optimization methods for high dose rate brachytherapy ... · modulated brachytherapy (embt), are...

Optimization methods for high dose rate brachytherapytreatment planning

by

Elodie R. Mok Tsze Chung

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Mechanical and Industrial EngineeringUniversity of Toronto

Copyright c© 2016 by Elodie R. Mok Tsze Chung

Abstract

Optimization methods for high dose rate brachytherapy treatment planning

Elodie R. Mok Tsze Chung

Master of Applied Science

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

2016

Optimization approaches for treatment planning in two novel high-dose-rate (HDR)

brachytherapy techniques, direction-modulation brachytherapy (DMBT) and energy-

modulated brachytherapy (EMBT), are investigated for cervical cancer and prostate

cancer. Brachytherapy is a form of radiation therapy where a radioactive source is placed

inside the body to irradiate the tumour internally. Conventionally, only one source is used

and it is unshielded, thus providing an isotropic dose distribution. DMBT makes use of

a new shielded applicator that is capable of delivering highly directional radiation distri-

butions. In EMBT, three HDR sources, 192Ir, 60Co, and 169Yb, are used in combination

to provide variety in dose profiles. To investigate the benefit of these two new techniques

over conventional brachytherapy, we use an inverse planning approach to generate the

treatment plans. We model the treatment planning problem as a quadratic program and

use an interior point constraint generation algorithm to generate the treatment plans.

ii

Acknowledgements

First and foremost, I would like to thank my two supervisors and mentors, Dr. Dionne

Aleman and Dr. William Song for patiently guiding and encouraging me throughout my

entire research. This work would not have been possible without their support and

expertise.

I am grateful to my colleagues and friends from the Medical Operations Research

Laboratory and Sunnybrook Health Sciences Centre for their guidance as I embarked on

my journey at U of T, as well as their insightful ideas and helpful discussions about my

research work. They have helped me overcome many obstacles and I have learned so

much from them.

Lastly, my biggest thanks go to my family for their everlasting love and support from

halfway across the globe. Without them, I would not be the person I am today. As hard

as it was to be away from them, their words of encouragement cheered me through the

hardships of my degree. I would also like to thank Cole for being my rock over the last

two years.

iii

Contents

1 Introduction 1

1.1 Brachytherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Inverse planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Publications and presentations . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Methodology 9

2.1 Treatment plan evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Interior point constraint generation algorithm . . . . . . . . . . . . . . . 14

3 Direction-modulated brachytherapy 16

3.1 DMBT optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Energy-modulated brachytherapy 29

4.1 EMBT optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

iv

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Conclusion 49

Bibliography 51

v

List of Tables

3.1 Patient information for cervical cancer cancer . . . . . . . . . . . . . . . 21

3.2 Plan quality comparison between conventional BT and DMBT . . . . . . 22

3.3 Homogeneity index and conformal index for cervical cancer plans . . . . 24

3.4 Computation time for cervical cancer plans . . . . . . . . . . . . . . . . . 26

4.1 HDR source information . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Patient information for prostate cancer . . . . . . . . . . . . . . . . . . . 34

4.3 Clinical protocols for prostate cancer . . . . . . . . . . . . . . . . . . . . 34

4.4 Plan quality summary for the target volume . . . . . . . . . . . . . . . . 35

4.5 Plan quality summary for the OARs . . . . . . . . . . . . . . . . . . . . 36

4.6 Homogeneity index for the prostate cancer plans . . . . . . . . . . . . . . 37

4.7 Conformal index for the prostate cancer plans . . . . . . . . . . . . . . . 38

4.8 Comparison of source combinations to 192Ir plan . . . . . . . . . . . . . . 39

4.9 Treatment time for prostate cancer plans . . . . . . . . . . . . . . . . . . 43

4.10 Breakdown of TRAK for prostate cancer plans . . . . . . . . . . . . . . . 44

4.11 Computation time for prostate cancer plans . . . . . . . . . . . . . . . . 45

vi

List of Figures

1.1 HDR brachytherapy treatment . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 DTO penalty function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Linear approximations of a convex function . . . . . . . . . . . . . . . . . 14

2.3 morDiRECT evaluation window . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Tandem and ring applicator . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 DMBT tandem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Comparison of tandem design . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Comparison of conventional BT and DMBT plans . . . . . . . . . . . . . 23

3.5 Comparison of plan quality for a representative cervical case . . . . . . . 25

3.6 Computation time for DMBT . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Radial dose functions of 192Ir, 60Co, and 169Yb . . . . . . . . . . . . . . . 32

4.2 Depth dose functions of 192Ir, 60Co, and 169Yb . . . . . . . . . . . . . . . 32

4.3 Comparison of source combinations to 192Ir plan . . . . . . . . . . . . . . 40

4.4 Comparison of plan quality for a representative prostate case without DILs 41

4.5 Comparison of plan quality for a representative prostate case with DILs . 42

4.6 Computation time for EMBT . . . . . . . . . . . . . . . . . . . . . . . . 46

vii

Chapter 1

Introduction

Cancer is the leading cause of death in Canada [9]. In 2015, the Canadian Cancer Society

estimated 196,900 new cancer diagnoses and 78,000 deaths [10]. There are several ways

to treat cancer, including radiation therapy, chemotherapy, and surgery. In radiation

therapy, high energy radiation is directed to the tumour to kill the cancerous cells. The

goal is to deliver enough radiation to the tumour while minimizing the dose delivered to

the healthy tissues and critical structures surrounding the tumour, known as organs-at-

risk (OARs).

Radiation therapy can be performed externally or internally. In external beam ra-

diation therapy (EBRT), a machine directs beams of radiation from different directions

outside the patient towards the tumour. The radiation must travel through the skin and

healthy tissues before reaching the tumour. In internal radiation therapy, also known as

brachytherapy, radioactive sources are placed inside the body to deliver radiation to the

tumour internally.

1.1 Brachytherapy

Different types of brachytherapy can be defined according to three characteristics. The

first characteristic is the source placement. In intracavitary brachytherapy, the appli-

1

Chapter 1. Introduction 2

cators and the source are placed inside a body cavity near the tumour. This type of

brachytherapy is usually used for the treatment of cervical cancer, where the source is

placed in the vagina. In interstitial brachytherapy, the applicators in which the source

travels are inserted directly into the tumour tissue. Interstitial brachytherapy is com-

monly used to treat prostate or breast cancer. Other types include intralumenal and

intravascular brachytherapy, where the applicators are inserted inside a body lumen (a

tubular-shaped structure) and an artery, respectively.

The second characteristic is the duration of dose delivery. In temporary brachyther-

apy, radioactive sources are temporarily implanted inside the body. The time can range

from a few minutes to several days. On the other hand, in permanent brachytherapy,

small low dose rate radioactive seeds or pellets are permanently placed inside the body

and are left to decay. The radiation will decrease with time until it is insignificant and

the seeds are safe to remain in the body.

The third characteristic is the dose rate, which depends on the energy of the source

being used. Brachytherapy is divided into three modalities: high-dose-rate (HDR), low-

dose-rate (LDR), and pulse-dose-rate (PDR). HDR brachytherapy sources have a dose

rate greater than 12 Gray per hour (Gy/h), while LDR sources have a dose rate smaller

than 2 Gy/h. HDR brachytherapy treatments typically lasts for a few minutes and are

done in one or several sessions, called fractions. In LDR brachytherapy, the radioactive

sources (seeds) are implanted inside the tumour for a few days or permanently. Finally,

in PDR brachytherapy, the treatment is delivered in shorter “pulses”.

HDR brachytherapy offers many advantages. Due to the high dose rate of the source,

the treatment time is very short and can be mainly carried out on an outpatient basis.

Furthermore, it is minimally invasive compared to other treatment types such as surgery.

Unlike EBRT, HDR brachytherapy, as well as LDR and PDR brachytherapy, has the

advantage of reducing the dose delivered outside the tumour. Hence, it allows for higher

amounts of radiation to be prescribed to the tumour with limited exposure of the OARs.


HDR brachytherapy is also robust to tumour movement inside the body; since the source

is placed inside or near the tumour, its position relative to the tumour is generally

maintained.

Due to the high radioactivity of the source, usually the isotope Iridum-192 (192Ir),

treatment cannot be done manually and is instead delivered through a technique called

remote afterloading. After the applicators are inserted into the patient (Figure 1.1a),

they are connected to a computer-controlled machine, called an afterloader, using guid-

ing tubes (Figure 1.1b). The source is mounted at the end of a wire which is stored

in a shielded safe within the afterloader. Once the clinical staff leaves the room, the

afterloader is programmed to first send a dummy wire through the applicators to ensure

that the path is unobstructed (Figure 1.1c) and then send the HDR source through the

guiding tubes to pre-determined points in the applicator known as dwell positions (Figure

1.1d). The source sequentially stays at these dwell positions for a pre-specified amount

of time (Figures 1.1e, 1.1f, and 1.1g), known as the dwell time, after which it is pulled

out and returned to the shielded safe (Figure 1.1h).

