

by the ability of using a class of solution methods that

we consider to be suited to successfully address the

BAO problem: pattern search methods. The FMO

model used is presented next.

2.2 FMO Model

For a given beam angle set, an optimal IMRT plan

is obtained by solving the FMO problem - the prob-

lem of determining the optimal beamlet weights for

the ﬁxed beam angles. Many mathematical optimiza-

tion models and algorithms have been proposed for

the FMO problem, including linear models (Romeijn

et al., 2003), mixed integer linear models (Lee et al.,

2006), nonlinear models (Aleman et al., 2008), and

multiobjective models (Craft et al., 2006).

Radiation dose distribution deposited in the pa-

tient, measured in Gray (Gy), needs to be assessed

accurately in order to solve the FMO problem, i.e., to

determine optimal ﬂuencemaps. Eachstructure’s vol-

ume is discretized in voxels (small volume elements)

and dose is computed for each voxel using the super-

position principle, i.e., considering the contribution

of each beamlet. Typically, a dose matrix D is con-

structed from the collection of all beamlet weights, by

indexing the rows of D to each voxel and the columns

to each beamlet, i.e., the number of rows of matrix D

equals the number of voxels (V) and the number of

columns equals the number of beamlets (N) from all

beam directions considered. Therefore, using matrix

format, we can say that the total dose received by the

voxel i is given by

∑

j=1

, with w

the weight of

beamlet j. Usually, the total number of voxels con-

sidered reaches the tens of thousands, thus the row di-

mension of the dose matrix is of that magnitude. The

size of D originates large-scale problems being one of

the main reasons for the difﬁculty of solving the FMO

problem.

Here, we will use a convexpenalty function voxel-

based nonlinear model (Aleman et al., 2008). In

this model, each voxel is penalized according to the

square difference of the amount of dose received by

the voxel and the amount of dose desired/allowed for

the voxel. This formulation yields a quadratic pro-

gramming problem with only linear non-negativity

constraints on the ﬂuence values:

min

∑

i=1

∑

j=1

s.t. w

≥ 0, j = 1, . . . , N,

(2)

with F

deﬁned as asymmetric quadratic penalty func-

tions (Romeijn et al., 2003):

∑

j=1





−

∑

j=1

∑

j=1

− T





ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

356

where T

is the desired dose for voxel i, λ

and

are the penalty weights of underdose and overdose of

voxel i, and (·)

= max{0, ·}. Although this formula-

tion allows uniqueweights for each voxel, similarly to

the implementation in (Aleman et al., 2008), weights

are assigned by structure only so that every voxel in a

given structure has the weight assigned to that struc-

ture divided by the number of voxels of the structure

). This nonlinear formulation implies that a very

small amount of underdose or overdose may be ac-

cepted in clinical decision making, but larger devia-

tions from the desired/allowed doses are decreasingly

tolerated.

3 PATTERN SEARCH METHODS

Pattern search methods are directional direct search

methods that belong to a broader class of derivative-

free optimization methods. Pattern search methods

are iterative methods generating a sequence of iter-

ates {x

} using positive bases (or positive spanning

sets) and moving in the direction that would produce

a functiondecrease. A positive basis forR

can be de-

ﬁned as a set of nonzero vectors of R

whose positive

combinations span R

, but no proper set does.

One of the main features of positive bases (or pos-

itive spanning sets), that is the motivation for direc-

tional direct search methods, is that, unless the current

iterate is at a stationary point, there is always a vec-

tor v

in a positive basis (or positive spanning set) that

is a descent direction (Davis, 1954), i.e., there is an

α > 0 such that f(x

+ αv

) < f(x

). This is the core

of directional direct search methods and in particular

of pattern search methods.

