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A Fully-Automated

Intensity-Modulated Radiation Therapy

Planning System

Shabbir Ahmed, Ozan Gozbasi, Martin Savelsbergh

Georgia Institute of Technology

Ian Crocker, Tim Fox, Eduard Schreibmann

Emory University

Abstract

We designed and implemented a linear programming based IMRT treatment plan generation

technology that effectively and efficiently optimizes beam geometry as well as beam intensities.

The core of the technology is a fluence map optimization model that approximates partial dose

volume constraints using conditional value-at-risk constraints. Conditional value-at-risk con-

straints require careful tuning of their parameters to achieve desirable plan quality and therefore

we developed an automated search strategy for parameter tuning. Another novel feature of our

fluence map optimization model is the use of virtual critical structures to control coverage and

conformity. Finally, beam angle selection has been integrated with fluence map optimization.

The beam angle selection scheme employs a bi-criteria weighting of beam angle geometries and

a selection mechanism to choose from among the set of non-dominated geometries. The tech-

nology is fully automated and generates several high-quality treatment plans satisfying dose

prescription requirements in a single invocation and without human guidance. The technology

has been tested on various real patient cases with uniform success. Solution times are an order

of magnitude faster than what is possible with currently available commercial planning systems

and the quality of the generated treatment plans is at least as good as the quality of the plans

produced with existing technology.

1 Introduction

External beam irradiation is used to treat over 500,000 cancer patients annually in the United

States. External beam radiation therapy uses multiple beams of radiation from different directions

to cross-fire at a cancerous tumor volume. Thus, radiation exposure of normal tissue is kept at

low levels, while the desired dose to the tumor volume is delivered. The key to the effectiveness

of radiation therapy for the treatment of cancer lies in the fact that the repair mechanisms for

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cancerous cells are less efficient than those of normal cells. Intensity-modulated radiation therapy

(IMRT) is a relatively recent advance in treatment delivery in radiation therapy. In IMRT, the

beam intensity is varied across each treatment field. Rather than being treated with a large uniform

beam, the patient is treated instead with many small pencil beams (referred to as beamlets), each

of which can have a different intensity. For many types of cancer, such as prostate cancer and

head and neck cancer, the use of intensity modulation allows more concentrated treatment of the

tumor volume, while limiting the radiation dose to adjacent healthy tissue (see Veldeman et al. [20]

for comparisons of IMRT and non-IMRT treatments on different tumor sites). IMRT treatment

planning is concerned with selecting a beam geometry and beamlet intensities to produce the best

dose distribution. Because of the many possible beam geometries, the large number of beamlets, and

the range of beamlet intensities, there is an infinite number of treatment plans, and consistently and

efficiently generating high-quality treatment plans is beyond human capability. Consequently, it is

necessary to design and implement optimization-based decision support systems that can construct

and suggest high-quality treatment plans in a short period of time. That is the goal of our research

and the subject of this paper.

Constructing an IMRT treatment plan that radiates the cancerous tumor volume without im-

pacting normal structures is virtually impossible. Therefore, trade-offs have to be made. Radiation

oncologists have developed a variety of measures to evaluate these trade-offs and to assess the

quality of a treatment plan, such as the coverage of a target volume by a prescription dose, the

conformity of a prescription dose around a target volume, the highest and the lowest doses received

by a target volume. Furthermore, radiation oncologists review dose-volume histograms (DVHs)

depicting the dose distributions at various structures (both target volumes and organs at risk) of

treatment plans. Typically, a radiation oncologist specifies a set of dose-related minimum require-

ments (a set of prescriptions) that have to be satisfied in any acceptable treatment plan. The

measures and DVHs are then used to choose among acceptable treatment plans. Further com-

plicating the evaluation of treatment plans is the fact that the minimum requirements and their

relative importance are subjective as are the underlying trade-offs they are trying to capture. As

a result, most existing IMRT treatment planning systems are iterative in nature and necessitate

human evaluation and guidance throughout the treatment plan construction process. This makes

the process time-consuming and costly. One of the primary goals of our research is the develop-

ment of fully-automated treatment plan generation technology that completely eliminates human

intervention, thereby saving valuable time for the radiation oncologist and physicist.

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Natural models of dose-volume requirements result in nonlinear and/or mixed integer programs,

which are notoriously hard to solve. Recently, Romeijn et al. ([18]) observed that conditional

value-at-risk constraints (C-VaR constraints), which are popular and commonly used in financial

engineering, can be used to approximate dose-volume constraints. The use of C-VaR constraints

has significant computational advantages as they can be handled using linear programming models,

which are easily and efficiently solved. However, the parameters controlling the C-VaR constraints

have to be chosen carefully to get an accurate approximation of the dose-volume constraints. Our

treatment plan generation technology, which is based on an optimization model using C-VaR con-

straints, embeds an effective and efficient parameter search scheme to ensure high-quality approx-

imations of the dose-volume constraints. Another innovation in our treatment plan generation

technology is the use of virtual critical structures. Virtual critical structures surround target vol-

umes and are implemented to control the dose deposits specifically at the boundary of the target

volume.

Our treatment plan generation technology incorporates an effective and efficient scheme for

the selection of beam geometries, which is ignored in many commercial planning systems as it

adds another layer of complexity to the search for high-quality treatment plans. Most commercial

planning systems consider only beam geometries involving a small number of equi-distant beam

angles around a circle (on the order of 5 to 8). Because of the computational efficiency of our core

plan generation technology, we are able to embed a scheme that optimizes the beam geometry. At

the heart of the scheme is a multi-attribute beam scoring mechanism based on a treatment plan

constructed for a beam geometry involving a large number of beam angles (on the order of to 24).

In summary, we have developed treatment plan generation technology that optimizes both beam

geometry and beamlet intensities in a short amount of time. The technology is fully automated

and generates several high-quality treatment plans satisfying the provided minimum requirements

in a single invocation and without human guidance. The technology has been tested on various real

patient cases with great success. Solution times range from a few minutes to a quarter of an hour,

which is an order of magnitude faster than what is possible with currently available commercial

planning systems. Furthermore, the quality of the generated treatment plans is at least as good as

those produced with existing technology.

The remainder of the paper is organized as follows. In Section 2, we introduce IMRT treatment

plan generation and evaluation in more detail. In Section 3, we describe the core components of

the IMRT treatment plan generation technology that we have developed. Finally, in Section 4, we

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present the results of an extensive computational study.

2 Problem Description

Intensity-modulated radiation therapy (IMRT) is an advanced mode of high-precision radiotherapy

that utilizes computer-controlled mega-voltage x-ray accelerators to deliver precise radiation doses

to a cancerous tumor or specific areas within the tumor. The radiation dose is designed to conform

to the three-dimensional (3-D) shape of the tumor by modulating – or controlling - the intensity of

the radiation beam to focus a higher radiation dose to the tumor while minimizing radiation expo-

sure to surrounding normal tissues. Typically, combinations of several intensity-modulated fields

coming from different beam directions produce a custom tailored radiation dose that maximizes

tumor dose while also protecting adjacent normal tissues. IMRT is being used to treat cancers of

the prostate, head and neck, breast, thyroid, lung, liver, brain tumors, lymphomas and sarcomas.