The treatment potential of two novel HDR brachytherapy techniques are explored

using optimization methods to develop treatment plans. Conventionally, only one HDR

source is used during treatment and it is unshielded. The dose distribution about the

source is thus isotropic. The first technique is called direction modulated brachytherapy

(DMBT) and makes use of a shielded applicator to produce anisotropic dose distributions.

The second technique, called energy modulated brachytherapy (EMBT), makes use of

three HDR sources, 192Ir, Cobalt-60 (60Co), and Ytterbium-169 (169Yb), in combination.

1.2 Inverse planning

In radiation therapy, including brachytherapy, radiation kills both cancerous and healthy

cells. Therefore, treatments must be designed carefully for each patient to achieve an


(a) Applicators inserted in tumour (b) Applicators connected to afterloader

(c) Dummy wire sent through applicators (d) Dwell positions along catheters

(e) HDR source at the first dwell position (f) HDR source at another dwell position

(g) HDR source at the last dwell position (h) HDR source sent back to afterloader

Figure 1.1: HDR brachytherapy treatment procedure (Source: https://www.youtube.

com/watch?v=myxl4HeCcN4)

https://www.youtube.com/watch?v=myxl4HeCcN4

https://www.youtube.com/watch?v=myxl4HeCcN4


accurate treatment plan and a successful outcome. In brachytherapy, the amount of

radiation delivered by the source from a dwell position is determined by the correspond-

ing dwell time. Therefore, the selection of dwell positions and dwell times, known as

treatment planning, is a critical part of HDR brachytherapy.

Treatment plans are typically generated manually, called forward planning. That

is, the dwell positions and dwell times are iteratively changed until the desired dose

distribution is achieved. The trial-and-error nature of forward planning is time-consuming

and the quality of the treatment plans is heavily dependent on the experience and skill

of the planner.

Alternatively, inverse planning optimization can be used to develop treatment plans to

ensure that the best set of dwell positions and dwell times are selected. Inverse planning

starts with a set of dosimetric criteria and the anatomical information of the patient

obtained from ultrasound (US), computed tomography (CT) or magnetic resonance (MR)

images, and then uses optimization techniques to find the optimal set of dwell positions

and dwell times that satisfies the clinical objectives.

Inverse treatment planning has gained popularity over the last decade [13]. Several

mathematical models and solution techniques have been proposed to optimize brachyther-

apy treatment plans. The optimization techniques can be classified as heuristics or exact

methods. Heuristics produce a solution that is assumed to be “good enough” but can-

not be guaranteed to be optimal. Conversely, exact methods provide certainty in the

optimality of the solution, but may be computationally intensive.

Linear and integer programming are often used to model the brachytherapy treatment

planning problem as they can be solved by exact techniques. Linear programming is

usually used to optimize the dwell times given fixed dwell positions [5, 27, 28, 35]. They

are typically solved using the simplex method [5, 35] or commercial softwares, such as

CPLEX [27, 28]. For interstitial brachytherapy, the need to determine the insertion or

non-insertion of an applicator requires the use of integer programs to model the treatment


planning problem. Integer programs and mixed integer programs for brachytherapy have

been solved using branch-and-bound algorithms [41, 42, 76] and commercial softwares

[17, 23].

While these models can be solved exactly to produce an optimal solution, they be-

come increasingly hard to solve as the problem increases in size, especially with in-

teger programs [75]. As an alternative, heuristics can be used in HDR brachyther-

apy optimization. They can further be broken down into stochastic or deterministic

categories. Stochastic heuristics used in brachytherapy include simulated annealing

[15, 30, 32, 37, 43, 44, 74, 77], evolutionary algorithms [38–40, 52, 53, 70], and harmony

search [58]. Deterministic heuristics include gradient methods, such as the projected gra-

dient algorithm [25, 81, 82, 85], the Broyden-Fletcher-Goldberg-Shanno algorithm [54],

or the Fletcher-Reeves-Polak-Ribiere algorithm [54]. Non-gradient methods include the

modified Powell algorithm [54] and an attraction-repulsion model [79].

To combine the guaranteed optimality of an exact method with the speed of a heuris-

tic, we adapt the sector duration optimization (SDO) problem for stereotactic radio-

surgery (SRS) [4, 14, 18–20, 56] for our brachytherapy treatment planning problem. This

model is similar to the fluence map optimization (FMO) problem for intensity-modulated

radiation therapy (IMRT) [1–4, 51, 67–69]. We model the problem as a quadratic pro-

gram and solve it using an interior point constraint generation (IPCG) algorithm, which

was successfully implemented on a SRS inverse planning problem [56]. In SRS, beams

of radiation are directed to a point in the tumour, called an isocenter. Meanwhile in

brachytherapy, radiation diverges from the source at a dwell position. Dwell positions

and dwell times are therefore similar to isocenters and the time of radiation delivery from

the beams in SRS, respectively. Since brachytherapy and SRS are analogous, we expect

the IPCG algorithm to perform well on brachytherapy inverse planning problems.


1.3 Contributions

We contribute to the field of brachytherapy by exploring two new forms of treatment

delivery: DMBT for cervical cancer and EMBT for prostate cancer. We show that the

DMBT applicator, with its modulating capacity to produce anisotropic dose profiles,

allows for superior treatment plans compared to conventional brachytherapy. DMBT

especially proves useful in cases where the tumour is asymmetric or extends laterally.

Additionally, we demonstrate that the combination of different HDR sources in EMBT

allows for better OAR sparing without compromising target coverage. With the intro-

duction of afterloaders equipped with additional wires (i.e., capable of handling a second

HDR source) in the market, EMBT is now clinically viable.

We also contribute to the literature on HDR brachytherapy inverse planning. We

use an exact algorithm to solve a dwell time optimization problem to ε-optimality in a

finite number of iterations. This algorithm is able to handle large-scale convex problems,

which is desirable since the DMBT and EMBT problems are more complex than their

conventional counterparts.

1.4 Publications and presentations

The following contributions were made to the literature.

Publications

1. E. Mok Tsze Chung, H. Sagholi, A. Nicolae, M. Davidson, A. Ravi, D. Aleman,

W. Song. Evaluation of 192Ir, 60Co, and 169Yb sources for high dose rate prostate

brachytherapy inverse planning using an interior point constraint generation algo-

rithm. Work in progress.

Presentations

The underlined text indicate the presenter(s) of the work.


1. E. Mok Tsze Chung, H. Sagholi, A. Nicolae, M. Davidson, A. Ravi, D. Aleman, W.

Song. Evaluation of 192Ir, 60Co, and 169Yb sources for HDR prostate brachyther-

apy using an interior point constraint generation algorithm. INFORMS Annual

Conference, Nashville, TN, November 2016.



brachytherapy inverse planning using an interior point constraint generation algo-

rithm. AAPM Annual Conference, Washington DC. August 2016.



brachytherapy inverse planning using an interior point constraint generation al-

gorithm. Mechanical and Industrial Engineering Graduate Research Symposium.

University of Toronto, Canada. June 2016.

Chapter 2

Methodology

In our optimization model, we consider every dwell position along the applicator(s) as a

dwell position to be used, thereby ensuring the best possible treatment quality (though

potentially at the expense of treatment time). We then use a dwell time optimization

(DTO) model to optimize the dwell times of each dwell position so that the final dose

distribution meets the clinical objectives as much as possible. Then, we assess the treat-

ment plan quality with common evaluation metrics to determine whether the plans are

clinically acceptable or not.

2.1 Treatment plan evaluation

Once a patient is diagnosed and scheduled for HDR brachytherapy, s/he is imaged using

US, CT or MRI scanners. After the images are obtained, the structures (tumour volume

and surrounding OARs) are contoured by a radiation oncologist. The prescription dose

and the dose thresholds are obtained from the radiation therapist. The structure volumes

are then broken down into 3D pixels, called voxels. Voxels have a size of 1 mm x 1

mm x 1 mm, which depends on the image resolution. The applicators are inserted in

the patient and contoured on the planning scans, after which the location of the dwell

positions relative to the structures are obtained. The corresponding dwell times can then

9

Chapter 2. Methodology 10

be optimized.

The most common method to evaluate a treatment plan is to use the cumulative

dose-volume histogram (DVH) [62]. A DVH shows the percentage of a structure volume

that receives a certain amount of dose or more. The structures can be the target or the

OARs. From the DVH curve, clinically relevant DVH parameters are obtained: (1) Vx,

the structure volume that receives x% of the prescription dose, and (2) Dx, the dose

received by x% of the structure volume. Ideally, 100% of the tumour volume should

receive 100% of the prescription dose (V100 = 100%), while 100% of the OARS should

receive no dose at all. Isodose lines, which are curved lines joining points that receive

the same amount of radiation through the target volume overlaid on structure images,

are also used to assess the plan quality.

There are also a variety of dosimetric indices to assess the quality of a treatment plan.

To measure how well the isodose corresponding to the prescription dose covers the target

volume, we use the conformal index (COIN) [8]. We also use the homogeneity index (HI)

to describe the homogeneity of the dose delivered to the target volume [84]. Both COIN

and HI have an ideal value of 1, with values less than 1 indicating worse conformity and

homogeneity, respectively. COIN and HI are calculated using the following formulas:

COIN =PTVref

PTV× PTVref

BVref

HI =V100 − V150

V100

where PTVref is the target volume that is covered by the prescription dose (100% isodose

line), and BVref is the body volume that receives the prescription dose.