Pattern search methods are iterative methods gen-

erating a sequence of non-increasing iterates {x

Given the current iterate x

, at each iteration k, the

next point x

k+1

is chosen from a ﬁnite number of can-

didates on a given mesh M

(deﬁned using the vectors

forming a positive spanning set) aiming to provide a

decrease on the objective function: f(x

k+1

) < f(x

Pattern search methods consider two steps at every it-

eration. The ﬁrst step consists of a ﬁnite search on

the mesh, with the goal of ﬁnding a new iterate that

decreases the value of the objective function at the

current iterate. This step, called the search step, has

the ﬂexibility to use any strategy, method or heuris-

tic, or take advantage of a priori knowledge of the

problem at hand, as long as it searches only a ﬁnite

number of points in the mesh. The search step pro-

vides the ﬂexibility for a global search since it allows

searches away from the neighborhood of the current

iterate, and inﬂuences the quality of the local mini-

mizer or stationary point found by the method. If the

search step is unsuccessful a second step, called the

poll step, is performed around the current iterate with

the goal of decreasing the objective function. The poll

step follows stricter rules and appeals to the concepts

of positive bases. The poll step attempts to perform

a local search in a mesh neighborhood that, for a suf-

ﬁciently small mesh-size parameter ∆

, is guaranteed

to provide a function reduction, unless the current it-

erate is at a stationary point (Alberto et al., 2004). So,

if the poll step also fails, the mesh-size parameter ∆

must be decreased. The most common choice for the

mesh-size parameter update is to half the mesh-size

parameter at unsuccessful iterations and to keep it or

double it at successful ones. The purposeof the mesh-

size parameter is twofold: to bound the size of the

minimization step and also to control the local area

where the function is sampled around the current it-

erate. Most derivative-free methods couple the mesh-

size (or step-size) with the size of the sample set (or

the search space). The initial mesh-size parameter

value deﬁned in (Mor´e and Wild, 2009) for compari-

son of several derivative-free optimization algorithms

is ∆

= max{1, kx

∞

}. This choice of initial mesh-

size parameter is commonly used as default in many

derivative-free algorithms such as implementations of

the Nelder-Mead method. However, this choice of ini-

tial mesh-size parameter might not be adequate as we

will illustrate along this paper.

For driving the resolution of the BAO problem,

we will use the last version of SID-PSM (Cust´odio

and Vicente, 2007; Cust´odio et al., 2010) which is a

MATLAB implementation of the pattern search meth-

ods that incorporate improvementsforthe searchstep,

with the use of minimum Frobenius norm quadratic

models to be minimized within a trust region, and im-

provements for the poll step, where efﬁciency on the

number of function value computations improved sig-

niﬁcantly by reordering the poll directions according

to descent indicators. The default initial mesh-size

parameter of SID-PSM is also ∆

= max{1, kx

∞

The beneﬁts of using this particular implementation

of pattern search methods in the optimization of the

BAO problem and the inﬂuence of the initial mesh-

size parameter choice on the quality of the solution

obtained are illustrated using a clinical example of

head and neck case that is presented next.

4 NUMERICAL TESTS AND

DISCUSSION

A clinical example of a retrospective treated case of

head and neck tumor at the Portuguese Institute of

BEAM ANGLE OPTIMIZATION IN IMRT USING PATTERN SEARCH METHODS: INITIAL MESH-SIZE

CONSIDERATIONS

357

Table 1: Prescribed doses for all the structures considered

for IMRT optimization.

Structure Mean dose Max dose Prescribed dose

Spinal cord – 45 Gy –

Brainstem – 54 Gy –

Left parotid 26 Gy – –

Right parotid 26 Gy – –

PTV left – – 59.4 Gy

PTV right – – 50.4 Gy

Body – 70 Gy –

Oncology of Coimbra is used to verify the beneﬁts

and issues of using pattern search methods in the op-

timization of the BAO problem. The patients’ CT set

and delineated structures were exported via Dicom

RT to a freeware computational environment for ra-

diotherapy research. In general, the head and neck

region is a complex area to treat with radiotherapy

due to the large number of sensitive organs in this re-

gion (e.g. eyes, mandible, larynx, oral cavity, etc.).

For simplicity, in this study, the OARs used for treat-

ment optimization were limited to the spinal cord, the

brainstem and the parotid glands. The tumor to be

treated plus some safety margins is called planning

target volume (PTV). For the head and neck case in

study it was separated in two parts: PTV left and PTV

right. The prescribed doses for all the structures con-

sidered in the optimization are presented in Table 1.

Our tests were performed on a 2.66Ghz Intel Core

Duo PC with 3 GB RAM. In order to facilitate con-

venient access, visualization and analysis of patient

treatment planning data, the computational tools de-

veloped within MATLAB and CERR (Deasy et al.,

2003) (computational environment for radiotherapy

research) were used as the main software platform

to embody our optimization research and provide the

necessary dosimetry data to perform optimization in

IMRT. The dose was computed using CERR’s pencil

beam algorithm (QIB). To address the convex non-

linear formulation of the FMO problem we used a

trust-region-reﬂective algorithm (fmincon) of MAT-

LAB 7.4.0 (R2007a) Optimization Toolbox.