Radiation therapy, including IMRT, stops cancer cells from dividing and growing, thus slowing

tumor growth. In many cases, radiation therapy is capable of killing cancer cells, thus shrinking or

eliminating tumors. For a more detailed description, review, history, and physical basis of IMRT,

see Bortfeld ([3]), Boyer et al. ([4]), Shepard ([19]), and Webb ([21]).

An IMRT treatment plan has to specify a beam geometry (a set of beam directions) and for each

beam direction a fluence map (a set of beamlet intensities; a beam can be thought of as consisting of

a number of beamlets that can be controlled individually). Typically, a small number of equi-spaced

coplanar beam directions are used. Coplanar beam directions are obtained by rotating the gantry

while keeping the treatment couch in a fixed position parallel to the gantry axis of rotation. There

are practical reasons for limiting the number of beam directions (between 5 and 8) as it reduces

patient positioning times, chances for patient positioning errors, and delivery time, but considering

only a restricted set of beam geometries is driven primarily by the computational complexity it

introduces in the treatment planning process.

For modeling purposes, the structures (target and critical structures) are discretized into cubes

called voxels (e.g., cubes of 5 × 5 × 5 mm) and the dose delivered to each voxel per intensity of each

beamlet is calculated (see Ahnesjo [1], Mackie et al. [12], and Mohan et al. [13] for dose calculation

methods). The total dose received by a voxel is assumed to be the sum of doses deposited from each

beamlet. The goal is to build a treatment plan that creates a dose distribution that will destroy, or

at least damage, target cells while sparing healthy tissue by choosing proper beam directions and

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beamlet intensities.

In order to guide the construction of a treatment plan, a radiation oncologist specifies a set of

requirements that have to be satisfied in any acceptable treatment plan. These requirements are in

the form of a minimum prescription dose for target structure voxels and a maximum tolerance dose

for nearby critical structure voxels. A prescription dose is the dose level necessary to destroy or

damage target cells, while a tolerance dose is the dose level above which complications for healthy

tissues may occur. The requirements are grouped into two types: (1) full-volume constraints and

(2) partial-volume constraints. Full-volume constraints specify dose limits that have to be satisfied

by all voxels of a structure. Partial-volume constraints, on the other hand, specify dose limits that

have to be satisfied by a specified fraction of the voxels of a structure. The choice of a full-volume

or a partial-volume constraint for a healthy structure depends on the functionality of the organ.

For example, a partial-volume constraint is used when the organ at risk will not be damaged as

long as a specified volume receives a dose below the tolerance dose.

In addition to the set of dose requirements defining acceptable treatment plans, clinical oncolo-

gists use a set of evaluation metrics to assess the quality of a treatment plan. The cold spot and hot

spot metrics are evaluation measures that consider the minimum and maximum dose received by

a voxel in a structure relative to the prescription dose. The coverage metric considers the fraction

of voxels in the target structure receiving a dose greater than the prescription dose. The confor-

mity metric considers the ratio of the number of voxels in the target region and surrounding tissue

receiving a dose greater than the prescription dose to the number of voxels in the target structure

itself receiving a dose greater than the prescription dose.

More formally, these evaluation metrics are defined as follows. Let N denote the set of beamlets,

S the set of critical structures, T the set of target structures, Vs the number of voxels in structure

s, and zjs the dose received by voxel j of structure s. (Of course zjs depends on the treatment

plan.)

The cold spot of target structure s ∈ T is defined as the ratio of the minimum dose received by

any of the voxels of the structure to the prescription dose for that structure, i.e.

cold spot(s) =min{zjs : j = 1, . . . , Vs}

PDs, (1)

where PDs is the prescription dose for target structure s. Similarly, the hot spot of target structure

s ∈ T is defined as the ratio of the maximum dose received by any of the voxels of the structure to

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the prescription dose, i.e.

hot spot(s) =max{zjs : j = 1, . . . , Vs}

PDs. (2)

Ideally, every voxel in a target structure receives exactly the prescription dose. The cold spot

metric and hot spot metric measure the deviation from this ideal situation by examining the voxel

receiving the smallest dose and the voxel receiving the largest dose.

The coverage of target structure s ∈ T is the proportion of voxels receiving a dose greater than

or equal to the prescription dose PDs, i.e.

coverage(s) =

Vs∑

j=1

I+(zjs − PDs)

Vs, (3)

where I+ is the indicator function for the non-negative real line, i.e., I+(a) is equal to 1 if a ≥ 0

and 0 otherwise. For simplicity assume that T = {s}. Note that 0 ≤ coverage(s) ≤ 1 and that

values closer to 1 are preferable. The conformity of target structure s is the ratio of the number

of voxels in the structure and its surrounding tissue receiving a dose greater than or equal to the

prescription dose PDs to the number of voxels in the structure itself receiving greater than or equal

to the prescription dose, i.e.,

conformity(s) =

∑

s′∈S∪T

Vs′∑

j=1

I+(zjs′ − PDs)

Vs∑

j=1

I+(zjs − PDs)

= 1 +

∑

s′∈S

Vs′∑

j=1

I+(zjs′ − PDs)

Vs∑

j=1

I+(zjs − PDs)

. (4)

Note that 1 ≤ conformity(s) and that values closer to 1 are preferable. The coverage and conformity

metrics are illustrated in Figure 1. In the schematic on the left, the set of voxels receiving a dose

greater than or equal to the prescription dose is exactly the set of voxels in target structure s and

thus coverage(s) = conformity(s) = 1; this is the ideal solution. In the schematic in the center the

set of voxels receiving a dose greater than or equal to the prescription dose is twice the size of the

set of voxels in target structure s (and includes all the voxels of s) and thus coverage(s) = 1 and

conformity(s) = 2. Finally, in the schematic on the right, the set of voxels receiving a dose greater

than or equal to the prescription dose is exactly half the set of voxels in target structure s and

thus coverage(s) = 0.5 and conformity(s) = 1. Ideally, every voxel in a target structure receives

exactly the prescription dose and all the voxels in the surrounding tissue (e.g., in the nearby critical

structures) receive no dose at all. The coverage metric measures how well we cover the voxels in

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TARGET TARGET

Coverage = 1

Conformity = 2

Coverage = 1

Conformity = 1

TARGET

Coverage = 0.5

Conformity = 1

Prescription Dose Region

Figure 1: Coverage and conformity metrics

the target structure. The conformity metric measures, in a sense, what happens at the boundary

of the target structure. It measures the price that has to be paid, in terms of exposure of healthy

tissue, to achieve high coverage. Due to the fact that coverage and conformity consider the target

as a whole, these metrics are typically considered of higher importance than the cold spot and hot

spot metrics. (See Lee et al. ([8, 9]) for earlier use of these metrics for plan evaluation.)

As we will see in the next section, the evaluation metrics are optimized implicitly; the evaluation

metrics are not incorporated explicitly in the optimization problem.