The treatment time is an important factor to consider in evaluating a treatment plan,

since longer treatment times mean that the patient needs to stay in the brachytherapy

unit and under anaesthesia for longer. Treatment time is assumed to be a linear sum

of the individual dwell times. This definition is not entirely accurate because transition


time between dwell positions is not accounted for. However, the dwell positions are

evenly spaced and hence the transition time should remain constant across plans, and

can therefore be ignored for comparison purposes.

2.2 Optimization model

Similar to existing optimization models for IMRT [1–4, 51, 67–69] and SRS [4, 14, 18–

20, 56], we formulate the brachytherapy treatment planning problem as a quadratic

program that minimizes the sum of penalties incurred by deviating from the desired

dose per voxel. The only constraints are that the dwell times must be non-negative

and bounded. Quadratic programs generally reflect clinical attitudes towards overdose

and underdose, i.e., small deviations are more acceptable than large deviations [3, 69].

The DTO model is a basic formulation for brachytherapy treatments using conventional

applicators and only one HDR source.

Define P as the set of all possible dwell positions along the applicator(s), S as the set

of structures, and Vs as the voxels in structure s ∈ S. The set S consists of the target(s)

and all or a subset of the OARs surrounding the target. The decision variables are xi,

the dwell time at dwell position i ∈ P . The dose delivered to voxel j in structure s, zjs,

is calculated as

zjs =∑i∈P

Dijsxi ∀j ∈ Vs,∀s ∈ S (2.1)

where Dijs is the amount of dose delivered from dwell position i to voxel j in structure

s per unit time. These dose coefficients were obtained using the Monte Carlo N-Particle

(MCNP) code [22] to simulate the dose distributions around the source in water. All

recommendations of the AAPM TG-43 [55, 66] and AAPM-ESTRO [60] reports for HDR

brachytherapy sources were considered in the simulation.

Each voxel is assigned a penalty for any overdose or underdose it receives. The


T_u T_oz

js

Fs(z

js)

Figure 2.1: The penalty function of the DTO problem is convex and non-smooth. Notethat no penalty is incurred between the lower and upper dose thresholds, Tu and To.

penalties are weighted according to the structure to which the voxel belongs so that

some structures have a higher priority than others. Additionally, some structures may

benefit from an underdose but not an overdose. For example, the penalty weight for

underdosing an OAR could be zero. To this end, the penalties for an underdose may be

different from the penalties for an overdose. The penalty function for voxel j in structure

s is

Fs(zjs) =1

|Vs|[ws(zjs − T s)

2+ + ws(T s − zjs)2+] (2.2)

where (·)+ is max{0, ·}; ws and ws are the overdose and underdose penalties for structure

s ∈ S, respectively; and T s and T s are the upper and lower dose thresholds for structure

s ∈ S, respectively. These dose thresholds provide flexibility to the model: If zjs lies

between T s and T s, then there is no penalty. The penalty function is normalized with

respect to the number of voxels in the structure to remove any bias towards structure

size. Figure 2.1 illustrates the shape of Fs.


The DTO model is then to minimize the total penalty over all voxels:

minimize∑s∈S

∑j∈Vs

Fs(zjs) (DTO)

subject to zjs =∑i∈P

Dijsxi ∀j ∈ Vs,∀s ∈ S

0 ≤ xi ≤ tmax ∀i ∈ P

where tmax is the upper bound on the dwell times.

Since the penalty functions Fs are convex, the DTO model can be reformulated as

a semi-infinite linear optimization (SILO) problem and solved using an interior point

constraint generation (IPCG) algorithm developed by Oskoorouchi et al. [56]. A SILO

problem is an optimization problem in which there is an infinite number of variables or

an infinite number of constraints, but not both. Our DTO-SILO problem has a linear

objective and infinitely many linear constraints:

minimize δ (DTO-SILO)

subject to∑s∈S

∑j∈Vs

Fs(zjs) ≤ δ

zjs =∑i∈P

Dijsxi ∀j ∈ Vs,∀s ∈ S

0 ≤ xi ≤ tmax ∀i ∈ P

The infinite number of constraints come from the constraints used to approximate the

convex functions Fs(zjs), as shown in Figure 2.2.

A graphical user interface (GUI) called morDiRECT (the Medical Operations Reseach

Laboratory’s Display for Ranking and Evaluating Customized Treatments) [65] was used

to generate treatment plans with ranges of parameter values automatically (Figure 2.3).

The best plan was then chosen for each patient according to target coverage and OAR


Figure 2.2: Linear approximations (blue) of a convex function (black)

sparing. morDiRECT is a multi-criteria decision support system that allows the decision-

maker to easily generate and choose a high-quality plan without the iterative process

of identifying suitable model parameters, which is a common characteristic of inverse

treatment planning in brachytherapy.

2.3 Interior point constraint generation algorithm

The IPCG algorithm was developed to solve a similar model for a SRS inverse planning

problem [4, 14, 56]. The algorithm is guaranteed to find an ε-optimal solution, unlike

heuristics and gradient descent methods, and it was shown to converge to an ε-optimal

solution in a finite number of iterations. For simplicity, we only present the main idea

of the IPCG algorithm. The theoretical aspects and mathematical proofs behind the

algorithm can be found in Oskoorouchi et al. [56].

We start with a simpler version of the original problem that only considers a small

subset of the constraints, called the reduced problem. An optimal solution to the reduced

problem is found and multiple constraints violated by that optimal solution are identified.


Figure 2.3: morDiRECT’s evaluation window

These constraints are added to the reduced problem and at the same time, the barrier

function is updated by reducing the barrier parameter. The feasibility of the solution

is then recovered, and an optimal solution is found for the new reduced problem. The

process is repeated until the duality gap is within ε distance, that is, the solution is

ε-optimal.

Chapter 3

Direction-modulated brachytherapy

Using conventional intracavitary applicators, such as the tandem and ring applicator

(Figure 3.1), with isotropic sources limits the maximal dose delivered to the tumour,

especially in cases where the target volume is laterally extended or non-symmetric. To

prevent the overdose of OARs, parts of the target volume must be underdosed, leading

to less conformal plans.

To address the lack of shielded intrauterine tandem applicators for cervical cancer

brachytherapy and build on the concept of anisotropic dose profiles [15], a novel tandem

applicator, called the DMBT tandem (Figure 3.2), was proposed that is able to produce

anisotropic dose distributions [25]. The tandem can generate directional radiation dose

profiles through its intelligent shielding design to achieve superior target coverage (Figure

3.3). DMBT was theoretically studied on rectal cancer [81, 82], breast cancer [83], and

cervical cancer [25, 26, 71–73].

The DMBT tandem is symmetric along the transverse and longitudinal axes. It has

six peripheral holes of width 1.3 mm grooved along a non-magnetic tungsten alloy (95%

tungsten, 3.5% nickel, and 1.5% copper, ρ = 18 g/cm3), enclosed in a 0.3 mm thick

plastic sheath. The DMBT tandem diameter is no larger than 6 mm, the dimension

of a conventional tandem. Thus, it can be readily used with existing tandem-and-ring

16

Chapter 3. Direction-modulated brachytherapy 17

Tandem

Ring

Interstitial/needles(optional)

Figure 3.1: Tandem and ring applicator. The HDR source can travel through the ringand the tandem. (Adapted from Viswanathan et al. [80])

Figure 3.2: DMBT tandem (Source: Han et al. [26])

applicators. Furthermore, the paramagnetic tungsten alloy renders the DMBT tandem

MRI-safe.

The high density of the tungsten alloy allows the rod to be used as a shield to block

part of the radiation. With the grooves equally spaced at 60◦, highly directional beams

of radiation can be delivered in six different directions, which is a sharp contrast to the

near-circular, or isotropic, dose distribution obtained from a conventional tandem that

has no shielding.

To evaluate the modulating capacity of the DMBT tandem, we use our inverse plan-

ning approach to develop treatment plans for both conventional brachytherapy (conven-


6.0 mm

(a) Conventional tandem

60°

6.0 mm

1.3mm

(b) DMBT tandem

(c) Isotropic dose distribution fromconventional tandem

(d) Anisotropic dose distribution fromDMBT tandem

Figure 3.3: Cross-sections of conventional and DMBT tandems


tional BT) and DMBT for cervical cancer, and then compare the plan quality. We use

the DTO formulation to model the conventional BT problem and then solve it using the

IPCG algorithm.

3.1 DMBT optimization model

The optimization model for DMBT is a slight modification of the DTO model. In addition

to the parameters previously defined, let C be the set of channels grooved along the

tandem applicator, where |C| = 6. The dwell positions (in the ring and in the tandem)

are fixed to the positions used in the clinical treatments. The decision variables for

DMBT-DTO are xic, the dwell time at dwell position i ∈ P in channel c ∈ C, as opposed

to xi for conventional brachytherapy. The dose delivered to voxel j in structure s, zjs, is

calculated as

zjs =∑c∈C

∑i∈P

Dicjsxic ∀j ∈ Vs,∀s ∈ S (3.1)

where Dicjs is the amount of dose delivered from dwell position i to voxel j in structure

s per unit time along channel c.