The last version of SID-PSM was used as our pat-

tern search methods framework. In order to initial-

ize the algorithm we need to choose an initial point

, a positive spanning set, and an initial mesh-size

parameter ∆

> 0. Typically, in head and neck can-

cer cases, patients are treated with 5 to 9 equispaced

beams in a coplanar arrangement. Here, we will con-

sider the equispaced 5 beamconﬁgurationwith angles

0, 72, 144, 216 and 288, and with 0 collimator angle.

Since our goal is to improve the typically used treat-

ment plans, this is a good starting point and thus we

will consider x

as the previous 5-beam equispaced

angle conﬁguration. The choice of this initial point

and the non-increasing property of the sequence of

iterates generated by SID-PSM imply that each suc-

cessful iteration correspond to an effective improve-

ment with respect to the usual equispaced beam con-

ﬁguration. The spanning set used was the positive

spanning set ([e − e I − I], with I being the identity

matrix and e = [1 1]

). Each of these directions cor-

responds to, respectively, the rotation of all incidence

directions clockwise, the rotation of all incidence di-

rections counter-clockwise, the rotation of each in-

dividual incidence direction clockwise, and the rota-

tion of each individual incidence direction counter-

clockwise.

The default initial mesh-size parameter, as men-

tioned before, is ∆

= max{1, kx

∞

}. For the con-

sidered initial point this would give an initial mesh-

size of ∆

= 288. For this “cyclic” problem such ini-

tial mesh-size is too large implying in practice that

huge rotation of angles would occur. Moreover, con-

vergence would take too long leading to an exces-

sive number of function evaluations. Obtaining the

optimal solution for a beam angle set is time costly

and even if only a beam angle is changed in that set,

a complete dose computation is required in order to

compute and obtain the corresponding optimal FMO

solution. Therefore, few function value evaluations

should be used to tackle the BAO problem within a

clinically acceptable time frame.

Note that if the initial mesh-size parameter is a

power of 2, (∆

= 2

, p ∈ N), and the initial point

is a vector of integers, using the default mesh up-

date, i.e., to half the mesh parameter at unsuccess-

ful iterations and to keep it at successful ones, all

iterates will be a vector of integers until the mesh

parameter size becomes inferior to 1. This possi-

bility is rather interesting for our BAO problem at

hand and, for initial mesh-size parameter, we tested

(∆

= 2

, p = 1, 2, . . .). For the initial point selected,

the distance between two consecutive beam angle di-

rections is 72, thus, the maximum ∆

we considered

was 64. The history of the beam angle optimization

process using SID-PSM, for each initial mesh-size

parameter considered, is presented in Figure 1. By

simple inspection we can verify that only mesh-size

parameters 32 and 64 originate a sequence of iter-

ates that are reasonably well distributed by amplitude

in R

, while mesh-size parameters inferior to 32 fail

to cover in amplitude all the search space. At ﬁrst

sight, larger mesh-size parameters, which obtain it-

erates better distributed by amplitude in R

, should

be desirable for an improved global search in con-

junction with the search step. One of the main ad-

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

358

(a) (b)

(e) (f)

Figure 1: History of the beam angle optimization process

using SID-PSM for mesh-size parameters 2 to 64, 1(a) to

1(f) respectively. Initial angle conﬁguration, optimal angle

conﬁguration and intermediate angle conﬁgurations are dis-

played with solid, dashed and dotted lines, respectively.

vantages of this pattern search methods framework

is the ﬂexibility provided by the search step, where

any strategy can be applied as long as only a ﬁnite

number of points is tested. This allows the inser-

tion of previously used and tested strategies/heuristics

that successfully address the BAO problem and en-

hance for a global search by inﬂuencing the qual-

ity of the local minimizer or stationary point found

by the method. In the last version of SID-PSM, the

search step computes a single trial point using min-

imum Frobenius norm quadratic models to be min-

imized within a trust region, which enhanced a sig-

niﬁcant improvement of direct search for black-box

non-smooth functions (Cust´odio et al., 2010) similar

to the BAO problem at hand. The size of the trust re-

gion is coupled to radius of the sample set. Thus, for

an effective global search, the sample points should

span all the search space. However, since the BAO

problem has many local minima and the number of

sample points is scarce, the polynomial interpolation

or regression models (usually quadratic models) used

within the trust region struggle to ﬁnd the best local

minima. Therefore, starting with larger mesh-size pa-

rameters have the advantage of a better coverage of

Table 2: Results obtained using x

= (0, 72, 144, 216,288)

for different initial mesh-size parameters.