3 Methodology

In this section, we describe the key components of the proposed IMRT treatment plan generation

technology. The core of the technology is a fluence map optimization (FMO) model based on

linear programming. This model is similar to the one proposed by Romeijn et al. ([18]) and

approximates partial volume constraints using conditional value-at-risk (C-VaR) constraints. C-

VaR constraints require careful tuning of their parameters to achieve desirable plan quality and

therefore an automated search strategy for parameter tuning has been developed. (See Preciado-

Walters et al. [14] and Lee et al. [8] for exact mixed integer programming formulations of partial

volume constraints.) A novel feature of the fluence map optimization model is the use of virtual

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critical structures to control coverage and conformity. Finally, beam angle selection has been

integrated with fluence map optimization. The beam angle selection scheme employs a bi-criteria

weighting of beam angle geometries and a selection mechanism to choose from among the set of

non-dominated geometries.

3.1 Fluence Map Optimization Model

Recall that N denotes the set of beamlets, S the set of critical structures, T the set of target

structures, Vs the number of voxels in structure s, and Dijs the dose received by voxel j of structure

s per unit intensity of beamlet i. Let xi be the intensity of beamlet i, i.e., a decision variable, then

the dose zjs received by voxel j of structure s, is

zjs =∑

i∈N

Dijsxi ∀j = 1, ..., Vs; s ∈ S ∪ T. (5)

Full volume constraints

Let Ls and Us be the prescribed lower and upper dose limits for structure s, respectively. The

full-volume constraints for dose limits are

Ls ≤ zjs ≤ Us ∀j = 1, ..., Vs; s ∈ S ∪ T. (6)

Partial volume constraints

The dose-volume constraints for target structures are formulated as

cs − 1(1− αs).Vs

Vs∑

j=1

(cs − zjs)+ ≥ Lαs s ∈ T. (7)

This C-VaR constraint enforces that the average dose in the (1 − αs)-fraction of voxels of target

structure s receiving the lowest amount of dose is greater than or equal to Lαs . When satisfied at

least αs × 100 per cent of the voxels will receive a dose greater than or equal to Lαs . For example,

with αs = 0.9 and Lαs = 30 at least 90% of the voxels will receive a dose greater than 30 Gy. Here

cs is a free variable associated with the dose-volume constraint of structure s.

Similarly, the partial-volume constraints for critical structures are

cs +1

(1− αs).Vs

Vs∑

j=1

(zjs − cs)+ ≤ Uαs s ∈ S. (8)

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For critical structure s the average dose in the subset of size (1−αs) receiving the highest amount

of dose is required to be less than or equal to Uαs . When satisfied at least αs percent of the voxels

will receive a dose less than or equal to Uαs .

Cold spot and hot spot requirements

As mentioned before the cold spot metric (1) for a target structure s ∈ T is defined as the ratio

of the minimum dose received by the voxels of the target structure to the prescription dose, and

the hot spot metric (2) for a target structure s ∈ T is defined as the ratio of the maximum dose

received by the voxels of the target structure to the prescription dose. Prescribed requirements on

cold spot and hot spot can be enforced by the bounds Ls and Us in the full volume constraints (6).

Coverage requirements

The coverage metric (3) of a target structure s ∈ T is the proportion of voxels receiving a dose

greater than or equal to the prescription dose PDs. Due to the indicator function in the expression

(3), requirements on coverage are hard to model using convex constraints. Instead, we use partial

volume constraints to enforce the coverage requirements using the following result.

Proposition 1 Given αs ∈ (0, 1), if there exists a cs such that

cs − 1(1− αs).Vs

Vs∑

j=1

(cs − zjs)+ ≥ PDs (9)

then

coverage(s) ≥ αs. (10)

A proof of this proposition can be found in the appendix. Thus coverage(s) can be increased by

increasing αs in the partial volume constraints (7) with Lαs = PDs. Of course αs = 1 is most

preferable but such an aggressive value may lead to problem infeasibility. Note that in this case

(i.e. when αs = 1) the partial volume constraint reduces to a full volume constraint.

Conformity requirements

The conformity metric (4) of a target structure s ∈ T is the ratio of the total number of voxels in

the body receiving a dose greater than or equal to the target prescription dose PDs to the number

of voxels in the target receiving greater than or equal to prescription dose in the target structure.

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Unlike cold spot, hot spot and coverage; conformity requirements cannot be directly enforced

by full or partial volume constraints on the dose on the target structure. This is because the

conformity metric needs to consider dose deposited outside the target. In order to keep track of the

intended dose for the target s that is deposited outside the target, we introduce a virtual critical

structure around the target structure s. This is typically a 30 mm band around the target structure

s, denoted by b(s), which is included in S. Price et al. ([15]) and Lee et al. ([9]) showed on clinical

cases that limiting or minimizing the dose on virtual critical structures around the target structures

can improve the conformity of the plans. Girinsky et al. ([6]) also used virtual critical structures

to avoid high doses around the tumor volume.

Assuming that all of the dose intended for the target s ∈ T will be deposited in the voxels in

s and in b(s), the only structures in S that can have more than the prescribed dose PDs will be

b(s). Then from (4)

conformity(s) = 1 +

Vb(s)∑

j=1

I+(zjs′ − PDs)

Vs∑

j=1

I+(zjs − PDs)

= 1 +coverage(b(s)) · Vb(s)

coverage(s) · Vs,

where the second equality follows from the definition of coverage.

We know that by enforcing the partial volume constraint (7) on s with Lαs = PDs and given

αs ∈ (0, 1) we ensure coverage(s) ≥ αs. Similarly, it can be shown that by enforcing the partial

volume constraint (8) on b(s) with Uαb(s) = PDs and given αb(s) ∈ (0, 1), we ensure coverage(b(s)) ≤

(1− αb(s)). Thus

conformity(s) ≤ 1 +(1− αb(s)) · Vb(s)

αs · Vs. (11)

Therefore we can reduce conformity(s) by increasing αs as well as by increasing αb(s). Setting these

parameters at their maximum value of 1 typically leads to problem infeasibility, and so a careful

selection has to be made. This is further discussed in Section 3.2.

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The LP model

The final formulation of our FMO model is

min∑

s∈S

1Vs

Vs∑

j=1

zjs −∑

s∈T

1Vs

Vs∑

j=1

zjs (12)

s.t. zjs =∑

i∈N

Dijsxi ∀j = 1, ..., Vs; s ∈ S ∪ T (13)

Ls ≤ zjs ≤ Us ∀j = 1, ..., Vs; s ∈ S ∪ T (14)

cs +1

(1− αs)Vs

Vs∑

j=1

(zjs − cs)+ ≤ Uαs s ∈ S (15)

cs − 1(1− αs)Vs

Vs∑

j=1

(cs − zjs)+ ≥ Lαs s ∈ T (16)

xi ≥ 0 i = 1, ..., N (17)

zjs ≥ 0 j = 1, ..., Vs; s ∈ S ∪ T (18)

cs free s ∈ S ∪ T . (19)

The objective function (12) attempts to decrease the average dose on the critical structures while

increasing the average dose on the target structures. (See Kessler et al. [7] and Yang and Xing

[23] for objective functions used in IMRT formulations). Constraint (13) calculates the total dose

zjs deposited on to voxel j of structure s from all beamlets. Constraints (14) are the full volume

constraints on the structures according to the prescribed dose limits and the cold spot and hot

spot considerations. Constraints (15) are the partial volume constraints on the critical structures.