Using the same penalty function Fs(zjs) (Equation 2.2), the DMBT-DTO problem is

then

minimize∑s∈S

∑j∈Vs

Fs(zjs) (DMBT-DTO)

subject to zjs =∑c∈C

∑i∈P

Dicjsxic ∀j ∈ Vs,∀s ∈ S

0 ≤ xic ≤ tmax ∀i ∈ P, ∀c ∈ C



3.2 Results

Twenty-seven clinical cervical cancer cases obtained from Aarhus University Hospital

(Aarhus, Denmark) are studied retrospectively. The target volume and the OARs (blad-

der, rectum, and sigmoid) were contoured on T2w MR images and treatment plans were

generated using the BrachyVisionTM (Varian Medical Systems, Palo Alto, CA, USA)

treatment planning system. The prescription dose was 15 Gy or 17.5 Gy. All patients

were treated using a tandem and ring applicator and an 192Ir pulsed-dose-rate source,

with source strength normalized to one Curie (Ci). The results can be converted back to

match a 10 Ci source, which is typical in HDR brachytherapy. The clinical details of the

cases are shown in Table 3.1.

To ensure that any improvement solely resulted from the modulating capacity of the

DMBT tandem, the ring was left untouched in both the conventional BT and DMBT

setup and only the tandem was replaced. To evaluate the quality of treatment plans, all

plans were normalized to receive their respective clinical target volume D90 values. D2cc,

the dose to the hottest 2 cm3 of the structure volume, was calculated for the three OARs,

as well as COIN and HI.

On average, DMBT improved the sparing of all three OARs (Table 3.2). The per-

cent improvement in OAR D2cc is illustrated in Figure 3.4. The dose delivered to the

bladder, rectum, and sigmoid was reduced by 6.4%, 12.5%, and 2.6%, respectively. The

corresponding maximum decrease was 17.2%, 43.3%, and 16.7%, respectively. In 23 out

of 27 cases (85%), DMBT plans were superior to the conventional BT plans for all three

OARs.

The COIN and HI values are shown in Table 3.3. In terms of conformity, 25 of 27

(92.5%) DMBT plans are more conformal than the conventional BT plans. The mean

COIN values for conventional BT and DMBT plans were 0.47 ± 0.07 (mean ± standard

deviation) and 0.55 ± 0.08, respectively. With regards to homogeneity, the DMBT plans

and conventional BT plans exhibit no clear relationship, but they are comparable on


Table 3.1: Patient information

Number of dwell positions

Patient ID Target vol. (cc) Conventional BT DMBT Rx dose (Gy)

1 12.2 27 67 17.5

2 14.0 29 79 15.0

3 16.1 29 79 17.5

4 16.8 27 67 17.5

5 17.8 34 94 15.0

6 19.6 29 79 15.0

7 19.9 31 91 17.5

8 21.1 27 67 17.5

9 21.6 34 94 17.5

10 22.0 31 91 17.5

11 22.1 31 91 17.5

12 22.4 31 91 15.0

13 22.4 28 88 17.5

14 22.5 31 91 17.5

15 24.6 34 94 17.5

16 25.0 32 82 15.0

17 28.0 31 91 15.0

18 28.5 34 94 15.0

19 30.7 29 79 15.0

20 31.8 29 79 17.5

21 35.6 31 91 17.5

22 36.1 33 103 17.5

23 38.5 38 118 15.0

24 38.8 34 94 17.5

25 49.6 30 80 15.0

26 54.5 36 106 15.0

27 81.1 38 118 15.0

Mean 28.6 31 89 16.4

Stdev 14.6 3 13 1.3


Table 3.2: OAR dose metrics for the conventional BT plans and the DMBT plans. Anegative value (shaded) means that the DMBT plan improves on the conventional BTplan. The maximum decrease is bolded.

D2cc Bladder (Gy) D2cc Rectum (Gy) D2cc Sigmoid (Gy)

Patient

ID

Conventional

BTDMBT % Diff

Conventional

BTDMBT % Diff

Conventional

BTDMBT % Diff

1 8.6 8.2 -4.8 6.8 5.9 -13.2 11.1 10.8 -2.3

2 6.6 6.5 -0.5 4.4 4.0 -9.8 7.5 7.1 -4.8

3 13.2 13.7 3.5 10.7 9.8 -8.4 7.4 7.6 3.2

4 13.3 12.8 -3.6 10.1 8.6 -14.9 12.9 12.9 0.6

5 11.9 10.9 -8.9 4.5 4.0 -11.5 11.2 9.9 -11.0

6 6.6 6.1 -7.2 7.6 7.2 -5.7 9.7 9.0 -7.3

7 13.5 13.2 -2.5 4.6 4.5 -2.5 9.1 9.7 6.7

8 7.5 6.5 -13.8 8.3 6.9 -16.7 7.2 6.7 -6.0

9 13.6 12.8 -5.9 4.4 4.1 -7.5 8.3 8.2 -1.1

10 11.7 9.8 -15.9 9.0 7.3 -19.3 6.5 6.4 -2.4

11 13.0 13.0 -0.2 7.7 7.1 -8.1 13.3 13.4 0.6

12 9.5 8.3 -12.5 9.7 7.1 -27.2 8.8 7.9 -10.6

13 11.5 10.4 -9.6 11.1 9.3 -16.8 11.2 11.4 1.3

14 11.5 11.4 -1.5 6.4 5.9 -8.2 12.3 11.4 -7.4

15 6.4 5.3 -17.1 4.1 3.5 -16.6 11.3 11.3 -0.5

16 9.8 9.3 -5.3 6.2 5.8 -6.2 11.6 12.0 3.4

17 9.1 8.1 -11.0 4.0 3.5 -13.3 10.0 9.7 -2.8

18 12.2 12.1 -0.6 7.8 6.5 -16.4 11.6 11.9 2.2

19 8.2 6.8 -17.2 6.4 5.1 -19.9 11.3 10.8 -4.4

20 12.4 11.7 -6.2 8.0 7.9 -2.0 6.6 6.7 1.6

21 12.4 10.8 -13.1 8.5 4.8 -43.3 11.9 9.9 -16.7

22 11.3 11.2 -1.0 4.9 4.9 0.2 11.6 12.0 3.7

23 13.0 12.9 -0.7 6.8 6.4 -6.3 9.8 10.2 4.5

24 9.8 9.4 -4.5 8.4 6.6 -21.0 16.9 16.9 0.0

25 14.4 13.9 -3.4 9.3 8.9 -4.4 7.5 6.7 -10.5

26 11.4 11.1 -2.1 6.1 5.8 -5.5 12.8 11.7 -8.1

27 13.9 12.9 -7.3 11.0 9.4 -14.1 8.3 8.3 -1.0

Mean 11 10.3 -6.4 7.3 6.3 -12.5 10.3 10 -2.6

Stdev 2.4 2.6 5.7 2.2 1.9 9 2.4 2.5 5.6


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27−50

−40

−30

−20

−10

0

10

Case number

Pe

rce

nt

ch

an

ge

Bladder Rectum Sigmoid

Figure 3.4: Pairwise difference between the conventional BT and DMBT plans for all 27cases

.

average. The mean HI was of 0.31 ± 0.08 for the conventional BT plans and 0.30 ± 0.06

for the DMBT plans.

The DVH and isodose lines for a representative case (Patient 4) are shown in Figure

3.5 to illustrate the benefits of the DMBT tandem. The DVHs (Figure 3.5a) show that the

target receives the required prescription dose in both plans while the dose to the three

OARs decreases in the DMBT plans. The corresponding slices (Figure 3.5b) further

illustrate the OAR sparing, as well as the superior conformity of the DMBT plans.

The average IPCG computation time for conventional BT plans was 0.4 min for a

mean of 31 dwell positions and 1.1 min for a mean of 89 dwell positions for DMBT plans

(Table 3.4). As expected, the DMBT computation times are consistently longer than

their corresponding conventional BT times since the number of variables increases when

the DMBT tandem applicator is used. Figure 3.6 shows that the computation time is

quadratic with the number of dwell positions, despite the exponential complexity of the

algorithm [18].


Table 3.3: Homogeneity index and conformal index. The better index value is shaded.

HI COIN

Patient IDConventional

BTDMBT

Conventional

BTDMBT

1 0.30 0.31 0.42 0.49

2 0.30 0.25 0.43 0.45

3 0.16 0.15 0.42 0.41

4 0.23 0.20 0.39 0.46

5 0.24 0.25 0.46 0.55

6 0.33 0.34 0.40 0.44

7 0.14 0.14 0.50 0.44

8 0.42 0.35 0.42 0.48

9 0.33 0.35 0.44 0.53

10 0.30 0.29 0.46 0.56

11 0.38 0.41 0.47 0.53

12 0.27 0.26 0.33 0.47

13 0.33 0.33 0.42 0.58

14 0.27 0.28 0.42 0.45

15 0.28 0.28 0.47 0.59

16 0.33 0.38 0.47 0.57

17 0.27 0.28 0.48 0.57

18 0.27 0.28 0.49 0.58

19 0.28 0.30 0.43 0.58

20 0.33 0.35 0.62 0.65

21 0.29 0.25 0.42 0.58

22 0.34 0.43 0.57 0.67

23 0.26 0.26 0.51 0.54

24 0.36 0.37 0.53 0.65

25 0.33 0.35 0.60 0.67

26 0.41 0.48 0.53 0.66

27 0.35 0.37 0.55 0.73

Mean 0.30 0.31 0.47 0.55

Stdev 0.06 0.08 0.07 0.08


0 20 40 60 80 100 120 140 160 180 200 220 2400

10

20

30

40

50

60

70

80

90

100

Percent dose (%)

Pe

rce

nt

vo

lum

e (

%)

HRCTVBladder

RectumSigmoid

(a) Dose-volume histogram

(b) Slices with 100% and 50% isodose lines

Figure 3.5: Comparison between a conventional BT plan (dashed lines) and a DMBTplan (solid lines) for a representative case.