∆

f init. f opt. f evals. time (s)

2 90.21 79.70 51 3162

4 90.21 79.60 65 4183

8 90.21 79.26 113 7159

16 90.21 79.36 115 7216

32 90.21 83.16 137 8645

64 90.21 79.26 139 8649

the search space but may cause the algorithm to jump

over lower local minima than the obtained one. That

was the case for ∆

= 32 which originated the worst

result. The results obtained for the different initial

mesh-size parameters are presented in Table 2. The

quality of the treatment plan obtained is directly pro-

portional to the correspondent ﬁnal objective func-

tion value. For this initial point the treatment plans

obtained for all initial mesh-size considered except

∆

= 32 are equivalent. This means that larger initial

mesh parameters do not lead to better local minima

despite the improved search space coverage.

An alternative popular approach to keep small

mesh-size parameters and still have a good coverage

of the search space is to use a multi-start approach.

The multi-start approach has the disadvantage of in-

creasing the total number of function evaluations and

with that the overall computational time. Moreover,

despite the better span of R

in amplitude, that is only

obtained by overlapping all the iterates which might

be fallacious for this particular problem. In future

work we aim to use a single starting point, a small

initial mesh-size parameter, and obtain a good span in

amplitude of R

by incorporatingan additional global

strategy in the search step such as response surface

approach or radial basis functions interpolation.

Let us now illustrate the beneﬁts of using a treat-

ment plan with the best optimal angle conﬁguration

obtained by SID-PSM (5 PSM) compared with the

usual treatment plan with equispaced beam directions

(5 equi). Typically, results are judged by their cumu-

lative dose-volume histogram (DVH). The DVH dis-

plays the fraction of a structure’s volume that receives

at least a given dose. An ideal DVH for the tumor

would present 100% volume for all dose values rang-

ing from zero to the prescribed dose value and then

drop immediately to zero, indicating that the whole

target volume is treated exactly as prescribed. Ideally,

the curves for the organs at risk would instead drop

immediately to zero, meaning thatno volume receives

radiation. Another metric usually used for plan eval-

uation is the volume of PTV that receives 95% of the

BEAM ANGLE OPTIMIZATION IN IMRT USING PATTERN SEARCH METHODS: INITIAL MESH-SIZE

CONSIDERATIONS

359

0 10 20 30 40 50 60 70

100

Dose (Gy)

Percent Volume (%)

PTV left − 5 PSM

Left parotid − 5 PSM

PTV left − 5 equi

Left parotid − 5 equi

0 10 20 30 40 50 60

100

Dose (Gy)

Percent Volume (%)

PTV right − 5 PSM

Right parotid − 5 PSM

PTV right − 5 equi

Right parotid − 5 equi

Figure 2: Cumulative dose volume histogram comparing

the treatment plans 5 PSM and 5 equi.

prescribed dose. Typically, 95% of the PTV volume

is required. DVH results are displayed in Figure 2.

Since parotids are the most difﬁcult organs to spare,

for clarity, the DVH only includes the targets and the

parotids. The asterisk indicates 95% of PTV volume

versus 95% of the prescribed dose. By observing Fig-

ure 2 we conﬁrm that both treatment plans fulﬁll the

goal of having 95% of the prescribed dose for 95%

of the volume for both PTV right and PTV left. Fo-

cusing in parotid sparing we can observe that a better

parotid sparing can be obtained using the beam angle

solution obtained by SID-PSM.

5 CONCLUSIONS

The BAO problem is a continuous global highly non-

convex optimization problem known to be extremely

challengingand yet to be solved satisfactorily. Pattern

search methods framework is a suitable approach for

the resolution of the non-convex BAO problem due

to their structure, organized around two phases at ev-

ery iteration. The poll step, where convergence to a

local minima is assured, and the search step, where

ﬂexibility is conferred to the method since any strat-

egy can be applied. We have shown that a beam angle

set can be locally improved in a continuous manner

using pattern search methods. The initial mesh-size

parameter importance and other strategies for a better

coverage and exploration of the BAO problem search

space were tested and debated. In future work, pat-

tern search methods improvement will be tested with

the incorporation of additional global strategies in the

search step such as response surface approaches or ra-

dial basis functions interpolation. We have to high-

light the low number of function evaluations required

by SID-PSM to obtain locally optimal solutions. The

efﬁciency on the number of function value computa-

tion is ofthe utmost importance, particularlywhen the

BAO problem is modeled using the optimal values of

the FMO problem. Thus, the global strategies to be

incorporated must comply with this requirement.

ACKNOWLEDGEMENTS

This work has been partially supported by FCT under

project grant PEst-C/EEI/UI0308/2011. The work of

H. Rocha was supported by the European social fund

and Portuguese funds from MCTES.

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