Since S includes the virtual critical structure b(s) for each target s ∈ T , these constraints serve

to enforce conformity requirements. Constraints (16) are the partial volume constraints on the

target structures and enforce coverage requirements. Finally, the beamlet intensities and total

dose deposited are required to be non-negative while the artificial variables cs are free. Note that

the nonlinear function (·)+ in the partial volume constraint can be linearized in the standard way

leading to a linear programming formulation of the above FMO model.

3.2 Parameter Search

As mentioned before, the coverage and conformity corresponding to a target structure s ∈ T in a

treatment plan resulting from the FMO model (12)-(19) depends on the values of the parameters

αb(s) and αs in the upper and lower partial volume constraints (15) and (16) for b(s) and s ∈T , respectively. In this section, we describe a search technique for appropriate values of these

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parameters. For simplicity of exposition we assume that there is only one target structure, i.e.,

T = {t}.Recall that, for good coverage and conformity for the target t, high values of αb(t) and αt are

preferable, however this may lead to problem infeasibility. Let

Ft = {(αb(t), αt) ∈ (0, 1)2 : The FMO model (12)-(19) is feasible}.

Note that if (α∗t , α∗b(t)) ∈ Ft, αt ≤ α∗t and αb(t) ≤ α∗b(t), then (αt, αb(t)) ∈ Ft. Assuming Ft is

non-empty we would like to maximize αt and αb(t) over Ft. Such solutions will lie on the upper

boundary of Ft. Our search technique starts with an initial solution and goes through several

search phases to identify solutions on the upper boundary of Ft. Figure 2 illustrates the search

phases, which are now detailed. Initial values of the parameters αt and αb(t) are chosen based on

a minimum coverage requirement MinCov and a maximum allowed conformity value MaxConf .

In particular

α0t = MinCov × β (20)

α0b(t) = (1− MinCov(MaxConf − 1)Vt

Vb(t))× β (21)

where 0 < β < 1 is scaling factor to account for the fact that the bounds (10) and (11) are

conservative. In our implementation for case studies we used β = 0.9.

We solve the FMO model (12)-(19) with the initial parameter values (α0b(t), α

0t ). If the problem

is infeasible, i.e. (α0b(t), α

0t ) 6∈ Ft, then we execute Phase 0. Here we decrease both parameters by λ,

where 0 < λ < 1, and resolve the FMO model. This is continued until a feasible set of parameters

is found. The structure of the set Ft guarantees that if Ft is non-empty, then Phase 0 will produce

a feasible set of parameters.

Once we have initial feasible parameters, we execute Phase 1, where we increase both αb(t)

and αt by λ and resolve the FMO model. This is repeated until we reach an infeasible set of

parameters. The last set of feasible parameters, denoted by (α1b(t), α

1t ), then is a solution near the

upper boundary of Ft.

Next we execute Phase 2 where, starting from (α1b(t), α

1t ), we increase α1

t by λ until we reach an

infeasible set of parameters. The motivation here is that, since increasing αt improves both coverage

and conformity, we want to obtain additional solutions with good coverage and conformity values.

The last feasible solution from Phase 2 is denoted by (α2b(t), α

2t ).

In order to find additional candidate solutions on the boundary of Ft, we execute Phase 3. Here

we first reduce αb(t) by λ, and then start Phase 2 to increase αt. Note that in the solutions produced

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a t

a b(t)

Initial Values

(Until a Feasible Solution)

P h a s e

0

P h a s e

1

P h

a s e

2

Phase 3

Phase 4

(Until Infeasibility)




Final Values

F t (a

b(t) , a

t )

F t ( a b ( t ) , a t )

(a b(t)

1 , a t 1 )

(a b(t)

2 , a t 2 )

(a b(t)

3 , a t 3 )

(a b(t)

0 , a t 0 )

Figure 2: Parameter Search for αt and αb(t)

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in this phase the conformity values may be worse than those from Phase 2, however coverage will

not worsen.

Finally, if Phases 2 and 3 are not able to produce feasible parameters that are greater than

(α1b(t), α

1t ), then we execute Phase 4. Here we increase αb(t) by λ in order to find additional solutions

near the upper boundary of Ft.

The step size λ is an important component of the search process. From initial experimentations,

we chose this to be λ = 0.01.

3.3 Beam Selection

Careful selection of beam angles is an important part of creating IMRT plans as these angles

provide the candidate beamlets for which the intensities are optimized. Holder et al. ([5]) provide

a rigorous presentation of the beam selection problem and some comparisons of current algorithms

on phantom cases. (See Webb [2] for a classification of beam selection algorithms proposed in the

literature.) Clinical use of the beam selection algorithms are rare and most commercial planning

systems still rely on manual selection of beam angles by planners (Pugachev and Xing [17]). In this

section, we describe a scheme for selecting a set of beam angles (geometry/configuration) to deliver

the treatment from. Suppose there are 18 candidate angles and we wish to select 8 delivery angles,

then there are 43,758 choices making complete enumeration clinically impossible. (See Liu et al.

[11] for a complete enumeration based method.) Instead of evaluating all possible combinations

using fluence optimization, we use a bi-criteria scoring approach to limit our choices to a relatively

small number of non-dominated combinations. We compare our beam selection methodology with

scoring-based approach of Pugachev and Xing ([16]) in Section 4.

The first step is to use the FMO model and parameter search to determine a treatment plan

satisfying prescription criteria assuming that all candidate beam angles (say there are M of these)

can be used. Let x∗i be the intensity of beamlet i in beam (angle) B (where B = 1, . . . , M) and z∗jt

be the dose delivered to voxel j of the target structure t (PTV) in this treatment plan. Each beam

(angle) B is then assigned the following two weights:

Total Dose Delivered to PTV. The total dose delivered to PTV by beam B is computed as

DPTVB =∑

i∈B

Vt∑

j=1

Dijtx∗i . (22)

Preferring beams with a higher value of DPTVB favors beams that deliver higher dose to

the target. The disadvantage of this measure is that it does not directly consider the dose

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received by PTV voxels in the lower tail of the dose distribution, i.e., cold spots, and cold

spots have a significant effect on treatment quality.

Weighted Sum of Dose Delivered to the Cold Region of PTV. We define the cold region Ct of PTV

as the set of voxels which receive a dose less than 10% above the minimum dose observed in

PTV, i.e.,

Ct = {j = 1, . . . , Vt : z∗jt ≤ 1.10minj

(z∗jt)}. (23)

The weighted sum of dose delivered to the cold region of PTV from beam angle B is computed

as

WPTVB =∑

i∈B

∑

j∈Ct

Dijtx∗i

1max{z∗jt, ε}

, (24)

where ε > 0 is included to avoid division by zero. Preferring beams with higher value of

WPTVB favors beams that deliver more dose to the cold region of the target. Note that

we distinguish voxels in the cold region by assigning weights inversely proportional to the

actual dose received by these voxels. Thus, a beam delivering the same amount of dose as

another beam but to colder voxels will have a higher score. The advantage of this measure

is that it can help to identify beams, the blocking of which may result in a significant loss of

dose delivery to the cold region of PTV and thus lower tumor control. However, a scoring

mechanism purely based on WPTVB values may lead to formation of new cold spots, because

these beams are likely to intersect at the same part of PTV and may be ineffective in other

parts of PTV.