Table 3.4: Computation time in minutes for the conventional BT and DMBT plans

Conventional BT DMBT

Patient ID # dwell positions Comp. time # dwell positions Comp. time

1 27 0.29 67 0.70

2 29 0.29 79 0.68

3 29 0.38 79 1.00

4 27 0.45 67 1.02

5 34 0.50 94 1.46

6 29 0.34 79 0.82

7 31 0.55 91 1.42

8 27 0.30 67 0.71

9 34 0.52 94 1.32

10 31 0.26 91 0.61

11 31 0.28 91 0.80

12 31 0.36 91 0.89

13 28 0.32 88 0.95

14 31 0.36 91 0.92

15 34 0.43 94 1.06

16 32 0.37 82 0.90

17 31 0.29 91 0.80

18 34 0.68 94 1.56

19 29 0.48 79 1.17

20 29 0.29 79 0.74

21 31 0.39 91 1.15

22 33 0.46 103 1.62

23 38 0.45 118 1.42

24 34 0.52 94 1.40

25 30 0.36 80 0.97

26 36 0.54 106 1.82

27 38 0.48 118 1.48

Mean 31 0.41 89 1.09

Stdev 3 0.11 13 0.34


0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0


Co

mp

uta

tio

n t

ime

(m

in)

Conventional BT DMBT Best fit: Quadratic

Figure 3.6: Computation time for DMBT as a function of number of dwell positions.

3.3 Discussion

Treatment planning for cervical cancer can be challenging, especially with the presence

of asymmetric or bulky tumours. We have shown that the DMBT tandem, with its

ability to generate highly directional beams of radiation, allows for clinically significant

reduction in dose to the OARs without compromising the target coverage. Furthermore,

the results are independent of the tumour size. The dose homogeneity is maintained

from the conventional BT plans to the DMBT plans, while conformity is significantly

improved clinically in the DMBT plans. Since a prototype has already been constructed,

the next step is to obtain approval for DMBT clinical trials.

Our results also indicate that the inverse planning approach used in SRS [4, 14, 56]

can be successfully implemented in HDR brachytherapy. High quality treatment plans

were obtained in less than two minutes. While these times are comparable to, say,

simulated annealing run times previously used in brachytherapy inverse planning [43],

we are able to guarantee the optimality of our results. If the planner is not satisfied


with the plan obtained, new plans can be quickly generated within a few minutes. To

avoid such situations, we used morDiRECT [65] as a complementary tool to choose the

best treatment plans according to our criteria. The short computation time of the IPCG

algorithm is advantageous when running several trials through morDiRECT.

3.4 Conclusion

We investigated the dosimetric benefits of DMBT on 27 cervical cancer cases with differ-

ent tumour sizes. The DMBT plans achieved lower OAR doses than the conventional BT

plans while maintaining similar target coverage. An alternative interpretation of these

results is that for the same OAR exposure to radiation, the dose to the target can be

safely escalated, thus potentially improving tumour control and treatment outcome.

In terms of algorithm performance, we showed that a quadratic penalty optimiza-

tion approach combined with IPCG [56], similar to approaches in radiosurgery inverse

planning [4, 14, 18–20, 56], performs well for HDR brachytherapy inverse planning prob-

lems. Good quality treatment plans were generated within a few minutes of run time.

Additionally, the treatment time obtained were all clinically acceptable.

Chapter 4

Energy-modulated brachytherapy

The radionuclides 192Ir and 60Co are commonly used as sources in HDR brachytherapy,

but other nuclides, such as 169Yb, have also been previously used as HDR sources [50].

While the use of these individual radionuclides as HDR sources has been studied, the use

of two or more radionuclides in combination, or EMBT, has yet to be investigated on

prostate cancer. If used together, the different dose profiles of these radioactive sources

can potentially improve OAR sparing. The dosimetric benefits of EMBT were successfully

investigated on cervical cancer using 192Ir, 60Co, and 169Yb [71, 73]. A quadratic penalty

model (with same penalty for overdose and underdose of a structure) and a projected

gradient algorithm [25] were used to develop the treatment plans for single- and dual-

source combinations and only the DMBT tandem was used (instead of the conventional

tandem). Since the DMBT tandem alone allows for superior plan quality, we assess the

potential advantages of EMBT only on prostate cancer. Furthermore, we also consider

using all three sources in combination.

Recently, in addition to irradiating the whole prostate gland, regions of the prostate

with the highest tumour concentration have been prescribed higher doses [29]. These

regions are called dominant intraprostatic lesions (DILs) and they are now a new focus

of HDR prostate brachytherapy [29]. The expectation is that recurrence is less likely to

29

Chapter 4. Energy-modulated brachytherapy 30

happen when DILs are identified and boosted [33]. EMBT may allow safer dose escalation

to the DILs while limiting the dose to the OARs.

192Ir is the most common radioactive source for HDR brachytherapy [64, 78]. Due to

its high specific activity, which is defined as the rate at which unstable nuclei decay per

unit mass, the radioactive source can be made small enough to be inserted in the body

while keeping a significantly high activity, which is essential for HDR purposes. However,

the source has a short half-life of 74 days, which means that it needs to be replaced every

three to four months to maintain acceptable treatment times.

60Co sources are now commercially available in the same geometric dimensions as

192Ir, and are comparable with 192Ir with respect to clinical aspects for HDR prostate

brachytherapy [31, 64, 78]. However, 60Co’s longer half-life of approximately five years

means that it does not need to be replaced frequently, leading to fewer source exchanges.

Thus, 60Co has a lower operating cost, and can be a cheaper alternative for developing

countries [7, 64]. 169Yb, with a half-life of 32 days, has also been investigated as a potential

HDR source [11, 47, 59]. It has been shown to be at least equivalent to 192Ir in terms of

dosimetry [36, 45, 46], with reduced radiation protection and shielding requirements due

to its lower energy [24].

Afterloaders that can handle two sources, such as the Multisource R© afterloader from

Eckert & Ziegler BEBIG [16], have already been introduced on the market. However,

such afterloaders cannot handle both sources at the same time [57]. Recently, a new

afterloader capable of handling two different sources simultaneously was proposed by

Elekta (Flexitron R©, Elekta Brachytherapy, Veenendaal, The Netherlands). Furthermore,

the machine is equipped to handle 192Ir, 60Co, or 169Yb. We therefore investigate the

dosimetric benefits of EMBT for every possible combination of 192Ir, 60Co, and 169Yb,

and then compare treatment quality to the 192Ir-only plan. The DTO model is used in

combination with the IPCG algorithm to generate the single-source plans.


Table 4.1: HDR source information

Source Model ManufacturerLength

(mm)

Diameter

(mm)

Energy

(keV)

Activity

(Ci)

Half-life

(days)

192IrmicroSelectron

v2Nucletron 4.5 0.9 380 10 73.83

60Co Co0.A86Eckert & Ziegler

BEBIG5.0 1.0 1250 2 1925.00

169Yb 4140Implant Sciences

Corporation4.8 0.9 93 10 32.02

4.1 EMBT optimization model

Based on the DTO model, we formulate the EMBT-DTO problem as follows. In addition

to the previously defined parameters, let R be the set of sources, where |R| = 3 in our

study. The dwell positions, evenly spaced at intervals (which vary per case), are fixed

to the positions used in the clinical treatments.The decision variables are xir, the dwell

time of source r ∈ R at dwell position i ∈ P . The dose delivered to voxel j in structure

s, zjs, is calculated as

zjs =∑r∈R

∑i∈P

Dirjsxir ∀j ∈ Vs,∀s ∈ S (4.1)

where Dirjs is the amount of dose delivered from dwell position i to voxel j in structure

s per unit time from source r. Details about the sources are shown in Table 4.1. Their

radial and depth dose functions are plotted against the distance from the middle of each

source in Figures 4.1 and 4.2, respectively.


0 2 4 6 8 100.7

0.8

0.9

1

1.1

1.2

Distance (cm)

Radia

l dose function

Ir−192

Co−60

Yb−169

Figure 4.1: Radial dose function of 192Ir, 60Co, and 169Yb normalized at 1 cm.

0 2 4 6 8 10

10−2

10−1

100

Distance (cm)

De

pth

do

se

Ir−192

Co−60

Yb−169

Figure 4.2: Depth dose function of 192Ir, 60Co, and 169Yb normalized at 1 cm.


Using the same penalty function Fs(zjs) (Equation 2.2), the EMBT-DTO problem is

minimize∑s∈S

∑j∈Vs

Fs(zjs) (EMBT-DTO)

subject to zjs =∑r∈R

∑i∈P

Dirjsxir ∀j ∈ Vs,∀s ∈ S

0 ≤ xir ≤ tmax ∀i ∈ P, ∀r ∈ R


4.2 Results

We test our approach retrospectively on 12 anonymized HDR prostate cases treated at

the Odette Cancer Centre, Sunnybrook Health Sciences Centre (Toronto, ON, Canada)

(Table 4.2). The structures (prostate, DILs, urethra, and rectum) were contoured on

the planning scans and treatment plans were generated using the Oncentra Brachy R©

(Nucletron, Veenendaal, The Netherlands) treatment planning system. For planning

purposes, the cases were separated into two groups: Group A patients have the prostate

as the main target volume, while Group B patients have DILs as secondary targets in

addition to the prostate. DILs are prescribed 150% of the prescription dose. All patients

were treated with the Nucletron microSelectron HDR-version 2 (mHDR-V2) 192Ir source

and followed the clinical protocol shown in Table 4.3.

The plans were separated into three categories: (1) single-source with Co, Ir, and Yb

individually; (2) double-source with the pairs Co-Ir, Co-Yb, and Ir-Yb; and (3) triple-

source with Co-Ir-Yb (assuming future developments in afterloader technology allows for

triple-source delivery). There were a total of seven treatment plans per patient. For fair

comparison, all plans were normalized to the clinical prostate V100 and where applicable,

the clinical DIL V150. In addition to reporting the DVH parameters presented in Table

4.3, COIN and HI are also calculated.


Table 4.2: Patient information. Group A patients have the prostate as main targetvolume, and Group B patients have DILs as secondary targets.

Group Patient IDTarget vol.

(cc)

#

Applicators

# Dwell

positions

Rx dose

(Gy)

A

1 24.3 12 165 15.0

2 28.2 12 152 13.5

3 28.7 12 257 15.0

4 34.7 16 189 15.0

5 35.4 16 226 15.0

6 39.6 12 283 15.0

7 40.9 12 166 15.0

8 53.6 16 233 19.0

Mean 38.1

Stdev 11.4

B

9 22.0 16 231 19.0

10 25.5 16 350 19.0

11 34.1 16 447 19.0

12 40.7 16 330 19.0

Mean 30.6

Stdev 8.5

Table 4.3: Clinical protocols

Target DILs Urethra Rectum

V100 ≥ 95 % V150 = 100 % Dmax ≤ 130 % Dmax ≤ 90 %

V150 < 35 % - D10 ≤ 110 % V80 ≤ 0.5 cc

V200 < 12 % - - -


Table 4.4: Average values ± standard deviation of target DVH parameters and indices.Group A patients have the prostate as main target volume, and Group B patients haveDILs as secondary targets.

Group Plan V150 (%) V200 (%) HI COIN

A

Clinical 37.6 ± 3.4 11.6 ± 2.5 0.61 ± 0.04 0.74 ± 0.04

Ir 27.3 ± 4.4 11.9 ± 2.1 0.72 ± 0.05 0.78 ± 0.03

Co 29.1 ± 4.5 12.8 ± 2.3 0.70 ± 0.05 0.78 ± 0.03

Yb 24.8 ± 4.6 10.5 ± 1.9 0.75 ± 0.05 0.79 ± 0.03

Co-Ir 29.8 ± 4.3 13.2 ± 2.2 0.70 ± 0.05 0.77 ± 0.03

Co-Yb 29.0 ± 4.2 13.0 ± 2.2 0.70 ± 0.05 0.78 ± 0.03

Ir-Yb 26.7 ± 4.0 11.6 ± 2.1 0.73 ± 0.04 0.77 ± 0.04

Co-Ir-Yb 28.6 ± 4.1 12.4 ± 2.2 0.71 ± 0.04 0.77 ± 0.02

B

Clinical 34.1 ± 3.2 9.7 ± 2.2 0.65 ± 0.04 0.58 ± 0.04

Ir 32.1 ± 2.6 13.7 ± 0.6 0.66 ± 0.05 0.75 ± 0.06

Co 33.7 ± 3.9 14.8 ± 1.6 0.65 ± 0.05 0.74 ± 0.05

Yb 30.9 ± 1.7 12.8 ± 0.6 0.68 ± 0.04 0.76 ± 0.06

Co-Ir 33.9 ± 3.5 14.8 ± 1.4 0.64 ± 0.05 0.74 ± 0.06

Co-Yb 33.9 ± 3.5 14.6 ± 1.2 0.64 ± 0.05 0.74 ± 0.06

Ir-Yb 31.9 ± 2.9 13.7 ± 0.8 0.66 ± 0.05 0.75 ± 0.06

Co-Ir-Yb 33.5 ± 3.8 14.6 ± 1.3 0.65 ± 0.04 0.73 ± 0.05

On average, treatment plans generated from all combinations achieve the clinical

objectives for both targets (Table 4.4) and OARs (Table 4.5). As expected, Group B

patients have higher V150 and V200 values than Group A patients due to the presence of

DILs. Additionally, source combinations that include 60Co generate plans with higher

V150 and V200 values than the conventional Ir-only plans, due to the higher energy of 60Co.

Yb-only plans have the lowest V150 and V200 values since 169Yb has the lowest energy.

The HI and COIN values are presented in Tables 4.6 and 4.7, respectively. The column

with the heading “Clinical 192Ir” shows the clinical treatment values. With the exception

of Patient 12, all optimized Ir-only plans have better HI values than the clinical Ir plans.

Source combinations that include 60Co generate plans that are slightly less homogeneous


Table 4.5: Average values ± standard deviation of OARs DVH parameters. Group Apatients have the prostate as main target volume, and Group B patients have DILs assecondary targets.

Urethra Rectum

Group Plan Dmax (%) D10 (%) Dmax (%) V80 (cc)

A

Clinical 123.9 ± 6.9 116.4 ± 0.5 88.2 ± 10.2 0.2 ± 0.2

Ir 125.6 ± 8.7 112.1 ± 2.2 83.8 ± 8.9 0.3 ± 0.5

Co 127.2 ± 9.8 111.7 ± 2.9 83.6 ± 8.5 0.2 ± 0.2

Yb 130.0 ± 8.7 113.8 ± 2.7 84.5 ± 8.1 0.4 ± 0.7

Co-Ir 124.2 ± 7.0 111.2 ± 3.0 83.5 ± 9.7 0.4 ± 0.6

Co-Yb 124.8 ± 6.3 110.7 ± 1.9 83.6 ± 9.6 0.5 ± 0.7

Ir-Yb 123.6 ± 5.4 110.9 ± 1.9 83.7 ± 8.5 0.2 ± 0.2

Co-Ir-Yb 122.6 ± 6.3 109.5 ± 1.2 82.8 ± 8.7 0.2 ± 0.2

B

Clinical 121.5 ± 1.9 116.0 ± 1.0 85.0 ± 2.8 0.1 ± 0.1

Ir 121.8 ± 7.8 112.9 ± 5.7 88.7 ± 7.7 0.4 ± 0.6

Co 118.6 ± 8.0 110.8 ± 6.5 85.3 ± 7.2 0.2 ± 0.3

Yb 121.6 ± 9.9 114.4 ± 6.8 90.7 ± 8.1 0.6 ± 0.7

Co-Ir 117.5 ± 9.1 111.1 ± 6.5 84.9 ± 8.0 0.2 ± 0.3

Co-Yb 117.6 ± 8.2 110.8 ± 6.2 85.3 ± 7.0 0.2 ± 0.3

Ir-Yb 119.5 ± 7.1 112.2 ± 4.8 88.0 ± 8.7 0.4 ± 0.5

Co-Ir-Yb 118.0 ± 8.6 110.9 ± 6.4 84.7 ± 6.6 0.2 ± 0.3


Table 4.6: Homogeneity index. Any metric in the inverse plans that achieved a bettervalue than the clinical Ir-only plans and the inverse Ir-only plans are bolded and shaded,respectively.