Next, we consider each possible choice of L delivery angles from the total M angles. (Note

there are(ML

)possibilities.) For each of the possible configurations, i.e., a set of beam angles

C ⊂ {1, . . . , M} such that |C| = L, we compute the configuration weights, i.e.,

DPTV (C) =∑

B∈C

DPTVB, and

WPTV (C) =∑

B∈C

WPTVB.

Next, we identify a set K of non-dominated configurations, i.e.,

K = {C :6 ∃ D s.t. DPTV (D) > DPTV (C) and WPTV (D) > WPTV (C)}.

Ideally, to choose between the non-dominated configurations in K, we would generate an optimized

treatment plan for each (using the FMO model with parameter search) and select the one with the

15

highest quality. However this may be too time consuming. Instead, we perform a “greedy” search.

We start with an arbitrary configuration C ∈ K and execute Phase 1 and Phase 2 of the parameter

search. Let the resulting set of parameter values be (α2b(t), α

2t ). Next, we search for a configuration

C ∈ K for which (α2b(t), α

2t + λ) is feasible. If no such configuration C exists, then configuration

C is selected. On the other hand, if such a configuration C exists, we iterate and execute Phase 1

and Phase 2 of the parameter search for C starting from (α2b(t), α

2t ).

4 Computational Study

In this section, we present the results of a computational study conducted using three real-life cases:

a pediatric brain case, a head and neck case, and a prostate case. The computational study focuses

on the value of the techniques and solution schemes introduced above: (1) the use of virtual critical

structures to construct high-quality treatment plans in terms of coverage and conformity, (2) the

development of an automated, systematic parameter search scheme for treatment plan construction

models using conditional value at risk constraints to model partial dose volume constraints, and

(3) the development of an efficient and effective beam angle selection scheme.

4.1 Case Descriptions

The computational study is conducted using three real-life cases: a pediatric brain case, a head-

and-neck case, and a prostate case. As these cases represent different parts of the body, and

thus a variety of challenges in terms of treatment plan generation, their use will demonstrate the

robustness of our approach; no tuning is necessary for the specific cases.

Information concerning the prescription requirements for the tumor as well as for the critical

structures (organs at risk) in terms of full volume and partial volume dose constraints for the three

cases can be found in Tables 1, 2, and 3. Partial dose volume constraints for the tumor (PTV) and

the virtual critical structure (VCS) are computed using the formulas presented in Section 3 using

a minimum coverage requirement of 0.95 and a maximum conformity requirement of 1.2. Observe

that there are no partial dose-volume constraints for critical structures in the pediatric brain case.

To be able to analyze and judge the value of the different techniques and solution schemes we

have developed, we start with base settings in each of the three cases. With base settings radiation

is delivered from 8 equi-spaced beams in the pediatric brain case, from 8 equi-spaced beams in the

head and neck case, and from 6 equi-spaced beams in the prostate case. Furthermore, the full and

16

Constraint Type Structure #Voxels Percentage LB (cGy) UB (cGy)

PTV 1,620 - 0 3,300

Hypothalamus 88 - 0 2,160

Right cochlea 26 - 0 1,260

Full volume Left cochlea 21 - 0 1,260

Pituitary 11 - 0 2,160

Left eye 158 - 0 500

Right eye 163 - 0 500

VCS 6,581 - 0 3,300

Partial volume PTV 1,620 95 3,060 -

VCS 6,581 95 - 3,060

Table 1: Brain Case - Structures and Constraints

partial dose volume constraints as specified in Tables 1, 2, and 3 are used except for those related

to the virtual critical structure.

4.2 Value of the Virtual Critical Structure

The virtual critical structure was introduced to improve conformity. Our first computational ex-

periment focuses on whether the introduction of the virtual critical structure indeed improves

conformity and, if so, by how much. This simply means that we compare the base setting with and

without the full and partial dose volume constraints related to the virtual critical structure. The

results are presented in Table 4. We see that the effect of introducing the virtual critical structure

is dramatic. For all three cases, the conformity is significantly improved. We also observe that the

coverage is improved in all three cases, although only slightly. We pay a small price in terms of the

coldspot; for both the head and neck case and the prostate case, the minimum dose has decreased

slightly.

4.3 Value of the Parameter Search

As the treatment plan construction model only indirectly attempts to optimize coverage and confor-

mity, we may be able to improve coverage and conformity by judiciously adjusting the parameters

αPTV and αV CS . In Section 3.2, we have outlined an automated, systematic parameter search

17


PTV 2,647 - 0 6,120

Brainstem 1,075 - 0 5,400

Spinal cord 387 - 0 4,500

Globe LT 388 - 0 2,000

Globe RT 355 - 0 2,000

Full volume Optic chiasm 13 - 0 5,400

Optic nerve LT 26 - 0 5,400

Optic nerve RT 10 - 0 5,400

Parotid LT 811 - 0 5,400

Parotid RT 832 - 0 5,400

VCS 7,634 - 0 6,120

PTV 2,647 95 5,100 -

Partial volume Parotid LT 811 50 - 2,600

Parotid RT 832 50 - 2,600

VCS 7,634 93 - 5,100

Table 2: Head & Neck Case - Structures and Constraints

scheme to produce treatment plans with better coverage and conformity than the plans produced

with the initial parameter values. In the next computational experiment, we focus on the improve-

ment in coverage and conformity that can be achieved through the parameter search. We compare

the base settings (with full and partial dose volume constraints for the virtual critical structure in-

cluded) with and without parameter search. The results are presented in Table 5. We observe that

there are significant improvements for coverage and conformity for the brain and the prostate case.

However, for the head-and-neck case the slight improvement in coverage is offset by a deterioration

in conformity. The parameter search for the brain case and for the head and neck case end with a

Phase 3 iteration, in which the virtual critical structure constraint is relaxed to see if that allows

for treatment plans with higher coverage. As a result, the value of conformity increases slightly in

the last iteration. More details are provided in Figure 3 where we plot the change in the evaluation

metrics during the course of the parameter search.

18


PTV 239 - 0 3,300

Full volume Rectum 1,267 - 0 2,160

Bladder 1,513 - 0 1,260

VCS 2,544 - 0 3,300

PTV 239 95 7,560 -

Rectum 2,544 70 - 7,560

Partial volume Rectum 2,544 50 - 4,500

Bladder 1,513 50 - 4,500

VCS 2,544 98 - 7,560

Table 3: Prostate Case - Structures and Constraints

Brain Head & Neck Prostate

without VCS with VCS without VCS with VCS without VCS with VCS

Coverage 0.954 0.957 0.953 0.956 0.933 0.950

Conformity 1.533 1.274 1.742 1.230 1.601 1.264

Coldspot 0.669 0.704 0.328 0.311 0.853 0.829

Hotspot 1.078 1.078 1.200 1.200 1.150 1.150

Table 4: Value of Virtual Critical Structure

4.4 Beam Angle Selection

Carefully selecting beam angles may significantly enhance the ability to construct high-quality

treatment plans. However, due to the large number of possible beam angles and the complexity

of generating high-quality treatment plans given a set of beam angles, beam angle selection has

not been extensively investigated and few effective schemes for beam angle selection exist. In the

next computational experiments, we focus on the effect of the beam angle selection scheme we have

developed. A core ingredient of our scheme is identifying non-dominated beam configurations with

respect to two quantities computed for a treatment plan in which all candidate beam angles are

available for use: (1) the dose delivered to the PTV and (2) the dose delivered to cold regions of

the PTV. In Figures 4, 5, and 6, we plot the scores for all 8-beam configurations for the brain

19


without with without with without with

parameter parameter parameter parameter parameter parameter

search search search search search search

Coverage 0.957 0.984 0.956 0.959 0.950 1.000

Conformity 1.274 1.212 1.230 1.261 1.264 1.046

Coldspot 0.704 0.822 0.311 0.313 0.829 1.000

Hotspot 1.078 1.078 1.200 1.200 1.150 1.150

Table 5: Value of Parameter Search

and head and neck cases and all 6-beam configurations for the prostate case selected from 18 equi-

spaced candidate beams and highlight the configurations that are non-dominated. The figures show

that few non-dominated configurations exist. In Table 6, we provide additional information to

support the observation that very few non-dominated configurations exist. For configurations of

different sizes, we report the total number of configurations as well as the number of non-dominated

configurations.

Next, we explore the effect of carefully selecting beam angles, and thus of integrating all the

techniques and solution schemes developed. The results are presented in Table 7. The results

are mostly self-explanatory. They clearly demonstrate the value of the various techniques and

solution schemes we have introduced. However, these evaluation metrics only tell part of the story.

Clinicians like to examine dose volume histograms as well. We show dose volume histograms in

Figures 7, 8, and 9 for the brain case, the head and neck case, and the prostate case, respectively.

Each of the figures displays dose volume histograms for three treatment plans: (1) a plan with equi-

spaced beams obtained by solving the model, but without parameter search, (2) a plan with equi-

spaced beams obtained by solving the model incorporating parameter search, and (3) a plan with

optimized beam angles obtained by solving the model incorporating parameter search. The dose

volume histograms show that improving coverage and conformity (i.e., by performing a parameter

search) can have a negative impact on the dose volume histograms of the organs at risk (increased

doses delivered), but that by carefully selecting the beam angles these undesirable effects can be

mostly negated. This can be observed especially well in the head and neck case; consider for

example the brainstem.

20

Configuration Size # Non-Dominated Configurations # Configurations


2 4 2 2 153

3 5 3 2 816

4 7 4 3 3,060

5 8 5 3 8,568

6 10 5 2 18,564

7 13 7 2 31,824

8 12 8 3 43,758

9 12 7 3 48,620

Table 6: Number of Non-dominated Configurations for Various Configuration Sizes

Comparison with another Beam Selection Algorithm

To further validate our approach, we compare our results to the results obtained by selecting the

beam angles according to the pseudo Beam’s-Eye-View (pBEV) scores introduced by Pugachev and

Xing ([16]). The steps for the pBEV calculation of a given beam angle are as follows:

1. Find the voxels crossed by each beamlet of the beam angle;

2. Assign each beamlet an intensity that delivers a dose equal to or higher than the prescription

dose in every target voxel;

3. For each OAR or normal tissue voxel crossed by the beamlet, calculate the factor by which

the beamlet intensity has to be multiplied to ensure the tolerance dose is not exceeded (factor

less than or equal to 1);

4. Find the minimum factor among the ones calculated in Step 3;

5. Adjust the beamlet intensities obtained in Step 2 using the minimum factor. The resulting

values, referred to as the “maximum” beam intensity profile, represent the maximum usable

intensity of the beamlet;

6. Perform a forward dose calculation using the “maximum” beam intensity profile;

21

Case Evaluation Equi-spaced Equi-spaced & Equi-spaced & Optimized angles &

measure VCS VCS & VCS &

Parameter search Parameter search

Coverage 0.954 0.957 0.984 1.000

Brain Conformity 1.533 1.274 1.212 1.125

Coldspot 0.669 0.704 0.822 1.000

Hotspot 1.078 1.078 1.078 1.078

Coverage 0.953 0.956 0.959 0.974

Head & Conformity 1.742 1.230 1.261 1.232

Neck Coldspot 0.328 0.311 0.313 0.310

Hotspot 1.200 1.200 1.200 1.200

Coverage 0.933 0.950 1.000 1.000

Prostate Conformity 1.601 1.264 1.046 1.033

Coldspot 0.853 0.829 1.000 1.000

Hotspot 1.150 1.150 1.150 1.150

Table 7: Evaluation Metrics for Different Schemes

7. Compute the score for chosen beam angle i according to an empiric score function as follows:

Si =1Vt

Vt∑

j=1

( dij

PDt

)2 (25)

where dij is the dose delivered to voxel j from beam angle i, Vt is the number of voxels in the

target, and PDt is the target prescription dose.

We selected the final configuration simply by picking the highest scoring beams. (In [16], Pugachev

and Xing manually pick the beams in the case examples when some high-scoring angles are too

close to each other to create a separation.)

We computed beam angle configurations for the cases using pBEV scores and our proposed

scores, and then performed fluence map optimization using our technology. Multiple solutions are

produced for each case by varying the number of beam angles. The configuration sizes are 6, 7, 8

and 9 for both brain and head and neck cases, and 4, 5, 6 for the prostate case.

The average values of the evaluation metrics (over the different configuration sizes) are presented

in Table 8. The values of coverage and conformity are better in all cases with our proposed scheme.

22

Case Evaluation pBEV Optimized Angles

meatric

Coverage 0.974 0.991

Brain Conformity 1.209 1.146

Coldspot 0.949 0.989

Hotspot 1.078 1.078


Head & Conformity 1.219 1.214

Neck Coldspot 0.344 0.311

Hotspot 1.200 1.200


Prostate Conformity 1.045 1.033

Coldspot 1.000 1.000

Hotspot 1.150 1.150

Table 8: Comparison of Evaluation Metrics for Beam Angles Selected with pBEV and Our Scheme

The improvement is significant especially for the brain case.

The main difference between our scoring scheme and the pBEV scoring scheme is that we score

beams based on their exact contributions to the delivered doses to target structures in an optimized

solution rather than just using their geometric properties. Another important difference is that we

consider multiple beam configurations in the parameter search phase, which allows us to adjust

dynamically in response to observed interaction effects between beams. In the pBEV algorithm

there is only one configuration which is chosen based on initial parameter values.

4.5 Beam Configuration Size

The results presented so far assumed that the desired number of beam angles was decided in

advance. We have seen that by carefully selecting beam angles substantially better treatment plans

can be generated. This suggests that it may be possible to get high-quality treatment plans with

fewer beam angles. Table 9 presents the values of the evaluation criteria for different sizes of beam

configurations. The same information is displayed graphically in Figure 10.