GroupPatient

ID

Clinical

IrIr Yb Co Co-Ir Co-Yb Ir-Yb Co-Ir-Yb

Inverse

plan

range

A

1 0.66 0.72 0.72 0.68 0.69 0.68 0.72 0.69 [0.68, 0.72]

2 0.60 0.72 0.73 0.69 0.69 0.70 0.72 0.69 [0.69, 0.73]

3 0.66 0.74 0.79 0.75 0.75 0.75 0.77 0.76 [0.74, 0.79]

4 0.58 0.77 0.80 0.75 0.74 0.75 0.77 0.75 [0.74, 0.80]

5 0.65 0.73 0.75 0.71 0.69 0.71 0.73 0.71 [0.69, 0.75]

6 0.57 0.61 0.65 0.61 0.60 0.61 0.64 0.62 [0.60, 0.65]

7 0.59 0.73 0.75 0.71 0.71 0.70 0.71 0.70 [0.70, 0.75]

8 0.62 0.74 0.77 0.73 0.71 0.73 0.75 0.74 [0.71, 0.77]

Mean 0.61 0.72 0.75 0.70 0.70 0.70 0.73 0.71

Stdev 0.04 0.05 0.05 0.05 0.05 0.05 0.04 0.04

B

9 0.67 0.67 0.68 0.66 0.66 0.66 0.67 0.66 [0.66, 0.68]

10 0.67 0.67 0.68 0.64 0.64 0.64 0.67 0.65 [0.64, 0.68]

11 0.60 0.70 0.70 0.70 0.69 0.69 0.70 0.70 [0.69, 0.70]

12 0.65 0.64 0.66 0.61 0.61 0.61 0.64 0.61 [0.61, 0.66]

Mean 0.65 0.67 0.68 0.65 0.65 0.65 0.67 0.65

Stdev 0.04 0.02 0.01 0.04 0.03 0.03 0.03 0.04

than Ir plans, which is expected since Co plans have larger hotspots. On the other hand,

Yb-only plans are more homogeneous than Ir-only plans since they have smaller hotspots.

For the COIN, with the exception of Patients 5 and 7, all optimized Ir-only plans have

superior values than the clinical Ir-only plans. All Group B patients benefit from an

improvement in conformity across all plans, while most of Group A plans have a better

COIN value than the one obtained in the clinic.

It is clear that using dual or triple sources result in more OAR sparing (Table 4.8).

The improvement is further illustrated in Figure 4.3. For Group A patients, 60Co is

comparable to 192Ir except for the urethral Dmax. However, Yb-only plans have larger

doses to the OARs, especially the urethra. In the dual-source category for Group A, the

Ir-Yb plans have the lowest dose to the urethra, whereas the pair Co-Ir spares the rectum

the most compared to 192Ir. In the same group, the combination Co-Ir-Yb has the largest


Table 4.7: Conformal index. Any metric in the inverse plans that achieved a bettervalue than the clinical Ir-only plans and the inverse Ir-only plans are bolded and shaded,respectively.

GroupPatient

ID

Clinical

IrIr Yb Co Co-Ir Co-Yb Ir-Yb Co-Ir-Yb

Inverse

plan

range

A

1 0.75 0.81 0.85 0.81 0.79 0.82 0.81 0.80 [0.79, 0.85]

2 0.69 0.76 0.78 0.73 0.75 0.76 0.77 0.76 [0.73, 0.78]

3 0.71 0.79 0.78 0.81 0.80 0.76 0.79 0.76 [0.76, 0.81]

4 0.72 0.79 0.77 0.79 0.72 0.75 0.78 0.77 [0.72, 0.79]

5 0.79 0.76 0.81 0.76 0.78 0.80 0.79 0.80 [0.76, 0.81]

6 0.80 0.80 0.81 0.79 0.79 0.78 0.69 0.75 [0.69, 0.81]

7 0.78 0.74 0.76 0.75 0.73 0.75 0.76 0.75 [0.73, 0.76]

8 0.71 0.81 0.78 0.80 0.80 0.78 0.80 0.78 [0.78, 0.81]

Mean 0.74 0.78 0.79 0.78 0.77 0.78 0.77 0.77

Stdev 0.04 0.03 0.03 0.03 0.03 0.03 0.04 0.02

B

9 0.52 0.67 0.67 0.67 0.67 0.67 0.67 0.67 [0.67, 0.67]

10 0.57 0.81 0.81 0.79 0.82 0.82 0.82 0.80 [0.79, 0.82]

11 0.60 0.77 0.77 0.73 0.73 0.74 0.76 0.73 [0.73, 0.77]

12 0.62 0.76 0.77 0.75 0.74 0.74 0.76 0.73 [0.73, 0.77]

Mean 0.58 0.75 0.76 0.74 0.74 0.74 0.75 0.73

Stdev 0.04 0.06 0.06 0.05 0.06 0.06 0.06 0.05


Table 4.8: Average difference of OARs DVH parameters between Ir plans and proposedplans. A negative value (shaded) means that the plan improves on the Ir plan. Themaximum decrease is bolded. Group A patients have the prostate as main target volume,and Group B patients have DILs as secondary targets.

Urethra Rectum

Group Plan Dmax (%) D10 (%) Dmax (%) V80 (%)

A

Yb 4.39 ± 3.25 1.70 ± 2.07 0.71 ± 4.56 0.37 ± 0.84

Co 1.57 ± 5.30 -0.35 ± 1.92 -0.16 ± 3.03 -0.28 ± 0.64

Co-Ir -1.36 ± 4.08 -0.82 ± 1.74 -0.28 ± 4.26 -0.28 ± 0.58

Co-Yb -0.77 ± 2.79 -1.32 ± 1.11 -0.21 ± 2.15 -0.23 ± 0.62

Ir-Yb -1.97 ± 3.61 -1.18 ± 2.21 -0.12 ± 2.23 -0.14 ± 0.61

Co-Ir-Yb -3.03 ± 4.77 -2.58 ± 1.48 -1.00 ± 1.39 -0.23 ± 0.66

B

Yb -0.21 ± 3.77 1.47 ± 1.30 2.03 ± 0.54 0.37 ± 0.39

Co -3.24 ± 2.91 -2.09 ± 1.25 -3.38 ± 2.78 -0.57 ± 0.73

Co-Ir -4.29 ± 4.74 -1.85 ± 2.01 -3.76 ± 1.76 -0.59 ± 0.75

Co-Yb -4.20 ± 5.16 -2.10 ± 1.66 -3.41 ± 1.77 -0.57 ± 0.78

Ir-Yb -2.29 ± 2.13 -0.66 ± 1.07 -0.63 ± 1.19 -0.11 ± 0.19

Co-IrYb -3.79 ± 3.85 -2.03 ± 1.07 -3.94 ± 1.82 -0.67 ± 0.74

decrease in dose to both the urethra and the rectum. For Group B patients, plans from

all source combinations, except Yb-only plans, have lower doses to both OARs compared

to Ir-only plans.

DVHs and isodose lines illustrate the improvement in plan quality between a multi-

souce plan and an Ir-only plan for representative patients without DILs (Figure 4.4) and

with DILs (Figure 4.5). In both DVHs (Figures 4.4a and 4.5a), the target receives the

required prescription dose while the rectal and urethral doses are clearly reduced. The

corresponding slices (Figures 4.4b and 4.5b) further illustrate the dose reduction to the

OARs in the Co-Ir plans.

The activity of the sources is normalized to what is currently commercially available

to show real treatment times. The treatment times for the 192Ir and 169Yb single-source

plans are within an acceptable window ranging from 8 to 18 min, which is typical for HDR


Yb Co Co−Ir Co−Yb Ir−Yb Co−Ir−Yb−4

−3

−2

−1

0

1

2

3

4

5

Source combination

Perc

enta

ge d

iffe

rence

Urethra DmaxUrethra D10

Rectum DmaxRectum V80

(a) Group A

Yb Co Co−Ir Co−Yb Ir−Yb Co−Ir−Yb−5

−4

−3

−2

−1

0

1

2

3

Source combination

Perc

enta

ge d

iffe

rence

Urethra Dmax

Urethra D10

Rectum Dmax

Rectum V80

(b) Group B

Figure 4.3: Pairwise difference between an Ir-plan and the proposed plans


0 20 40 60 80 100 120 140 160 180 200 220 2400

10

20

30

40

50

60

70

80

90

100

Percent dose (%)

Perc

ent volu

me (

%)

Prostate

Urethra

Rectum


(b) Slices with 150%, 100% and 50% isodose lines

Figure 4.4: Comparison between an Ir plan (dashed) and a Co-Ir (solid) plan for arepresentative case without DILs.


0 20 40 60 80 100 120 140 160 180 200 220 2400

10

20

30

40

50

60

70

80

90

100

Percent dose (%)

Perc

ent volu

me (

%)

Prostate

DIL

Urethra

Rectum


(b) Slices with 150%, 100% and 50% isodose lines

Figure 4.5: Comparison between an Ir plan (dashed) and a Co-Ir (solid) plan for arepresentative case with DILs.


Table 4.9: Treatment time in minutes. Group A patients have the prostate as maintarget volume, and Group B patients have DILs as secondary targets.

GroupPatient

IDIr Yb Co Co-Ir Co-Yb Ir-Yb Co-Ir-Yb

A

1 8 8 51 49 50 8 50

2 8 8 55 46 45 8 45

3 9 10 58 55 56 9 52

4 10 11 65 63 63 10 65

5 10 10 68 62 61 10 60

6 11 12 74 60 64 13 60

7 11 12 76 68 69 11 67

8 17 18 112 110 109 16 110

Mean 10 11 70 64 65 11 64

Stdev 3 3 19 20 20 3 20

B

9 12 13 83 80 80 12 80

10 17 18 115 106 103 17 100

11 16 17 109 102 95 16 109

12 14 15 98 98 97 14 98

Mean 15 16 101 97 94 15 97

Stdev 2 2 14 11 10 2 12

prostate brachytherapy (Table 4.9). The treatment times for plans with 60Co are higher

since the activity of the radionuclide was normalized to 2 Ci. To evaluate the contribution

of each source towards the final dose delivered, a metric called total reference air kerma

(TRAK) is calculated. TRAK is the product of the source strength and irradiation time

and it represents the total dose accumulated at a distance of 1 m from the source [61].