A number of observations can be made. As expected, there are diminishing returns when we

23

2 beams 3 beams 4 beams 5 beams 6 beams 7 beams 8 beams 9 beams

Coverage 0.872 0.898 0.939 0.955 0.963 1.000 1.000 1.000

Brain Conformity 1.443 1.400 1.283 1.222 1.214 1.144 1.125 1.099

Coldspot 0.636 0.736 0.912 0.936 0.956 1.000 1.000 1.000

Hotspot 1.078 1.078 1.078 1.078 1.078 1.078 1.078 1.078

Coverage 0.923 0.937 0.961 0.965 0.968 0.971 0.974 0.973

Head & Neck Conformity 1.420 1.350 1.286 1.252 1.219 1.243 1.232 1.162

Coldspot 0.331 0.341 0.326 0.317 0.301 0.320 0.310 0.314

Hotspot 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200

Coverage 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Prostate Conformity 2.515 1.556 1.029 1.038 1.033 1.042 1.042 1.038

Coldspot 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Hotspot 1.150 1.120 1.150 1.150 1.150 1.150 1.150 1.150

Table 9: Metric Values for Different Beam Configuration Sizes

continue to increase the size of the configuration. (This is especially clear for the prostate case

where treatment plans of almost equal quality are produced for configurations of size 4 and up.)

Furthermore, treatment plans with good quality can already be achieved with configurations of

relatively small size (which has many practical advantages). If we compare the results presented

in Table 7 to those presented in Table 9, we see that carefully selecting beam angles allows us to

construct treatment plans of equal or better quality than those that can be obtained with equi-

spaced beams with fewer beams.

The above results indicate that our beam angle selection scheme produces beam configurations

that allow the construction of high-quality treatment plans. However, it is possible that other

beam configurations may allow even better treatment plans. Therefore, to further assess the quality

of the beam angle configurations produced by our scheme, we compare the beam configurations

produced by our scheme to those produced by an optimization approach based on an integer

programming formulation (Lee et al. [8, 9], Yang et al. [22], and Lim et al. [10]). Due to the

computational requirements of the optimization approach this comparison can only be performed

for settings with a relatively small number of candidate beam angles. We experimented with

selecting 5 beam angles out of 8 equi-spaced candidate beam angles for the brain and the head

and neck case. Both times, our selection scheme produced the optimal configuration, i.e., the same

configuration that the optimization approach produced. It is informative to look at the difference

in required computation time; see Table 10. It is clear that the optimization approach will become

computationally prohibitive when 10 or more candidate beam angles are used.

24

Time (secs.) Ratio

Scheme MIP

Brain 348 5,676 1 : 16

Head & Neck 363 16,988 1 : 46

Table 10: Comparison with Mixed Integer Programming Approach

Voxels Beamlets #cons #vars primal dual barrier

simplex simplex

Brain 8,668 1,010 17,881 16,871 1,548 1,395 48

Head & Neck 14,178 1,599 27,705 26,106 31,118 4,912 58

Table 11: Run Time (seconds) for Various Linear Programming Algorithms

4.6 Solution Times

One of the primary goals of our research is to automate the treatment plan generation in order to

save clinicians valuable time. A related, equally important goal is to do so in a reasonably short

period of time, so that more plans can be generated in a short amount of time and thus, hopefully,

more patients can be treated. Therefore, our next set of experiments are aimed at assessing the

run time requirements of our solution scheme. As mentioned above, we chose to develop only

linear programming based technology, because linear programs can be solved efficiently. Several

methodologies exist for solving linear programs, e.g., primal simplex, dual simplex, and interior

point methods, and our first experiment examines the solution times for the different solution

methods for our class of linear programs. We have used the linear programming solver of XPRESS

(Xpress-Optimizer 2007). Table 11 presents the instance characteristics for the head and neck case

and the brain case when generating a treatment plan for 8 equi-spaced beams (using a virtual critical

structure) and the run times in seconds for the three linear programming algorithms (with default

parameter settings). All our experiments were run on a 2.66GHz Pentium Core 2 Duo processor

with 3GB of RAM under Windows XP Operating System. The differences are staggering; for this

class of linear programs, interior point methods are by far superior. All remaining computational

experiments therefore use XPRESS’s barrier algorithm to solve linear programs.

Our next set of results illustrates the impact on computation times of including a virtual critical

25

structure. The results are presented in Table 12. The results show that we pay a price in terms of

Voxels Beamlets #cons #vars time (secs)

without Brain 2,087 1,010 3,708 4,781 18

VCS Head & Neck 6,544 1,599 10,837 12,436 26

Prostate 3,019 382 7,309 7,691 2

with Brain 8,668 1,010 17,881 16,871 48

VCS Head & Neck 14,178 1,599 27,705 26,106 58

Prostate 5,563 382 12,398 12,780 3

Table 12: Impact of Virtual Critical Structure on Run Time

computing time, primarily due to the increase in number of voxels, for the introduction of the virtual

critical structure. However, as we have seen in earlier experiments, the value of the introduction of

critical structures is significant, so it seems a reasonable price to pay.

The first step of our scheme to select beam angles is to solve the problem assuming that all

candidate beam angles are used. In our computation experiments this meant solving the problem

using 18 candidate beam angles (equi-spaced). The solution is used to get the beam angle scores

WPTV and DPTV . Table 13 presents the instance characteristics and the run times for the

three cases. Once the beam angle scores are computed they are used to identify non-dominated

Voxels Beamlets #cons #vars time (secs)

Brain 8,668 2,306 16,871 19,177 123

Head & Neck 14,178 3,650 26,106 29,756 278

Prostate 5,563 1,146 12,398 13,544 24

Table 13: Run Time for 18 Equi-Spaced Beams for Scoring

configurations. Then treatment plans are generated for these configurations (using a virtual critical

structure and parameter search). In Table 14, we present the solution times for this component

of the scheme for different beam configuration sizes. The resulting total run times can be found

in Table 15. The maximum run time is about 20 minutes and observed for head and neck case to

choose 9 beams. Our proposed scheme was able to produce high quality solutions for the prostate

case in less than 2 minutes.

26


Brain 145 199 263 514 478 689 798 445

Head & Neck 147 141 231 311 365 646 863 1,003

Prostate 28 12 14 23 28 35 50 53

Table 14: Run Time (seconds) after Beam Scoring for different configuration sizes


Brain 268 322 386 637 601 812 921 568

Head & Neck 425 419 509 589 643 924 1,141 1,281

Prostate 52 36 38 47 52 59 74 77

Table 15: Total Run Time (seconds) for different configuration sizes

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29

Appendix: Proof of Proposition 1

For simplicity of exposition, we assume that αs is such that αsVs is a positive integer. Let Dαs be

such thatVs∑

j=1

I+(zjs −Dαs) = αsVs.

Then the coverage requirement (10) is equivalent to

Dαs ≥ PDs.

We prove the result by showing that if there exists cs satisfying (9) then we must have Dαs ≥ PDs.

Note that

Dαs ≥ Dαs − 1(1− αs).Vs

Vs∑

j=1

(Dαs − zjs)+,

which follows simply from the fact that (Dαs − zjs)+ is non-negative. We claim that

Dαs − 1(1− αs).Vs

Vs∑

j=1

(Dαs − zjs)+ ≥ cs − 1(1− αs).Vs

Vs∑

j=1

(cs − zjs)+ ∀ cs ∈ R.

Suppose this is not true, that is, there exists c∗s such that

Dαs − 1(1− αs).Vs

Vs∑

j=1

(Dαs − zjs)+ < c∗s −1

(1− αs).Vs

Vs∑

j=1

(c∗s − zjs)+, (26)

and consider the following two cases.

Case 1: suppose that c∗s ≤ Dαs . Then the right-hand-side (rhs) of (26) is

rhs = c∗s −1

(1− αs).Vs

∑

j: c∗s>zjs

(c∗s − zjs)

= c∗s −1

(1− αs).Vs

∑

j: Dαs>zjs

(c∗s − zjs)−∑

j: c∗s≤zjs<Dαs

(c∗s − zjs)

= c∗s −1

(1− αs).Vs

∑

j: Dαs>zjs

(c∗s − zjs) +1

(1− αs).Vs

∑

j: c∗s≤zjs<Dαs

(c∗s − zjs)

︸︷︷︸≤0

≤ c∗s −1

(1− αs).Vs

∑

j: Dαs>zjs

(c∗s −Dαs + Dαs − zjs)

= c∗s −1

(1− αs).Vs

Vs∑

j=1

(Dαs − zjs)+ − 1(1− αs).Vs

∑

j: Dαs>zjs

(c∗s −Dαs)

30

= c∗s −1

(1− αs).Vs

Vs∑

j=1

(Dαs − zjs)+ − (c∗s −Dαs)

= Dαs − 1(1− αs).Vs

Vs∑

j=1

(Dαs − zjs)+ = lhs,

where lhs is the left-hand-side of inequality (26), hence a contradiction. Note that the last step

in the above follows from the definition of Dαs which implies |{j : zjs < Dαs}| = (1− αs)Vs.

Case 2: suppose that c∗s > Dαs . Then the right-hand-side (rhs) of (26) is

rhs = c∗s −1

(1− αs).Vs

∑

j: c∗s>zjs

(c∗s − zjs)

= c∗s −1

(1− αs).Vs

∑

j: Dαs>zjs

(c∗s − zjs) +∑

j: Dαs≤zjs<c∗s

(c∗s − zjs)

= c∗s −1

(1− αs).Vs

∑

j: Dαs>zjs

(c∗s − zjs)− 1(1− αs).Vs

∑

j: Dαs≤zjs<c∗s

(c∗s − zjs)

︸︷︷︸≥0

≤ c∗s −1

(1− αs).Vs

∑

j: Dαs>zjs

(c∗s −Dαs + Dαs − zjs)

= Dαs − 1(1− αs).Vs

Vs∑

j=1

(Dαs − zjs)+ = lhs,

where the last step is identical to Case 1. Hence we again have a contradiction.

Thus if there exists cs satisfying (9) then

Dαs ≥ Dαs − 1(1− αs).Vs

Vs∑

j=1

(Dαs − zjs)+ ≥ cs − 1(1− αs).Vs

Vs∑

j=1

(cs − zjs)+ ≥ PDs

completing the proof.

31

0

1

85-85 87-87 89-89 91-91 89-92

αB30MM-αPTV (%)

Evalu

ati

on

Metr

ic

0

1

84-85 85-86 82-87

αB30MM-αPTV (%)

Evalu

ati

on

Metr

ic

Coverage

Conformity

Coldspot

Hotspot

0

1

88-85 90-87 92-89 94-91 96-93 98-95 99-97

αB30MM-αPTV (%)

Evalu

ati

on

Metr

ic

a)

b)

c)

Figure 3: Change in Evaluation Metrics with Parameter Search: (a) Brain Case 8 Equi-spaced

Beams, (b) Head & Neck Case 8 Equi-spaced Beams, (c) Prostate Case 6 Equi-spaced Beams

32

50

400

750

1100

1450

1800

7.43E+05 1.24E+06 1.74E+06 2.24E+06 2.74E+06 3.24E+06 3.74E+06 4.24E+06

DPTV

WPTV

Figure 4: Brain Case - Dominated and Non-dominated Configurations

500

1500

2500

3500

4500

5500

6500

7500

3.19E+06 4.19E+06 5.19E+06 6.19E+06 7.19E+06 8.19E+06 9.19E+06 1.02E+07 1.12E+07

DPTV

WPTV

Figure 5: Head & Neck Case - Dominated and Non-dominated Configurations

33

50

5050

10050

15050

20050

25050

70490 270490 470490 670490 870490 1070490 1270490 1470490

DPTV

WPTV

Figure 6: Prostate Case - Dominated and Non-dominated Configurations

34

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000 3500

Dose (cGy)

Vo

lum

e(%

)

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000 3500

Dose (cGy)

Vo

lum

e(%

)

PTV

B30MM

Lt_Eye

Pituitary

Lt_Cochlea

Rt_Cochlea

Hypothalamus

Rt_Eye0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000 3500

Dose (cGy)

Vo

lum

e(%

)

a)

b)

c)

Figure 7: Brain Case Dose-Volume-Histograms: (a) Basic 8 Equi-spaced Beams Solution, (b) Final

8 Equi-Spaced Beams Solution, (c) Final 8 Selected Beams Solution

35

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500

Dose (cGy)

Volu

me(%

)

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500

Dose (cGy)

Volu

me(%

)

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500

Dose (cGy)

Volu

me(%

)

PTV

B30MM

Spinal_cord

Globe_lt

Globe_rt

Optic_chiasm

Optic_nerve_lt

Optic_nerve_rt

Parotid_lt

Parotid_rt

Brainstem

a)

b)

c)

Figure 8: Head & Neck Case Dose-Volume-Histograms: (a) Basic 8 Equi-spaced Beams Solution,

(b) Final 8 Equi-Spaced Beams Solution, (c) Final 8 Selected Beams Solution

36

0

20

40

60

80

100

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Dose (cGy)

Vo

lum

e(%

)

0

20

40

60

80

100

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Dose (cGy)

Vo

lum

e(%

)

PTV

B30MM

bladder

rectum

0

20

40

60

80

100

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Dose (cGy)

Vo

lum

e(%

)

a)

b)

c)

Figure 9: Prostate Case Dose-Volume-Histograms: (a) Basic 6 Equi-spaced Beams Solution, (b)

Final 6 Equi-Spaced Beams Solution, (c) Final 6 Selected Beams Solution

37

0.0

0.5

1.0

1.5


Configuration

Evalu

ati

on

Metr

ic

0.0

0.5

1.0

1.5


Configuration

Evalu

ati

on

Metr

ic

Coverage

Conformity

Coldspot

Hotspot

0

1

2

3


Configuration

Evalu

ati

on

Metr

ic

a)

b)

c)

Figure 10: Final Results Returned by the Algorithm with Different Number of Beams Selected: (a)

Brain Case, (b) Head &Neck, (c) Prostate

38