We use TRAK since it is independent of source strength or source activity. Sources with

higher energy have a larger contribution to the total dose delivered (Table 4.10). In

particular, 60Co contributes the most to the total dose delivered, especially for Group B

cases.

The mean computation time is 4 min for an average of 505 dwell positions (Table


Table 4.10: TRAK breakdown (%) for each multi-source plan. Group A patients have theprostate as main target volume, and Group B patients have DILs as secondary targets.The largest contributions are shaded.

GroupPatient

IDCo-Ir Co-Yb Ir-Yb Co-Ir-Yb

60Co 192Ir 60Co 169Yb 192Ir 169Yb 60Co 192Ir 169Yb

A

1 87 13 100 0 85 15 97 1 2

2 77 23 74 26 79 21 76 2 21

3 92 8 86 14 88 12 75 1 24

4 86 14 87 13 96 4 96 0 3

5 90 10 89 11 90 10 88 0 11

6 73 27 77 23 81 19 65 6 30

7 79 21 86 14 74 26 80 0 20

8 97 3 92 8 96 4 96 0 4

Mean 85 15 86 14 86 14 84 2 14

Stdev 8 8 8 8 8 8 12 2 11

B

9 96 4 96 4 93 7 96 0 4

10 91 9 87 13 86 14 84 0 16

11 92 8 81 19 80 20 73 0 27

12 100 0 100 0 100 0 100 0 0

Mean 95 5 91 9 90 10 88 0 12

Stdev 4 4 8 8 8 8 12 0 12


Table 4.11: Computation time in minutes. Group A patients have the prostate as maintarget volume, and Group B patients have DILs as secondary targets.

GroupPatient

IDIr Yb Co Co-Ir Co-Yb Ir-Yb Co-Ir-Yb

A

1 0.5 0.5 0.5 1.5 1.3 1.3 2.5

2 0.3 0.5 0.3 1.3 1.0 1.0 2.0

3 1.0 1.1 1.4 2.9 2.9 3.0 7.5

4 0.6 0.6 0.6 1.6 1.7 1.8 3.6

5 0.8 0.9 0.9 2.3 2.4 2.3 4.9

6 1.2 1.2 1.3 3.7 3.6 6.5 10.1

7 0.6 0.6 0.6 1.6 1.8 1.5 3.2

8 1.2 1.0 1.0 3.1 3.5 3.1 6.6

Mean 0.8 0.8 0.8 2.2 2.3 2.6 5.0

Stdev 0.3 0.3 0.4 0.9 1.0 1.8 2.8

B

9 1.4 1.5 1.3 1.9 1.9 3.1 4.4

10 2.8 2.5 2.5 6.2 6.4 7.2 12.3

11 4.0 4.3 4.1 10.2 9.6 10.8 19.3

12 2.3 2.3 2.3 5.9 6.0 5.5 10.4

Mean 2.6 2.6 2.5 6.0 6.0 6.7 11.6

Stdev 1.1 1.2 1.1 3.4 3.2 3.2 6.1

4.11). In multi-source plans, the number of dwell positions increases by a factor equal

to the number of sources used. As with DMBT, the computation time is quadratic with

the number of dwell positions (Figure 4.6). Our results align with Ghaffari [18] who

reported that the IPCG algorithm converges comparably with polynomial algorithms

despite its exponential complexity. The addition of dwell positions in the multi-source

configurations increases the complexity of the problem, and thus the improvement in

plan quality is obtained at the cost of larger computation time. Still, even for 1,400

dwell positions, computation time is under 25 minutes.


0 500 1000 15000

5

10

15

20

25

30


Co

mp

uta

tio

n t

ime

(m

in)

Ir

Co

Yb

Co−Ir

Co−Yb

Ir−Yb

Co−Ir−Yb

Best fit: Quadratic

Figure 4.6: Computation time for EMBT as a function of number of dwell positions.

4.3 Discussion

The introduction of dual-source afterloaders can revolutionize the field of HDR brachyther-

apy. The afterloaders can be exploited in two ways. First, two 192Ir sources can be used

simultaneously to reduce treatment time, if only one type of radionuclide were to be

used. Second, different radionuclides can be used simultaneously or sequentially during

treatment.

When comparing single-source plans, two conclusions are drawn. First, our results

agree with the literature that 60Co is at least equivalent to 192Ir with respect to clinical

aspects [31, 64, 78]. For boost cases, 60Co actually outperforms 192Ir. This behaviour

can be attributed to the dose profile of 60Co. With the faster fall-off of 60Co’s radial and

depth dose profiles, the OARs can be better spared. Second, in Yb-plans, doses to the

OARs are higher compared to 192Ir, which contradicts the findings of Lymperopoulou

et al. [45] and Krishnamurthy et al. [36]. This increase can also be explained by the

higher depth dose of 169Yb. Because higher doses are delivered by 169Yb compared to


192Ir for the same distance, more dose is delivered to the OARs.

In EMBT plans, it is possible to spare the OARs while maintaining the same target

coverage with multi-source brachytherapy. On average, dual- and triple-source combina-

tions outperform their individual single sources, as expected. In particular, cases with

DILs benefit the most from energy-modulation. The combinations Co-Ir and Co-Yb are

comparable, but both are superior to the Ir-Yb pair. These results are particularly in-

teresting given that boosting the DILs results in lower recurrence probability and better

treatment outcome [6, 21, 34, 48, 49, 63].

If all radionuclides are normalized to 10 Ci, there is no large difference in terms of

treatment time among the treatment plans, with a slight increase when multiple sources

are used. Furthermore, the number and location of the active dwell positions (dwell time

> 0 seconds) do not change significantly across the plans. One important point to note

is that we assume that the sources are used sequentially during treatment. Should the

sources be used concurrently, treatment times would be greatly reduced but inter-seed

attenuation would have to be accounted for [12].

EMBT is still in its infancy and more theoretical studies need to be carried out. While

some of the dosimetric improvements are very small for regular prostate cases, they show

a positive trend that can be more thoroughly explored. More clinical scenarios must

be tested in order to find the ideal use of multi-source brachytherapy and selection of

HDR sources. From a feasibility perspective, using three sources is not yet possible. We

therefore propose the dual use of 60Co and 192Ir for EMBT.

4.4 Conclusion

We retrospectively investigated the dosimetric benefits of a novel brachytherapy technique

called EMBT with three different HDR sources, 192Ir, 60Co, and 169Yb. On average, the

plans generated from all source combinations satisfied the clinical protocols, showing


that EMBT is viable. Our results show that HDR prostate brachytherapy benefits from

dose reduction to the OARs when multiple sources are used, while keeping similar target

coverage. Cases with DILs benefit the most from EMBT. From a practical point of view,

we propose the dual use of 60Co and 192Ir. As with DMBT, we also show that our inverse

planning approach is effective for HDR brachytherapy.

Chapter 5

Conclusion

The HDR brachytherapy forward planning process is time-consuming and heavily relies

on the planner’s expertise, which can vary from one planner to another. Since several

cancer patients are treated in one day, the treatment planning process should be faster

and the quality of the treatment plans more consistent. Our inverse planning approach

has solved some of these problems. We have shown that the mathematical framework used

in SRS and IMRT can be successfully applied to HDR brachytherapy. The DTO model

and its modified versions are solved to ε-optimality by the IPCG algorithm [56] within

a few minutes. The resulting treatment plans satisfy the clinical objectives while having

acceptable treatment times. This fast and automated framework to generate treatment

plans relieves the load of the planners and allows for more plans to be generated within

a certain time frame.

Two novel brachytherapy techniques, DMBT and EMBT, were evaluated. They may

improve treatment outcome and save lives by providing superior plan quality than con-

ventional brachytherapy treatments. We showed that DMBT, which makes use of a

shielded tandem applicator, allows for more OAR sparing while keeping the same target

coverage, homogeneity, and conformity. The next step is to use the DMBT tandem in

clinical trials.

49

Chapter 5. Conclusion 50

We also investigated the effectiveness of EMBT using 192Ir, 60Co, and 169Yb. Our

results showed that multi-source combinations were, on average, as good or better than

their individual building blocks, as expected. Cases with DILs benefited the most from

EMBT in terms of OAR sparing. With the introduction of new dual-source afterloaders

on the market, EMBT is clinically viable. Our hypothetical study shows that EMBT is

a promising treatment modality, and should be investigated for other cancer sites and

different radionuclides.

There are several extensions to this work. In terms of the mathematical model, the

dwell positions can be optimized in addition to the dwell times. We can approach the

problem as a two-stage optimization problem, where the location of the dwell positions

are selected first, followed by the optimization of the dwell times. Alternatively, both

types of variables can be optimized at the same time in a mixed integer programming

problem. Another extension is the incorporation of a penalty term in the objective

function to lower the treatment time. For example, the penalty term could be the sum

of the dwell times. Regarding the selection of model parameters, machine learning could

be used to identify patient clusters that would match a single set of penalty weights.

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